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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 866, August 2006-Screen Reader Version*

Assessing Structural VARs^*

Lawrence J. Christiano^†, Martin Eichenbaum,^‡ and Robert Vigfusson^§

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

This paper analyzes the quality of VAR-based procedures for estimating the response of the economy to a shock. We focus on two key issues. First, do VAR-based 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 short-run or long-run restrictions. Confidence intervals are wider in the case of long-run restrictions. Even so, long-run 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

1  Introduction

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, VAR-based 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 VAR-based 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, VAR-based 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 VAR-based 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, VAR-based 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 long-run effects of shocks.² Other authors exploit short-run 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 short-run 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 short-run identification schemes to identify the dynamic effects of shocks to the economy. Of course, one can question the particular short-run identifying assumptions used in any given analysis. However, our results strongly support the view that if the relevant short-run 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 long-run results is that the sampling uncertainty associated with estimated impulse response functions is substantially larger in the long-run 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 long-run restrictions one requires an estimate of the sum of the VAR coefficients. The specification error involved in using a finite-lag 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 long-run restrictions tends to be large.

The preceding observations led us to develop an alternative to the standard VAR-based 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 zero-frequency 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 long-run 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 long-run restrictions. More recently Erceg, Guerrieri, and Gust (2005) and Chari, Kehoe, and McGrattan (2005b) (henceforth CKM) also examine the reliability of VAR-based inference using long-run identifying restrictions.⁴Our 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 VAR-based estimators of impulse response functions. Section 4 analyzes the differences between short and long-run 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.

2  A Simple RBC Model

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 long-run 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 VAR-based 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.

2.1  The Model

The representative agent maximizes expected utility over per capita consumption, $c_{t},$ and per capita hours worked, $l_{t}:$

$$E_{0}\sum_{t=0}^{\infty}\left( \beta\left( 1+\gamma\right) \right... ... \log c_{t}+\psi\frac{\left( 1-l_{t}\right) ^{1-\sigma}-1}{1-\sigma }\right] ,$$

subject to the budget constraint:

$$c_{t}+\left( 1+\tau_{x,t}\right) i_{t}\leq\left( 1-\tau_{l,t}\right) w_{t}l_{t}+r_{t}k_{t}+T_{t},$$

where

$$i_{t}=\left( 1+\gamma\right) k_{t+1}-\left( 1-\delta\right) k_{t}.$$

Here, $k_{t}$ denotes the per capita capital stock at the beginning of period $t,$ $w_{t}$ is the wage rate, $r_{t}$ is the rental rate on capital, $\tau_{x,t}$ is an investment tax, $\tau_{l,t}$ is the tax rate on labor income, $\delta\in(0,1)$ is the depreciation rate on capital, $\gamma$ is the growth rate of the population, $T_{t}$ represents lump-sum taxes and $\sigma>0$ is a curvature parameter.

The representative competitive firm's production function is:

$$y_{t}=k_{t}^{\alpha}\left( Z_{t}l_{t}\right) ^{1-\alpha},$$

where $Z_{t}$ is the time $t$ state of technology and $\alpha\in(0,1).$ The stochastic processes for the shocks are:

$$\log z_{t} =\mu_{z}+\sigma_{z}\varepsilon_{t}^{z}$$

$$\tau_{l,t+1} =\left( 1-\rho_{l}\right) \tau_{l}+\rho_{l}\tau _{l,t}+\sigma_{l}\varepsilon_{t+1}^{l}$$ (1)

$$\tau_{x,t+1} =\left( 1-\rho_{x}\right) \tau_{x}+\rho_{x}\tau _{x,t}+\sigma_{x}\varepsilon_{t+1}^{x},$$

where $z_{t}=Z_{t}/Z_{t-1}.$ In addition, $\varepsilon_{t}^{z}$, $\varepsilon_{t}^{l}$, and $\varepsilon_{t}^{x}$ are independently and identically distributed (i.i.d.) random variables with mean zero and unit standard deviation. The parameters, $\sigma_{z},$ $\sigma_{l},$ and $\sigma_{x}$ are non-negative scalars. The constant, $\mu_{z},$ is the mean growth rate of technology, $\tau_{l}$ is the mean labor tax rate, and $\tau_{x}$ is the mean tax on capital. We restrict the autoregressive coefficients, $\rho_{l}$ and $\rho_{x},$ to be less than unity in absolute value.

Finally, the resource constraint is:

$$c_{t}+\left( 1+\gamma\right) k_{t+1}-\left( 1-\delta\right) k_{t}\leq y_{t}.$$

We consider two versions of the model, differentiated according to timing assumptions. In the standard or nonrecursive version, all time $t$ decisions are taken after the realization of the time $t$ shocks. This is the conventional assumption in the RBC literature. In the recursive version of the model the timing assumptions are as follows. First, $\tau_{l,t}$ is observed, and then labor decisions are made. Second, the other shocks are realized and agents make their investment and consumption decisions.

2.2  Relation of the RBC Model to VARs

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 Fernandez-Villaverde, Rubio-Ramirez, 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 state-space, observer form. Throughout, we analyze the log-linear approximations to model solutions. Suppose the variables of interest in the RBC model are denoted by $X_{t}.$ Let $s_{t}$ denote the vector of exogenous economic shocks and let $\hat{k}_{t}$ denote the percent deviation from steady state of the capital stock, after scaling by $Z_{t}.$⁵ The approximate solution for $X_{t}$ is given by:

$$X_{t}=a_{0}+a_{1}\hat{k}_{t}+a_{2}\hat{k}_{t-1}+b_{0}s_{t}+b_{1}s_{t-1},$$ (2)

where

$$\hat{k}_{t+1}=A\hat{k}_{t}+Bs_{t}.$$ (3)

Also, $s_{t}$ has the law of motion:

$$s_{t}=Ps_{t-1}+Q\varepsilon_{t},$$ (4)

where $\varepsilon_{t}$ is a vector of i.i.d. fundamental economic disturbances. The parameters of (2)and (3) are functions of the structural parameters of the model.

The `state' of the system is composed of the variables on the right side of (2):

$$\xi_{t}=\left( \begin{array}[c]{c} \hat{k}_{t}\ \hat{k}_{t-1}\ s_{t}\ s_{t-1} \end{array}\right) .$$

The law of motion of the state is:

$$\xi_{t}=F\xi_{t-1}+D\varepsilon_{t},$$ (5)

where $F$ and $D$ are constructed from $A,$ $B,$ $Q,$ $P.$ The econometrician observes the vector of variables, $Y_{t}.$ We assume $Y_{t}$ is equal to $X_{t}$ plus iid measurement error, $v_{t}$, which has diagonal variance-covariance, $R.$ Then:

$$Y_{t}=H\xi_{t}+v_{t}.$$ (6)

Here, $H$ is defined so that $X_{t}=H\xi_{t}$, that is, relation (2) is satisfied. In (6) we abstract from the constant term. Hamilton (1994, section 13.4) shows how the system formed by (5) and (6) can be used to construct the exact Gaussian density function for a series of observations, $Y_{1},...,Y_{T}.$ We use this approach when we estimate versions of the RBC model.

We now use (5) and (6) to establish conditions under which the reduced form representation for $X_{t}$ implied by the RBC model is a VAR with disturbances that are linear combinations of the economic shocks. In this discussion, we set $v_{t}=0,$ so that $X_{t}=Y_{t}.$ In addition, we assume that the number of elements in $\varepsilon_{t}$ coincides with the number of elements in $Y_{t}.$

We begin by substituting (5) into (6) to obtain:

$$Y_{t}=HF\xi_{t-1}+C\varepsilon_{t},$$ $$C\equiv HD.$$

Our assumption on the dimensions of $Y_{t}$ and $\varepsilon_{t}$ implies that the matrix $C$ is square. In addition, we assume $C$ is invertible. Then:

$$\varepsilon_{t}=C^{-1}Y_{t}-C^{-1}HF\xi_{t-1}.$$ (7)

Substituting (7) into (5), we obtain:

$$\xi_{t}=M\xi_{t-1}+DC^{-1}Y_{t},$$

where

$$M=\left[ I-DC^{-1}H\right] F.$$ (8)

As long as the eigenvalues of $M$ are less than unity in absolute value,

$$\xi_{t}=DC^{-1}Y_{t}+MDC^{-1}Y_{t-1}+M^{2}DC^{-1}Y_{t-2}+...$$ (9)

Using (9) to substitute out for $\xi_{t-1}$ in (7), we obtain:

$$\varepsilon_{t}=C^{-1}Y_{t}-C^{-1}HF\left[ DC^{-1}Y_{t-1}+MDC^{-1} Y_{t-2}+M^{2}DC^{-1}Y_{t-3}+...\right] ,$$

or, after rearranging:

$$Y_{t}=B_{1}Y_{t-1}+B_{2}Y_{t-2}+... +u_{t},$$ (10)

where

$$u_{t} =C\varepsilon_{t}$$ (11)

$$B_{j} =HFM^{j-1}DC^{-1}, j=1,2,...$$ (12)

Expression (10) is an infinite-order VAR, because $u_{t}$ is orthogonal to $Y_{t-j},$ $j\geq1.$

Proposition 2.1. (Fernandez-Villaverde, Rubio-Ramirez, and Sargent) If C is invertible and the eigenvalues of M are less than unity in absolute value, then the RBC model implies:

$\bullet Y_{t}$ has the infinite-order VAR representation in (10)

$\bullet$ The linear one-step-ahead forecast error $Y_{t}$ given past $Y_{t}$'s is $u_{t}$, which is related to the economic disturbances by (11)

$\bullet$ The variance-covariance of $u_{t}$ is $CC^{\prime}$

$\bullet$ The sum of the VAR lag matrices is given by:

$B(1)\equiv\sum\limits_{j=1}^{\infty}B_{j}=HF\left[ I-M\right] ^{-1}DC^{-1}$

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 $C$ is square and invertible may not be satisfied. Whether $C$ satisfies these conditions depends on how $Y_{t}$ is defined. Third, significant measurement errors may exist. Fourth, the matrix, $M$, 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.

2.3  VARs in Practice and the RBC Model

We are interested in the use of VARs as a way to estimate the response of $X_{t}$ to economic shocks, i.e., elements of $\varepsilon_{t}.$ In practice, macroeconomists use a version of (10) with finite lags, say $q.$ A researcher can estimate $B_{1},...,B_{q}$ and $V=Eu_{t}u_{t}^{\prime}.$ To obtain the impulse response functions, however, the researcher needs the $B_{i}$'s and the column of $C$ corresponding to the shock in $\varepsilon_{t}$ that is of interest. However, to compute the required column of $C$ requires additional identifying assumptions. In practice, two types of assumptions are used. Short-run assumptions take the form of direct restrictions on the matrix $C$. Long-run assumptions place indirect restrictions on $C$ that stem from restrictions on the long-run response of $X_{t}$ to a shock in an element of $\varepsilon_{t}$. In this section we use our RBC model to discuss these two types of assumptions and how they are imposed on VARs in practice.

2.3.1  The Standard Version of the Model

The log-linearized equilibrium laws of motion for capital and hours in this model can be written as follows:

$$\log\hat{k}_{t+1}=\gamma_{0}+\gamma_{k}\log\hat{k}_{t}+\gamma_{z}\log z_{t}+\gamma_{l}\tau_{l,t}+\gamma_{x}\tau_{x,t},$$ (13)

and

$$\log l_{t}=a_{0}+a_{k}\log\hat{k}_{t}+a_{z}\log z_{t}+a_{l}\tau_{l,t} +a_{x}\tau_{x,t}.$$ (14)

From (2.13) and (2.14), it is clear that all shocks have only a temporary effect on $l_{t}$ and $\hat{k}_{t}.$⁷ The only shock that has a permanent effect on labor productivity, $a_{t}\equiv y_{t}/l_{t},$ is $\varepsilon_{t}^{z}.$ The other shocks do not have a permanent effect on $a_{t}.$ Formally, this exclusion restriction is:

$$\lim_{j\rightarrow\infty}\left[ E_{t}a_{t+j}-E_{t-1}a_{t+j}\right] =f\left( \varepsilon_{t}^{z}\text{ only}\right) .$$ (15)

In our linear approximation to the model solution $f$ is a linear function. The model also implies the sign restriction that $f$ is an increasing function. In (2.15), $E_{t}$ is the expectation operator, conditional on the information set $\Omega_{t}=\left( \log\hat{k}_{t-s},\log z_{t-s} ,\tau_{l,t-s},\tau_{x,t-s};\text{ }s\geq0\right)$ .

In practice, researchers impose the exclusion and sign restrictions on a VAR to compute $\varepsilon_{t}^{z}$ and identify its dynamic effects on macroeconomic variables. Consider the $N\times1$ vector, $Y_{t}.$ The VAR for $Y_{t}$ is given by:

$$Y_{t+1} =B\left( L\right) Y_{t}+u_{t+1}, Eu_{t}u_{t}^{\prime }=V,$$ (16)

$$B(L) \equiv B_{1}+B_{2}L+...+B_{q}L^{q-1},$$

$$Y_{t} =\left( \begin{array}[c]{c} \Delta\log a_{t}\\ \log l_{t}\\ x_{t} \end{array} \right) .$$

Here, $x_{t}$ is an additional vector of variables that may be included in the VAR. Motivated by the type of reasoning discussed in the previous subsection, researchers assume that the fundamental economic shocks are related to $u_{t}$ as follows:

$$u_{t}=C\varepsilon_{t} E\varepsilon_{t}\varepsilon_{t}^{\prime }=I, CC^{\prime}=V.$$ (17)

Without loss of generality, we assume that the first element in $\varepsilon_{t}$ is $\varepsilon_{t}^{z}.$ We can easily verify that:

$$\lim_{j\rightarrow\infty}\left[ \tilde{E}_{t}a_{t+j}-\tilde{E}_{t-1} a_{t+j}\right] =\tau\left[ I-B(1)\right] ^{-1}C\varepsilon_{t},$$ (18)

where $\tau$ is a row vector with all zeros, but with unity in the first location. Here:

$$B(1)\equiv B_{1}+...+B_{q}.$$

Also, $\tilde{E}_{t}$ is the expectation operator, conditional on $\tilde{\Omega}_{t}=\left\{ Y_{t},...,Y_{t-q+1}\right\} .$ As mentioned above, to compute the dynamic effects of $\varepsilon_{t}^{z},$ we require $B_{1},...,B_{q}$ and $C_{1},$ the first column of $C.$

The symmetric matrix, $V,$ and the $B_{i}$'s can be computed using ordinary least squares regressions. However, the requirement that $CC^{\prime}=V$ is not sufficient to determine a unique value of $C_{1}.$ Adding the exclusion and sign restrictions does uniquely determine $C_{1}.$ Relation (2.18) implies that these restrictions are:

   exclusion restriction: $$\left[ I-B(1)\right] ^{-1}C=\left[ \begin{array}[c]{cc} \text... ...ine{0}\ \text{numbers} & \text{numbers} \end{array}\right] ,$$

where $\underline{0}$ is a row vector and

   sign restriction: $$(1,1)$$ element of $$\left[ I-B(1)\right] ^{-1}C$$ is positive.

There are many matrices, $C,$ that satisfy $CC^{\prime}=V$ as well as the exclusion and sign restrictions. It is well-known that the first column, $C_{1},$ of each of these matrices is the same. We prove this result here, because elements of the proof will be useful to analyze our simulation results. Let

$$D\equiv\left[ I-B(1)\right] ^{-1}C.$$

Let $S_{Y}\left( \omega\right)$ denote the spectral density of $Y_{t}$ at frequency $\omega$ that is implied by the $q^{th}$-order VAR. Then:

$$DD^{\prime}=\left[ I-B(1)\right] ^{-1}V\left[ I-B(1)^{\prime}\right] ^{-1}=S_{Y}\left( 0\right) .$$ (19)

The exclusion restriction requires that $D$ have a particular pattern of zeros:

$$D=\left[ \begin{array}[c]{cc} \underset{1\times1}{d_{11}} & ... ...\right) \times\left( N-1\right) }{D_{22}} \end{array}\right] ,$$

so that

$$DD^{\prime}=\left[ \begin{array}[c]{cc} d_{11}^{2} & d_{11}D... ...eft( 0\right) & S_{Y}^{22}\left( 0\right) \end{array}\right] ,$$

where

$$S_{Y}\left( \omega\right) \equiv\left[ \begin{array}[c]{cc} ... ...a\right) & S_{Y}^{22}\left( \omega\right) \end{array}\right] .$$

The exclusion restriction implies that

$$d_{11}^{2}=S_{Y}^{11}\left( 0\right) , D_{21}=S_{Y}^{21}\left( 0\right) /d_{11}.$$ (20)

There are two solutions to (2.20). The sign restriction

$$d_{11}>0$$ (21)

selects one of the two solutions to (2.20). So, the first column of $D,$ $D_{1},$ is uniquely determined. By our definition of $C,$ we have

$$C_{1}=\left[ I-B(1)\right] D_{1}.$$ (22)

We conclude that $C_{1}$ is uniquely determined.

2.3.2  The Recursive Version of the Model

In the recursive version of the model, the policy rule for labor involves $\log z_{t-1}$ and $\tau_{x},_{t-1}$ because these variables help forecast $\log z_{t}$ and $\tau_{x},_{t}:$

$$\log l_{t}=a_{0}+a_{k}\log\hat{k}_{t}+\tilde{a}_{l}\tau_{l,t}+\tilde{a} _{z}^{\prime}\log z_{t-1}+\tilde{a}_{x}^{\prime}\tau_{x,t-1}.$$

Because labor is a state variable at the time the investment decision is made, the equilibrium law of motion for $\hat{k}_{t+1}$ is:

$$\log\hat{k}_{t+1} =\gamma_{0}+\gamma_{k}\log\hat{k}_{t}+\tilde{\gamma} _{z}\log z_{t}+\tilde{\gamma}_{l}\tau_{l,t}+\tilde{\gamma}_{x}\tau_{x,t}$$

$$+\tilde{\gamma}_{z}^{\prime}\log z_{t-1}+\tilde{\gamma}_{x}^{\prime} \tau_{x,t-1}.$$

As in the standard model, the only shock that affects $a_{t}$ in the long run is a shock to technology. So, the long-run identification strategy discussed in section 2.3.1 applies to the recursive version of the model. However, an alternative procedure for identifying $\varepsilon_{t}^{z}$ applies to this version of the model. We refer to this alternative procedure as the `short-run' identification strategy because it involves recovering $\varepsilon_{t}^{z}$ using only the realized one-step-ahead forecast errors in labor productivity and hours, as well as the second moment properties of those forecast errors.

Let $u_{\Omega,t}^{a}$ and $u_{\Omega,t}^{l}$ denote the population one-step-ahead forecast errors in $a_{t}$ and $\log l_{t},$ conditional on the information set, $\Omega_{t-1}.$ The recursive version of the model implies that

$$u_{\Omega,t}^{a}=\alpha_{1}\varepsilon_{t}^{z}+\alpha_{2}\varepsilon_{t} ^{l},$$  $$u_{\Omega,t}^{l}=\gamma\varepsilon_{t}^{l},$$

where $\alpha_{1}>0,$ $\alpha_{2},$ and $\gamma$ are functions of the model parameters. The projection of $u_{\Omega,t}^{a}$ on $u_{\Omega,t}^{l}$ is given by

$$u_{\Omega,t}^{a}=\beta u_{\Omega,t}^{l}+\alpha_{1}\varepsilon_{t}^{z}, \beta=\frac{cov(u_{\Omega,t}^{a},u_{\Omega,t}^{l})}{var\left( u_{\Omega,t}^{l}\right) }.$$ (23)

Because we normalize the standard deviation of $\varepsilon_{t}^{z}$ to unity, $\alpha_{1}$ is given by:

$$\alpha_{1}=\sqrt{var\left( u_{\Omega,t}^{a}\right) -\beta^{2}var\left( u_{\Omega,t}^{l}\right) }.$$

In practice, we implement the previous procedure using the one-step-ahead forecast errors generated from a VAR in which the variables in $Y_{t}$ are ordered as follows:

$$Y_{t}=\left( \begin{array}[c]{c} \log l_{t}\ \Delta\log a_{t}\ x_{t} \end{array}\right) .$$

We write the vector of VAR one-step-ahead forecast errors, $u_{t},$ as:

    $$u_{t}=\left( \begin{array}[c]{c} u_{t}^{l}\ u_{t}^{a}\ u_{t}^{x} \end{array}\right) .$$

We identify the technology shock with the second element in $\varepsilon_{t}$ in (2.17). To compute the dynamic response of the variables in $Y_{t}$ to the technology shock we need $B_{1},...,B_{q}$ in (2.17) and the second column, $C_{2},$ of the matrix $C$, in (2.17). We obtain $C_{2}$ in two steps. First, we identify the technology shock using:

$$\varepsilon_{t}^{z}=\frac{1}{\hat{\alpha}_{1}}\left( u_{t}^{a}-\hat{\beta }u_{t}^{l}\right) \text{,}$$

where

$$\hat{\beta}=\frac{cov(u_{t}^{a},u_{t}^{l})}{var\left( u_{t}^{l}\r... ...sqrt{var\left( u_{t}^{a}\right) -\hat{\beta} ^{2}var\left( u_{t}^{l}\right) }.$$

The required variances and covariances are obtained from the estimate of $V$ in (2.16). Second, we regress $u_{t}$ on $\varepsilon_{t}^{z}$ to obtain:⁸

$$C_{2}=\left( \begin{array}[c]{c} \frac{cov(u^{l},\varepsilon... ...\left( u_{t}^{x},u_{t}^{l}\right) \right) \end{array}\right) .$$

2.4  Parameterization of the Model

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:

$$\beta =0.98^{1/4}, \theta=0.33, \delta=1-(1-.06)^{1/4}, \psi=2.5, \gamma=1.01^{1/4}-1$$ (24)

$$\tau_{x} =0.3, \tau_{l}=0.242, \mu_{z}=1.016^{1/4}-1, \sigma=1.$$

2.4.1  Our MLE Parameterizations

We estimate two versions of our model. In the two-shock maximum likelihood estimation (MLE) specification we assume that $\sigma_{x}=0,$ so that there are two shocks, $\tau_{l,t}$ and $\log z_{t}.$ We estimate the parameters $\rho_{l}$, $\sigma_{l},$ and $\sigma_{z},$ by maximizing the Gaussian likelihood function of the vector, $X_{t}=\left( \Delta\log y_{t},\log l_{t}\right) ^{\prime},$ subject to (24 )$.$⁹ Our results are given by:

$$\log z_{t} =\mu_{z}+0.00953\varepsilon_{t}^{z},$$

$$\tau_{l,t} =\left( 1-0.986\right) \bar{\tau}_{l}+0.986\tau _{l,t-1}+0.0056\varepsilon_{t}^{l}.$$

The three-shock MLE specification incorporates the investment tax shock, $\tau_{x,t},$ into the model. We estimate the three-shock MLE version of the model by maximizing the Gaussian likelihood function of the vector, $X_{t}=\left( \Delta\log y_{t},\log l_{t},\Delta\log i_{t}\right) ^{\prime}$ , subject to the parameter values in (24)$.$ The results are:

$$\log z_{t} =\mu_{z}+0.00968\varepsilon_{t}^{z},$$

$$\tau_{l,t} =\left( 1-0.9994\right) \tau_{l}+0.9994\tau_{l,t-1} +0.00631\varepsilon_{t}^{l},$$

$$\tau_{x,t} =\left( 1-0.9923\right) \tau_{x}+0.9923\tau_{x,t-1} +0.00963\varepsilon_{t}^{x}.$$

The estimated values of $\rho_{x}$ and $\rho_{l}$ are close to unity. This finding is consistent with other research that also reports that shocks in estimated general equilibrium models exhibit high degrees of serial correlation.¹⁰

2.4.2CKM Parameterizations

The two-shock CKM specification has two shocks, $z_{t}$ and $\tau_{l,t}.$ These shocks have the following time series representations:

$$\log z_{t} =\mu_{z}+0.0131\varepsilon_{t}^{z},$$

$$\tau_{l,t} =\left( 1-0.952\right) \tau_{l}+0.952\tau_{l,t-1} +0.0136\varepsilon_{t}^{l}.$$

The three-shock CKM specification adds an investment shock, $\tau_{x,t}$, to the model, and has the following law of motion:

$$\tau_{x,t}=\left( 1-0.98\right) \tau_{x}+0.98\tau_{x,t-1}+0.0123\varepsilon _{t}^{x}.$$ (25)

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 two-shock CKM specification than in our MLE specification. Also, the ratio of $\sigma _{l}^{2}$ to $\sigma_{z}^{2}$ is nearly three times larger in the two-shock CKM specification than in our two-shock MLE specification. Section 5 below discusses the reasons for these differences.

2.5  The Importance of Technology Shocks for Hours Worked

Table 1 reports the contribution, $V_{h},$ 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 HP-filtered, log hours and (iii) the variance in the one-step-ahead 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 $V_{h}$ is measured according to (i), it is always below 4 percent. When $V_{h}$ is measured using (ii) or (iii) it is always below 8 percent. For both (ii) and (iii), in the CKM specifications, $V_{h}$ 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 two-shock MLE specification. The volatile time series shows how log hours worked evolve in the presence of shocks to both $z_{t}$ and $\tau_{l,t}.$ The other time series shows how log hours worked evolve in response to just the technology shock, $z_{t}.$ The bottom panel is the analog of the top figure when the data are generated using the standard two-shock CKM specification.

3  Results Based on RBC Data Generating Mechanisms

In this section we analyze the properties of conventional VAR-based 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 $\varepsilon_{t}^{z}$, $\varepsilon_{t}^{l},$ and possibly $\varepsilon_{t}^{x},$ are drawn from $i.i.d.$ standard normal distributions. For each artificial data set, we estimate a four-lag 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 percentile-based confidence interval and a standard-deviation 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 percentile-based 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 standard-deviation-based 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 VAR-based 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 two-shock MLE and CKM specifications, we set $Y_{t}=(\Delta\log a_{t},$ $\log l_{t})^{\prime}.$ When we generate data from the three-shock MLE and CKM specifications, we set $Y_{t}=(\Delta\log a_{t},\log l_{t},\log i_{t}/y_{t})^{\prime}.$

3.1  Short-Run Identification

Results for the two- and three- Shock MLE Specifications

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 $95$ percent probability interval for the estimated impulse response functions. The stars with no line indicate the average percentile-based confidence intervals across the 1,000 data sets. The circles with no line indicate the average standard-deviation-based confidence intervals.

Figures 3 and 4 graph the coverage rates for the percentile-based and standard-deviation-based 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 VAR-based estimator of the response of hours to a technology shock when the data are generated by the two-shock MLE specification. The 2,1 panel corresponds to the case when the data generating process is the three-shock 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, VAR-based confidence intervals include the true value of the impulse response coefficients.

Results for the CKM Specification

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 three-shock 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 VAR-based 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). VAR-based confidence intervals detect this fact.

3.2  Long-run Identification

Results for the two- and three- Shock MLE Specifications

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 percentile-based confidence intervals are very close to 95 percent (see Figure 3). The coverage rates for the standard-deviation-based 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.

Results for the CKM Specification

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 Two-shock CKM specification, the contemporaneous response of hours worked to a one-standard-deviation technology shock is $0.3$ percent, while the mean estimated response is $0.97$ 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 standard-deviation-based 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, VAR-based estimators of the impulse response function have a downward bias.

Figure 3 reveals that for the two- and three-shock CKM specifications, percentile-based coverage rates are reasonably close to 95 percent. Figure 4 shows that the standard deviation based coverage rates are lower than the percentile-based 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.

3.3  Confidence Intervals in the RBC Examples and a Situation in Which VAR-Based Procedures Go Awry

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, $\sigma_{l},$ and by considering alternative values of $\sigma,$ the utility parameter that controls the Frisch elasticity of labor supply.

Consider Figure 6, which focuses on the long-run identification schemes. The first and second columns report results for the two-shock MLE and CKM specifications, respectively. For each specification we redo our experiments, reducing $\sigma_{l}$ 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 two-shock CKM specification. The 1,1 and 2,1 panels show what happens when we vary the value of $\sigma$, the parameter that controls the Frisch labor supply elasticity. In the 1,1 panel we set $\sigma=6,$ which corresponds to a Frisch elasticity of 0.63. In the 2,1 panel, we set $\sigma=0,$ 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 VAR-based 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 two-shock 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 $V_{h},$ two are $0.46$ percent and the third is $0.66$ percent . The 3,1 panel of Figure 7 shows that the VAR-based 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 VAR-based 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.

3.4  Are Long-Run Identification Schemes Informative?

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 long-run identification schemes are large. One might be tempted to conclude that VAR-based long-run 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 wage-price 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 long-run 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 four-variable structural VAR on each synthetic data set and compute the dynamic response of hours worked to a technology shock using long-run 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 wage-price 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 wage-price 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 wage-price alternative.

4  Contrasting Short- and Long- Run Restrictions

The previous section demonstrates that, in the examples we considered, when VARs are identified using short-run 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 long-run identifying restrictions can exhibit noticeable bias. In this section we argue that the key difference between the two identification strategies is that the long-run strategy requires an estimate of the sum of the VAR coefficients, $B\left( 1\right) .$ 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, fixed-lag, finite-order VAR would find in population. Let $\hat{B}_{1},...,\hat{B}_{q}$ and $\hat{V}$ denote the parameters of the $q$th-order VAR fit by the econometrician. Then:

$$\hat{V}=V+\min_{\hat{B}_{1},...,\hat{B}_{q}}\frac{1}{2\pi}\int_{-... ... e^{i\omega}\right) -\hat{B}\left( e^{i\omega}\right) \right] ^{\prime}d\omega,$$ (26)

where

$$B\left( L\right) =B_{1}+B_{2}L+B_{3}L^{2}+...,$$

$$\hat{B}\left( L\right) =\hat{B}_{1}+\hat{B}_{2}L+...+\hat{B}_{4}L^{3}.$$

Here, $B\left( e^{-i\omega}\right)$ and $\hat{B}\left( e^{-i\omega }\right)$ correspond to $B\left( L\right)$ and $\hat{B}\left( L\right)$ with $L$ replaced by $e^{-i\omega}.$¹⁷ In (26), $B$ and $V$ are the parameters of the actual infinite-ordered VAR representation of the data (see (10)), and $S_{Y}\left( \omega\right)$ is the associated spectral density at frequency $\omega$.¹⁸ According to (26), estimation of a VAR approximately involves choosing VAR lag matrices to minimize a quadratic form in the difference between the estimated and true lag matrices. The quadratic form assigns greatest weight to the frequencies for which the spectral density is the greatest. If the econometrician's VAR is correctly specified, then $\hat{B}\left( e^{-i\omega }\right) =B\left( e^{-i\omega}\right)$ for all $\omega,$ and $\hat{V}=V,$ so that the estimator is consistent. If there is specification error, then $\hat{B}\left( e^{-i\omega}\right) \neq B\left( e^{-i\omega}\right)$ for some $\omega$ and $V>\hat{V}.$¹⁹ In our context, specification error exists because the true VAR implied by our data generating processes has $q=\infty,$ but the econometrician uses a finite value of $q.$

To understand the implications of (26) for our analysis, it is useful to write in lag-operator form the estimated dynamic response of $Y_{t}$ to a shock in the first element of $\varepsilon_{t}$

$$Y_{t}=\left[ I+\theta_{1}L+\theta_{2}L^{2}+...\right] \hat{C}_{1} \varepsilon_{1,t},$$ (27)

where the $\theta_{k}$'s are related to the estimated VAR coefficients as follows:

$$\theta_{k}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\left[ I-\hat{B}\left( e^{-i\omega}\right) e^{-i\omega}\right] ^{-1}e^{k\omega i}d\omega.$$ (28)

In the case of long-run identification, the vector $\hat{C}_{1}$ is computed using (22), and $\hat{B}\left( 1\right)$ and $\hat{V}$ replace $B\left( 1\right)$ and $V$ respectively. In the case of short-run identification, we compute $\hat{C}_{1}$ as the second column in the upper triangular Cholesky decomposition of $\hat{V}$.²⁰

We use (26) to understand why estimation based on short-run and long-run 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 $\hat{C}_{1},$ and the dynamic part summarized in the term in square brackets. We argue that when a bias arises with long-run restrictions, it is because of difficulties in estimating $C_{1}.$ These difficulties do not arise with short-run restrictions.

In the short-run identification case, $\hat{C}_{1}$ is a function of $\hat{V}$ only. Across a variety of numerical examples, we find that $\hat{V}$ is very close to $V.$²¹ This result is not surprising because (26) indicates that the entire objective of estimation is to minimize the distance between $\hat{V}$ and $V.$ In the long-run identification case, $\hat{C}_{1}$ depends not only on $\hat{V}$ but also on $\hat{B}\left( 1\right) .$ A problem is that the criterion does not assign much weight to setting $\hat{B}\left( 1\right) =B\left( 1\right)$ unless $S_{Y}\left( \omega\right)$ happens to be relatively large in a neighborhood of $\omega=0.$ But, a large value of $S_{Y}\left( 0\right)$ is not something one can rely on.²² When $S_{Y}\left( 0\right)$ is relatively small, attempts to match $\hat{B}\left( e^{-i\omega }\right)$ with $B\left( e^{-i\omega}\right)$ at other frequencies can induce large errors in $\hat{B}(1).$

The previous argument about the difficulty of estimating $C_{1}$ in the long-run identification case does not apply to the $\theta_{k}^{\prime}$s$.$ According to (28) $\theta_{k}$ is a function of $\hat{B}\left( e^{-i\omega }\right)$ over the whole range of $\omega$'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 $B\left( 1\right)$, the bias and much of the sampling uncertainty associated with the Two-shock CKM specification disappears. Second, we demonstrate that bias problems essentially disappear when we use an alternative to the standard zero-frequency 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 two-shock CKM specification in which the effect of technology shocks can be estimated using either short- or long-run restrictions.

A Numerical Example

Table 2 reports various properties of the two-shock CKM specification. The first six $B_{j}$'s in the infinite-order VAR, computed using (12), are reported in Panel A. These $B_{j}$'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 $M$ in (8), which is 0.957. Panel B displays the $\hat{B}_{j}$'s that solve (26) with $q=4.$ Informally, the $\hat{B}_{j}$'s look similar to the $B_{j}$'s for $j=1,2,3,4.$ In line with this observation, the sum of the true $B_{j}$'s, $B_{1}+...+B_{4}$ is similar in magnitude to the sum of the estimated $\hat{B}_{j}$'s,