Inattentive professional forecasters

∗

Philippe Andrade

Herv´e Le Bihan

Banque de France & CREM

Banque de France

First draft: January 2009 This draft: April 2013

Abstract We use the ECB Survey of Professional Forecasters to characterize the dynamics of expectations at the micro level. We emphasize the following two new facts: forecasters (i) fail to systematically update their forecasts and (ii) disagree even when updating. We also find, as previous work on other surveys does, that forecasters have predictable forecast errors and disagree. We argue that these facts are altogether qualitatively supportive of recent theories in which agents are inattentive when they form their expectations. More precisely, they are in line with a model where agents imperfectly process information due to both sticky information ` a la Mankiw-Reis, and noisy information ` a la Sims. However, building and estimating such an expectation model, we find that it cannot quantitatively replicate the error and disagreement observed in the SPF data. Given how inaccurate they are, professionals agree too much to be consistent with the inattention model we test. Keywords: Expectations, imperfect information, inattention, forecast errors, disagreement, business cycles JEL classification: D84, E3, E37 ∗ We thank our discussants Bartosz Ma´ckowiak, Ernesto Past´en and Gregor Smith as well as Carlos Carvalho, Olivier Coibion, Christian Hellwig, Anil Kashyap, Noburo Kiyotaki, Juan Pablo Nicolini, Giorgio Primiceri, Sergio Rebelo, Xuguang Sheng, Jonathan Willis, Alexander Wolman, Michael Woodford, Tao Zha and seminar participants at the Banque de France, ECB, New-York Fed, Philadelphia Fed, San-Francisco Fed, University Paris 1 and at the conferences ESEM 2009, AFSE 2010, CESIfo on “Macroeconomics and Survey Data” and SED 2010 for useful comments. All remaining errors are ours. We are also grateful to Sylvie Tarrieu for superb research assistance as well as to Claudia Marchini and Ieva Rubene for their help with the SPF data. This paper does not reflect necessarily the views of the Banque de France. emails: [email protected], [email protected]

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1

Introduction

Models in which imperfect information and the formation of expectations act as a transmission mechanism of economic fluctuations—in the spirit of Friedman (1968), Phelps (1968) and Lucas (1972)—have recently regained interest in the macroeconomic literature. Imperfect information, in particular, has been related to the inattention of agents to new information, a behavior that can be rationalized by costly access to information and limited processing capacities. One appeal of these models is to provide an alternative channel to sticky prices to explain the persistent effects of transitory shocks — and in particular, monetary shocks — on the economy. Moreover, this approach can parsimoniously account for patterns of individual expectations observed in survey data that are at odds with the standard perfect information, rational expectation framework, such as predictable forecast errors and forecasts differing across forecasters.1 In this paper, we exploit the panel dimension of such a survey of forecasts — namely the ECB Survey of Professional Forecasters (SPF) — to produce new micro facts characterizing the formation of expectations. The ECB SPF is a quarterly panel starting in 1999 surveying around 90 forecasting units in either public or private institutions and allows to track sequences of forecasts made by the same institution. We then elaborate on these new facts to assess whether models of inattention accurately describe the behavior of forecasters. We focus on two types of inattention models that have been discussed in the recent literature. On the one hand, sticky information models developed by Mankiw & Reis (2002) and Reis (2006a,b), in which agents update their information set infrequently but get perfect information once they do. On the other hand, noisy information models proposed by Woodford (2002), Sims (2003) and Ma´ckowiak & Wiederholt (2009), in which 1

Mankiw & Reis (2011) and Veldkamp (2011) provide recent surveys.

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agents continuously update their information but have an imperfect access to it at each period. As some related previous work, we use professional forecast data to test imperfect information models. Expert forecasters may not be representative of less sophisticated agents, since professionals obviously allocate substantially more time, human, collecting and computing resources to the task of forecasting macroeconomic variables. However, Carroll (2003) shows that the opinion of professional forecasters spreads to firms and households, and hence also influences their expectations and decisions. Furthermore, we expect professional forecasters to be the agents in the best position to pay attention to the relevant macroeconomic information. As a result, the extent of attention to news among professional forecasters can be seen as an upper bound for other agents’ attention to aggregate conditions. Our paper has two main contributions. The first one is to document two new facts related to forecast revisions, namely that (i) forecasters do not systematically update their forecasts even when new information is released, and that (ii) forecasters who update also disagree on their forecasts. The originality of our approach is to exploit the sequences of individual forecasts for a given event (say inflation at the end of a given year) provided by the ECB SPF to construct a direct micro-data estimate of the frequency of updating a forecast, which, to our knowledge, has not been documented in survey data before. The results show that, on average, each quarter only 75% of professional forecasters update their 1-year or 2-year ahead forecasts. This first result is in line with the predictions of a sticky-information model ` a la Mankiw-Reis. Furthermore, in this setup the frequency of updating has a structural interpretation and corresponds to one key parameter, the attention degree. In addition to unfrequent updating, we also uncover that forecasters who update their information sets disagree about their forecasts. Consequently, disagreement among experts

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is not only related to differences in the information sets of forecasters who updated and of those who did not, but also to the fact that, when they update, they use different information sets. This second result is in line with the predictions of a noisy-information model ` a la Sims. We also find, as previous work relying on survey data does, that forecasts of experts exhibit predictable errors and that forecasters disagree as they report different predictions for the same variable at the same horizon. These latter two characteristics are in line with both sticky and noisy information models. The second main contribution of this paper is to perform a formal empirical assessment of inattention models exploiting the cross-section dimension of the survey expectations. Guided by the two aforementioned facts, we first develop a model that features both sticky and noisy information. We then assess the empirical performance of this model by comparing it to some key properties of the SPF through a Minimum Distance Estimation (MDE). More precisely, we compare moments characterizing the forecast errors and the disagreement generated by this theoretical model with their empirical counterpart observed in the ECB SPF data. Estimation results point to a rejection of the proposed inattention model. Fitting the smoothness observed in the average SPF forecasts would require a much lower attention degree than our micro data estimates. Such a low attention would in turn lead to much more disagreement, and volatility of disagreement, than observed in the SPF data. Therefore, elements others than the type of inattention included in our expectation model are needed to reconcile the relatively low disagreement among professionals and the relatively high persistence of the aggregate forecasting error. Our paper is related to studies which compare the properties of survey forecasts with the implications of theoretical expectation models.2 Numerous work found systematic aggregate forecast 2

See Pesaran & Weale (2006) for a survey.

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errors and disagreement in these data, at odds with the perfect information rational expectation framework. We provide additional evidence of such predictable forecast errors and disagreement for the European SPF data and a recent sample period. We also complement these results by providing new evidence on the infrequency of individual expectations revision. Our paper is also closely linked to the literature relying on survey expectation data to assess inattention and, more generally, imperfect information theories. A prominent contribution to this literature is Coibion & Gorodnichenko (2012a). They use aggregated survey data to examine the conditional response of the average forecast error and of the disagreement across forecasters to various structural shocks in order to disentangle the sticky-information from the noisy-information models of inattention. They find mixed support in favor of the two, just as we do. Their empirical evidence covers a broader range of forecasters than ours as they exploit information from the US SPF as well as forecasts from survey of US firms and consumers. By contrast, we rely on the mere ECB SPF. We however exploit the individual data, which allows us to observe individual forecast updating, and to assess arguably more directly some implications of the two types of inattention. Coibion & Gorodnichenko (2012a) consider different variants of each type of inattention separately. A distinctive feature of our approach is that we devise and estimate a model featuring the two types of inattention simultaneously. Other papers that rely on survey data to assess imperfect information models include Mankiw et al. (2003), Branch (2007), Patton & Timmermann (2010), Sarte (2010), and Coibion & Gorodnichenko (2012b). Mankiw et al. (2003) and Branch (2007) focus on the cross-section distribution of forecasts to calibrate the sticky-information attention parameter mentioned above. By comparison, we underline the importance of investigating the consistency of this parameter values with both the

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cross-section dispersion of forecasts and the aggregate forecast errors. Furthermore, we improve on their approach by considering a model that can explain the disagreement among forecasters who update their information set. Lastly, rather than calibrate it, we estimate the attention parameter using alternatively direct micro-data estimates and a MDE procedure. Patton & Timmermann (2010) rely on the patterns of forecasts observed over different horizons to emphasize the importance of model disagreement rather than differences in information sets.3 The model we consider is an alternative approach that generates disagreement without relying on “deep” heterogeneity among forecasters and that can moreover account for unfrequent updating. Sarte (2010) derives an indirect estimate of the attention degree of firms by matching the balance of opinions in the US ISM business condition survey to the US manufacturing production index. Coibion & Gorodnichenko (2012b) find evidence that average forecast errors are positively related to the past average forecast revisions, as implied by both sticky and noisy information models. Finally recent contributions, e.g. D¨ opke et al. (2008) and Coibion (2010), also use survey forecasts together with a set of auxiliary auxiliary assumptions about the economy to estimate the attention degree in a sticky-information Phillips curve. By contrast, we provide a direct, arguably more reliable, micro-data estimate of this parameter which, although much higher than in these previous works, still remains remarkably lower than one. The remainder of the paper is organized as follows. In section 2, we describe the European SPF data and provide some basic facts. In section 3 we introduce our approach to estimate the Mankiw-Reis parameter directly on micro data, and present our two main new facts. In section 4, we develop a model of inattention that incorporates both sticky and noisy information and put it to a test, 3 Coibion & Gorodnichenko (2012a) provide alternative evidence against model disagreement as a primary driver of professional forecasters’ behavior.

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relying on a moment-matching method. We provide some concluding remarks in section 5.4

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The ECB Survey of Professional Forecasters: data and basic facts

2.1

The ECB SPF

The ECB’s survey of professional forecasters has been conducted every quarter since 1999. The survey covers around 90 institutions involved in forecasting the euro area economy. The sample used in this article ends with the survey round of 2012Q4, so that we have a sample of 56 time periods. Each institution is asked to report, inter alia, forecasts for the (year-on-year) euro area inflation rate, the (year-on-year) real GDP growth rate and the unemployment rate for forecasting horizons of one year and two years.5 These data are matched with the corresponding (final release) realizations of the forecasted variable. We use for this purpose aggregate data for the euro area in changing composition from Eurostat (see Appendix A.1 for details). The ECB SPF has been rarely used for research purposes so far. In spite of a rather short sample in the time dimension, the ECB SPF has some specific advantages compared to some other survey expectation data. To start with, the data base is a panel so that one can track the response of a particular individual institution over time. Moreover, the responses are quantitative rather than qualitative. By contrast, many of the surveys that cover firms or households are typically repeated cross-sections, and report qualitative data. Furthermore, the number of actual respondents is 4

Supplemental materials about the data, the robustness of our empirical findings, and the properties of the stickynoisy information model are presented in a separate Appendix. 5 See Bowles et al. (2007) for a thorough presentation and discussion of the survey.

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relatively high (typically around 60 in a given quarter) compared with other surveys of professional forecasters. This number is for example twice as large as the ypical number of actual respondents in the more widely used US-SPF. Finally, the ECB SPF tracks almost the same institutions over time. By contrast there are changes in the set of institutions sampled in the US-SPF. Respondents provide two types of forecasts. The first one is a ‘rolling horizon’ forecast, with a fixed horizon of one or two years ahead of the last available observation. To be specific, in 2010Q2, each forecaster was asked about his forecast for the inflation rate one year ahead of the last observation, i.e. for March 2011.6 Then in 2010Q3, forecasters were asked about their inflation rate forecast for June 2011. The second type of forecast is a ‘calendar horizon’ forecast: in each quarter, forecasters are also asked to report their forecast for two fixed events, namely the current and the next years. For instance, in both 2010Q2 and 2010Q3, forecasters were surveyed about their annual inflation forecast for the end of 2010 and the end of 2011. This information together with the fact that we can track the same institution over time allow us to observe individual forecast revisions of the same event, a feature that is key in our study. x individual i’s rolling forecasts for the variable We now introduce some notations. We denote fit,t+h

x at date t and h quarters ahead. The variable x is either π (the year-on-year inflation rate), u (the unemployment rate), or ∆y (the year-on-year GDP growth rate). The forecast horizon h is set to 4 or 8 quarters ahead of the last observation of variable x available at the date of the response to the survey. Importantly there is an observation lag between the date of the response to the survey t and the date of the last available figure of x. This lag varies across variables: inflation is observed with a one month lag, unemployment with a two month lag and GDP growth with a 6

This illustration takes into account a variable-specific observation lag that we discuss in further length below.

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x two quarter lag.7 We also denote fit,T the calendar forecast associated to the last quarter T of a

specific calendar year in the sample, with the calendar year being is either the current or the next year. The so-called consensus forecast, i.e. the average forecast, associated to rolling horizons is x = ft,t+h

1 nt

Pnt i

x fit,t+h , with nt the number of respondents to the survey at date t. Finally, letting

xt be the realization of the forecasted variable at date t, we denote exit,t+h individual’s i forecast x x and its average as ext,t+h = xt+h − ft,t+h . error at date t + h, namely exit,t+h = xt+h − fit,t+h

2.2

Predictable forecast errors and disagreement among forecasters

This section documents some basic facts about individual expectations, namely predictable forecast errors and disagreement among forecasters. Similar facts have been evidenced by previous research on other data sets than the ECB SPF, and interpreted as signs in favor of imperfect information models.8 Predictable forecast errors are indeed an implication of both sticky information and noisy information inattention models. In a sticky information model, agents update their information set infrequently with a given, constant, probability. As a result, at each date, only a fraction of the population has access to the last vintage of macroeconomic news. The forecast error at date t + 1 of agents who did not update their information set at date t will be predictable, based on the information available at date t. As a consequence, the aggregate forecast error also will be predictable. In a noisy information model agents update their information but know that the news they get is imperfect and therefore only partly pass it onto their forecast. The average forecast 7 To economize on notation, we do not mention the observation lag in the formulas. The reader should keep in x refers to year-on-year growth rate forecast of x 2 quarters ahead mind that for horizon h = 4, the notation fit,t+h of current date t when x is real GDP, 10 months ahead when x is unemployment, and 11 months ahead when x is inflation. 8 See in particular Mankiw, Reis & Wolfers (2003) and Coibion & Gorodnichenko (2012a,b).

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thus incorporates only part of this new information, which makes the forecast error predictable with respect to the (perfect) information available ex-post to the econometrician. Disagreement among forecasters can also be rationalized by both sticky-information or noisyinformation models of expectations. In both cases, disagreement is a consequence of imperfect information which implies that agents do not have the same information set. A distinguishing feature of the two models is that while sticky-information models can account for time variations in disagreement, a basic version of noisy-information models with constant noise in the signal and homogenous forecasters cannot. Indeed, in a sticky-information setup, when a large shock hits the economy, individuals who update their information set produce very different forecasts than individuals who do not. These differences are less pronounced in the wake of a small shock. Consequently, the extent of disagreement evolves over time due to the magnitude of the shocks. By contrast, in a noisy-information basic model, forecasters have different opinions because they randomly get different perception of reality. Whenever the variance of the noise is constant over time and across individuals, disagreement will not be affected by the size of the shocks hitting the economy.9 A first sense of the predictability of the average forecast errors in the ECB SPF sample is provided in Figure 1 which plots the time series of the 1-year ahead consensus forecast together with the realizations of the predicted variable.10 Strikingly, periods of under/overestimation of the target variable realizations are very persistent, and last for more than 1 year, i.e. over time periods that 9 Coibion & Gorodnichenko (2012a) emphasize this difference between the basic versions of the sticky and the noisy information models. It is however possible to generate time-varying disagreement with more refined versions of noisy-information models, for instance, by introducing conditional time variance of the noisy signal correlated with the size of the true shock, or heterogeneity in the noise among the population of forecasters. 10 The median forecast is very close to the average.

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are longer than the forecast horizon.11 Table 1 provides some descriptive statistics and tests on forecasts errors. Over the sample period, inflation has been underestimated by an average annual rate of .43%. Unemployment exhibits a small underestimation of about .15% over the period. Finally, real GDP growth rate has been overrated by an average of .25%, due mainly to the recent crisis. Systematic biases go along with persistent forecast errors. Their first order auto-correlations range from .743 for inflation to .914 for unemployment. The bottom panel of Table 1 displays the results of a regression of the average forecast error on first a constant and second a constant and the last error known at the date when the forecast was made (that is h quarters before, with, due to the observation delay, h = 2, 3 or 4 depending on the variable). For the three variables studied either the bias or the link with past errors is significant at the 10% level so that errors are predictable.12 A natural measure of disagreement among forecasters is the cross-section standard deviation of forer 2 P t  x x x casts at each date, namely, using notations introduced above: σt,h = n1t ni=1 . fit,t+h − ft,t+h Like in several other studies, in particular Mankiw et al. (2003) for the US, we observe that the cross section-distribution of forecasts never degenerates to a single peak i.e. that disagreement is non-zero.13 Table 2 reports that the time average disagreement for the 1-year horizon rolling forecasts is equal to .29 for inflation, .30 for unemployment and .38 for real GDP. Figure 2 presents the time series of disagreement for the 1-year horizon rolling forecasts of the three macroeconomic variables of interest.14 Disagreement also varies over time, and is strongly positively correlated 11 This is noticeably the case for inflation which, up to 2006, was systematically underestimated. Note that part of the visible overestimation of unemployment early in the sample is due do substantial revision as discussed by Bowles et al. (2007) 12 The test statistics use HAC standard errors in order to take into account the auto-correlation of residuals induced by overlapping forecasts. 13 See Appendix A.2 for a figure illustrating this feature. The distribution has in general several modes. 14 Relying on rolling-horizon forecasts to evaluate forecasters’ disagreement is important to avoid the seasonal patterns emerging from the resolution of uncertainty as time gets closer to the forecasted event one gets when relying

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across variables. Noticeably, disagreement increased markedly for all three variables at the onset of the Great Recession. It is worth investigating whether this disagreement is state-dependent and reflects a slow diffusion of new information, as would be the case in a sticky-information model. We proceed by regressing x , on several rough measures of the amplitude of shocks hitting the econthe disagreement σt,h

omy: the squared last variation in the forecasted variable (∆xt−1 )2 , the squared last forecast error x (ext−h−1,t−1 )2 , and the squared current change in the forecast (∆ft,t+h )2 . Results are presented in

Panel B of Table 2. Coefficients are all positive and significant in 5 cases out of 9. These results broadly support that disagreement is an increasing function of the amplitude of the shocks hitting the economy.15 This last evidence somehow contrasts with Coibion & Gorodnichenko (2012a) who find that the response of dispersion of inflation forecasts to structural shocks is in general not significant. However their results are mixed across types of forecasters and of structural shocks, and while they fail to reject the null, the response they obtain is often positive. Overall, predictable and biased forecast errors and disagreement among forecasters suggest that both sticky information and noisy information models may be good candidates to describe expectation formation. In the next section, we go further by documenting new facts, related to the unfrequent updating of forecasts, that can be directly related to each of the two types of inattention. on calendar-horizon forecasts instead. 15 Alternatively, these results could reflect the impact of uncertainty shocks affecting the size of the innovations to macroeconomic variables. If these shocks also contribute to the future level of macroeconomic variables, and if uncertainty co-moves with disagreement, the above relationship may also emerge.

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3

The frequency of forecast revision: micro estimates

In this section, we propose a direct measure of the degree of attention using the individual answers to the ECB SPF survey. We show that it differs from one, i.e. that there is evidence of inattention in this sample of data. We then assess the effects of rounding and measurement errors on this result. Finally, we show that the disagreement among forecasters that do revise their forecasts is non-zero, i.e. disagreement is not entirely driven by infrequent information updating.

3.1

Measuring the degree of attention

The frequency of updating a forecast provides a measure of the extent with which agents pay attention to new information by incorporating it in their forecasts. This implicitly assumes that there is a systematic link between information and forecast updating. Such an assumption is arguably acceptable since, absent any reason for inaction — which we discuss at the end of this section — a forecast revision is observed whenever the statistical innovation of the forecaster’s model is non-zero. x x 6= fit−1,t+h ). Thanks to structure of Formally, the probability we aim at estimating is P (fit,t+h x x the ECB SPF data, we can directly compare fit,t+h and fit−1,t+h for each individual in the sample.

Thus, assuming that the probability is homogeneous across agents, the panel dimension of the SPF data allows us to deliver several empirical counterpart to this attention degree that we denote λxt (h). A first indicator relies on the “calendar horizon” forecasts and gives the probability of updating after a quarter of new information. At each date, each forecaster is surveyed about his expectations

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for the current and next calendar years, as well as, in each third and the fourth quarter vintages of the survey, about their expectations two years ahead. Therefore, for each calendar year Y , ending in quarter T , we have a sequence of 10 forecasts: two sets of forecasts made at the third and the fourth quarters of year Y − 2, and 8 sets from the first quarter of year Y − 1 onwards. We can thus build a sequence of 9 forecast revisions, for the same date T event, and the different forecast horizons h = T − t = 1, . . . , 9. The degree of attention for the calendar year ending in T and the horizon h can the be estimated using the empirical frequency: nt X x x bx (h) = 1 I(fit,T 6= fit−1,T ), λ t,cal nt i=1

x x with h = T −t = 1, · · · , 9, nt the number of respondents to the survey at date t and I(fit,T 6= fit−1,T ) x x 6= fit−1,T and 0 otherwise. an indicator function equal to 1 if fit,T

A second measure of the probability to update a forecast exploits the “rolling horizon” forecasts and gives the probability of updating a forecast after a year of new information. Consider the 8x .16 This can be compared to quarter horizon forecast released at date t − 4 by forecaster i, fit−4,t+4 x the 4-quarter horizon forecast released 4 quarters later fit,t+4 so as to define an empirical estimate

bx of the probability of a updating on a yearly basis λ t,rol =

1 nt

Pnt

x i=1 I(fit,t+4

x 6= fit−4,t+4 ), with

x x x x I(fit,t+4 6= fit−4,t+4 ) an indicator function equal to 1 if fit,t+4 6= fit−4,t+4 and 0 otherwise. By

contrast with the previous measure, the horizon is bound to 4 quarters due to the design of the bx = λ bx (4). This survey. We therefore skip the horizon index in the notation, referring to λ t,rol t,rol bx probability of updating on a yearly basis can be converted to a quarterly adjustment rate, λ t,rol,q ,

if one assumes (consistently with Mankiw & Reis (2002)’s model) that it is constant over the 4 16

For simplicity of exposition this paragraph assumes an horizon of 4 quarters, a case that is nearly relevant for inflation

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quarters of a given year. In that case, the probability of not updating over the whole year is 1 bx bx 4 (1 − λxt (4)) = (1 − λxt (1))4 so that a quarterly attention rate estimate is λ t,rol,q = [1 − (1 − λt,rol ) ]

Finally, assuming the various λxt (h) measures are constant over time and across horizons, we can bx bx or λ also recover a direct micro-data based estimates of the average attention degrees λ cal rol,q by simply taking the time average of the empirical frequencies defined above.

One concern with interpreting λxt (h) as a measure of attention is that a forecaster may choose not to revise his forecast in spite of having updated his information set. However, given the vast information set available to professional forecasters we deem unlikely that updating the information set leads to an exactly unchanged optimal forecast after one quarter. Rather, such a situation could more plausibly correspond to cases where either the forecaster chooses to avoid the cost of processing the new information by running a full statistical exercise, or to avoid the cost of communicating the new forecast outside the institution. Both situations relate to the existence of information cost, and arguably this lack of reaction to news can be characterized as (potentially optimal) inattention.17 It may also be argued that forecasters can refrain to revise their forecast for strategic motives (say because they value commitment to a scenario). However, we are not aware of a model relating infrequent forecast updating to strategic considerations, so we here stick to the sticky information interpretation. 17

The first situation we consider would be consistent with the Reis (2006a) micro-foundation for infrequent updating. The second one could be related to Alvarez, Lippi & Paciello (2011) who develop a model where firms pay an information cost to calculate the optimal reset price (i.e. implement a price review) and then decide to pay or not the usual menu cost of changing the price. By analogy, the forecaster could be seen as incurring the cost of calculating the optimal forecast but not the one of changing the optimal forecast because of the cost of communicating the reasons of this change.

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3.2

Results

Our first result is that forecasters actually fail to update systematically their forecasts on a quarterly bx (h) for the HICP variable.18 Each line in the basis. Figure 3 presents the estimate of attention λ t,cal

figure reports a sequence of probability of forecast revision pertaining to the same calendar year Y ,

bπ (h) is the proportion of forecasters revising with Y = 2001 to 2012. At each point in time, t, λ t,cal their forecast for a given target calendar year Y ending with quarter T . The associated forecasting horizon is thus h = T − t. Sequences of forecast revisions partially overlap since, in each vintage of the survey, respondents are asked their forecast for both current and next calendar year. As Figure 3 illustrates, except for a few instances, attention is not complete: depending on the dates and the forecast horizon, the probability of forecast revision varies between 30% and 100%. Table bx across horizons, dates is 72% for HICP, 75% for the unemployment 3 shows that the average λ cal

rate and 80% for GDP growth. Averaging across variables the typical degree of attention is thus bcal ≃ 75%. This is much higher than the values provided by previous empirical studies. around λ

b = 10% for monthly data, i.e. a By comparison, Mankiw et al. (2003) calibrated a value of λ b = 27% when converted to quarterly data, to reproduce the disagreement in UScorresponding λ

SPF inflation rate forecasts. Other studies based on aggregate data (e.g. Kiley (2007), or D¨opke et al. (2008)) also have found degrees of attention lower than 50%. One possible explanation of the discrepancy between our micro estimates and these previous ones is that the latter have to rely on auxiliary assumptions. The Mankiw et al. (2003) calibration exercise relies on the auxiliary assumption that disagreement is only generated by the unfrequent updating of the information set by forecasters. Macro estimates of the “sticky information” Phillips curve typically assume flexible 18

Figures for the unemployment rate and real GDP growth are not reported to save space but exhibit a similar pattern.

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prices, which will presumably bias their estimation in favor of information stickiness if prices are actually sticky. An advantage of our direct measure is that it bypasses the need for these auxiliary assumptions.19 A second result that stands out from our estimation is that the average probability to revise a forecast increases when the forecast horizon is reduced: all lines in Figure 3 are upward sloping. Two factors can explain this pattern: first, mean reversion implies that long run forecasts are close to the unconditional average of the process. So that news that lead to revising short run forecast may leave the forecast at a long horizon unchanged. Second, it may be the case that forecasters put more attention on revising their forecast for closest forecast horizons. Experiments in the next section suggest both factors are present. A third pattern is that the degree of attention varies over time. This can be seen from the average level of each curves in Figure 3. This is even clearer when one looks at the alternative attention bx , which is built on rolling horizon forecasts. Indeed, looking at Figure 4, which plots indicator, λ t,rol

its time series for the three possible forecasted variables x, illustrates that there is a significant

degree of time variation in these probabilities. By contrast, the sticky information model of Mankiw & Reis (2002) postulates that λ is constant. Observing fluctuations in aggregate attention would not be direct evidence against the sticky information model of Mankiw-Reis if they were purely bx , for random. However, fitting an AR(1) or an MA(1) process to the attention degrees, λ t,rol inflation, unemployment and real GDP shows that this is not the case: autocorrelations are found to be significantly positive.20 There is thus some persistence in the fluctuations of the attention 19

We checked that our result of an estimated degree of attention lower than one is not driven by some outliers among forecasters. The cross–section distribution of attention that we report in Appendix A.3. shows attention in larger than 0.5 for all forecasters. 20 Results are not reported to save space, but available upon request.

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degree. Figure 4 also suggests that the degree of attention was larger for all three variables in late 2008 and 2009, that is after the large shocks associated with the ‘Great Recession’. Coibion & Gorodnichenko (2012b) also find an increase in attention in the wake of large shocks such as the 9/11 attack and more generally over the US business cycle. These results could be reconciled with the more general state-dependent setup developed by Reis (2006a) and in which the length between two updates of information is optimally chosen as a function of the shocks that can occur over the interval.21 bx estimate is that forecasters do not update systematically A fourth result stemming from the λ t,rol

their forecasts even on a yearly basis. This confirms our first result that attention is not equal bx (h) estimates. It is somewhat even more striking since here some to 100% when looking at λ t,cal

forecasters choose not updating their one-year ahead forecast even after one more year of macroe-

bx , for the conomic news became available. Table 3 reports the attention degree time average, λ rol three macroeconomic variables surveyed. It is equal, respectively, to 88% for inflation, 96% for

unemployment and 94% for real GDP. Converting these average probability of forecast revision to bx , of 41%, 55% and 51%. Averaging across variables quarterly figures gives average frequencies, λ rol,q brol,q ≃ 50%. The difference between this result and the average λ bcal ≃ 75% is further x one gets λ evidence that the frequency of updating is not constant over forecast horizons. Our assessment

bcal is the most reliable indicator. Indeed, in practice, 8-quarter rolling horizon forecasts is that λ

are usually not a conventional exercise implemented by professionals. Moreover, since calendar forecasts are available for adjacent quarters, it is more likely that they compare forecasts delivered by the same forecaster in the same institution. 21 Under some restrictions, this model aggregates to a Mankiw-Reis model in which inattention can be described by a single parameter. However, these conditions would preclude non-random time variations in the probability to update one’s forecast as well as the findings that this probability to update varies with the forecasting horizon as Figure 3 illustrate.

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A fifth and final result is that forecasts are not jointly updated across variables and across horizons. The middle panel of Table 3 reports the probability that a forecaster revise his forecast of either inflation, or GDP growth, or unemployment, conditional on having revised his forecast of another variable. Such conditional probabilities of revision range from 88.2% (probability of revising inflation forecast given that real GDP forecast was changed) to 95.6% (probability of revising unemployment forecast given that inflation forecast was changed). These figures are based on revisions of rolling horizon forecasts (not converted to quarterly figures). Finally, as reported in the last row of Table 3, a forecaster virtually always revise (on a yearly basis) at least one of his forecast out of the three variables considered. The bottom panel of Table 3 also shows when the 1-year forecast of a variable is revised (on a yearly basis), the probability that the 2-year ahead forecast for the same variable is revised is always less then one. By contrast, in the basic version of a sticky-information information model, whenever a forecasters update his information set he updates the forecasts of for all variables and all future horizons. To sum up, we find that forecasters fail to update their forecasts at each period, a fact that is consistent with a sticky-information model. We also document several departures from a “pure” version of such sticky information model. In particular, and as Coibion & Gorodnichenko (2012b) also emphasize, the degree of updating varies over time, across horizons, across variables or with the nature of the forecast. We furthermore underline another dimension in which a simple version of the sticky information model is at odds with the data, namely that forecasts are not jointly updated (across variables and across horizons).

19

3.3

Robustness: the influence of rounding and of measurement errors.

Almost all forecasts (97% of forecast figures) in our data set are reported with only one digit. One caveat in the failure to revise a forecast documented above is that it may merely reflect the fact that these institutions report only rounded figures. Rounding can be considered as a form of inattention. Indeed, this practice is not formally requested by the ECB SPF questionnaire. Providing rounded figures may reflect the costs of processing and communicating precise information as we previously discussed. Still, as there exists a widespread consensus that higher order digits are not economically meaningful so that the forecasters find natural to report rounded forecasts even when they update them, it is important to gauge to what extent rounding lead to underestimate the degree of attention. To do so, we perform a Monte Carlo experiment.22 We use an estimated VAR model as a simulation and forecasting device for the (year-on-year) inflation and GDP growth rates. We then round each forecast to the first digit and compute the frequency with which two adjacent rounded forecasts corresponding to the same target date, are revised. Crucially, in this simulation exercise, there is no inattention in the underlying forecasting model. Absent any rounding, one would observe that forecasts are updated every period. So this exercise gives us an estimate of the bias to our measurement of attention that is due to rounding. Looking for example at inflation, we obtain a simulated probability of updating a forecast of 83% for the horizon h = 9 quarters, which rises to 91% or the horizon h = 1. These figures are lower than one, which suggest there is actually a rounding bias. The probability also increases when the target date gets closer to the forecast date, consistent with the pattern of Figure 3. However these figures are at all horizons markedly above the estimates of attention we 22

See Appendix A.4. for details on our experiment.

20

recover from actual SPF micro data. Computation of standard errors show that the difference is statistically significant. We thus conclude that, independently of rounding effects, there is a degree of genuine inattention in professional forecasts. Another possible concern is measurement error: if a forecast that has not been updated is reported with an error, this may lead us to spuriously conclude that the forecast has been updated. In contrast with the case of rounding, our estimator would then overestimate the probability of forecast revision. To assess the extent of such a bias, we simulate the probability of revision obtained with forecasts derived from a sticky information model with forecasters answering the survey with errors.23 More precisely, we postulate the probability of updating the information set is equal to ∗ ∗ λ0 and that forecasters report forecasts ft,t+h = ft,t+h + et where ft,t+h is there true unobserved

forecast and et is an i.i.d. normally distributed measurement error. Remark that in this set-up, the measured probability of forecast updating should always be one, given that the added error is a continuous variable. Another ingredient is thus needed to generate infrequent updating under measurement error. We choose to rely on a source of discreteness present in the actual data, namely rounding: in the exercise carried out, forecasts are generated, then contaminated by measurement errors, then rounded to 1 digit.24 The main result of the experiment (detailed in Appendix A.5) is that the bias is sizeable as long as the measurement error standard deviation is larger than around 0.05 percentage point. For instance, with a standard deviation of measurement error of 0.15 in GDP forecast, the observed frequency of forecast revision of 0.80 can be generated in a model in which the true attention parameter is only λ = 0.25. Thus, to the extent that measurement 23

See Appendix A.5. for details on our experiment. Another one option would be to consider discrete measurement errors: for instance there is a probability p that the forecast is contaminated by measurement error. However, such approach is not informative since one may b with any sticky information parameter, say λ0 (lower than rationalize any observed frequency of forecast revision λ, b by simply positing that p = λ b − λ0 . λ) 24

21

error is present, our estimate of the probability of forecast revision could be severely overestimated. However we note that this outcome if anything reinforces our finding of evidence of infrequent updating. At the same time, that measurement error is a quantitatively important concern in the case of the SPF is debatable: professional forecasters are used to report their forecast, for many of them their forecast is public, and forecasters answer the survey by filling a written questionnaire they are familiar with. This contrasts with repeated cross-sections and with telephone interviews, typical of consumer surveys, often viewed as prone to measurement errors.

3.4

Disagreement among forecasters who revise

We further exploit the specificity of the data set to document the behavior of forecasters when they update their forecasts. This leads to a second main new fact: forecasters who update disagree. We compute at each date in the sample the disagreement among forecasters that do revise their forecast. Figure 5 compares it with the disagreement of the forecasters who did not update, for the same forecast and date. The disagreement among revisers is shown on the horizontal axis and the disagreement among non-revisers is on the vertical axis. It is apparent that the disagreement among revisers differs from zero. Moreover, it is greater than the disagreement among non-revisers as the scatter is mostly above the 45 degree line. Lastly, sometimes all forecasters (or all forecasters but one) do revise, so there is zero disagreement stemming from the non-revisers. Observing disagreement among forecasters who revise their forecasts has clear-cut implications for models of information rigidity. Indeed, sticky-information models predict that forecasters who revise their information set will derive the same optimal forecast and thus should not disagree. This approach cannot explain the large degree of disagreement among forecasters updating their 22

forecast we find. By contrast, noisy-information models generate disagreement between forecasters who update since every of them has a specific information due to the heterogenous signals on the true state they receive.

4

A model of inattention: quantitative assessment

The facts reported in the previous section suggest that modeling agents’ expectations requires both types of inattention: agents infrequently update their information sets and when they do, they get a noisy perception of the true information.25 In this section, we develop a hybrid model featuring both sticky and noisy information and assess its empirical performances through a formal testing procedure.

4.1

4.1.1

A sticky and noisy information model

DGP and information structure

We assume that the economy can be summarized by a state vector Zt made of p lags of n (centered) variables Xt with associated innovation ǫt . Its dynamics is described by a reduced form invertible VAR(p) model which can be written in the compact companion form: Zt = F Zt−1 + ηt , with ′ ′ · · · Xt−p+1 )′ , and where ηt = (ǫ′t 0 · · · 0)′ has a covariance matrix Ση . Zt = (Xt′ Xt−1

Let i be an individual in the population of forecasters, i = 1, . . . , m. Along the lines of Mankiw & Reis (2002), at each date, every forecaster may update his information set, with a constant 25 This approach shares some features with the price-setting model of Woodford (2009). In this setup, firms have to choose on implementing a price review knowing that they will get a noisy information on the true state of the economy. This leads to optimal non-systematic price review and, when it happens, to reset prices that are determined conditional on the noisy information.

23

probability, or degree of attention, λ. We index by j the generation of forecasters who last updated their information set j periods before the current one, i.e. in t − j. We extend Mankiw and Reis’ model to include imperfect perception of the information when updating. We assume that, when an agent i updates his information, he observes a noisy perception of the true state, Zt , namely a signal Yit that follows Yit = H ′ Zt + vit ,

vit ∼ iid(0, Σv ),

where H is a matrix that selects the state variables that are observed with such idiosyncratic noise and which we assume to be H = I 26 and with Σv a diagonal matrix. This idiosyncratic noise conveys the notion that forecasters pay attention to private information to assess current macroeconomic conditions. In particular, respondents to the ECB SPF are located in various European countries and thus likely to have a more precise and timely access to news about their national economy. In addition, a large share of respondents are forecast units in private banks that may have access to specific contacts with businesses.

4.1.2

Model properties

We now summarize some properties of the above hybrid sticky/noisy information model. We refer the reader to Appendix A.6 for details on their derivation as well as illustrating simulations. We focus on how the model can reproduce the empirical regularities documented in the previous section, namely predictable forecast errors and disagreement among forecasters updating their information, emphasizing their link with the two inattention parameters, λ and Σv . 26

In this typical case, forecasters have noisy perceptions of all the state variables.

24

Characterizing the average forecast errors

Let fit−j,t+h denotes the optimal forecast of the

vector X at date t + h made by an agent i within a given generation j using the information vintage t − j. In this generation, every forecaster i observes his own noisy signal and his optimal forecast is

!
derived from the Kalman recursion: fit−j,t+h = F h+j Zit−j|t−j−1 + F h+j Gt−j Yit−j − Zit−j|t−j−1 . Matrix Gt−j is the Kalman gain which depends on the variance of the noise in the signal Σv !
−1 as follows: Gt−j = Pt−j|t−j−1 Pt−j|t−j−1 + Σv , where Pt−j|t−j−1 denotes the variance of the perceived forecast error. The average forecast error involves the average of individual forecasts across the different individuals i in different generations j. Using, inter alia, that, with λ the frequency of updating, each generation j has a weight of λ(1 − λ)j in the whole population of forecasters, one shows that, in the simple case where H = I, the average forecast error follows:

Et,t+h = F h−1 (I − Gt )

∞ X j=0

h−1 ! 
 X (1 − λ)j λ Zt − Ei Zit|t−j |j + F h−k ηt+k+1 .

(1)

k=0

with Ei (·|j) the average across individuals i in generation j. Under imperfect information, i.e. when I 6= Gt and/or λ < 1, the average forecast error is predictable with respect to the true state Zt , since E (Et,t+h |Zt ) 6= 0.

Properties of the average forecast errors

An increase in the degree of information imperfec-

tion generates more persistence and variance of the forecast errors.27 More specifically, everything else being constant, (i) a decrease in the attention degree λ increases both the persistence and the variance of the forecast error. Moreover, (ii) an increase in a term on the first diagonal of the noise 27

See Appendix A.6 for details.

25

variance matrix Σv leads both to an increase in persistence and variance of the forecast error of the corresponding forecasted variable.

Characterizing the disagreement between forecasters

Let Vij (fit−j,t+h ) be the total cross-

section variance of point forecasts across individuals i using the different information vintages j. This cross-section variance has two sources. The first one stems from differences of opinions within a given generation j of forecasters generated by the noise in individuals’ signal. For a given generation j, one can show that it follows



  !  ! Vi (fit−j,t+h |j) = F h+j Vi Zit−j|t−j−1 |j + Gt Σv + Vi Zit−j|t−j−1 |j G′t (F h+j )′ . with Vi (·|j) the variance across individuals i in a generation j.

(2)

This cross section variance

evolves with the forecast horizon, shrinking progressively to zero with h. The contribution of this disagreement within a generation to the total is given by the share of this generation in the whole population of forecasters, λ(1 − λ)j , in the total population, so that Ej {Vi (fit−j,t+h |j)} = P∞

j=0 (1 − λ)

j λV (f i it−j,t+h |j).

The second source in total disagreement comes from the differences

in (average) opinion between the different generations j of forecasters using different information vintages. More precisely, we have

Vj {Ei (fit−j,t+h |j)} =

∞ X j=0

(1 − λ)j λ {Ei (fit−j,t+h |j) − Ej [Ei (fit−j,t+h |j)]}2 .

(3)

Noticeably, due to idiosyncratic noisy signals within generations, the model generates disagreement even under full information updating λ = 1. Another important feature is that disagreement is time-varying, even when there is no time conditional heteroscedasticity in the noise, vit . This comes 26

from the differences between generations of forecasters: as the degree of disagreement depends on the difference between the new vintage of information and the previous ones, when an innovation is large compared to the average, the difference of opinion between the individuals revising and the others will also be larger than on average.

Properties of the disagreement An increase in the degree of imperfect information has ambiguous effects on disagreement.28 Everything else being constant, (i) a decrease in λ has two conflicting effects on the disagreement between generations of forecasters. It entails a greater heterogeneity between the generations, since it increases the length between information updating. However, it also increases the share of generations sticking to an old information vintage and for which the forecast horizon (h + j) is very long. Within these generations, the disagreement tends to zero as, whatsoever its idiosyncratic signal, every forecaster’s optimal prediction is given by the unconditional mean of the process. Likewise, everything else being constant, (ii) an increase in any diagonal element of Σv has also two conflicting effects on the disagreement within a generation of forecasters. It increases the amount of noise, thus differences of opinion within each generation of forecasters. On the other hand, because individuals know that the signal is very imprecise, they incorporate less the news into their forecast. In the extreme case, when the signal is completely uninformative, the optimal forecast is the unconditional mean of the process for all forecasters, which implies zero disagreement. Lastly, everything else being constant, (iii) a drop in λ leads to a higher time variance of disagreement, since it increases the difference between the new vintage of information and the previous one, hence the impact of the size of the shock on disagreement. By contrast, (iv) an increase in any 28

See Appendix A.6 for details.

27

diagonal element of Σv implies that this time variance of disagreement decays. The less informative the news, the less they are incorporated in the optimal forecast, therefore the less disagreement there is between generations of forecasters.

4.2

Estimation procedure

We estimate the previous model by a Minimum Distance Estimation (MDE) procedure.29 The methodology amounts to minimizing the distance between first a vector of chosen K data-moments µ b, such as, in our case, the average forecast error or the disagreement computed from the SPF

data, and, second corresponding model-generated moments which, in our case, are a function of the inattention parameters µ(λ, Σv ). Formally, estimates of λ and Σv are obtained by minimizing:

b −1 [b µ − µ(λ, Σv )] , [b µ − µ(λ, Σv )]′ Ω b is a consistent estimator of the asymptotic variance of µ where Ω b defined by

√

T (b µ − µ) → N (0, Ω)

when T → ∞. Stacking the parameters of interest into θ = (λ, vec(Σv ))′ , the minimum distance  √    estimator θb satisfies T θb − θ → N 0, (D′ Ω−1 D)−1 when T → ∞, with D ≡ D(θ) = ▽θ µ(θ)

b An estimator of the standard deviation of θb the Jacobian of µ(θ) with respect to θ evaluated at θ.

b b −1 D b where D b ≡ D(θ). b ′Ω is thus given by D

In addition to estimating parameters, MDE also allow for testing over-identifying restrictions, i.e. the null hypothesis that the set of K moments µ can accurately be described with the P parameters b −1 [b to be estimated, i.e. in our case λ and Σv . The test statistic is [b µ − µ(λ, Σv )]′ Ω µ − µ(λ, Σv )]. Here, rather than relying on the asymptotic Chi-square distribution, we use Monte Carlo simulations 29

See e.g. Section 14.6 in Wooldridge (2002) for a description of this method.

28

of the model to approximate the exact small sample distribution of the test statistics. To estimate the model, we select moments related to the forecast errors and disagreement. These are two key features of expectations that imperfect information models try to rationalize and which, in our model, are functions of the attention rate and the variance of the noise parameters. More precisely, we select two moments related to the average forecast errors: the mean square of forecast errors (MSE), E[(ext,t+h )2 ] and its first-order autocorrelation, ρxe (1); and two moments associated with disagreement: its average level, E(σt,h ) and its time variance, V(σt,h ). This latter moment is particularly crucial in order to entail the econometric identification of the model. Indeed an increase in both types of inattention (lower λ, higher diagonal components of Σv ) often impacts the first three moments in the same direction. By contrast, a lower frequency of updating increases the time variance of disagreement while noisier information lowers it. The estimation involves four steps. First, we estimate as an auxiliary model, a VAR model for the year-on-year inflation rate, change in unemployment rate, and real GDP growth. The VAR includes 4 lags of the vector of observations X = (π

u

∆y)′ with all variables centered prior to

estimation. In our baseline estimation, we use quarterly euro area data over the 1987Q1 to 2008Q2 period.30 The choice of the sample period reflects a trade-off between conflicting objectives: first, having a sufficiently long time series and second, considering an homogenous period in terms of monetary policy and the inflation regime. Second, taking the VAR parameters as given, and for a given set of structural parameters (λ, Σv ), we simulate the targeted moments using the Kalman filter and the hybrid model. Third, we compute the distance between these simulated moments and the actual ones. Fourth, we minimize the objective function over the space of parameters using a 30

Data sources are detailed in Appendix A.1. We also consider a larger sample estimation to 2012Q4

29

numerical routine.

4.3

Results

Table 4 gives the estimation results obtained for a baseline specification of the sticky/noisy information model in which the variance of the measurement error is constrained to be the same for inflation, unemployment and output growth, i.e. Σv = σv2 I3 . Those results are qualitatively robust to more flexible versions of the estimated model.31 We first consider the results obtained when trying to match the moments of inflation over a precrisis sample. Column 1 provides our baseline case. One main result is that the estimated value for b = .340, is well below the micro estimate of around .75 obtained from the the attention degree, λ micro data. Thus, along this first dimension, the model is at variance with the SPF micro facts.32

The measurement noise is σv = .325, which represents about one half of the inflation variance over this period. The tests for over-identifying restrictions rejects the null that the distance between the estimated moments and the observed ones is zero, with a p−value of .018. We assess how the results change when one tries to match the moments of real GDP (Column 2) or of the two variables jointly (Column 3). Compared to the baseline case, the results in Column b = .352 and a larger degree of noise (2) present a very similar estimate for the attention degree λ σ bv = .785. Noticeably, the over-identifying restrictions are not rejected by the data (the p−value

of the test is .956). Still this is at the cost of a frequency of updating much lower than in the micro 31

See Appendix A.7. for details on robustness checks including, noise variances that differ across variables, systematic bias in errors and recursive estimation of the auxiliary VAR. 32 By analogy with Carvalho (2006) in the case of the frequency of price adjustment, our estimation of the average attention rate may be prone to an aggregation effect if forecasters have heterogeneous attention rates. However, the lowest individual average frequency of updating is equal to 50%. Aggregation bias issues cannot thus not close the gap between the MDE results and the micro data estimates.

30

data. Trying to match the moments of the two variables together (Column 3)we obtain an even b = .180, and a standard deviation of the lower attention degree than in the two previous cases, λ idiosyncratic noise equal to σ bv = .832. The estimated value for λ is thus even farther than our micro-data estimate and the model is rejected by the formal test at the 10% level, with a p−value

of .059. We finally assess how the crisis affects the results. Column (4) presents the results when estimating the model jointly on inflation and GDP moments, including the crisis period. The attention degree b = .060, and the noise variance is substantially larger (b becomes λ σv = 1.193). A larger degree

of information imperfection is needed in order to match the formation of expectation during the crisis. This conveys the fact that the adjustment of expectations was moderate during the crisis and that at the same time disagreement spiralled up. Interestingly, this seems at odds with the fact

documented in Section 3.2 that forecasters adjusted more the forecasts than usual over the 2008-09 period. However, it can be that at the same time, λ increases and forecast stay unreactive if the signal becomes very noisy. However, and as the over-identifying test makes clear, such parameter estimates cannot account for the whole set of moments we try to replicate. Overall, these estimates and tests suggest that our flexible noisy information/sticky information model fails to quantitatively fit the data. Table 5 illustrates what lies behind the formal rejection of the model. It provides a comparison of the value of targeted moments in the SPF data for inflation (Column 1) with model based moments of inflation obtained under different values of the parameters λ and σv . Column (2) reports model-based moments obtained with the estimated values of parameters. The mean squared inflation forecast error generated by the model is strikingly lower than that observed in the data. The over-identifying restriction test indicates that the differences

31

in moments are significant. The experiment reported in Column (3) further illustrates that the model is at odds with micro data moments by looking at the model-based moments when keeping σv at its estimated value, but setting the attention degree λ to the value observed in the micro data λ = .75. The mean squared forecast error generated by this degree of attention becomes even lower. Moreover, the autocorrelation of the error, the level of disagreement about future inflation and, more strikingly, its time-variance are lower compared to the SPF data moments. In Columns 4 and 5, we assess to what extent an increase in the degree of imperfect information increases the model based mean squared error. In Column 4, we report results obtained with a higher variance of the noise σv = 1.2, i.e. that is four time the values from baseline estimate, keeping λ unchanged. As expected, the MSE of forecasts increases –to .336– and gets closer to the data values. The intuition suggests that increasing σv would also increase disagreement because each forecaster typically gets a fuzzier signal of the state variable. However, another mechanism counteracts this effect: with a higher variance of the noise, forecasters tend to shrink more their estimate of the state variable and put less emphasis on the signal they perceive (see Appendix A.6 for further discussion of model properties). It turns out that, in the present case, the latter effect dominates. Indeed, disagreement decreases from .223 under MDE estimates to .144. Another gap with the data that emerges which such a high variance of the noise is that it lowers too much the time-variance of disagreement, whereas it was close to the data in the estimated values of Columns (2). Alternatively, in Column 5, we reports results obtained with a lower degree of attention λ = .05, keeping σv unchanged. With this very low degree of the attention rate, the MSE gets closer to the data (but still below with .411) while the time variance of disagreement is extremely high compared to the micro data.

32

To summarize, the sticky/noisy information model is rejected by our quantitative analysis because whenever parameters fit the level and time-variation in disagreement in the data they then fail to generate enough variance and persistence in forecast errors.

5

Conclusion

In this paper, we analyze an original data set, the European SPF, to characterize the formation of expectations with a particular emphasis on providing micro facts relating to the sticky and noisy imperfect information models recently introduced in the macroeconomic literature. In particular, we provide an estimate for the degree of attention based on individual observations, and find that professionals are, albeit mildly, inattentive. We also document that, to a large extent, the disagreement among forecasters is similar whether they revise their forecasts or not. A formal test rejects a model of expectations featuring both sticky and noisy information. Indeed, this model is not able to account for the strong persistence of the forecast errors together with the relatively low level of disagreement and its variability over time observed in the data. There is more stickiness in experts’ expectations than the one the mere inattention is able to generate. Several avenues for future research are worth considering. More elaborate versions of inattention models could be investigated. One could for example consider a degree of attention that varies across individuals and over forecasting horizons, two features of the SPF data we highlight. One could also introduce noisy signals that are common to every forecasters, as a way to account for the low relatively low disagreement in the data. Moreover, and beyond the mere framework of inattention theories, another avenue is to investigate whether alternative forms of deviations from

33

the perfect information rational expectation setup, for instance model uncertainty or strategic interactions between forecasters, provide a better match of the empirical patterns documented in this paper.

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[15] Mankiw, N. G. and Reis, R. (2002). Sticky information versus sticky prices: a proposal to replace the New-Keynesian Phillips curve. Quarterly Journal of Economics, 117:1295–328. [16] Mankiw, N. G., Reis, R. and Wolfers, J. (2003). Disagreement about inflation expectations. NBER Macoeconomic Annuals, 209–247. [17] Mankiw, N. G. and Reis, R. (2011). Imperfect information and aggregate supply. Handbook of Monetary Economics vol. 3A Chapter 5, 183–230 B. Friedman and M. Woodford (eds), Elsevier-North Holland [18] Patton A. and Timmermann, A. (2010) Why do forecasters disagree? Lessons from the term structure of cross-sectional dispersion. Journal of Monetary Economics, 57:803–820. [19] Pesaran, M. H. and Weale, M. (2006). Survey Expectations. Handbook of Economic Forecasting. C. Elliot, C.W.J. Granger and A. Timmermann eds. North-Holland Press. [20] Phelps, E. S. (1968). Money-wage dynamics and labor market equilibrium. Journal of Political Economy, 76:678–711. [21] Reis, R. (2006a). Inattentive producers. Review of Economic Studies, 73:793–821. [22] Reis, R. (2006b). Inattentive consumers. Journal of Monetary Economics, 53:1761–1800. [23] Sarte, P.D. (2010). When is sticky information more information? mimeo, FRB Richmond. [24] Sims, C. (2003). Implications of rational inattention. Journal of Monetary Economics, 50:665– 690. [25] Veldkamp, L. (2011). Information choice in macroeconomics and finance. Princeton University Press. [26] Woodford, M. (2002). Imperfect common knowledge and the effect of monetary policy. Knowledge, Information and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps. Aghion, P., Frydman, R., Stiglitz, J. and Woodford, M. eds. Princeton University Press. [27] Woodford, M. (2009). Information-constrained state-dependent pricing. Journal of Monetary Economics, 56:S100–24. [28] Wooldridge, J. (2002). The econometrics of individual and panel data. MIT press.

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Average forecast errors, e Sample period: 2000Q1–2012Q4 HICP UNEM a. descriptive statistics Mean(e) .42 .15 p Mean(e2 ) .90 .84 ρe (1) .741 .914 b. tests Bias: e = α + u α b .426 *** (.117)

.154

(.199)

Efficiency: e = α + βe−h + u .447 * βb −.370 (.453)

(.312)

RGDP −.26 1.57 .849

−.258 (.383)

.522 **

(.227)

Table 1: Average forecast errors – Descriptive statistics and tests The Table provides descriptive statistics and tests for the 1-year ahead average (across forecasters) forecast error (e = x−f ) for different macroeconomic variables observed in the European SPF data. The variables that are forecasted are the euro area Harmonized Index of Consumer Prices year-on-year inflation rate (HICP), the euro area unemployment rate (UNEM) and the euro area real GDP year-on-year growth rate p (RGDP). All variables are in %. Mean(e) denotes the mean of the average forecast errors across dates, and Mean(e2 ) the Root Mean Square Error of the average forecast. ρe (h) stands for the h−order autocorrelation of the error. Numbers in brackets are standard errors of estimates using a robust Newey-West procedure. (***), (**), and (*) indicates significance respectively at the 2.5%, 5% and 10% levels.

36

Disagreement, σ Sample period: 1999Q1–2012Q4 HICP UNEM a. descriptive statistics Mean(σ) .29 .30 Std-Dev(σ) .08 .11

RGDP .38 .16

b. bivariate regressions, σ on (∆x−1 )2 .078 ** .669 .049 * (e−1 )2 (∆f )2

(.047)

(1.224)

.031 **

.055 **

(.018)

.145

(.034)

.876

(.558)

(1.158)

(.036)

.007

(.006)

.141 ***

(.053)

Table 2: Disagreement among forecasters – Descriptive statistics and bivariate regressions The Table presents statistics and regression results for the disagreement across forecasters (σ) on their 1-year ahead forecast (f ) for various macroeconomic variables (x) observed in the European SPF data. Disagreement is measured as the cross-section standard deviation of individual point forecasts at each date of the sample. The variables forecasted are the euro area Harmonized Index of Consumer Prices year-on-year inflation rate (HICP), the euro area unemployment rate (UNEM) and the euro area real GDP year-on-year growth rate (RGDP). All variables are expressed in %.Mean(σ) and Std-Dev(σ) denote, respectively, the average and the standard deviation of the disagreement over time. ∆x−1 is the last period change in the forecasted variable, ∆f the current change in the average forecast, and e−1 the last period average forecast error. All regressions include a constant term and numbers in brackets are standard errors of estimates using a robust Newey-West procedure. (***), (**), and (*) indicates significance respectively at the 2.5%, 5% and 10% levels.

37

Probability of updating a forecast, λ Sample period: 1999Q1–2012Q4

HICP

x UNEM

RGDP

.72 .88 .41

.75 .96 .55

.80 .94 .51

1 .956 .940

.884 1 .941

.882 .951 1

.876

.851

.924 1

revising f x Mean(λcal ) Mean(λrol ) Mean(λrol,q ) revising f y if revising f x y

HICP UNEM RGDP

others Rev. next year if rev. current year Rev. at least one macro variable Table 3: Probability of updating a forecast The Table reports different measures of the probability (λ) of updating a forecast (f ) for various macroeconomic variables (x) as observed in the European SPF data. The variables forecasted are the euro area Harmonized Index of Consumer Prices year-on-year inflation rate (HICP), the euro area unemployment rate (UNEM) and the euro area real GDP year-on-year growth rate (RGDP). λcal is the frequency of revising a calendar horizon forecast between two subsequent quarters. λrol is the frequency of revising a rolling horizon forecast between two subsequent years. λrol,q derives a frequency of updating a forecast on a quarterly basis from the previous λrol .

38

MDE – Results (1) specification Sample Pre-Crisis Moments INF

(2)

(3)

(4)

Pre-Crisis GDP

Pre-Crisis INF/GDP

Incl.-crisis INF/GDP

parameter estimates b λ .340 (.629)

.352

(.081)

.180

.060

(.016)

(.352)

(.011)

(.006)

.832

1.193

specification test (j-stat) p-value .018 .956

.059

.006

σ bv

.325

(.099)

.785

(.151)

Table 4: Minimum Distance Estimation The Table gives the Minimum Distance Estimation results of the two parameters characterizing the b is the estimated frequency sticky/noisy expectation model described in Section 4. More specifically, λ of updating one’s information set and σ b the estimated variance of the noise in the signal. Targeted moments are the mean squared forecast error Mean(e2t ), the first-order autocorrelation of the forecast error ρe (1), the average disagreement Mean(σt ), and the time variance of the disagreement V(σt ), of either the euro area inflation rate (INF), or the euro area real GDP growth rate (GDP), or both. All moments are calculated with 1-year horizon forecasts. Two samples are considered: a pre-crisis sample, that goes from 1999Q1 to 2008Q3, and a sample including the Great Recession, that covers 1999Q1 to 2012Q4. p-values of the J-stat (over-identification test) are obtained by Monte-Carlo simulations.

Moments under various configurations inflation, π pre-crisis sample (1) (2) (3) (4) b b Data λ = λ λ = .75 λ=λ σ=σ b σ=σ b σ = 1.2 Mean(e2t ) .555 .165 .134 .336 ρe (1) .799 .739 .656 .883 Mean(σt ) .250 .223 .170 .144 V(10 × σt ) .248 .204 .014 .021

(5) λ = .05 σ=σ b .411 .897 .232 1.109

Table 5: Comparing data and model-based moments The Table compares the targeted MDE moments for euro area HICP inflation, π, observed in the SPF data (column (1)), obtained with the MDE parameter estimates (column (2)), or under alternate parameters values (columns (3) to (5)). Moments are the 1-year ahead mean squared forecast error Mean(e2t ), the firstorder autocorrelation of the forecast error ρe (1), the average disagreement Mean(σt ), and the time variance of the disagreement V(σt ).

39

40

The Figure shows the average of individual 1-year ahead point forecasts for the three different target variables (solid line) — inflation (HICP), unemployment rate (UNEM) and real GDP growth rate (RGDP) — together with their realized values (dashed line) over the 2000Q1–2012Q4 sample.

Figure 1: Average of (1Y ahead) forecasts and realizations.

Figure 2: Time series of disagreement among forecasters. The Figure presents the cross-sectional standard deviation of individual 1-year ahead mean point forecasts for the three different target variables (solid line) — inflation (HICP), unemployment rate (UNEM) and real GDP growth rate (RGDP). The sample is 1999Q1–2012Q4.

41

Figure 3: Probability of updating after one quarter of information – Calendar horizon forecasts. The Figure reports the probability of updating a calendar (end-of-year) inflation forecast between two subsequent quarters as estimated from the SPF data. Each line corresponds to a given year in the sample. Each point on a given line corresponds to a date at which forecasters are asked to report their forecast for the end of the associated year. Hence, each line is made of a sequence of probabilities to update a given end-of-year forecast and is thus associated to a sequence of (decreasing) forecast horizons.

42

Figure 4: Probability of updating after one year of information – Rolling horizon forecasts. The Figure shows the probability of updating a rolling (1-year ahead) forecast between two subsequent years as estimated from the SPF data. For each individual and each date in the sample, a revision of a forecast is derived by comparing its current 1-year ahead rolling forecast with the 1-year in 1-year ahead rolling forecast observed 1 year before the current date.

43

Figure 5: Disagreement among non-revisers vs. among revisers. The Figure plots, for each date in the sample, the cross-sectional standard deviation of individual 1-year ahead mean point forecasts among the subpopulation of forecasters who do not update their forecast against the same cross-sectional standard deviation obtained for the subpopulation of forecasters who update their forecast. Forecast revisions are revisions of a 1-year ahead forecast on a yearly basis. The sample is 1999Q1– 2012Q4.

44

Inattentive professional forecasters Appendix: Supplemental material not intended for publication Philippe Andrade

Herv´e Le Bihan

Banque de France & CREM

Banque de France

April 2013

1

A.1

Data sources

SPF individual data.

These data are available from the ECB website:

http://www.ecb.int/stats/prices/indic/forecast/html/index.en.html

Macroeconomic data.

We compare the individual forecasts with realizations of inflation, un-

employment and real gross domestic product. The series we use are chosen to be consistent with information required from respondents to the SPF. The source is Eurostat (some of the series have been downloaded from the ECB website). More precisely

• Inflation is defined on the basis of the Harmonized Index of Consumer Prices (HICP). We use the series ICP.M.U 2.N.000000.4.IN X of the ECB’s Statistical Data Warehouse database. To be consistent with the SPF questionnaire design, we chose as the inflation rate the yearon-year percentage change in the HICP observed at the month for which the forecasters are ask to forecast inflation. • Unemployment rate is measured using the standardized ESA definition produced by Eurostat (in percentage of the labor force). We use the series ST S.M.U 2.U N EH.RT T 000.4.000 of the ECB’s Statistical Data Warehouse database. • Real gross domestic product is defined using the standardized ESA definition of GDP. We use the series N AM Q GDP K from the Eurostat database.

We rely on pre-1999 back-casted series using the ECB’s AWM database (see Fagan, Henry & Mestre, 2001) when estimating a VAR model of the three variables and generating forecasts from such model. This gives us a full sample of macro-variable that starts in 1970Q1. Note that the back-casted HICP index series is available at a quarterly frequency, so that the inflation rate we rely on in our VAR modeling exercises is needed is a year-on-year inflation rate based on quarterly

2

indices. We checked that the gap with the year-on-year monthly inflation rate observed with a timing consistent with the one of the survey is minor. Another important point is that we use the “changing composition” of the euro-area HICP, unemployment and RGDP series. The “changing composition” definition is consistent with the fact that at each date, SPF respondents are asked to forecast the euro area variable given its current composition. We recall that the euro area, which has 17 countries at the time of writing this paper, was initially made of 11 countries. It grew up to 12 countries after Greece joined in January 2001. Subsequent enlargements happened in January of 2007 (Slovenia), 2008 (Cyprus, Malta), 2009 (Slovakia) and 2011 (Estonia). A drawback of the changing composition series is that they exhibit level shifts at the date new countries joined, which reflects a pure size effect. This shift can be of significant order, in particular for GDP, with for instance, the entrance of Greece accounting for a shift of nearly 3% in the EA real GDP growth rate in 2001. However, forecasters do not incorporate these effects in their forecasts of inflation or real GDP growth. We thus correct for such level shifts in the real GDP growth rate series by replacing it by the “fixed composition” rate of change for each of the 5 quarters in which there were an enlargement of the euro area.

3

A.2

The cross-section dispersion of forecasts: an illustration

As Figure 2 in the main text makes clear, the cross-section distribution of individual point forecasts is never single peaked. There is always some heterogeneity in these forecasts, i.e. disagreement is non-zero. The histograms in Figure A-1 below illustrates this point further by looking at the cross-section distribution of individual forecasts, for each variable, and at a specific date in the sample. We picked up 2009-Q2, which is a date that closely follows the big shock of the Great Recession. We notice that the distribution of individual forecasts is spread-out on a large range of values (in line with the spike in disagreement observed for the same date on the time series of disagreement presented in Section 2.2 of the main text. Furthermore, it is remarkable that the distributions exhibit several modes, consistent with the findings of Mankiw et al. (2003).

Figure A-1: Disagreement among forecasters in 2009-Q2 The Figure displays the cross-section distribution of individual 1-year ahead point forecasts observed in 2009-Q2 for the three different target variables: inflation (HICP), unemployment rate (UNEM) and real GDP growth rate (RGDP).

4

A.3

The cross-section distribution of attention

We compute the average frequency of updating a (1-year) forecast on a yearly basis for each forecasters in the sample. The Figure A-2 below shows the cross-section distribution of the these individual average frequencies. The three histograms underline that the non-zero probability of updating that we obtain is not driven by just a few forecasters who would update very infrequently. While there is a large mass of forecasters who update systematically on a yearly basis, there is also a substantial mass of forecasters who do not. Moreover, the figure underlines some heterogeneity in the degree of attention in the latter subpopulation.

Figure A-2: The cross section distribution of the frequency of updating a forecast The Figure plots the cross-section distribution of the individual average frequency of updating a 1-year ahead forecast at a yearly frequency for each of the three target variables: inflation (HICP), unemployment rate (UNEM) and real GDP growth rate (RGDP).

5

A.4

Assessing the influence of rounding: details

We conduct a quantitative assessment of the influence of rounding on our estimation results by a simulation experiment. We compute the frequency of updating a forecast that would be empirically observed if forecasters were to use a model without any information rigidity but rounded their optimal forecasts to the first digit. We rely on an estimated VAR model of the joint dynamics of the euro area inflation, unemployment and real GDP growth rates as a simulation and forecasting device, drawing shocks from a multivariate distribution with a covariance matrix equal to that of the estimated innovations. As in Section 4, the VAR is made of the year-on-year changes in HICP, in the unemployment rate and in the real RGDP and has a constant term and 4 lags. Using this estimated VAR model as a DGP, we simulate S = 1000 draws of artificial observations of length Tsim = 200 which roughly corresponds to the sample size on which the VAR is estimated. For each draws, we discard the first 50 simulated data, we then re-estimate the VAR, and we generate, for each date in the simulated sample, a sequence of optimal forecasts, up to a 10 quarter forecast horizon, derived to the estimation. We then round each forecast to the first digit. Using the data set of rounded forecasts, we compute the probability that two adjacent forecasts, corresponding to the same target date, are different. For a given draw, we are able to compute an average quarterly frequency of forecast revision associated to each horizon h = 1 to h = 9. We can compute the same average frequency of updating derived from rounded VAR forecasts for each draw S. So, we eventually obtain a whole distribution of these average frequencies of updating a forecast implied by the mere rounding, the variance of which conveys the extent of sample uncertainty in the simulation exercise. Crucially, in our simulation exercise, inattention is absent from the forecasting model: absent rounding effects, and in line with the VAR model implied forecasts, we would observe that actual forecasts are updated at every period and for every horizon with probability one. The probability of not updating the forecast is here an estimate of the bias to our measurement of attention that is due

6

to the mere rounding. The magnitude of this bias will obviously depend on several parameters of the exercise: the horizon considered (we expect more bias at longer horizons due to mean reversion), the size of the innovation variance (larger shock will imply lower bias since forecast revision will be too large to be wiped out by rounding) and the persistence of the process (low persistence of shocks will imply fast mean reversion thus less forecast revision).

Figure A-3: The effect of rounding on the estimated probability of updating – Calendar horizon (HICP) forecasts. The Figure compares, for a set of forecasting horizons, the probability of updating (on a quarterly basis) inflation (HICP) forecasts derived from the SPF data with the one obtained by simulations of forecasts generated from a VAR model and rounded to the first digit (see Section A.4 for details). The dark solid line corresponds to the mean in the distribution of the simulated probabilities, while the grey dashed lines show the associated 10% and the 90% quantiles.

Figure (A-3) illustrates the results obtained for HICP inflation.1 It allows a comparison of the frequency of updating a forecast, for the horizons h = 9 to h = 1, derived from the micro SPF data with the one implied by the pure rounding effect simulation exercise. We provide both the 1

We obtain comparable results for unemployment and RGDP.

7

average of the model implied frequencies of updating together with the 90% and 10% quantiles in the simulated distribution. The estimated probability of updating a forecast is 83% for the horizon h = 9 quarters and rises to 91% for the horizon h = 1. These figures are lower than one, which suggest there is actually a substantial rounding bias. The probability also increases when the target date gets closer, hence the forecast horizon shrinks to zero, a pattern that is present in the micro data as discussed in Section 3.2 of the main text and as the Figure also illustrate. However, for all horizons, and as Figure (A-3) makes clear, the simulated probabilities of updating due to rounding are markedly above the estimates of attention we recover from actual SPF micro data, which are below the 10% quantile. Thus, independently of rounding effects, there is a degree of supplementary inattention in professional forecasts.

8

A.5

Assessing the influence of measurement errors: details

Our estimation of the probability to update one’s forecast might be biased upward by the presence of measurement errors. Indeed, it can be that a forecaster who did not update her/his forecast for the same event between two subsequent quarters reported a different number when answering the survey, hence leading to a spurious observation of a forecast revision. In this section, we provide an assessment of such an upward bias in the frequency of updating a forecast. More precisely, we consider that the data generating process of the joint dynamics of the euro area inflation, unemployment and real GDP growth rates is the same VAR model that the one we rely on in the MDE and in the assessment of rounding exercise. We draw a realization of size Tsim = 150 of this DGP. We then calculate optimal forecast series for inflation, unemployment and real GDP for each date in this sample (and for a given horizon of 4 quarters). From this sequence of optimal forecast, we also generate for each date nsim = 50 individual forecasts (of inflation, unemployment and real GDP) implied by a sticky information model with a probability of updating λ0 . Moreover, we add to each individual forecasts a measurement error eit with σe2 . Individual forecasts are then rounded to the first digit.2 The observed forecast is then i h x x fit|t+h = round (fit|t+h )⋆ + eit x x x with eit ∼ iidN (0, σu2 ), and (fit|t+h )⋆ = EL (xt+h |Xt ) with probability λ0 or (fit|t+h )⋆ = (fit−1|t+h )⋆

with probability 1 − λ0 . We replicate this experiment for a number of S = 1000 replications. We then recover estimates of b using these simulated individual observed forecasts that are the frequency of updating a forecast λ

associated to the postulated frequency λ0 and standard deviation of the measurement error, σe ,   b {f x namely λobs = λ }; λ , σ 0 e . it|t+h 2

Note that absent an effect of rounding, any measurement error in SPF data will imply that observed individual observed forecast are always updated with probability one.

9

Figure A-4: The estimated probability of updating under measurement errors. The Figure shows the estimated probability of updating a forecast as a function of the standard deviation of measurement errors in the answers to the surveys. More precisely, the vertical axis gives the value of our estimator when applied to simulated forecasts generated by a VAR model with measurement error and when the forecasts are rounded to the first digit. The horizontal axis reports the standard deviation of the measurement error in the simulations. The black dotted line gives the average probability of updating estimate on the SPF data. Forecasts are average calendar real GDP growth rate forecasts.

Figure A-4 presents the results obtained for the real GDP growth rate under different values of λ0 and of σe . Due to the order of rounding (first digit) a measurement error with a standard deviation of .1 adds about .5 in the observed frequency of updating. This order of magnitude is thus potentially sufficient to explain the gap between the frequency of updating recovered from micro data (.8 for RGDP) with the MDE estimates needed in order to replicate the moments of the SPF data (.35 for RGDP). We now briefly discuss the implication of measurement errors on the average forecast error and the disagreement across forecasters. We first note, as it is symmetrically distributed around zero, our “classical” measurement error will not affect the average forecast. By contrast, since it is is

10

non-correlated with the true unobserved individual forecasts, measurement error will increase the cross-section variance of point forecasts, i.e. the disagreement across forecasters. Hence, including measurement error in the model would affect our MDE estimation results. We deem the full estimation of such a model to be beyond the scope of this paper. However, we remark that measurement errors will reduce the level of disagreement associated to the mere imperfect information mechanisms while at the same time leaving the properties of the average forecast, hence error, unchanged. Therefore we can conjecture that this will reinforce the difficulty of our expectation model to jointly replicate the moments of the forecast error and of the disagreement across forecasters discussed in Section 4.3 of the main text.

11

A.6

Properties of the sticky/noisy information model

This appendix provides more details on the Sticky and Noisy information model estimated in the main text. For convenience we start by replicating the model set-up. We then derive some properties of the model that we illustrate graphically using results from simulations.

A.6.1

The sticky and noisy information model

The economy is described by a reduced form VAR(p) model

A(L)Xt = ǫt ,

t ∈ Z,

where Xt is a set of n macroeconomic variables, centered on their average, L is the lag opPp k erator, A(L) ≡ k=0 Ak L has all its roots outside the unit-circle and A0 = I and where E {ǫt |Xt−1 , . . . , Xt−p } = 0. The VAR(p) model can be rewritten in the usual more compact firstorder VAR companion form Zt = F Zt−1 + ηt , ′ ′ with Zt = (Xt′ Xt−1 · · · Xt−p+1 )′ , and where ηt = (ǫ′t 0 · · · 0)′ is an innovation with variance matrix

Ση . Let i be an individual in the population of forecasters, i = 1, . . . , n. At each date, every forecaster may update his information set or not. Along the lines of Mankiw & Reis (2002), we model this updating as a Poisson random variable, P(λ). For each individual λ is the probability to update his forecast. Assuming that the population of forecasters is large, by the law of large numbers, each date t a fraction λ of the total population updates its forecast. Consequently, at each point in time, the whole population of forecasters is split into groups, generations say, within which each forecaster refers to the same vintage of information. We let j denote the generation of forecasters that last updated their information set j periods before the current one, i.e. in t − j. Thus j is also an index 12

of the vintage of information they use. The fraction of generation j in the whole population is given by λ(1 − λ)j . In Mankiw & Reis (2002) model, every agent updating his information set gets a perfect signal on the state of the economy Zt . The optimal forecast at horizon h is given by the expectation conditional on this perfect information, E(Zt+h |Zt ) = F h Zt . It is therefore identical for every forecaster who updates. Likewise, the forecasters who last updated in t − j receive a perfect signal on Zt−j and their optimal forecast for t + h is the conditional expectation with respect to that information vintage, E(Zt+h |Zt−j ) = F h+j Zt−j . So, the Mankiw & Reis (2002) information structure implies no disagreement within a generation j of forecasters. Our set-up extends Mankiw and Reis’ model to include imperfect perception of the information when updating. We assume that, when an agent i updates his information, he gets a noisy perception of the true state, Zt , namely a signal Yit that follows

Yit = H ′ Zt + vit ,

vit ∼ iid(0, Σv ),

where H is a matrix that selects the state variables that are observed with a noise. A typical case is H = I, in which case forecasters have noisy perceptions of all the state variables. We assume this is the case here. Remark that the average over forecasters gives the right signal: Ei (Yit ) = H ′ Zt .

Optimal individual forecasts

Let fit,t+h denotes agent’s i optimal forecast for the vector X

x at date t + h, with respect to date t information (and fit,t+h the corresponding expectation for the

specific component x in X). Every individual updating his information set faces the problem of inferring the true state of the economy, Zt , from his imperfect signal Yit . Let Zit|t denotes the date t state of the economy perceived by agent i at date t: Zit|t = E(Zt |Yit , Yit−1 , . . .). Its optimal vector of forecasts is given by fit,t+h = E(Zt+h |Zit|t ) = F h Zit|t .

13

A solution to this signal extraction problem is given by the Kalman filter as follows: !
Zit|t = Zit|t−1 + Git Yit − ′Zit|t−1 , with Zit|t−1 = E(Zt |Yit−1 , Yit−2 , . . .) = F Zit−1|t−1 , and Git the gain of the Kalman filter, defined by !
−1 , Git = Pit|t−1 Pit|t−1 + Σv where Pit|t−1 denotes the variance of the perceived forecast error,   Pit|t−1 = E (Zt − Zit|t−1 )(Zt − Zit|t−1 )′ . Noticeably, the gain is common across agents, Git = Gt , as soon as one postulates the usual initial conditions of the recursion, namely that Zi1|0 = E(Z1 ) and Pi1|0 = E {[Z1 − E(Z1 )] [Z1 − E(Z1 )′ ]}, for every agent in the population. Indeed, the conditional variance of the forecast error in a   Kalman filter follows Pit+1|t = F Pit|t−1 − Pit|t−1 (Pit|t−1 + Σv )−1 Pit|t−1 F ′ + Ση , and the recursion is therefore identical across agents when they start from the same a priori Pi1|0 .3 Individuals who cannot update their information set at date t stick to their old one, like in a Mankiw-Reis type model, but with the difference that past information vintages are noisy. Let fit−j,t+h be the optimal expectation of an agent i using information vintage j, his optimal forecast for date t + h is given by fit−j,t+h = E(Zt+h |Zit−j|t−j ) with Zit−j|t−j = E(Zt |Yit−j , Yit−j−1 , . . .) the state Zt−j perceived by an agent i who sticks to the information vintage j. The optimal forecast of individuals who updated j periods ago is therefore given by

fit−j,t+h = E(Zt+h |Zit−j|t−j ) = F h+j Zit−j|t−j . 3 That would not be the case if, in particular, one considered a “deep” disagreement involving differences in the perception of the parameters value underlying the true state DGP, for instance E(Zi1|0 ) 6= E(Z1 ). See Lahiri & Sheng (2008) or Patton & Timmerman (2010) for expectation models where agents have different priors.

14

A.6.2

Model properties

Average forecast and error

Let Eij (·) be the expectation over i and j. The average forecast

over the whole population of forecasters, also called the consensus, is given by

Eij (fit−j,t+h ) = Ej [Ei (fit−j,t+h |j)] .

That forecast average can be split into the average over individuals using the same information vintage j, and the average over the different generations of forecasters or information vintages. We consider, without loss of generality,4 individuals using the information vintage j = 0 (i.e. the generation of forecasters able to update information at the current date). Combining the Kalman filter expressions above, we can rewrite the optimal vector of forecasts of individuals in this generation as !
fit,t+h = F h Zit|t−1 + F h Gt Yit − Zit|t−1 .

(A-1)

Remarking that Eij (Yit |j = 0) = H ′ Zt , this leads to the following forecast average within this generation !
 !
 Eij (fit−j,t+h |j = 0) = F h+1 Ei Zit−1|t−1 + F h Gt Zt − F Ei Zit−1|t−1 .

We now compare with the perfect information case. In that case every individual would have an optimal forecast given by

∗ = F h+1 Zt−1 + F h (Zt − F Zt−1 ) = F h+1 Zt−1 + F h ηt . ft,t+h

(A-2)

Comparing equations (A-1) and (A-2), the difference between the noisy information and the perfect information cases stems from two channels. First, the perceived state one period backward is not 4

Forecasts that were generated at date t − j for the t + h horizon date can be rewritten as forecast at date τ for the τ + l horizon date, with τ = t − j and l = h + j.

15

!
the true one: Zt−1 = 6 Ei Zit−1|t−1 . This also implies that the perceived innovation is not the !
true one; Zt − F Ei Zit−1|t−1 6= ηt . Second, because it is acknowledged to be noisy, the perceived innovation is not completely incorporated into the forecast: Gt 6= I. Consequently, the forecast error associated with the average within a generation is predictable with respect to the information j = Zt+h − Ei (fit−j,t+h ) be that forecast error, we have available at date t. Indeed, let Et,t+h

j=0 = F h Zt + Et,t+h

h X k=0

h !
!
i F h−k ηt+k − F h Ei Zit|t−1 + F h Gt (Zt − Ei Zit|t−1

h  !
 X F h−k ηt+k , = F h (I − Gt ) Zt − Ei Zit|t−1 + k=0

  j=0 which implies E Et,t+h |Zt 6= 0 as long as I 6= Gt . The optimal forecast for an individual updating his information set at time t is given by fit,t+h and by fit−j,t+h if the information set was last updated j periods ago. The average of individuals’ across generations of forecasters then follows

Ej [Ei (fit−j,t+h |j)] =

∞ X

(1 − λ)j λEi (fit−j,t+h ) .

j=0

  j Let Et,t+h = Ej Et,t+h , the average forecast error, i.e. the error associated with the consensus forecast, we have

Et,t+h = Zt+h − Ej [Ei (fit−j,t+h |j)] = F h−1 (I − Gt )

∞ X j=0

h−1 !
 X  (1 − λ)j λ Zt − Ei Zit|t−j + F h−k ηt+k+1 . k=0

Taking the expectation with respect to the true state, Zt leads to

E (Et,t+h |Zt ) = F h−1 (I − Gt )

∞ X j=0

16

!
  (1 − λ)j λ Zt − Ei Zit|t−j ,

(A-3)

which shows that the average forecast error is predictable, i.e. E (Et,t+h |Zt ) 6= 0, as long as I 6= Gt and/or λ < 1.

Properties of the average forecast errors

The average forecast does not fully incorporate the

news released at date t, ηt . This leads to persistent forecast errors, as can be seen from equation (A-3). A decrease in the attention degree, λ, increases both the persistence and the variance of forecast error. An increase in a term on the first diagonal of the noise variance matrix, Σv , increases the persistence of the forecast error of the corresponding forecasted variable. It also increases the variance of the forecast errors. All in all, less attention generates more persistence and variance of the forecast errors. Figure A-5 illustrates this properties using simulations relying on the estimated values of our model. It shows how average squared forecast error varies with the size of the noise (for various values of the attention parameters).

Disagreement between forecasters

The extent of differences in opinion can be assessed by

the cross-section variance of point forecasts over individuals, i, and information vintages, j, that we denote Vij (fit−j,t+h ). Using the standard variance decomposition formula leads to

Vij (fit−j,t+h ) = Ej {Vi (fit−j,t+h |j)} + Vj {Ei (fit−j,t+h |j)} .

(A-4)

This expressing underlies that the disagreement across individuals stems from two sources. The first source is the noise in individuals’ signal leading to differences in perception within a given generation of forecasters j, i.e. Vi (fit−j,t+h |j) 6= 0. For a given generation of forecasters, disagreement is only generated by the variance of the individual noise, Σv . Indeed, it holds that  !  !

 Vi (fit−j,t+h |j) = F h+j Vi Zit−j|t−j−1 |j + Gt Σv + Vi Zit−j|t−j−1 |j G′t (F h+j )′ .

17

(A-5)

Figure A-5: Model based mean square errors as a function of idiosyncratic noise variance σv (for different value the attention rate λ)

18

This cross section variance evolves with the forecast horizon, shrinking progressively to zero with h. The first term in total disagreement averages the vintage-j specific ‘within’ components of disagreement described by equation (A-5). Each generation j is weighted by its relative share, λ(1 − λ)j , in the total population, so that

Ej {Vi (fit−j,t+h |j)} =

∞ X

(1 − λ)j λVi (fit−j,t+h |j) .

(A-6)

j=0

It is important to remark that this component of disagreement is not equal to zero, even when λ = 1. The second source comes from the differences in average opinion due to the different information vintages used by the forecasters, that is differences between generations of forecasters. Indeed, the heterogeneity within a generation averages out in Ei (fit−j,t+h |j). So, the only cross-section dispersion remaining in the second term of equation (A-4) is due to differences in information vintage, j. More precisely, we have

Vj {Ei (fit−j,t+h |j)} =

∞ X

(1 − λ)j λ {Ei (fit−j,t+h |j) − Ej [Ei (fit−j,t+h |j)]}2 .

(A-7)

j=0

Unlike the first component of disagreement of equation (A-6), this second component is equal to zero when λ = 1.

Properties of the disagreement

As equations (A-6) and (A-7) above show, the model generates

disagreement even under full information updating, λ = 1. In line with Mankiw-Reis model, when σ = 0, disagreement increases when λ decreases. As Figure A-6 illustrates, when the two sources of imperfect information coexist, disagreement can also decrease when λ decreases. This comes from the fact that an decrease in λ increases the relative weight given to generation of forecasters who stick to an old information set, hence have a long forecast horizon (h + j) and thus tend to agree as their forecast gets closer to the unconditional mean of the process whatever the noise in their

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Figure A-6: Model based disagreement as a function of the attention rate λ (for different values of the idiosyncratic noise variance σv ) signal. An increase in any diagonal element of Σv has two different effects on the disagreement within a generation of forecasters. On the one hand, it increases the amount of noise, thus raising the differences of opinions within the subgroup of forecasters that refer to the same vintage of information. On the other hand, because individuals know that the signal is very imprecise, they incorporate less the news to their forecast. In the extreme case, when the signal is completely uninformative, the optimal forecast is the unconditional mean of the process for all forecasters, implying zero disagreement. Figure A-7 below illustrates how disagreement varies with the standard deviation of the idiosyncratic noise. Another feature of the model is to generate time-varying disagreement even when there is no time

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Figure A-7: Model based Disagreement as a function of the idiosyncratic noise variance σv (for different value the attention rate λ)

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conditional heteroscedasticity in the measurement error, vit so that Σv is constant across dates. This comes from the differences across generations of forecasters. The degree of disagreement depends on the difference between the new vintage of information and the previous one. If the last innovation is large compared to the average, the difference of opinion between the individuals revising and the others, therefore the disagreement, will be larger than the average one. This time variance of disagreement increases when λ decreases. Lastly, the model structure also implies that the volatility of disagreement over time decays with the variance of the noise. The less informative the news, the less they are incorporated in the optimal forecast, therefore the less disagreement there is between generations of forecasters. In the extreme case where the precision of the signal approaches zero, news are not reflected in the forecasts, and the time variance of the disagreement shrinks to zero. Figure A-8 below illustrates how the time variance of disagreement varies with the standard deviation of the idiosyncratic noise.

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Figure A-8: Model based time variance of disagreement as a function of the idiosyncratic noise variance σv (for different value the attention rate λ)

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A.7

Sensitivity analysis of Minimum Distance Estimation results

We investigate that the MDE results are robust to more flexible versions of the expectation model that the one presented in the main text. More precisely, we consider • (MSIG): a (multi-sigma) version of the model where the variance of the idiosyncratic observation noise differs across predicted variables.

Specifically we allow for the variance

of the noise to be proportional to the variance of the underlying forecasted variables, i.e. Σv = σv2 [V (π)V (u)V (y)]. (We recall that the baseline assumption is of Σv = σv2 I3 .) • (CE): a (centered errors) version of the model correcting for the fact that in the SPF data, forecasters exhibit forecast errors that have non-zero mean. This amounts to removing such systematic error from the SPF forecast errors moments we try to match. Put it differently, we postulate that such systematic bias is related to other mechanisms than the specific imperfect information ones considered in our setup. As a matter of fact, while the two types of inattention considered in our model can generate average forecast errors that differ from zero in finite samples, in population such errors should have a zero mean. • (REC): a (recursive VAR) version of the model where agents do not know the parameters of the DGP but rather the latest estimate of the VAR model up to the date where data are available. A motivation for this extension is that our reference estimation, which assumes that individuals know the parameters of the VAR, gives the model-based forecasters some information advantage compared to the ones in the SPF. Considering that agents have to recursively estimate the parameters of the VAR potentially opens the possibility that forecasters disagree on the parameters of the DGP itself since strictly speaking, each agents should rely on its own information and thus agents should have different VAR estimates. We deem this extension to be outside of the scope of our paper and we assume the fiction that an econometric agency has access to the true observations, estimates the VAR recursively at each period and communicate the updated parameters to the different forecasters of the sample. The 24

difference between the recursive and the full-sample (FS) VAR estimate probably matters more for samples that include the Great Recession. We thus conducted the estimation based on such recursive VAR for both the “pre-crisis” sample and the sample “including the crisis”.

Results are reported in Table A-1. One first remarks that none of the extension can reconcile the properties of the forecast error moments with the ones of disagreement. These more flexible versions of the expectation model are still rejected by the data. A second striking result is that the degree of inattention stays roughly the same when one allows for different variance of the noise (MSIG) or when one centers the forecast errors of the SPF data to zero (FE). The results associated to recursive estimations of the VAR model describing the underlying DGP of the forecasted data leads to a striking increase in the estimates of attention. The reason is that the recursive estimation generates larger forecast errors of the optimal forecast. So less inattention is needed in order to reproduce the error observed in the SPF micro data. This b increasing from .060 with the is even more striking for the sample that include the crisis with λ

VAR estimated over the full-sample (FS) including the crisis to .393 (REC) and σ bv decreasing from

1.193 (FS) to .672 (REC).

In addition to these three extensions, we also present the results obtained with a full-sample (FS) VAR for inflation and real GDP growth on a sample that includes the crisis. This allows to better understand the result of Table 5 in the main text which shows that, over the very shaking period of the crisis, the idiosyncratic noise in the signal increased, but the frequency of updating a forecast decreased, in contrast to the intuition and the results presented in Figures 4 and 5 in the main text. As a matter of fact, the decomposition provided here underline that such a result is mostly driven by the properties of the real GDP growth forecasts. Still it remains that the stability of the SPF forecasts over the crisis, hence the persistence of the forecast error and the lower estimates for λ, is at odds with the increasing frequency of updating that we observe in the micro SPF data over the same period.

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MDE – Additional Results (5) (6) specification Sample Pre-Crisis Moments INF/GDP Specif MSIG parameter estimates b λ .230 (.021)

Pre-Crisis INF/GDP CE

.264

(.007)

(7)

(8)

(9)

(10)

Pre-Crisis INF/GDP REC

Incl.-Crisis INF/GDP REC

Incl.-Crisis INF FS

Incl.-Crisis GDP FS

.335

(.000)

.393

(.001)

.426

.267

(.099)

(.216)

(.001)

(.003)

(.048)

(.064)

.562

1.604

specification test (j-stat) p-value .026 .280

.000

.003

.016

.977

σ bv

.766

(.003)

.550

.306

.672

(.064)

Table A-1: Minimum Distance Estimation and Tests The Table gives the Minimum Distance Estimation results of the two parameters characterizing the b is the estimated frequency sticky/noisy expectation model described in Section 4. More specifically, λ of updating one’s information set and σ b the estimated variance of the noise in the signal. Targeted moments are the mean squared forecast error Mean(e2t ), the first-order autocorrelation of the forecast error ρe (1), the average disagreement Mean(σt ), and the time variance of the disagreement V(σt ), of either the euro area inflation rate (INF), or the euro area real GDP growth rate (GDP), or both. All moments are calculated with 1-year horizon forecasts. Two samples are considered: a pre-crisis sample, that goes from 1999Q1 to 2008Q3, and a sample including the Great Recession, that covers 1999Q1 to 2012Q4. In addition, the estimations are run under different specifications of the auxiliary VAR model used to generate the forecasts: a VAR estimated over the full-sample considered (FS), a VAR estimated on the full-sample with multiple variance for the of the different variables (MSIG), a VAR estimated on the full-sample with centered errors (CE), a VAR estimated recursively from 1999Q1 onwards (REC). p-values of the J-stat (over-identification test) are obtained by Monte-Carlo simulations.

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References [1] Bowles C., Friz, R., Genre, V., Geoff, K., Meyler, A. and Rautanen, T. (2007). The ECB survey of professional forecasters (SPF). A review after eight years’ experience. ECB Occasional Paper Series, n. 59. [2] Fagan G., Henry J., and Mestre, R. (2001) , An area-wide model (AWM) for the euro area, European Central Bank, Working Paper Series, n. 42. [3] Lahiri K., and Sheng, X. (2008) Evolution of forecast disagreement in a Bayesian learning model. Journal of Econometrics, 144:325–40. [4] Patton A. and Timmermann, A. (2010) Why do forecasters disagree? Lessons from the term structure of cross-sectional dispersion. Journal of Monetary Economics, 57:803–820.

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