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Information Rigidity and the Expectations Formation Process

Author(s):
Olivier Coibion, and Yuriy Gorodnichenko
Published Date:
December 2012
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I. Introduction

Expectations matter. How much to consume or save, what price to set, and whether to hire or fire workers are just some of the fundamental decisions underlying macroeconomic dynamics that hinge upon agents’ expectations of the future. Yet how those expectations are formed, and how best to model this process, remains an open question. From the simple automatons of adaptive expectations to the all-knowing agents of modern full-information rational expectations models, macroeconomists have considered a wide variety of frameworks to model the expectations formation process, yielding radically different results for macroeconomic dynamics and policy implications. Recent work on rational expectations models with information frictions such as Mankiw and Reis (2002), Woodford (2001), and Sims (2003) has emphasized how information rigidities can account for otherwise puzzling empirical findings but these same frictions can also lead to policy prescriptions that differ from those under models with full information.2 Despite a growing body of work studying the implications of possible departures from full-information rational expectations, the empirical evidence against this assumption underlying most modern macroeconomic models has been limited. In particular, while statistical evidence against the null is commonly uncovered, the economic significance of these rejections remains unclear.

Building from the predictions of rational expectations models with information rigidities, we propose a novel approach to test the null of full-information rational expectations in a way that sheds new light on possible departures from the null. Our baseline specification relates ex-post mean forecast errors to the ex-ante revisions in the average forecast across agents and possesses multiple advantages over traditional tests of full-information rational expectations (FIRE).3 First, we rely on the predictions of theoretical models of information rigidities to guide our choice of the relevant regressors. Second, models of information rigidities make specific predictions about the sign of the coefficient on forecast revisions, so that our specification provides guidance not only about the null of FIRE but also about alternative models. As a result, our framework can help determine whether rejections of the null should be interpreted as rejecting either the rationality of expectations or the full-information assumption. Third, we show that the coefficient on forecast revisions maps one-to-one into the underlying degree of information rigidity and therefore our approach can provide a metric by which to assess the economic significance of departures from the null of FIRE.

Two theoretical rational expectations models of information frictions motivate our empirical specification. In the sticky-information model of Mankiw and Reis (2002), agents update their information sets infrequently as a result of fixed costs to the acquisition of information. The degree of information rigidity in this model is then the probability of not acquiring new information each period. The second class of models we consider consists of noisy-information models such as Woodford (2001), Sims (2003), and Mackowiak and Wiederholt (2009). Here, agents continuously update their information sets but, because they can never fully observe the true state, they form and update beliefs about the underlying fundamentals via a signal extraction problem. Forecasts are a weighted average of agents’ prior beliefs and the new information received, where the weight on prior beliefs can be interpreted as the degree of information rigidity. Strikingly, both models predict the same relationship between the average ex-post forecast errors across agents and the average ex-ante forecast revision such that the coefficient on forecast revisions depends only on the degree of information rigidity in each model. This predictability of the average forecast error across agents from forecast revisions is an emergent property in both models, i.e. a property which arises only from the aggregation process and not at the individual level.

The resulting empirical specification can be applied to study information rigidities for a variety of economic agents such as consumers, firms, and financial market participants for whom forecast data are available. As a first step, we focus on inflation forecasts from the U.S. Survey of Professional Forecasters (SPF) for two reasons. First, inflation forecasts have received the most attention in the literature so that these results are more readily comparable to previous work. Second, because professional forecasters are some of the most informed economic agents, they can provide a conservative benchmark for assessing potential deviations from full-information rational expectations. From 1969-2010, we can strongly reject the null of FIRE and find that the estimated coefficient on forecast revisions is positive, consistent with the prediction of rational expectations models incorporating information rigidities. Additional coefficient restrictions implied by these models cannot be rejected and past information incorporated in other economic variables loses much of its predictive power for ex-post mean forecast errors once we control for the forecast revision. This indicates that rejections of the null are unlikely to be driven by departures from rationality (such as adaptive expectations) and instead reflect deviations from the assumption of full-information. Furthermore, the implied degree of information rigidity is high: in the context of sticky-information models, it implies an average duration of six to seven months between information updates, while in noisy-information models it implies that new information receives less than half of the weight that it would under full-information relative to prior beliefs.

In addition, we document that qualitatively similar results obtain for different kinds of economic agents, such as academics, commercial banks, and non-financial businesses, as well as for consumers and financial market-based inflation expectations. This implies that information rigidities are present not just amongst professional forecasters but also for firms and consumers. Given that the estimated degree of information rigidity in inflation forecasts is relatively high across different types of economic agents, this suggests that information rigidities are likely to play a pervasive role in macroeconomic dynamics. The prevalence of information rigidities across agents also suggests that the estimated levels of information rigidities are unlikely to be driven by either strategic behavior on the part of professional forecasters or reputational considerations. As a result, our empirical estimates provide a new set of stylized facts which can be used for the calibration of models with information rigidities.

To further verify that our results are indeed driven by information rigidities, we derive testable predictions from a number of competing hypotheses which could potentially account for the predictability of forecast errors. For example, if forecasters are heterogeneous in the degree of loss-aversion with respect to their forecast errors, then predictability in forecast errors can arise even in the absence of information rigidities as in Capistran and Timmermann (2009). However, we show that such a model would imply a negative correlation—rather than positive as observed in the data—between ex-post forecast errors and ex-ante forecast revisions. We similarly derive testable predictions from models in which agents place a different weight on new information, as they would if they held different beliefs about underlying parameter values, or hold heterogeneous views about long-run means of macroeconomic variables. In each case, we find that these models yield counterfactual predictions about the predictability of forecast errors.

Using professional forecasts for a number of additional macroeconomic variables, both in the U.S. and across eleven additional countries, we provide further evidence of pervasive information rigidity. First, pooled estimates across macroeconomic variables confirm the finding of predictability of forecast errors coming from ex-ante forecast revisions with the signs predicted by models of information rigidities. Across datasets, we also find robust evidence that the degree of information rigidity varies systematically across macroeconomic variables and that this cross-sectional variation is consistent with the predicted determinants of noisy-information models: the persistence of a variable and measures of the signal-noise ratio can account for about 20–30 percent of the variation in the estimated degree of information rigidity across countries and macroeconomic variables in the Consensus Economics dataset. Since the canonical sticky-information model assumes a common rate of information updating across variables, these results suggest that subsequent work with the sticky-information model should explore how such heterogeneity can arise in the context of infrequent information updating.

Because our empirical specification allows us to recover estimates of the underlying degree of information rigidity, we can also characterize whether the degree of information rigidity varies in response to economic conditions, as the incentives for agents to collect and process additional information change. For example, macroeconomic volatility declined significantly after the early to mid-1980s during the Great Moderation. According to models with information rigidities, such a decline in volatility should result in a higher degree of inattention. We study the low-frequency time variation in the estimated degree of information rigidity among U.S. professional forecasters and find evidence that accords remarkably well with this intuition: the degree of information rigidity fell consistently throughout the 1970s and early 1980s when macroeconomic volatility was high, reaching a minimum around 1983–84. Since then, the degree of information rigidity has been consistently rising as macroeconomic volatility has been subdued. We also document higher frequency endogenous variation in information rigidities. For example, the degree of information rigidity declines significantly during U.S. recessions, which points to state-dependence in the expectations formation process. Additional evidence of state-dependence comes from the response of economic forecasts to the 9/11 attacks: the large forecast revisions in the immediate aftermath of this highly visible shock are well-characterized as full-information rational expectations updates, unlike the results under normal conditions. Hence, agents appear to adjust the resources devoted to the collection and processing of information in response to economic conditions, consistent with state-dependence in the information updating process as in Gorodnichenko (2008).

This paper is closely related to recent empirical work trying to ascertain the nature of the expectations formation process. For example, Mankiw, Reis and Wolfers (2004) assess whether a sticky-information model can replicate some stylized facts about the predictability of forecast errors by professional forecasters while Andolfatto, Hendry and Moran (2007) consider whether noisy information with respect to the inflation target of the central bank can account for observed deviations from FIRE. Khan and Zhu (2006), Kiley (2007), and Coibion (2010) assess the validity of sticky information using estimates of its predicted Phillips curve. One advantage of our approach is that we can directly recover an estimate of the degree of information rigidity without having to make auxiliary assumptions about the model, such as the nature of price-setting decisions. Sarte (2010) also uses surveys to quantify sticky information in firm forecasts, but our approach allows us to assess both sticky-information and noisy-information models. Coibion and Gorodnichenko (2012) study the evidence for sticky-information and noisy-information models but do so by estimating the response of forecast errors and disagreement to structural shocks whereas our approach does not require the identification of any shock. In the same spirit, Andrade and LeBihan (2010) provide evidence for both sticky and noisy information in the European Survey of Professional forecasters, Branch (2007) compares the fit of sticky-information and model-switching characterizations of the expectations formation process while Carroll (2003) tests an epidemiological model of expectations in which information diffuses over time from professional forecasters to consumers. However, these papers focus almost exclusively on inflationary expectations whereas we utilize forecasts for a wide variety of macroeconomic variables as well as cross-country data, allowing us to more fully characterize the nature and quantitative importance of information rigidities faced by economic agents. Finally, our paper is closely related to the long literature on the rationality of both individual and consensus forecasts. In this context, our results using models of information rigidities provide a new rationalization for the otherwise puzzlingly weak evidence against rational expectations found at the individual level relative that observed in average forecasts (Pesaran and Weale 2006). If agents form their expectations rationally subject to information frictions, predictability in forecast errors will follow from the aggregation of forecasts across agents, even if no such predictability exists at the individual level.

The paper is structured as follows. Section II presents the predicted relationship between ex-post mean forecast errors and ex-ante mean forecast revisions in sticky-information and noisy-information models, baseline results from professionals’ forecasts of inflation, as well as tests of competing explanations. Section III expands the set of forecasts used to different kinds of agents, macroeconomic variables and countries. Section IV presents evidence on the extent to which the degree of information rigidity varies in response to low-frequency, business cycle, and highfrequency changes in macroeconomic conditions. Section V concludes.

II. Forecast Errors, Forecast Revisions and Information Rigidities

In this section, we present two models of information rigidities and derive their respective predictions for the relationship between ex-post mean forecast errors and ex-ante mean forecast revisions. We document evidence consistent with these predictions using U.S. inflation forecasts of professional forecasters and argue that alternative explanations are unlikely to account for these findings.

A. Sticky-Information Model

Mankiw and Reis (2002) propose a model of inattentive agents who update their information sets each period with probability (1 − λ) but acquire no new information with probability λ, so that λ can be interpreted as the degree of information rigidity and 1/(1 − λ) is the average duration between information updates. When agents update their information sets, they acquire full-information and have rational expectations. Reis (2006) shows how this time-dependent updating of information sets obtains when firms face a fixed cost to updating their information. The average time t forecast across agents (Ft) of a variable x at time t + h is a weighted average of current and past full-information rational expectations forecasts (Et–j) of the variable being forecasted such that

The average forecast at time t − 1 can similarly be written as

which implies that the current average forecast is just a weighted average of the previous period’s average forecast and the current rational expectation of variable x at time t + h

Full-information rational expectations are such that

where vt+h,t is the full-information rational expectations error and is thus uncorrelated with information dated t or earlier.

Combining equations (3) and (4) yields the predicted relationship between the ex-post mean forecast error across agents and the ex-ante mean forecast revision

Importantly, the coefficient on the forecast revision depends only the degree of information rigidity λ. In the special case of no information frictions, λ = 0 and the specification collapses to equation (4), i.e. the average forecast error is unpredictable using information dated t or earlier. Because this canonical sticky-information model implies a single rate of information acquisition, equation (5) holds for any macroeconomic variable and any forecasting horizon. In addition, this specification holds regardless of the structure of the rest of the model.

B. Noisy-Information Model

We also consider models in which agents know the structure of the model and underlying parameter values, continuously update their information sets, but never fully observe the state. This class of models includes most famously the Lucas (1972) islands model but also a wide variety of limited information settings considered in the literature. For example, Kydland and Prescott (1982) assume that the level of technology reflects both permanent and transitory shocks but that agents cannot separately identify these two components. More recently, Woodford (2001) considers an environment in which firms observe aggregate demand subject to idiosyncratic errors, which combined with strategic complementarity in price-setting, can account for the persistent effect of monetary policy shocks. Suppose that a macroeconomic variable follows an AR(1) process:4

where vt is an i.i.d. normally distributed innovation to xt. Agents cannot directly observe xt but instead receive a signal yit such that

where ωit represents i.i.d. normally distributed mean-zero noise. Each agent i then generates forecasts (conditional expectations) Fitxt+h given their information sets via the Kalman filter

where G is the Kalman gain which represents the relative weight placed on new information relative to previous forecasts. When the signal is perfectly revealing about the true state, G = 1; while the presence of noise induces G < 1. Thus, (1 − G) can be interpreted as the degree of information rigidity in this model.5

After averaging across agents and rearranging, the following relationship between ex-post mean forecast errors and ex-ante mean forecast revisions holds:

where vt+h,t=Σj=1hρhjvt+j is the rational expectations error and Ft denotes the average forecast across agents at time t.6 Thus, while individuals form their forecasts rationally conditional on their information set, the ex-post mean forecast error across agents is systematically predictable using ex-ante mean forecast revisions. This specification is identical to equation (5), when (1 − G) is interpreted as the degree of information rigidity. In contrast to equation (5) derived under sticky information, the coefficient on forecast revisions need not be the same for different macroeconomic variables or forecast horizons in noisy-information models. Instead, the coefficient will vary with the determinants of the Kalman gain, for example, the persistence of the series and the signal-noise ratio.

C. A New Approach for Assessing the Nature of the Expectations Formation Process

The sticky-information and noisy-information models both point to the same relationship between ex-post mean forecast errors and ex-ante mean forecast revisions such that the coefficient on forecast revisions maps one to one into the underlying degree of information rigidities. This relationship can be readily estimated for a given macroeconomic variable x, mean forecasts across agents Fx and forecasting horizon h using the following empirical specification:

While this is just a special case of the more general test of FIRE commonly employed in the literature in which the forecast error is regressed on a subset of the information available to agents at the time the forecast was made, it addresses several important shortcomings of traditional tests. First, the relevant regressor to use for testing the predictability of forecast errors is specified by the theory. Second, when traditional tests identify a rejection of the null hypothesis of FIRE, this rejection is not directly informative about other theories of the expectations formation process in the absence of a clear theoretical mapping from the theory to the empirical tests. In contrast, our specification is informative not just about the null hypothesis of FIRE but also about models with information rigidities. Third, statistical rejections of the null hypothesis of FIRE in the standard test do not directly address the economic significance of departures from FIRE. Specification (10), on the other hand, allows us to map estimates of β directly into the underlying degrees of information rigidity (λ under sticky information and 1 − G under noisy information) and, hence, can help assess the economic significance of any rejections of the null hypothesis of FIRE.

As a first step to applying our approach, we follow much of the literature on survey measures of expectations and focus first on historical forecasts of U.S. annual inflation from the Survey of Professional Forecasters (SPF). Inflation expectations have received disproportionate attention because of their importance in measuring ex-ante real interest rates, their role in expectational Phillips curves, and for monetary policy. Professional forecasts, while not typically included in macroeconomic models, are useful not only due to their historical availability but also because, as some of the most informed economic agents in the economy, they provide a conservative benchmark for assessing possible deviations from the null of FIRE. The SPF is a survey currently run by the Philadelphia Fed of approximately 30–40 professional forecasters each quarter who provide forecasts for a variety of macroeconomic variables at different horizons. GDP/GNP deflator inflation forecasts are available starting in 1968Q4 at horizons ranging from the current quarter to four quarters ahead. We focus for now on forecasts of year-on-year annual inflation, where e.g. πt + 3, t refers to the average inflation rate over the current (t) and next three quarters. Forecast errors are constructed using forecasts made at time t and real-time data available one year after the period being forecasted over. We use real-time data to measure ex-post variables because final data may reflect reclassifications and redefinitions such that the final values are not directly comparable to the historical forecasts made by agents (Croushore 2010).

Because the predictability of ex-post forecast errors from ex-ante forecast revisions in the sticky-information and noisy-information models obtains only when averaging across agents, we focus on mean forecasts across professional forecasters. It is important to emphasize that the predicted relationship between forecast errors and forecast revisions should not be expected at the individual level. Under sticky information, agents either do not update their information, and hence do not revise their forecasts, or else update their forecasts to the FIRE, in which case their forecast errors are uncorrelated with their forecast revision. Since the decision to update information is time-dependent and therefore orthogonal to current economic conditions, there should be no predictability in an individual’s forecast errors arising from their forecast revisions. Under noisy information, the use of the Kalman gain by agents ensures that individual forecast errors should also be unpredictable on average given the agent’s information set, since the latter includes their forecast revision. Hence, the predictability of the average ex-post forecast errors across agents using ex-ante forecast revisions is an emergent property of the aggregation across individuals, not a property of the individual forecasts.

The relationship between average year-ahead inflation forecast errors across agents and average forecast revisions in both sticky-information and noisy-information models can be expressed as

where β > 0 if information rigidities are present and errort is the rational expectations error which is orthogonal to information dated t and earlier, so equation (11) can be estimated by OLS. From 1969-2010, we find β^ = 1.23 (s. e. = 0.50) as shown in Panel B of Table 1. As a result, we can reject the null of FIRE at the 5% level of statistical significance in a manner that is directly informative about the expectations formation process. First, the rejection of the null goes exactly in the direction predicted by models of information rigidities, so that this finding presents direct evidence in favor of these models. Second, because β maps into the degree of information rigidity from each model, we can extract an estimate of information frictions. In the context of sticky—information models, λ^=β^/(1+β^)0.55 would imply that agents update their information sets every six to seven months on average. This magnitude of sticky information should significantly affect macroeconomic dynamics and optimal policy decisions, as documented in Reis (2009). Alternatively, one can interpret this estimate of β under noisy-information models as implying that agents put a weight of less than one-half on new information and more than one-half on their previous forecasts (i.e.,Ĝ=1/(1+β^)0.45). This is in line with the calibrated rational inattention model of Mackowiak and Wiederholt (2011) in which such magnitudes of inattention can account for the slow response of inflation to aggregate shocks. Thus, our approach implies that information frictions are economically and statistically significant.

Table 1.Tests of the Inflation Expectations Process
Additional Control:zt − 1
Dependent variable

Forecast error

πt + 3,tFtπt + 3,t
NoneInflationAverage

quarterly

3-month

Tbill rate
Quarterly

change

in the log

of the

real oil

price
Average

unemployment

rate
(1)(2)(3)(4)(5)
Panel A: πt + 3,tFt πt + 3, t = c + β Ftπt + 3, t + γZt − 1 + errort
Constant−0.25−0.110.01−0.192.21***
(0.27)(0.25)(0.28)(0.24)(0.71)
Ftπt + 3, t0.07−0.210.25**0.050.21**
(0.09)(0.14)(0.11)(0.08)(0.09)
Additional Control:
Zt − 10.25*−0.16**1.71**−0.49***
(0.15)(0.07)(0.85)(0.13)
Observations158158158158158
R20.0130.0620.0770.0500.259
Panel B: πt + 3, tFtπt + 3, t = c + β(Ft πt + 3, tFt − 1 πt + 3, t) + γZt −1 + errort
Constant0.04−0.010.250.031.77***
(0.15)(0.21)(0.23)(0.15)(0.52)
Ftπt + 3, tFt − 1 πt + 3, t1.23**1.21**1.23**1.17**1.02**
(0.50)(0.49)(0.51)(0.50)(0.48)
Additional Control:
Zt − 10.01−0.040.63−0.28***
(0.05)(0.04)(0.72)(0.07)
Observations157157157157157
R20.1950.1890.1970.1940.282
Notes: The table reports coefficient estimates for the specified equations at the top of each panel. The additional controls (z) are lagged by one quarter. Newey-West standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.
Notes: The table reports coefficient estimates for the specified equations at the top of each panel. The additional controls (z) are lagged by one quarter. Newey-West standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

We can also test theoretical restrictions implied by these models. For example, both sticky-information and noisy-information models predict a constant of zero in equation (11), which we cannot reject in the data. If we estimate equation (11) omitting the constant, the estimated value of β and standard errors are essentially unchanged. Second, these models predict that the coefficients on the contemporaneous forecast and on the lagged forecast are equal in absolute value. To implement this additional test, we decompose the forecast revision into two terms as follows

Under models of information rigidities, we expect β1 > 0, β2 < 0, and β1 + β2 = 0. Estimating equation (12) from 1969-2010, we find β^1=1.24 (s.e. = 0.51) and β^2=1.27 (s.e. = 0.51). The signs on both coefficients conform to the theoretical predictions of models of information rigidities, and we cannot reject the null that the sum of the two coefficients is equal to zero (p- value=0.6). The results thus provide additional evidence consistent with the notion that the expectations formation process of professional forecasters is subject to information constraints.

A third restriction from models of information rigidities is that, while the average ex-post forecast error should be predictable using ex-ante average forecast revisions across agents, no other variable should have any predictive power for forecast errors. This is in the same spirit as traditional tests of FIRE but now conditional on forecast revisions. To assess this prediction, we focus on four specific macroeconomic variables which previous work (for example, Mankiw, Reis and Wolfers 2004, Pesaran and Weale 2006) has identified as having significant predictive power for ex-post inflation forecast errors: lagged annual inflation (πt − 1, t − 4), lagged quarterly interest rates (3-month Tbills), lagged quarterly changes in real oil prices (WTI spot price deflated by the GDP deflator), and the lagged quarterly unemployment rate. As a first step, we present estimates of the traditional test, i.e.

in which ex-post forecast errors are regressed on observable variables. We first include the contemporaneous forecast of inflation as a RHS variable in equation (13) then augment this with each of the additional variables discussed. The results, presented in Panel A of Table 1, confirm that these variables have predictive power for average ex-post inflation forecast errors over this time sample: lagged inflation, interest rates, changes in real oil prices and unemployment rates all have statistically significant coefficients pointing to predictability of ex-post forecast errors.

We then assess whether these variables retain their predictive power for ex-post forecast errors after conditioning on forecast revisions, i.e. we estimate equation (11) augmented with each of these variables. When controlling for either inflation, interest rates, or changes in real oil prices, the coefficient on forecast revisions is qualitatively unchanged, while the coefficients on these additional variables are no longer statistically different from zero. Hence, once one controls for forecast revisions, the predictive power of these three variables is eliminated, as predicted by models of information rigidities. In the case of unemployment, however, there is additional predictive power even after controlling for forecast revisions, although the coefficient on the unemployment rate is approximately cut in half. This finding suggests that deviations from FIRE may exist above and beyond those captured by simple models of information rigidities and further exploration of these deviations is a fruitful avenue for future research.

It should also be emphasized that the degree of information rigidity, just like the degree of nominal rigidity in typical New Keynesian models, is not a structural parameter. Rather, it should depend on underlying economic conditions. Reis (2006), for example, shows that the rate of information updating in sticky-information models depends on the volatility of macroeconomic variables. The same result applies with respect to the Kalman gain in noisy information models. As a result, we investigate in Sections III and IV the extent to which the degree of information rigidity varies cross-sectionally as well as over time. Our baseline estimates should therefore be interpreted as capturing the average degree of information rigidity over this time period, which provides a useful benchmark both for the calibration of economic models as well as for subsequently assessing the extent of cross-sectional and time-variation in information rigidity. However, as discussed in Croushore (2010), previous work has documented that rejections of the null of full-information rational expectations are much more common over short samples in which specific episodes, such as the Volcker disinflation, can have a disproportionate influence on measuring the predictability of forecast errors. The predictability of ex-post forecast errors from ex-ante forecast revisions over the longer sample of 1969 to 2010 is therefore particularly notable.

Finally, the predictability of average ex-post forecast errors across agents from ex-ante forecast revisions should not be interpreted as a form of irrationality on the part of agents. Both sticky-information and noisy-information models have, as their foundation, agents forming rational expectations subject to information constraints (for example, Reis 2006, Sims 2003, Mackowiak and Wiederholt 2009). The predictability of average ex-post forecast errors that results from the aggregation process across agents is an emergent property that does not obtain at the individual level. Indeed, previous tests of the rationality of forecasts have commonly found much weaker rejections, if any, of the null of full-information rational expectations at the individual level relative to the average forecast (Pesaran and Weale 2006). This is precisely what one would expect from models of information rigidities.

D. Extensions and Alternative Interpretations

We consider a number of extensions of models with information rigidities to assess whether these qualitatively and quantitatively affect our baseline predictions. Specifically, we first extend the noisy-information model along three dimensions: a more general process for the variable being forecasted, heterogeneity in priors about long-run means, and heterogeneity in signal strength. In addition, we consider alternative explanations proposed in the literature to account for the predictability of forecast errors that do not appeal to information rigidity: heterogeneity in loss aversion and forecast smoothing on the part of professional forecasters. None of these extensions and alternative explanations finds support in the data.

Generalized Noisy-Information Model

While the predictability of average ex-post forecast errors from forecast revisions in the sticky-information model does not depend on the specific data-generating process for the variable being forecasted, the equivalent prediction in the context of the noisy-information model requires the additional assumption of an AR(1) process. In this section, we consider the implications of a more general process. First, suppose that the variable being forecasted x follows an AR(p) such that Zt = [xtxt − p + 1]′ and Zt = BZt − 1 + H′vt where H = [1 0 … 0] and vtiid×N(0,σv2). Each agent i observes signal yit = Hzt + ωit with vit ~ iid N(0, Σω) is the agent-specific shock which is uncorrelated across agents. We assume that E(vtωit)=0, that is shocks to fundamentals vt and measurement error shocks ωit are independent. Agent i’s forecast for the unobserved state is Fitzt = Fi,t–1Zt + G(yit − Fi,t − 1yt) where G is the (p × 1) gain of the Kalman filter. We show in Appendix B that the average forecast error for x at horizon h will follow

where βij (ith,jth) element of βBh[{GH+(I − GH)](Bh)–1 and errort is the rational expectations error. Note that this expression is quite close to the baseline prediction, except for the fact that predictability in forecast errors now obtains not just from the revision in the forecasts at horizon h but also from the contemporaneous revisions to forecasts at shorter horizons due to the higher-order dynamic process. An additional difference is that the coefficients on forecast revisions may reflect not just the degree of information rigidity but also the specific AR(p) parameters in B. While the effect of the latter will tend to be quantitatively small since information rigidities are premultiplied by Bh then post-multiplied by its inverse, coefficient estimates on forecast revisions may not depend only on information rigidities as was the case with AR(1).

An appealing feature of the generalized noisy-information model is that we can therefore test empirically whether higher-order dynamics are important in characterizing the nature of the expectations formation process. To this end, we estimate equation (14) with forecast revisions for different horizons. Because SPF forecasts are at quarterly horizons ranging from t to t + 4, we focus on forecasts of quarterly inflation two quarters ahead, which allows us to consider up to AR(3) specifications. We rely on Bayesian information criterion (BIC) to identify the best-fitting specification because, if e.g. an AR(1) approximates the true model, forecast revisions at other horizons will be highly correlated, and point estimates at different horizons will be imprecise. The results, shown in Table 2, point to an AR(1) representation of inflation as the preferred specification, consistent with the description of the model in Section II.B.

Table 2.Testing for Higher Order Dynamics
Order, p
123
AR(p)1.1481.1631.171
VAR(p)
Output growth rates, GY1.1821.2151.264
Housing starts, HS1.1831.2221.267
Unemployment rate, UE1.1831.2361.278
Industrial production, IP1.1851.2261.272
GY, HS, UE, IP1.2841.4271.571
Notes: The table reports BIC statistics associated with different specifications of equation (14) in the text for AR(p) dynamics or equation (15) for VAR(p) dynamics. Lower values of the BIC indicate preferred specifications. See Section II.D.1 in the text for details.
Notes: The table reports BIC statistics associated with different specifications of equation (14) in the text for AR(p) dynamics or equation (15) for VAR(p) dynamics. Lower values of the BIC indicate preferred specifications. See Section II.D.1 in the text for details.

One can also consider VAR(p) representations of the data-generating process, such that the dynamics of e.g. inflation also depend on the dynamics of other macroeconomic variables. The predictability of forecast errors for one variable will then depend not just on forecast revisions for that variable but also upon forecast revisions of the other variables in the dynamic system. For example, if we are interested in forecasts of variable xa whose dynamics are determined jointly with xb in a two-variable VAR(1), Appendix B shows that the predictability of ex-post forecast errors for xa would follow

Because the SPF includes forecasts of additional variables since 1969 (output growth, unemployment, housing starts and industrial production), we can use the same procedure as with the AR(p) to assess whether the predictability of inflation forecast errors is better represented as a VAR process. The results in Table 2 illustrate that the simpler AR(1) specification is preferred to any VAR representation of the data-generating process, again consistent with the representation of the noisy-information model in Section II.B. Of course, these results should not be interpreted as implying that an AR(1) is the best representation of inflation dynamics. Rather, the results suggest that this parsimonious description of the data is sufficient to adequately characterize the predictability of forecast errors in terms of forecast revisions.

Heterogeneity in Signal-Noise Ratios

Another extension of the noisy-information model allows agents to have different signal-noise ratios and therefore place different weight on new information received. An appealing feature of this setup is that two agents can receive the exact same signal but, because of heterogeneity in their Kalman gains, will adjust their forecasts by different amounts. Because heterogeneity in Kalman gains quickly reduces the tractability of the model, we impose the following restrictions to simplify exposition: 1) ρ = 1 such that the variable being forecasted is a random walk, 2) Kalman gains are distributed normally across agents with variance σG2 and mean G. In such a setting, we show in Appendix C that the predictability of the forecast error is given by

where G is the average gain across agents and Ak(G) are coefficients defined in Appendix C. The key prediction that follows from heterogeneity in signal strength is that average forecast errors will be predictable using not only forecast revisions but also lagged values of the variable being forecasted. But as documented in Table 1, there is no evidence that average inflation forecast errors in the U.S. are predictable using lagged inflation once one conditions on forecast revisions. Hence, heterogeneity in signal strength does not appear to be a quantitatively significant source of information rigidity in our data.

Heterogeneity in Beliefs about Long-Run Means

Finally, we consider a variation on the baseline noisy-information model in which agents hold different priors about long-run values for economic variables, as in Patton and Timmermann (2010). Specifically, given the same setup as in Section II.B, we follow Patton and Timmermann (2010) and assume agents report forecasts Fitxt+h = ωμi + (1 − ω)E[xt+hit] where conditional expectations E[xt+hit] are formed using the Kalman filter and the agent specific signals, as in Section II.B, ω ∊ [0,1] is the shrinkage factor, and μi is agent i’s belief about the long-run value of x with μi being zero mean across agents. We show in Appendix D that in this case the predictability of average forecast errors is given by

There are two differences relative to the baseline prediction from the noisy-information model in Section II.B. First, ex-post forecast errors should be predictable using not only the ex-ante forecast revisions but also past values of the variable being forecasted. Second, the error term now includes a time-t component, such that OLS is invalid. To assess the quantitative importance of heterogeneity about long-run means, we regress annual year-ahead inflation forecast errors on forecast revisions and lagged annual inflation using two quarters of lagged changes in real oil prices as instruments. As documented in Panel A of Table 3, these instruments are strong predictors of forecast revisions. The results from estimating equation (16) by IV yield no evidence of predictability in forecast errors coming from lagged inflation once one conditions on forecast revisions. This suggests that heterogeneity about long-run means is not playing a quantitatively significant role in accounting for the predictability of inflation forecast errors.

Table 3.Tests of Alternative Interpretations of Forecast Error Predictability
Dependent variable

Forecast error
OLSIV
(1)(2)(3)
Panel A: Test Heterogeneity in Beliefs about Long-Run Means
Ftπt+3,tFt − 1πt + 3, t1.23**1.77***1.97***
(0.50)(0.80)(1.08)
πt − 1−0.06
(0.08)
Observations157157157
1st stage F-stat13.139.53
Panel B: Test Forecast Smoothing
Ftπt+hFt–1π+h0.64**2.14***2.15***
(0.33)(0.56)(0.74)
Ft–1πt+hFt–2πt+h−0.03
(0.31)
Observations483483483
1st stage F-stat21.3414.52
Notes: The table reports estimates of equation (10) augmented with additional regressors and estimated by OLS and instrumental variables (IV). In Panel A, the dependent variable is year-ahead forecast error for inflation πt+3,tFtπt+3,t. In Panel B, the dependent variable is h-period ahead forecast error for inflation πt+hFtπt+h. Panel B pools across horizons h=0 through h=2. In both panels, the instrumental variables are the first two lags of log change in oil prices. 1ststage F-stat shows the first stage fit. Column (3) in Panel A reports estimates of specification (16). Column (3) in Panel B reports estimates of specification (17). Driscoll-Kraay (1998) standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.
Notes: The table reports estimates of equation (10) augmented with additional regressors and estimated by OLS and instrumental variables (IV). In Panel A, the dependent variable is year-ahead forecast error for inflation πt+3,tFtπt+3,t. In Panel B, the dependent variable is h-period ahead forecast error for inflation πt+hFtπt+h. Panel B pools across horizons h=0 through h=2. In both panels, the instrumental variables are the first two lags of log change in oil prices. 1ststage F-stat shows the first stage fit. Column (3) in Panel A reports estimates of specification (16). Column (3) in Panel B reports estimates of specification (17). Driscoll-Kraay (1998) standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

Heterogeneity in Loss-Aversion

While each of the preceding interpretations of the predictability of ex-post forecast errors stemmed from the presence of information rigidities, previous research has suggested that such predictability could arise even under full-information rational expectations. Capistran and Timmermann (2009), for example, present a model in which forecasters have asymmetric loss functions with heterogeneity in the degree of loss-aversion. In the presence of GARCH dynamics, Capistran and Timmermann (2009) show that such heterogeneity in loss-aversion can account for predictable forecast errors despite forecasters having identical and complete information. To assess whether this theory can account for the predictability of forecast errors in terms of forecast revisions, we derive the covariance between the two predicted by this theory.

Agents face a LINEX loss function over their forecast errors F Et,t–1 = πt − Ft − 1πt given by

Positive ϕi imply that agents dislike positive forecast errors more than negative ones, and vice-versa, while ϕi = 0 yields the standard mean-squared-error (MSE) objective. Suppose that inflation follows: πt = ρπt − 1 + vt where t is not serially correlated but potentially heteroskedastic. Specifically, σt=α0+α1vt12+βσt1 so that vt ~ (0, σt) as in Capistran and Timmermann (2009). In this setting, we show in Appendix E that the sign of the covariance between the average forecast error and average forecast revision across agents is negative. This is because large innovations to inflation (in absolute value) will lead agents to significantly raise (lower) their inflation forecast when the average φ > 0 (φ < 0) relative to the conditional expectation because of the asymmetry in the loss function. As a result, an upward forecast revision will tend to be associated with a subsequent negative forecast error. Thus, the model implies a predictability of average forecast errors from ex-ante forecast revisions, but in the wrong direction relative to our empirical estimates.

Forecast Smoothing

An alternative explanation for predictable forecast errors by professional forecasters is that they engage in forecast smoothing for reputational considerations. For example, forecasters typically provide not just a numerical forecast to clients but also a qualitative interpretation of recent economic developments. To preserve their reputation, forecasters may try to avoid drastic short-run changes in their forecasts. As a result, their forecast errors could be predictable even in the absence of information frictions.

Suppose forecasters face the following problem: at time t, given a forecast from time t − 1, the forecaster needs to choose a sequence of forecasts (in expectation) of a variable x at time t + h

where γ is the discount factor. As we show in Appendix F, the first-order condition, after imposing FIRE, can be written as

where the error term errort is correlated with information at time t but not earlier. Forecast smoothing also yields predictability of ex-post forecast errors, but the coefficient on the contemporaneous forecast revision is negative while that on the lagged forecast revision is positive. These coefficients reflect the fact that, while forecasters will smooth the change in their forecasts to minimize adjust costs, they must also take into account how their choice of forecasts will affect the subsequent period’s adjustment cost. As a result, forecast errors should be predictable using both contemporaneous and past forecast revisions for a given horizon. In this respect, forecast smoothing yields predictions which differ from those of information rigidities and which can be assessed empirically.

We regress ex-post inflation forecast errors on both current and lagged forecast revisions using two lags of log changes in real oil prices as instruments to account for the correlation between the error term and time-t variables. However, because the horizons in the SPF data are not sufficient to construct lagged forecast revisions of year-on-year inflation at overlapping horizons, we use quarterly forecasting horizons ranging from the current quarter (h = 0) to two quarters ahead (h = 2) and present pooled estimates over these three horizons in Panel B of Table 3. We find no evidence of predictive power for the lagged forecast revision, and the coefficient on the contemporaneous forecast revision is positive, as predicted by models of information rigidities, rather than negative as predicted by forecast smoothing. Hence, the empirical evidence again supports the notion that the predictability of forecast errors across agents is driven by the presence of information rigidities such as sticky information or noisy information.7

III. Information Rigidities Across Agent Types, Macroeconomic Variables, and Countries

The results using U.S. inflation forecasts of professional forecasters are consistent with persistent deviations from full-information rational expectations and with the presence of economically significant information rigidities. In this section, we expand the scope of these results by examining forecasts of other economic agents, forecasts of additional macroeconomic variables, as well as cross-country evidence on information rigidities.

A. Information Rigidity across Agents

While professional forecasters provide a useful benchmark to assess the null of full-information rational expectations, the quantitative importance of their expectations is unclear. For example, most macroeconomic models do not include professional forecasters and there is little evidence on how their forecasts map into other agents’ beliefs (Carroll (2003) being a notable exception). Thus, we turn to surveys of other agents to assess whether the information rigidities identified for professional forecasters appear to be a more general phenomenon. The Livingston Survey provides biannual individual inflation forecasts from academic institutions, commercial banks, and non-financial firms, among others.8 Thus, we can use this alternative source of data to assess information rigidities for academics and agents in the private sector, both financial and non-financial. The Livingston survey includes individual forecasts of the CPI in 6 months and in 12 months so we can apply our empirical specification at the 6-month forecasting horizon. Table 2 presents the results from 1969 to 2010 for the mean forecasts across all forecasters as well as using the mean forecasts across subsets of professional forecasters. Academics have the smallest estimated coefficient on forecast revisions—different from zero only at the 10 percent significance level—while those of forecasters from commercial banks and non-financial institutions are slightly larger. For forecasts of firms, as well as when we average across all forecasters, the estimated level of information rigidity is positive and statistically significant, indicating that information rigidities are not limited to professional forecasters in the SPF. Furthermore, when converted to quarterly equivalents, the quantitative magnitudes are close to those reported for the SPF in the previous section.

We also consider two additional types of agents whose expectations are of economic importance: consumer expectations and expectations derived from asset prices. For the former, we rely on the Michigan Survey of Consumers. Each month, the University of Michigan surveys 500-1,500 households and asks them about their expectation of price changes over the course of the next year. For the latter, we use the inflation expectations data from the Cleveland Fed based on the method developed in Haubrich, Pennacchini, and Ritchken (2008) who rely on the term structure of interest rates and inflation swaps to extract measures of market expectations of CPI inflation at multiple yearly horizons starting in 1982. A drawback of both sources of expectations data is that they are only available at a forecasting horizon of one year and therefore revisions in forecasts over identical horizons are not available. Thus, we replace the forecast revision with the change in the year-ahead forecast, yielding the following quarterly specification

where πt + 4, t + 1 denotes the inflation rate between t + 4 and t + 1. In this case, the error term will consist of the rational expectations forecast error, as in equation (11), and β(Ft − 1 πtFt − 1 πt + 4) because the forecasts do not have perfectly overlapping time horizons across periods. As a result, this specification cannot be estimated by OLS. Instead, we estimate this specification by GMM, using as instruments contemporaneous innovations to oil prices. Specifically, we run an AR(2) on the first difference of the log of nominal oil prices and define the residuals as oil price innovations. These innovations are valid instruments because they are uncorrelated with both past information (t − 1 and earlier) as well as the rational expectations error arising from shocks occurring after time t. Furthermore, because oil prices have significant effects on CPI inflation, these oil price innovations are statistically significant predictors of contemporaneous changes in inflation forecasts for all three measures of inflation expectations and can account for an important share of their volatility (Table 4).

Table 4.Information Rigidity in Inflation Forecasts by Forecaster Types
Panel A: Livingston survey
Dependent

variable

Forecast error

πt+1,t − Ftπt+1,t
Academic

Institutions
Commercial

Banks
Non-Financial

Businesses
All

Forecasters
(1)(2)(3)(4)
Forecast revision0.448*0.820***0.616**1.023***
(0.245)(0.202)(0.228)(0.250)
Sample1969-20101969-20101969-20101969-2010
Observations83838383
Panel B: Instrumental variable regression
Dependent variable

Forecast error

πt+4,t+1Ftπt+4,t+1
Survey of

Professional

Forecasters

(SPF)
Michigan

Survey of

Consumers

(MSC)
Financial

markets

(FIN)
(1)(2)(3)
Forecast revision1.255**0.738***2.013*
(0.486)(0.194)(1.069)
s.e.e.1.1551.2581.651
1st stage F-stat15.70734.0925.421
Observations111111111
Notes: Panel A reports estimates of equation (10) using inflation forecasts from the Livingston Survey of Professional Forecasters at the biannual frequency. Columns (1)-(3) report OLS estimates using subsets of the forecasters while column (4) reports estimates using all forecasters in the survey. Panel B reports quarterly estimates of specification (18) with inflation forecasts by instrumental variables (IV). The instrumental variable is oil price shocks defined as residuals from projecting changes in oil prices on its own two lags. 1st stage F-stat shows the first stage fit. Newey-West robust standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.
Notes: Panel A reports estimates of equation (10) using inflation forecasts from the Livingston Survey of Professional Forecasters at the biannual frequency. Columns (1)-(3) report OLS estimates using subsets of the forecasters while column (4) reports estimates using all forecasters in the survey. Panel B reports quarterly estimates of specification (18) with inflation forecasts by instrumental variables (IV). The instrumental variable is oil price shocks defined as residuals from projecting changes in oil prices on its own two lags. 1st stage F-stat shows the first stage fit. Newey-West robust standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

In addition to estimates using consumer and financial market forecasts, we also report estimates of equation (18) using SPF forecasts at equivalent forecasting horizons. This allows us to assess the extent to which the additional error from non-overlapping horizons—and the need to use instruments—affects the empirical estimates. Using a common time sample of 1982 to 2009, we find a coefficient on forecast revisions of 1.3 for professional forecasters, a finding that closely mirrors our previous results. This indicates that our GMM procedure and instruments can adequately address the additional complications introduced by non-overlapping horizons. For consumers, the point estimate is smaller but also highly statistically significant which conforms to the findings of Coibion and Gorodnichenko (2012). For market expectations, the point estimate is even higher than for professional forecasters, but it is also less precisely estimated. This is likely a result of the reduced predictive power of oil price innovations on market-based forecast revisions, as can be seen in the first-stage fit. In short, estimates of the predictability of inflation errors from forecast revisions point to the presence of information rigidities across consumers, financial market participants, and firms as well as professional forecasters.9

B. Information Rigidity across Variables and Horizons

While much of the empirical literature on the expectations formation process has focused on inflation forecasts, our approach is readily applicable to any variable. We can also use the multiple forecasting horizons available in the data to expand the power of our tests. Furthermore, exploiting this cross-sectional dimension allows for additional tests of the two models of information rigidities.

In the canonical sticky-information model, agents update their information sets infrequently, but when they do so, they acquire full-information rational expectations. As a result, there is a single parameter governing the frequency of updating information which is common across macroeconomic variables and forecasting horizons. Thus, a testable implication of the canonical sticky-information model is that the estimated degree of information rigidity is invariant to the forecasting horizon and the variable being forecasted. In noisy-information models, on the other hand, the coefficient on forecast revisions for a given macroeconomic variable will be governed by the Kalman gain associated with that variable, which will depend on factors such as the persistence of the series and the strength of the signal observed with respect to that macroeconomic variable. At the same time, the degree of information rigidity in noisy-information models should be invariant to the forecasting horizon as long as the signal-to-noise ratio of individual information is constant for different horizons. One way to assess these two models in accounting for the expectations formation process of professional forecasters is therefore to compare the estimated degrees of information rigidity across macroeconomic variables being forecasted as well as across forecasting horizons.

We exploit the fact that the Survey of Professional Forecasters (SPF) contains quarterly forecasts for four additional macroeconomic variables going back to 1968Q4. Besides the GDP price deflator, these include real GDP, industrial production, housing starts, and the unemployment rate.10 Furthermore, each of these variables is available at multiple forecasting horizons, ranging from forecasts of the current quarter to 4 quarters ahead. While the SPF includes forecasts up to 4 quarters ahead, the horizon is limited to 3 quarters in the empirical specification because forecast revisions call for an additional forecasting horizon, e.g. when h=3, the forecast revision is Ftxt + 3 − Ft − 1xt + 3. To construct forecast errors, we use real-time values available one year after the relevant time horizon. For the first three series, forecasts of annualized quarterly percent changes are constructed from the underlying mean forecasts of the levels. Starting in 1981, the SPF also includes forecasts of 8 additional macroeconomic variables: the 3-month Treasury bill (Tbill) rate, the AAA interest rate, real consumption expenditures, real residential investment, real non-residential investment, real federal government expenditures, real state/local government expenditures, and the overall CPI. For each NIPA series and CPI inflation, we construct forecasts of annualized quarterly percent changes and use real-time data to construct forecast errors, while the two interest rates are measured in levels. The forecast horizons usable in our approach again run from h = 0 to h = 3.

We can estimate the degree of information rigidity pooled across variables, pooled across horizons, or pooled across both variables and horizon to assess the extent to which there is heterogeneity in estimated degrees of information rigidity across each dimension. Specifically, we estimate pooled regressions

where xk indicates which macroeconomic variable is included and h denotes the specific forecasting horizon ranging from 0 (forecasts of the current quarter) to 3 (forecasts for 3 quarters ahead). The results when we pool across variables for each forecasting horizon as well as when we pool across horizons for each variable are presented in Figure 1. In each case, we employ either the five variables consistently available since 1968 or the thirteen variables consistently available since 1981. We also show the estimated level of information rigidity pooled across all variables and horizons within each time period to provide a benchmark for assessing the extent of cross-sectional heterogeneity. Standard errors are constructed as in Driscoll and Kraay (1998), which allows for both cross-sectional and serial correlation in the errors as well as heteroskedasticity in the errors. In each case, the estimate of information rigidity pooling across all variables and horizons is positive and statistically different from zero.11

Figure 1.Estimates of Information Rigidity by Horizon and Macroeconomic Variable

Panel A: U.S. SPF Variables Available 1968-2010

Panel B: U.S. SPF Variables Available 1981-2010

Notes: The figure plots estimated coefficient β on forecast revisions for different forecast horizons (left column) and macroeconomic variables (right column) in specification (10). Each circle presents a point estimate and whiskers show the 95% confidence interval. The solid red line is the point estimate of the coefficient on forecast revisions in specification (14) on pooled (across variables) sample with the shaded region showing the associated 95% confidence interval. All standard errors are Driscoll and Kraay (1998). GY = real GDP growth rate, HS = Housing starts, IP = Growth rate of industrial production index, DEFL = Inflation rate for GDP deflator, UE = Unemployment rate, 3TB = 3 month treasure bill interest rate, AAA = Interest rate on AAA debt, CPI = Inflation rate for the consumer price index, C = Consumption growth rate, GF = growth rate of federal government consumption expenditures, GS = Growth rate of state government consumption expenditures, NRI = Growth rate of non-residential investment; RI = growth rate of residential investment.

When pooling across variables for each forecasting horizon, we find that one cannot reject the null hypothesis that the estimated degree of information rigidity is invariant across forecasting horizons (p-values of 0.15 and 0.23 for 1968 variables and 1981 variables respectively). Hence, the null hypothesis implied by both sticky-information and noisy-information models in terms of forecasting horizons cannot be rejected. When pooling across horizons for each variable, we can reject the null hypothesis that the degree of information rigidity is the same for all variables: p- values for the null are 0.06 for the five variables since 1968 and 0.0001 for the thirteen variables available since 1981. Rejection of the null hypothesis of equal degrees of information rigidity across variables in the post-1980 sample is driven primarily by the low estimate for state and local government expenditures, as well as the high estimated levels of information rigidity for housing starts and non-residential investment. The presence of heterogeneity in estimated levels of information rigidity across variables is inconsistent with the baseline sticky-information model in which agents are assumed to update their expectations about all macroeconomic variables simultaneously. This suggests that future work employing the sticky-information model should allow for differential updating rates across macroeconomic variables, in the same spirit as Mackowiak and Wiederholt’s (2008) model of rational inattention.

C. Information Rigidity across Countries

In addition to the U.S. Survey of Professional Forecasters, we have constructed a dataset of quarterly forecasts from the international survey of professional forecasters done by Consensus Economics. This dataset covers twelve countries: the G-7 countries of U.S., U.K., France, Germany, Italy, Japan and Canada as well as Spain, Norway, the Netherlands, Sweden and Switzerland. Data for the G-7 countries span 1989 to 2010 while data for other countries begin primarily in 1994. For each country, forecasts for five macroeconomic variables are available: consumer price inflation, real GDP growth, interest rates, industrial production growth and real consumption growth. Forecasts are available for the current quarter and for the subsequent 5-6 quarters.

This dataset therefore provides cross-sectional variation over countries, horizons and macroeconomic variables. Thus, we can assess how the degree of information rigidities varies not just along horizons and variables, but also between countries. Figure 2 presents estimates of information rigidity, by country, pooling across all horizons and macroeconomic variables available for that country. The countries with the highest degrees of information rigidities are Spain and Sweden, while the lowest are Canada and Norway. All of the estimates are statistically significantly positive so we can reject the null of FIRE for every country and this rejection of the null goes exactly in the direction predicted by models of information rigidities. Figure 2 also plots the estimated degree of information rigidity pooled across all countries, variables and horizons to provide a benchmark for assessing the amount of cross-country heterogeneity. This cross-country pooled estimate is almost identical to that obtained for the U.S.12 Furthermore; the country-specific estimate for the U.S. using Consensus Economic forecasts is remarkably close to those found using the SPF forecasts.

Figure 2.International Evidence on Information Rigidities

Notes: Each figure plots estimated coefficient β on forecast revisions in specification (10) from pooled specifications. In Panel A, estimates are for each country, pooled across variables and horizons. In Panel B, estimates are across forecast horizons, pooled across countries and variables. In Panel C, estimates are for each macroeconomic variable, pooled across countries and horizons. Each circle presents a point estimate for a given country and whiskers show the 95% confidence interval. The solid red line is the point estimate of the coefficient on forecast revisions pooled across all countries, variables and horizons with the shaded region showing the associated 95% confidence interval. All standard errors are Driscoll and Kraay (1998). In Panel A: CA = Canada, CH = Switzerland, DE = Germany, FR = France, IT = Italy, JP = Japan, ND = Netherlands, NW = Norway, SP = Spain, SW = Sweden, UK = United Kingdom, US = USA.

We can also use the Consensus Economics forecasts to verify our previous results regarding heterogeneity in information rigidity across variables. Figure 2 plots the estimated coefficients on forecast revisions, pooled across countries and horizons, for each of the five variables in this data. We can reject the null hypothesis of equality across variables with a p-value of 0.0003. This confirms our finding from the SPF that heterogeneity in the degree of information rigidity exists across variables.

We can perform a similar decomposition by forecast horizon, pooling across countries and variables. The results, presented in Figure 2, point to rising levels of information rigidity at longer horizons, and we can now reject the null of equality across horizons at the 1 percent level. As presented in Section II, both models predict that the estimated coefficient on forecast revisions should be identical for different forecasting horizons when forecasts are for quarterly changes, a hypothesis that could not be rejected with SPF forecasts of quarterly values. However, the forecasts of GDP, consumption and industrial production growth in the Consensus Economics survey are for year-on-year percent changes. While the distinction between forecasts of quarterly and annual changes does not matter under sticky information, it is significant for noisy-information models. Consider, for example, the forecast for the current quarter year-on-year GDP growth: forecasters have observed values for at least two of the four quarters over which they are forecasting. Hence, they have already received very strong signals about the value of current year-on-year GDP growth. When, on the other hand, they must forecast year-on-year GDP growth in four quarters, they will not have observed any of the quarterly values over which the forecast is made and therefore the available signals will be much weaker. Thus, the strength of the signal is falling over the first four forecasting horizons (h= 0 to 3) so that one would expect the estimated coefficient on forecast revisions to be rising over these horizons, which is exactly the pattern observed in Figure 2.13 Further evidence that the large increase in estimated coefficients with the forecasting horizon is driven by this feature of the Consensus Economics forecasts is that if we estimate the coefficient on forecast revisions at different horizons specifically for interest rate forecasts, which are not measured in year-on-year changes, the rising pattern of estimated coefficients is substantially dampened. Thus, the difference in results by forecast horizon in SPF and Consensus Economics data is consistent with what one would expect from noisy-information models.

D. Cross-Sectional Variation and the Determinants of Information Rigidity

Clearly, the degree of information rigidity is not equal across macroeconomic variables, which is at odds with canonical sticky-information models. On the other hand, the fact that heterogeneity in information rigidity exists across macroeconomic variables does not imply that noisy-information models can account for this cross-sectional variation. In the simple noisy-information model of Section II.B, the degree of information rigidity depends on the Kalman gain, which is itself a function of the persistence of the underlying macroeconomic process as well as the precision of the signal received by economic agents. More persistent processes imply, holding all else constant, that agents should put a higher weight (G) to current signals. A more precise signal naturally implies that agents should place relatively more weight on the current signal than on past forecasts. Thus, noisy-information models imply that the degree of information rigidity should be decreasing in the persistence of the series being forecasted and increasing in the amount of noise in the signal.

We can assess these predictions by exploiting the cross-sectional variation in the Consensus Economics cross-country data. First, for each country j and macroeconomic variable i in the Consensus Economics survey of professional forecasters, we fit an autoregressive process which yields an estimate of both the persistence of the variable (ρi,j) and the volatility of its innovations (σi,j). Second, we generate a measure of the noise associated with each series from a) the standard deviation of revisions to this series or b) the standard deviation of forecast disagreement for this series.14 Third, we construct a measure of the noise-signal ratio (κi,j) by taking the ratio of a measure of the noise to the standard deviation of the innovations to the variable from the first step. Given these measures of the predictors of information rigidity, we assess their importance by regressing our estimates of the coefficients on forecast revisions for each country-macroeconomic variable pair, pooled across forecasting horizons, in the cross-country Consensus Economics dataset set

where i denotes a specific variable, j denotes the country, and βi,j is the estimated coefficient on forecast revisions for each country-variable pair in the cross-country data-set.

The results are presented in Table 5. The coefficients on the persistence are consistently negative across specifications.15 When using the noise-signal ratio measured based on data-revisions (i.e. exploiting only common sources of noise), the coefficient is positive, as expected, but not significantly different from zero. Figure 7 shows that this is sensitive to outliers. As a result, we also consider estimates of (20) based on robust S-regressions, which automatically identify and account for outliers, and the results point to a positive and statistically significant effect of the noise-signal ratio, as predicted by the noisy-information model. When we use the cross-sectional dispersion of forecasts, which includes both common and idiosyncratic information among forecasters, to measure noise, the coefficient on the noise-signal ratio of each country-variable pair is positive and statistically significant whether or not we control for outliers. Strikingly, this simple specification can account for about 20-30 percent of the heterogeneity in information rigidities. Thus, not only are the theoretical predictions of noisy-information models qualitatively consistent with the observed heterogeneity in information rigidities across countries and variables, but this model can also quantitatively account for a considerable share of the observed cross-sectional variation.

Table 5.The Macroeconomic Determinants of Information Rigidities
Dependent variable:

estimated coefficient on

forecast revisions for

country-variable pairs
Revisions in data releases as

a measure of noise
Forecast disagreement as

a measure of noise
OLSOLS

exclude

outliers
Robust

regression
OLSRobust

regression
(1)(2)(3)(4)(5)
Persistence of Series, ρi,j−0.802***−0.833***−0.628*−1.068***−1.146***
(0.282)(0.248)(0.317)(0.269)(0.354)
Noise-Signal Ratio, ki,j0.1530.464**0.506*0.871**0.790*
(0.304)(0.201)(0.255)(0.425)(0.476)
Observations6056604848
R-squared0.1530.3260.196
Notes: The table reports estimated specification (20). The persistence of each series (ρi,j) is estimated as the sum of AR(4) coefficients. In columns (1)-(3), standard deviation of the difference between first and final data releases is taken as a measure of noise in the series. In columns (4) and (5), the average standard deviation of forecast disagreement is taken as a measure of noise in the series. In column (1), four observations are identified as outliers: consumption growth rates for Italy, France, Germany and Japan. These outliers are dropped in estimation in column (2). In columns (3) and (5), robust S-regressions are run with no dummies for outliers and all available observations included. Robust standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.
Notes: The table reports estimated specification (20). The persistence of each series (ρi,j) is estimated as the sum of AR(4) coefficients. In columns (1)-(3), standard deviation of the difference between first and final data releases is taken as a measure of noise in the series. In columns (4) and (5), the average standard deviation of forecast disagreement is taken as a measure of noise in the series. In column (1), four observations are identified as outliers: consumption growth rates for Italy, France, Germany and Japan. These outliers are dropped in estimation in column (2). In columns (3) and (5), robust S-regressions are run with no dummies for outliers and all available observations included. Robust standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

Figure 3.Information Rigidity and the Great Moderation

Notes: The figure plots the time series of two variables. The first is the standard deviation of the U.S. real GDP growth rate (annualized) over a five year moving window (red dash line; right axis). The second is the smoothed coefficient βt on forecast revisions in specification (10) estimated for each quarter separately on the SPF data (black thick solid line; left axis). The shaded region is the 95% confidence interval. The smoother is a local average which uses Epanechnikov kernel with bandwidth equal to five.

Figure 4.Information Rigidity during a Business Cycle

Notes: The figure plots the response of the coefficient βt on forecast revisions in specification (10) estimated for each quarter separately on the SPF data. The response is estimated as in specification (21). The response is normalized to be at the average value of the coefficient βt one period before a recession starts. The shaded region is the 95% confidence interval. The horizontal, thin, dashed line shows the average value of the coefficient βt. The vertical, thin, dashed line shows the time when economy moves into a recession.

Figure 5.Forecasts of U.S. Production before and After the September 11th, 2001 Attacks

Note: The figure plots consensus forecasts of real GDP growth rates (Panel A) and industrial production growth rates (Panel B) from thee different surveys of professional forecasters by Consensus Economics.

IV. State-Dependence in Information Rigidities

The previous section presents evidence that the varying degrees of information rigidity associated with macroeconomic variables are well-explained by the persistence and noise-signal ratios of these variables. However, the precision of an agent’s signal can be thought of as choice variable when agents have the ability to devote more resources to collecting and processing information as in the rational inattention models of Sims (2003) and Mackowiak and Wiederholt (2009). In this section, we investigate whether the degree of information rigidity responds to changing economic conditions. We focus on three particular dimensions. First, we assess whether the degree of information rigidity responds to low-frequency variation in economic volatility, as exemplified by the Great Moderation. Second, we characterize the extent to which information rigidity responds to business cycle fluctuations. Third, we focus on even higher-frequency variation in information rigidity as illustrated by changes in economic forecasts immediately following the 9/11 attacks. In each case, we document evidence of state-dependence in the degree of information rigidity, consistent with the reallocation of information collection and processing resources in light of economic conditions as suggested by rational inattention theories.

A. Information Rigidities and the Great Moderation

McConnell and Perez-Quirós (2000) and others have documented a substantial decrease in macroeconomic volatility both in the U.S. and other developed countries since the early to mid-1980s. Figure 3 plots the time-varying standard deviation of real GDP growth for the U.S., for example, which is rising throughout the 1970s, peaks in the very early 1980s, then exhibits a very sharp decline in the mid-1980s, declining by more than half relative to the average level during the 1970s. While the source of this phenomenon remains a point of contention, one explanation emphasizes the changes in monetary policy put in place under Volcker, either in terms of a stronger endogenous response to macroeconomic fluctuations as in Clarida, Gali, and Gertler (2000) or because of the Volcker disinflation as in Coibion and Gorodnichenko (2011). At the same time, there is only mixed evidence that microeconomic volatility declined over this time period. For example, Davis and others (2006) report that the volatility of employment has fallen since the 1970s for non-publicly traded firms while Comin and Mulani (2004) and Comin and Philippon (2005) show that volatility increased for publicly traded firms over the same period. Furthermore, volatility at the household level appears to have been trending up over time (see Davis and Kahn (2008) for a review). As a result of the reduction in the volatility of macroeconomic variables relative to microeconomic variables, one might expect that economic agents would choose to allocate relatively more resources to tracking micro rather than macro-level shocks since these shocks became quantitatively more important for profits and utility as in Mackowiak and Wiederholt (2009). Thus information rigidity should have increased with the arrival of the Great Moderation.

To explore this hypothesis, we estimate equation (10) for each quarter separately using SPF data and then compute non-parametrically a local average of the estimated β’s to provide a sense of the low frequency variation in the degree of information rigidities. Figure 3 plots the dynamics of the local averages of β as well as associated standard errors. The figure shows that information rigidities were falling from the late 1960s to the early 1980s as the volatility of macroeconomic variables was rising. The minimum level of information rigidity is reached in 1983-84, which closely matches the start of the Great Moderation identified in McConnell and Perez-Quirós (2000), and since then the estimated degree of information rigidity has consistently been increasing. The changes in the level of information rigidities over time are statistically and economically significant, especially when one compares the mid-1980s to the late 2000s.

This significant low-frequency variation in the estimated coefficients on forecast revisions suggests that one should be wary of treating information rigidities at the macroeconomic level as a structural parameter since these rigidities can vary over time in response to changes in macroeconomic conditions. Specifically, more tranquil times should be ceteris paribus associated with greater information rigidities, as suggested by the endogenous inattention model of Branch and others (2009). The rising degree of inattention over the course of the 1990s and 2000s also implies that the same sized shock would have larger real effects towards the end of the sample because information rigidities, like nominal rigidities, amplify the response of the economy to a given set of shocks. Thus, this observation suggests an additional mechanism, along with increased risk-taking on the part of financial market participants, through which the Great Moderation may have contributed to the severity of the Great Recession. The figure also suggests that the high volatility of the 1970s was associated with a gradual increase in attention on the part of economic agents, making the economy become progressively less sensitive to any given shock. This decline in information rigidity may therefore also have contributed to onset of the Great Moderation. More generally, the figure points to the possibility of low-frequency cycles in volatility arising from the endogenous response of information rigidities to volatility and the feedback effect of changing information rigidity on volatility.

B. Information Rigidities over the Business Cycle

Our results in the previous section indicate that calm times are associated with stronger information rigidities. In light of this evidence, one may expect that recessions, as periods of increased volatility, should be times when economic agents update and process information faster than in expansions since the (relative) cost of ignoring macroeconomic shocks in recession rises. Gorodnichenko (2008), for example, shows in a theoretical model that the acquisition of information endogenously increases shortly after the occurrence of an aggregate shock as economic agents face increased uncertainty about the current state of the economy and consequently find it beneficial to devote more resources to learning about current macroeconomic conditions. Using the U.S. estimates of β computed for each quarter separately as in Section IV.A, we consider the following econometric specification

where ItREC is a dummy variable equal to one in the first quarter of each recession, as identified by the NBER, and zero otherwise. By varying index j, we construct a sequence of estimated ϕj which may be interpreted as an impulse response of information rigidities to a recession. To smooth the path of coefficients ϕj, we fit a polynomial distributed lag model with the polynomial order equal to 4 and J = 20. Figure 4 shows the path of the estimated ϕj over four years after the economy slides into a recession. We assume that the economy starts at an average level of information rigidity which is equal to α^. At the start of a recession, information rigidities are and remain relatively high. However, as time passes, information rigidities become less severe to the point where we cannot reject the null of FIRE one to two years after the start of the recession. The degree of information rigidity stays low about two years and then starts to recover to the level observed before the start of a recession.16

These dynamics of information rigidity pose a challenge for popular models of information frictions such as the sticky-information and noisy-information models. In both types of models the choice of frequency of updates or allocation of attention is typically made given the “average” behavior of the economy rather than a specific contingency. Agents in these models do not reoptimize every period how much attention should be allocated to tracking macroeconomic conditions and the degree of information rigidity does not vary over the business cycle. Accounting for the business-cycle variation in information rigidity observed in Figure 4 will require models with state-dependent acquisition of information, such as Gorodnichenko (2008), to qualitatively generate variation of information rigidities over the business cycle and, more generally, in response to aggregate shocks.17

C. Information Rigidities after Large Visible Shocks

Both models of information rigidities considered in Section II imply that the current mean forecast is a weighted average of the previous mean forecast and the current full-information rational expectation forecast, where the weights used in computing the average current forecast in the classical formulations of these models are fixed and thus the diffusion of information is time-dependent. For example, under the sticky-information model of Reis (2006), if firms face a fixed cost to acquiring new information, then information updates will be infrequent and time-dependent if no new information can be acquired without paying the fixed cost. But if a large shock occurs which is visible to all economic agents, this would induce a state-dependent response and synchronized updating of information as in the state-dependent model of Gorodnichenko (2008) or Woodford (2009). As a result, the degree of information rigidity should be much lower after a large and visible shock than during normal periods.

The time period of our analysis includes one such unambiguously visible and economically potent shock: the attacks of September 11, 2001. As shown in Figure 5, the 9/11 attacks were followed by very large downward revisions to U.S. macroeconomic forecasts. For example, in the survey done in August 2001, the consensus forecast for the growth rate of the year-on-year real GDP for 2002Q1 was approximately 2 percent. In the special October forecasts of professional forecasters organized by Consensus Economics in response to the September 11th attacks, the consensus forecast for the same time period was revised down to -0.5%. Forecasts of industrial production were similarly substantially lowered as a result of the attacks. However, by February 2002, forecasters had raised their projected growth rates of real GDP back up substantially whereas forecasts of industrial projection growth remained very similar to the initial post-9/11 forecasts. The latter points to a rapid adjustment of expectations in line with the FIRE assumption, whereas the former actually points to overshooting expectations.

To quantitatively assess whether the degree of information rigidity varied during the periods immediately following the 9/11 attacks, we create a dummy variable It9/11 equal to one in the fourth quarter of 2001 and the first two quarters of 2002. We then consider the following specification

The coefficient ψ on the interaction term of forecast revisions and the 9/11 dummy indicates the difference in the degree of information rigidity associated with the forecast revisions during these three quarters. Results from applying this method to U.S. professional forecasters for those macroeconomic variables available since 1968 as well as the estimates using the larger set of variables from 1982 are presented in Table 6 (columns (1)-(4)) as are results from applying this test to the cross-country data-set, pooled across all countries, variables, and forecasting horizons (columns (5) and (6)). In each case, the coefficient ψ is negative and statistically significant, indicating that the degree of information rigidity was lower during the forecast revisions following the 9/11 attacks, as expected in the face of a large and visible shock. In fact, point estimates of ψ are, if anything, larger in absolute value than the point estimates of β, so that the forecast revisions can be characterized as either FIRE or even overshooting expectations in the case of the U.S. Thus, consistent with the predictions of state-dependent models of information rigidities, large and highly visible economic shocks will lead to much more rapid adjustment of expectations than during run-of-the-mill periods. The fact that our empirical approach can identify and differentiate between these types of forecast revisions lends further credence to the notion that information frictions are the source of the underlying rigidity in the expectations formation process.

Table 6.Forecast Revisions after the 9/11 Attacks
Dependent variable

Forecast error

xt+hFtxt+h
U.S. SPF

1968-2010

5 Variables
U.S. SPF

1982-2010

13 Variables
Cross-Country

Professional

Forecasters, 1989–2010

5 Variables, 12

countries
OLS

(1)
FE

(2)
OLS

(3)
FE

(4)
OLS

(5)
FE

(6)
Ftxt+hFt–1xt+h0.414***0.407***0.713***0.732***0.736***0.681***
(0.049)(0.045)(0.139)(0.148)(0.060)(0.058)
(Ftxt+hFt–1xt+h) × It9/11−0.894**−0.851**−0.826***−0.828***
1.041***1.011***
(0.222)(0.219)(0.274)(0.267)(0.129)(0.117)
p-value (β + γ) = 0< 0.01< 0.01< 0.01< 0.010.4670.209
Observations3,2403,2405,7935,79322,34122,341
R-squared0.0220.0210.0350.0360.0510.047
Notes: The table reports estimated specification (22). It9/11 is a dummy variable equal to one in 2001Q4, 2002Q1, and 2002Q2 and zero otherwise. p-value (β + γ) = 0 shows the probability value for the null that the coefficients on the contemporaneous forecast revisions (Ftxt+hFt–1xt+h) and the interaction between forecast revisions and the 9/11 dummy sum up to zero. Driscoll-Kraay (1998) standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.
Notes: The table reports estimated specification (22). It9/11 is a dummy variable equal to one in 2001Q4, 2002Q1, and 2002Q2 and zero otherwise. p-value (β + γ) = 0 shows the probability value for the null that the coefficients on the contemporaneous forecast revisions (Ftxt+hFt–1xt+h) and the interaction between forecast revisions and the 9/11 dummy sum up to zero. Driscoll-Kraay (1998) standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

V. Concluding Remarks

Building from the predictions of models of information rigidities, we provide a new test of the null of full-information rational expectations which is informative about the economic significance of departures from the null as well as the models that can account for these departures. The core of the proposed approach is a tight theoretical link between ex-post mean forecast errors and ex-ante mean forecast revisions. Applying this approach to professional forecasters in the U.S. and other industrialized countries, we document widespread rejections of full-information rational expectations in exactly the direction predicted by models of information rigidities. Consistent with these models, when one takes into account forecast revisions, other macroeconomic variables lose much of their ability to predict forecast errors. One interpretation of our results is that commonly observed rejections of the null of full-information rational expectations most likely reflect deviations from full-information rather than departures from rational expectations. Indeed, economic agents in the sticky-information or noisy-information models are fully rational but operate subject to information frictions. The estimates also point to economically significant estimates of information rigidities, thereby providing support for the recent body of work studying the integration of information frictions into macroeconomic models.

While we have focused primarily on professional forecasters, this approach can be applied to other economic agents. For example, we document qualitatively similar results using the inflation forecasts from the Michigan Survey of Consumers, businesses in the Livingston Survey, as well as forecasts extracted from financial market prices. The pervasiveness of information rigidity across agents suggests that future work on imperfect information should strive to incorporate information frictions on the part of different agents, as in Reis (2009). Our empirical results can provide a disciplining device for such models, in terms of stylized facts to be matched, and can be helpful in calibrating the degrees of information rigidity for different kinds of agents.

In addition, one can apply our approach to study the implications of different policies on the expectations formation process. For example, we document that the Great Moderation, frequently attributed to the monetary policy changes enacted by Fed Chairman Paul Volcker, was associated with a pronounced and persistent increase in the degree of information rigidity for professional forecasters. This finding suggests a new mechanism through which, along with increased risk-taking behavior on the part of financial market participants, the Great Moderation may have played a role in generating the Great Recession. Our empirical specification can also help quantify the effect of policy changes on the expectations formation process, thereby providing a more theoretically grounded notion of otherwise vague concepts such as “anchored” expectations. Importantly, this approach can be applied to study a wide variety of policies such as inflation targeting or exchange rate regimes to shed light on one of the key mechanisms via which these policies are supposed to affect dynamics: the expectations formation process.

Appendix A: Bias in OLS Estimates under Common Noise

Fundamental/State equation: πt = ρπt − 1 + υt where υt ∼ iid N (0, Σ υ)

Measurement equation: yt = πt + et + ωit where ωid ~ iid N (0, Σω) is the idiosyncratic noise, et ∼ iid N (0, Σe) is the common noise shock. We assume that all shocks are uncorrelated contemporaneously and at all leads and lads.

Denote the one step-ahead forecast error for the forecast πt|t − 1 (i) ≡ E (πt | yi,t − 1, yi,t− 2,…) in the Kalman filter with

We can find Ψ from the Ricatti equation Ψ=ρ2{ΨΨ(Ψ+Σω+Σω+Σe)1Ψ}+Σv. Denote the gain of the Kalman filter with G=Ψ(Ψ+Σω+Σe)1. The forecast for the unobserved state πt evolves as follows:

with yt|t1(i)=πt|t1(i).Letπt+h|t¯Ei(πt+h|t(i)) be the average forecast across agents. Using Ei(ωit) = 0, we can re-write equation (A.1) as

where υt + h,t is a linear combination of shocks υt occurring between t and t + h which constitutes a rational expectations error.

Suppose one runs an OLS regression of the following form

Hence, β^OLS is biased downward and if we find β^OLS > 0 then the true β=1GG must be even larger.

Appendix B: Noisy-Information Model under Generalized Dynamics

Consider the following state-space representation of the AR(p) inflation process (i.e., the fundamental) and observed signal about inflation:

where H=[10…0],vt~iidN(0,σv2) is a shock to fundamental with Σv=σv2HH.

Measurement: yit=Hzt+ωit where ωitiidN(0,Σω) is the agent specific shock which is uncorrelated across agents. Without loss of generality we omit signals contaminated with noise shocks common across agents. We assume that E(vtωit)=0, that is shocks to fundamentals vt and measurement error shocks ωit are independent.

Denote the one step-ahead forecast error for the forecast

Zt|t1(i)E(ztYi,t1,Yi,t2,) in the Kalman filter with Ψ=Σz(tt1)E((ztt1(i)zt)(ztt1(i)zt)). We can find Ψ from the Ricatti equation: Ψ=B{ΨΨH(HΨH+Σω)1HΨ}B+Σv. Denote the gain of the Kalman filter with G=ΨH(HΨH+Σω)1.

The forecast for the unobserved state πt evolves as follows:

where Ytt1(i)=Hztt1(i). After taking averages across agents (and hence dropping index i) and using Hzt=πt,Hztt1=πtt1, and Hztt=πtt, we note that

After re-arranging (B.2) averaged across agents and using generalized inverses (denoted as “+” in the power), we obtain

Note that (B.5) nests the AR(1) case in which β11=(1G)/G. The only difference in (B.4) relative to the AR(1) case is that we have (I − GH) as a matrix (rather than a scalar) measure of information rigidity but there is no difference in interpretation. In summary, the forecast error depends on how agents revise their forecast of inflation that period as well as previous p periods.

This generalizes to VAR(p) process by expressing the joint dynamics of variables in the state-space equation. Under VAR dynamics, the relevant forecast revisions on the RHS also include revisions in forecasts for other variables included in the VAR. As an example consider VAR(1) process (i.e., the fundamental is VAR(1)) with inflation being the variable of interest:

where xt(1),,xt(k1) are macroeconomic variables other than inflation, vtiidN(0,Σv) is a vector shock to fundamentals.

Measurement: Yit=zt+ωit where ωitiidN(0,Σω) with E(vtωit)=0. Given that Hzt=πt,Hztt1=πtt1, and Hztt=πtt, we immediately reproduce (B.3) and (B.4).

Consequently,

It follows that VAR(p) equivalent of (B.7) is

Appendix C: Noisy-Information Model with Heterogeneous Signal-Noise Ratios

For simplicity, assume a random walk process for x s.t. ρ = 1. Assume Kalman gains are distributed normally across population with cross-sectional variance σG2 and mean G. Then Coibion and Gorodnichenko (2012) show that

where xt+h,t is the average forecast of xt+h made at time t across the population and where G is the average gain. This can be rewritten as

so forecast errors should be predictable using lagged values of the variable being forecasted in addition to forecast revisions.

Appendix D: Noisy-Information Model with Heterogeneous Priors about Long-Run Means

The setup is the same as in the noisy-information model of Section II.B, but forecasters report forecasts

where, following Patton and Timmermann (2010), the shrinkage factor ω is common across agents, and xt+ht(i)=E(xt+hΩit) is agent i’s conditional expectation of x from the Kalman filter given private information Ωit.

As a result, the average reported forecast across agents is given by (assuming Ei(μi)=0):

Averaging across Kalman forecasts of agents, we previously had:

where REerrort is the rational expectations error.

After substituting for reported forecasts, we get

Hence, the ex-post forecast error now depends on ex-ante forecast revisions as well as the lagged forecast level. In addition, the error now includes a time t component which will generally be correlated with current forecasts.

Appendix E: Heterogeneity in Loss Aversion

Following Capistran and Timmermann (2009), suppose that inflation follows: πt=ρπt1+vt Where vt is not serially correlated but potentially heteroskedastic. Specifically, Σt=α0+α1vt12+βσt1 so that vt∼(0, σt) The mean forecast is given by

where φi measures loss-aversion for agent i based on the LINEX function in Section II.D.4, Ē(φi)

is the average value of φi across agents.

Note that

Forecast error is given by

The revision of the forecast is given by

That is revision in the forecast is a function of the shock received at time t+1. More specifically, the level of the shock (vt+1) and how its volatility is different from expected volatility ({vt+12[α0+α1vt2+βσt]}).

Consider regressing forecast error πt+2πt+2t+1¯

on forecast revision πt+2t+1¯πt+2t¯.
. Note that if the distribution of shocks is symmetric then (vt+1,vt+12)=0.
. Also, by assumption, cov (vt+1,vt+2) =0. Finally σt+1 is predetermined at time t and thus E[vt+12σt+1]=E(Et(vt+12σt+1))=E(σt+1Et(vt+12))=E(σt+1σt+1)=E(σt+12)S

Thus the sign of the regression coefficient is given by

Therefore, forecast errors and forecast revisions should be negatively correlated.

Appendix F: Dynamic Forecast Smoothing

We want to consider the optimization problem of a forecaster who seeks to minimize the mean squared forecast error but also wishes to minimize changes in the forecasts for reputational considerations. At time t, given a forecast from time t − 1, the forecaster needs to choose a sequence of forecasts (in expectation) of outcomes of a variable x at time t + h

where γ is the discount factor and the problem is bounded in time by t + h, since in subsequent periods the actual data will be revealed.

The first-order condition (FOC) is

Etxt+hEtFt+jxt+h=α(EtFt+jxt+hEtFt+j1xt+h)αγ(EtFt+j+1xt+hEtFt+jxt+h) for all j < h. At horizon j = h, the FOC is the same as in the static model.

Consider the case of j = O, i.e. choosing current forecast, we have

such that if we impose full-information rational expectations, we get

or using the same timing as for other specifications in the text

where REerrort-1 is the rational expectations error such that E[REerrort-1Zt-1]= 0, i.e. the error is orthogonal to information dated t − 1 and earlier but not time t.

With α > 0, β > 0, dynamic forecast smoothing implies a negative coefficient on contemporaneous forecast revision and positive on lagged forecast revision. The error term will in general be correlated with information dated time t so OLS is inapplicable.

Table 7.Properties of Inflation Forecasts
Panel A: Comparison of Mean Squared Error (MSE)
Survey of

Professional

Forecasters

(SPF)
Michigan Survey of

Consumers

(MSC)
Financial markets

(FIN)
(1)(2)(3)
MSE1.190

(0.212)
2.437

(0.561)
1.420

(0.211)
p-value of equality0.0390.443
Observations112112112
Panel B: Predictability of ex-post CPI inflation πt + 1, t + 4
(1)(2)(3)
SPF, Ftπt + 1, t + 40.750**

(0.364)
0.957***

(0.281)
0.789*

(0.404)
MSC, Ftπt + 1, t + 40.139

(0.310)
0.138

(0.312)
FIN, Ftπt + 1, t + 4−0.041

(0.270)
−0.039

(0.283)
R-squared0.2970.2900.297
Observations112112112
Notes: In Panel A, figures in parentheses are the standard errors of the MSE estimates. The last row in Panel A reports the p-value of the t-test of equality of MSE for SPF and an alternative source of forecasts. In Panel B, standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.
Notes: In Panel A, figures in parentheses are the standard errors of the MSE estimates. The last row in Panel A reports the p-value of the t-test of equality of MSE for SPF and an alternative source of forecasts. In Panel B, standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.
Table 8.Pooled Estimates of the Expectations Formation Process
Dependent variable

Forecast error
U.S. SPF 1968-2010

5 Variables
U.S. SPF 1982-2010

13 Variables
Cross-Country Professional

Forecasters 1989-2010

5 Variables 12 countries
xt + hFtxt + hOLSFEFE + time

dummies
OLSFEFE + time dummiesOLSFEFE + time dummies
(1)(2)(3)(4)(5)(6)(7)(8)(9)
Panel A
Forecast revision0.387**0.382**0.307***0.653***0.653***0.634***0.690***0.635***0.512***
(Ft + h − Ftxt + h)(0.178)(0.177)(0.118)(0.188)(0.188)(0.134)(0.143)(0.139)(0.072)
Observations3,2403,2403,2405,7935,7935,79322,34122,34122,341
R-squared0.0190.0180.1290.0300.0320.0800.0470.0430.234
Panel B
Ftxt + h0.421**0.429**0.359**0.651***0.670***0.631***0.721***0.663***0.525***
(0.173)(0.174)(0.111)(0.179)(0.179)(0.154)(0.140)(0.131)(0.106)
Ft − 1xt + h−0.481**−0.530***−0.483***−0.576***−0.506**−0.491**−0.782***−0.736***−0.541***
(0.171)(0.184)(0.074)(0.181)(0.202)(0.180)(0.119)(0.096)(0.082)
p-value (β + γ) = 00.1350.2340.2570.2580.1040.2070.0730.2330.625
Observations3,2403,2403,2405,7935,7935,79322,34122,34122,341
R-squared0.0220.0230.1350.0320.0390.0860.07310.0460.234
Notes: The table reports estimates of specification (10) in Panels A and B respectively. Driscoll-Kraay (1998) standard errors are in parentheses in columns (1), (2), (4), (5), (7), (8). Robust standard errors clustered by forecasted variable are in parentheses in columns (3), (6) and (9). Fixed effects in columns (2), (3), (5), (6), (8) and (9) are for each combination of country, variable, and forecast horizon. Time dummies in columns (3), (6) and (9) are for each time period (calendar quarter). ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.
Notes: The table reports estimates of specification (10) in Panels A and B respectively. Driscoll-Kraay (1998) standard errors are in parentheses in columns (1), (2), (4), (5), (7), (8). Robust standard errors clustered by forecasted variable are in parentheses in columns (3), (6) and (9). Fixed effects in columns (2), (3), (5), (6), (8) and (9) are for each combination of country, variable, and forecast horizon. Time dummies in columns (3), (6) and (9) are for each time period (calendar quarter). ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

Figure 6:Inflation Forecasts from Professional Forecasters, Consumers and Financial Markets

Note: The figure plots the one-year ahead CPI forecasts from the Survey of Professional Forecasters, the Michigan Survey of Consumers, and financial markets. See Section III.A for details.

Figure 7:Noise-Signal Ratios and Estimated Coefficients on Forecast Revisions

Note: The table plots the noise/signal ratio for each country/variable pair (horizontal axis) where noise is measured using the size of revisions to the data, as discussed in Section III.D. The vertical axis indicates the coefficient on forecast revisions from estimating (19) for each country/macroeconomic variable pair. The empty circles are outliers as identified by robust S-regression of (20) in the text.

References

Visiting Scholar, Research Department; University of Texas, Austin; National Bureau of Economic Research (NBER) and University of California, Berkeley; National Bureau of Economic Research (NBER) respectively. We are grateful to Bob Archibald, Christopher Crowe, Zeno Enders, Ulrich Fritsche, Pierre-Olivier Gourinchas, Ed Knotek, Javier Reyes, David Romer and Chris Sims for helpful comments as well as seminar participants at the Bank of France, CESifo/LMU Conference on Macroeconomics and Survey Data, College of William and Mary, Duke University, George Washington University, Columbia University, IMF, Michigan, Federal Reserve Bank of Minneapolis, Michigan State, NBER Summer Institute, NC State, Notre Dame, Federal Reserve Bank of Richmond, University of Arkansas, UC Berkeley, UC Irvine, UNC, University of Richmond, UT Austin, University of Virginia, and Yale. We thank Yury Yatsynovich for excellent research assistance.

For example, Ball and others (2005) show that price-level targeting is optimal in sticky-information models whereas inflation targeting is optimal in a sticky-price model. Paciello and Wiederholt (2010) document how rational inattention as in Sims (2003) alters optimal monetary policy. Likewise, Mankiw and Reis (2002) argue that the observed delayed response of inflation to monetary policy shocks is not readily matched by New Keynesian models without the addition of informational rigidities or the counterfactual assumption of price indexation. Roberts (1997, 1998) and Adam and Padula (2003) demonstrate that empirical estimates of the slope of the New Keynesian Phillips Curve have the correct sign when conditioning on survey measures of inflation expectations while this is typically not the case under the assumption of full-information rational expectations. Piazzesi and Schneider (2008), Gourinchas and Tornell (2004) and Bacchetta and others (2009) all identify links between systematic forecast errors in survey forecasts and puzzles in various financial markets.

See Pesaran and Weale (2006) for a survey of this literature.

We consider a much more general data-generating process in Section II.D.1 but focus here on the simpler case for analytical tractability.

We do not explicitly allow for public signals, multiple signals, or the ability of forecasters to observe past average forecasts but relaxing these assumptions does not qualitatively alter the predictions of the model (Crowe 2010).

The presence of common noise would introduce another component to the error term, dated time t and uncorrelated with information from t − 1 and earlier. In this case, our baseline empirical specification cannot be estimated by OLS. However, one can show that the bias in OLS will be downward (see Appendix A), such that our estimates will present lower bounds on the degree of information rigidity.

Two additional pieces of evidence support the notion that professional forecasters are not significantly smoothing their forecasts because of reputational considerations. First, we show in Table 7 that SPF forecasts are no worse at forecasting inflation (in terms of mean squared error) than households or financial markets. This is important because neither households not financial markets should have reputational incentives as household surveys are anonymous while forecast smoothing would leave money on the table for financial market participants. Second, when we regress ex-post inflation on the ex-ante forecasts of SPF and either household or financial market forecasts (or both), we cannot reject the hypothesis that the coefficient on SPF forecasts is one while that on other forecasts is zero as documented in Table 7. Hence, professional forecasters do not seem to be underutilizing information in the interest of protecting their reputations relative to other agents for whom reputational concerns should be less important.

The categories of forecasters also include investment banks, government forecasters, the Federal Reserve, labor organizations, and “other.” We do not look at these in detail because of how few forecasters there are in each of these groups over time while the theoretical predictions from sticky- and noisy-information models apply specifically to the mean forecasts across agents. The Livingston data is available on the website of the Federal Reserve Bank of Philadelphia.

One can also estimate these IV regressions at the monthly frequency using forecasts for consumers and financial market forecasts. The results are qualitatively similar.

Output is measured by GNP prior to 1992 and GDP thereafter. The price deflator is the implicit GNP deflator before 1992, implicit GDP deflator from 1992 to 1996, and the chained GDP deflator thereafter.

Table 8, we present additional results for the specifications pooling across all variables and horizons. We show for example that the estimates are invariant to controlling for cross-sectional fixed effects, as well as controlling for both cross-sectional and time fixed-effects. Furthermore, we also present estimates of equation (11) in which we allow for different coefficients on contemporaneous and lagged forecasts. In each case, we cannot reject the null hypothesis implied by models of information rigidities that the sum of the two coefficients is equal to zero.

The estimate pooled across countries is almost identical to the average across country-specific estimates. In Table 8, we also show that the degree of information rigidity pooled across countries is largely invariant to controlling for cross-sectional and time fixed effects. Furthermore, we again cannot reject at the 5 percent level (and at the 10% when we control for fixed effects) the null that the coefficients on current and lagged forecasts sum to zero. Controlling for the introduction of the Euro also does not affect the estimates of information rigidity.

The drop in the estimated coefficients at longer forecasting horizons in Figures 1 and 2, which occurs in both the SPF and Consensus data, appears to be driven by finite sample issues combined with some variables not being very persistent. This is because, with low persistence, forecasts of distant values will be near constant, so that contemporaneous forecast revisions will have very little explanatory power for ex-post forecast errors, pushing the estimated coefficient toward zero in small samples. When we reproduce the decomposition across forecasting horizons for variables measured in changes (GDP growth, consumption growth, etc.) which are not very persistent versus those variables measured in levels which are much more persistent on average, we find that the decline in estimated coefficients at longer forecasting horizons is non-existent for the latter but particularly pronounced for the former.

Specifically, for each time period, we take the difference between measures of the variable available two quarters and four quarters later, then compute the standard deviation of these revisions across the entire sample. Alternative time horizons for measuring revisions yield the same qualitative results. Real-time data, including revisions over the course of a year, are included in the Consensus Economics dataset. Data on cross-section dispersion of forecasts (forecast disagreement) are available for the growth rate of GDP, consumption and industrial production as well as inflation.

Under the null hypothesis that γ1and γ2 are zero, Pagan (1984) shows that standard errors on generated regressors are asymptotically valid. We found that our inference is not affected when we correct standard errors as in Murphy and Topel (1985). Furthermore, to the extent that the persistence of macroeconomic variables depends on the degree of information rigidity, this potential endogeneity will tend to bias our estimate of γ1 upward. This reflects the fact that if the persistence of macroeconomic variables is increasing in the degree of information rigidity, then this would tend to push γ1up.

Loungani, Stekler and Tamirisa (2011) apply this methodology to real GDP growth forecasts of professional forecasters in 46 countries and similarly find reduced rates of information rigidities after recessions and banking crises.

The substantial decrease in information rigidity during recessions could also be consistent with models in which agents have ambiguity aversion, as in Epstein and Schneider (2008) and Ilut (2010). In these models, rational agents face uncertainty about the true data generating process and place more weight on the less favorable data generating processes. As a result, these models point to information rigidity being insensitive to good news and decreasing strongly in response to bad news, such as recessions.

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