Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 866, August 2006Screen Reader Version*
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Abstract:
This paper analyzes the quality of VARbased procedures for estimating the response of the economy to a shock. We focus on two key issues. First, do VARbased confidence intervals accurately reflect the actual degree of sampling uncertainty associated with impulse response functions? Second, what is the size of bias relative to confidence intervals, and how do coverage rates of confidence intervals compare with their nominal size? We address these questions using data generated from a series of estimated dynamic, stochastic general equilibrium models. We organize most of our analysis around a particular question that has attracted a great deal of attention in the literature: How do hours worked respond to an identified shock? In all of our examples, as long as the variance in hours worked due to a given shock is above the remarkably low number of 1 percent, structural VARs perform well. This finding is true regardless of whether identification is based on shortrun or longrun restrictions. Confidence intervals are wider in the case of longrun restrictions. Even so, longrun identified VARs can be useful for discriminating among competing economic models.
Keywords: Vector autoregression, dynamic stochastic general equilibrium model, confidence intervals, impulse response functions, identification, long run restrictions, specification error, sampling
JEL Classification: C1
Sims's seminal paper Macroeconomics and Reality (1980) argued that procedures based on vector autoregression (VAR) would be useful to macroeconomists interested in constructing and evaluating economic models. Given a minimal set of identifying assumptions, structural VARs allow one to estimate the dynamic effects of economic shocks. The estimated impulse response functions provide a natural way to choose the parameters of a structural model and to assess the empirical plausibility of alternative models.^{1}
To be useful in practice, VARbased procedures must have good sampling properties. In particular, they should accurately characterize the amount of information in the data about the effects of a shock to the economy. Also, they should accurately uncover the information that is there.
These considerations lead us to investigate two key issues. First, do VARbased confidence intervals accurately reflect the actual degree of sampling uncertainty associated with impulse response functions? Second, what is the size of bias relative to confidence intervals, and how do coverage rates of confidence intervals compare with their nominal size?
We address these questions using data generated from a series of estimated dynamic, stochastic general equilibrium (DSGE) models. We consider real business cycle (RBC) models and the model in Altig, Christiano, Eichenbaum, and Linde (2005) (hereafter, ACEL) that embodies real and nominal frictions. We organize most of our analysis around a particular question that has attracted a great deal of attention in the literature: How do hours worked respond to an identified shock? In the case of the RBC model, we consider a neutral shock to technology. In the ACEL model, we consider two types of technology shocks as well as a monetary policy shock.
We focus our analysis on an unavoidable specification error that occurs when the data generating process is a DSGE model and the econometrician uses a VAR. In this case the true VAR is infinite ordered, but the econometrician must use a VAR with a finite number of lags.
We find that as long as the variance in hours worked due to a given shock is above the remarkably low number of 1 percent, VARbased methods for recovering the response of hours to that shock have good sampling properties. Technology shocks account for a much larger fraction of the variance of hours worked in the ACEL model than in any of our estimated RBC models. Not surprisingly, inference about the effects of a technology shock on hours worked is much sharper when the ACEL model is the data generating mechanism.
Taken as a whole, our results support the view that structural VARs are a useful guide to constructing and evaluating DSGE models. Of course, as with any econometric procedure it is possible to find examples in which VARbased procedures do not do well. Indeed, we present such an example based on an RBC model in which technology shocks account for less than 1 percent of the variance in hours worked. In this example, VARbased methods work poorly in the sense that bias exceeds sampling uncertainty. Although instructive, the example is based on a model that fits the data poorly and so is unlikely to be of practical importance.
Having good sampling properties does not mean that structural VARs always deliver small confidence intervals. Of course, it would be a Pyrrhic victory for structural VARs if the best one could say about them is that sampling uncertainty is always large and the econometrician will always know it. Fortunately, this is not the case. We describe examples in which structural VARs are useful for discriminating between competing economic models.
Researchers use two types of identifying restrictions in structural VARs. Blanchard and Quah (1989), Gali (1999), and others exploit the implications that many models have for the longrun effects of shocks.^{2} Other authors exploit shortrun restrictions.^{3} It is useful to distinguish between these two types of identifying restrictions to summarize our results.
We find that structural VARs perform remarkably well when identification is based on shortrun restrictions. For all the specifications that we consider, the sampling properties of impulse response estimators are good and sampling uncertainty is small. This good performance obtains even when technology shocks account for as little as 0.5 percent of the variance in hours. Our results are comforting for the vast literature that has exploited shortrun identification schemes to identify the dynamic effects of shocks to the economy. Of course, one can question the particular shortrun identifying assumptions used in any given analysis. However, our results strongly support the view that if the relevant shortrun assumptions are satisfied in the data generating process, then standard structural VAR procedures reliably uncover and identify the dynamic effects of shocks to the economy.
The main distinction between our short and longrun results is that the sampling uncertainty associated with estimated impulse response functions is substantially larger in the longrun case. In addition, we find some evidence of bias when the fraction of the variance in hours worked that is accounted for by technology shocks is very small. However, this bias is not large relative to sampling uncertainty as long as technology shocks account for at least 1 percent of the variance of hours worked. Still, the reason for this bias is interesting. We document that, when substantial bias exists, it stems from the fact that with longrun restrictions one requires an estimate of the sum of the VAR coefficients. The specification error involved in using a finitelag VAR is the reason that in some of our examples, the sum of VAR coefficients is difficult to estimate accurately. This difficulty also explains why sampling uncertainty with longrun restrictions tends to be large.
The preceding observations led us to develop an alternative to the standard VARbased estimator of impulse response functions. The only place the sum of the VAR coefficients appears in the standard strategy is in the computation of the zerofrequency spectral density of the data. Our alternative estimator avoids using the sum of the VAR coefficients by working with a nonparametric estimator of this spectral density. We find that in cases when the standard VAR procedure entails some bias, our adjustment virtually eliminates the bias.
Our results are related to a literature that questions the ability of longrun identified VARs to reliably estimate the dynamic response of macroeconomic variables to structural shocks. Perhaps the first critique of this sort was provided by Sims (1972). Although his paper was written before the advent of VARs, it articulates why estimates of the sum of regression coefficients may be distorted when there is specification error. Faust and Leeper (1997) and Pagan and Robertson (1998) make an important related critique of identification strategies based on longrun restrictions. More recently Erceg, Guerrieri, and Gust (2005) and Chari, Kehoe, and McGrattan (2005b) (henceforth CKM) also examine the reliability of VARbased inference using longrun identifying restrictions.^{4}Our conclusions regarding the value of identified VARs differ sharply from those recently reached by CKM. One parameterization of the RBC model that we consider is identical to the one considered by CKM. This parameterization is included for pedagogical purposes only, as it is overwhelmingly rejected by the data.
The remainder of the paper is organized as follows. Section 2 presents the versions of the RBC models that we use in our analysis. Section 3 discusses our results for standard VARbased estimators of impulse response functions. Section 4 analyzes the differences between short and longrun restrictions. Section 5 discusses the relation between our work and the recent critique of VARs offered by CKM. Section 6 summarizes the ACEL model and reports its implications for VARs. Section 7 contains concluding comments.
In this section, we display the RBC model that serves as one of the data generating processes in our analysis. In this model the only shock that affects labor productivity in the longrun is a shock to technology. This property lies at the core of the identification strategy used by King, et al (1991), Gali (1999) and other researchers to identify the effects of a shock to technology. We also consider a variant of the model which rationalizes short run restrictions as a strategy for identifying a technology shock. In this variant, agents choose hours worked before the technology shock is realized. We describe the conventional VARbased strategies for estimating the dynamic effect on hours worked of a shock to technology. Finally, we discuss parameterizations of the RBC model that we use in our experiments.
The representative agent maximizes expected utility over per capita consumption, and per capita hours worked,
The representative competitive firm's production function is:
Finally, the resource constraint is:
We consider two versions of the model, differentiated according to timing assumptions. In the standard or nonrecursive version, all time decisions are taken after the realization of the time shocks. This is the conventional assumption in the RBC literature. In the recursive version of the model the timing assumptions are as follows. First, is observed, and then labor decisions are made. Second, the other shocks are realized and agents make their investment and consumption decisions.
We now discuss the relation between the RBC model and a VAR. Specifically, we establish conditions under which the reduced form of the RBC model is a VAR with disturbances that are linear combinations of the economic shocks. Our exposition is a simplified version of the discussion in FernandezVillaverde, RubioRamirez, and Sargent (2005) (see especially their section III). We include this discussion because it frames many of the issues that we address. Our discussion applies to both the standard and the recursive versions of the model.
We begin by showing how to put the reduced form of the RBC model into a statespace, observer form. Throughout, we analyze the loglinear approximations to model solutions. Suppose the variables of interest in the RBC model are denoted by Let denote the vector of exogenous economic shocks and let denote the percent deviation from steady state of the capital stock, after scaling by ^{5} The approximate solution for is given by:
The `state' of the system is composed of the variables on the right side of (2):
We now use (5) and (6) to establish conditions under which the reduced form representation for implied by the RBC model is a VAR with disturbances that are linear combinations of the economic shocks. In this discussion, we set so that In addition, we assume that the number of elements in coincides with the number of elements in
We begin by substituting (5) into (6) to obtain:
Proposition 2.1. (FernandezVillaverde, RubioRamirez, and Sargent) If C is invertible and the eigenvalues of M are less than unity in absolute value, then the RBC model implies:
has the infiniteorder VAR representation in (10)
The linear onestepahead forecast error given past 's is , which is related to the economic disturbances by (11)
The variancecovariance of is
The sum of the VAR lag matrices is given by:
We will use the last of these results below.
Relation (10) indicates why researchers interested in constructing DSGE models find it useful to analyze VARs. At the same time, this relationship clarifies some of the potential pitfalls in the use of VARs. First, in practice the econometrician must work with finite lags. Second, the assumption that is square and invertible may not be satisfied. Whether satisfies these conditions depends on how is defined. Third, significant measurement errors may exist. Fourth, the matrix, , may not have eigenvalues inside the unit circle. In this case, the economic shocks are not recoverable from the VAR disturbances.^{6} Implicitly, the econometrician who works with VARs assumes that these pitfalls are not quantitatively important.
We are interested in the use of VARs as a way to estimate the response of to economic shocks, i.e., elements of In practice, macroeconomists use a version of (10) with finite lags, say A researcher can estimate and To obtain the impulse response functions, however, the researcher needs the 's and the column of corresponding to the shock in that is of interest. However, to compute the required column of requires additional identifying assumptions. In practice, two types of assumptions are used. Shortrun assumptions take the form of direct restrictions on the matrix . Longrun assumptions place indirect restrictions on that stem from restrictions on the longrun response of to a shock in an element of . In this section we use our RBC model to discuss these two types of assumptions and how they are imposed on VARs in practice.
The loglinearized equilibrium laws of motion for capital and hours in this model can be written as follows:
In practice, researchers impose the exclusion and sign restrictions on a VAR to compute and identify its dynamic effects on macroeconomic variables. Consider the vector, The VAR for is given by:
The symmetric matrix, and the 's can be computed using ordinary least squares regressions. However, the requirement that is not sufficient to determine a unique value of Adding the exclusion and sign restrictions does uniquely determine Relation (2.18) implies that these restrictions are:
In the recursive version of the model, the policy rule for labor involves and because these variables help forecast and
Let and denote the population onestepahead forecast errors in and conditional on the information set, The recursive version of the model implies that
In practice, we implement the previous procedure using the onestepahead forecast errors generated from a VAR in which the variables in are ordered as follows:
We consider different specifications of the RBC model that are distinguished by the parameterization of the laws of motion of the exogenous shocks. In all specifications we assume, as in CKM , that:
We estimate two versions of our model. In the twoshock maximum likelihood estimation (MLE) specification we assume that so that there are two shocks, and We estimate the parameters , and by maximizing the Gaussian likelihood function of the vector, subject to (24 )^{9} Our results are given by:
The threeshock MLE specification incorporates the investment tax shock, into the model. We estimate the threeshock MLE version of the model by maximizing the Gaussian likelihood function of the vector, , subject to the parameter values in (24) The results are:
The twoshock CKM specification has two shocks, and These shocks have the following time series representations:
As in our specifications, CKM obtain their parameter estimates using maximum likelihood methods. However, their estimates are very different from ours. For example, the variances of the shocks are larger in the twoshock CKM specification than in our MLE specification. Also, the ratio of to is nearly three times larger in the twoshock CKM specification than in our twoshock MLE specification. Section 5 below discusses the reasons for these differences.
Table 1 reports the contribution, of technology shocks to three different measures of the volatility in the log of hours worked: (i) the variance of the log hours, (ii) the variance of HPfiltered, log hours and (iii) the variance in the onestepahead forecast error in log hours.^{11} With one exception, we compute the analogous statistics for log output. The exception is (i), for which we compute the contribution of technology shocks to the variance of the growth rate of output.
The key result in this table is that technology shocks account for a very small fraction of the volatility in hours worked. When is measured according to (i), it is always below 4 percent. When is measured using (ii) or (iii) it is always below 8 percent. For both (ii) and (iii), in the CKM specifications, is below 2 percent.^{12} Consistent with the RBC literature, the table also shows that technology accounts for a much larger movement in output.
Figure 1 displays visually how unimportant technology shocks are for hours worked. The top panel displays two sets of 180 artificial observations on hours worked, simulated using the standard twoshock MLE specification. The volatile time series shows how log hours worked evolve in the presence of shocks to both and The other time series shows how log hours worked evolve in response to just the technology shock, The bottom panel is the analog of the top figure when the data are generated using the standard twoshock CKM specification.
In this section we analyze the properties of conventional VARbased strategies for identifying the effects of a technology shock on hours worked. We focus on the bias properties of the impulse response estimator, and on standard procedures for estimating sampling uncertainty.
We use the RBC model parameterizations discussed in the previous section as the data generating processes. For each parameterization, we simulate 1,000 data sets of 180 observations each. The shocks , and possibly are drawn from standard normal distributions. For each artificial data set, we estimate a fourlag VAR. The average, across the 1,000 data sets, of the estimated impulse response functions, allows us to assess bias.
For each data set we also estimate two different confidence intervals: a percentilebased confidence interval and a standarddeviation based confidence interval.^{13} We construct the intervals using the following bootstrap procedure. Using random draws from the fitted VAR disturbances, we use the estimated four lag VAR to generate 200 synthetic data sets, each with 180 observations. For each of these 200 synthetic data sets we estimate a new VAR and impulse response function. For each artificial data set the percentilebased confidence interval is defined as the top 2.5 percent and bottom 2.5 percent of the estimated coefficients in the dynamic response functions. The standarddeviationbased confidence interval is defined as the estimated impulse response plus or minus two standard deviations where the standard deviations are calculated across the 200 simulated estimated coefficients in the dynamic response functions.
We assess the accuracy of the confidence interval estimators in two ways. First, we compute the coverage rate for each type of confidence interval. This rate is the fraction of times, across the 1,000 data sets simulated from the economic model, that the confidence interval contains the relevant true coefficient. If the confidence intervals were perfectly accurate, the coverage rate would be 95 percent. Second, we provide an indication of the actual degree of sampling uncertainty in the VARbased impulse response functions. In particular, we report centered 95 percent probability intervals for each lag in our impulse response function estimators.^{14} If the confidence intervals were perfectly accurate, they should on average coincide with the boundary of the 95 percent probability interval.
When we generate data from the twoshock MLE and CKM specifications, we set When we generate data from the threeshock MLE and CKM specifications, we set
Figure 2 reports results generated from four different parameterizations of the recursive version of the RBC model. In each panel, the solid line is the average estimated impulse response function for the 1,000 data sets simulated using the indicated economic model. For each model, the starred line is the true impulse response function of hours worked. In each panel, the gray area defines the centered percent probability interval for the estimated impulse response functions. The stars with no line indicate the average percentilebased confidence intervals across the 1,000 data sets. The circles with no line indicate the average standarddeviationbased confidence intervals.
Figures 3 and 4 graph the coverage rates for the percentilebased and standarddeviationbased confidence intervals. For each case we graph how often, across the 1,000 data sets simulated from the economic model, the econometrician's confidence interval contains the relevant coefficient of the true impulse response function.
The 1,1 panel in Figure 2 exhibits the properties of the VARbased estimator of the response of hours to a technology shock when the data are generated by the twoshock MLE specification. The 2,1 panel corresponds to the case when the data generating process is the threeshock MLE specification.
The panels have two striking features. First, there is essentially no evidence of bias in the estimated impulse response functions. In all cases, the solid lines are very close to the starred lines. Second, an econometrician would not be misled in inference by using standard procedures for constructing confidence intervals. The circles and stars are close to the boundaries of the gray area. The 1,1 panels in Figures 3 and 4 indicate that the coverage rates are roughly 90 percent. So, with high probability, VARbased confidence intervals include the true value of the impulse response coefficients.
The second column of Figure 2 reports the results when the data generating process is given by variants of the CKM specification. The 1,2 and 2,1 panels correspond to the two and threeshock CKM specification, respectively.
The second column of Figure 2 contains the same striking features as the first column. There is very little bias in the estimated impulse response functions. In addition, the average value of the econometrician's confidence interval coincides closely with the actual range of variation in the impulse response function (the gray area). Coverage rates, reported in the 1,2 panels of Figures 3 and 4, are roughly 90 percent. These rates are consistent with the view that VARbased procedures lead to reliable inference.
A comparison of the gray areas across the first and second columns of Figure 2, clearly indicates that more sampling uncertainty occurs when the data are generated from the CKM specifications than when they are generated from the MLE specifications (the gray areas are wider). VARbased confidence intervals detect this fact.
The first and second rows of column 1 in Figure 5 exhibit our results when the data are generated by the two and three shock MLE specifications. Once again there is virtually no bias in the estimated impulse response functions and inference is accurate. The coverage rates associated with the percentilebased confidence intervals are very close to 95 percent (see Figure 3). The coverage rates for the standarddeviationbased confidence intervals are somewhat lower, roughly 80 percent (see Figure 4). The difference in coverage rates can be seen in Figure 5, which shows that the stars are shifted down slightly relative to the circles. Still, the circles and stars are very good indicators of the boundaries of the gray area, although not quite as good as in the analog cases in Figure 2.
Comparing Figures 2 and 5, we see that Figure 5 reports more sampling uncertainty. That is, the gray areas are wider. Again, the crucial point is that the econometrician who computes standard confidence intervals would detect the increase in sampling uncertainty.
The third and fourth rows of column 1 in Figure 5 report results for the two and three  shock CKM specifications. Consistent with results reported in CKM, there is substantial bias in the estimated dynamic response functions. For example, in the Twoshock CKM specification, the contemporaneous response of hours worked to a onestandarddeviation technology shock is percent, while the mean estimated response is percent. This bias stands in contrast to our other results.
Is this bias big or problematic? In our view, bias cannot be evaluated without taking into account sampling uncertainty. Bias matters only to the extent that the econometrician is led to an incorrect inference. For example, suppose sampling uncertainty is large and the econometrician knows it. Then the econometrician would conclude that the data contain little information and, therefore, would not be misled. In this case, we say that bias is not large. In contrast, suppose sampling uncertainty is large, but the econometrician thinks it is small. Here, we would say bias is large.
We now turn to the sampling uncertainty in the CKM specifications. Figure 5 shows that the econometrician's average confidence interval is large relative to the bias. Interestingly, the percentile confidence intervals (stars) are shifted down slightly relative to the standarddeviationbased confidence intervals (circles). On average, the estimated impulse response function is not in the center of the percentile confidence interval. This phenomenon often occurs in practice.^{15} Recall that we estimate a four lag VAR in each of our 1,000 synthetic data sets. For the purposes of the bootstrap, each of these VARs is treated as a true data generating process. The asymmetric percentile confidence intervals show that when data are generated by these VARs, VARbased estimators of the impulse response function have a downward bias.
Figure 3 reveals that for the two and threeshock CKM specifications, percentilebased coverage rates are reasonably close to 95 percent. Figure 4 shows that the standard deviation based coverage rates are lower than the percentilebased coverage rates. However even these coverage rates are relatively high in that they exceed 70 percent.
In summary, the results for the MLE specification differ from those of the CKM specifications in two interesting ways. First, sampling uncertainty is much larger with the CKM specification. Second, the estimated responses are somewhat biased with the CKM specification. But the bias is small: It has no substantial effect on inference, at least as judged by coverage rates for the econometrician's confidence intervals.
Here we show that the more important technology shocks are in the dynamics of hours worked, the easier it is for VARs to answer the question, `how do hours worked respond to a technology shock'. We demonstrate this by considering alternative values of the innovation variance in the labor tax, and by considering alternative values of the utility parameter that controls the Frisch elasticity of labor supply.
Consider Figure 6, which focuses on the longrun identification schemes. The first and second columns report results for the twoshock MLE and CKM specifications, respectively. For each specification we redo our experiments, reducing by a half and then by a quarter. Table 1 shows that the importance of technology shocks rises as the standard deviation of the labor tax shock falls. Figure 6 indicates that the magnitude of sampling uncertainty and the size of confidence intervals fall as the relative importance of labor tax shocks falls.^{16}
Figure 7 presents the results of a different set of experiments based on perturbations of the twoshock CKM specification. The 1,1 and 2,1 panels show what happens when we vary the value of , the parameter that controls the Frisch labor supply elasticity. In the 1,1 panel we set which corresponds to a Frisch elasticity of 0.63. In the 2,1 panel, we set which corresponds to a Frisch elasticity of infinity. As the Frisch elasticity is increased, the fraction of the variance in hours worked due to technology shocks decreases (see Table 1). The magnitude of bias and the size of confidence intervals are larger for the higher Frisch elasticity case. In both cases the bias is still smaller than the sampling uncertainty.
We were determined to construct at least one example in which the VARbased estimator of impulse response functions have bad properties, i.e., bias is larger than sampling uncertainty. We display such an example in the 3,1 panel of Figure 7. The data generating process is a version of the twoshock CKM model with an infinite Frisch elasticity and double the standard deviation of the labor tax rate. Table 1 indicates that with this specification, technology shocks account for a trivial fraction of the variance in hours worked. Of the three measures of two are percent and the third is percent . The 3,1 panel of Figure 7 shows that the VARbased procedure now has very bad properties: the true value of the impulse response function lies outside the average value of both confidence intervals that we consider. This example shows that constructing scenarios in which VARbased procedures go awry is certainly possible. However, this example seems unlikely to be of practical significance given the poor fit to the data of this version of the model.
Up to now, we have focused on the RBC model as the data generating process. For empirically reasonable specifications of the RBC model, confidence intervals associated with longrun identification schemes are large. One might be tempted to conclude that VARbased longrun identification schemes are uninformative. Specifically, are the confidence intervals so large that we can never discriminate between competing economic models? Erceg, Guerrieri, and Gust (2005) show that the answer to this question is `no'. They consider an RBC model similar to the one discussed above and a version of the sticky wageprice model developed by Christiano, Eichenbaum, and Evans (2005) in which hours worked fall after a positive technology shock. They then conduct a series of experiments to assess the ability of a longrun identified structural VAR to discriminate between the two models on the basis of the response of hours worked to a technology shock.
Using estimated versions of each of the economic models as a data generating process, they generate 10,000 synthetic data sets each with 180 observations. They then estimate a fourvariable structural VAR on each synthetic data set and compute the dynamic response of hours worked to a technology shock using longrun identification. Erceg, Guerrieri, and Gust (2005) report that the probability of finding an initial decline in hours that persists for two quarters is much higher in the model with nominal rigidities than in the RBC model (93 percent versus 26 percent). So, if these are the only two models contemplated by the researcher, an empirical finding that hours worked decline after a positive innovation to technology will constitute compelling evidence in favor of the sticky wageprice model.
Erceg, Guerrieri, and Gust (2005) also report that the probability of finding an initial rise in hours that persists for two quarters is much higher in the RBC model than in the sticky wageprice model (71 percent versus 1 percent). So, an empirical finding that hours worked rises after a positive innovation to technology would constitute compelling evidence in favor of the RBC model versus the sticky wageprice alternative.
The previous section demonstrates that, in the examples we considered, when VARs are identified using shortrun restrictions, the conventional estimator of impulse response functions is remarkably accurate. In contrast, for some parameterizations of the data generating process, the conventional estimator of impulse response functions based on longrun identifying restrictions can exhibit noticeable bias. In this section we argue that the key difference between the two identification strategies is that the longrun strategy requires an estimate of the sum of the VAR coefficients, This object is notoriously difficult to estimate accurately (see Sims, 1972).
We consider a simple analytic expression related to one in Sims (1972). Our expression shows what an econometrician who fits a misspecified, fixedlag, finiteorder VAR would find in population. Let and denote the parameters of the thorder VAR fit by the econometrician. Then:
To understand the implications of (26) for our analysis, it is useful to write in lagoperator form the estimated dynamic response of to a shock in the first element of
We use (26) to understand why estimation based on shortrun and longrun identification can produce different results. According to (27), impulse response functions can be decomposed into two parts, the impact effect of the shocks, summarized by and the dynamic part summarized in the term in square brackets. We argue that when a bias arises with longrun restrictions, it is because of difficulties in estimating These difficulties do not arise with shortrun restrictions.
In the shortrun identification case, is a function of only. Across a variety of numerical examples, we find that is very close to ^{21} This result is not surprising because (26) indicates that the entire objective of estimation is to minimize the distance between and In the longrun identification case, depends not only on but also on A problem is that the criterion does not assign much weight to setting unless happens to be relatively large in a neighborhood of But, a large value of is not something one can rely on.^{22} When is relatively small, attempts to match with at other frequencies can induce large errors in
The previous argument about the difficulty of estimating in the longrun identification case does not apply to the s According to (28) is a function of over the whole range of 's, not just one specific frequency.
We now present a numerical example, which illustrates Proposition 1 as well as some of the observations we have made in discussing (26). Our numerical example focuses on population results. Therefore, it provides only an indication of what happens in small samples.
To understand what happens in small samples, we consider four additional numerical examples. First, we show that when the econometrician uses the true value of , the bias and much of the sampling uncertainty associated with the Twoshock CKM specification disappears. Second, we demonstrate that bias problems essentially disappear when we use an alternative to the standard zerofrequency spectral density estimator used in the VAR literature. Third, we show that the problems are attenuated when the preference shock is more persistent. Fourth, we consider the recursive version of the twoshock CKM specification in which the effect of technology shocks can be estimated using either short or longrun restrictions.
Table 2 reports various properties of the twoshock CKM specification. The first six 's in the infiniteorder VAR, computed using (12), are reported in Panel A. These 's eventually converge to zero, however they do so slowly. The speed of convergence is governed by the size of the maximal eigenvalue of the matrix in (8), which is 0.957. Panel B displays the 's that solve (26) with Informally, the 's look similar to the 's for In line with this observation, the sum of the true 's, is similar in magnitude to the sum of the estimated 's, (see Panel C). But the econometrician using longrun restrictions needs a good estimate of This matrix is very different from Although the remaining 's for are individually small, their sum is not. For example, the 1,1 element of is 0.28, or six times larger than the 1,1 element of
The distortion in manifests itself in a distortion in the estimated zerofrequency spectral density (see Panel D). As a result, there is distortion in the estimated impact vector, (Panel F).^{23} To illustrate the significance of the latter distortion for estimated impulse response functions, we display in Figure 8 the part of (27) that corresponds to the response of hours worked to a technology shock. In addition, we display the true response. There is a substantial distortion, which is approximately the same magnitude as the one reported for small samples in Figure 5. The third line in Figure 8 corresponds to (27) when is replaced by its true value, . Most of the distortion in the estimated impulse response function is eliminated by this replacement. Finally, the distortion in is due to distortion in as is virtually identical to (panel E).
This example is consistent with our overall conclusion that the individual 's and are well estimated by the econometrician using a fourlag VAR. The distortions that arise in practice primarily reflect difficulties in estimating . Our shortrun identification results in Figure 2 are consistent with this claim, because distortions are minimal with shortrun identification.
A natural way to isolate the role of distortions in is to replace by its true value when estimating the effects of a technology shock. We perform this replacement for the twoshock CKM specification, and report the results in Figure 9. For convenience, the 1,1 panel of Figure 9 repeats our results for the twoshock CKM specification from the 3,1 panel in Figure 5. The 1,2 panel of Figure 9 shows the sampling properties of our estimator when the true value of is used in repeated samples. When we use the true value of the bias completely disappears. In addition, coverage rates are much closer to percent and the boundaries of the average confidence intervals are very close to the boundaries of the gray area.
In practice, the econometrician does not know . However, we can replace the VARbased zerofrequency spectral density in (19) with an alternative estimator of . Here, we consider the effects of using a standard Bartlett estimator:^{24}
We now assess the effect of our modified longrun estimator. The first two rows in Figure 5 present results for cases in which the data generating mechanism corresponds to our two and threeshock MLE specifications. Both the standard estimator (the left column) and our modified estimator (the right column) exhibit little bias. In the case of the standard estimator, the econometrician's estimator of standard errors understates somewhat the degree of sampling uncertainty associated with the impulse response functions. The modified estimator reduces this discrepancy. Specifically, the circles and stars in the right column of Figure 5 coincide closely with the boundary of the gray area. Coverage rates are reported in the 2,1 panels of Figures 3 and 4. In Figure 3, coverage rates now exceed 95 percent. The coverage rates in Figure 4 are much improved relative to the standard case. Indeed, these rates are now close to 95 percent. Significantly, the degree of sampling uncertainty associated with the modified estimator is not greater than that associated with the standard estimator. In fact, in some cases, sampling uncertainty declines slightly.
The last two rows of column 1 in Figure 5 display the results when the data generating process is a version of the CKM specification. As shown in the second column, the bias is essentially eliminated by using the modified estimator. Once again the circles and stars roughly coincide with the boundary of the gray area. Coverage rates for the percentilebased confidence intervals reported in Figure 3 again have a tendency to exceed 95 percent (2,2 panel). As shown in the 2,2 panel of Figure 4, coverage rates associated with the standard deviation based estimator are very close to 95 percent. There is a substantial improvement over the coverage rates associated with the standard spectral density estimator.
Figure 5 indicates that when the standard estimator works well, the modified estimator also works well. When the standard estimator results in biases, the modified estimator removes them. These findings are consistent with the notion that the biases for the two CKM specifications reflect difficulties in estimating the spectral density at frequency zero. Given our finding that is an accurate estimator of , we conclude that the difficulties in estimating the zerofrequency spectral density in fact reflect problems with
The second column of Figure 7 shows how our modified VARbased estimator works when the data are generated by the various perturbations on the Twoshock CKM specification. In every case, bias is substantially reduced.
Formula (26), suggests that, other things being equal, the more power there is near frequency zero, the less bias there is in and the better behaved is the estimated impulse response function to a technology shock. To pursue this observation we change the parameterization of the nontechnology shock in the twoshock CKM specification. We reallocate power toward frequency zero, holding the variance of the shock constant by increasing to 0.998 and suitably lowering in (1). The results are reported in the 2,1 panel of Figure 9. The bias associated with the twoshock CKM specification almost completely disappears. This result is consistent with the notion that the bias problems with the twoshock CKM specification stem from difficulties in estimating
The previous result calls into question conjectures in the literature (see Erceg, Guerrieri, and Gust, 2005). According to these conjectures, if there is more persistence in a nontechnology shock, then the VAR will produce biased results because it will confuse the technology and nontechnology shocks. Our result shows that this intuition is incomplete, because it fails to take into account all of the factors mentioned in our discussion of (26). To show the effect of persistence, we consider a range of values of to show that the impact of on bias is in fact not monotone.
The 2,2 panel of Figure 9 displays the econometrician's estimator of the contemporaneous impact on hours worked of a technology shock against . The dashed line indicates the true contemporaneous effect of a technology shock on hours worked in the twoshock CKM specification. The dotdashed line in the figure corresponds to the solution of (26), with using the standard VARbased estimator.^{26} The star in the figure indicates the value of in the twoshock CKM specification. In the neighborhood of this value of the distortion in the estimator falls sharply as increases. Indeed, for essentially no distortion occurs. For values of in the region, the distortion increases with increases in
The 2,2 panel of Figure 9 also allows us to assess the value of our proposed modification to the standard estimator. The line with diamonds displays the modified estimator of the contemporaneous impact on hours worked of a technology shock. When the standard estimator works well, that is, for large values of the modified and standard estimators produce similar results. However, when the standard estimator works poorly, e.g. for values of near , our modified estimator cuts the bias in half.
A potential shortcoming of the previous experiments is that persistent changes in do not necessarily induce very persistent changes in labor productivity. To assess the robustness of our results, we also considered what happens when there are persistent changes in These do have a persistent impact on labor productivity. In the twoshock CKM model, we set to a constant and allowed to be stochastic. We considered values of in the range, holding the variance of constant. We obtain results similar to those reported in the 2,2 panel of Figure 9.
We conclude this section by considering the recursive version of the twoshock CKM specification. This specification rationalizes estimating the impact on hours worked of a shock to technology using either the short or the longrun identification strategy. We generate 1,000 data sets, each of length 180. On each synthetic data set, we estimate a four lag, bivariate VAR. Given this estimated VAR, we can estimate the effect of a technology shock using the short and longrun identification strategy. Figure 10 reports our results. For the longrun identification strategy, there is substantial bias. In sharp contrast, there is no bias for the shortrun identification strategy. Because both procedures use the same estimated VAR parameters, the bias in the longrun identification strategy is entirely attributable due to the use of
In the preceding sections we argue that structural VARbased procedures have good statistical properties. Our conclusions about the usefulness of structural VARs stand in sharp contrast to the conclusions of CKM. These authors argue that, for plausibly parameterized RBC models, structural VARs lead to misleading results. They conclude that structural VARs are not useful for constructing and evaluating structural economic models. In this section we present the reasons we disagree with CKM.
CKM's critique of VARs is based on simulations using particular DSGE models estimated by maximum likelihood methods. Here, we argue that their key results are driven by assumptions about measurement error. CKM's measurement error assumptions are overwhelmingly rejected in favor of alternatives under which their key results are overturned.
CKM adopt a stateobserver setup to estimate their model. Define:
To demonstrate the sensitivity of CKM's results to their specification of the magnitude of we consider the different assumptions that CKM make in different drafts of their paper. In the draft of May 2005, CKM set the diagonal elements of to In the draft of July 2005, CKM set the diagonal element of equal to times the variance of the element of
The 1,1 and 2,1 panels in Figure 11 report results corresponding to CKM's twoshock specifications in the July and May drafts, respectively.^{27} These panels display the log likelihood value (see ) of these two models and their implications for VARbased impulse response functions (the 1,1 panel is the same as the 3,1 panel in Figure 5). Surprisingly, the loglikelihood of the July specification is orders of magnitude worse than that of the May specification.
The 3,1 panel in Figure 11 displays our results when the diagonal elements of are included among the parameters being estimated.^{28} We refer to the resulting specification as the `` CKM free measurement error specification''. First, both the May and the July specifications are rejected relative to the free measurement error specification. The likelihood ratio statistic for testing the May and July specifications are 428 and 6,266, respectively. Under the null hypothesis that the May or July specification is true, these statistics are realizations of a chisquare distribution with 4 degrees of freedom. The evidence against CKM's May or July specifications of measurement error is overwhelming.
Second, when the data generating process is the CKM free measurement error specification, the VARbased impulse response function is virtually unbiased (see the 3,1 panel in Figure 11). We conclude that the bias in the twoshock CKM specification is a direct consequence of CKM's choice of the measurement error variance.
As noted above, CKM's measurement error assumption has the implication that is roughly equals to To investigate the role played by this peculiar implication, we delete from and reestimate the system. We present the results in the right column of Figure 11. In each panel of that column, we reestimate the system in the same way as the corresponding panel in the left column, except that is excluded from Comparing the 2,1 and 2,2 panels, we see that, with the May measurement error specification, the bias disappears after relaxing CKM's assumption. Under the July specification of measurement error, the bias result remains even after relaxing CKM's assumption (compare the 1,1 and 1,2 graphs of Figure 11). As noted above, the May specification of CKM's model has a likelihood that is orders of magnitude higher than the July specification. So, in the version of the CKM model selected by the likelihood criterion (i.e., the May version), the assumption plays a central role in driving the CKM's bias result.
In sum, CKM's examples which imply that VARs with longrun identification display substantial bias, are not empirically interesting from a likelihood point of view. The bias in their examples is due to the way CKM choose the measurement error variance. When their measurement error specification is tested, it is overwhelmingly rejected in favor of an alternative in which the CKM bias result disappears.
CKM argue that there is considerable uncertainty in the business cycle literature about the values of parameters governing stochastic processes such as preferences and technology. They argue that this uncertainty translates into a wide class of examples in which the bias in structural VARs leads to severely misleading inference. The right panel in Figure 12 summarizes their argument. The horizontal axis covers the range of values of considered by CKM. For each value of we estimate, by maximum likelihood, four parameters of the twoshock model: and .^{29} We use the estimated model as a data generating process. The left vertical axis displays the small sample mean of the corresponding VARbased estimator of the contemporaneous response of hours worked to a onestandard deviation technology shock.
Based on a review the RBC literature, CKM report that they have a roughly uniform prior over the different values of considered in Figure 12. The figure indicates that for many of these values, the bias is large (compare the small sample mean, the solid line, with the true response, the starred line). For example, there is a noticeable bias in the 2shock CKM specification, where
We emphasize three points. First, as we stress repeatedly, bias cannot be viewed in isolation from sampling uncertainty. The two dashed lines in the figure indicate the 95 percent probability interval. These intervals are enormous relative to the bias. Second, not all values of are equally likely, and for the ones with greatest likelihood there is little bias. On the horizontal axis of the left panel of Figure 12, we display the same range of values of as in the right panel. On the vertical axis we report the loglikelihood value of the associated model. The peak of this likelihood occurs close to the estimated value in the twoshock MLE specification. Note how the loglikelihood value drops sharply as we consider values of away from the unconstrained maximum likelihood estimate. The vertical bars in the figure indicate the 95 percent confidence interval for ^{30} Figure 12 reveals that the confidence interval is very narrow relative to the range of values considered by CKM, and that within the interval, the bias is quite small.
Third, the right axis in the right panel of Figure 12 plots the percent of the variance in log hours due to technology, as a function of The values of for which there is a noticeable bias correspond to model economies where is less than percent. Here, identifying the effects of a technology shock on hours worked is tantamount to looking for a needle in a haystack.
CKM emphasize comparisons between the true dynamic response function in the data generating process and the response function that an econometrician would estimate using a fourlag VAR with an infinite amount of data. In our own analysis in section 4, we find population calculations with four lag VARs useful for some purposes. However, we do not view the probability limit of a four lag VAR as an interesting metric for measuring the usefulness of structural VARs. In practice econometricians do not have an infinite amount of data. Even if they did, they would certainly not use a fixed lag length. Econometricians determine lag length endogenously and, in a large sample, lag length would grow. If lag lengths grow at the appropriate rate with sample size, VARbased estimators of impulse response functions are consistent. The interesting issue (to us) is how VARbased procedures perform in samples of the size that practitioners have at their disposal. This is why we focus on small sample properties like bias and sampling uncertainty.
The potential power of the CKM argument lies in showing that VARbased procedures are misleading, even under circumstances when everyone would agree that VARs should work well, namely when the econometrician commits no avoidable specification error. The econometrician does, however, commit one unavoidable specification error. The true VAR is infinite ordered, but the econometrician assumes the VAR has a finite number of lags. CKM argue that this seemingly innocuous assumption is fatal for VAR analysis. We have argued that this conclusion is unwarranted.
CKM present other examples in which the econometrician commits an avoidable specification error. Specifically, they study the consequences of over differencing hours worked. That is, the econometrician first differences hours worked when hours worked are stationary.^{31} This error gives rise to bias in VARbased impulse response functions that is large relative to sampling uncertainty. CKM argue that this bias is another reason not to use VARs.
However, the observation that avoidable specification error is possible in VAR analysis is not a problem for VARs per se. The possibility of specification error is a potential pitfall for any type of empirical work. In any case, CKM's analysis of the consequences of over differencing is not new. For example, Christiano, Eichenbaum and Vigfusson (2003, hereafter, CEV) study a situation in which the true data generating process satisfies two properties: Hours worked are stationary and they rise after a positive technology shock. CEV then consider an econometrician who does VARbased longrun identification when in (16) contains the growth rate of hours rather than the log level of hours. CEV show that the econometrician would falsely conclude that hours worked fall after a positive technology shock. CEV do not conclude from this exercise that structural VARs are not useful. Rather, they develop a statistical procedure to help decide whether hours worked should be first differenced or not.
We argue that VARbased shortrun identification schemes lead to remarkably accurate and precise inference. This result is of interest because the preponderance of the empirical literature on structural VARs explores the implications of shortrun identification schemes. CKM are silent on this literature. McGrattan (2006) dismisses shortrun identification schemes as `` hokey.'' One possible interpretation of this adjective is that McGrattan can easily imagine models in which the identification scheme is incorrect. The problem with this interpretation is that all models are a collection of strong identifying assumptions, all of which can be characterized as `` hokey''. A second interpretation is that in McGrattan (2006)'s view, the type of zero restrictions typically used in short run identification are not compatible with dynamic equilibrium theory. This view is simply incorrect (see Sims and Zha (2006)). A third possible interpretation is that no one finds shortrun identifying assumptions interesting. However, the results of shortrun identification schemes have had an enormous effect on the construction of dynamic, general equilibrium models. See Woodford (2003) for a summary in the context of monetary models.
CKM argue that VARs are very sensitive to the choice of data. Specifically, they review the papers by Francis and Ramey (2004), CEV, and Gali and Rabanal (2004), which use longrun VAR methods to estimate the response of hours worked to a positive technology shock. CKM note that these studies use different measures of per capita hours worked and output in the VAR analysis. The bottom panel of Figure 13 displays the different measures of per capita hours worked that these studies use. Note how the low frequency properties of these series differ. The corresponding estimated impulse response functions and confidence intervals are reported in the top panel. CKM view it as a defect in VAR methodology that the different measures of hours worked lead to different estimated impulse response functions. We disagree. Empirical results should be sensitive to substantial changes in the data. A constructive response to the sensitivity in Figure 13 is to carefully analyze the different measures of hours worked, see which is more appropriate, and perhaps construct a better measure. It is not constructive to dismiss an econometric technique that signals the need for better measurement.
CKM note that the principle differences in the hours data occur in the early part of the sample. According to CKM, when they drop these early observations they obtain different impulse response functions. However, as Figure 13 shows, these impulse response functions are not significantly different from each other.
In this section we use the model in ACEL to assess the accuracy of structural VARs for estimating the dynamic response of hours worked to shocks. This model allows for nominal rigidities in prices and wages and has three shocks: a monetary policy shock, a neutral technology shock, and a capitalembodied technology shock. Both technology shocks affect labor productivity in the long run. However, the only shock in the model that affects the price of investment in the long run is the capitalembodied technology shock. We use the ACEL model to evaluate the ability of a VAR to uncover the response of hours worked to both types of technology shock and to the monetary policy shock. Our strategy for identifying the two technology shocks is similar to the one proposed by Fisher (2006). The model rationalizes a version of the shortrun, recursive identification strategy used by Christiano, Eichenbaum and Evans (1999) to identify monetary shocks. This strategy corresponds closely to the recursive procedure studied in section 2.3.2.
The details of the ACEL model, as well as the parameter estimates, are reported in Appendix A of the NBER Working Paper version of this paper. Here, we limit our discussion to what is necessary to clarify the nature of the shocks in the ACEL model. Final goods, are produced using a standard DixitStiglitz aggregator of intermediate goods, To produce a unit of consumption goods, one unit of final goods is required. To produce one unit of investment goods, units of final goods are required. In equilibrium, is the price, in units of consumption goods, of an investment good. Let denote the growth rate of let denote the nonstochastic steady state value of and let denote the percent deviation of from its steady state value:
We now turn to the monetary policy shock. Let denote where denotes the monetary base. Let denote the percentage deviation of from its steady state, i.e., We suppose that is the sum of three components. One, represents the component of reflecting an exogenous shock to monetary policy. The other two, and represent the endogenous response of to the neutral and capitalembodied technology shocks, respectively. Thus monetary policy is given by:
Table 3 summarizes the importance of different shocks for the variance of hours worked and output. Neutral and capitalembodied technology shocks account for roughly equal percentages of the variance of hours worked ( percent each), while monetary policy shocks account for the remainder. Working with HPfiltered data reduces the importance of neutral technology shocks to about percent. Monetary policy shocks become much more important for the variance of hours worked. A qualitatively similar picture emerges when we consider output.
It is worth emphasizing that neutral technology shocks are much more important in hours worked in the ACEL model than in the RBC model. This fact plays an important role in determining the precision of VARbased inference using longrun restrictions in the ACEL model.
We use the ACEL model to simulate 1,000 data sets each with 180 observations. We report results from two different VARs. In the first VAR, we simultaneously estimate the dynamic effect on hours worked of a neutral technology shock and a capitalembodied technology shock. The variables in this VAR are:
The 1,1 panel of Figure 14 displays our results using the standard VAR procedure to estimate the dynamic response of hours worked to a neutral technology shock. Several results are worth emphasizing. First, the estimator is essentially unbiased. Second, the econometrician's estimator of sampling uncertainty is also reasonably unbiased. The circles and stars, which indicate the mean value of the econometrician's standarddeviationbased and percentilebased confidence intervals, roughly coincide with the boundaries of the gray area. However, there is a slight tendency, in both cases, to understate the degree of sampling uncertainty. Third, confidence intervals are small, relative to those in the RBC examples. Both sets of confidence intervals exclude zero at all lags shown. This result provides another example, in addition to the one provided by Erceg, Guerrieri, and Gust (2005), in which longrun identifying restrictions are useful for discriminating between models. An econometrician who estimates that hours drop after a positive technology shock would reject our parameterization of the ACEL model. Similarly, an econometrician with a model implying that hours fall after a positive technology shock would most likely reject that model if the actual data were generated by our parameterization of the ACEL model.
The 2,1 panel in Figure 14 shows results for the response to a capitalembodied technology shock as estimated using the standard VAR estimator. The sampling uncertainty is somewhat higher for this estimator than for the neutral technology shock. In addition, there is a slight amount of bias. The econometrician understates somewhat the degree of sampling uncertainty.
We now consider the response of hours worked to a monetary policy shock. We estimate this response using a VAR with the following variables:
In this paper we study the ability of structural VARs to uncover the response of hours worked to a technology shock. We consider two classes of data generating processes. The first class consists of a series of real business cycle models that we estimate using maximum likelihood methods. The second class consists of the monetary model in ACEL. We find that with shortrun restrictions, structural VARs perform remarkably well in all our examples. With longrun restrictions, structural VARs work well as long as technology shocks explain at least a very small portion of the variation in hours worked.
In a number of examples that we consider, VARbased impulse response functions using longrun restrictions exhibit some bias. Even though these examples do not emerge from empirically plausible data generating processes, we find them of interest. They allow us to diagnose what can go wrong with longrun identification schemes. Our diagnosis leads us to propose a modification to the standard VARbased procedure for estimating impulse response functions using longrun identification. This procedure works well in our examples.
Finally, we find that confidence intervals with longrun identification schemes are substantially larger than those with shortrun identification schemes. In all empirically plausible cases, the VARs deliver confidence intervals that accurately reflect the true degree of sampling uncertainty. We view this characteristic as a great virtue of VARbased methods. When the data contain little information, the VAR will indicate the lack of information. To reduce large confidence intervals the analyst must either impose additional identifying restrictions (i.e., use more theory) or obtain better data.
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This appendix describes the ACEL model used in section 6. The model economy is composed of households, firms, and a monetary authority.
There is a continuum of households, indexed by The household is a monopoly supplier of a differentiated labor service, and sets its wage subject to Calvostyle wage frictions. In general, households earn different wage rates and work different amounts. A straightforward extension of arguments in Erceg, Henderson, and Levin (2000) and in Woodford (1996) establishes that in the presence of state contingent securities, households are homogeneous with respect to consumption and asset holdings.^{33} Our notation reflects this result. The preferences of the household are given by:
The household is a monopoly supplier of a differentiated labor service, . It sells this service to a representative, competitive firm that transforms it into an aggregate labor input, using the technology:
In each period, a household faces a constant probability, of being able to reoptimize its nominal wage. The ability to reoptimize is independent across households and time. If a household cannot reoptimize its wage at time it sets according to:
where The presence of implies that there are no distortions from wage dispersion along the steady state growth path.At time a final consumption good, is produced by a perfectly competitive, representative final good firm. This firm produces the final good by combining a continuum of intermediate goods, indexed by using the technology
Intermediate good is produced by a monopolist using the following technology:
Intermediate good firms hire labor in perfectly competitive factor markets at the wage rate, . Profits are distributed to households at the end of each time period. We assume that the firm must borrow the wage bill in advance at the gross interest rate,
In each period, the intermediate goods firm faces a constant probability, of being able to reoptimize its nominal price. The ability to reoptimize prices is independent across firms and time. If firm cannot reoptimize, it sets according to:
Let denote the physical stock of capital available to the firm at the beginning of period The services of capital, are related to stock of physical capital, by:
Here is firm capital utilization rate. The cost, in investment goods, of setting the utilization rate to is where is increasing and convex. We assume that in steady state and These two conditions determine the level and slope of in steady state. To implement our loglinear solution method, we must also specify a value for the curvature of in steady state,There is no technology for transferring capital between firms. The only way a firm can change its stock of physical capital is by varying the rate of investment, over time. The technology for accumulating physical capital by intermediate good firm is given by:
The present discounted value of the intermediate good's net cash flow is given by:
The monetary policy rule is defined by (35) and (36). Financial intermediaries receive from the household. Our notation reflects the equilibrium condition, Financial intermediaries lend all of their money to intermediate good firms, which use the funds to pay labor wages. Loan market clearing requires that:
The aggregate resource constraint is:
We refer the reader to ACEL for a description of how the model is solved and for the methodology used to estimate the model parameters. The data and programs, as well as an extensive technical appendix, may be found at the following website:
www.faculty.econ.northwestern.edu/faculty/christiano/research/ACEL/acelweb.htm.
This appendix generalizes the strategy for longrun identification of one shock to two shocks, using the strategy of Fisher (2006). As before, the VAR is:
To construct our modified VAR procedure, simply replace in (45) by (29).
Time 
TwoShock MLE Specification: Both shocks, ln(hours)  TwoShock MLE Specification: Only Technology Shocks, ln(hours)  TwoShock CKM Specification: Both Shocks, ln(hours)  TwoShock CKM Specification: Only Technology Shocks, ln(hours) 

1  1.57346  1.56093  1.54288  1.55774 
2  1.58337  1.5605  1.57193  1.55714 
3  1.58809  1.55865  1.58755  1.5546 
4  1.59002  1.55497  1.59738  1.54955 
5  1.58387  1.55489  1.57951  1.54944 
6  1.59784  1.55268  1.6205  1.5464 
7  1.59196  1.55952  1.59292  1.5558 
8  1.60947  1.5581  1.64233  1.55385 
9  1.6063  1.56053  1.62777  1.55719 
10  1.60639  1.56131  1.62512  1.55826 
11  1.59256  1.55975  1.58758  1.55612 
12  1.60073  1.55848  1.61142  1.55437 
13  1.60544  1.55861  1.62289  1.55455 
14  1.61011  1.56303  1.62795  1.56062 
15  1.60276  1.56192  1.60768  1.5591 
16  1.59233  1.56456  1.57447  1.56272 
17  1.5918  1.56665  1.57059  1.56559 
18  1.58706  1.56767  1.55683  1.56699 
19  1.59596  1.56688  1.5836  1.5659 
20  1.59523  1.56682  1.58186  1.56582 
21  1.59475  1.56936  1.57728  1.56931 
22  1.58534  1.56511  1.5578  1.56348 
23  1.59005  1.56434  1.57283  1.56242 
24  1.60655  1.56916  1.61197  1.56904 
25  1.60882  1.56731  1.61997  1.5665 
26  1.60803  1.57046  1.61234  1.57082 
27  1.60814  1.57055  1.61182  1.57094 
28  1.60282  1.56997  1.59733  1.57015 
29  1.61685  1.5716  1.63348  1.57239 
30  1.60214  1.57278  1.59006  1.57401 
31  1.60858  1.57194  1.60919  1.57285 
32  1.59613  1.56909  1.57843  1.56894 
33  1.60963  1.56958  1.61537  1.56961 
34  1.61503  1.56856  1.63079  1.56821 
35  1.62583  1.57122  1.65539  1.57186 
36  1.62128  1.5716  1.6402  1.57238 
37  1.61537  1.57097  1.62327  1.57152 
38  1.61601  1.56937  1.62624  1.56933 
39  1.60415  1.57001  1.59165  1.5702 
40  1.61106  1.57231  1.60766  1.57336 
41  1.61268  1.57713  1.60519  1.57998 
42  1.61087  1.57651  1.60111  1.57912 
43  1.60225  1.57822  1.57523  1.58148 
44  1.60623  1.57869  1.5867  1.58212 
45  1.59319  1.5763  1.55494  1.57884 
46  1.58248  1.57832  1.52456  1.58161 
47  1.58525  1.57772  1.53598  1.58079 
48  1.57366  1.57358  1.51233  1.5751 
49  1.56778  1.57494  1.49744  1.57697 
50  1.55676  1.57448  1.4715  1.57634 
51  1.56001  1.577  1.48156  1.5798 
52  1.54993  1.57305  1.46357  1.57438 
53  1.55138  1.57187  1.47384  1.57276 
54  1.55105  1.56798  1.48248  1.56741 
55  1.53253  1.56954  1.43309  1.56956 
56  1.53606  1.57279  1.44378  1.57402 
57  1.52905  1.57079  1.43246  1.57127 
58  1.54096  1.57531  1.46436  1.57748 
59  1.53165  1.57021  1.4502  1.57048 
60  1.52313  1.56594  1.43727  1.56461 
61  1.52199  1.56768  1.43657  1.567 
62  1.52176  1.57016  1.43738  1.5704 
63  1.5289  1.57336  1.4575  1.5748 
64  1.51957  1.57329  1.4362  1.5747 
65  1.51828  1.57658  1.43313  1.57922 
66  1.51638  1.57635  1.43341  1.5789 
67  1.52266  1.57556  1.45683  1.57782 
68  1.51537  1.57303  1.44442  1.57435 
69  1.51154  1.57133  1.44064  1.57202 
70  1.5123  1.56899  1.45037  1.5688 
71  1.51228  1.56548  1.459  1.56398 
72  1.51973  1.56489  1.48361  1.56317 
73  1.52839  1.56613  1.50807  1.56487 
74  1.52732  1.56806  1.50399  1.56752 
75  1.52123  1.5703  1.48589  1.5706 
76  1.51916  1.57298  1.479  1.57428 
77  1.53729  1.57217  1.53275  1.57316 
78  1.5302  1.5716  1.51484  1.57238 
79  1.5178  1.57183  1.48187  1.57271 
80  1.51569  1.57467  1.47477  1.5766 
81  1.50516  1.57259  1.45165  1.57375 
82  1.50613  1.56997  1.46158  1.57015 
83  1.50712  1.57208  1.46455  1.57304 
84  1.49811  1.56965  1.44623  1.56971 
85  1.50877  1.56817  1.48112  1.56768 
86  1.50747  1.56568  1.48318  1.56425 
87  1.50255  1.56441  1.47336  1.56252 
88  1.50314  1.56195  1.4806  1.55913 
89  1.50444  1.5606  1.48784  1.55728 
90  1.49878  1.56041  1.474  1.55703 
91  1.49284  1.55987  1.46038  1.55627 
92  1.48933  1.56291  1.44892  1.56045 
93  1.48503  1.56036  1.44355  1.55695 
94  1.47852  1.561  1.42773  1.55783 
95  1.49108  1.56141  1.46518  1.55839 
96  1.47872  1.55997  1.43533  1.55642 
97  1.47965  1.55819  1.44344  1.55397 
98  1.48673  1.56367  1.45806  1.56149 
99  1.4972  1.56264  1.49064  1.56008 
100  1.51597  1.56349  1.54213  1.56125 
101  1.50403  1.56616  1.50495  1.56491 
102  1.51362  1.56411  1.53487  1.5621 
103  1.52539  1.56377  1.56723  1.56163 
104  1.52507  1.56415  1.5642  1.56216 
105  1.52725  1.56393  1.56903  1.56185 
106  1.54459  1.564  1.61489  1.56195 
107  1.54746  1.5661  1.61661  1.56483 
108  1.53566  1.5643  1.58344  1.56236 
109  1.53471  1.56212  1.58176  1.55937 
110  1.52884  1.56617  1.55797  1.56493 
111  1.5382  1.56551  1.58361  1.56402 
112  1.55043  1.57099  1.60775  1.57155 
113  1.53006  1.57225  1.54752  1.57328 
114  1.52914  1.57379  1.54266  1.57539 
115  1.52378  1.57118  1.53153  1.57181 
116  1.52202  1.56656  1.53333  1.56547 
117  1.52992  1.56825  1.55271  1.56778 
118  1.51915  1.5676  1.52338  1.5669 
119  1.51676  1.56891  1.51546  1.56869 
120  1.52094  1.57114  1.52465  1.57176 
121  1.51878  1.56908  1.52209  1.56892 
122  1.51218  1.56876  1.50489  1.56848 
123  1.50779  1.57257  1.48874  1.57372 
124  1.51508  1.57581  1.5062  1.57816 
125  1.5309  1.578  1.54802  1.58118 
126  1.52722  1.58007  1.53502  1.58402 
127  1.53126  1.58039  1.54622  1.58446 
128  1.51024  1.5784  1.49134  1.58172 
129  1.49767  1.57788  1.45956  1.58101 
130  1.50036  1.57574  1.47301  1.57807 
131  1.51285  1.57925  1.50502  1.58289 
132  1.50385  1.5805  1.48006  1.5846 
133  1.50698  1.57414  1.49983  1.57588 
134  1.51393  1.57665  1.51681  1.57933 
135  1.511  1.5743  1.51286  1.57609 
136  1.51532  1.57135  1.52964  1.57204 
137  1.50779  1.5742  1.50512  1.57595 
138  1.51453  1.57344  1.52582  1.57492 
139  1.50526  1.5711  1.50388  1.5717 
140  1.50822  1.57238  1.51124  1.57346 
141  1.50994  1.57299  1.51592  1.57429 
142  1.50369  1.57157  1.50134  1.57235 
143  1.50943  1.56816  1.52287  1.56766 
144  1.50481  1.56693  1.51199  1.56597 
145  1.5158  1.56562  1.54442  1.56417 
146  1.52346  1.56365  1.56736  1.56146 
147  1.52042  1.56534  1.55495  1.56378 
148  1.52118  1.56572  1.55534  1.56431 
149  1.51791  1.56697  1.54346  1.56603 
150  1.52661  1.56629  1.56765  1.5651 
151  1.51524  1.5637  1.53842  1.56154 
152  1.52396  1.56218  1.56386  1.55945 
153  1.52228  1.56315  1.55632  1.56079 
154  1.52871  1.56171  1.57469  1.5588 
155  1.52939  1.56025  1.57658  1.55679 
156  1.5448  1.56237  1.61391  1.55971 
157  1.54997  1.56442  1.62197  1.56252 
158  1.52977  1.56648  1.56016  1.56535 
159  1.53532  1.56663  1.57413  1.56556 
160  1.53089  1.5629  1.56551  1.56044 
161  1.51606  1.56063  1.52647  1.55732 
162  1.52319  1.56314  1.54252  1.56078 
163  1.51984  1.56239  1.53381  1.55974 
164  1.52843  1.56419  1.55467  1.5622 
165  1.51793  1.56372  1.52554  1.56157 
166  1.52436  1.56611  1.54005  1.56484 
167  1.52852  1.56447  1.55343  1.5626 
168  1.53991  1.56534  1.58267  1.56379 
169  1.54662  1.56552  1.599  1.56403 
170  1.53981  1.56344  1.58071  1.56119 
171  1.53925  1.56352  1.57723  1.56129 
172  1.54097  1.56901  1.57273  1.56883 
173  1.5352  1.57327  1.54979  1.57468 
174  1.52825  1.57468  1.52862  1.57662 
175  1.53071  1.57565  1.53474  1.57795 
176  1.53992  1.57869  1.55638  1.58211 
177  1.53134  1.57532  1.5373  1.5775 
178  1.53506  1.57776  1.54455  1.58085 
179  1.53675  1.57903  1.54768  1.58259 
180  1.54383  1.57625  1.57115  1.57878 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distribution: Lower Bound  Sampling Distribution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0  0  0  0  0  0  0  0 
1  0.224817  0.220434  0.098599  0.337069  0.097945  0.342924  0.090327  0.330482 
2  0.215228  0.2079  0.039281  0.383645  0.037362  0.378438  0.02747  0.361311 
3  0.206048  0.200182  0.01286  0.400865  0.0032  0.403564  0.01473  0.383115 
4  0.197259  0.18902  0.05212  0.417773  0.03497  0.41301  0.04918  0.389581 
5  0.188845  0.179063  0.04676  0.400474  0.0301  0.388226  0.04557  0.364014 
6  0.18079  0.169726  0.04336  0.383369  0.02888  0.368333  0.04555  0.344396 
7  0.173079  0.162209  0.04279  0.371952  0.02709  0.351512  0.04393  0.328352 
8  0.165696  0.152485  0.04527  0.352999  0.02816  0.333134  0.04659  0.308549 
9  0.158629  0.145963  0.03891  0.341741  0.02664  0.318566  0.04382  0.294886 
10  0.151863  0.13846  0.03827  0.328068  0.02669  0.303609  0.04329  0.280302 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distribution: Lower Bound  Sampling Distribution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0  0  0  0  0  0  0  0 
1  0.308693  0.314592  0.04254  0.673878  0.01805  0.647234  0.02779  0.623804 
2  0.295526  0.295759  0.15234  0.783033  0.15649  0.74801  0.16729  0.717305 
3  0.282921  0.281186  0.25529  0.845505  0.25329  0.815666  0.26638  0.78063 
4  0.270853  0.260686  0.31303  0.866041  0.29426  0.815632  0.31242  0.773652 
5  0.2593  0.238182  0.284  0.793435  0.26321  0.739571  0.28485  0.69847 
6  0.24824  0.218456  0.25879  0.751611  0.24347  0.680386  0.26833  0.640736 
7  0.237652  0.202869  0.25245  0.714697  0.22241  0.62815  0.24698  0.593162 
8  0.227515  0.181005  0.24496  0.666628  0.20977  0.571779  0.23657  0.535228 
9  0.217811  0.168637  0.22516  0.635865  0.19168  0.528953  0.21241  0.499599 
10  0.20852  0.155345  0.20497  0.590961  0.17709  0.487783  0.19451  0.462818 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distribution: Lower Bound  Sampling Distribution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0  2.26E18  7E17  7.76E17  5.3E17  5.71E17  5.9E17  6.14E17 
1  0.221812  0.21271  0.028786  0.400301  0.032151  0.39327  0.022389  0.375947 
2  0.212351  0.197342  0.06132  0.445025  0.05106  0.445744  0.06409  0.421712 
3  0.203294  0.185881  0.13224  0.492192  0.10853  0.480296  0.12355  0.453656 
4  0.194622  0.183614  0.17631  0.513344  0.13496  0.502187  0.14997  0.473984 
5  0.186321  0.169369  0.18177  0.507313  0.13459  0.47333  0.15165  0.443525 
6  0.178374  0.151232  0.19338  0.478514  0.14119  0.443653  0.16079  0.413848 
7  0.170765  0.134955  0.19966  0.45798  0.1497  0.41961  0.16993  0.390676 
8  0.163482  0.116794  0.21185  0.442209  0.16213  0.395715  0.18421  0.365182 
9  0.156509  0.104316  0.22515  0.430158  0.16943  0.378062  0.18921  0.350317 
10  0.149833  0.091458  0.23089  0.434803  0.177  0.359915  0.1959  0.333271 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distribution: Lower Bound  Sampling Distribution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0  7.9E18  9.5E17  0  6.4E17  4.82E17  8.3E17  3.92E17 
1  0.308693  0.302079  0.06073  0.692116  0.04332  0.647474  0.05725  0.621147 
2  0.295526  0.288764  0.2043  0.800781  0.17862  0.756148  0.1972  0.717048 
3  0.282921  0.279528  0.30843  0.81611  0.26766  0.826717  0.28859  0.783652 
4  0.270853  0.258725  0.34187  0.843412  0.30685  0.824295  0.32684  0.781213 
5  0.2593  0.241813  0.29098  0.77331  0.27768  0.76131  0.30129  0.720517 
6  0.24824  0.226933  0.3035  0.732467  0.25613  0.709991  0.27994  0.671858 
7  0.237652  0.214981  0.28693  0.699766  0.24037  0.670333  0.26315  0.635784 
8  0.227515  0.197313  0.27941  0.658455  0.23445  0.629075  0.25843  0.593517 
9  0.217811  0.187768  0.26733  0.627373  0.224  0.599539  0.24221  0.569403 
10  0.20852  0.176062  0.25425  0.605437  0.21766  0.569781  0.2331  0.542569 
Period After Shock  Recursive MLE: .95 line  Recursive MLE: 2 Shock Standard  Recursive MLE: 3 Shock Standard  Recursive CKM: .95 line  Recursive CKM: 2 Shock Standard  Recursive CKM: 3 Shock Standard 

0  0.95  0.95  
1  0.95  0.932  0.941  0.95  0.933  0.932 
2  0.95  0.943  0.942  0.95  0.943  0.939 
3  0.95  0.936  0.926  0.95  0.943  0.934 
4  0.95  0.928  0.938  0.95  0.936  0.944 
5  0.95  0.92  0.916  0.95  0.937  0.937 
6  0.95  0.91  0.912  0.95  0.931  0.937 
7  0.95  0.903  0.895  0.95  0.925  0.93 
8  0.95  0.891  0.871  0.95  0.903  0.917 
9  0.95  0.887  0.864  0.95  0.892  0.92 
10  0.95  0.879  0.853  0.95  0.869  0.908 
Period After Shock  Nonrecursive MLE: .95 line  Nonrecursive MLE: 2 Shock Standard  Nonrecursive MLE: 2 Shock Bartlett  Nonrecursive MLE: 3 Shock Standard  Nonrecursive MLE: 3 Shock Bartlett  Nonrecursive CKM: .95 line  Nonrecursive CKM: 2 Shock Standard  Nonrecursive CKM: 2 Shock Bartlett  Nonrecursive CKM: 3 Shock Standard  Nonrecursive CKM: 3 Shock Bartlett 

0  0.95  0.949  0.991  0.963  0.993  0.95  0.888  0.999  0.911  0.998 
1  0.95  0.941  0.989  0.961  0.988  0.95  0.895  0.999  0.913  1 
2  0.95  0.943  0.987  0.962  0.993  0.95  0.91  0.997  0.929  0.999 
3  0.95  0.939  0.989  0.96  0.991  0.95  0.911  0.997  0.94  0.999 
4  0.95  0.94  0.984  0.961  0.989  0.95  0.924  0.997  0.945  0.999 
5  0.95  0.94  0.976  0.959  0.986  0.95  0.934  0.996  0.945  0.998 
6  0.95  0.94  0.972  0.961  0.984  0.95  0.946  0.995  0.947  0.998 
7  0.95  0.935  0.962  0.959  0.983  0.95  0.954  0.993  0.959  0.995 
8  0.95  0.928  0.956  0.962  0.981  0.95  0.959  0.987  0.967  0.993 
9  0.95  0.926  0.946  0.965  0.978  0.95  0.965  0.984  0.975  0.992 
10  0.95  0.922  0.937  0.967  0.977  0.95  0.963  0.982  0.978  0.992 
Period after shock  Recursive MLE: .95 percent  Recursive MLE: 2 Shock Standard  Recursive MLE: 3 Shock Standard  Recursive CKM: .95 percent  Recursive CKM: 2 Shock Standard  Recursive CKM: 3 Shock Standard 

0  0.95  0.95  
1  0.95  0.936  0.938  0.95  0.931  0.928 
2  0.95  0.946  0.943  0.95  0.942  0.93 
3  0.95  0.937  0.934  0.95  0.945  0.935 
4  0.95  0.927  0.926  0.95  0.938  0.934 
5  0.95  0.926  0.912  0.95  0.935  0.936 
6  0.95  0.927  0.907  0.95  0.934  0.927 
7  0.95  0.914  0.897  0.95  0.924  0.926 
8  0.95  0.907  0.882  0.95  0.91  0.916 
9  0.95  0.899  0.871  0.95  0.892  0.912 
10  0.95  0.885  0.861  0.95  0.874  0.902 
Period after shock  Nonrecursive MLE: .95 percent  Nonrecursive MLE: 2 Shock Standard  Nonrecursive MLE: 2 Shock Bartlett  Nonrecursive MLE: 3 Shock Standard  Nonrecursive MLE: 3 Shock Bartlett  Nonrecursive CKM: .95 percent  Nonrecursive CKM: 2 Shock Standard  Nonrecursive CKM: 2 Shock Bartlett  Nonrecursive CKM: 3 Shock Standard  Nonrecursive CKM: 3 Shock Bartlett 

0  0.95  0.814  0.912  0.808  0.957  0.95  0.713  0.929  0.739  0.943 
1  0.95  0.811  0.912  0.799  0.95  0.95  0.714  0.934  0.746  0.937 
2  0.95  0.816  0.904  0.814  0.942  0.95  0.728  0.931  0.76  0.938 
3  0.95  0.816  0.905  0.815  0.939  0.95  0.758  0.93  0.77  0.936 
4  0.95  0.818  0.892  0.82  0.931  0.95  0.754  0.923  0.778  0.935 
5  0.95  0.81  0.883  0.822  0.922  0.95  0.779  0.913  0.784  0.927 
6  0.95  0.822  0.873  0.823  0.921  0.95  0.798  0.913  0.797  0.921 
7  0.95  0.822  0.859  0.827  0.917  0.95  0.819  0.899  0.821  0.918 
8  0.95  0.818  0.844  0.835  0.914  0.95  0.842  0.895  0.843  0.915 
9  0.95  0.82  0.825  0.836  0.909  0.95  0.858  0.885  0.851  0.916 
10  0.95  0.818  0.811  0.84  0.903  0.95  0.873  0.874  0.864  0.918 
Period After Shock  TwoShock MLE specification, Standard: Response: TRUE  TwoShock MLE specification, Standard: Response: Estimated  TwoShock MLE specification, Standard: Sampling Distribution: Lower Bound  TwoShock MLE specification, Standard:Sampling Distribution: Upper Bound  TwoShock MLE specification, Standard: Average CI Standard Deviation Based: Lower Bound  TwoShock MLE specification, Standard: Average CI Standard Deviation Based: Upper Bound  TwoShock MLE specification, Standard: Average CI Percentile Based: Lower Bound  TwoShock MLE specification, Standard: Average CI Percentile Based: Upper Bound  Twoshock MLE specification, Bartlett: Response: TRUE  Twoshock MLE specification, Bartlett: Response: Estimated  Twoshock MLE specification, Bartlett: Sampling Distribution: Lower Bound  Twoshock MLE specification, Bartlett: Sampling Distribution: Upper Bound  Twoshock MLE specification, Bartlett: Average CI Standard Deviation Based: Lower Bound  Twoshock MLE specification, Bartlett: Average CI Standard Deviation Based: Upper Bound  Twoshock MLE specification, Bartlett: Average CI Percentile Based: Lower Bound  Twoshock MLE specification, Bartlett: Average CI Percentile Based: Upper Bound 

0  0.231057  0.289676  0.56289  0.810931  0.24935  0.828706  0.36018  0.65647  0.231057  0.105155  0.48134  0.632083  0.43523  0.645539  0.42096  0.633054 
1  0.221202  0.281535  0.54349  0.804135  0.2481  0.811174  0.35252  0.651572  0.221202  0.100249  0.46359  0.60489  0.41372  0.614216  0.3997  0.60453 
2  0.211766  0.271501  0.52456  0.800842  0.24725  0.790255  0.34876  0.639333  0.211766  0.095507  0.42724  0.585615  0.39252  0.583536  0.37927  0.576247 
3  0.202734  0.262708  0.54301  0.799717  0.24628  0.771695  0.34153  0.630416  0.202734  0.092642  0.42826  0.564985  0.3716  0.556884  0.35959  0.551648 
4  0.194086  0.249743  0.52404  0.775162  0.25133  0.750821  0.3414  0.615344  0.194086  0.085304  0.41533  0.543163  0.35536  0.525964  0.34106  0.519757 
5  0.185808  0.239338  0.50002  0.762952  0.23762  0.716298  0.32426  0.590341  0.185808  0.081458  0.39563  0.517728  0.32766  0.490577  0.31405  0.48332 
6  0.177883  0.230296  0.48517  0.741558  0.22866  0.689257  0.3112  0.571273  0.177883  0.077931  0.37354  0.492491  0.30705  0.462909  0.29344  0.45667 
7  0.170295  0.220451  0.47291  0.727716  0.21755  0.658456  0.29606  0.548969  0.170295  0.074566  0.36034  0.470608  0.2838  0.432935  0.27098  0.427365 
8  0.163032  0.210401  0.45182  0.717407  0.20899  0.629787  0.28339  0.527136  0.163032  0.070735  0.33841  0.453983  0.26434  0.405814  0.25275  0.400616 
9  0.156078  0.201652  0.4373  0.700829  0.19994  0.603249  0.27079  0.507401  0.156078  0.06781  0.31753  0.443913  0.24569  0.38131  0.23473  0.376393 
10  0.14942  0.193404  0.41767  0.689355  0.19247  0.57928  0.2598  0.489705  0.14942  0.065028  0.30679  0.425891  0.2296  0.359658  0.21978  0.35503 
Period After Shock  ThreeShock MLE specification, Standard: Response: TRUE  ThreeShock MLE specification, Standard: Response: Estimated  ThreeShock MLE specification, Standard: Sampling Distribution: Lower Bound  ThreeShock MLE specification, Standard:Sampling Distribution: Upper Bound  ThreeShock MLE specification, Standard: Average CI Standard Deviation Based: Lower Bound  ThreeShock MLE specification, Standard: Average CI Standard Deviation Based: Upper Bound  ThreeShock MLE specification, Standard: Average CI Percentile Based: Lower Bound  ThreeShock MLE specification, Standard: Average CI Percentile Based: Upper Bound  Threeshock MLE specification, Bartlett: Response: TRUE  Threeshock MLE specification, Bartlett: Response: Estimated  Threeshock MLE specification, Bartlett: Sampling Distribution: Lower Bound  Threeshock MLE specification, Bartlett: Sampling Distribution: Upper Bound  Threeshock MLE specification, Bartlett: Average CI Standard Deviation Based: Lower Bound  Threeshock MLE specification, Bartlett: Average CI Standard Deviation Based: Upper Bound  Threeshock MLE specification, Bartlett: Average CI Percentile Based: Lower Bound  Threeshock MLE specification, Bartlett: Average CI Percentile Based: Upper Bound 

0  0.227968  0.321596  0.94455  1.177234  0.52003  1.163218  0.62267  0.953856  0.227968  0.171726  0.65892  0.998247  0.69831  1.041767  0.66269  1.039794 
1  0.218245  0.30471  0.93827  1.205732  0.53238  1.141802  0.62544  0.950611  0.218245  0.162159  0.65379  0.971568  0.65786  0.982179  0.62595  0.976519 
2  0.208936  0.288413  0.94373  1.169734  0.53548  1.11231  0.62259  0.936964  0.208936  0.150499  0.65214  0.929636  0.6225  0.9235  0.59274  0.919375 
3  0.200024  0.282122  0.92895  1.187192  0.52843  1.092675  0.61174  0.927821  0.200024  0.145215  0.62126  0.892868  0.58637  0.876797  0.55756  0.874256 
4  0.191492  0.271278  0.90445  1.175605  0.52425  1.066803  0.60779  0.904703  0.191492  0.13654  0.58727  0.850243  0.56055  0.833629  0.5337  0.828337 
5  0.183324  0.255028  0.87923  1.160452  0.50895  1.019009  0.58744  0.866951  0.183324  0.128196  0.55787  0.800487  0.52537  0.781759  0.4988  0.77528 
6  0.175505  0.242805  0.8703  1.147439  0.5012  0.986812  0.57542  0.844939  0.175505  0.121769  0.5323  0.768648  0.50253  0.746066  0.47486  0.740424 
7  0.168019  0.230056  0.8571  1.118676  0.49071  0.950822  0.56084  0.817033  0.168019  0.115281  0.51396  0.735898  0.47683  0.707392  0.44928  0.702422 
8  0.160852  0.217086  0.84291  1.107298  0.48596  0.920136  0.5517  0.793303  0.160852  0.108222  0.48765  0.706562  0.45791  0.674356  0.43089  0.669156 
9  0.153991  0.206461  0.83631  1.07805  0.47837  0.891297  0.53947  0.773561  0.153991  0.102808  0.46985  0.684111  0.43875  0.644366  0.41228  0.640125 
10  0.147423  0.196914  0.81458  1.07255  0.47234  0.866163  0.52921  0.757129  0.147423  0.097773  0.45007  0.666064  0.4229  0.618448  0.3978  0.616195 
Period After Shock  TwoShock CKM specification, Standard: Response: TRUE  TwoShock CKM specification, Standard: Response: Estimated  TwoShock CKM specification, Standard: Sampling Distribution: Lower Bound  CKMShock MLE specification, Standard:Sampling Distribution: Upper Bound  TwoShock CKM specification, Standard: Average CI Standard Deviation Based: Lower Bound  TwoShock CKM specification, Standard: Average CI Standard Deviation Based: Upper Bound  TwoShock CKM specification, Standard: Average CI Percentile Based: Lower Bound  TwoShock CKM specification, Standard: Average CI Percentile Based: Upper Bound  Twoshock CKM specification, Bartlett: Response: TRUE  Twoshock CKM specification, Bartlett: Response: Estimated  Twoshock CKM specification, Bartlett: Sampling Distribution: Lower Bound  Twoshock CKM specification, Bartlett: Sampling Distribution: Upper Bound  Twoshock CKM specification, Bartlett: Average CI Standard Deviation Based: Lower Bound  Twoshock CKM specification, Bartlett: Average CI Standard Deviation Based: Upper Bound  Twoshock CKM specification, Bartlett: Average CI Percentile Based: Lower Bound  Twoshock CKM specification, Bartlett: Average CI Percentile Based: Upper Bound 

0  0.317261  0.965227  0.5737  2.041564  0.23723  2.167689  0.54776  1.754532  0.317261  0.401947  1.06463  1.77546  0.95284  1.756733  0.95562  1.734179 
1  0.303728  0.914762  0.5517  2.052182  0.24142  2.07094  0.52804  1.698675  0.303728  0.37738  0.99964  1.659215  0.87072  1.625484  0.869  1.611061 
2  0.290773  0.862236  0.58651  1.896666  0.25437  1.978844  0.51244  1.641089  0.290773  0.358631  0.89623  1.56193  0.79641  1.51367  0.8029  1.501026 
3  0.27837  0.813542  0.62769  1.83466  0.2612  1.888288  0.49405  1.584659  0.27837  0.336563  0.84174  1.445324  0.7307  1.40383  0.73075  1.402262 
4  0.266497  0.757098  0.57787  1.773265  0.26403  1.778225  0.48408  1.497654  0.266497  0.313335  0.80083  1.3124  0.65894  1.285612  0.66792  1.275868 
5  0.25513  0.696839  0.5213  1.695599  0.2468  1.640481  0.44142  1.390243  0.25513  0.287873  0.71307  1.212097  0.57857  1.15432  0.58406  1.144804 
6  0.244248  0.646706  0.49512  1.605174  0.23834  1.531754  0.40723  1.309435  0.244248  0.266012  0.67691  1.118517  0.52158  1.053608  0.52553  1.046025 
7  0.23383  0.59587  0.4539  1.560015  0.22609  1.417834  0.37322  1.222341  0.23383  0.244787  0.59762  1.040434  0.46167  0.95125  0.4631  0.945924 
8  0.223856  0.547017  0.42835  1.49176  0.22499  1.319028  0.3532  1.143076  0.223856  0.223118  0.55751  0.9644  0.41996  0.866198  0.42055  0.862044 
9  0.214308  0.505281  0.38415  1.418986  0.21635  1.226913  0.32337  1.073589  0.214308  0.205925  0.50252  0.902687  0.37653  0.78838  0.37433  0.789189 
10  0.205167  0.467497  0.35197  1.361024  0.21042  1.145417  0.29882  1.010757  0.205167  0.190531  0.46913  0.863785  0.34116  0.722227  0.33682  0.727048 
Period After Shock  ThreeShock CKM specification, Standard: Response: TRUE  ThreeShock CKM specification, Standard: Response: Estimated  ThreeShock CKM specification, Standard: Sampling Distribution: Lower Bound  ThreeShock CKM specification, Standard:Sampling Distribution: Upper Bound  ThreeShock CKM specification, Standard: Average CI Standard Deviation Based: Lower Bound  ThreeShock CKM specification, Standard: Average CI Standard Deviation Based: Upper Bound  ThreeShock CKM specification, Standard: Average CI Percentile Based: Lower Bound  ThreeShock CKM specification, Standard: Average CI Percentile Based: Upper Bound  Threeshock CKM specification, Bartlett: Response: TRUE  Threeshock CKM specification, Bartlett: Response: Estimated  Threeshock CKM specification, Bartlett: Sampling Distribution: Lower Bound  Threeshock CKM specification, Bartlett: Sampling Distribution: Upper Bound  Threeshock CKM specification, Bartlett: Average CI Standard Deviation Based: Lower Bound  Threeshock CKM specification, Bartlett: Average CI Standard Deviation Based: Upper Bound  Threeshock CKM specification, Bartlett: Average CI Percentile Based: Lower Bound  Threeshock CKM specification, Bartlett: Average CI Percentile Based: Upper Bound 

0  0.317261  0.851112  0.90259  2.127806  0.40053  2.102756  0.63095  1.76735  0.317261  0.404461  1.16389  1.837508  1.07281  1.881728  1.03476  1.897182 
1  0.303728  0.801645  0.87738  2.109122  0.39827  2.001562  0.60661  1.703234  0.303728  0.37956  1.05439  1.701848  0.95791  1.717034  0.9304  1.72348 
2  0.290773  0.770071  0.84958  2.07164  0.38991  1.930049  0.58181  1.651805  0.290773  0.365329  0.94598  1.572458  0.86518  1.595842  0.84098  1.603683 
3  0.27837  0.731267  0.78884  2.009485  0.39109  1.853628  0.56648  1.598786  0.27837  0.342985  0.91632  1.513652  0.7957  1.481667  0.77472  1.493743 
4  0.266497  0.693385  0.72295  1.92564  0.36666  1.75343  0.5314  1.516108  0.266497  0.326585  0.86883  1.439891  0.71703  1.370203  0.70772  1.375656 
5  0.25513  0.656799  0.60293  1.815994  0.32063  1.634232  0.46767  1.422802  0.25513  0.311235  0.75075  1.322529  0.62412  1.246591  0.61411  1.249644 
6  0.244248  0.625136  0.57637  1.733189  0.29391  1.544177  0.41964  1.357026  0.244248  0.298135  0.65295  1.288795  0.56011  1.156377  0.54712  1.163023 
7  0.23383  0.591828  0.50264  1.651405  0.27084  1.454497  0.38222  1.287398  0.23383  0.283388  0.60395  1.223977  0.49973  1.066505  0.48668  1.075609 
8  0.223856  0.559127  0.43887  1.572853  0.26133  1.379589  0.35995  1.22807  0.223856  0.268636  0.53633  1.156231  0.4575  0.994768  0.44405  1.002715 
9  0.214308  0.530744  0.41011  1.471036  0.2494  1.310886  0.33421  1.174993  0.214308  0.256049  0.48587  1.082434  0.41687  0.928965  0.40242  0.938054 
10  0.205167  0.504108  0.38807  1.404788  0.24266  1.250874  0.31447  1.130154  0.205167  0.244043  0.4462  1.036987  0.3856  0.873685  0.36862  0.885256 
Period After Shock  MLE specification, : Response: TRUE  MLE specification, : Response: Estimated  MLE specification, : Sampling Distrbution: Lower Bound  MLE specification, : Sampling Distrbution: Upper Bound  MLE specification, : Average CI Standard Deviation Based: Lower Bound  MLE specification, : Average CI Standard Deviation Based: Upper Bound  MLE specification, : Average CI Percentile Based: Lower Bound  MLE specification, : Average CI Percentile Based: Upper Bound  CKM specification, : Response: TRUE  CKM specification, : Response: Estimated  CKM specification, : Sampling Distrbution: Lower Bound  CKM specification, : Sampling Distrbution: Upper Bound  CKM specification, : Average CI Standard Deviation Based: Lower Bound  CKM specification, : Average CI Standard Deviation Based: Upper Bound  CKM specification, : Average CI Percentile Based: Lower Bound  CKM specification, : Average CI Percentile Based: Upper Bound 

0  0.231057  0.289676  0.56289  0.810931  0.24935  0.828706  0.36018  0.65647  0.317261  0.965227  0.5737  2.041564  0.23723  2.167689  0.54776  1.754532 
1  0.221202  0.281535  0.54349  0.804135  0.2481  0.811174  0.35252  0.651572  0.303728  0.914762  0.5517  2.052182  0.24142  2.07094  0.52804  1.698675 
2  0.211766  0.271501  0.52456  0.800842  0.24725  0.790255  0.34876  0.639333  0.290773  0.862236  0.58651  1.896666  0.25437  1.978844  0.51244  1.641089 
3  0.202734  0.262708  0.54301  0.799717  0.24628  0.771695  0.34153  0.630416  0.27837  0.813542  0.62769  1.83466  0.2612  1.888288  0.49405  1.584659 
4  0.194086  0.249743  0.52404  0.775162  0.25133  0.750821  0.3414  0.615344  0.266497  0.757098  0.57787  1.773265  0.26403  1.778225  0.48408  1.497654 
5  0.185808  0.239338  0.50002  0.762952  0.23762  0.716298  0.32426  0.590341  0.25513  0.696839  0.5213  1.695599  0.2468  1.640481  0.44142  1.390243 
6  0.177883  0.230296  0.48517  0.741558  0.22866  0.689257  0.3112  0.571273  0.244248  0.646706  0.49512  1.605174  0.23834  1.531754  0.40723  1.309435 
7  0.170295  0.220451  0.47291  0.727716  0.21755  0.658456  0.29606  0.548969  0.23383  0.59587  0.4539  1.560015  0.22609  1.417834  0.37322  1.222341 
8  0.163032  0.210401  0.45182  0.717407  0.20899  0.629787  0.28339  0.527136  0.223856  0.547017  0.42835  1.49176  0.22499  1.319028  0.3532  1.143076 
9  0.156078  0.201652  0.4373  0.700829  0.19994  0.603249  0.27079  0.507401  0.214308  0.505281  0.38415  1.418986  0.21635  1.226913  0.32337  1.073589 
10  0.14942  0.193404  0.41767  0.689355  0.19247  0.57928  0.2598  0.489705  0.205167  0.467497  0.35197  1.361024  0.21042  1.145417  0.29882  1.010757 
Period After Shock  MLE specification, /2: Response: TRUE  MLE specification, /2: Response: Estimated  MLE specification, /2: Sampling Distrbution: Lower Bound  MLE specification, /2: Sampling Distrbution: Upper Bound  MLE specification, /2: Average CI Standard Deviation Based: Lower Bound  MLE specification, /2: Average CI Standard Deviation Based: Upper Bound  MLE specification, /2: Average CI Percentile Based: Lower Bound  MLE specification, /2: Average CI Percentile Based: Upper Bound  CKM specification, /2: Response: TRUE  CKM specification, /2: Response: Estimated  CKM specification, /2: Sampling Distrbution: Lower Bound  CKM specification, /2: Sampling Distrbution: Upper Bound  CKM specification, /2: Average CI Standard Deviation Based: Lower Bound  CKM specification, /2: Average CI Standard Deviation Based: Upper Bound  CKM specification, /2: Average CI Percentile Based: Lower Bound  CKM specification, /2: Average CI Percentile Based: Upper Bound 

0  0.231057  0.210906  0.23928  0.44873  0.07263  0.494439  0.1528  0.38436  0.317261  0.426006  0.4685  1.00459  0.20548  1.057492  0.33593  0.869008 
1  0.221202  0.20237  0.22924  0.441947  0.0769  0.48164  0.1519  0.380285  0.303728  0.406312  0.44723  0.981755  0.19992  1.012548  0.3219  0.841089 
2  0.211766  0.192385  0.22934  0.439963  0.0832  0.467969  0.15361  0.373439  0.290773  0.388226  0.43328  0.952395  0.19792  0.974369  0.30987  0.817935 
3  0.202734  0.182589  0.22297  0.445458  0.08847  0.453648  0.15611  0.364797  0.27837  0.37193  0.41554  0.940841  0.194  0.937861  0.29692  0.795577 
4  0.194086  0.175374  0.2445  0.436871  0.09073  0.441477  0.1544  0.357013  0.266497  0.348006  0.38115  0.908067  0.19679  0.892798  0.29209  0.759203 
5  0.185808  0.167271  0.2316  0.42552  0.08582  0.420367  0.14623  0.341343  0.25513  0.320667  0.3409  0.869498  0.18249  0.823825  0.26749  0.705256 
6  0.177883  0.160808  0.21755  0.412886  0.0825  0.404112  0.13921  0.329518  0.244248  0.298194  0.3103  0.822821  0.17264  0.769026  0.2467  0.664448 
7  0.170295  0.153599  0.20534  0.403814  0.07836  0.385556  0.13158  0.315702  0.23383  0.275265  0.2935  0.779299  0.16176  0.712288  0.22655  0.619661 
8  0.163032  0.147148  0.19547  0.397502  0.07454  0.36883  0.12436  0.303493  0.223856  0.253617  0.27301  0.753587  0.15391  0.661145  0.21043  0.579449 
9  0.156078  0.14063  0.18522  0.394737  0.07166  0.352918  0.11861  0.291455  0.214308  0.234932  0.2597  0.715045  0.14539  0.615257  0.19282  0.543699 
10  0.14942  0.134519  0.17494  0.387264  0.06939  0.338426  0.11354  0.280469  0.205167  0.217668  0.23863  0.690665  0.13917  0.574505  0.17854  0.512139 
Period After Shock  MLE specification, /4: Response: TRUE  MLE specification, /4: Response: Estimated  MLE specification, /4: Sampling Distrbution: Lower Bound  MLE specification, /4: Sampling Distrbution: Upper Bound  MLE specification, /4: Average CI Standard Deviation Based: Lower Bound  MLE specification, /4: Average CI Standard Deviation Based: Upper Bound  MLE specification, /4: Average CI Percentile Based: Lower Bound  MLE specification, /4: Average CI Percentile Based: Upper Bound  CKM specification, /4: Response: TRUE  CKM specification, /4: Response: Estimated  CKM specification, /4: Sampling Distrbution: Lower Bound  CKM specification, /4: Sampling Distrbution: Upper Bound  CKM specification, /4: Average CI Standard Deviation Based: Lower Bound  CKM specification, /4: Average CI Standard Deviation Based: Upper Bound  CKM specification, /4: Average CI Percentile Based: Lower Bound  CKM specification, /4: Average CI Percentile Based: Upper Bound 

0  0.231057  0.202356  0.03268  0.307991  0.052635  0.352076  0.00977  0.276081  0.317261  0.318734  0.1179  0.602859  0.00947  0.646935  0.10575  0.522908 
1  0.221202  0.192648  0.02878  0.313275  0.042578  0.342717  0.01603  0.273672  0.303728  0.305368  0.10075  0.588801  0.01315  0.623883  0.10211  0.512215 
2  0.211766  0.183896  0.03732  0.319046  0.032986  0.334805  0.02203  0.270267  0.290773  0.289948  0.11947  0.585012  0.01929  0.59919  0.09995  0.497898 
3  0.202734  0.174894  0.04674  0.313968  0.024074  0.325714  0.02728  0.26557  0.27837  0.277487  0.12012  0.558601  0.02454  0.579511  0.09921  0.486508 
4  0.194086  0.165596  0.0332  0.307502  0.017589  0.313602  0.02964  0.257551  0.266497  0.262405  0.12845  0.53979  0.03048  0.555293  0.09789  0.468251 
5  0.185808  0.156648  0.0404  0.300271  0.01484  0.298456  0.02919  0.245466  0.25513  0.244072  0.11009  0.513599  0.02919  0.517338  0.09023  0.437761 
6  0.177883  0.149376  0.03609  0.293449  0.012269  0.286482  0.02828  0.236537  0.244248  0.227821  0.11109  0.494903  0.02993  0.485572  0.08368  0.413597 
7  0.170295  0.141602  0.03059  0.277448  0.009863  0.273341  0.02776  0.225762  0.23383  0.211666  0.08877  0.473493  0.02984  0.453171  0.07687  0.388479 
8  0.163032  0.135085  0.02771  0.272083  0.008232  0.261938  0.02592  0.217155  0.223856  0.197013  0.07723  0.454074  0.02999  0.42402  0.07049  0.366278 
9  0.156078  0.128097  0.02637  0.265127  0.00592  0.250275  0.02567  0.207617  0.214308  0.182936  0.07005  0.439053  0.03116  0.397028  0.06591  0.344898 
10  0.14942  0.121609  0.0245  0.259728  0.003647  0.239571  0.02523  0.199064  0.205167  0.170125  0.06049  0.417588  0.03271  0.37296  0.06197  0.326313 
Period After Shock 
Standard: Response: TRUE  Standard: Response: Estimated  Standard: Sampling Distribution: Lower Bound  Standard: Sampling Distribution: Upper Bound 
Standard: Average CI Standard Deviation Based: Lower Bound 
Standard: Average CI Standard Deviation Based: Upper Bound  Standard: Average CI Percentile Based: Lower Bound  Standard: Average CI Percentile Based: Upper Bound  Bartlett: Response: TRUE  Bartlett: Response: Estimated  Bartlett: Sampling Distribution: Lower Bound  Bartlett: Sampling Distribution: Upper Bound  Bartlett: Average CI Standard Deviation Based: Lower Bound  Bartlett: Average CI Standard Deviation Based: Upper Bound  Bartlett: Average CI Percentile Based: Lower Bound  Bartlett: Average CI Percentile Based: Upper Bound 

0  0.122083  0.530941  0.76794  1.454371  0.43084  1.492717  0.61339  1.220703  0.122083  0.168674  1.03915  1.274497  0.86488  1.202232  0.82025  1.223388 
1  0.117775  0.507677  0.78177  1.411787  0.42204  1.437398  0.5918  1.187444  0.117775  0.160606  0.97209  1.193033  0.80239  1.123596  0.75818  1.147591 
2  0.113618  0.487617  0.70523  1.395412  0.41134  1.386573  0.56543  1.15754  0.113618  0.1554  0.92088  1.101551  0.74323  1.05403  0.70118  1.080689 
3  0.109609  0.464566  0.72735  1.347642  0.40532  1.334449  0.54673  1.124306  0.109609  0.14859  0.88088  1.041175  0.69222  0.9894  0.65537  1.014775 
4  0.105741  0.433406  0.67353  1.277555  0.40311  1.269926  0.53894  1.071706  0.105741  0.13693  0.83113  0.964751  0.64105  0.914911  0.60919  0.930973 
5  0.102009  0.404719  0.59728  1.22969  0.37146  1.180896  0.4933  1.003061  0.102009  0.128985  0.73438  0.915514  0.57157  0.829534  0.54275  0.844298 
6  0.09841  0.380428  0.56737  1.181033  0.34841  1.109261  0.45832  0.950038  0.09841  0.122118  0.66284  0.863069  0.51845  0.762689  0.49313  0.776887 
7  0.094937  0.354807  0.52225  1.12587  0.3241  1.033715  0.42299  0.891757  0.094937  0.114538  0.617  0.788495  0.46479  0.693863  0.44325  0.708376 
8  0.091586  0.328211  0.4816  1.08809  0.30878  0.965202  0.39868  0.83684  0.091586  0.105248  0.56279  0.737689  0.42369  0.634185  0.40502  0.647067 
9  0.088354  0.30685  0.4523  1.04743  0.28906  0.902758  0.36788  0.788929  0.088354  0.099128  0.52149  0.693609  0.38266  0.580915  0.36563  0.595468 
10  0.085236  0.287438  0.42084  1.004767  0.27299  0.84787  0.34214  0.74673  0.085236  0.093473  0.49107  0.660393  0.3485  0.535443  0.33308  0.551339 
Period After Shock 
Standard: Response: TRUE  Standard: Response: Estimated  Standard: Sampling Distribution: Lower Bound  Standard: Sampling Distribution: Upper Bound 
Standard: Average CI Standard Deviation Based: Lower Bound 
Standard: Average CI Standard Deviation Based: Upper Bound  Standard: Average CI Percentile Based: Lower Bound  Standard: Average CI Percentile Based: Upper Bound  Bartlett: Response: TRUE  Bartlett: Response: Estimated  Bartlett: Sampling Distribution: Lower Bound  Bartlett: Sampling Distribution: Upper Bound  Bartlett: Average CI Standard Deviation Based: Lower Bound  Bartlett: Average CI Standard Deviation Based: Upper Bound  Bartlett: Average CI Percentile Based: Lower Bound  Bartlett: Average CI Percentile Based: Upper Bound 

0  0.475978  1.886516  0.36851  3.219048  0.230943  3.542089  0.31862  2.8741  0.475978  0.713306  1.59699  2.57335  1.26399  2.690598  1.19064  2.739748 
1  0.452677  1.778356  0.50613  3.146697  0.174209  3.382503  0.32836  2.786861  0.452677  0.67716  1.36983  2.483957  1.13299  2.48731  1.06844  2.532761 
2  0.430518  1.681075  0.27372  3.098105  0.121613  3.240538  0.33032  2.701988  0.430518  0.641294  1.29145  2.414946  1.02863  2.31122  0.95812  2.359559 
3  0.409443  1.586166  0.45737  2.980229  0.067145  3.105187  0.33912  2.613931  0.409443  0.613208  1.21342  2.2316  0.93201  2.158423  0.86542  2.2152 
4  0.389399  1.457312  0.40756  2.862199  0.009759  2.904865  0.35874  2.460654  0.389399  0.557095  1.11997  2.12092  0.84944  1.963631  0.79024  2.0132 
5  0.370337  1.329944  0.33519  2.695049  0.02083  2.680721  0.34272  2.28272  0.370337  0.50662  1.00843  1.994265  0.74546  1.758701  0.69285  1.797438 
6  0.352208  1.22469  0.30175  2.510417  0.04916  2.498538  0.32285  2.143125  0.352208  0.466244  0.92188  1.842566  0.66617  1.598654  0.61022  1.637799 
7  0.334967  1.118379  0.24402  2.376348  0.07401  2.310766  0.30481  1.991026  0.334967  0.425467  0.8204  1.703178  0.58852  1.439451  0.53472  1.473908 
8  0.318569  1.021517  0.21816  2.261767  0.10753  2.150562  0.29953  1.864883  0.318569  0.385281  0.74664  1.596839  0.53877  1.309335  0.48632  1.342865 
9  0.302974  0.93353  0.19285  2.138047  0.1271  1.994159  0.28001  1.74181  0.302974  0.352355  0.67026  1.506221  0.48207  1.186781  0.42832  1.224433 
10  0.288143  0.854252  0.17209  2.033628  0.14542  1.85392  0.2636  1.633743  0.288143  0.322297  0.58746  1.382422  0.43701  1.081603  0.38097  1.123652 
Period After Shock  Standard: Response: TRUE  Standard: Response: Estimated  Standard: Sampling Distrbution: Lower Bound  Standard: Sampling Distrbution: Upper Bound  Standard: Average CI Standard Deviation Based: Lower Bound  Standard: Average CI Standard Deviation Based: Upper Bound  Standard: Average CI Percentile Based: Lower Bound  Standard: Average CI Percentile Based: Upper Bound  Bartlett: Response: TRUE  Bartlett: Response: Estimated  Bartlett: Sampling Distrbution: Lower Bound  Bartlett: Sampling Distrbution: Upper Bound  Bartlett: Average CI Standard Deviation Based: Lower Bound  Bartlett: Average CI Standard Deviation Based: Upper Bound  Bartlett: Average CI Percentile Based: Lower Bound  Bartlett: Average CI Percentile Based: Upper Bound 

0  0.475978  4.924118  1.601438  6.764953  2.228082  7.620155  1.052018  6.289814  0.475978  1.594761  2.18835  5.348079  2.25614  5.445659  2.03097  5.556197 
1  0.452677  4.634205  1.310125  6.694006  1.965557  7.302854  0.908367  6.134902  0.452677  1.526345  1.95883  5.045293  1.90161  4.954301  1.66101  5.102808 
2  0.430518  4.341193  1.135471  6.372285  1.695023  6.987364  0.751508  5.945333  0.430518  1.431536  1.70615  4.520173  1.66031  4.523381  1.38588  4.715821 
3  0.409443  4.046291  0.869467  6.266529  1.427161  6.665421  0.576703  5.727348  0.409443  1.350616  1.4484  4.227534  1.4418  4.143037  1.16705  4.352615 
4  0.389399  3.719431  0.779246  5.940579  1.172084  6.266779  0.428105  5.424505  0.389399  1.242075  1.27752  4.020461  1.30359  3.787737  1.05858  3.978987 
5  0.370337  3.387582  0.615105  5.61329  0.960154  5.81501  0.284942  5.023251  0.370337  1.130387  1.19981  3.651212  1.13853  3.399309  0.92935  3.550816 
6  0.352208  3.113108  0.446215  5.34771  0.773683  5.452533  0.180888  4.712459  0.352208  1.039995  1.05734  3.395965  1.02034  3.100329  0.82255  3.231222 
7  0.334967  2.836992  0.374049  5.063051  0.599909  5.074076  0.078683  4.383642  0.334967  0.948949  0.9829  3.166817  0.90445  2.802346  0.71642  2.914035 
8  0.318569  2.600488  0.25392  4.792389  0.441773  4.759203  0.000989  4.121149  0.318569  0.86565  0.91253  2.973106  0.83197  2.563268  0.65275  2.658017 
9  0.302974  2.368904  0.204032  4.530136  0.303669  4.434139  0.07069  3.8451  0.302974  0.791006  0.8161  2.73385  0.74755  2.329564  0.57178  2.418752 
10  0.288143  2.162351  0.133858  4.259095  0.183914  4.140787  0.12201  3.605635  0.288143  0.72367  0.73247  2.498902  0.68093  2.128266  0.50928  2.22007 
Period After Shock  Standard LongRun Identification  Response Using Estimated B(L) and True C  True Response 

0  1.208229  0.317261  0.317261 
1  1.142129  0.305724  0.303728 
2  1.079728  0.294777  0.290773 
3  1.020835  0.284401  0.27837 
4  0.957999  0.268881  0.266497 
5  0.898085  0.252375  0.25513 
6  0.841749  0.236737  0.244248 
7  0.788776  0.221917  0.23383 
8  0.739076  0.207956  0.223856 
9  0.692495  0.194857  0.214308 
10  0.648844  0.182578  0.205167 
11  0.607941  0.171069  0.196415 
12  0.569617  0.160285  0.188038 
13  0.533708  0.150181  0.180017 
14  0.500063  0.140714  0.172339 
15  0.468539  0.131843  0.164988 
16  0.439002  0.123532  0.15795 
17  0.411327  0.115744  0.151213 
18  0.385397  0.108448  0.144764 
19  0.361101  0.101611  0.138589 
20  0.338337  0.095205  0.132677 
21  0.317008  0.089204  0.127018 
22  0.297024  0.08358  0.1216 
23  0.278299  0.078311  0.116414 
24  0.260755  0.073374  0.111448 
25  0.244317  0.068749  0.106695 
26  0.228915  0.064415  0.102144 
27  0.214484  0.060354  0.097787 
28  0.200963  0.056549  0.093616 
29  0.188294  0.052984  0.089623 
30  0.176424  0.049644  0.0858 
31  0.165302  0.046515  0.08214 
32  0.154881  0.043582  0.078637 
33  0.145117  0.040835  0.075283 
34  0.135969  0.038261  0.072072 
35  0.127398  0.035849  0.068997 
36  0.119366  0.033589  0.066054 
37  0.111841  0.031471  0.063237 
38  0.104791  0.029487  0.06054 
39  0.098185  0.027628  0.057957 
40  0.091995  0.025887  0.055485 
41  0.086196  0.024255  0.053119 
42  0.080762  0.022726  0.050853 
43  0.075671  0.021293  0.048684 
44  0.0709  0.019951  0.046607 
45  0.066431  0.018693  0.044619 
46  0.062243  0.017515  0.042716 
47  0.058319  0.016411  0.040894 
48  0.054643  0.015376  0.03915 
49  0.051198  0.014407  0.03748 
50  0.04797  0.013498  0.035881 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.317261  0.965227  0.5737  2.041564  0.23723  2.167689  0.54776  1.754532 
1  0.303728  0.914762  0.5517  2.052182  0.24142  2.07094  0.52804  1.698675 
2  0.290773  0.862236  0.58651  1.896666  0.25437  1.978844  0.51244  1.641089 
3  0.27837  0.813542  0.62769  1.83466  0.2612  1.888288  0.49405  1.584659 
4  0.266497  0.757098  0.57787  1.773265  0.26403  1.778225  0.48408  1.497654 
5  0.25513  0.696839  0.5213  1.695599  0.2468  1.640481  0.44142  1.390243 
6  0.244248  0.646706  0.49512  1.605174  0.23834  1.531754  0.40723  1.309435 
7  0.23383  0.59587  0.4539  1.560015  0.22609  1.417834  0.37322  1.222341 
8  0.223856  0.547017  0.42835  1.49176  0.22499  1.319028  0.3532  1.143076 
9  0.214308  0.505281  0.38415  1.418986  0.21635  1.226913  0.32337  1.073589 
10  0.205167  0.467497  0.35197  1.361024  0.21042  1.145417  0.29882  1.010757 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.317261  0.344718  0.020846  0.67241  0.011769  0.677666  0.000704  0.652684 
1  0.303728  0.333181  0.14261  0.754398  0.121  0.787361  0.13555  0.754117 
2  0.290773  0.327444  0.21514  0.857402  0.20789  0.862774  0.22325  0.824153 
3  0.27837  0.329395  0.28215  0.963986  0.266  0.92479  0.28616  0.878412 
4  0.266497  0.303896  0.32419  0.912992  0.29518  0.902972  0.31942  0.85161 
5  0.25513  0.279276  0.29983  0.840631  0.27156  0.830114  0.29984  0.781466 
6  0.244248  0.258055  0.29424  0.779502  0.24854  0.764652  0.27755  0.717966 
7  0.23383  0.236308  0.26314  0.731748  0.2292  0.701817  0.25936  0.657068 
8  0.223856  0.212396  0.23971  0.680789  0.216  0.640793  0.24779  0.597042 
9  0.214308  0.196027  0.22142  0.636223  0.19896  0.591013  0.22446  0.555093 
10  0.205167  0.180867  0.19873  0.592398  0.18409  0.545822  0.20501  0.515099 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.317261  0.263405  0.24606  0.491359  0.02353  0.550339  0.12231  0.420437 
1  0.303728  0.251407  0.25086  0.492732  0.03466  0.53747  0.12763  0.417374 
2  0.290773  0.239092  0.24817  0.495833  0.04422  0.522403  0.13175  0.411784 
3  0.27837  0.227771  0.26604  0.491281  0.05372  0.509263  0.13626  0.40424 
4  0.266497  0.216715  0.26456  0.481165  0.05953  0.492956  0.13784  0.393589 
5  0.25513  0.207098  0.24643  0.475581  0.05813  0.472328  0.13282  0.378479 
6  0.244248  0.199403  0.23667  0.472004  0.05806  0.456865  0.12874  0.367708 
7  0.23383  0.190687  0.23351  0.457838  0.05738  0.438758  0.12425  0.353905 
8  0.223856  0.183543  0.22656  0.447867  0.05584  0.422927  0.11865  0.342563 
9  0.214308  0.175769  0.22348  0.441573  0.0554  0.406934  0.11503  0.330545 
10  0.205167  0.168449  0.21398  0.436209  0.05531  0.39221  0.11151  0.319526 
True Response  Standard  Bartlett  PopulationBased Standard Estimate  

0.5  0.317261  1.205044  1.47882  1.669926 
0.42857  0.317261  1.375361  1.382143  1.778329 
0.35714  0.317261  1.316122  1.400827  1.883894 
0.28571  0.317261  1.564127  1.276274  1.986631 
0.21429  0.317261  1.697806  1.361238  2.086397 
0.14286  0.317261  1.77251  1.278426  2.182922 
0.07143  0.317261  1.784865  1.293939  2.275806 
0  0.317261  2.055606  1.251349  2.364507 
0.071429  0.317261  2.154164  1.118277  2.448307 
0.142857  0.317261  2.078208  1.352746  2.526263 
0.214286  0.317261  2.241181  1.168981  2.597132 
0.285714  0.317261  2.408541  1.222079  2.659251 
0.357143  0.317261  2.584617  1.267463  2.710358 
0.428571  0.317261  2.345757  1.054461  2.747304 
0.5  0.317261  2.462397  1.093749  2.765572 
0.55  0.317261  2.59961  1.042513  2.763781 
0.571053  0.317261  2.399501  1.053214  2.758575 
0.592105  0.317261  2.651187  1.194388  2.750305 
0.613158  0.317261  2.483105  0.920886  2.73862 
0.634211  0.317261  2.462684  0.927314  2.723112 
0.655263  0.317261  2.576047  1.111044  2.703303 
0.676316  0.317261  2.33515  0.890941  2.678632 
0.697368  0.317261  2.344389  0.884775  2.648435 
0.718421  0.317261  2.30162  1.050025  2.611918 
0.739474  0.317261  2.321482  0.80601  2.568127 
0.760526  0.317261  2.252083  0.944605  2.515901 
0.781579  0.317261  2.138282  0.79084  2.453816 
0.802632  0.317261  2.189468  0.86457  2.380103 
0.823684  0.317261  2.114965  0.840188  2.292535 
0.844737  0.317261  1.976487  0.769628  2.188273 
0.865789  0.317261  1.82915  0.753758  2.063639 
0.886842  0.317261  1.728714  0.639389  1.913797 
0.907895  0.317261  1.472864  0.528816  1.7323 
0.928947  0.317261  1.212567  0.523036  1.510522 
0.95  0.317261  0.951562  0.335352  1.237312 
0.955  0.317261  0.917915  0.323633  1.163505 
0.961286  0.317261  0.855739  0.318611  1.065421 
0.967571  0.317261  0.757558  0.316248  0.961321 
0.973857  0.317261  0.577478  0.226781  0.851329 
0.980143  0.317261  0.507929  0.198865  0.73602 
0.986429  0.317261  0.358135  0.120174  0.616685 
0.992714  0.317261  0.27998  0.087576  0.495002 
0.999  0.317261  0.265755  0.094928  0.356907 
NOTE: Twoshock CKM specification is = 0.952 . Contemporaneous Impact of Technology on Hours is 1.2373.
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0  0  0  0  0  0  0  0 
1  0.308693  0.314592  0.04254  0.673878  0.01805  0.647234  0.02779  0.623804 
2  0.295526  0.295759  0.15234  0.783033  0.15649  0.74801  0.16729  0.717305 
3  0.282921  0.281186  0.25529  0.845505  0.25329  0.815666  0.26638  0.78063 
4  0.270853  0.260686  0.31303  0.866041  0.29426  0.815632  0.31242  0.773652 
5  0.2593  0.238182  0.284  0.793435  0.26321  0.739571  0.28485  0.69847 
6  0.24824  0.218456  0.25879  0.751611  0.24347  0.680386  0.26833  0.640736 
7  0.237652  0.202869  0.25245  0.714697  0.22241  0.62815  0.24698  0.593162 
8  0.227515  0.181005  0.24496  0.666628  0.20977  0.571779  0.23657  0.535228 
9  0.217811  0.168637  0.22516  0.635865  0.19168  0.528953  0.21241  0.499599 
10  0.20852  0.155345  0.20497  0.590961  0.17709  0.487783  0.19451  0.462818 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0  0.670222  0.94239  1.816276  0.58068  1.921127  0.84911  1.542768 
1  0.308693  0.906562  0.61077  1.93652  0.26774  2.080859  0.5537  1.7015 
2  0.295526  0.850878  0.63225  1.864488  0.28111  1.982867  0.54025  1.640374 
3  0.282921  0.801137  0.6031  1.815329  0.2944  1.896672  0.53501  1.584783 
4  0.270853  0.745823  0.59488  1.749627  0.29764  1.789283  0.52528  1.497591 
5  0.2593  0.688751  0.55676  1.645804  0.27605  1.653549  0.48148  1.391739 
6  0.24824  0.640131  0.53077  1.573238  0.26495  1.545208  0.44428  1.311457 
7  0.237652  0.593338  0.47181  1.49637  0.24848  1.435154  0.404  1.22964 
8  0.227515  0.545079  0.43449  1.411705  0.24502  1.335175  0.38203  1.150828 
9  0.217811  0.505683  0.39825  1.347316  0.23203  1.243398  0.34736  1.083199 
10  0.20852  0.468209  0.36423  1.286965  0.22472  1.161139  0.32201  1.021709 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.317261  0.965227  0.5737  2.041564  0.23723  2.167689  0.54776  1.754532 
1  0.303728  0.914762  0.5517  2.052182  0.24142  2.07094  0.52804  1.698675 
2  0.290773  0.862236  0.58651  1.896666  0.25437  1.978844  0.51244  1.641089 
3  0.27837  0.813542  0.62769  1.83466  0.2612  1.888288  0.49405  1.584659 
4  0.266497  0.757098  0.57787  1.773265  0.26403  1.778225  0.48408  1.497654 
5  0.25513  0.696839  0.5213  1.695599  0.2468  1.640481  0.44142  1.390243 
6  0.244248  0.646706  0.49512  1.605174  0.23834  1.531754  0.40723  1.309435 
7  0.23383  0.59587  0.4539  1.560015  0.22609  1.417834  0.37322  1.222341 
8  0.223856  0.547017  0.42835  1.49176  0.22499  1.319028  0.3532  1.143076 
9  0.214308  0.505281  0.38415  1.418986  0.21635  1.226913  0.32337  1.073589 
10  0.205167  0.467497  0.35197  1.361024  0.21042  1.145417  0.29882  1.010757 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.224883  0.575659  0.42037  1.289111  0.19393  1.34525  0.34138  1.143311 
1  0.216063  0.52498  0.35842  1.168939  0.17959  1.229552  0.30781  1.054731 
2  0.207589  0.484193  0.34112  1.094027  0.17116  1.139547  0.28383  0.988839 
3  0.199448  0.452953  0.31454  1.050045  0.16222  1.068127  0.26147  0.936934 
4  0.191625  0.398474  0.31214  0.94166  0.16179  0.958734  0.25044  0.841497 
5  0.18411  0.344402  0.24103  0.846034  0.14779  0.836599  0.21757  0.74127 
6  0.176889  0.300029  0.21319  0.774089  0.14137  0.741431  0.19244  0.667184 
7  0.169952  0.258434  0.16916  0.710066  0.13466  0.651526  0.17024  0.593785 
8  0.163286  0.221221  0.14686  0.655411  0.13438  0.576818  0.15828  0.531692 
9  0.156882  0.192283  0.11521  0.59036  0.12838  0.512942  0.13974  0.481051 
10  0.150729  0.16772  0.08477  0.549879  0.12377  0.459207  0.12661  0.43803 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.139153  0.638739  0.21024  1.237154  0.05003  1.327507  0.24037  1.086476 
1  0.133652  0.590606  0.18849  1.187589  0.07239  1.253601  0.24056  1.040708 
2  0.128368  0.556629  0.21385  1.131858  0.08031  1.193569  0.23355  1.001089 
3  0.123293  0.521656  0.23195  1.11703  0.09214  1.135452  0.2286  0.960592 
4  0.118419  0.475092  0.22841  1.035591  0.10283  1.05301  0.22676  0.895025 
5  0.113737  0.431351  0.19396  0.962195  0.09933  0.96203  0.20582  0.823215 
6  0.109241  0.394902  0.17564  0.906732  0.09908  0.888887  0.18898  0.767874 
7  0.104922  0.35799  0.15538  0.865995  0.09851  0.814495  0.1727  0.709581 
8  0.100774  0.3245  0.14483  0.815208  0.10279  0.751785  0.1638  0.659918 
9  0.09679  0.295299  0.12635  0.777582  0.10173  0.692325  0.14972  0.614484 
10  0.092963  0.269269  0.10676  0.733809  0.10143  0.639965  0.13773  0.573729 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.297739  0.368935  0.50685  0.938599  0.23406  0.971932  0.36366  0.777643 
1  0.28548  0.354956  0.50203  0.929779  0.23401  0.94392  0.35725  0.764858 
2  0.273725  0.339732  0.46649  0.919791  0.23839  0.917853  0.35412  0.751634 
3  0.262455  0.324077  0.46835  0.912223  0.24138  0.889532  0.34772  0.734661 
4  0.251648  0.307428  0.44181  0.903257  0.2438  0.858659  0.34322  0.71092 
5  0.241287  0.28999  0.42708  0.876545  0.22909  0.809069  0.32129  0.674221 
6  0.231352  0.276281  0.41049  0.844849  0.21807  0.770636  0.30284  0.645586 
7  0.221826  0.261221  0.39002  0.816886  0.20553  0.727972  0.28353  0.6139 
8  0.212692  0.246627  0.37169  0.792009  0.19548  0.688734  0.26772  0.583941 
9  0.203935  0.233669  0.35903  0.773063  0.18539  0.652726  0.25177  0.556859 
10  0.195538  0.22156  0.34448  0.747201  0.17716  0.620281  0.23785  0.532471 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.227192  0.328107  0.50411  0.833439  0.22834  0.884557  0.35934  0.692193 
1  0.217835  0.317488  0.46793  0.846076  0.22922  0.864192  0.35306  0.683634 
2  0.208863  0.302676  0.43559  0.842378  0.23125  0.836606  0.34732  0.669465 
3  0.200261  0.291022  0.45691  0.838207  0.23433  0.816371  0.34348  0.661812 
4  0.192012  0.281761  0.42403  0.823603  0.23304  0.796565  0.33921  0.645742 
5  0.184104  0.269061  0.40279  0.804648  0.2191  0.757217  0.32032  0.615924 
6  0.176522  0.258202  0.38687  0.783289  0.20957  0.725978  0.30414  0.594048 
7  0.169251  0.246983  0.36499  0.761979  0.19802  0.691984  0.28754  0.569092 
8  0.16228  0.235228  0.34501  0.741904  0.18934  0.659794  0.27457  0.544595 
9  0.155597  0.225087  0.32736  0.727828  0.18012  0.630296  0.26019  0.522985 
10  0.149188  0.215514  0.31204  0.71708  0.1726  0.603632  0.24819  0.503128 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.236913  0.329729  0.51553  0.869305  0.23797  0.897424  0.37483  0.697272 
1  0.227124  0.318216  0.51792  0.838851  0.23773  0.874163  0.36669  0.688972 
2  0.217739  0.31017  0.48901  0.852339  0.23692  0.857259  0.35935  0.682316 
3  0.208743  0.302858  0.47291  0.842614  0.23554  0.841256  0.35378  0.675196 
4  0.200118  0.290317  0.47958  0.8246  0.23884  0.819473  0.35236  0.660786 
5  0.191849  0.277828  0.45997  0.80927  0.22582  0.781476  0.33388  0.633079 
6  0.183922  0.267526  0.44724  0.799297  0.21761  0.75266  0.32007  0.613904 
7  0.176323  0.255779  0.42994  0.782939  0.20677  0.718328  0.30336  0.588055 
8  0.169037  0.243989  0.41619  0.76863  0.19877  0.686743  0.29041  0.564067 
9  0.162053  0.234057  0.39814  0.749878  0.18975  0.657861  0.27699  0.542746 
10  0.155357  0.224551  0.3808  0.73939  0.18227  0.631376  0.26501  0.523254 
Ratio of Innovation Variances  Concentrated Likelihood Function  95 percent Critical Value 

0.05  1476.373  1543.417 
0.094898  1511.671  1543.417 
0.139796  1527.342  1543.417 
0.184694  1535.538  1543.417 
0.229592  1540.097  1543.417 
0.27449  1542.672  1543.417 
0.319388  1544.103  1543.417 
0.364286  1544.864  1543.417 
0.409184  1545.228  1543.417 
0.454082  1545.338  1543.417 
0.49898  1545.099  1543.417 
0.543878  1544.565  1543.417 
0.588776  1543.81  1543.417 
0.633673  1542.886  1543.417 
0.678571  1541.835  1543.417 
0.723469  1540.686  1543.417 
0.768367  1539.463  1543.417 
0.813265  1538.184  1543.417 
0.858163  1536.864  1543.417 
0.903061  1535.513  1543.417 
0.947959  1534.14  1543.417 
0.992857  1532.753  1543.417 
1.037755  1531.356  1543.417 
1.082653  1529.956  1543.417 
1.127551  1528.556  1543.417 
1.172449  1527.158  1543.417 
1.217347  1525.765  1543.417 
1.262245  1524.378  1543.417 
1.307143  1523.001  1543.417 
1.352041  1521.633  1543.417 
1.396939  1520.277  1543.417 
1.441837  1518.931  1543.417 
1.486735  1517.599  1543.417 
1.531633  1516.279  1543.417 
1.576531  1514.972  1543.417 
1.621429  1513.678  1543.417 
1.666327  1512.398  1543.417 
1.711224  1511.132  1543.417 
1.756122  1509.88  1543.417 
1.80102  1508.641  1543.417 
1.845918  1507.416  1543.417 
1.890816  1506.204  1543.417 
1.935714  1505.007  1543.417 
1.980612  1503.822  1543.417 
2.02551  1502.651  1543.417 
2.070408  1501.494  1543.417 
2.115306  1500.349  1543.417 
2.160204  1499.217  1543.417 
2.205102  1498.097  1543.417 
2.25  1496.99  1543.417 
Percent Response of Hours to a Tech Shock: Ratio of Innovation Variances  Percent Response of Hours to a Tech Shock: Average Estimated Response  Percent Response of Hours to a Tech Shock: True Estimated Response  Percent Response of Hours to a Tech Shock: Lower Bound of Sampling Distribution  Percent Response of Hours to a Tech Shock: Upper Bound of Sampling Distribution  Percent of Variance in log employment due to technology: V 

0.05  0.490982  0.520715  0.18412  0.895891  33.43882 
0.094898  0.42254  0.405239  0.32109  0.873227  19.52765 
0.139796  0.384992  0.35548  0.37328  0.872422  13.14735 
0.184694  0.386058  0.327338  0.44897  0.897673  9.437397 
0.229592  0.375604  0.309406  0.50556  0.922912  7.001525 
0.27449  0.362851  0.297594  0.52792  0.936802  5.203693 
0.319388  0.378986  0.289488  0.61108  0.987897  3.817391 
0.364286  0.375893  0.283833  0.69193  0.999453  2.703472 
0.409184  0.365809  0.280231  0.72747  1.035829  1.748769 
0.454082  0.348576  0.275795  0.83297  1.053788  1.201214 
0.49898  0.376282  0.269932  0.7709  1.074675  1.057111 
0.543878  0.389481  0.26492  0.85989  1.112762  0.93695 
0.588776  0.378523  0.260487  0.84761  1.113969  0.838642 
0.633673  0.379952  0.256709  0.90797  1.143577  0.750639 
0.678571  0.366501  0.253283  0.96213  1.148361  0.680405 
0.723469  0.411943  0.250355  0.91118  1.173002  0.619446 
0.768367  0.404405  0.24763  0.91199  1.194228  0.564904 
0.813265  0.398782  0.245122  0.93964  1.22046  0.518893 
0.858163  0.402729  0.242979  1.05685  1.249878  0.476484 
0.903061  0.40631  0.240964  1.02801  1.242471  0.439432 
0.947959  0.453106  0.23908  1.01082  1.268871  0.406911 
0.992857  0.455546  0.237382  1.0335  1.293059  0.377172 
1.037755  0.464204  0.235902  1.06133  1.323753  0.350019 
1.082653  0.441687  0.234339  1.08738  1.322863  0.326567 
1.127551  0.456615  0.233112  1.0727  1.353671  0.305175 
1.172449  0.480116  0.23179  1.1405  1.36508  0.283672 
1.217347  0.511076  0.230658  1.01194  1.391692  0.266425 
1.262245  0.469738  0.229473  1.1445  1.393845  0.251685 
1.307143  0.50242  0.228499  1.12004  1.420262  0.233554 
1.352041  0.491346  0.227496  1.16841  1.437776  0.220994 
1.396939  0.494005  0.226603  1.15969  1.444383  0.207644 
1.441837  0.529517  0.225706  1.09348  1.473961  0.19542 
1.486735  0.54601  0.2249  1.18044  1.491641  0.183891 
1.531633  0.551853  0.224151  1.16964  1.501809  0.173679 
1.576531  0.556199  0.223429  1.20543  1.525675  0.16473 
1.621429  0.55712  0.222744  1.22243  1.53937  0.155051 
1.666327  0.558141  0.222013  1.27507  1.533489  0.148084 
1.711224  0.543631  0.221403  1.23846  1.546864  0.139547 
1.756122  0.582436  0.22083  1.29837  1.582841  0.133212 
1.80102  0.584248  0.22022  1.2734  1.581234  0.126651 
1.845918  0.599786  0.219702  1.2981  1.610683  0.11955 
1.890816  0.607398  0.219123  1.30324  1.632166  0.113715 
1.935714  0.608834  0.218647  1.35647  1.629151  0.10821 
1.980612  0.607198  0.218184  1.3957  1.663735  0.102738 
2.02551  0.586875  0.217668  1.39551  1.672081  0.098305 
2.070408  0.595955  0.217289  1.36485  1.671275  0.093173 
2.115306  0.627049  0.216793  1.41  1.696596  0.089314 
2.160204  0.619913  0.216427  1.33211  1.705346  0.085538 
2.205102  0.652623  0.216017  1.39618  1.710424  0.081858 
2.25  0.631667  0.215662  1.48893  1.714363  0.077595 
Period after the Shock  FR Data: Estimate  FR Data: Lower Bound  FR Data: Upper Bound  CEV Data: Estimate  CEV Data: Lower Bound  CEV Data: Upper Bound  GR Data: Estimate  GR Data: Lower Bound  GR Data: Upper Bound 

0  0.58554  0.7546  0.14166  0.341745  0.27165  0.547913  0.036124  0.44695  0.369653 
1  0.79645  1.09668  0.07533  0.617813  0.26154  0.896  0.146489  0.64894  0.646549 
2  0.81707  1.24083  0.108136  0.893252  0.10601  1.185429  0.327783  0.66217  0.882233 
3  0.76764  1.29647  0.239548  1.100975  0.089929  1.410154  0.490029  0.58166  1.075763 
4  0.68261  1.26594  0.328973  1.187992  0.165609  1.486961  0.564906  0.49593  1.134192 
5  0.59429  1.19652  0.359794  1.164059  0.209707  1.4868  0.584557  0.40021  1.116595 
6  0.51262  1.09529  0.346046  1.089292  0.201536  1.405928  0.563863  0.33827  1.035938 
7  0.44789  1.01748  0.304811  0.999599  0.195448  1.305092  0.517692  0.2946  0.948204 
8  0.4007  0.9349  0.251231  0.919414  0.159009  1.234999  0.459793  0.25213  0.861177 
9  0.36677  0.87681  0.197898  0.859382  0.127279  1.177883  0.404048  0.21632  0.778468 
10  0.34142  0.82393  0.156657  0.820531  0.102613  1.136092  0.35676  0.18544  0.703819 
11  0.32032  0.77825  0.12997  0.797644  0.085448  1.111785  0.319842  0.17322  0.657733 
12  0.30085  0.73503  0.113272  0.783456  0.064028  1.099125  0.292421  0.15994  0.624285 
Period after the Shock  FR Data: Estimate  FR Data: Lower Bound  FR Data: Upper Bound  CEV Data: Estimate  CEV Data: Lower Bound  CEV Data: Upper Bound  GR Data: Estimate  GR Data: Lower Bound  GR Data: Upper Bound 

0  0.22381  0.5621  0.195027  0.134698  0.33045  0.377046  0.017911  0.48499  0.370122 
1  0.2187  0.7767  0.373328  0.330302  0.34948  0.6463  0.115792  0.69167  0.650732 
2  0.16165  0.84415  0.513474  0.482289  0.31357  0.833209  0.267378  0.73654  0.840562 
3  0.03455  0.84501  0.706112  0.705134  0.16653  1.063178  0.473565  0.69118  1.097026 
4  0.071211  0.8237  0.828733  0.8135  0.06653  1.173525  0.541576  0.66966  1.167376 
5  0.138179  0.76106  0.882716  0.856311  0.01078  1.205038  0.577328  0.61218  1.168548 
6  0.178836  0.6888  0.904233  0.839028  0.020444  1.183153  0.581903  0.52194  1.175323 
7  0.197271  0.60203  0.861805  0.796053  0.051306  1.155959  0.56753  0.46611  1.097845 
8  0.197792  0.55531  0.798708  0.730453  0.058606  1.069207  0.526376  0.40929  1.044721 
9  0.186363  0.50144  0.719698  0.660278  0.055957  0.99339  0.480347  0.35779  0.972255 
10  0.168712  0.45598  0.661679  0.592914  0.035419  0.922226  0.437265  0.31742  0.918565 
11  0.149361  0.41384  0.602678  0.535464  0.025478  0.855224  0.399069  0.28571  0.853028 
12  0.13086  0.3816  0.556149  0.487841  0.010515  0.818939  0.365382  0.25207  0.799213 
FR  CEV  GR  

1948  0.86933  1.106904  0.95778 
1948.25  0.868239  1.102799  0.952289 
1948.5  0.874798  1.111018  0.957686 
1948.75  0.869141  1.102484  0.947663 
1949  0.856552  1.085609  0.930932 
1949.25  0.848126  1.07124  0.910119 
1949.5  0.832183  1.051572  0.899186 
1949.75  0.820706  1.031904  0.889897 
1950  0.82004  1.030552  0.893983 
1950.25  0.836879  1.05256  0.91487 
1950.5  0.864313  1.078353  0.942819 
1950.75  0.87607  1.087708  0.953543 
1951  0.892024  1.102342  0.972901 
1951.25  0.902774  1.107626  0.981328 
1951.5  0.892723  1.093945  0.970946 
1951.75  0.895366  1.096792  0.969055 
1952  0.899368  1.100544  0.976321 
1952.25  0.889683  1.085801  0.968322 
1952.5  0.890441  1.085637  0.971879 
1952.75  0.910726  1.108245  0.995057 
1953  0.918202  1.104695  0.997502 
1953.25  0.914963  1.099039  0.995331 
1953.5  0.904352  1.085273  0.981714 
1953.75  0.887138  1.065434  0.963741 
1954  0.879527  1.055302  0.947427 
1954.25  0.864871  1.036306  0.936708 
1954.5  0.858697  1.028952  0.929566 
1954.75  0.864067  1.0337  0.939508 
1955  0.878754  1.04955  0.950977 
1955.25  0.886699  1.056792  0.960257 
1955.5  0.899215  1.067662  0.966754 
1955.75  0.907699  1.078929  0.976808 
1956  0.90389  1.072879  0.980516 
1956.25  0.905546  1.0715  0.979402 
1956.5  0.900078  1.062103  0.970231 
1956.75  0.898126  1.059075  0.976888 
1957  0.895953  1.056332  0.974359 
1957.25  0.891089  1.045512  0.968408 
1957.5  0.888375  1.040383  0.96042 
1957.75  0.867244  1.014814  0.938111 
1958  0.841833  0.98233  0.911857 
1958.25  0.833937  0.970586  0.895609 
1958.5  0.838626  0.97554  0.903897 
1958.75  0.851501  0.987018  0.915754 
1959  0.864157  0.996945  0.929095 
1959.25  0.88477  1.015911  0.942388 
1959.5  0.874893  1.001763  0.936002 
1959.75  0.874394  0.998588  0.933036 
1960  0.871349  0.988853  0.932721 
1960.25  0.879866  0.994984  0.931594 
1960.5  0.87794  0.990331  0.922662 
1960.75  0.866861  0.973581  0.909028 
1961  0.860852  0.964162  0.899802 
1961.25  0.850982  0.950805  0.895432 
1961.5  0.857042  0.955194  0.898571 
1961.75  0.864436  0.96097  0.907424 
1962  0.872911  0.96964  0.90892 
1962.25  0.881341  0.971876  0.917262 
1962.5  0.875941  0.963114  0.911627 
1962.75  0.868012  0.949107  0.899294 
1963  0.873022  0.951614  0.899216 
1963.25  0.876491  0.953202  0.902503 
1963.5  0.874908  0.949294  0.901655 
1963.75  0.878847  0.949527  0.903539 
1964  0.887984  0.945541  0.905405 
1964.25  0.894455  0.951568  0.907433 
1964.5  0.89863  0.951657  0.90774 
1964.75  0.902795  0.955805  0.913536 
1965  0.91317  0.961487  0.923634 
1965.25  0.923392  0.969983  0.928212 
1965.5  0.921339  0.964023  0.926515 
1965.75  0.932383  0.973107  0.934433 
1966  0.941753  0.980039  0.945651 
1966.25  0.948456  0.982737  0.951221 
1966.5  0.95028  0.98265  0.953972 
1966.75  0.947024  0.97837  0.948408 
1967  0.946085  0.972442  0.942559 
1967.25  0.934682  0.95942  0.933638 
1967.5  0.939567  0.961389  0.93309 
1967.75  0.941849  0.963685  0.934222 
1968  0.941146  0.959943  0.931928 
1968.25  0.945354  0.964615  0.939407 
1968.5  0.949474  0.968579  0.943527 
1968.75  0.952053  0.969218  0.944322 
1969  0.961844  0.977912  0.949284 
1969.25  0.961739  0.975237  0.953263 
1969.5  0.960051  0.974947  0.954394 
1969.75  0.952074  0.965109  0.945914 
1970  0.942571  0.955005  0.938448 
1970.25  0.928105  0.941023  0.921932 
1970.5  0.917789  0.926837  0.909065 
1970.75  0.905021  0.913903  0.897448 
1971  0.904423  0.911319  0.896232 
1971.25  0.90512  0.912362  0.893649 
1971.5  0.898835  0.903622  0.887476 
1971.75  0.903709  0.909579  0.894838 
1972  0.913627  0.910317  0.894594 
1972.25  0.914142  0.911592  0.89721 
1972.5  0.918457  0.912786  0.897311 
1972.75  0.923873  0.918009  0.902674 
1973  0.93255  0.924013  0.911229 
1973.25  0.938189  0.928733  0.91608 
1973.5  0.939742  0.928669  0.917301 
1973.75  0.942487  0.93102  0.917397 
1974  0.93957  0.9251  0.908014 
1974.25  0.933899  0.916923  0.904575 
1974.5  0.925497  0.907726  0.897733 
1974.75  0.90792  0.891704  0.879367 
1975  0.876207  0.857664  0.846332 
1975.25  0.865407  0.846789  0.834317 
1975.5  0.868998  0.851038  0.83756 
1975.75  0.876795  0.85925  0.848192 
1976  0.884263  0.865244  0.856566 
1976.25  0.881252  0.863595  0.852724 
1976.5  0.880301  0.862771  0.850858 
1976.75  0.878223  0.862272  0.853774 
1977  0.882118  0.867317  0.858916 
1977.25  0.897453  0.882982  0.871448 
1977.5  0.900267  0.885217  0.87598 
1977.75  0.905172  0.888886  0.879743 
1978  0.905441  0.890485  0.881419 
1978.25  0.927483  0.912469  0.90362 
1978.5  0.930521  0.913666  0.904894 
1978.75  0.935873  0.919007  0.911416 
1979  0.937595  0.923388  0.91589 
1979.25  0.936462  0.921622  0.91417 
1979.5  0.939331  0.924789  0.918482 
1979.75  0.938158  0.923168  0.914807 
1980  0.929295  0.916123  0.910925 
1980.25  0.908004  0.894899  0.889599 
1980.5  0.90104  0.888098  0.881757 
1980.75  0.910761  0.898423  0.891159 
1981  0.910802  0.901477  0.89428 
1981.25  0.905886  0.89442  0.889279 
1981.5  0.902534  0.8914  0.886277 
1981.75  0.894167  0.885339  0.879189 
1982  0.875476  0.871575  0.865366 
1982.25  0.873409  0.864884  0.859683 
1982.5  0.86407  0.855317  0.850082 
1982.75  0.852959  0.846699  0.840427 
1983  0.854742  0.848466  0.843247 
1983.25  0.861645  0.855252  0.851115 
1983.5  0.875403  0.869777  0.864766 
1983.75  0.8892  0.882188  0.879295 
1984  0.902808  0.89436  0.892585 
1984.25  0.91141  0.902947  0.900258 
1984.5  0.915465  0.906449  0.903808 
1984.75  0.91778  0.907626  0.906008 
1985  0.922897  0.913465  0.911905 
1985.25  0.92497  0.913242  0.912682 
1985.5  0.92408  0.911825  0.913236 
1985.75  0.926306  0.913995  0.916413 
1986  0.923932  0.908532  0.909958 
1986.25  0.91953  0.903144  0.904547 
1986.5  0.92255  0.906339  0.907777 
1986.75  0.927759  0.909442  0.911881 
1987  0.937869  0.917724  0.920233 
1987.25  0.942277  0.919843  0.922379 
1987.5  0.946907  0.924981  0.928519 
1987.75  0.955207  0.929339  0.933868 
1988  0.955248  0.93248  0.937036 
1988.25  0.966336  0.941657  0.946281 
1988.5  0.967066  0.94124  0.946812 
1988.75  0.978109  0.949462  0.955093 
1989  0.987294  0.955228  0.960902 
1989.25  0.987942  0.955015  0.96257 
1989.5  0.988815  0.956569  0.962257 
1989.75  0.989635  0.954445  0.96012 
1990  0.987769  0.953294  0.959897 
1990.25  0.980563  0.947613  0.955102 
1990.5  0.969596  0.939004  0.946431 
1990.75  0.962471  0.931886  0.93834 
1991  0.948231  0.91894  0.924384 
1991.25  0.939173  0.913442  0.918851 
1991.5  0.934771  0.911217  0.916612 
1991.75  0.929252  0.905979  0.912254 
1992  0.921232  0.89855  0.904773 
1992.25  0.923542  0.901951  0.907288 
1992.5  0.92424  0.902233  0.908482 
1992.75  0.924389  0.903966  0.910227 
1993  0.928243  0.908846  0.916042 
1993.25  0.934633  0.913893  0.922922 
1993.5  0.935441  0.916838  0.925879 
1993.75  0.940606  0.921354  0.930419 
1994  0.941452  0.92449  0.931786 
1994.25  0.954841  0.937471  0.943964 
1994.5  0.958808  0.941133  0.949428 
1994.75  0.963652  0.946427  0.953867 
1995  0.966338  0.951229  0.958702 
1995.25  0.962562  0.948368  0.955821 
1995.5  0.966434  0.953797  0.961284 
1995.75  0.964187  0.952063  0.960413 
1996  0.959991  0.950041  0.958375 
1996.25  0.963101  0.954596  0.963829 
1996.5  0.966021  0.957943  0.967193 
1996.75  0.971565  0.963681  0.97383 
1997  0.977548  0.9678  0.977964 
1997.25  0.98036  0.972558  0.981882 
1997.5  0.980241  0.972581  0.983619 
1997.75  0.982963  0.975834  0.987742 
1998  0.983334  0.979592  0.992372 
1998.25  0.983967  0.981395  0.993317 
1998.5  0.985586  0.983711  0.996486 
1998.75  0.989609  0.989209  1.002853 
1999  0.98662  0.986327  1.001618 
1999.25  0.990236  0.990388  1.004841 
1999.5  0.992987  0.992487  1.007774 
1999.75  0.995103  0.995385  1.009828 
2000  1  1  1 
2000.25  0.995467  0.99447  0.99526 
2000.5  0.992455  0.992625  0.991275 
2000.75  0.984314  0.986377  0.986658 
2001  0.979356  0.987508  0.985801 
2001.25  0.966784  0.978582  0.976367 
2001.5  0.954096  0.968418  0.965947 
2001.75  0.939873  0.956607  0.953145 
2002  0.930679  0.945061  
2002.25  0.927911  0.941101  
2002.5  0.923694  0.938233  
2002.75  0.921035  0.936935 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.164924  0.170414  0.079132  0.240328  0.091007  0.249821  0.071522  0.226678 
1  0.252601  0.261306  0.096684  0.38755  0.121724  0.400888  0.09089  0.363504 
2  0.290485  0.300386  0.087488  0.475297  0.116671  0.4841  0.081144  0.44032 
3  0.297014  0.317087  0.070429  0.524872  0.112351  0.521824  0.075996  0.47634 
4  0.285931  0.315598  0.072961  0.541264  0.10595  0.525246  0.071281  0.481302 
5  0.266221  0.299862  0.066738  0.522521  0.09814  0.501583  0.067204  0.461245 
6  0.243445  0.275497  0.06555  0.487065  0.087687  0.463307  0.061502  0.428138 
7  0.220924  0.246138  0.062033  0.441184  0.073465  0.418811  0.050384  0.387663 
8  0.200539  0.2151  0.036718  0.389677  0.054982  0.375218  0.032707  0.345536 
9  0.183244  0.184483  0.002214  0.352335  0.032863  0.336102  0.00828  0.305832 
10  0.169388  0.155708  0.02335  0.322844  0.008868  0.302548  0.01859  0.270835 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.01409  2.8E18  2.7E16  3.08E16  2.2E16  2.11E16  2.3E16  2.5E16 
1  0.176465  0.201027  0.168963  0.233957  0.169511  0.232544  0.162747  0.224445 
2  0.274404  0.307237  0.242938  0.373709  0.244794  0.36968  0.230246  0.352443 
3  0.314261  0.350107  0.255772  0.454162  0.2582  0.442015  0.237444  0.417562 
4  0.316416  0.354562  0.230257  0.485811  0.237807  0.471317  0.212851  0.441629 
5  0.294069  0.330354  0.183237  0.48566  0.195186  0.465522  0.168453  0.433204 
6  0.256372  0.284442  0.115489  0.452502  0.136744  0.432139  0.110834  0.399705 
7  0.210029  0.226903  0.04551  0.407455  0.070954  0.382853  0.048645  0.353634 
8  0.160114  0.165439  0.01353  0.34439  0.005512  0.325366  0.01219  0.300727 
9  0.110479  0.107691  0.06246  0.285254  0.0523  0.267687  0.06419  0.248934 
10  0.063988  0.05637  0.10743  0.235284  0.10069  0.213431  0.10721  0.200454 
Period After Shock  Response: TRUE  Response: Estimated  Sampling Distrbution: Lower Bound  Sampling Distrbution: Upper Bound  Average CI Standard Deviation Based: Lower Bound  Average CI Standard Deviation Based: Upper Bound  Average CI Percentile Based: Lower Bound  Average CI Percentile Based: Upper Bound 

0  0.068992  0.04652  0.09862  0.167191  0.07106  0.164098  0.08307  0.143077 
1  0.191457  0.148478  0.0827  0.328883  0.03678  0.333734  0.06625  0.289508 
2  0.290109  0.231806  0.06314  0.444067  0.00914  0.454472  0.03347  0.395344 
3  0.352507  0.283542  0.02501  0.52443  0.039861  0.527222  0.00945  0.461074 
4  0.381505  0.309443  0.01458  0.563606  0.05714  0.561746  0.006813  0.494244 
5  0.383138  0.306318  0.01366  0.562008  0.056364  0.556272  0.009286  0.492992 
6  0.363798  0.284096  0.02117  0.546402  0.044011  0.524181  0.00399  0.468556 
7  0.329509  0.251071  0.02563  0.515953  0.025405  0.476737  0.00534  0.431896 
8  0.285698  0.213111  0.02887  0.463615  0.00311  0.423112  0.01832  0.389828 
9  0.23707  0.174659  0.03187  0.413083  0.02016  0.369482  0.03267  0.3464 
10  0.187542  0.138315  0.04741  0.360549  0.0429  0.319533  0.04798  0.30491 
Model specification  Measure of Variation: Unfiltered: (  Measure of Variation: Unfiltered:  Measure of Variation: HPfiltered:  Measure of Variation: HPfiltered:  Measure of Variation: Onestepahead forecast error :  Measure of Variation: Onestepahead forecast error :  

MLE: Base: Nonrecursive  3.73  67.16  7.30  67.14  7.23  67.24  
MLE: Base: Recursive  3.53  58.47  6.93  64.83  0.00  57.08  
MLE: : Nonrecursive  13.40  89.13  23.97  89.17  23.77  89.16  
MLE: : Recursive  12.73  84.93  22.95  88.01  0.00  84.17  
MLE: : Nonrecursive  38.12  97.06  55.85  97.10  55.49  97.08  
MLE: : Recursive  36.67  95.75  54.33  96.68  0.00  95.51  
MLE: : Nonrecursive  3.26  90.67  6.64  90.70  6.59  90.61  
MLE: : Recursive  3.07  89.13  6.28  90.10  0.00  88.93  
MLE: : Nonrecursive  4.11  53.99  7.80  53.97  7.73  54.14  
MLE: : Recursive  3.90  41.75  7.43  50.90  0.00  38.84  
MLE: Three: Nonrecursive  0.18  45.67  3.15  45.69  3.10  45.72  
MLE: Three: Recursive  0.18  36.96  3.05  43.61  0.00  39.51  
CKM: Base: Nonrecursive  2.76  33.50  1.91  33.53  1.91  33.86  
CKM: Base: Recursive  2.61  25.77  1.81  31.41  0.00  24.93  
CKM: : Nonrecursive  10.20  66.86  7.24  66.94  7.23  67.16  
CKM: : Recursive  9.68  58.15  6.88  64.63  0.00  57.00  
CKM: : Nonrecursive  31.20  89.00  23.81  89.08  23.76  89.08  
CKM: : Recursive  29.96  84.76  22.79  87.91  0.00  84.07  
CKM: : Nonrecursive  0.78  41.41  0.52  41.33  0.52  41.68  
CKM: : Recursive 
0.73  37.44  0.49  40.11  0.00  37.42  
CKM: : Nonrecursive  2.57  20.37  1.82  20.45  1.82  20.70  
CKM: : Recursive  2.44  13.53  1.73  18.59  0.00  12.33  
CKM:

0.66  6.01  0.46  6.03  0.46  6.12  
CKM:

0.62  3.76  0.44  5.41  0.00  3.40  
CKM: Three: Nonrecursive  2.23  30.73  1.71  31.11  1.72  31.79  
CKM: Three: Recursive  2.31  23.62  1.66  29.67  0.00  25.62 
Note: (a) corresponds to the columns denoted by ln (b) In each case, the results report the ratio of two variances:the numerator is the variance for the system with only technology shocksand the denominator is the variance for the system with all shocks. All statistics are the averages of the ratios and are based on based on 300 simulations of 5000 observations for each model. (c) `Base' means the twoshock specification, whether MLE or CKM, as indicated. `Three' means the threeshock specification (d) For a description of the procedure used to calculate the forecast error variance, see footnote 13. MLE Maximum Likelihood Estimate. CKM Chari, Kehoe, and McGrattan (2005b).
Statistic  Types of shock: Monetary Policy  Types of shock: Neutral Technology  Types of shock: CapitalEmbodied 

variance of logged hours  22.2  40.0  38.5 
variance of HP filtered logged hours  37.8  17.7  44.5 
variance of  29.9  46.7  23.6 
variance of HP filtered logged output  31.9  32.3  36.1 
Note: Results are average values based on 500 simulations of 3100 observations each.
ACEL: Altig Christiano, Eichenbaum and Linde (2005).
* The first two authors are grateful to the National Science Foundation for Financial Support. We thank Lars Hansen and our colleagues at the Federal Reserve Bank of Chicago and the Board of Governors for useful comments at various stages of this project. The views in this paper are solely those of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System. Return to text
† Northwestern University, the Federal Reserve Bank of Chicago, and the NBER. Return to text
‡ Northwestern University, the Federal Reserve Bank of Chicago, and the NBER. Return to text
§ Federal Reserve Board of Governors. Return to text
1. See for example Sims (1989), Eichenbaum and Evans (1995), Rotemberg and Woodford (1997), Gali (1999), Francis and Ramey (2004), Christiano, Eichenbaum, and Evans (2005), and Del Negro, Schorfheide, Smets, and Wouters (2005). Return to text
2. See, for example, Basu, Fernald, and Kimball (2004), Christiano, Eichenbaum, and Vigfusson (2003, 2004), Fisher (2006), Francis and Ramey (2004), King, Plosser, Stock and Watson (1991), Shapiro and Watson (1988) and Vigfusson (2004). Francis, Owyang, and Roush (2005) pursue a related strategy to identify a technology shock as the shock that maximizes the forecast error variance share of labor productivity at a long but finite horizon. Return to text
3. This list is particularly long and includes at least Bernanke (1986), Bernanke and Blinder (1992), Bernanke and Mihov (1998), Blanchard and Perotti (2002), Blanchard and Watson (1986), Christiano and Eichenbaum (1992), Christiano, Eichenbaum and Evans (2005), Cushman and Zha (1997), Eichenbaum and Evans (1995), Hamilton (1997), Rotemberg and Woodford (1992), Sims (1986), and Sims and Zha (2006). Return to text
4. See also FernandezVillaverdez, RubioRamirez, and Sargent (2005) who investigate the circumstances in which the economic shocks are recoverable from the VAR disturbances. They provide a simple matrix algebra check to assess recoverability. They identify models in which the conditions are satisfied and other models in which they are not. Return to text
5. Let Then, where denotes the value of in nonstochastic steady state. Return to text
6. For an early example, see Hansen and Sargent (1980, footnote 12). Sims and Zha (forthcoming) discuss the possibility that, although a given economic shock may not lie exactly in the space of current and past it may nevertheless be `close'. They discuss methods to detect this case. Return to text
7. Cooley and Dwyer (1998) argue that in the standard RBC model, if technology shocks have a unit root, then per capita hours worked will be difference stationary. This claim, which plays an important role in their analysis of VARs, is incorrect. Return to text
8. We implement the procedure for estimating by computing where is the lower triangular Cholesky decomposition of and setting equal to the second column of . Return to text
9. We use the standard Kalman filter strategy discussed in Hamilton (1994, section 13.4). We remove the sample mean from prior to estimation and set the measurement error in the Kalman filter system to zero, i.e. in (6). Return to text
10. See, for example, Christiano (1988), Christiano, et al. (2004), and Smets and Wouters (2003). Return to text
11. We compute forecast error variances based on a four lag VAR. The variables in the VAR depend on whether the calculations correspond to the two or three shock model. In the case of the twoshock model, the VAR has two variables, output growth and log hours. In the case of the threeshock model, the VAR has three variables: output growth, log hours and the log of the investment to output ratio. Computing requires estimating VARs in artificial data generated with all shocks, as well as in artificial data generated with only the technology shock. In the latter case, the onestep ahead forecast error from the VAR is well defined, even though the VAR coefficients themselves are not well defined due to multicollinearity problems. Return to text
12. When we measure according to (i), drops from 3.73 in the twoshock MLE model to in the threeshock MLE model. The analogous drop in is an order of magnitude smaller when is measured using (ii) or (iii). The reason for this difference is that goes from in the twoshock MLE model to in the threeshock MLE model. In the latter specification there is a nearunit root in which translates into a nearunit root in hours worked. As a result, the variance of hours worked becomes very large at the low frequencies. The nearunit root in has less of an effect on hours worked at high and business cycle frequencies. Return to text
13. Sims and Zha (1999) refer to what we call the percentilebased confidence interval as the `otherpercentile bootstrap interval'. This procedure has been used in several studies, such as Blanchard and Quah (1989), Christiano, Eichenbaum, and Evans (1999), Francis and Ramey (2004), McGrattan (2006), and Runkle (1987). The standarddeviation based confidence interval has been used by other researchers, such as Christiano Eichenbaum, and Evans (2005), Gali (1999), and Gali and Rabanal (2004). Return to text
14. For each lag starting at the impact period, we ordered the 1,000 estimated impulse responses from smallest to largest. The lower and upper boundaries correspond to the 25 and the 975 impulses in this ordering. Return to text
15. An extreme example, in which the point estimates roughly coincide with one of the boundaries of the percentilebased confidence interval, appears in Blanchard and Quah (1989). Return to text
16. As falls, the total volatility of hours worked falls, as does the relative importance of labor tax shocks. In principle, both effects contribute to the decline in sampling uncertainty. Return to text
17. The minimization in (26) is actually over the trace of the indicated integral. One interpretation of (26) is that it provides the probability limit of our estimators  what they would converge to as the sample size increases to infinity. We do not adopt this interpretation, because in practice an econometrician would use a consistent laglength selection method. The probability limit of our estimators corresponds to the true impulse response functions for all cases considered in this paper. Return to text
18. The derivation of this formula is straightforward. Write (10) in lag operator form as follows:
where Let the fitted disturbances associated with a particular parameterization, be denoted Simple substitution implies:19. By we mean that is a positive definite matrix. Return to text
20. In the earlier discussion it was convenient to adopt the normalization that the technology shock is the second element of Here, we adopt the same normalization as for the longrun identification  namely, that the technology shock is the first element of Return to text
21. This result explains why laglength selection methods, such as the Akaike criterion, almost never suggest values of greater than 4 in artificial data sets of length 180, regardless of which of our data generating methods we used. These lag length selection methods focus on Return to text
22. Equation (26) shows that corresponds to only a single point in the integral. So other things equal, the estimation criterion assigns no weight at all to getting right The reason is identified in our setting is that the functions we consider are continuous at Return to text
23. A similar argument is presented in Ravenna (2005). Return to text
24. Christiano, Eichenbaum and Vigfusson (2006) also consider the estimator proposed by Andrews and Monahan (1992). Return to text
25. The rule of always setting the bandwidth, equal to sample size does not yield a consistent estimator of the spectral density at frequency zero. We assume that as sample size is increased beyond the bandwidth is increased sufficiently slowly to achieve consistency. Return to text
26. Because (26) is a quadratic function, we solve the optimization problem by solving the linear firstorder conditions. These are the YuleWalker equations, which rely on population second moments of the data. We obtain the population second moments by complex integration of the reduced form of the model used to generate the data, as suggested by Christiano (2002). Return to text
27. To ensure comparability of results we use CKM's computer code and data, available on Ellen McGrattan's webpage. The algorithm used by CKM to form the estimation criterion is essentially the same as the one we used to estimate our models. The only difference is that CKM use an approximation to the Gaussian function by working with the steady state Kalman gain. We form the exact Gaussian density function, in which the Kalman gain varies over dates, as described in Hamilton (1994). We believe this difference is inconsequential. Return to text
28. When generating the artificial data underlying the calculations in the 3,1 panel of Figure 11, we set the measurement error to zero. (The same assumption was made for all the results reported here.) However, simulations that include the estimated measurement error produce results that are essentially the same. Return to text
29. We use CKM's computer code and data to ensure comparability of results. Return to text
30. The bounds of this interval are the upper and lower values of where twice the difference of the loglikelihood from its maximal value equals the critical value associated with the relevant likelihood ratio test. Return to text
31. For technical reasons, CKM actually consider `quasi differencing' hours worked using a differencing parameter close to unity. In small samples this type of quasi differencing is virtually indistinguishable from first differencing. Return to text
32. Our strategy differs somewhat from the one pursued in Fisher (2006), who applies a version of the instrumental variables strategy proposed by Shapiro and Watson (1988). Return to text
33. Erceg, Christopher J., Dale W. Henderson, and Andrew T. Levin (2000). `` Optimal Monetary Policy with Staggered Wage and Price Contracts,'' Journal of Monetary Economics, vol. 46 (October), pp. 281313.
Woodford, Michael M. (1996). `` Control of the Public Debt: A Requirement for Price Stability?'' NBER Working Paper Series 5684. Cambridge, Mass.: National Bureau of Economic Research, July. Return to text
34. Similar specifications have been used by authors such as Sims (1994) and SchmittGrohe and Uribe (2004). (SchmittGrohé, Stefanie, and Martin Uribe (2004). `` Optimal Fiscal and Monetary Policy under Sticky Prices,'' Journal of Economic Theory, vol. 114 (February), pp. 198230. Sims, Christopher, (1994), `` A Simple Model for Study of the Determination of the Price Level and the Interaction of Monetary and Fiscal Policy,'' Economic Theory, vol. 4 (3), 38199.) Return to text
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