I estimate stickyprice and stickyinformation models of price setting for the United States via maximumlikelihood techniques, reaching several conclusions. First, the stickyprice model fits best, and captures inflation dynamics as well as reducedform equations once hybridbehavior is allowed. Second, the importance of hybrid behavior in stickyprice models is potentially consistent with a role for some information imperfections, such as sticky information, as a complement to nominal price rigidities. Finally, the favorable results herein for the hybrid stickyprice model when evaluated by statistics that summarize the relative fit of different models is consistent with the existing literature that is both supportive and dismissive of such models, as this literature has largely ignored fit in evaluating such models. Many previous studies have focused on ancillary issues, such as the standard errors associated with certain parameters or Grangercausality tests that may not provide much information about stickyprice models.
JEL Codes: E3
There is a great deal of disagreement over how to best model inflation dynamics, and rival views are often only weakly contrasted empirically. Our analysis provides a quantitative evaluation of two leading structural models of inflation dynamics  the stickyprice model and the stickyinformation model  and considers their performance relative to reducedform regressions. The analysis differs from much previous work in its focus on measures of fit and explicit consideration of alternative models. Three conclusions are reached: The stickyprice model appears to fit best, particularly over the recent sample and if allowance is made for what has been termed "hybrid" behavior; in addition, the presence of "hybrid" behavior is consistent with the types of information imperfections emphasized in the stickyinformation model, implying that the data's preference for a stickyprice hybrid model provides support to some role for such information imperfections as a complement to sticky prices; and, finally, the results herein are consistent with previous research that has reached quite diverse conclusions. For example, authors have used similar results to support rational expectations stickyprice models (e.g., Gali and Gertler (1999) or Sbordone (2005)), or to criticize such models (e.g., Rudd and Whelan (2003, 2005a, 2005b, 2005c)). Our discussions will take pains to illustrate how emphasis on particular stylized facts by other authors has led to the range of conclusions they have offered.
Before turning to the substantive analysis, it is useful to compare our analysis to the themes covered thirtyfive years ago in a conference at the Federal Reserve Board on a broadly similar theme, The Econometrics of Price Determination (Board of Governors of the Federal Reserve System (1971)). The research at that earlier conference largely focused on the "wageprice" sectors of a number of macroeconometric models; our focus on stickyprice and stickyinformation models of price setting, increasingly employed as the price and/or wage specification in dynamic general equilibrium models, falls directly within that earlier tradition. But significant differences in approach, reflecting the results of decades of research, are also apparent. Most importantly, the research at the earlier conference made at most ad hoc attempts to distinguish between dynamics induced by expectations and those intrinsic to price or wage setting (although the contributions of Robert Lucas and Peter von zur Muehlen emphasized the importance of separating the impact of expectations from other factors in a manner immediately familiar to the modern reader). This distinction is the hallmark of rational expectations models and remains the central area of disagreement in empirical work on inflation dynamics.
The next section discusses our empirical strategy. The analysis then turns to baseline and hybrid versions of a standard stickyprice model. The stickyinformation model is considered next, with a special focus on comparing the stickyinformation and stickyprice models. The final two sections discuss the interpretation of our results, with particular emphasis on linking the findings herein to other research.
The empirical analysis will focus on the price level for the nonfarm business sector of the United States. The primary estimation sample is 1965Q1 to 2002Q4; I will also consider the more recent 1983Q1 to 2002Q4 sample separately. The price level (P) is given by the nonfarm output deflator. In each model below, prices will depend upon a measure of marginal cost. In practice, I measure nominal marginal cost (P*MC) by unit labor costs.^{1} Unit labor costs are the theoretically correct measure of nominal marginal cost under some reasonable assumptions and are the standard in models of pricesetting (e.g., Gali and Gertler (1999), Sbordone (2002)). The literature on reducedform price equations has traditionally focused on output gap measures as the cost measure (e.g., Fuhrer and Moore (1995), Roberts (1997), Rudd and Whelan (2005c)), and our discussion later will examine the consistency of the estimated models with the reducedform correlation between detrended output and inflation. Real marginal cost is nominal marginal labor cost divided by the price level, or the labor share of income (MC). Other data used in the estimation include the level of real output (Y) in the nonfarm business sector and the nominal effective federal funds rate (r); the manner in which these variables enter the system is discussed below. The data on prices (P) and the labor share (MC) are logged and linearly detrended prior to estimation; the data on real output is logged and linearlydetrended with a break in the trend in 1973Q1; and the nominal federal funds rate is demeaned.
The persistence of inflation will prove important in model comparisons. Figure 1 graphs the autocorrelation functions of inflation and real marginal cost (the labor share) over the sample period (for the first 12 lags). Inflation is quite persistent, and the autocorrelations decay very slowly. Real marginal cost is also very highly autocorrelated. As marginal cost is the key driver of inflation in the models examined, such models will have no difficulty predicting persistent inflation, a point emphasized for the stickyprice model in, for example, Rudd and Whelan (2005a) and Fuhrer (2005).
Each of the models presented below is estimated via maximum likelihood (ML). The system of equations consists of the structural pricing equation and three reducedform equations for (log) nominal marginal cost (p+mc), the nominal funds rate (r), and (log) real output (y)
A reducedform system for marginal cost and any of its determinants is necessary to form the expectations that enter the structural pricing equation for ML estimation. I assume that the reducedform process governing (the change in) nominal marginal cost is well described by lags of real marginal cost, output, the nominal interest rate, and inflation. Output is similarly wellcaptured by reducedform equations containing lags of each variable in the system, while the reducedform for the nominal interest rate contains contemporaneous values as well, consistent with the literature on monetary policy reaction functions and the typical identifying assumption in vector autoregressions used for monetary policy analysis.^{2} A reducedform system can quite accurately summarize the information in past data for currentperiod realizations and is the correct reduced form in many structural models, limiting the degree of concern regarding misspecification.^{3} Combining the reducedform equations for marginal cost, output, and the nominal interest rate with a structural pricing equation results in a restricted reducedform representation for inflation, in which inflation depends upon lags of itself and all the other variables.
In the estimation, the likelihood was formed solely based on the onestepahead inflation forecast errors derived via the Kalman filter using the AndersonMoore AIM algorithm (see Anderson (2000)); the reducedform coefficients in the system and the disturbances were estimated separately and held fixed during the maximization of the likelihood constructed from the onestepahead inflation errors.^{4} Consequently, the differences in estimated likelihoods reflect only the different fits of the inflation equations, facilitating their comparison. The onestepahead forecast errors ignore any information in contemporaneous variables that would be useful in forecasting inflation and hence are comparable to the errors from reducedform equations relating inflation to its own lags and lags of other variables.^{5}
An alternative estimation strategy would be to instrument for expectations and find the structural parameters via an instrumental variables (IV) estimator (e.g., the Generalized Method of Moments). However, limitedinformation estimators have been shown to have some poor properties in the context of forwardlooking equations like the pricesetting models examined herein; examples include Ma (2002), Jondeau and Le Bihan (2003), Eichenbaum and Fisher (2003), Fuhrer and Olivei (2004), and Rudd and Whelan (2005a, 2005b, 2005c).
ML estimation provides a measure of fit through the likelihood function, facilitating model comparisons. In the current analysis, the set of models examined are nonnested. It is also likely that none of the models considered is literally true. Even when the models are not true, the likelihood
will concentrate in the neighborhood of the "best" parameter vector and provide a summary measure of the congruence between the data and model under reasonable regularity conditions, as discussed in the literature on pseudoML estimation (e.g., White (1994)). The comparison across models can be
made by penalizing the individual likelihoods by the number of parameters as in the Bayesian information criterion (BIC). BIC for model j equals
Consistent with this latter observation on the similarity between BIC and a Bayesian approach in large samples, a pseudoposterior odds measure can be found by using the BIC in place of the marginal likelihood, yielding the datadetermined probability of model j, (j),
Specifications
The stickyprice models will follow relatively standard specifications. For our current purposes, I only present the equations that are estimated; a discussion of the behavioral practices underlying the baseline and hybrid specifications is deferred to section 5 and the appendix.^{6} Following common practice, our baseline stickyprice model is derived from the Calvo stickyprice model and relates current inflation to expected inflation next period and to real marginal cost,
Baseline SP .^{7}
In the hybrid stickyprice model, inflation is related to a lead of inflation, a lag of inflation, and real marginal cost  with the sum of the coefficients on the lead and lag of inflation equaling one (see footnote 7):
Hybrid SP
As discussed later, the presence of a lag can be justified through a number of amendments to the baseline model  such as the presence of ruleofthumb pricesetters as in Gali and Gertler (1999) or dynamic indexation as in Christiano et al (2005). This hybrid model is the natural extension that allows for indexing or ruleofthumb behavior to a measure of inflation that smoothes through some of the quarterly volatility in inflation. Roberts (2005) also considers hybrids with a moving average of inflation lags, although he does not consider the behavioral assumptions that would generate such a specification that I discuss later. For now, the inclusion of the movingaverage term is motivated solely by the observation that reducedform Phillips curves that fit well are often specified to include long moving averages of inflation (e.g., Gordon (1998)).
Results
Table 1 presents estimates of the baseline and hybrid stickyprice models for the full sample, 1965Q1 to 2002Q4. Two hybrid models are considered: a onelag model (e.g., that of Gali and Gertler (1999)), and a fourlag model. For each model, I present estimates of the parameters of the model and their standard errors; the Qstatistic that examines the degree of serial correlation over 12 lags in the onestepahead forecast errors (and its pvalue); the loglikelihood and BIC for the model; and the R^{2} for the model, computed as one minus the sum of squared forecast errors divided by the sum of squared deviations of inflation from its sample mean.
Turning first to the baseline stickyprice model, the estimated sensitivity to marginal cost ( falls in the range typical of the literature, and the R^{2} for the model is about 0.4 (line 1). Nonetheless, it is clear the model fails to track inflation very well, as the Qstatistic reveals a very large degree of autocorrelation in the forecast errors or costpush shocks. This does not imply that the model does not predict very persistent inflation even in the absence of seriallycorrelated costpush shocks. Because the baseline model essentially has inflation track real marginal cost (or the labor share) and real marginal cost is very persistent (with a firstorder autocorrelation near 0.9), the baseline model predicts persistent inflation. The poor performance of the model and extreme serial correlation of the residuals rather reflects the fact that the predictions of the model result in large and persistent forecast errors. In economic terms, the baseline model expects inflation to track real marginal cost closely, and it does not. This failure is the source of Fuhrer's (2005) critique, in which he emphasizes that the errors in baseline and simple hybrid stickyprice models are large and persistent.
One way to quickly see how poorly the baseline stickyprice model performs is to compare it to a reducedform regression. In particular, rows 4 through 6 of table 1 report the likelihood, BIC, and R^{2} for regressions of the following form:
The hybrid stickyprice models fit substantially better than the baseline models over the 1965Q1 to 2002Q4 sample (as indicated by the likelihood and BIC values), with the model using a fourquarter average of lagged inflation (N equal to four) dominating the onelag model. The parameter estimates show that the weight on the sum of inflation lags ( falls to just above as N increases from one to four. Interpretation of this parameter value is tricky and depends on the behavioral assumptions used to motivate the inclusion of additional lags. However, this value suggests that in a model with ruleofthumb pricesetters (like that presented in the appendix) about of pricesetters use a ruleofthumb, similar to the value implied by the onelag hybrid model. This seems large enough to suggest that a search for microfoundations motivating such behavior is valuable.
Turning back to the empirical results, the R^{2} for the bestfitting model lies very close to that of the reducedform regressions reported in table 1. In addition, there is only marginal evidence of serial correlation in the residuals of the bestfitting model; this suggests that the inclusion of long lags allows for sufficient intrinsic inflation inertia to account for inflation dynamics, in contrast to the baseline and oneperiodlag hybrid critiqued in Fuhrer (2005). The Bayesian information criterion indicates that the hybrid stickyprice model (with N equal to four) fits the data as well or better than the reducedform regressions in table 1, and the pseudoodds measure places nearly all the weight on the hybrid stickyprice model with four lags.
Table 2 presents estimation results for the stickyprice models over the 1983Q1 to 2002Q4 period. Some may view this period as more appropriate for evaluation of each structural model, as the monetary policy regime has arguably been more stable during this period. With regard to which structural model fits the data best, the change in sample period has no effect; the stickyprice hybrid model with a fourquarter moving average of inflation (N equal to four) remains the bestfitting model. It also remains the case that the bestfitting stickyprice model is preferred to the reducedform regressions by the BIC criterion. On other dimensions, the change in sample period alters the results a bit. In particular, the baseline and oneperiod indexation stickyprice hybrids both fit as well or better than reducedform models over the 1983Q1 to 2002Q4 period according to the BIC criterion. In addition, the estimated sensitivity of inflation to marginal cost in the stickyprice models is much lower in the recent sample.
Combining the results from both sample periods, I reach three conclusions. First, hybrid stickyprice models of price setting fit the data as well as simple reduced forms, after correcting for degrees of freedom by the BIC criterion, if long lags are included in the hybrid model; this begs the question, considered in section 5, of what behavioral assumptions could justify such long lags. Second, it is clear that the hybrid stickyprice model with a oneperiod lag (N equal to 1)  the standard specification of Christiano et al (2005) and similar to the specification of Gali and Gertler (1999)  performs poorly relative to a reducedform regression over the last four decades. Finally, this model  and the baseline model  performs relatively better over the last two decades  a period of greater stability in monetary policy behavior.
Specifications
Stickyinformation models assume that prices are reset optimally every period, but that the information gathering activities of firms  on which prices are based  are costly and occur only infrequently. Sims (2003) discusses an optimization problem confronting a firm that faces such costs. Our analysis follows a more ad hoc, but empirically tractable, direction. In the baseline stickyinformation model, firms reset price in every period, subject to their constrained information set. In particular, some firms have access only to stale, i.e., lagged, information. An informationsetupdating rule analogous to the Calvo pricesetting rule is employed, as in Mankiw and Reis (2002). Each firm faces a probability 1 in period t that it will update its information set through the previous period, t1; with probability , the firm does not update its information set. This process repeats itself for each period t, implying a geometric distribution of periods since last updating of the information set across the population of firms. The implied average information lag equals 1/(1.
As shown in Mankiw and Reis (2002), inflation is governed by the following equation (ignoring constants)
Baseline SI .^{8}
where the parameter D measures the sensitivity of the desired relative price absent information imperfections to real marginal cost..
In the hybrid stickyinformation model, there exist two types of firms. The first set, a fraction 1 of firms, choose the optimal price given their information set (which is determined in the same manner as that in the baseline model). A second set follow a "ruleofthumb", setting their periodt price equal to the aggregate price level last period, plus the rate of inflation over the previous N quarters (expressed at a quarterly rate). Allowance for such indexation may appear odd, in that it assumes that the "ruleofthumb" firms have information on recent inflation and the aggregate price level when some of the optimizing firms do not. However, the information problem associated with estimating currentperiod marginal cost, which depends on productivity, input costs, and potentially other factors, may be more severe than that of finding recent inflation data. More importantly, the pricing scheme provides a simple hybrid extension of the stickyinformation model that can be taken to the data; if such an amendment to the baseline stickyinformation model finds empirical support, further work could focus more finely on the consistency of the specification with plausible assumptions regarding information acquisition and firm behavior.
Given these assumptions, inflation is governed by the following equation
Hybrid SI
Results
Table 3 presents estimates of the stickyinformation models for the full sample, 1965Q1 to 2002Q4. As for the stickyprice models, I present estimates of the parameters of the model and their standard errors; the Qstatistic that examines the degree of serial correlation over 12 lags in the onestepahead forecast errors (and its pvalue); the loglikelihood and BIC for the model; and the R^{2} for the model, computed as one minus the sum of squared forecast errors divided by the sum of squared deviations of inflation from its sample mean.^{9}
The baseline stickyinformation model (line 1) fits much better than the baseline stickyprice model reported in table 1 and matches the fit of a onelag reducedform for inflation (line 4). The residuals are only moderately seriallycorrelated (see the Qstatistic). And the estimated information lag appears reasonable at between one and two quarters (column 3), a bit below the four quarters assumed by Mankiw and Reis (2002) and the estimates presented by Carroll (2003) and Khan and Zhu (2004), all obtained via quite different methods. However, the baseline stickyinformation model provides a poor fit to the data compared to wellfitting reducedform regressions with two or more lags, as can be seen by comparing the likelihood, BIC, and R^{2} for the reducedform regressions reported in lines 5 and 6 of table 3.
The hybrid models, lines 2 and 3, fit only modestly better than the baseline model (as indicated by the likelihood and BIC values reported in the righthand columns). The bestfitting stickyinformation model uses a four lags of inflation for ruleofthumb price setters (N equal to four). In this case, the estimated average information lag is about 11/2 quarters. The estimated parameters suggest that the share of ruleofthumb price setters is high at near . This result illustrates that the stickyinformation model has the same problem as the stickyprice model in generating the importance of lagged inflation for forecasting inflation in the absence of some ad hoc adjustment to the model. The importance of hybrid dynamics for both the stickyprice and stickyinformation models highlights the need for sources of richer dynamics, a point pursued in section 5 below.
The Bayesian information criterion indicates that the hybrid stickyinformation models (with N equal to one) fit the data worse than the reducedform regression in table 1, although the degreesoffreedom adjusted fits are pretty close to each other. However, comparison to table 1 indicates that the hybrid stickyprice model appears to fit better than the hybrid stickyinformation model. This highlights how the typical empirical strategy  look at one model and if it performs acceptably along some metric, stop  will not help discriminate across different model specifications that both perform relatively well.
Table 4 presents estimation results for the stickyinformation models over the 1983Q1 to 2002Q4 period. The stickyinformation models fit much better (relative to reduced forms) in this sample. The bestfitting model is again the fourlag hybrid. Interestingly, in this sample the baseline stickyinformation model fits slightly worse than the baseline stickyprice model from table 2.
Combining the results from the stickyprice and stickyinformation models, I conclude that the stickyprice hybrid model with a longindexation lag matches the data on inflation better than the stickyinformation alternatives. The next section considers behavioral assumptions that may help us understand the role of sticky prices and sticky information in the hybrid stickyprice model.
The empirical results show a preference for hybrid stickyprice models both over the full sample and in the more recent period. One interpretation of this finding is that the stickyprice model fails to account for the behavior of inflation, as ad hoc amendments to the baseline stickyprice model are needed to match the data. Moreover, the findings in favor of stickyprice models seem to contradict the need for "sticky information" emphasized in analyses of the costs of disinflations or the slow, humpshaped response of inflation to certain economic shocks.
Our interpretation differs from this pessimistic assessment: in our view, the results support the importance of both stickyinformation and stickyprices. As discussed in the appendix, the stickyprice hybrid with four lags can be rationalized by different behavioral assumptions. One plausible set of assumptions leading to this model is the following modification of the "ruleofthumb" story from Gali and Gertler (1999):
Of course, the specific manner in which "sticky information" is incorporated into the stickyprice model in our Nlag hybrid remains ad hoc. Future research integrating the stickyprice and stickyinformation models should focus on plausible behavioral stories motivated from first principles. At least two approaches appear promising. Kumhof and Laxton (2005) integrate information and priceadjustment costs in an optimizing model to explain inflation inertia; while their implementation does not have infrequent nominal price changes as in the stickyprice literature, the basic thrust of their research appears promising.
Finally, a comparison of the post1982 results with the full sample results may provide a clue as to an important source of imperfect information over the last forty years. The results for the most recent sample period for a baseline stickyprice model were more favorable  although the hybrid models were still preferred. This suggests that imperfect information  a source of hybridtype behavior  may have been less important over the past two decades. One potential explanation is that monetary policy has been more stable  especially with respect to its inflation goal  over the more recent period and that this has lowered the importance for inflation dynamics of learning about the inflation goal. Erceg and Levin (2003) formalize this intuition.
The set of results from estimates of stickyprice models and, to a lesser extent, stickyinformation models, has exploded in recent years. Nonetheless, there seems to have been only modest convergence of opinion, and a researcher lacking direct contact with the data and range of techniques applied by different authors could find summarizing the results of this research difficult. The remainder of our discussion relates the findings herein to those of previous authors.
Gali, Gertler, and LopezSalido (2005) and Sbordone (2005) (building on Gali and Gertler (1999) and Sbordone (2002)) both suggest that a hybrid stickyprice model adequately describes inflation dynamics in the United States. While these authors use different estimation techniques, their conclusions are partially driven by an informal consideration of the tracking performance of their models. In particular, Gali and Gertler (1999) and subsequent researchers have presented a graph like figure 2, which plots (demeaned) inflation and the onestepahead forecast of inflation from the estimated hybrid (N equal to one) model over 1965Q1 to 2002Q4. As is clear, the series move together, especially before 1998  as should have been apparent from the R^{2} statistic in table 1. Nonetheless, this model was soundly rejected by the BIC criterion. As shown in figure 3, the graph of actual and predicted inflation for the hybrid model with N equal to four is hard to compare visually to figure 2; at least to this author's eye, the figures look pretty similar. One lesson from this exercise is that "ocular" regressions of the form emphasized by previous research provide little information.
Rudd and Whelan (2003, 2005b, 2005c) also examine the hybrid stickyprice model; in contrast to the previous authors, they conclude that it provides a poor approximation to inflation dynamics. This different conclusion does not arise because Rudd and Whelan estimate parameters of the hybrid stickyprice model that are very different from those presented by supporters of such models. Rather, they also emphasize informal measures of fit like figure 2, and note that the marginal improvement in fit from real marginal cost (the labor share) in such equations is small. This interpretation is completely consistent with the results reported by supporters of the stickyprice model; it is also consistent with our previous conclusion that such informal metrics provide little information.
The different conclusions of these supporters and detractors of the stickyprice model are partly related to three other considerations. First, there has been some debate over whether the coefficient on real marginal cost ( in the hybrid stickyprice model) is different from zero in the statistical sense. The estimated value of this parameter is (largely) consistent across studies, but standard errors differ. Our estimates are statistically different from zero, using maximum likelihood; estimates via GMM using the same data tend to have larger standard errors and to not differ from zero in a statistical sense. We performed a Monte Carlo experiment in which our preferred stickyprice model, the fourlag hybrid, was simulated to create artificial samples of 152 periods (with 500 replications), drawing from the shocks estimated for the model over 19652002. The hybrid model was then estimated via GMM, with statistics from the simulations reported in table 5.^{10} As shown in the upper rows, the simulations tend to estimate fairly accurately the coefficient on lags of inflation (. While the coefficient on real marginal cost ( lies quite close to its true value on average, its standard error is sizable. This suggests that a focus on the standard error of this parameter is not a useful way to assess this model.
There has also been debate about whether inflation Grangercauses real marginal cost (the labor share), with Rudd and Whelan (2005a, 2005c) differing from, for example, Sbordone (2005). Results from our Monte Carlos exercises suggest this is not a useful test; as reported in the middle panel of table 5, the simulations failed to find Grangercausality at the 10percent significance level in 41.4 percent of the simulations. This occurs largely because the costpush shocks ( from the equations in section 2) cause inflation to move in ways independent of the labor share and lower the ability of the regressions to find Granger causality from inflation to the labor share. Alternative Monte Carlo exercises (not reported) in which the costpush shocks are made much smaller almost always detected Granger Causality of the type emphasized in previous research.
Finally, Rudd and Whelan (2005c) emphasize that detrended output enters reducedform regressions for inflation with a highlysignificant coefficient. This is completely consistent with our stickyprice hybrid model. As reported in the lower panel of table 5, estimation of an inflation regression including four lags of the dependent variable and lagged detrended output on the simulated data yielded a coefficient significantly greater than zero (at the 10percent level) in 99.8 percent of the simulations. This occurs because detrended output is an important factor in the reducedform equations for marginal cost (the labor share) and the federal funds rate.
To summarize, most of the reported results in Gali, Gertler, and LopezSalido (2005), Sbordone (2005), and Rudd and Whelan (2005c) echo those reported herein. Estimated parameters for the onelag hybrid are in the same neighborhood, and tracking exercises like figure 2 are very similar in each study. However, none of these earlier studies focus on formal metrics of fit; rather, they focus on ancillary issues like standard errors of parameters, Grangercausality of the labor share by inflation, and the partial correlation of inflation and lagged detrended output. None of these ancillary issues is likely to provide much information about the ability of a hybrid stickyprice model to track inflation according to our Monte Carlo simulation. And perhaps more importantly, only Gali and Gertler (1999) emphasize the economic story behind a hybrid model  an emphasis of our section 5  and none of these earlier investigations formally considers a more complex, but betterfitting, hybrid with four inflation lags.
Our focus on the fit of alternative models is most closely related to a long series of papers by Jeff Fuhrer. However, his evaluations do not consider relative fit as measured by BIC. A typical strategy, pursued, for example, in Fuhrer (1997), is to consider whether the likelihoodratio statistic rejects the null hypothesis that the restrictions imposed by the model under consideration are satisfied by a baseline reduced form. Table 6 considers tests of this type: it reports the likelihoods for the baseline and hybrid stickyprice models for each sample period as well as that of a reduced form with four lags of inflation, detrended output, the nominal federal funds rate, and real marginal cost^{11}; the likelihood ratio statistic; and the pvalue of the null hypothesis that the restrictions of the structural model considered are satisfied. Turning first to the set of results for the 1965Q1 to 2002Q4 period, the likelihood ratio statistics clearly reject the null hypothesis that the restrictions of the stickyprice models are satisfied for the baseline and oneperiod hybrid models, with pvalues essentially equal to zero. In the fourlag case, the null is not rejected at the onepercent level and barely rejected at the fivepercent level. On balance, these results echo those in Fuhrer, who argues that classical tests of this type suggest failures of stickyprice models.
Our interpretation is quite different for two reasons. First, even the baseline model's restrictions are not rejected at the fivepercent level for the 1983Q1 to 2002Q4 period. More importantly, I have been assessing the relative performance by the BIC criterion, not likelihoodratio statistics. To see the different implications of these approaches, consider first the likelihoodratio statistic; with a critical value of fivepercent, this criterion chooses the reducedform model over the stickyprice model  i.e., rejects the restrictions of the stickyprice model  when the likelihood ratio statistic exceeds 23.7. By contrast, the BIC criterion prefers the reducedform model when the LR statistic exceeds 67.6.^{12} Obviously, the BIC criterion requires much larger "violations" of the stickyprice model's restrictions in order to prefer the reducedform model. This occurs for two reasons: first, the BIC criterion is derived as an asymptotic approximation to a Bayesian analysis that prefers parsimony; and second, this approximation requires stronger evidence from the data as the sample size increases  as more data should lead to stronger rejections of false hypotheses. The first reason seems a desirable feature. The second, as emphasized by Leamer (1983) and Sims (2002), corresponds to the notion that the significance level of a classical test should decrease with the sample size because, for any prior views regarding null hypotheses, the data should speak more loudly with a larger sample.
Our discussion of previous work has focused largely on stickyprice models, as stickyinformation models have received far less attention.^{13} Consistent with the research that has motivated the stickyinformation model, we find that the baseline version of this model can generate intrinsically persistent inflation and hence fits better than the pure stickyprice model over the 19652002 period, but this advantage disappears in the recent period. With regard to empirical work, Khan and Zhu (2004) estimate a stickyinformation model via limitedinformation methods and find plausible parameters; however, they do not assess fit relative to other models of inflation dynamics. Korenok (2004) compares baseline stickyprice and stickyinformation models using Bayesian methods, and finds that the baseline stickyprice model fits better than the baseline stickyinformation model. Laforte (2006) compares hybrid stickyprice models to a stickyinformation model in a setup that assumes that the remainder of the system is given by a simple dynamic general equilibrium model. His estimation strategy uses Bayesian methods; as in this research, the stickyprice models he considers dominate the stickyinformation models.
I have estimated stickyprice and stickyinformation models for the United States via maximumlikelihood techniques. While the baseline stickyinformation model generates greater serial correlation in inflation and hence dominates the baseline stickyprice model over the 19652002 sample, this advantage disappears in the recent period. In addition, hybrid stickyprice models empirically dominate stickyinformation alternatives  particularly when hybrid behavior is tied to lags up to one year.
The finding that hybrid stickyprice models provide a better summary of inflation dynamics than a stickyinformation model does not imply that stickyinformation stories, or more generally information imperfections, are unimportant. Rather, it confirms the importance for inflation dynamics of factors in addition to nominal price rigidities, perhaps including sticky information. For example, the hybrid stickyprice model, under some assumptions, relies on an imperfect information element to generate a form of backwardlooking behavior. But clearly stronger microfoundations and a more careful understanding of any changes in the importance of price rigidities or information imperfections across time periods are needed in future research.
Finally, our discussion has attempted to show the links between the results reported herein and those in the very large related literature. While it is impossible to summarize all previous work, it is noteworthy that our findings for stickyprice models are very similar to those in earlier work that has reached a range of different conclusions. The range of conclusions in other work stems to an important extent from a focus in such research on issues unrelated to model fit. Our comparison of stickyprice models to stickyinformation and reducedform models of inflation suggest that hybrid stickyprice models can match inflation dynamics in the United States over the past forty years.
There are a large number of symmetric, monopolisticallycompetitive intermediategoods firms that set nominal prices; preferences over varieties take the DixitStiglitz form. Firm j's marginal cost in period t is denoted by MC, its nominal price by , its demand by , and the economywide average price index by . The firm's profit in period t is then
Ruleofthumb firms are themselves split into N groups. Members of the kth group set their nominal price equal to the economywide average price reset in period tk ( plus the change in the aggregate price level between period tk1 and period t1. In this sense, the firms have different amounts of "stickyinformation". Gali and Gertler (1999) assume N equals one.
Following the same steps as in Gali and Gertler (1999), the loglinearized equations for the aggregate price level, the economywide average reset price, the reset price of forwardlooking firms, and the average reset price of ruleofthumb firms are given by
Price index p (A.1)
Index of newly reset prices p* (A.2)
Forwardlooking reset price pf (A.3)
Average of ruleofthumb reset prices: (A.4)
where is the fraction of ruleofthumb price setters and a is the fraction of ruleofthumb price setters in the kth group.
Combining (A.1) through (A.4), and some algebra, yields
(A.5)
This equation is of the basic form of the N lag hybrid stickyprice model; if the fraction of firms in the kth group of ruleofthumb firms declines by the appropriate amount between k and k+1, the hybrid presented in the main text arises when the discount factor equals one.
Anderson, Gary (2000) The AndersonMoore Algorithm. Federal Reserve Board Occasional Staff Study 4, http://www.federalreserve.gov/pubs/oss/oss4/aimindex.html.
Brock, William, Steven Durlauf and Kenneth West (2003) Policy Evaluation in Uncertain Economic Environments. Brookings Papers on Economic Activity.
Calvo, G.A. (1983) Staggered Prices in a Utility Maximizing Framework. Journal of Monetary Economics 12:383398.
Carroll, Christopher (2003) The Epidemiology of Macroeconomic Expectations. In Larry Blume and Steven Durlauf, eds., The Economy as an Evolving Complex System, III, Oxford University Press, forthcoming.
Christiano, Lawrence J., Martin Eichenbaum and Charles Evans (2005) Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy. Journal of Political Economy.
Corrado, Carol and Laurence Slifman, "Decomposition of Productivity and Costs" American Economic Review, 89 (1999), pp. 32832.
Dopplehoffer, Gernot, Ronald I. Miller and Xavier SalaiMartin (2000) Determinants of LongRun Growth: A Bayesian Averaging of Classical Estimates (BACE) Approach. NBER Working Paper No. 7750, June.
Eichenbaum, Martin and Jonas D.M. Fisher (2003) Time Series Implications of the Calvo Model of Sticky Prices. Federal Reserve Bank of Chicago Working Paper 200323.
Erceg, Christopher J. and Andrew T. Levin (2003) Imperfect Credibility and Inflation Persistence. Journal of Monetary Economics, v. 50, iss. 4, pp. 91544
Fuhrer, Jeffrey (1997) "The (Un)Importance of ForwardLooking Behavior in Price Specifications," Journal of Money, Credit, and Banking 29, 33850.
Fuhrer, Jeffrey (2002) Habit Formation in Consumption and Its Implications for Monetary Policy Models. American Economic Review, 90, 367390.
Fuhrer, Jeffrey (2005) Intrinsic and Inherited Inflation Persistence. Federal Reserve Bank of Boston Working Paper 0508.
Fuhrer, Jeffrey and Geoffrey Moore (1995) Inflation Persistence. Quarterly Journal of Economics.
Fuhrer, Jeffrey and Giovanni P. Olivei (2004) Estimating ForwardLooking Euler Equations with GMM and Maximum Likelihood Estimators: An Optimal Instruments Approach. Mimeo, March 2004.
Galí, Jordi and Mark Gertler (1999) Inflation Dynamics: A Structural Econometric Approach. Journal of Monetary Economics, 44(2), October, 195222.
Gali, J., Gertler, M. and J.D. LopezSalido (2005) Robustness of the Estimates of the Hybrid New Keynesian Phillips Curve, Journal of Monetary Economics.
Gordon, Robert J. (1998) Foundations of the Goldilocks Economy: Supply Shocks and the TimeVarying NAIRU. Brookings Papers on Economic Activity, 1998 Issue 2, p297333
Jondeau, Eric and Hervé Le Bihan (2003) ML VS GMM Estimates of Hybrid Macroeconomic Models (with an Application to the New Phillips Curve). Banque de France Working Paper NER#103, October 2003.
Khan, Hashmat and Zhenhua Zhu (2004) Estimates of the StickyInformation Phillips Curve for the United States, Canada, and the United Kingdom. Revised Version of Bank of Canada Working paper 200219, forthcoming Journal of Money, Credit and Banking.
Kiley, Michael T. (2002) Partial Adjustment and Staggered Price Setting. Journal of Money, Credit, and Banking 34(2), May, pages 283298
Kiley, Michael T. (2004) Is ModeratetoHigh Inflation Inherently Unstable? Federal Reserve Board FEDS Working Paper 200443
Korenok, Oleg (2004) Empirical Comparison of Sticky Price and Sticky Information Models. Department of Economics, Rutgers University. Mimeo.
Kumhof, Michael and Douglas Laxton (2005) A Rational Expectations Model of Optimal Inflation Inertia. Mimeo, International Monetary Fund.
Laforte, JeanPhilippe (2006) Pricing Models: A Bayesian DSGE Approach to the U.S. Economy. Federal Reserve Board.
Leamer, Edward (1983) Model Choice and Specification Analysis. In Griliches, Zvi and Michael D. Intriligator, eds., Handbook of Econometrics, Volume 1. Elsevier.
Ma, Adrian (2002) GMM Estimation of the New Phillips Curve. Economics Letters, 76, 411417.
Mankiw, N. Gregory and Ricardo Reis (2002) Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve. Quarterly Journal of Economics, vol. 117 (4), pp. 12951328.
Mankiw, N. Gregory, Ricardo Reis and Justin Wolfers (2003) Disagreement About Inflation Expectations. National Bureau of Economic Research Macroeconomics Annual.
Roberts, John (1997) Is Inflation Sticky? Journal of Monetary Economics, vol. 39 (July 1997), pp. 17396.
Roberts, John (2005) How Well Does the New Keynesian Phillips Curve Fit the Data? Mimeo, Federal Reserve Board.
Rudd, Jeremy and Karl Whelan (2003) "Can Rational Expectations StickyPrice Models Explain Inflation Dynamics?" Forthcoming in the American Economic Review.
Rudd, Jeremy and Karl Whelan (2005a) "Does Labor's Share Drive Inflation?" Journal of Money, Credit, and Banking, 37, 297312.
Rudd, Jeremy and Karl Whelan (2005b) "Modelling Inflation Dynamics: A Critical Review of Recent Research." Federal Reserve Board Finance and Economics Discussion Papers 200566.
Rudd, Jeremy and Karl Whelan (2005c) "New Tests of the NewKeynesian Phillips Curve," Journal of Monetary Economics, 52, 11671181.
SbordoneArgia M. (2002) Prices and Unit Labor Costs: A New Test of Price Stickiness. Journal of Monetary Economics, March 2002, v. 49, iss. 2, pp. 26592
SbordoneArgia M. (2005) Do Expected Future Marginal Costs Drive Inflation Dynamics? Journal of Monetary Economics.
Sims, Christopher A. (2002) Testing Restrictions and Comparing Models. Mimeo, Princeton University.
Sims, Christopher A. (2003) Implications of Rational Inattention. Journal of Monetary Economics, 50, 665690.
White, Halbert (1994) Estimation, inference, and specification analysis. Cambridge; New York: Cambridge University Press.


Q Statistic
(pvalue) 
Loglikelihood  BIC  R^{2}  

1. Baseline (N=0)  0.0477 (0.006)  0.00  371.62
(0.000) 
409.94  404.92  0.41 
2. Hybrid with N=1  0.0112 (0.003)  0.42 (0.019)  39.02 (0.000)  468.52  460.99  0.73 
3. Hybrid with N=4  0.0168 (0.003)  0.26 (0.016)  23.75 (0.022)  483.66  476.13  0.78 
4. Reducedform model, N equals 1 
464.19  451.63  0.71  
5. Reducedform model, N equals 2 
490.26  467.66  0.80  
6. Reducedform model, N equals 3 
492.87  460.22  0.80 
7. Baseline StickyPrice model  0.00 
8. Hybrid StickyPrice model with N = 1  0.00 
9. Hybrid StickyPrice model with N = 4  1.00 
10. Reducedform model (N equals 2)  0.00 


Q Statistic
(pvalue) 
Loglikelihood  BIC  R^{2}  

1. Baseline (N=0)  0.0063 (0.0016)  0.00  66.39 (0.000)  309.61  305.23  0.26 
2. Hybrid with N=1  0.0036 (0.0014)  0.24
(0.059) 
24.27 (0.019)  314.25  307.68  0.34 
3. Hybrid with N=4  0.0036 (0.0017)  0.17
(0.039) 
10.83 (0.543)  316.49  309.91  0.38 
4. Reducedform model, N equals 1 
313.58  302.63  0.33  
5. Reducedform model, N equals 2 
317.06  297.34  0.38  
6. Reducedform model, N equals 3 
319.67  291.18  0.42 
7. Baseline StickyPrice model  0.01 
8. Hybrid StickyPrice model with N = 1  0.10 
9. Hybrid StickyPrice model with N = 4  0.90 
10. Reducedform model (N equals 1)  0.00 



Q Statistic
(pvalue) 
Loglikelihood  BIC  R^{2}  

1. Baseline (N=0)  0.408
(0.063) 
0  0.050
(0.001) 
25.50
(0.013) 
462.26  457.24  0.70 
2. Hybrid with N=1  0.577
(0.105) 
0.344
(0.128) 
0.235
(0.119) 
27.04
(0.008) 
463.37  455.83  0.71 
3. Hybrid with N=4  0.319
(0.128) 
0.456
(0.128) 
0.090
(0.058) 
18.62
(0.098) 
468.00  460.46  0.73 
4. Reducedform model, N equals 1 
464.19  451.63  0.71  
5. Reducedform model, N equals 2 
490.26  467.66  0.80  
6. Reducedform model, N equals 3 
492.87  460.22  0.80 
7. Baseline StickyInformation model  0.00 
8. Hybrid StickyInformation model with N = 1  0.00 
9. Hybrid StickyInformation model with N = 4  0.00 
10. Reducedform model (N equals 2)  1.00 


D

Q Statistic
(pvalue) 
Loglikelihood  BIC  R^{2}  

1. Baseline (N=0)  0.610
(0.086) 
0  0.199
(0.133) 
12.84 (0.381)  308.70  304.32  0.25 
2. Hybrid with N=1  0.694
(0.073) 
0.280
(0.112) 
0.376
(0.122) 
20.83 (0.053)  311.24  304.67  0.29 
3. Hybrid with N=4  0.647
(0.088) 
0.337
(0.118) 
0.355
(0.128) 
12.86
(0.379) 
312.29  305.72  0.34 
4. Reducedform model, N equals 1 
313.58  302.63  0.33  
5. Reducedform model, N equals 2 
317.06  297.34  0.38  
6. Reducedform model, N equals 3 
319.67  291.18  0.42 
7. Baseline StickyInformation model  0.15 
8. Hybrid StickyInformation model with N = 1  0.21 
9. Hybrid StickyInformation model with N = 4  0.61 
10. Reducedform model (N equals 2)  0.03 
w  

Mean  0.28  0.020 
Standard Error  0.07  0.014 
Percent of simulations in which inflation does not Grangercause real marginal cost at the 10percent significance level or better  41.4% 
Percent of simulations in which detrended output is significant at the 10percent level in the reducedform inflation equation  99.8% 
Loglikelihood  LR statistic  Pvalue  

Baseline StickyPrice Model  409.94  171.18  0.000 
Hybrid StickyPrice Model with N = 1  468.52  54.02  0.000 
Hybrid StickyPrice Model with N = 4  483.66  23.74  0.049 
ReducedForm with four lags  495.53     
Loglikelihood  LR statistic  Pvalue  

Baseline StickyPrice Model  309.61  24.36  0.059 
Hybrid StickyPrice Model with N = 1  314.25  15.08  0.373 
Hybrid StickyPrice Model with N = 4  316.49  10.61  0.716 
ReducedForm with four lags  321.79     
Notes: LR statistic equals the difference between the likelihood of the reduced form and the structural model multiplied by 2. The Pvalue is the probability that a random variable with the appropriate degrees of freedom is greater than or equal to the LR statistic. The reduced form for inflation contains four lags of inflation, output, the nominal federal funds rate, and real marginal cost.