NBER WORKING PAPER SERIESEXPECTATIONS FORMATION AND FORWARD INFORMATION
Nathan Goldstein
Yuriy Gorodnichenko
Working Paper 29711
http://www.nber.org/papers/w29711

NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
January 2022, Revised August 2022

We thank Gaetano Gaballo, Filip Mat jka, Dmitriy Sergeyev and conference and seminar
participants at NASMES2022, UC Davis, UC Berkeley, HEC Paris, Tel-Aviv University, BenGurion University, University of Haifa, Reichman University, Bar-Ilan University, Bank of Israel
for comments. We thank Eric Sims for sharing code for the Barsky-Sims approach. The views
expressed herein are those of the authors and do not necessarily reflect the views of the National
Bureau of Economic Research.
NBER working papers are circulated for discussion and comment purposes. They have not been
peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies
official NBER publications.
© 2022 by Nathan Goldstein and Yuriy Gorodnichenko. All rights reserved. Short sections oftext, not to exceed two paragraphs, may be quoted without explicit permission provided that fullcredit, including © notice, is given to the source.

Expectations Formation and Forward Information
Nathan Goldstein and Yuriy Gorodnichenko
NBER Working Paper No. 29711
January 2022, Revised August 2022
JEL No. C83,D84,E31
ABSTRACT
We propose a model where forecasters have access to noisy signals about the future (forward
information). In this setting, information varies not only across agents but also across horizons.
As a result, the estimated persistence of forecasts deviates from the persistence of fundamentals
and the ability of forecasts at shorter horizons to explain forecasts at longer horizons is limited.
These properties tend to diminish as the forecast horizon increases. We document that this novel
pattern is consistent with survey data for professional forecasters. We provide further evidence
that time-series and cross-sectional variation in professional forecasts are driven by forward
information. We propose a simple method for extracting the forward information component
from a survey and provide several applications of forward information.

Nathan Goldstein
Department of Economics
Building 504
Bar-Ilan University
Ramat-Gan 5290002
IsraelNathan.Goldstein@biu.ac.il
Yuriy Gorodnichenko
Department of Economics
530 Evans Hall #3880
University of California, Berkeley
Berkeley, CA 94720-3880
and IZA
and also NBER
ygorodni@econ.berkeley.edu

1. Introduction
Yogi Berra famously summarized challenges in forecasting with “it’s tough to make predictions, especiallyabout the future.” Yet economic agents routinely engage in forecasting because almost every decision they
make depends on what they think inflation, income and other variables are going to be in a week, month,year, or longer. But how are expectations formed? What information do economic agents have about thecurrent and future conditions? How useful are signals about the future? To shed more light on thesequestions, we propose a new approach that incorporates "forward information" in expectations formation.
We start with the observation that forecasts are more than just a model iterated forward. Indeed,using a special survey conducted among participants in the Survey of Professional Forecasters (SPF), Stark(2013) documents that a vast majority of professional forecasters (80%) describe their forecasting method
as a combination of “model” and “subjective adjustment”. For example, this practice means that when
forecasters try to predict inflation, they use an econometric model to generate a first pass at the inflation
projection, but then they adjust the prediction due to whatever information (e.g., news about futuremonetary/fiscal policy, technology) they have which is beyond their model.1 This “forward” information isuseful because it may reduce forecast errors. Since forecasters can face different and likely imperfect signals
about the future, forward information can also be a key source of forecast dispersion and apparent deviations
from full-information rational expectations (FIRE) documented in the literature. Our objective is to extract
this forward information from forecasts and then study its properties and effects.
In a nutshell, consider the standard noisy information framework as a benchmark: agents faceimperfect signals about realized fundamentals and use the Kalman filter or similar tools to filter out the noise.
Once the forecast for the current period is formed, the forecasts for longer horizons obey the state equation
and, hence, persistence in expectations should reflect persistence in the fundamental. For example, if thefundamental 𝑥 is an AR(1) process with persistence 𝜌, the forecast for horizon ℎ (𝑥the belief about the current value of the fundamental (𝑥 | ): 𝑥
forecast for horizon ℎ

1 (𝑥

|

|

is simply 𝜌 times

𝜌 𝑥 | . As a result, when we regress the| ) on the forecast for horizon ℎ (𝑥| ), the slope coefficient in thisregression is 𝜌, i.e., persistence in the fundamental, for any horizon ℎ. Furthermore, the 𝑅 in this regression
should be one because there is no new information beyond the current estimate of the fundamental. We test
these predictions using SPF forecasts for multiple horizons. In contrast to the predictions, we find that boththe slope coefficient and the fit of the regression gradually increase and tend to “converge” as we apply the
regression to forecasts with longer horizons. This pattern points to some additional forecasting component,

1

A common technical tool for subjective adjustment is known in practice as “add factoring”, where the forecasteradds some subjective factor to the estimated specification used for prediction. For an early discussion of add factoring,see Fair (1986).

1beyond the application of the state equation, which produces a variation in correlation across the forecasthorizons.
We interpret this forecasting component as reflecting the “subjective adjustment.” Specifically,
forecasters adjust the state-equation prediction in light of forward information (i.e., signals about the
future). Intuitively, when forecasters adjust their “econometric” predictions to incorporate forward
information, they introduce an “error term” (which captures forward information) in the state equation:
𝑥
ℎ

|

𝜌𝑥

|

𝑒𝑟𝑟𝑜𝑟. Because forward information can influence not only beliefs about horizon

1 but also beliefs about horizon ℎ, the error term is correlated with 𝑥

estimate of the slope away from 𝜌 and result in 𝑅|

and hence can bias the

1. However, for long enough horizons, for which

forward signals are no longer informative, the forecasts would only be based on the state equation withouta biasing adjustment. Thus, the forward information setup should augment the standard noisy information
framework in two important directions: First, there will be informative signals not only about the past but
also about future fundamentals. Second, information will vary not only across agents, as in the standard
framework, but also across future horizons.
We present a formal model of expectations formation with forward information along these lines.
Agents receive multiple noisy signals referring to several periods in the future. They exploit these signals
and optimally filter the noise when forming forecasts for multiple periods ahead. We show that forecast
adjustment due to forward information can vary across forecasting horizons since agents need to optimallyreweight the multiple forward signals when moving from one horizon to another. This reweighting
mechanism diminishes and eventually stops when forecasting goes far enough into the future, at which
point there are no further informative signals. We use simulations to show that this setup can qualitatively
reproduce the patterns documented in the SPF data.
Our approach offers several additional insights. First, we can assess the importance of forward
information. In particular, our generalized framework allows us to directly test the restricted form impliedby the standard noisy information model, which includes only signals about the current and past
fundamentals, against the unrestricted form, which includes forward information. We show that thecoefficients in the augmented specification are directly mapped to the optimal weights placed on themultiple forward signals in our framework. Based on SPF data, our results point to a significant role offorward information and rule out a significant amount of noisy information in the sense of the standard
model. These findings support the view that SPF forecasts are driven by dispersed information about the
future, rather than about the past.
Second, our approach offers a straightforward way to extract forward information directly fromexpectations data. According to our forecast decomposition (𝑥

|

𝜌𝑥

|

𝑒𝑟𝑟𝑜𝑟), forward

information is the component that leads forecasters to make an adjustment to their “econometric” prediction2

(i.e., 𝑒𝑟𝑟𝑜𝑟 in the state equation iterated forward). We apply this decomposition to SPF inflation forecasts.
We show that forward information is helpful for predicting inflation at short horizons. Furthermore, forward
information in the Greenbook forecasts from the Federal Reserve predicts policy rate, thus suggesting thatthe central bank acts on news about future inflation.
Finally, we apply our approach to other variables and survey data. When we examine SPF forecasts
for additional macroeconomic variables, we find both indirect and direct evidence for the presence of
forward information that varies across forecasting horizons. Similar evidence is found for the Greenbooks.
We also apply inflation forecast data from the SPF run by the European Central Bank (ECB). We exploit
the availability of both calendar-year and rolling-year forecasts in this survey and find a pattern of variation
across calendar quarters, which is consistent with the utilization of forward information that varies acrossforecasting horizons.
Our study is related to several strands of literature. First, we contribute to the vast and growing
literature on the expectations formation process (Coibion, Gorodnichenko and Kamdar (2018) provide a
review of this literature). Survey forecasts are widely used to test and quantify models with information
rigidities or behavioral frictions (e.g. Mankiw, Reis, and Wolfers, 2004; Capistran and Timmermann, 2009;
Patton and Timmermann, 2010; Andrade et al., 2013; Coibion and Gorodnichenko, 2012, 2015; Fuhrer,
2018; Angeletos et al., 2020; Bordalo et al., 2020; Giacomini et al., 2020; Kohlhas and Walther, 2021;
Goldstein, 2021). Building on this literature, we explicitly model the “subjective adjustment” of forecasts
which we interpret as a practice driven by the availability of forward information. We emphasize thatforward information plays a key role in how professional forecasters form their macroeconomic projections.
Our paper is also related to the literature about news shocks (Beaudry and Portier (2014) provide asurvey of this literature and Chahrour and Jurado (2018) connect news shocks and noise in signals). Unlike
the literature on expectations formation, this literature stresses the role of news about the future in driving
the business cycle, but it largely ignores microfoundations of expectational assumptions (Chahrour andJurado (2021) is a recent exception). Furthermore, with some exceptions (e.g., Barsky and Sims, 2012;
Nguyen and Miamoto, 2019), it tends to extract news from realized macroeconomic series rather than
expectations data. Our approach bridges these two literatures by linking forward information directly to the
expectations formation process. Furthermore, we emphasize and exploit heterogeneity across both agentsand forecast horizons. Our findings can be used to quantify forward information systematically using
forecast data and to assess its role in predicting macroeconomic dynamics and affecting policy.
Our paper also contributes to the literature that studies the persistence of inflation and documents
a notable decline in inflation persistence in the U.S. and other countries since the 2000s (Fuhrer (2011)
surveys this literature). Jain (2019), which is the closest in spirit to the starting point of our analysis, utilizes
revisions in SPF forecasts to estimate the size and time variation in the perceived persistence of inflation.
3

In contrast to Jain (2019), we allow persistence to vary with the forecast horizon. As explained above, thisnovel type of variation that we document motivates our idea of forward information. Based on our results,
we argue that persistence estimated on long horizons can provide a better metric of perceived persistenceof the fundamental because forward information is less likely to be quantitively important at long horizons.
We also offer a real-time estimate of persistence based on cross-sectional/across-horizon variation in
forecasts, which contrasts with the focus on time-series variation in actual data.
The remainder of the paper is organized as follows. Section 2 provides a simple noisy-information
benchmark model to build intuition and testable predictions. We show that these predictions are rejectedby stylized facts documented for SPF inflation forecasts. Section 3 presents our model of forward
information and rationalizes the stylized facts. Section 4 reports further tests of forward information.
Section 5 describes how one can extract forward information from forecast data and documents theproperties of forward information, including the ability of forward information to predict policy choices
and future inflation. Section 6 summarizes evidence from forecasts of other key macroeconomic variablesin the U.S. SPF and the ECB SPF. Section 7 concludes.

2. Noisy information: The standard framework
To fix ideas, we use the standard noisy-information model as a starting point. Following Woodford (2002)
and Coibion and Gorodnichenko (2015), assume that fundamental 𝑥 follows an AR(1) process. Whileagents know the underlying process, they cannot perfectly observe 𝑥 due to idiosyncratic noise in signals
about 𝑥 . Their forecasting problem can be described by a state-space representation:
State:
𝑥

𝜌𝑥

𝜔,

(1)

𝜈,

(2)

Measurement:𝑦where 𝜔 ∼ 𝑖𝑖𝑑 𝑁 0, 𝜎𝑥is a shock to the fundamental, 𝜈 ∼ 𝑖𝑖𝑑 𝑁 0, 𝜎

the persistence of the fundamental. An optimal forecast 𝑥

is idiosyncratic noise, and 𝜌 is

| , for ℎ steps ahead, is formed by each agent

𝑖 with a Kalman filter:
𝑥

|

𝑥

|

G 𝜌 𝑦

𝑥

|

,

where G is the Kalman gain. Thus, agents revise their former forecast, 𝑥(3)|, by placing some weight on

the prediction 𝜌 𝑦 , which utilizes the new signal.

4

An important implication of equation (3) is that forecasts made at time 𝑡 for various horizons shouldobey the process of the state equation:
𝑥

|

𝜌𝑥

| .

(4)

Equation (4) posits several predictions which can be tested with data on survey forecasts for multiple
horizons. In particular:
When estimating a regression of the forecast 𝑥
I.

|on the forecast 𝑥

The fit of the regression should be perfect (𝑅1) for any ℎ

1| :

1.

II. The regression coefficient recovers 𝜌, the persistence parameter, for any ℎ1.

Note that these predictions apply to both forecaster-level projections and consensus projections.
We test these predictions with CPI inflation forecasts from the Survey of Professional Forecasters.
Quarterly forecasts are available since 1981Q3 for multiple horizons, running from the current quarter tofour quarters ahead. In addition, there is a backcast provided in each quarter for the inflation rate in theprevious quarter. Thus, we estimate five horizon-specific regressions for each ℎ between 0 to 4.2 The results
for the full sample period (1981Q3-2017Q4) are described in Panel A of Figure 1. We observe several
patterns, which deviate from the above predictions. First, the estimated persistence and 𝑅 vary with theforecast horizon significantly. Second, both the estimated persistence and 𝑅 increase in the forecasthorizon. The estimated persistence rises from ≈0.4 for ℎ

0 to ≈0.9 for the longest horizon ℎ

4.

Likewise, 𝑅 increases from ≈0.3 to ≈0.8. Finally, the fit of the regressions is far from perfect for the shorter
horizons (𝑅 ≪ 1).
To explore the robustness of these patterns, we split the sample into four decades from the 1980s to
the 2010s and repeat the estimation. We report results in Panel B of Figure 1. Overall, the coefficient estimatesand R-squared show the same pattern of increase with the forecast horizon for each sub-period. The only
difference is that the range of variation is getting narrower in recent decades. This could reflect the decline ininflation persistence in the US over the years, which was documented by the previous literature (e.g., Fuhrer,2011). Nevertheless, differences across horizons are still highly significant as indicated by the p-values.3ℎ 0 in the data is equivalent to ℎ 1 in the above model, since SPF respondents receive a signal on realization in
the previous quarter.
3
We also preform the estimation with consensus forecasts and find the same pattern documented in Figure 1. See thefirst row in Panel A of Appendix Table G.3.
2

5

Because equation (4) holds for any period, we can estimate the corresponding regression quarterby quarter separately and examine time-series variation in the estimates.4 Using a non-parametric smootherto remove high-frequency noise in the estimates, we report the results in Figure 2. We find that both theestimated persistence and 𝑅 are higher at longer horizons (Panels A and B, respectively). Furthermore, as
the estimated persistence declines over the years, so does the 𝑅 of the cross-sectional regressions. Thissuggests that the dynamics of persistence and 𝑅 may be related phenomena.
Specification (4) reflects a simple AR(1) process, whereas agents may use more sophisticatedprocesses. In Figure 3, we examine more general processes, such as AR(4), VAR(4), and AR(1) in termsof the inflation gap (the difference between inflation and trend inflation). Because we do not have enoughforecast horizons to explore how 𝜌 varies in ℎ, we focus on the 𝑅 from a quarter-by-quarter estimation
corresponding to each specification for ℎ

3. For the AR(4) process, we regress the forecast 𝑥|

on theforecasts for the 4 preceding quarters. For the VAR(4), we add forecasts of “lagged” interest rate andunemployment from the SPF. The inflation gap specification utilizes long-run inflation forecasts, which areprovided in the SPF more recently, and computes the forecast for the inflation gap as the difference between
the CPI inflation forecast and the long-run forecast.5 The figure describes a fit of the different regressions
over the sample period which is very similar to the fit obtained with the simple AR(1). This is in line with
Coibion and Gorodnichenko (2015), who find that a univariate AR(1) approximates well the expectationsin the SPF data.6
Another potential explanation for the results in Figures 1 and 2 is that forecasters use different
models, i.e., each forecaster applies her own (perceived) 𝜌 , which may be different from the true
persistence parameter 𝜌. Hence, the forecast of an agent 𝑖 for time 𝑡
𝑥

|

𝜌𝑥

|

𝜌𝑥

|

𝜌

𝜌 𝑥

ℎ, denoted by 𝑥
| .

| , follows:(5)4

Notice that this estimation is also in line with the well-known unobserved-component modelling of inflation.
𝜇𝑥 , where 𝜇 is a trend component following a random walkSpecifically, suppose that inflation follows 𝑥
is a transitory component following AR(1) with persistence 0 𝜌 1. Then, the process of 𝑥 can be
and 𝑥
𝑐
𝜌𝑥
𝜀 , where 𝜀 captures shocks to both components and 𝑐
1 𝜌 𝜇behaves
written as AR(1), 𝑥
like a "constant" that varies over time due to the trend component. When applying our AR(1) regression to inflation
forecasts quarter by quarter, the regression constant can vary over time in a similar way.
5
From the previous footnote, it is clear that the inflation-gap autoregression arises from the same unobserved𝑥
𝜇 . The difference in the current estimation is that we allow
component modelling of inflation, since 𝑥
heterogeneity in forecasters' views on trend inflation, by using their long-run forecasts from the survey.
6
Another robustness check for our stylized pattern is reported in Appendix Figure G.1. As for the AR(1), we applied
estimations across different horizons for higher order auto-regressions. Each additional “lag” implies one lessregression (i.e., instead of 5 regressions for the different horizons with AR(1), we can ran 4 regressions with AR(2)and so on). The figure reports persistence estimates as the sum of coefficients and the 𝑅 values. In all the panels theyclearly increase with the horizon, consistent with the pattern documented for AR(1).

6

If we regress 𝑥

|

on 𝑥

|

as in specification (4), the error term corresponding to 𝜌is clearly correlated with the regressor 𝑥

𝜌 𝑥|| . Furthermore, the cross-sectional estimate of average(perceived) persistence could depend on the distribution of 𝜌

, thus creating variation in ℎ. Finally,

differences in 𝜌 generate disagreement in forecasts.7
To assess this alternative explanation, we estimate the persistence in expectations across horizons
for a specific forecaster 𝑖, using OLS:
𝑥

|

𝜌 𝑥

|

𝑒𝑟𝑟𝑜𝑟 .

(6)

For each horizon ℎ, we compute the average (across forecasters) value of the estimated 𝜌 . If heterogeneityin 𝜌 were driving our results in Figures 1 and 2, we should observe that the average value of 𝜌 should notincrease in horizon ℎ. Panel A of Figure 4 documents that forecasters indeed have different perceived
persistence of inflation: the standard deviation of 𝜌

is above zero and it is generally increasing overtime. However, this heterogeneity does not have a materially important effect on how inflation persistence
varies across horizons and over time. Similar to Panel A in Figure 2, Panel B in Figure 4 documents thatpersistence (average of 𝜌 across 𝑖) is larger for longer horizons and there is a broad decline in persistenceover time. Thus, model heterogeneity does not explain why persistence increases in horizon ℎ.
In summary, our evidence suggests that the standard noisy information framework misses some
important forecasting component, which varies both cross-sectionally and across horizons. In what follows,
we argue that this component is associated with the practice of “subjective adjustment” of forecasts, or “addfactoring”. This subjective adjustment can capture forecasters’ information about future fundamentals.

3. A model of forward noisy information
To build intuition, we begin with a tractable example using two forward signals. The example is used toderive some analytical results that illustrate how forward information augments the forecast in the standard
noisy information framework and introduces an additional key component to equation (4). We will thenpresent a more general framework and use simulations to show how one can rationalize the patterns
documented in the previous section.

3.1.

A case of two forward signals

7

The correlation between the regressor and the error term can arise for other, potentially behavioral reasons. In
Appendix D, we show how such correlation can arise in the model of asymmetric loss function by Elliott, Komunjer
and Timmermann (2008) and Capistran and Timmermann (2009). However, this model does not explain why
estimated persistence should increase with the forecast horizon.

7

Consider a state-space representation of a fundamental 𝑥 , which follows an AR(1) process. However,unlike the one-signal state-space in equations (1) and (2), here agents receive three informative signals attime 𝑡: A perfect signal about realization (the current value of the fundamental) and two forward signals
referring to two subsequent periods, 𝑡

1 and 𝑡2. Accordingly, the state-space representation isState:
𝑥
𝒙 ≡ 𝑥
𝑥

𝜌
1
0

0
0
1

0
0 𝒙
0

1
0 𝜔
0

𝑥
𝑥
𝑥

𝜈,

Ρ𝒙

𝑆′𝜔

(7)

where 𝜔 ∼ 𝑖𝑖𝑑 𝑁 0, 𝜎 .
Measurement:𝑦,𝒚 ≡ 𝑦,𝑦,∼ 𝑖𝑖𝑑 𝑁 0, 𝜎

where 𝜈 ,𝑦,

and 𝑦 ,and 𝜈 ,𝒙

𝜈,0

∼ 𝑖𝑖𝑑 𝑁 0, 𝜎𝝂(8)are idiosyncratic noise in the two forward signals,, respectively. Notice that signal 𝑦 , is perfect (it does not contain noise) so that 𝑥 is
and 𝑦 ,

perfectly known at time 𝑡. Only the forward signals 𝑦 ,are imperfect. We use this example todemonstrate how the empirical patterns in the previous section can be driven only by forward signals, evenin the absence of noise about the realized fundamental, as in equation (2).
Because 𝑥 |𝑥 , we can simplify the analytical derivation of the weights in the optimal forecasts

for the subsequent horizons. One can show (Appendix B) that the one-step ahead forecast is given by:
𝑥

|

𝑊 𝜌𝑥

𝑊𝑦

𝑊𝑦,

,

𝑊𝜌

𝑦,

,

(9)

where 𝑊 , 𝑊 , 𝑊 and 𝑊 are the optimal weights placed on each informative signal, which minimizes
𝐸 𝑥𝑥

and obeys 𝑊

|

𝑊

𝑊

𝑊

1. As derived in Appendix B.1., the optimal weights

depend on the noise-to-signal ratios (𝜎 /𝜎 and 𝜎 /𝜎 ) and the persistence parameter 𝜌.
In a similar way, the two-step-ahead forecast is a weighted sum of the same four signals, which areshifted by one period to the future:
𝑥

|

𝑤 𝜌 𝑥

𝑤 𝜌𝑦

,

𝑤 𝜌𝑦 ,

𝑤𝑦,

.

(10)

If the corresponding optimal weights in equations (9) and (10) are equal to each other, that is, if 𝑊𝑤

for each 𝑘| ,1,2,3,4, then the relationship between the forecasts would simply obey 𝑥|

𝜌𝑥which follows the AR(1) process in the state equation. Hence, our setup can nest the standard noisy
information model. However, this simple relationship 𝑥

|

𝜌𝑥

|

does not hold when we introduce

forward signals. As shown in Appendix B.1., the optimal weights, when minimizing the two-step-aheadsquared forecast error, are smaller than the corresponding weights in equation (9), except for 𝑤

𝑊.
8refers directly to 𝑥

Intuitively, because the fourth signal (𝑦 ,, it is given an extra weight in the two-step-ahead forecast.
As a result, the relationship between forecasts at different horizons is more complex:𝑥
where 𝑆𝑖𝑔𝑛𝑎𝑙 ,𝑆𝑖𝑔𝑛𝑎𝑙 ,

|

≡ 𝜌𝑦

𝜌𝑥

|

𝑤

|

𝑊 𝑆𝑖𝑔𝑛𝑎𝑙 ,

(11)

|

corresponds to each of the four signals used in equation (10) (𝑆𝑖𝑔𝑛𝑎𝑙 ,|≡𝜌 𝑥 ,,, etc.). Equation (11) illustrates that forward signals induce an adjustment to the“standard” forecast 𝜌𝑥| , and that this adjustment depends on how the optimal weights change across|forecasting horizons. From an econometric perspective, this adjustment introduces an “error term” inspecification (4), which can account for the empirical patterns from the previous section. For example, thesignals in this error term are correlated with the regressor 𝑥| , thus, potentially explaining why estimatedpersistence can vary with the horizon.
Importantly, equation (11) could also be interpreted as a decomposition of the forecast 𝑥
two components. The first component, 𝜌𝑥

|

into

| , is a standard prediction based on the state process, while

the second component is an adjustment due to forward information, which is beyond the information
already included in the forecast for the previous step. In Section 5, we use this interpretation to quantify thesecond component as a measure of news. We further note that the decomposition in equation (11) provides
a formal description of the common forecasting practice, that applies some “subjective adjustment” to themodel-based forecast (Stark, 2013).
We now examine what happens to the adjustment component when increasing the horizon of theforecast to ℎ

3 (i.e., 𝑥| ), for which the forecaster does not have any forward information. As shownin Appendix B.1., the optimal weights do not change and remain 𝑤 as in equation (10) so that:𝑤 𝜌𝑦 ,

𝜌𝑥

Since the forward signals do not refer to future periods beyond 𝑡

2, 𝑥

𝑥

|

forecast for 𝑡

𝑤 𝜌 𝑥

𝑤 𝜌 𝑦

,

𝑤 𝜌 𝑦,

| .
|

(12)

is sufficient for an optimal

3 and there is no adjustment component as we have in equation (11). More generally, thisresult illustrates that the relationship between forecasts with two consecutive horizons would obey thesimple relation of 𝑥

𝜌𝑥

|

|

for any horizon beyond the horizons of the forward signals (ℎ3 inour example).
We could use 𝑥 |shortest horizons, 𝑥|

𝑥 and equation (9) to express the relation between the forecasts with theand 𝑥 | :

9

𝑥

𝜌𝑥 |

|

𝑊

1 𝜌𝑥

𝑊𝑦

,

𝑊𝑦,

𝑊𝜌

𝑦,

The interpretation is similar to equation (11), where the simple relation 𝑥.

(13)
𝜌𝑥 | , based on the state|process, is modified due to variations in the optimal weights placed on the signal when the forecast horizon
changes from ℎ

0 to ℎ

1. The optimal weights in 𝑥|

are 𝑊 , while for 𝑥 | the whole weight isplaced on the first signal, which is the perfectly observed realization of 𝑥 (i.e., 𝑥 |𝑥 ). These weight

differentials are multiplied by the corresponding signals as in equation (11).
This simple model with forward information can thus shed light on the empirical patterns when we
examine regressions of 𝑥

|

on 𝑥

| . Specifically, if we estimate a cross-sectional regression of 𝑥|

and 𝑥 | generated by our model, the estimated persistence and 𝑅 would be biased towards zero, since 𝑥 |has no cross-sectional variation. At the other extreme, if we estimate a cross-sectional regression of 𝑥
on 𝑥𝑥

| , the coefficient will be exactly 𝜌 with a perfect fit. The “middle”-horizon regression of 𝑥
|

|

|

and

will provide intermediate estimates. More generally, changes in the optimal weights across forecasthorizons induce a “deviation” from the state process, manifested as a regression error which is also
correlated with the regressor 𝑥| .

Consequently, the OLS coefficient estimate could be biased awayfrom the underlying persistence parameter 𝜌 and the fit of the regression could be poor. However, as wemove to the longer horizons and estimate the regression, the variation in the optimal weights across horizons
would diminish, thus shrinking the error term of the regression. As a result, the relation between forecasts
horizons converges to 𝑥

|

𝜌𝑥

|

as ℎ increases.

Our example can provide further intuition for this pattern of gradual convergence. Specifically,
instead of increasing ℎ, we can focus on the middle horizon ℎ2, and examine how the relation between

𝑥

|

|

and 𝑥

| , as specified in equation (11), converges to 𝑥

𝜌𝑥|when increasing the noise in

the more forward-looking signal, namely, when increasing 𝜎 . We show in Appendix B.3. how the propertiesof the regression (estimated 𝜌 and 𝑅 ) change when increasing 𝜎 , in line with the empirical pattern acrosshorizons. We also analyze how changes in 𝜌 affect the regression properties and we find that one can generate
co-movement of the estimated persistence coefficient and 𝑅 that mimics the empirical pattern. A summaryof the results is presented in Table 1. More generally, our key insight from this analysis is that the empiricalpatterns are driven by the gap between optimal weights at different horizons 𝑤

𝑊, which tends todiminish in both the forecast horizon (or 𝜎 ) and the underlying persistence 𝜌.
Finally, similar to the standard noisy information model, the augmented model with forward
information can generate predictability of forecast errors, in the spirit of Coibion and Gorodnichenko

10

(2015). Consider the one step-ahead forecast error. After plugging equations (7) and (8) into equation (9)and taking the average across agents (and thus eliminating all the idiosyncratic terms), we get:𝑥where 𝑥

𝑥

|

𝑊𝜔

𝑊

𝜌

𝜔

,

(14)

(without superscript 𝑖) denotes the cross-sectional average (consensus) forecast. Using|

equations (9) and (10), and averaging across agents, we find (see Appendix B.5 for derivations) that therevision of the average forecast is:
𝑥

|

𝑥

|

𝑤 𝜌𝜔

𝑤

𝑤

𝑤

𝑊 𝜔

𝑊𝜌

Notice that both the forecast error and forecast revision contain the shocks 𝜔

𝜔

.

(15)

and 𝜔

, which producesa correlation between the forecast error and forecast revision. In the standard noisy information setup, thecorrelation arises from shocks up to time 𝑡, due to the gradual processing of imperfect information about
the past. By contrast, in the forward information setup, the correlation between forecast errors and revisions
arises from future shocks, due to the gradual processing of forward information. In Appendix B.5., we
derive the explicit expression for the OLS coefficient in a regression of the (mean-level) forecast error on
forecast revision and show that it should be positive, in line with Coibion and Gorodnichenko (2015).
Hence, the forward information model brings a different interpretation to the well-documented
predictability of forecast errors.8

3.2.

General Framework

At time 𝑡, agents receive multiple signals denoted by 𝑦 ,, which refers to a time 𝑡ℎ in the future,

where ℎ is the horizon of the signal, running from 0 to 𝐻 periods ahead. The state-space model could simply
be written as:State:
𝑥

𝜌𝑥

𝜔,

(16)

where 𝜔 ∼ 𝑖𝑖𝑑 𝑁 0, 𝜎 .

8Recent studies have documented forecast error predictability even at the individual level, mostly with the opposite
sign. Thus, forecasts underreact to information at the aggregate level but overreact at the individual level. Motivatedby this evidence, they advocated a hybrid approach that combines informational frictions and behavioral frictions(Broer and Kohlhas, 2019; Bordalo et al., 2020; Angeletos et al., 2020). Such a combination is still required when
information frictions are due to forward information. However, note that according to equation (14) the impulse
response function of forecast errors to macroeconomic shocks may flip signs. This property speaks to the evidence on
IRF of forecast errors, studied in Angeletos et al. (2020), which is explained there by over-extrapolation. Forward
information may also call for a reinterpretation of individual-level results as well. For example, do forecasters
overreact to all types of information in the same way? Or do they tend to do so only with respect to past information
rather than forward information or vice versa?

11Measurement:𝑦,where ℎ0, … , 𝐻 and 𝜈 ,𝑥𝜈,∼ 𝑖𝑖𝑑 𝑁 0, 𝜎

,

(17)

is an idiosyncratic noise. Thus, agents have some forward

noisy information referring to future values of the fundamental up until 𝐻 periods ahead. There are nouseful signals from 𝐻

1 period and onward. We could view this as if the sequence of 𝜎 goes to infinity

is so large that a forward signal for horizon 𝐻

with ℎ and 𝜎

1 is too noisy to be practically useful.

Although it is natural to assume that 𝜎 increases in ℎ (which is consistent with the data as we describe
below), we do not restrict our framework to such monotonicity. For instance, some forward guidance
relating to the path of inflation a year from now may represent an improved forward signal with at a horizonof a year ahead.
It is also interesting to consider a case where the current fundamental is observed perfectly, by
0. In this case, all heterogeneity in expectations is driven by signals referring to the future

imposing 𝜎

and not because of imprecise current/past data, as it is assumed by the standard noisy-information
framework. As before, the standard model is nested in our general framework by imposing 𝜎 to be infinite
0. Under these restrictions, there is no valuable forward information andfor any ℎ, except for 𝜎

realizations are not perfectly observed. Section 4 will use these restrictions to test our framework against
the standard model.
The augmented state-space model, corresponding to equations (16) and (17), takes the following
form:State:
𝒙 ≡where 𝑆

𝑥
𝑥

𝜌 0
1 0
⋱ ⋱
0 ⋱

⋮
𝑥
1

0

⋯

⋯
0
⋱
1

0
0
𝒙
⋮
0

𝑆′𝜔

Ρ𝒙

𝑆′𝜔

,

(18)

0 , so that the variance-covariance matrix of 𝑆′𝜔 is Σ

𝑆′𝑆𝜎 .

Measurement:𝑦⎡ ,⎢𝒚 ≡ ⎢𝑦 ,
⋮
⎢
𝑦
⎣
,

⎤
⎥
⎥
⎥
⎦

𝑥
𝑥
⋮
𝑥

𝜈
⎡ ,
⎢𝜈 ,
⎢
⋮
⎢
𝜈
⎣
,

where the variance-covariance matrix of 𝝂 is Σ

⎤
⎥
⎥
⎥
⎦

𝒙

𝑑𝑖𝑎𝑔 𝝈𝝂,(19)with 𝝈𝜎𝜎⋯

𝜎 , i.e., theidiosyncratic noise in the forward signals is uncorrelated across horizons. We also assume that the noise is12uncorrelated across agents and that shocks to the fundamentals and noise in the forward signal areuncorrelated (i.e., 𝐸 𝜔

𝝂

𝟎).9

Based on this representation of the state-space, the Kalman filter can be applied to derive theoptimal forecast of an agent 𝑖:10
𝒙|

𝒙|

G 𝒚

𝒙|

,

(20)

where 𝒙 | is a vector of forecasts made at time 𝑡, with horizons running from 0 to 𝐻 steps ahead, and G is
the gain matrix (with dimension 𝐻

1).

Importantly, the ℎ step-ahead forecast, 𝑥
𝑥| , when ℎ𝜌𝑥|𝐻, should simply obey

| ,

(21)

which resembles equation (4) in the standard noisy information framework. However, we can show
(Appendix A) that the forecasts for shorter horizons (0
𝑥

|

𝜌𝑥

𝑥

|

𝜌𝑥

|

𝜌𝐆

𝐆

ℎ

𝐻) in vector 𝒙 | has a different dynamic:

|

𝒙

𝒙|

where 𝐆 denotes the vector of elements in row 𝑗

𝐆

𝐻

ℎ

𝜌𝐆

𝝂,

(22)

1 of matrix G. Equation (22) corresponds toequation (11) in our simple example above. As stressed earlier, introducing forward information that varies
across horizons creates a variation across horizons in the optimal weights which did not exist in the standard
noisy information framework. In equation (11), the variation in the weights was captured by 𝑤
Here it is captured by the term 𝐆𝜌𝐆, where 𝐆 and 𝐆

the forecasts for consecutive horizons (𝐆𝑊 .

summarize the optimal weights applied inis adjusted to the subsequent horizon ℎ by the loading 𝜌).

Also similar to equation (11), the weight differential is multiplied by the signals. In the more generalframework, the signals are divided into three components:
1. The lag component 𝑥

|

𝜌𝑥

|

, capturing signals from the past. This recursive formof the Kalman filter was absent in our above example, since the example above assumed thateventually the fundamental was perfectly observed.
2. The ex-post “errors” of lagged forecasts 𝒙

𝒙|, representing the additional information in

the new forward signals.

9
Although we rule out these types of correlations to simplify the analysis, our framework still imposes a structural
is correlated
correlation of the signals across horizons, due to the correlation between future fundamentals (i.e., 𝑥). Alternatively, the signals can refer to the future shocks 𝜔 s and therefore be uncorrelated. However,with 𝑥we will demonstrate that these types of forward signals could not account for the empirical patterns in the previous
section.
10
See Appendix A. The analysis there also covers the case with common noise in the signals.

13

3. The noise in the new signals (𝝂 ).
Equation (22) is key for understanding the empirical patterns documented in the previous section. It defines
the missing error term in the regression of 𝑥

on 𝑥

|

as in specification (4). Furthermore, it showsthat the error-term is correlated with the regressor 𝑥|

since the forecast 𝑥||

applies the same

signals that are captured by the error term. Consequently, the estimated coefficient would be biased, while
the sign of the bias depends on the signs of the elements in the weight differential vector 𝐆

𝜌𝐆

.11

When the horizon is beyond 𝐻, the weight differential would shrink so that the bias goes to zero and the 𝑅converges to 1. As in the simple model above, equation (22) further allows us to extract the forward signals
about future fundamentals which we cover in Section 5.

3.3.Simulation
The purpose of the simulation is to examine whether our forward information framework can qualitatively
replicate the empirical patterns documented in the previous section. We perform simulations that use
different degrees of persistence in the state process. Each simulation is based on 1,000 draws and includes
the four following steps:
I.

Simulating the state equation: The state process is simulated with a certain degree of persistence(𝜌) for a period similar to the SPF survey (about forty years of quarterly data). The variance of theshocks to the fundamental is set to 𝜎

1.

II. Simulating the measurement equation: Forward noisy signals are simulated for a group of 40
forecasters (similar to the number of participants in the SPF survey). The horizon of the signals
runs from 0 to 7, and the vector of the noise variance is set to
10000

100

4

3 2

1

𝝈

0.5 0.2 . This structure assumes that noise increases in thehorizon, and signals become extremely uninformative for horizons ℎ

6 and ℎ7, where thenoise variance goes to 100 and 10000, respectively (and then to infinity). To examine if the patternof increasing noise is essential, we also conduct a second set of simulations in which all the varianceparameters in 𝝈

are set to 2.

III. Computing the forecasts: The gain matrix G is computed, based on the Kalman filter, and then used
to calculate optimal forecasts, with horizons running from 0 to 7, for the 40 simulated forecasters,
using equation (20).

11

By contrast, if forward signals refer purely to future shocks, as mentioned in footnote 9, the error-term would not
be correlated with 𝑥| and the estimated coefficient would not be biased.

14

IV. Estimating regressions: For a certain “quarter” in the middle of the simulated sample, we run crosssectional regressions of 𝑥

|

on 𝑥

| , for each ℎ between 1 to 7, and obtain the coefficientestimate and 𝑅 statistic.12
Figure 5 reports the estimated persistence 𝜌 (Panel A) and 𝑅 statistics (Panel B), measured on thevertical axis. The value of 𝜌, applied in each simulation, is indicated by the first horizontal axis. The second
horizontal axis indicates the horizon of each regression (i.e. ℎ𝑥|

3 corresponds to a regression of 𝑥|on

. The description is therefore equivalent to Figure 1, with a further dimension of variation in 𝜌.
The simulations qualitatively reproduce all patterns documented in the previous section.

Specifically, both the estimated persistence and 𝑅 demonstrate the pattern of convergence across horizons.
For short horizons their values are low. As the regression is estimated for longer horizons the coefficientgets closer to 𝜌 and the fit of the regression gets stronger. In addition, for each horizon, 𝜌 and 𝑅 get higher
when 𝜌 is higher. Appendix C provides more details of the simulation results, which are in line with the
SPF evidence. It also makes a comparison with another set of simulations, in which the variance of thenoise in the signals is fixed across the horizons.
Besides rationalizing the empirical patterns, our model and simulations are instructive about thedetection of horizon 𝐻, namely, the point in the future from which forward signals are no longer
informative. The simulation results imply a convergence of the coefficient estimate to the value of 𝜌. On
one hand, we have seen that estimating the persistence of inflation with specification (4) could lead to abiased measure, mainly downward. On the other hand, this bias would vanish if we use forecasts with longenough horizons, for which forward signals are almost uninformative. Similarly, in Figures 1 and 2, weobserve signs of convergence for 𝜌 at horizons ℎ3,4. This convergence thus indicates that the signals atthese horizons should be weak. Hence, SPF forecasters obtain useful forward information until about a year
ahead on average. Furthermore, at these horizons, the estimated persistence is less likely to be biased.

4. Forward information: Direct evidence
Building on Goldstein (2021), we can have a more direct test of forward information by focusing on thedeviation of an individual forecast from the consensus forecast. Note that the optimal vector of forecasts inequation (20) could be expressed as follows (using the measurement equation (19)):

12

As in Figure 2, we apply a narrow window of 8 cross-sections ending at the chosen “quarter”. Similar qualitative
results are obtained with a single cross-section.

15

G 𝒚𝒊

𝒙|

𝒙|

𝐼

𝒙|

𝒙|

G 𝒙

𝐺 𝒙|

𝐺 𝒙

𝝂𝒊 .

𝝂𝒊

𝒙|

(23)

Next, we take the average across individuals (hence we drop superscript 𝑖) and obtain:𝐼𝒙|where 𝒙 | and 𝒙 |

𝐺 𝒙|

𝐺𝒙 ,

(24)

are cross-sectional averages (consensus forecasts) of 𝒙 | and 𝒙 |, respectively.

Subtracting equation (24) from equation (23) we get
𝒙|

𝒙|

𝐼

𝐺 𝒙|

𝒙|

𝐺𝝂 .

(25)

Equation (25) describes a simple relationship between the deviation of an individual forecast from theconsensus forecast in period 𝑡 and the lagged deviation from period 𝑡
is uncorrelated with forecasts made at time 𝑡

1. Because idiosyncratic noise 𝝂

1,13 we can use OLS to directly estimate the elements in thegain matrix, row by row, by running a regression of the deviation from the mean on lagged deviation, foreach forecast horizon:
𝑥

𝑥

|

𝛽 𝑥

|

𝛽 𝑥
𝛽

𝑥

|

|

𝑥

|

𝑥|

where the 𝛽 coefficients are elements of row 𝐻

⋯

|

𝑥|

𝑒𝑟𝑟𝑜𝑟 ,ℎ1 in the matrix 𝐼(26)

𝐺 . Note that the set of regressorsin equation (26) is the same for all ℎ.
Specification (26) can be viewed as an augmented version of the standard noisy information model
where agents receive noisy signals about the current value of the fundamental. Indeed, Goldstein (2021)
has proposed the following specification for the standard model:𝑥where 𝛽|𝑥is equal to 1

𝛽

|

𝐺

𝑥

𝑥

|

and 𝐺

𝑒𝑟𝑟𝑜𝑟 ,|

(27)

is the Kalman gain, representing the weight placed on

the single noisy signal. The standard framework imposes a restriction on the estimated matrix 𝐼

𝐺 ,requiring that all the off-diagonal elements should be equal to zero. This provides a straightforward test of
our model with forward information against the null of the standard noisy-information version without
forward signals: We simply need to estimate specification (26) and test the significance of all thecoefficients, other than the coefficient on 𝑥

|

𝑥

|

.

13

A similar specification is obtained, when adding a common noise besides the individual-specific noise. As
demonstrated in appendix A, by taking the deviation of the individual forecast from the mean, the term with thecommon noise will be dropped out, since it appears in the same way in both forecasts.

16

Specification (26) is also useful for testing the null that heterogeneity in information may only be
due to forward signals, while information about realized inflation is not noisy as in the standard framework.
Under the null, the coefficients in the regression for ℎ0 (i.e., backcasts of the last quarter are also

provided by SPF participants) should all be zeroes. Intuitively, when all forecasters observe realized
inflation in the same way, the deviation of their backcasts from the mean should not be persistent.14 We
implement this test in Table 2. We find that none of the coefficients are significant when the regression isestimated on the full sample. Although some coefficients are statistically significant when we estimate
specification (26) decade by decade, the estimates are economically small and R2 stays close to zero. These
results suggest that information is noisy because of forward signals rather than signals about the current
state of the fundamental.
To explore the importance of forward information further, we estimate specification (26) for longer
horizons (ℎ0) and test the restrictions implied by specification (27). As reported in Table 3, thecoefficients on 𝑥

|

𝑥

|

, which represent diagonal elements, are strongly significant. But ineach estimation, there is at least one additional coefficient that is highly significant, despite the fact that
regressors tend to be correlated. In other words, we have at least one non-zero off-diagonal element in eachrow of the gain matrix (columns in Table 3 correspond to rows in the matrix 𝐼

𝐺). This is consistent with

SPF forecasters utilizing forward signals. Furthermore, information criteria suggest that including otherhorizons in specification (26), as opposed to (27), improves the fit considerably, which is consistent with
important forward information. Although it is hard to provide economic interpretation for the estimatedcoefficients, we estimate specification (26) on simulated data to check if the estimates in Table 3 areplausible. We find (Appendix C) that in the more realistic case, where the variance of the noise increases
in the horizon, the simulation provides patterns that are similar to Table 3.15 For example, the dominantcoefficients are those that correspond to the diagonal elements in 𝐼

𝐺 and they tend to increase along

the diagonal, namely, when the specification is estimated for longer horizons. The standard noisy
information model cannot reproduce this pattern. In summary, forward information appears to bequantitatively important.

5. Forward information: Measurement and applications

More formally, the null imposes a restriction on the last row of the gain matrix, specifically, 𝐆
0 0 ⋯ 1 . Hence, the corresponding row in 𝐼 𝐺 , estimated by (23), should be a vector of zero coefficients.
The tractable example in Section 3.1. implies such a restriction. See also the analysis in Appendix B.2.
15
In Appendix C, we also verify that truncation of longer horizons with additional information, which can happen inpractice, does not raise concerns about the available coefficient estimates.
14

17

Measurement of news about the future is usually a challenging task that requires structural restrictions oradditional variables. We propose an alternative approach to recover forward information from expectations
data. We first illustrate how the quantification of forward information directly follows from our framework,and then use several applications to demonstrate our method, focusing on SPF inflation forecasts.

5.1.

Quantifying forward information

We can re-write equation (22) in the following form:
𝑥

|

𝜌𝑥

𝜌𝑥

|

𝐹𝐼

|

| ,

(28)

where
𝐹𝐼
and 𝐹𝐼
𝐹𝐼

|

|

|

≡ 𝑥

|

𝐆

𝜌𝐆

𝒙

𝒙|

𝜌𝐆 𝝂𝒊

𝐆

represents the adjustment to the forecast induced by forward information. The expression for

has three terms. The first term in the expression is forward information inherited from the previous

period. The second term is new information from observing 𝒙 . The last term is the noise contained in thesignal about 𝒙 . One can also use equation (11) to re-write 𝐹𝐼
𝐹𝐼

𝑊,

|

𝑊,

𝑆𝑖𝑔𝑛𝑎𝑙 ,

|

as:

| ,

(29)

which underscores that forward information is a function of signals about the future.
Although signals in equation (29) are not observed, one can readily recover forward information
using equation (28):
𝐹𝐼

|

𝑥

|

𝜌𝑥

| .

(30)

By taking the average across forecasters, we obtain an aggregate time series of forward information:
𝐹𝐼

|

𝑥

|

𝜌𝑥

| .

(31)

Equation (31) suggests a simple method for quantifying forward information. To illustrate how it works,
suppose we are interested in constructing forward information at horizon ℎ
𝐹𝐼 |

𝑥|

𝑐̂

𝜌

𝑥

,

0. To this end, we compute:(32)

where 𝑐 and 𝜌 are time-varying intercept and slope that generalize equation (31) by applying the following
steps:(i)

Forecast data: For 𝑥 | , we use the average inflation forecasts for the current quarter. The
backcast 𝑥|that refers to inflation in the previous quarter is replaced by actual lag 𝑥

. As

18

we show above, the information on realized inflation is almost perfect, making 𝑥

equivalent

to the backcast.
(ii)

Persistence parameter: Because forecasters may use persistence 𝜌 that it is different from theactual value, we need to recover 𝜌 from forecasts. As we show above, 𝜌 estimated by OLS
varies with the horizon because OLS is biased when forward information is present. However,as we increase the forecast horizon, one may expect forward information to become less precise
and thus the bias in the OLS estimate of 𝜌 should decline. Consistent with this notion, we
observe that 𝑅 and 𝜌 increase with the horizon and stabilize at long horizons. Because some
forecasts are missing at ℎ4, we use ℎ

3 to estimate 𝜌, that is, we regress 𝑥

and we essentially use the time-varying estimates in Figure 2. Note that we use 𝜌

|

on 𝑥

|

rather than𝜌 to recover 𝐹𝐼 | because the time-varying 𝜌 may contain some news with respect to changes
in the persistence parameter (𝜌 will be used to evaluate 𝐹𝐼
(iii)

|

for ℎ

0).

Constant parameter: The AR(1) state equation in equation (16) (and equation (31)) did not include
a constant. In practice, we include a constant that can capture time-varying trend inflation. The
constant is estimated by the time-varying regressions from the previous step, and we take again
a lag 𝑐̂, due to news about changes in trend inflation that are embedded in 𝑐̂ . 16

Using the same logic, we can recover forward information for other horizons from:
𝐹𝐼

|

𝑥

|

𝑐̂

𝜌𝑥

|

.

(33)

Note that this equation measures only the “marginal” forward information across horizons.17
Figure 6 plots the series of forward information produced by equations (32) and (33) for CPI
inflation and the corresponding projections in the SPF. Panel A shows a strong co-movement of 𝐹𝐼 | andactual inflation. In particular, times of high variation in forward information corresponds to times of highvariation in inflation. Other panels show series for ℎ

1, … ,4. The series of forward information fluctuatequite closely to each other and variation diminishes as the horizon increases.
More generally, equations (28) and (29) suggest several properties for forward information. First,
the variation of forward information over time decreases in the horizon. This pattern is in line withdiminishing information in forward signals for longer horizons. It is also driven by the decay in weight

The results are quite similar though, when employing 𝑐̂ and 𝜌 , as well as when employing the survey backcasts
instead of the actual lag 𝑥 .
17
Although we stick here to a simplified AR(1) process, our method can be easily extended to processes of higher𝑥|order by the same principles. For example, the representation of equation (32) for VAR (1) would be 𝐹𝐼 |
𝝆 𝒙
is a vector of variables and 𝝆
is a vector of coefficient estimates. The coefficientswhere 𝒙𝑐̂will again be estimated using forecast data, by regressing 𝑥 | on a vector of forecasts 𝒙
| .
16

19

differentials 𝑊 ,𝑊,across horizons. As illustrated in Section 3, 𝑊 ,approaches 𝑊 , , as thehorizon increases. Second, the series for forward information are correlated across horizons because thesame signals are applied at each horizon. The correlation should eventually decay due to the diminishingvariation. Third, series for forward information should be serially correlated due to the overlap in forward
signals over time. That is, previous forward signals which look beyond time 𝑡 are still useful for the forecast
made at time 𝑡. These properties are broadly supported by the summary statistics presented in Table 4.18

5.2.

Forward information and predictability of inflation

If useful, forward information should help to predict future inflation. To assess the quality of forward
information, we regress future values of inflation on lagged inflation and forward information. We reportresults in Table 5. Column (2) provides estimates for the nowcast. This regression has two predictors: thelagged inflation and forward information 𝐹𝐼 | . The coefficient on forward information is highly significant
and 𝑅 rises dramatically by ≈0.6 compared to a regression without 𝐹𝐼 | (column (1)). Thus, forward
information can explain a large share of current inflation.
Each subsequent column reports estimates from a regression of inflation at time 𝑡ℎ on lagged

inflation and all forward information components available at time 𝑡, which refer to future periods up to𝑡ℎ. We find a significant coefficient for 𝐹𝐼| . For longer horizons the effect is not significant, but so

is the effect of lagged inflation. Overall, these findings are in line with the evidence in Section 2. As
demonstrated above, the estimates of persistence regressions in Figure 2 were biased due to forward
information, which is more dominant at the shorter horizons. The results in Figure 2 illustrate that thehorizons with the main biases are ℎ0,1. Thus, the main predictability of forward information should beobtained for these horizons.19
Another way to gauge the importance of forward information is to examine the ratio
∑ 𝐹𝐼

|

∑ 𝑥

|

where the numerator is the variation in marginal forward information (thatis, information that applies only to horizon ℎ) and the denominator is the variation in the consensus forecast𝑥| . A higher value of the ratio represents a larger contribution of forward information to forecasts. Table

18

Another related implication of our measure of forward information, based on the theoretical analysis in Section 3,
is that variation in news over time may induce variations in expectations' persistence, especially at the shorter horizons.
In times of big events, with potentially high provision of forward information, this biasing effect can be particularly
strong, even at longer horizons. Appendix E explores such events and reports evidence in this direction, especiallyfollowing the outbreak of COVID-19.
19
In Appendix E we present another related application in which we analyze the impulse response to our inflation
forward information in VARs. We also compare our approach to the approach of news shocks in Barsky and Sims(2011).

20

4 reports that the variance of 𝐹𝐼 | is relatively close to the variance of actual inflation which points towarda high contribution of forward information. We find that the ratio for the nowcast is 0.356 for the full sample
but the ratio has been increasing over time.20 For longer horizons, the ratio is 0.157 for ℎ
to 0.033 for ℎ

1 and declines

4. The results suggest that professional forecasters have a lot of additional information

about current and near-future inflation but the quality of (marginal) forward information rapidly declines
in forecast horizon.

5.3.

Forward information and disagreement

As we discussed above, variation in forward information across agents can be a source of disagreement inforecasts. Although previous subsections focus on consensus forecasts, we can apply the same algorithm tothe data at the forecaster level. In particular, we can estimate persistence 𝜌 and intercept 𝑐 for each
forecaster 𝑖 and then compute the “model” component of the forecast with 𝑐̂
express forward information as 𝐹𝐼|

cross-sectional variation in forecasts (𝑥
(𝐹𝐼

𝑥

𝑐̂

|

𝜌𝑥

|

𝜌𝑥

|

and hence.21 In the next step, we compare the| ) with the cross-sectional variation in forward information| ). We report results in Table 6.

We find that forward information accounts for a large fraction of cross-sectional variation in thedata. For example, nowcasts and one-step-ahead forecasts are characterized by 𝑣𝑎𝑟 𝐹𝐼
𝑣𝑎𝑟 𝑥
term”
𝜌𝑥

|

|

. This pattern also underscores the negative correlation between forward information (“the errorinequation

(28))

and

the

model-implied

forecast(𝑐̂| ), which rationalizes the bias in the estimates of inflation persistence. As we increase thehorizon, the cross-sectional variation in forward information declines faster than the cross-sectionalvariation in forecasts, which yields 𝑣𝑎𝑟 𝐹𝐼

|

𝑣𝑎𝑟 𝑥

|

for ℎ

2. However, even at longer

horizons ("marginal") forward information accounts for more than a third of the variation in forecast
disagreement and thus remains a meaningful contributor to forecast dispersion.

5.4.

Forward information and the Taylor rule

20

Appendix Figure G.5. presents time series for the ratio for the nowcast. The ratio fluctuates between 18% to 66%
over the sample period. The contribution of forward information deteriorates following the years of high inflation
which can result from a growing inattention to inflation following the Great Moderation and the decline in trend
inflation. However, the ratio recovers in the recent years which can result from higher provision of informative forward
signals in recent years, potentially due to forward guidance provided by the central bank.
21
We also follow the analysis in Figure 4 and estimate time-varying persistence and intercept for each forecaster.

21

If forward information helps predict inflation, one may expect policymakers to act on this information.
Indeed, given that monetary policy affects the economy with significant lags, policymakers should react tonews about changes in future inflation rather than wait until changes in inflation materialize. Following
Romer and Romer (2004) and Coibion and Gorodnichenko (2011), we apply the following specification totest this prediction:
𝑟

𝑐

𝛾𝜋 |

𝜃 𝑔𝑎𝑝 |

𝜃 𝑔𝑟 |

𝜌 𝑟

𝜌 𝑟

𝜀,

(34)

where 𝑟 is the federal funds rate, 𝜋 | is expected CPI inflation (nowcast) and 𝑔𝑎𝑝 | and 𝑔𝑟 | areexpectations (nowcast) of the output gap and GDP growth, respectively. We use forecast data from the Fed
Greenbooks (hence, the superscripts GB), which are prepared by the Fed's staff before FOMC meetings.
Our focus is on the effect of 𝐹𝐼| , which we construct following equations (32) and (33).

Table 7 presents estimation results of specification (34) for 1983Q1-2015Q4.22 Each column
applies different measures of expected inflation. Column (1) presents results for the standard policy reaction
function estimated in the literature. The estimates are broadly consistent with the results reported in earlier𝜌work. For example, the policy response is highly inertial (𝜌inflation is generally consistent with determinacy (𝛾/ 1
In column (2), we use 𝜋 |𝑐̂𝜌contributions due to the “model” component 𝑐̂𝜌𝜌rate is largely explained by macroeconomic conditions (𝑅1.05). The dynamics of the fed funds

0.982).
𝐹𝐼 |𝜋𝜌0.95) and the long-run response to𝜋(i.e., equation (32)) to separate theand forward information (𝐹𝐼 | ). We

modify equation (34) to have different responses to these two components. We find that only the forwardinformation component is significant, while the “model” component of 𝜋 | is not. These results suggest
that, like nowcasts in the SPF, the Greenbook nowcasts for inflation contain information useful forpolicymakers and they respond to this information strongly.
Column (3) presents results for an augmented specification where in addition to 𝐹𝐼 | we control
for forward information at longer horizons. We find that policymakers significantly respond to 𝐹𝐼 | and𝐹𝐼| . The estimated coefficients are positive for longer horizons (ℎ1 in column (3)) but they are

imprecisely estimated. These findings are in line with our earlier results for the SPF, i.e., forward
information is useful at short horizons but its ability to forecast at longer horizons is limited.
22

The Greenbooks are published with a lag of five years. To apply (32) and (33) we use the time-varying estimates of𝑐̂ and 𝜌 that were obtained above using cross-sections of SPF forecasts. For this reason, the sample period starts at
1983 which is also right after the Volcker disinflation. For lagged inflation, we take backcasts of the Greenbooks toalign with the Greenbook's definition of quarterly inflation. There are 8 Greenbooks every year which precede themeetings of the FOMC, usually two in a quarter. In each quarter we take the forecasts from the earlier Greenbook
which are provided around the same time of the SPF forecasts. The Greenbooks are available at the website of the
Philadelphia Fed.

22

To assess the stability of this pattern, we split the sample roughly in halves (1983-1999 and 20002015) and report results in columns (4) and (5). We find that forward information can predict policy in bothsubsamples. However, forward information at longer horizons appears to be more important in the second
subsample: the coefficient on 𝐹𝐼|

is now also statistically significant. We conclude that forward

information plays a significant role in shaping monetary policy.

6. Evidence from additional macro forecasts
To preserve space, our empirical analysis has been focused on inflation expectations from the SPF. Thissection briefly reports results for additional macroeconomic expectations. For a detailed description of theresults, see Appendix F.
More SPF forecasts. We examine forecast data of other measures of inflation (core, PCE, and GDP deflator
inflation) and additional key variables (GDP growth, unemployment and interest rates). We first estimate
the persistence regressions from Section 2 across the forecast horizons. The results broadly confirm thestylized patterns documented in Section 2: both the persistence estimate and the fit of the regression tend
to increase with the forecast horizon. Some variables show more sensitivity to the underlying process sothat the pattern is more clearly demonstrated for higher-order AR processes. We also apply the direct
approach proposed in Section 4. We again find evidence consistent with forward information. We thenextract the forward information component for some variables, applying the method from Section 5. We
find that this measure of forward information has significant predictive power for future movements in thecorresponding variables.
Fed forecasts. In Section 5.4, we estimate a Taylor rule using projections from the Fed’s Greenbooks. The
availability of Greenbook forecasts for multiple horizons enables us to examine if the stylized pattern of
persistence that was documented in the SPF (Section 2) also exists in forecasts of the Fed. Since there is nodispersion in Greenbook forecasts, we estimated time-series regressions. Yet, the pattern of increasing
persistence across horizons is detected in Greenbook forecasts for several key variables (for comparison wealso estimated similar time-series regressions with mean-level SPF forecasts). Thus, beyond the specificapplication of the Taylor rule, this evidence points to a more general utilization of forward information by
the Fed staff.
ECB SPF inflation forecasts. Another Survey of Professional Forecasters is managed by the European
Central Bank. We investigate the presence of forward information in forecasts of European CPI inflation.
In the U.S. SPF we used quarterly forecasts for multiple horizons which are not available in the ECB SPF.
23

Instead, we took advantage of two types of annual forecasts that are provided in each quarter: calendar
forecasts referring to current and next calendar years and rolling forecasts referring to a year and two years
ahead. We estimate persistence in both types of forecasts and document a new type of variation in the first
type. Specifically, persistence in calendar-year forecasts decreases over the calendar quarters of the year.
This pattern is consistent with the presence of forward information that varies across horizons. In thecalendar forecasts, the horizon changes when moving from quarter to quarter so that forward information
increases over the year with a biasing effect on the estimated persistence. This pattern does not appear in
the rolling-year forecasts, for which the horizon is fixed in each quarter. Interestingly, we find the same
effect in annual forecasts of the U.S. SPF. These findings provide additional support for the presence of
forward information in both surveys.

7. Conclusion
Although there has been an explosion of research on how expectations are formed and departures from fullinformation rational expectations are increasingly clear, much remains to be learned. There is also a new senseof urgency to shed more light on expectations as central bankers and other government bodies must
increasingly rely on tools based on the management of expectations (e.g., forward guidance) in the hopes ofappropriately changing the beliefs about future macroeconomic aggregates and hence stabilize the economy.
We propose a new approach to characterize the expectations formation process. We formalize thefact that projections are a combination of model-based prediction and add-factoring, which uses forward
information that is not incorporated into the model and current/past values of macroeconomic variables.
Specifically, we extend the canonical noisy-information model by introducing signals about future
fundamentals (forward information) at multiple horizons. Thus, we emphasize information variation not
only across agents but also across future horizons. We find that cross-sectional dispersion in SPF forecasts
is driven by forward information. That is, forecasters disagree about the future while holding (almost)perfect information about the realized values. The presence of forward information can also rationalize why
perceived persistence of fundamentals increases in the forecast horizon, an empirical pattern that we
document in the SPF data. We propose a simple method to extract the forward information component inforecast data and demonstrate the usefulness of this direct measure for forecasting and policymaking.
We view these findings as a first step in utilizing forward information to understand how economic
agents form expectations. For example, we use forward information to directly measure the impact of central
bank communication on the beliefs of economic agents. We focused on professional forecasters who are someof the most informed players in the economy and who likely place a lot of weight on forward information.

24

Consumers and firms may be less sophisticated and more inattentive to incoming macroeconomic data. As aresult, they could be better characterized by a mix of noisy information about past, current and future
fundamentals. To determine what type of information is a good description of consumers and firms, one mayneed to run customized surveys of these agents that mimic the structure of the SPF.

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26

Figure 1: Persistence Across Forecast Horizons
Panel A: 1981-2017 (full sample)

Panel B. Results by decade

Notes: The figure plots persistence coefficients and R-squared statistics, based on estimating specification (4) for different forecasthorizons, using different CPI inflation forecasts from the SPF survey. The whiskers show the confidence interval around the point
estimate, based on Driscoll and Kraay (1998) standard errors. P-values refer to the test of equality of coefficients across forecastinghorizons.

27

Figure 2: Expectation-Based Persistence by Forecasting Horizon
Panel A: Persistence Estimates

Panel B: R-Squared of Persistence Regressions

Notes: The figure plots smoothed estimates of persistence 𝜌 (panel A) and R-squared measures (panel B) based on estimating
specification (4) for different forecast horizons in the SPF survey. Each quarterly point is based on OLS estimation using theforecasts data from the last 8 quarters for a specific horizon. The smoother is a local mean which uses an Epanechnikov kernel.

28

Figure 3: Different Specifications

Notes: The figure plots R-squared measures from four different specifications which were estimated for each quarter of the sample
period, using SPF inflation forecasts. In panel A, inflation expectations follow an AR(1) process as in specification (4). In panel B
inflation expectations follow an AR(4) process, by including forecasts for additional quarters as regressors. In panel C expectationsof the inflation gap follow an AR(1) process. The forecasted inflation gap is computed as the difference between forecasts of inflationand 10-year inflation (sample starts on 1990Q2 because of 10-year forecasts). In panel D inflation expectations follow a VAR(4)process which augments the specification in panel B with four “lags” of unemployment and 3-month interest rate forecasts. All
specifications are estimated for ℎ 3. Each point on the black lines is based on OLS estimation using the forecasts data from thelast 8 quarters. The brown line is a local mean smoother which uses an Epanechnikov kernel.

29

Figure 4: Disagreement about PersistencePanel A: Disagreement about Persistence1

Panel B: Persistence Estimates.2.4

Inflation persistence
.6

.8

h=4
h=3
h=2
h=1
h=0

1995q1

2000q1

2005q1

2010q1

2015q1

Notes: The figure plots smoothed estimates of persistence 𝜌 (Panel B) and cross-sectional standard deviations of persistence (PanelA) based on estimating specification (6) for individuals in the SPF survey. The specification is estimated for each forecaster who
deliver at least 20 observations in a rolling window of 40 quarters. For each time window, the mean across forecasters of persistence
estimates for a certain horizon is displayed in Panel B. The smoother is a local mean which uses an Epanechnikov kernel. The
standard deviation of persistence estimates (at ℎ 4) across forecasters is displayed in Panel A.

30

Figure 5: Simulation Results
Panel A: Persistence Estimates

Panel B: R-Squared of Persistence Regressions

Notes: The two panels in the figure show estimation results of specification (4) applied to a simulated data of forecasts, according
to the model presented in section 3. Estimated persistence (panel A) and R-squared (panel B) are averaged across 1000 draws of thesimulation. Each simulation applies a different value of persistence in the fundamental. The variance of the shock in the state process
is standardized to one. Regressions were estimated for seven forecast horizons out of eight horizons for which noisy signals areavailable (two consecutive horizons in each regression). The noise-to-signal ratio at horizons ℎ 0, … ,7 is set to 0.2, 0.5, 1 ,2 ,3 ,4,100, 10000, respectively.

31

Figure 6: Inflation Forward Information

Notes: The figure plots time series of forward information for inflation. Forward information is computed according to equations(32) and (33).

32

TABLE 1
Summary of regression properties in the example of section 5.2
Forecast Horizon
ℎ 1
ℎ 2
ℎ 3
𝜌

0

0

Persistence
𝜌 1

𝜌

1

Estimated
persistence0

Positive but Biased.
Bias decreases in 𝜎
𝜎
𝜎 .
if 𝜎

𝜌

0

Increases in 𝜌 if𝜎𝜎 .
𝜎

Positive but
BiasedR-squared

0

Between 0 and 1.
Increases in 𝜎 if
𝜎
𝜎 .
𝜎

1

0

Increases in 𝜌

Between 0
and 1

Notes: The table summarizes the theoretical predictions by the model analyzed in section 3.3. Model predictions
refer to the estimated persistence 𝜌 and R-squared of a regression of the forecast 𝑥 | on 𝑥
| .

33

TABLE 2
Regressions of the deviation from the mean backcast
Dependent variable:
Full
1980s
𝑥 | (backcasts)𝑥|
Sample
𝑥|
𝑥|
0.018
0.118
𝑥
𝑥
𝑥
𝑥

𝑥

|

𝑥

|

𝑥

|

𝑥

|

Constant
Obs.
𝑅

|

|

|

|

1990s

2000s

2010s

0.017**-0.0000.002**(0.014)(0.082)(0.008)(0.000)(0.001)

0.009

0.014

0.032**

0.000

0.002

(0.016)(0.084)(0.015)(0.000)(0.001)-0.029-0.202**-0.034***

0.001

0.000

(0.018)(0.085)(0.013)(0.001)(0.003)

0.009

0.078

0.013

-0.0000.002(0.018)(0.123)(0.015)(0.001)(0.002)0.001-0.007-0.009-0.0000.002**(0.014)(0.067)(0.009)(0.001)(0.001)

0.004

0.027

0.002

0.000

0.001

(0.005)(0.037)(0.002)(0.000)(0.002)

3,849
0.004

559
0.023

1,068
0.023

1,272
0.003

950
0.002

Notes: The table reports coefficient estimates from regressions of the individual deviation from the mean backcast, based on
specification (26) with ℎ 0. Each column reports results for a specified sample period. Each panel refers to a different horizon.
Driscoll-Kraay standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

34

TABLE 3
Regressions of the deviation from the mean forecast
Dependent variable:
𝑥|

𝑥|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

-0.013-0.013-0.063***-0.062***(0.025)(0.021)(0.012)(0.020)0.220***

0.003

0.021

0.032

(0.052)(0.040)(0.044)(0.034)0.130***

0.458***-0.095**-0.056*(0.050)(0.057)(0.046)(0.032)-0.126**-0.120**0.486***

0.103*(0.061)(0.057)(0.069)(0.059)

0.071

0.056

0.037

0.362***(0.066)(0.043)(0.038)(0.044)-0.008-0.001

0.005

0.007

(0.009)3,854(0.006)3,856(0.008)3,855(0.008)
3,853

0.053

0.146

0.213

0.178

BIC

10,515

8,565

7,434

7,323

BIC for specification (27)

10,822

8,763

7,635

7,484

𝑥
𝑥
𝑥
𝑥

|

|

𝑥
𝑥

|

𝑥

|

𝑥

|

|

|

|

Constant
Obs.
𝑅|

Notes: The table reports coefficient estimates from regressions of the individual deviation from the mean forecast, based on
specification (26). Each column reports results for a specified dependent variable, using the whole sample period. BIC reports
the value of Bayes Information criterion for each specification as well as the BIC corresponding to specification (27). DriscollKraay standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

35

TABLE 4
Summary statistics of series for actual inflation and forward information about inflation
Variable:
𝐹𝐼 |

Mean

Standard deviation

Serial correlation-0.1501.122-0.275

Correlation
between horizons𝐹𝐼|-0.078

0.454

0.204

-0.258𝐹𝐼|-0.0290.182-0.073

0.324

𝐹𝐼

|

-0.0170.134-0.209

0.069

𝐹𝐼

|

-0.010

0.104

0.011

-0.094

2.704

1.992

0.350

Actual inflation

Notes: The table reports summary statistics for the time series of (marginal) forward information computed according toequations (32) and (33). The correlation between horizons in the right column shows the correlation between 𝐹𝐼 | and𝐹𝐼| . The sample period is 1983Q3-2007Q4. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

36

TABLE 5
Inflation predictability and forward information 1983-2017
Dependent variable:

𝜋

𝜋

𝜋

𝜋

𝜋

𝜋

(1)

(2)

(3)

(4)

(5)

(6)

1.554***

0.670***-0.0130.411

0.405**(0.146)(0.140)(0.191)(0.257)(0.197)1.275***

0.405

0.415

0.853

(0.397)(0.437)(0.380)(0.605)

0.097

0.373

0.512

(0.936)(0.584)(1.112)-0.1340.191(1.035)(1.814)

𝐹𝐼 |
𝐹𝐼
𝐹𝐼
𝐹𝐼
𝐹𝐼

|

|

|

-0.782|(1.033)𝜋Constant
𝑅

0.349***

0.758***

0.368**

0.186

0.211

0.345**(0.083)(0.095)(0.164)(0.193)(0.175)(0.165)1.757***

0.877***

1.898***

2.210***

2.192***

1.856***(0.221)(0.295)(0.417)(0.432)(0.405)(0.405)

0.122

0.721

0.109

0.035

0.038

0.082

Notes: The table reports coefficient estimates for regressions of current and future inflation on realized inflation and forward
information. 𝑥 ≡ 𝜋 is CPI inflation. The forward information variables are computed according to equations (32) and (33).
Newey-West standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

37

TABLE 6
Cross-sectional variation in forecasts and forward information
𝑣𝑎𝑟 𝑥

|

𝑣𝑎𝑟 𝑐̂

𝜌𝑥|𝑣𝑎𝑟 𝐹𝐼|𝑣𝑎𝑟 𝐹𝐼

|

𝑣𝑎𝑟 𝑥

|

ℎ

0

0.889

0.398

1.143

1.134

ℎ

1

0.610

0.533

0.700

1.071

ℎ

2

0.498

0.336

0.324

0.807

ℎ

3

0.481

0.308

0.279

0.762

ℎ

4

0.459

0.304

0.178

0.623

𝜌𝑥
and
Notes: The table reports cross-sectional variation in inflation forecasts and their two components: model 𝑐̂|
forward information (𝐹𝐼 | ). The components are computed according to equations (32) and (33) at the individual level ineach quarter. Individual-level perceived parameters of inflation are estimated as in Figure 4 for each quarter, using rolling
windows of 40 quarters. Within a window, the cross-section includes forecasters with at least 15 forecasts. The entries in thetable report the mean of cross-sectional variation over the sample period.

38

TABLE 7
Taylor Rule and Forward Information 1983-2015
Full sample

Dependent variable: 𝑟(1)𝜋|

19831999

20002015

(2)

(3)

(4)

(5)

0.031

0.124

0.310***-0.021(0.053)(0.083)(0.085)(0.078)0.058***

0.151***

0.286***

0.088***(0.018)(0.039)(0.079)(0.028)0.176***

0.266**

0.096**(0.059)(0.107)(0.044)

0.069

0.112

0.208**(0.087)(0.136)(0.089)

0.134

0.165

0.276

(0.191)(0.196)(0.440)

0.205

0.223

-0.515*(0.276)(0.425)(0.302)0.051**(0.021)

𝜌

𝑐̂

𝜋

𝐹𝐼 |
𝐹𝐼
𝐹𝐼
𝐹𝐼
𝐹𝐼

|

|

|

|

𝑔𝑎𝑝 |
𝑔𝑟 |
𝑟
𝑟
𝑅

0.025*

0.024

0.027

0.070***

0.026(0.014)(0.015)(0.017)(0.021)(0.023)0.149***

0.149***

0.143***

0.216***

0.135***(0.039)(0.039)(0.038)(0.026)(0.046)

1.134***

1.151***

1.089***

0.889***

1.268***(0.099)(0.111)(0.128)(0.155)(0.093)-0.184**-0.196**-0.185*-0.028-0.323***(0.089)(0.097)(0.103)(0.116)(0.092)

0.982

0.982

0.984

0.962

0.980

Notes: The table reports coefficient estimates for regressions based on specification (34). The quarterly
forecasts are taken from the Fed Greenbooks. The “model” component of inflation forecasts,𝜌 𝜋, uses time-varying persistence estimates from cross-sections of SPF forecasts, as𝑐̂explained in the text. The forward information variables are computed according to equations (32) and (33).
Newey-West standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

39

Online Appendix

1

Appendix A: Forward information with a common noiseThis appendix introduces a common signal to the forward information framework, presented in section 3, and shows
how it affects the main results derived in sections 3 and 4.
Consider the following state-space representation:
State:

𝒙 ≡

𝑥
𝑥

𝜌 0
1 0
⋱ ⋱
0 ⋱

⋮
𝑥

where 𝜔 ∼ 𝑖𝑖𝑑 𝑁 0, 𝜎
𝑆′𝑆𝜎 .

⋯
0
⋱
1

and 𝑆

0
0
𝒙
⋮
0
1 0

𝑆′𝜔

Ρ𝒙

𝑆′𝜔

(A.1)

0 , so that the variance-covariance matrix of 𝑆′𝜔⋯

would be Σ

Measurement:𝑦⎡ ,𝒚𝒊 ≡ ⎢⎢𝑦 ,
⋮
⎢
⎣ 𝑦,

𝜈
𝑥
𝑒
⎤
⎡ ,
⎤
𝑥
𝑒
⎥
⎢𝜈 ,
⎥
𝝂𝒊 𝒆
(A.2)
𝒙
⎥
⎢
⎥
⋮
⋮
⋮
⎥
⎢
⎥
𝑥
𝑒
⎦
⎣ 𝜈, ⎦𝑰 𝝈 with 𝝈where 𝝂𝒊 is an iid normally distributed idiosyncratic noise and Σ𝜎𝜎⋯ 𝜎 . We also
𝑰 𝝈 where the variance may also vary
introduce the common iid noise 𝒆 with a variance-covariance matrix Σacross horizons. All types of shocks are uncorrelated contemporaneously and at all leads and lags.
To solve for the optimal forecast by the Kalman filter, we use the following Riccati equation:
Ψ

ΡΨ

Ψ Ψ

Σ

Σ

Ψ Ρ′

Σ

(A.3)

where Ψ is the variance of the one-step ahead forecast error. The Kalman gain matrix, denoted by G (with dimension
𝐻 1), would then be obtained by:
G

Ψ Ψ

Σ

Σ

(A.4)

Thus, the optimal forecast of an agent 𝑖 would be:
𝒙|

G 𝒚𝒊

𝒙|

𝒙|

(A.5)

where 𝒙 | is a vector of forecasts made at time 𝑡, with horizons running from 0 to 𝐻 steps ahead. The forecast takes thesame form as the forecast in equation (20) in the main text, except that the gain matrix would be different and signals in
𝒚𝒊 also contain common noises.
| , which is an element in 𝒙 | , can be written as:

From (A.5), the ℎ step-ahead forecast 𝑥
𝑥

|

𝑥

|

G, 𝑦,

𝑥

𝑦,

𝑥|

G,
where the coefficients G , , G , , …, G ,

|

𝑥

G, 𝑦,

are elements of row 𝑗

𝐻

ℎ

|

|

𝜌𝑥

|

(A.6)

1 in the gain matrix G.

As in the case without a common noise, when forecasting more than 𝐻 steps ahead (ℎ
be:
𝑥

⋯

𝐻), the forecast would simply

(A.7)
2

which resembles equation (4) in the standard noisy information framework. We now examine the relationship between
𝑥 | and 𝑥ℎ 𝐻). From (A.5) the forecast 𝑥| would be| for shorter horizons (0
𝑥

𝑥

|

|

G

G

,

𝑦,

G

,

|

𝜌𝑥

(A.8)

𝑥|

𝑦,

|

G,

are the elements of row 𝑗

Comparing to (A.7), the term 𝑥
𝜌𝑥

𝜌𝑥

|

G,

𝜌G

,

𝑦,

𝑥

|

𝜌G
𝜌G

G,
|

⋯

|

1 in the gain matrix G. Multiplying (A.8)|

𝑥

the relation 𝑥|𝑥where the coefficients G , , G , , …, G ,by 𝜌 and subtracting it from (A.6), we obtain:
𝑥

𝑥

𝑦,

,

𝜌𝑥

|

|

|

⋯

(A.9)

𝑥|

𝑦,

,

𝑥

𝑦,

,

does not equal to zero. The RHS of (A.9) expresses the deviation from| , for each horizon in 0ℎ𝐻.

To get a compact version of (A.9) that corresponds to the key result in equation (22) in the main text, denote the vector
that is equal to row 𝑗 in matrix G by 𝐆 . Moving 𝜌𝑥| to the RHS and using substitutes from (A.2.) we obtain:
𝑥

|

𝜌𝑥

𝑥

|

𝐆
Thus, the forecast 𝑥

|

𝜌𝑥

|

𝜌𝐆

𝝂

𝒊

𝟏

𝐆

𝜌𝐆

𝒆

|

𝐆

𝜌𝐆

𝒙

𝒙|

is a composition of a prediction based on the underlying process (𝜌𝑥(A.10)| ), and “subjective

adjustment” which account for forward information. In the standard noisy information framework, as in section 2, the𝜌𝑥second component would be absent, so that the optimal forecast would obey 𝑥 || . The need for an
adjustment component in the presence of forward signals is driven by the variation across horizons in the optimal𝜌𝐆
are consecutive
weights placed on the signals. This variation is captured by the term 𝐆, where 𝐆 and 𝐆
rows in the gain matrix.
Hence, equation (A.10) has the same interpretation as equation (22). The only difference is the presence of an additional
𝜌𝐆term 𝐆𝒆 in the adjustment component, which represents the effect of the common noise. The empiricalimplications for the estimation of the parameter of persistence by specification (4) would also be similar: the adjustment
component introduces an error to the regression. The error would bias the coefficient estimate from 𝜌 because it iscorrelated with the RHS forecast 𝑥| . The common noise introduces another source of correlation, but only in thetime dimension. In a cross-sectional estimation of specification (4), common noise can be captured by including a
constant in the regression. In a panel estimation, the time correlation between 𝒆 and 𝑥| can also be captured bytime fixed-effects. Thus, the only remaining source of bias would be the idiosyncratic noise 𝝂𝒊 , leading to the same
empirical pattern that is addressed in the main text. Specifically, when estimating persistence by specification (4) for
different horizons, the coefficient estimate would converge to 𝜌 and the fit of the regression will get close to perfect, forlonger horizons, due to the decay in forward information at farther horizons.
Next, we turn to specification (26) from Section 4 which is used to directly estimate the gain matrix in the forward
information framework and to test this framework against the standard setup of noisy information. Specification (26) is
derived by subtracting the mean forecast from the individual forecast. With common noise, the individual forecast from
(A.5) could be written as (using a substitute from (A.2)):
𝒙|

𝐼

𝐺 𝒙|

𝐺 𝒙

𝝂𝒊

𝒆

(A.11)Taking the average across individuals (denoted by dropping superscript 𝑖) we obtain:

3

𝒙|

𝐼

𝐺 𝒙|

𝐺 𝒙

𝒆

(A.12)

Notice that only 𝝂𝒊 is vanished in the mean forecast while the common noise 𝒆 remains. Thus, when subtracting themean from the individual forecast, the common noise disappears, obtaining:
𝒙|

𝒙|

𝐼

𝐺 𝒙|

𝒙|

𝐺𝝂𝒊

(A.13)

which is similar to (25) in the main text. It follows that even in the presence of a common noise, specification (26) could
be applied in the same way to estimate 𝐼 𝐺 and to preform tests. The error component in the regression is only due
to the individual-specific noise.
In a mean-level estimation such as a regression of average forecast errors on average forecast revisions, which is
discussed in section 3.1, the results would be affected by the presence of a common noise. Coibion and Gorodnichenko(2015) who proposed this mean level specification to estimate the Kalman gain in the standard noisy information setup,show that common noise introduces estimation bias. Similarly, common noise will affect the estimation properties underforward information. This highlights an additional advantage in using specification (26): since we take the deviation ofindividual forecasts from the mean, the specification is not sensitive to the presence of common noise. This advantage
is also highlighted by Goldstein (2021) in the context of quantifying information frictions under the standard noisy
framework, as in equation (27).

4Appendix B: Forward information model: Derivations
B.1. Deriving the optimal forecasts
In section 3.2., we consider the optimal forecasting for following state-space representation:
State:
𝑥
𝜌 0 0
1
𝑥
𝒙 ≡
Ρ𝒙
𝑆′𝜔
(B.1)
0 𝜔
1 0 0 𝒙𝑥00 1 0
where 𝜔 ∼ 𝑖𝑖𝑑 𝑁 0, 𝜎 .
Measurement:
𝑦,

𝑥
𝑥
𝑥

𝒊

𝜈,

(B.2)
𝝂𝒊
𝒙
𝜈,
𝑦,
0
where 𝜈 ,∼ 𝑖𝑖𝑑 𝑁 0, 𝜎 and 𝜈 ,∼ 𝑖𝑖𝑑 𝑁 0, 𝜎 .
Because𝑥isperfectly
observed,
theforecast
for
time
𝑡
would
𝑥|
𝑥 . The one step-ahead forecast 𝑥 | should apply four useful signals:
1. 𝜌𝑥 , with ex-post forecast error equal to 𝜔 .
2. 𝑦 ,(forward signal from the previous period), with ex-post forecast error equal to 𝜈 ,𝒚 ≡ 𝑦,

3. 𝑦 ,, with ex-post forecast error equal to 𝜈 ,simplybe.

.

4. 𝜌 𝑦 , , with ex-post forecast error equal to 𝜌 𝜔𝜈,.
All other noisy signals from previous periods are no longer useful after 𝑥 is perfectly observed.
Accordingly, the one step-ahead forecast could be represented as a weighted sum of the four signals:
𝑥

|

𝑊 𝜌𝑥

where ∑

𝑊

1. The forecaster should choose the weights which minimize the following expected squared error:

𝐸 𝑥

𝑥

|

𝑊𝑦

𝐸 𝑊𝜔

𝑊𝑦,

,

𝑊

𝜈

𝑊𝜌

,

𝑊 𝜎
𝑊 𝜌 𝜎
𝜎
𝑊 𝜎
Thus, the optimization problem can be written as:
𝑚𝑖𝑛
𝑊 𝜎
𝑊
𝑊 𝜎

𝑦,

𝑊

𝜈,

𝜎

1

(B.3)

𝑊

𝑊

𝜌

𝜔

𝑊

𝑊

𝜌

𝜈,

𝜎

𝑊

𝜎

𝜎

After setting 𝜎
1 for normalization, the FOCs are:2 1 𝑊
𝑊𝑊 𝜌 1 𝜎2𝑊
2 1 𝑊
𝑊
𝑊 𝜌 1 𝜎
2𝑊 𝜎
2 1 𝑊
𝑊
𝑊 𝜌 1 𝜎2𝑊 𝜎The solution to this system would obtain the following optimal weights:
𝜎 𝜎 1 𝜎
𝑊
𝑚
𝜎 1 𝜎
𝑊
𝑚
(B.4)
𝜎 1 𝜎
𝑊
𝑚
𝜌 𝜎 𝜎
,
𝑊
𝑚
𝜎 1 𝜎
𝜎 1 𝜎
𝜌 𝜎 𝜎 .
where 𝑚 𝜎 𝜎 1 𝜎
Next, we follow the same steps to derive the two steps-ahead optimal forecasts 𝑥 | . The same four signals used in the
one-step-ahead forecast apply for 𝑥 | , after multiplying by 𝜌 to adjust to the new horizon. Specifically, the four
available predictions for two steps ahead are:.
𝜔
1. 𝜌 𝑥 , with ex-post forecast error equal to 𝜌𝜔52. 𝜌𝑦3. 𝜌𝑦 ,

𝜌𝜈, with ex-post forecast error equal to,, with ex-post forecast error equal to𝜌𝜈 ,4. 𝑦 , , with ex-post forecast error equal to 𝜈 ,Hence, we write the optimal forecast as
𝑥 |
𝑤 𝜌 𝑥
𝑤 𝜌𝑦 ,𝑤 𝜌𝑦 ,

𝜔

,

𝜔

.

.

.
𝑤 𝑦,

(B.5)

The expected squared error could therefore be expressed as follows:
𝐸 𝑥𝑥

𝐸 𝑤 𝜌𝜔

|

𝜔

𝑤

𝜌𝜈

𝑤 𝜌
𝑤
𝑤
𝑤
𝑤 𝜌 𝜎
𝜈,
𝑤
and the optimization problem is consequently:
𝑤 𝜌
𝑤
𝑤
𝑤
𝑤
𝑚𝑖𝑛
The FOCs are:
2 1 𝑤
2 𝑤
𝑤
𝑤
𝑤
𝑤 𝜎
2𝑤 𝜌
2𝑤 𝜌 𝜎
2 1 𝑤
2 𝑤
𝑤
𝑤
𝑤
𝑤 𝜎
2 𝑤
𝑤
𝑤
𝑤
𝑤 𝜎
2𝑤 𝜌 𝜎2 1 𝑤
and the solution for the optimal weights is:
𝑤

𝜎 𝜎 𝜎
𝑚

𝑤

𝜎 𝜎
𝑚

𝑤

𝜎 𝜎
𝑚

1

𝑤

𝜌 𝜎 𝜎
𝑚

𝜔

,

𝑤

𝑤

𝜌 𝜎

𝜌 𝜎

𝑤
𝑤

𝜔

𝜌𝜈 ,
𝜎

𝜌 𝜎

𝑤

𝜎

(B.6)

𝜎

𝜎

When moving further to forecast three steps ahead, there is no further change in the optimal weights, and thus the optimal
𝜌𝑥 | . The optimal weights do not change for ℎ 3, since there are no forward signals
forecast obeys 𝑥 |referring to these horizons. To see that, we repeat the same steps as above to derive 𝑥
signals, the forecast 𝑥 | uses the following predictions:1. 𝜌 𝑥 , with ex-post forecast error equal to 𝜌 𝜔
2. 𝜌 𝑦 , , with ex-post forecast error equal to3. 𝜌 𝑦 ,, with ex-post forecast error equal to4. 𝜌𝑦 , , with ex-post forecast error equal to
The optimal forecast follows:
𝑥

|

𝑤 𝜌 𝑥

𝑤 𝜌 𝑦

𝑤 𝜌 𝑦,

,

𝜌𝜔
𝜌 𝜈 ,
𝜌 𝜈,

𝜌𝜈 ,

𝜔
.
𝜌𝜔

𝜔

𝜔

.

𝜌𝜔
𝜔

| . Based on the same four.

.

𝑤 𝜌𝑦 ,(B.7)

Note that in this case the expected squared error is expressed as:
𝐸 𝑥

𝑥

|

𝐸 𝜌 𝑥

So, minimizing 𝐸 𝑥

𝑥

𝑥
|

|

𝜔

𝜌 𝐸 𝑥

𝑥

is equivalent to the minimizing 𝐸 𝑥

1

|

𝑥

|

and produces the same optimal

weights, 𝑤 , derived above. The same reasoning holds for any horizon beyond the range of forward signals. Thus, the𝜌𝑥3.
relationship 𝑥 || applies for any ℎ

B.2. Kalman filter representation
In this section, we represent the optimal forecast in the Kalman filter framework. The filter is given by:
𝒙|

𝒙|

G 𝒚𝒊

𝒙|

(B.8)

where the gain matrix G must be specified. Expanding the matrix notation, we have:
6

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥|

G,
G ,
G ,

G,
G ,
G ,

𝑥|

G,
G ,
G ,

𝑦,

𝑥

|

𝑦,

𝑥

|

𝑦,

(B.9)

𝑥|

Each element in G corresponds to a weight placed on one of the current signals, in each of the three forecasts. Hence,
we could use the optimal weights derived above to guess the elements in the gain matrix, using the fact that that thefilter algorithm is also based on the minimization of the squared error.
𝑦,𝑥 , which is due to theSpecifically, the last row in the matrix should include 0, 0 and 1 in order to obtain 𝑥 |perfect signal of the realized fundamental at period 𝑡. The second row corresponds to the weights in the forecast 𝑥 | .
According to (B.3), the elements of this row should be G ,𝑊𝜌 , G ,𝑊 and G ,𝑊 𝜌. Similarly, theelements in the first row of the gain matrix should correspond to the optimal weights in (B.5). Thus, we get that G ,𝑤 ,G ,𝑤 𝜌 and G ,𝑤 𝜌 and the gain matrix is therefore:𝑤
𝑤 𝜌 𝑤 𝜌
𝑊𝜌
𝑊
𝑊𝜌
0
0
1
To validate this result, we derive the variance-covariance matrix Ψ of the one-step-ahead forecast error, by using G, and then verify that Ψ solves the Riccati equation.
Ψ Ψ ΣIt should be noted that in the Kalman filter representation of the optimal forecast there is no explicit reference for thesignal 𝑦 , , as in (B.3) and (B.5). However, this signal is implicit in the lagged forecasts in (B.9). For instance, theone-step-ahead forecast in the Kalman filter representation of (B.9) follows:
𝐺

𝑥

|

𝑥

𝑊𝜌 𝑦,

|

𝑥|

𝑊 𝑦,

The lagged forecasts on the right-hand-side follow𝑥|𝑊𝜌 𝑦 ,
𝑥 |
𝑥|
𝑊 𝑦
𝑥
𝑥

|

𝑥

|

𝜌𝑥

𝑤 𝜌 𝑦

|

,

𝑥

|

𝑥

|

,

𝑥|

𝑤 𝜌 𝑦

𝑊𝜌

,

𝑊𝜌
𝑥|

𝑥

𝑦,
𝑦

𝑥

,

𝑤 𝑦

(B.10)

|

,

|

𝑥

|

|

Plugging into (B.10) and rearranging terms, we obtain:𝑥 |𝑊 𝜌𝑦 ,1 𝑊
𝑊 𝑤
𝑊𝑊 𝑦 ,

𝑊𝑦,

𝑊𝜌

𝑦,

𝑜𝑡ℎ𝑒𝑟 𝑠𝑖𝑔𝑛𝑎𝑙𝑠
where 𝑜𝑡ℎ𝑒𝑟 𝑠𝑖𝑔𝑛𝑎𝑙𝑠 represent all the signals in the lagged forecasts, which are not the signal 𝑦(B.11),. The signal𝑦 ,is the only lagged signal which is still informative in period 𝑡. All other lagged signals, which refer to period 𝑡
and before, are not informative after 𝑥 is perfectly observed.
By using the expressions for the optimal weights from (B.4) and (B.6), it can be verified that the term 𝑜𝑡ℎ𝑒𝑟 𝑠𝑖𝑔𝑛𝑎𝑙𝑠 in(B.11) is equal to zero. Furthermore, the coefficient on the lagged signal 𝑦 ,in (B.11), which is given by
1 𝑊
𝑊 𝑤
𝑊 𝑊 , is just equal to 𝑊 , which is the same weight placed on this signal in (B.3). Thus, the optimal
forecast in (B.11), based on the Kalman filter, is the same optimal forecast derived in (B.3) by directly optimizing the weights
to minimize the squared forecast error. In a similar way, it can be shown that the optimal forecast 𝑥 | according to the
Kalman filter is the same forecast derived in (B.5). Thus, our guessed solution for the gain matrix is verified again.

B.3. Patterns of regression properties across forecast horizons
The simulation results, as presented in Figure 5, show that in a cross-sectional regression of 𝑥

|

on 𝑥

|

with lowℎ (short horizon), the coefficient estimate and R-squared would be low, and as ℎ increases they would converge to thevalues of 𝜌 and 1, respectively. In our tractable example, because there are only three signals (one perfect signal of
realized fundamental and two forward signals), this pattern is demonstrated in a compact form by increasing the horizon
from ℎ 1 to ℎ 3.
𝒉 𝟏: Suppose that we run a cross-sectional regression of 𝑥 | on 𝑥 | . Based on (13), the coefficient on 𝑥 | is 𝜌.
However, the OLS coefficient estimate would be zero. This is easily verified, when recalling that 𝑥 |𝑥 , due to the7perfect signal 𝑦 , in (B.2). so that there is no cross-sectional variation in 𝑥 | . It also follows that all the cross-sectionalvariation in 𝑥|

is determined by the “error term” specified in (13), and the 𝑅 should therefore be 0.

In the simulation presented in Figure 5, this limiting case was avoided by introducing some low degree of noise in the0.2), thereby, allowing some variation in 𝑥 | . Consequently, the simulated coefficient estimates andsignal 𝑦 , (𝜎𝑅 are positive but still very low.
𝒉 𝟑: At the other extreme, if we regress 𝑥

on 𝑥

|

| , we should get a perfect fit where the OLS coefficientestimate is equal to 𝜌. As shown in Section B.1., the optimal weights in the two forecasts 𝑥
so 𝑥

is exactly equal to 𝜌𝑥|| . This applies to any ℎ

|

and 𝑥

|

are the same,

3. More generally, this second limiting case would apply

when ℎ is sufficiently high so that there is no informative signal referring to that horizon. Also, recall that according toour simulation results, even if the noise in the forward signal is very high (but not infinite), the coefficient estimate
would be very close to 𝜌, while the fit of the regression could be considerably far from a perfect fit.
𝒉 𝟐: This is the intermediate case, which is described by equation (11) in the main text. The coefficient on 𝑥 | shouldbe 𝜌, but the OLS estimate would be biased since 𝑥

is correlated with 𝑆𝑖𝑔𝑛𝑎𝑙 ,|in the error term. The 𝑅 would be

between 0 and 1.
It is also interesting to examine how regression properties for ℎ 2 vary when increasing the noise of the two steps aheadsignal, that is, when increasing 𝜎 . This is, in a sense, a way to imitate our above simulation in which the noise in forward
signals increases when moving to longer horizons. As shown in section 3.1 in the main text, by combining (B.3) and (B.5),
the relation between the optimal forecasts with consecutive horizons 𝑥 | and 𝑥 | can be expressed as:
𝑥

𝜌𝑥

|

𝑤

|

Thus, a deviation from the simple state relation of 𝑥

𝑊 𝑆𝑖𝑔𝑛𝑎𝑙 ,
𝜌𝑥

|

|

(B.12)is due to changes in the optimal weights 𝑤

𝑊across forecast horizons.
become meaningless. As a
It is useful to begin with the special case where 𝜎 goes to infinity so that the signal 𝑦 ,consequence, forecasts at time 𝑡 only apply two signals: the perfect signal about realized 𝑥 and the forward signal
𝑦 , . The one-step-ahead forecast is then:𝑥
where 𝑊

𝑊

𝐸 𝑥

𝑊

|

𝜌𝑥

𝑊

𝑦,

(B.13)

1 .The expected squared forecast error is:
𝑥

𝐸 𝑊

|

𝜔

𝜈,

𝑊

𝑊

𝑊

𝜎
𝜎 1

so that optimal weights which minimize the expected squared error are simply given by 𝑊𝑊
1 𝜎.
Similarly, the two-step-ahead forecast is also a weighted average of two signals:𝑥where 𝑤
𝐸 𝑥
𝑤

𝑤
𝑥

𝑤

𝑤

|

𝜌 𝑥

𝑤

𝜌𝑦 ,

,

𝜎

and

(B.14)

1 .The expected squared forecast error is:
𝐸 𝑤

|

𝑤

𝜌𝜔
𝜌

𝑤

𝜔

𝑤

𝜌 𝜎

1

𝜌𝜈 ,

𝜔

𝑤

𝜌

𝑤

𝜌 𝜎

Hence, the optimal weights which minimize the squared error should again be 𝑤
𝜎 1 𝜎
and 𝑤
. This corresponds to the previous result that going beyond informative horizons, there will be no variation
1 𝜎
𝜌𝑥 | .
in the optimal weights across consecutive horizons and we obtain the simple relation of 𝑥 |More generally, we now show that, under the plausible assumption 𝜎𝜎
𝜎 =1, the gap between corresponding
optimal weights 𝑊 and 𝑤 is getting closer to zero when 𝜎 increases, leading to the empirical patterns observed for thecoefficient estimate and R-squared across horizons. From (B.4) and (B.6), it is easy to see that 𝑤
𝑊 for 𝑘 1,2,3,
𝑊 . Thus, we need to show that 𝑤
𝑊 increases in 𝜎 for 𝑘 1,2,3, while 𝑤
𝑊 decreases in 𝑘:
while 𝑤
8𝑘 1:
From (B.4) and (B.6) the difference between 𝑤 and 𝑊 is:𝜎 𝜎 1 𝜎
𝜎 𝜎 𝜎
𝜎 𝜎
𝑤
𝑊
𝑚
𝑚
where the expression for 𝑚, as specified above, is:𝑚 𝜎 𝜎 1 𝜎
2 𝜌 𝜎 𝜎
𝜎
𝜎 𝜎
𝜎
𝜎
𝜎 1 𝜎
𝜎 1 𝜎
𝜌 𝜎 𝜎
Taking the derivative with respect to 𝜎 we obtain
𝜕 𝑤
𝑊
2𝜎
2𝜎 𝜎
1
𝜎 𝑚 𝜎 𝜎 2 𝜌 𝜎𝑚𝜕𝜎Plugging 𝑚 into the derivative and rearranging we get:
𝑊
𝜎
𝜎
𝜎 𝜎
𝜎 𝜎
𝜎
1 𝜎
𝜎
𝜕 𝑤
0
𝑚
𝑚
𝜕𝜎
𝜎
𝜎 =1.
under the assumption of 𝜎
𝑘 2:
From (B.4) and (B.6) the difference between 𝑤 and 𝑊 is:
𝜎 𝜎
𝜎
𝜎 1 𝜎
𝑊
𝑤
𝑚
𝑚
Taking the derivative with respect to 𝜎 we obtain:
𝜎 2 𝜌 𝜎
𝑊
2𝜎
2𝜎 𝜎
1
𝜕 𝑤
0
𝑚
𝜕𝜎
𝑘 3:
From (B.4) and (B.6) the difference between 𝑤 and 𝑊 is:
𝜎 𝜎
𝜎
𝜎 1 𝜎
𝑊
𝑤
𝑚
𝑚
Taking the derivative with respect to 𝜎 we obtain:𝜕 𝑤
𝑚 𝜎 2 𝜌 𝜎
1 𝜕 𝑤
𝑊
2𝜎
2𝜎 𝜎
1
𝑊
⋅
0
𝑚
𝜕𝜎
𝜕𝜎
𝜎
𝜎
𝜎 =1, as demonstrated for the case of 𝑘 1.
under the assumption of 𝜎𝑘 4:∑Recall that ∑𝑊𝑤
1. Accordingly, we obtain:
𝜕 𝑤
𝑊
𝑊
𝑊
𝑊
𝑊
𝑊
𝑊
𝜕
𝑤
𝜕 𝑤
𝜕 𝑤
𝜕 𝑤
𝑤
𝑤
0
𝜕𝜎
𝜕𝜎
𝜕𝜎
𝜕𝜎
𝜕𝜎
𝜎
𝜎
1, by using our above results for 𝑘 1,2,3.
under the assumption 𝜎
At the limit, all the weight-differentials converge to zero when 𝜎 goes to infinity, which again demonstrates why theestimated coefficient and the R-squared should converge to 𝜌 and 1, respectively, when the horizon becomes uninformative.
Finally, we note that the sign of the bias in the coefficient estimate can change from negative to positive before theestimate eventually convergences to 𝜌. This possibility can be noticed in the simulation results presented in Figure 5.
Here it is demonstrated using our tractable case, by increasing the noise 𝜎 . Using equations (B.3) – (B.6), we express
the OLS coefficient estimate (for ℎ 2) as follows:9

𝑊2 𝑤2 𝜌𝜎22

𝐶𝑜𝑣 𝑥𝑡 2|𝑡 , 𝑥𝑡 1|𝑡𝛽𝑊2 2 𝜎22

𝑉𝑎𝑟 𝑥𝑡 1|𝑡
𝜌

𝜌

𝜌

𝜌

𝜌

𝜌

𝑊3 𝑤3 𝜌𝜎21

𝑊2 𝑤2 𝜎22

𝑊3 𝑤3 𝜎21

𝑊4 𝑤4 𝜌 2 𝜎22

𝑊2 2 𝜎22

𝑊3 2 𝜎21

𝑊4 2 𝜌 2 𝜎22

𝑊2 𝑤2

𝑊2 𝜌𝜎22

𝑊3 𝑤3

𝑊2 2 𝜎22
𝜎21 𝜎21 1

𝜎22 𝜌𝜎22𝜎21 1

𝜎22 𝜎21 1

𝜎21 1

𝜎22 𝜌

𝜎21 1

𝜎21 1

𝜎21

𝜎22 1

𝑊3 2 𝜎21

𝜎22 𝜎22𝜎22 1

𝜎22𝑊3 2 𝜎21𝑊3 𝜌𝜎21𝜎22 𝜎22 1

𝜎22 1

𝜎22 1

𝜎22 𝜎21𝜎22 𝜎22 1𝜎22 𝜎21

𝜎22 1
𝜎21

𝜎22 1

𝜎22

𝜎22 1

𝜎22 1

𝜌 𝜎22 𝜎22 𝜎21

𝜎21
2 2
𝜎2

𝜎22

1

𝑊4 𝜌 1 𝜎22

𝑊4 2 𝜌 2 𝜎22

𝜌 𝜎22 𝜎22 𝜎21

𝜎22

𝑊4 2 𝜌 2 𝜎22𝑊4 𝑤4

𝜎22 𝜌𝜎21

𝜎22 𝜌𝑊4 𝑤4 𝜌 1 𝜎22

𝜎21

𝜎21

𝜎22 𝜌2 𝜎22 𝜎21 𝜌 1 𝜎22𝜎22 𝜎21

𝜌2 𝜎22 𝜎21 𝜎22 𝜎21 𝜎22(B.15)𝜎22 𝜌𝜎22𝜎22

𝜌2 𝜎22 𝜎21 𝜎22𝜎22

𝜌2 𝜎22 𝜎21 𝜎22𝜎22

𝜌2 𝜎22 𝜎21 𝜎22Thus, we obtain an expression for the bias which consists of competing negative and positive terms:
𝛽

𝜌

𝜌 𝜎21
𝜎21

𝜎22 1

𝜎22

𝜎22 2

𝜌𝜎22 𝜎22 𝜎21
𝜌2 𝜎22 𝜎21 𝜎22

𝜎21

𝜎22 1

𝜎22 2

𝜌2 𝜎22 𝜎21 𝜎22

Specifically, for low values of 𝜎 , the negative bias dominates, whereas the positive bias dominates for high values of
𝜎 . Nevertheless, it is easy to see that both biases eventually converge to zero when 𝜎 goes to infinity.

B.4. Patterns of regression properties when changing persistence in the state
Another empirical pattern in the regression of 𝑥

|

on 𝑥

| , documented in section 2 and demonstrated by thesimulation results in Figure 5, is the co-movement of the coefficient and R-squared. Apparently, this pattern is also
associated with the differences in the optimal weights across forecasting horizons, as we illustrate with our tractableversion of our model (see the right side of Table 1). We first look at the two extreme cases of 𝜌 0 and 𝜌 1.
𝝆 𝟎 : the lack of persistence in the state process implies that the forecasts are uncorrelated across horizons. That is, aforecast 𝑥 | relies only on signals referring to 𝑡 h, and is uncorrelated with 𝑥| (nor with 𝑥| ). In termsof our system of equations, the optimal forecasts would be:
𝑥|
𝑥

𝑥

𝑊𝑦

|

𝑥

𝑊𝑦,

,

𝑤𝑦,

|

𝑥

(B.16)

|

0

1 𝑊
𝜎 𝜎
𝜎
and 𝑤
1. The forecasts are uncorrelated with each other since thewhere 𝑊
fundamental is uncorrelated across different periods and the noise in forward signals is uncorrelated across horizons.
It follows that in a cross-sectional regression of 𝑥 | on 𝑥
| , the coefficient-estimate and R-squared should be
zero. In terms of equation (B.12), for example, all cross-sectional variation in 𝑥|

is driven by the regression error

term, which collapses to 𝑤 𝑦 ,(according to (B.16)).
𝝆 𝟏 : When the fundamental follows a random walk, the general results obtained in Section B.3. would hold. This is
not a limiting case where the coefficient and R-squared should converge to 1 (𝜌 1). Rather, the regression properties
10

would depend on the forecast horizon in the same way described above. Specifically, for the short horizon ℎ 1 thecoefficient-estimate and R-squared should still be zero, while for the long horizon ℎ 3 the fit becomes perfect. Theinteresting case for investigating the response to change in persistence is the middle horizon, ℎ 2. As implied byequation (B.12), the coefficient on 𝑥 | should be 𝜌 1, but the OLS estimate would be biased, due to the correlationof 𝑥with the error term, and the R-squared would be between zero and one. Nevertheless, the difference in|regression properties between zero persistence and random walk still demonstrates an increase in the coefficient estimate
and R-squared from zero to positive values, following a rise in persistence.
More generally, we now show how the relation between 𝑥 | and 𝑥 | , as specified in (B.12) would vary whenchanging the persistence of the state process. First, notice that the difference between optimal weights placed on the𝑊 , tends to diminish when the persistence 𝜌 increases. From oursame signals in 𝑥 | and 𝑥 | , which is 𝑤
𝑊
𝑎 𝑚 . For 𝑘previous results, the weight differential can be expressed as weights can be expressed as 𝑤
1,2,3, 𝑎 (numerator) is negative and only 𝑚 (denominator) is a function of 𝜌. Hence, it is sufficient to show that thederivative of 𝑚 with respect to 𝜌 is negative:𝜕𝑚𝜕 2 𝜌 𝜎 𝜎
2𝜌𝜎 𝜎
𝜎
𝜎 𝜎
𝜎
𝜎
0
𝜕𝜌
𝜕𝜌
𝑚
𝑊 , 𝑘 1,2,3, diminishes as persistence increases. For 𝑘 4 the difference 𝑤
Thus, the negative difference 𝑤
𝑊 is positive. Following the same above argument, this gap should decrease in 𝜌 because the weights 𝑤 and 𝑊 sum
to one. Overall, this implies that the correlation between 𝑥 | and 𝑥 | will tend to be higher with more persistence
since the error term in (B.12) would diminish due to the diminishing weight differential.
Second, we show more formally that the (biased) OLS coefficient estimate and fit of the regression are increasing in 𝜌,even in the presence of a non-diminishing bias. Based on (B.16), we begin with:
𝛽

𝜌

𝜌 𝜎21
𝜎21

𝜎22 1

𝜎22

𝜎22 2

𝜌𝜎22 𝜎22 𝜎21
𝜌2 𝜎22 𝜎21 𝜎22

𝜎21

𝜎22 1

Thus, the downward bias is no greater than 𝜌, so that 𝛽
Taking the derivative with respect to 𝜌, we get:
𝜕𝛽
𝜕𝜌

1

𝜎22 𝜎22 𝜎21

𝜎21

𝜎22

𝜎21

𝜎21

𝜎22 1

𝜎22 1

𝜎21

𝜎22 2
2 2 2
𝜎2 𝜎2 𝜎1 𝜎21
𝜎22 1 𝜎22 2

1

𝛽

𝜌 𝜌 1

𝛽

𝜌 1

𝛽

𝛽

𝜌 1

1

𝜌

𝜎21

𝜎22

𝜎22 2

𝜌 𝜎21
𝜌2 𝜎22 𝜎21 𝜎22

𝜎21

𝜎22 1

𝜎22

𝜎22 2

𝜌2 𝜎22 𝜎21 𝜎22

𝜌should always be positive.

𝜎22 2

𝜌2 𝜎22 𝜎21 𝜎22

2𝜌2 𝜎22 𝜎21 𝜎22 𝜎22 𝜎22 𝜎21

𝜎21

𝜎22

𝜌2 𝜎22 𝜎21 𝜎22 2

𝜎21 𝜎22 1 𝜎22 2 𝜌2 𝜎22 𝜎21 𝜎22 2
2 2 2 22𝜌 𝜎2 𝜎1 𝜎2 𝜎22 𝜎22 𝜎21 𝜎21 𝜎22𝜎21 𝜎22 1 𝜎22 2 𝜌2 𝜎22 𝜎21 𝜎22 22𝜌𝜎22 𝜎21 𝜎22𝜌𝛽𝜎21 𝜎22 1 𝜎22 2 𝜌2 𝜎22 𝜎21 𝜎22
2𝜌𝜎22 𝜎21 𝜎22
2
2
𝜎1 𝜎2 1 𝜎22 2 𝜌2 𝜎22 𝜎21 𝜎22
2𝜌𝜎22 𝜎21 𝜎22
2𝜌𝜎22 𝜎21 𝜎22
𝜌
0
2 2
2
2
2
2
2
1 𝜎2
𝜌2 𝜎2 𝜎1 𝜎2
𝜎1 𝜎2 1 𝜎22 2 𝜌2 𝜎22 𝜎21 𝜎22𝜎22
𝜌2 𝜎22 𝜎21 𝜎22

It is easy to see that the term in brackets is positive under the assumption 𝜎
𝜎 . Since 𝛽
is also positive as shown
above, the derivative is positive. Thus, we can conclude that the coefficient-estimate 𝛽
increases in 𝜌.
The R-squared follows a similar pattern. The R-squared between 𝑥 | and 𝑥 | is given by:𝑅

𝐶𝑜𝑣 𝑥 | , 𝑥 |𝑉𝑎𝑟 𝑥 | 𝑉𝑎𝑟 𝑥

𝑊 𝑤 𝜌𝜎𝑊 𝑤 𝜌𝜎

𝑊𝑤 𝜌

𝜎

𝑊 𝜎
𝑊 𝜌 𝜎 𝑤 𝜌 𝜎
𝑤 𝜌 𝜎
𝑤 𝜎
𝑊 𝜎
After plugging in the optimal weights from (B.4) and (B.6) and rearranging, we can express the R-squared in thefollowing way:
Β
,
𝑅
Β
Δ
Where:
Β 𝑚 𝐶𝑜𝑣 𝑥 | , 𝑥 |𝜌 1 𝜎 𝜎
𝜎
𝜎
𝜎
𝜌 𝜎 𝜎
𝜌 𝜎 𝜎
𝜎 𝜎
Δ 𝜎 𝜎
𝜎 𝜌 𝜎
2𝜌 𝜎 𝜎 1 𝜎 𝜎 𝜎
𝜎
𝜎
𝜎
𝜎
𝜎
𝜎
1 𝜎
𝜎 𝜎
|

11

Taking the derivative with respect to 𝜌 yields:
𝜕Δ
𝜕𝐵
𝜕𝐵
𝜕𝐵 𝜕Δ
Β
Δ
Β 2Β
Β
2Β
Β 2Δ
𝜕𝐵
𝜕Δ
𝜕𝑅
𝜕𝜌
𝜕𝜌
𝜕𝜌
𝜕𝜌 𝜕𝜌
Κ 2Δ
Β
Β
Β
𝜕𝜌
Δ
Δ
𝜕𝜌
𝜕𝜌
Where Κ Β⁄ Β
0. The derivative, 𝜕𝐵⁄𝜕𝜌, can be expressed as:
Δ
𝜕𝐵
𝐵𝜌
2𝜌 𝜎
𝜎
𝜕𝜌
Plugging this into the derivative of 𝑅 , we obtain:
𝜕𝑅
𝜕Δ
4Δ𝜌 𝜎
Κ Β 2Δ𝜌𝜎𝜕𝜌𝜕𝜌Κ Β 2𝜌 𝜎𝜎
𝜎
2𝜌 𝜎 𝜎
𝜎
𝜎 𝜎
𝜎
𝜎 1 𝜎
𝜎
𝜎
4Δ𝜌 𝜎
𝜎
Κ Β𝜎 𝜎𝜎2𝜌 1 𝜎
𝜎
𝜎
𝜎
2𝜌 𝜎
𝜎
𝜎 𝜎
𝜎
4Δ𝜌 𝜎
The term in the squared brackets is positive, and consequently the whole derivative. It follows that the R-squared
increases in 𝜌. Hence, the R-squared and the coefficient estimate should demonstrate a pattern of co-movement inresponse to a change in 𝜌.

B.5. Predictability of forecast errors
This section shows that forecast errors are predictable by forecast revisions at the aggregate level due to forward signals.
Consider the one-step-ahead forecast error. Using (B.3), we obtain:
𝑥

𝑥

|

𝑊𝜔

𝑊

𝜈

𝑊

,

𝜈,

𝑊

𝜌

𝜔

𝜈,

By taking the average across agents, all terms with idiosyncratic noise drop out. Hence, we get:
𝑥
𝑥 |
𝑊𝜔
𝑊 𝜌 𝜔
where 𝑥 | (without superscript 𝑖) denotes the cross-sectional average.
The forecast 𝑥 | revises the forecast 𝑥 | , which is the two-step-ahead forecast from the last period. Using (B.3)
and (B.5), and averaging across agents, the revision to the average forecast could be expressed as (exploiting the property
that the optimal weights amount to 1):
𝑥 |
𝑊 𝜌𝑥
𝑊𝑥
𝑊𝑥
𝑊𝜌 𝑥
𝑤 𝜌𝑥
𝑤 𝜌𝑥
𝑤𝑥
𝑤 𝜌 𝑥
𝑥 |
𝑊 𝜔
𝑤 𝜌𝜔
𝜔
𝑤 𝜔
𝑤 𝜌𝜔
𝑤
𝑤
𝑤
𝑊 𝜔
𝑊𝜌 𝜔
𝑤
𝑊𝜌 𝜔
Consider a regression of forecast error with the forecast revision as the explanatory variable. Then, the expected OLS
coefficient estimate would be (setting 𝜎 =1 and assuming 𝐸 𝜔𝑡 𝜔𝑡 1 ) :
𝐶𝑜𝑣 𝑥
𝐸 𝑥
𝑥 | ,𝑥 |𝑥 |𝑥 | 𝑥 |𝑥 |𝛽
2𝑥 |𝑉𝑎𝑟 𝑥 |
𝐸 𝑥 |
𝑥 |
𝐸 𝑊𝜔

𝑊

𝜌

𝐸 𝑤 𝜌𝜔

𝑤

𝑊 𝑤
𝜌

𝑤
𝑤

𝑤
𝑤

𝜔

𝑤 𝜌𝜔

𝑤

𝑤

𝑤
𝑤
𝑤
𝑊 𝜔
𝑊 𝜌
𝑊
𝑤
𝑊
𝑊 𝜌

𝑤

𝑊 𝜔
𝑊𝜌

𝑊𝜌

𝜔

2

𝜔

We can further verify that the expected coefficient is positive by showing that the numerator is positive. Using theexpressions for the optimal weights from (B.4) and (B.6), we obtain (𝑚 is defined in (B.4)):
𝜎 𝜎 1 𝜎 𝜎 𝜎
𝜌 𝜎 𝜎 𝜎 𝜎
𝑊 𝑤
𝑤
𝑤
𝑊 𝜌
𝑊
𝑚
𝜌 𝜎
1 𝜎 𝜎
𝜎 𝜎 𝜎
The expression in brackets

1

𝜎 𝜎

𝑚
𝜌 𝜎 is positive due to 𝜎𝜎

1 and 𝜌numerator is positive, making the coefficient on the forecast revision positive as well (𝛽1. Consequently, the whole0).
12Appendix C: Simulations
This appendix provides more details on the simulation results reported in Section 3.3. Figure 5 replicates the persistence
regression of 𝑥 | on 𝑥
| that was estimated in Section 2, using SPF inflation forecasts. The results in Figure 5
are in line with the patterns documented in the SPF data along several dimensions.
First, we observe in Panel A that the coefficient estimate is substantially lower than the true persistence for the shorthorizons. For example, when the true persistence is 0.5 the coefficient estimate is below 0.2 in the regression applied toℎ 1. However, as we estimate the regression using forecasts with longer horizons (higher noise to signal ratios), thecoefficient estimate gets closer to the persistence value and finally converges to it. It is important to note that theconvergence is not monotonic, but rather takes somewhat hump-shaped path. This result illustrates the possibility of
upward bias as well, although the bias in this direction is quite small in the simulation (the positive bias is also possible
in our above tractable example. See Appendix B.3.). The hump shape is also consistent with the evidence in Figure 2,
in which the lines corresponding to the two longest horizons are flipping over time.
Second, there is a similar pattern of convergence for the R-squared statistic (Panel B). Interestingly, the fit is very poor
in regressions with forecasts at short horizons, representing the higher role of forward information in the error term. The
values of the R-squared start below 0.1 for each persistence value, but eventually increase towards a perfect fit.
Furthermore, the convergence could be quite slow, especially in cases where the degree of persistence is low. For
instance, with 𝜌 0.2, the R-squared is below 0,5, even when estimating the regression for ℎ 6, for which the signal
is effectively uninformative. This property is consistent with the low R-squared values documented in recent years, evenfor the longer horizons in the SPF (Figure 2, Panel B).
Third, the simulation results in Figure 5 also demonstrate a dependence of the regression properties on the underlying
degree of persistence. Importantly, not only does the coefficient-estimate increases in the degree of persistence (PanelA), but so does the R-squared statistic (Panel B). This pattern is consistent across the different horizons (horizontal axis),
for which we estimate the regressions. The co-movement of the regression estimates and persistence resembles thepattern documented in Section 2, where the coefficient and the fit of the regression deteriorate over time along with the
decline in inflation persistence.
Appendix Figure G.2 describes results from another set of simulations, in which the variance of the noise is constantinstead of monotonically increasing in the horizon. Overall, it seems that the patterns of the regression properties are
similar to those in Figure 5. However, notice that the rate of convergence is different. Due to the concavity of the lines,
the convergence rate is slow for the short horizons and becomes faster for the longer horizons. This type of convergenceseems less consistent with the evidence in Figure 2. The next simulation provides further evidence that supports themore realistic pattern of increasing noisiness in forward signals.
As mentioned in Section 4, we further simulate the estimation of specification (26) on simulated forecasts as in Section
3.3, with 𝜌 set to 0.5. Appendix Table G.1. resembles the structure of Table 3, where each column is a simulated
estimation for a certain horizon. The coefficient estimates are in bold, while the regular numbers report the true
parameters from the matrix 𝐼 𝐺 . Although 𝐻 7 in the simulation, the regressions were estimated as if there were
only 5 available horizons in the data, thereby, checking the sensitivity of the results to this practical limitation. The
results confirm that such truncation is not a concern. However, the estimates in Appendix Table G.1. are still very close
to the true values. Thus, the estimation of specification (26) seems very reliable despite a truncation. In addition, as inour results for the SPF the diagonal coefficients are very dominant. Interestingly, the simulation provides a good
approximation even to the off-diagonal elements. In practice, as evident in Table 3 these estimates could still suffer from
imprecision due to the small values, and some multicollinearity due to measurement error.
Another useful point is demonstrated by the results in Appendix Table G.2., which are based on the simulation as inAppendix Figure G.2, where the variance of the noise stays constant across horizons. Compared to the previous
simulation with increasing noise, there is a notable difference in the pattern of the diagonal elements across horizons.
when noise increases in the horizon (Appendix Table G.1.), the diagonal estimates also increase in the horizon, while
for fixed noise there is a decline. Our results in Table 3 for the SPF are in line with the first pattern. Thus, this result isanother form of support for a general pattern of deterioration in forward information as the horizon gets longer.
This result also sheds light on the estimation of the restricted specification in (27). Because this version focuses on the
diagonal elements of 𝐼 𝐺 in the forward information framework, the coefficients should tend to increase when thespecification is estimated for longer horizons. This pattern was documented in the SPF data by Goldstein (2021). Thus,
according to our simulation, this variation provides another form of evidence against the standard noisy information
framework, in which the coefficient in (27) should not vary across horizons.
13Appendix D: Estimating Persistence with Asymmetric Loss Function
According to Elliot et al. (2008) and Capistrán and Timmerman (2009), biased forecast errors observed in surveys could
result from asymmetric preference of positive over negative forecast errors or vise-versa. Different asymmetricaltendencies across individuals would explain forecast disagreement. More formally, following Capistrán andTimmermann (2009), this asymmetry is modeled by a LINEX loss-function over forecast errors:
𝐿 𝐹𝐸 𝑥 ; 𝜃
1 ⁄𝜃
𝑒𝑥𝑝 𝜃 𝐹𝐸 𝑥
𝜃 ℎ 𝐹𝐸 𝑥≡𝑥
𝑥 | is the forecast error of forecaster 𝑖 and 𝜃 is the asymmetry parameter. A positive value
where 𝐹𝐸 𝑥of 𝜃 corresponds to positive error loss-aversion, while a negative 𝜃 corresponds to the opposite. As 𝜃 shrinks to zerothe function converges to the regular (symmetric) squared error loss-function.
The optimal individual forecast which minimizes the specified loss-function is:
1
(D.1)
𝐸𝑥
𝜃𝜎 |
𝑥 |
2
where the variable 𝑥 is assumed to be normally distributed with conditional mean, represented by the rational expectation
term 𝐸 𝑥 , and with conditional variance 𝜎 | . Thus, the individual forecast is biased relative to the rational
expectation by a term that depends on the asymmetric tendency parameter 𝜃 , and the variance of the 𝑥.
Similarly, the forecast for ℎ 1 steps ahead is:
1
(D.2)
𝐸𝑥
𝜃𝜎
𝑥
|
|
2
Suppose that the fundamental follows an AR(1) process 𝑥
𝜌𝑥
𝜔 , where 𝜔 ∼ 𝑖𝑖𝑑 𝑁 0, 𝜎 |. The implied
rational expectation would therefore be:
1
𝐸𝑥
𝜌 𝑥
𝜌𝐸 𝑥
𝜌𝑥
𝜌𝜃 𝜎
|
|
2
where we use (D.2) to substitute for 𝐸 𝑥. We then substitute this expression in (D.1) and rearrange to obtain
1
(D.3)
𝜌𝑥
𝜃 𝜎 |
𝜌𝜎
𝑥 |
|
|
2
The last term on the right-hand side would correspond to the error term in a cross-sectional regression of the forecast
𝑥 | on the forecast 𝑥| . The mean of the error term would be zero only if 𝜃 has a zero mean (symmetry). In
the general case, where the mean of 𝜃 is 𝜃, which is different from zero, the regression would include a constant term
and the error term would be 𝜃 𝜃 𝜎 |𝜌𝜎| .
In any case, it is clear from (D.2) that 𝜃 is positively correlated with 𝑥| , so that the OLS estimate of the coefficienton 𝑥| would be a biased estimate of the persistence parameter 𝜌. In fact, the OLS coefficient in a cross-sectional
regression is unrelated to 𝜌, because heterogeneity in forecasts is driven only by the asymmetric bias component, while
the persistence component, 𝜌 𝑥 , is the same across agents. More formally:
1
1
1
1
𝐶𝑜𝑣 𝜌 𝑥𝐶𝑜𝑣 𝜃 𝜎 | , 𝜃 𝜎
𝜃 𝜎 | ,𝜌 𝑥
𝜃𝜎
𝐶𝑜𝑣 𝑥𝑡 ℎ|𝑡 , 𝑥𝑡 ℎ 1|𝑡
|
|
2
2
2
2
𝛽
1
1
𝑉𝑎𝑟 𝑥𝑡 ℎ 1|𝑡
𝑥
𝜃𝜎
𝑉𝑎𝑟 𝜌
𝑉𝑎𝑟 𝜃 𝜎
|
|
2
2
𝜎 |
𝜎
|
Thus, if the conditional variance of the fundamental is time-independent, the OLS coefficient in a cross-sectional
regression of 𝑥 | on 𝑥
| equals 1. Capistrán and Timmermann (2009) have assumed that the conditional variancefollows a GARCH(1,1) process:
𝛼
𝛼 𝜔
𝛽 𝜎|
𝜎 |
Thus, the resulting OLS coefficient (𝛼 ⁄𝜎𝛼𝛽)depends
on the conditional variance. Still, it does not|depend on 𝜌, nor does it follow a particular pattern of variation across forecast horizons.

14Appendix E: More Applications of Forward Information
In Section 6, we proposed a simple measure of forward information, based on the forward information component inforecast data. This appendix complements Section 6 with two additional applications of our measure, which aredescribed briefly in Sections 6.2.1. and 6.2.3.

E.1. Big news
As demonstrated in the main text, our method for extracting forward information from forecasts and the forecast persistence
regressions from Section 2 are closely related. The forward information component shifts the persistence estimates
downward at the shorter horizons. Thus, it is expected that times of big news will induce big shifts in forecast persistence.
A recent prominent example of a major event that should deliver a high amount of forward information is the outbreak of
COVID-19. In terms of our framework, forecasters would interpret the outbreak of the pandemic as a series of multihorizon shocks that will hit the economy in the following quarters. According to our analysis, the adjustment of forecasts
to such significant news may lead to a decline in the persistence of their forecasts at the various horizons.
The results in Panel A of Appendix Figure G.3 support this conjecture. The figure shows estimates of forecast
persistence before and after the outbreak of COVID-19, by applying specification (4) to the SPF waves of 2020Q1 and2020Q2. The forecasts in these waves were collected in February and May 2020, respectively, which is before and after theoutbreak of the pandemic outside China. Interestingly, there is a substantial decline in the estimates of persistence in thesurvey wave of 2020Q2 relative to 2020Q1, for all four major macro variables presented in the figure (inflation, GDP growth,interest rate and unemployment). Moreover, the decline is observed not only in short-horizon estimates (ℎ 1, left graph),
but also in estimates for the longer horizon (ℎ 3, left graph). This finding suggests that the scope of forward information
embedded in the outbreak of COVID-19 is long enough to have a pervasive biasing effect even on the year-ahead estimates.
Still, in line with the decaying effect of forward information across horizons, as demonstrated above, the decline in theestimates at the longer horizon is more moderate. Specifically, as shown in the figure, estimated persistence has declined byaround 0.5 (!) in 2020Q2 relative to 2020Q1 for ℎ 1, and by around a half of this size for ℎ 3.
As a reference to the evidence from COVID-19, Appendix Figure G.3 presents the results from a similar exercise
applied to two other major events: the financial collapse on the eve of the Great Recession and the 9/11 terror attack(Panels B and C, respectively). For the collapse of the big US financial firms in September 2008, we compare the SPF
waves of 2008Q3 and 2008Q4. For the terror attack of September 2001, we compare the SPF waves of 2001Q3 and2001Q4. In contrast to the COVID-19 results, the pattern of decline in persistence is much weaker, following these
events, and it is not consistent across variables and horizons. Hence, the sharp pattern of decline in persistence following
COVID-19, which was obtained despite a small sample of forecasters in single waves, points to an exceptional amountof forward information brought by the burst of this unprecedented crisis.

E.2. Comparison with a methodology of news shocks
This application provides a comparison of the forward-information measure of news with news shocks identified by afamiliar method in the literature. For this purpose, we extend the analysis in Section 5.2 to a VAR framework. We
estimate the effect of the series of forward information about inflation, 𝐹𝐼 | , in a VAR with four macroeconomic
variables, including the inflation rate, the unemployment rate, 3-month and 10-year treasury bill rates. The estimatedimpulse responses of the four variables to 𝐹𝐼 | are presented in panel A of Appendix Figure G.4. Again, we obtain astrong response of inflation to its forward information, despite taking into account the effect of other variables. The
initial response is close to the estimates in Table 6 (column (2)) and then dies out quite quickly. The response of theother variables to forward information about inflation is quite weak and insignificant.
We then identify inflation news shocks, using the influential method suggested in Barsky and Sims (2011). The
idea of their approach is to identify a news shock as a shock orthogonal to the current innovation in the fundamental,
which best explains future movements in that fundamental. In line with the literature on news shocks, they apply their
method to identify technology shocks. Here, we implement their method to identify inflation news shocks, in a VAR
framework that includes the same variables used above (inflation, unemployment, short and long-run interest rates).
The estimated impulse responses to the identified inflation news shocks are reported in Panel B of Appendix Figure
G.4. In contrast to our previous findings (Panel A), the news shocks identified by the Barsky-Sims method do not induce
any significant response to it, even by inflation.
Based on a forecast error variance decomposition, Appendix Table G.4 reports, for each method, the share of
inflation volatility at various horizons which is explained by the news variable. The difference between the two methods
is again very clear: While according to the Barsky-Sims method news shocks account for no more than 10% of thevariation in inflation, the component quantified by the forward information approach can explain more than a half ofthe same variation. This comparison illustrates how the straightforward application of the notion of forward information
to forecast data can be useful for quantifying predictive components that are important for explaining future movementsin the fundamental beyond the standard VAR.
15

Appendix F: Forward information in forecasts of additional macro variables
This appendix describes in more detail the evidence that was summarized in Section 6 in the main text.

F.1. SPF forecasts
The empirical evidence through the paper was based on inflation expectations, on which there is a particular focus by
the literature, due to their main role in macroeconomic analysis. We examine how forward information is incorporated
in expectations more generally, by first investigating SPF forecasts of additional key macroeconomic variables, as well
as forecasts of other measures of inflation.
We first examine if the patterns of persistence documented in Section 2, characterize forecasts of othermacroeconomic variables as well. Thus, we regress the forecast x | on the forecast x| , using multi-horizon
forecasts of other macroeconomic variables, available in the SPF. A pattern of increase in the coefficient estimates andR-squared across forecast horizons, as in Figure 2, would point to a utilization of forward information in expectationsof other variables. Appendix Figure G.6 presents the results for SPF forecasts of unemployment rate, interest rate andreal GDP growth. The results are presented in the same form as in Figure 2. Regressions are estimated quarter-byquarter, using the cross-sections of forecasts from the last eight quarters. For each variable, the figure describes thecoefficient estimate (left side) and R-squared statistic (right side), estimated over time and across the different horizons
available in the survey (ℎ 0,1,2,3,4. For GDP growth forecasts ℎ 0 is not available due to the conversion of level
forecasts from the survey to growth forecasts). Appendix Figure G.7 presents results for forecasts of additional measuresof inflation: GDP deflator, PCE inflation and core inflation (due to a similar conversion of prices-level forecasts to
inflation forecasts, ℎ 0 estimates are also not available for the GDP deflator). The sample period varies depending on
the availability of forecast data for each variable: For unemployment, GDP growth and GDP inflation, the sample starts
at the beginning of the 1970s. For the interest rate, the sample starts at the beginning of the 1980s and for PCE and core
inflation, it begins in 2009 (eight quarters after the first available forecasts due to the small window size).
For most variables, both the coefficient-estimate and R-squared demonstrate patterns of variation acrosshorizons and convergence that are similar to the baseline results in Figure 2. For unemployment and interest rate
forecasts (panels A and B in Appendix Figure G.6), the estimated persistence in the “convergence” region (longer
horizons), is quite steady over the years, around the level of 1. Thus, SPF participants consistently associate a random
walk process with movements in unemployment and interest rate. Notably, compared to inflation, the convergence of
estimates across horizons for unemployment and interest rate forecasts is quite fast: the graphs for ℎ 2,3,4 are quite
similar, whereas only for ℎ 0 the coefficient-estimates and R-squared are noticeably lower. The pattern of fasterconvergence suggests that utilization of forward information with respect to these variables might be lower, compared
to inflation. This result may reflect a higher degree of attention to inflation by the forecasters or a greater availability ofvaluable forward signals about inflation, due to measures of central bank communication, especially in recent years.
The convergence pattern could also be affected by the noise in realized data. Recall, that according to our above
evidence, survey participants are well-informed about realized inflation. This, however, could be different for variables
with more data releases such as unemployment, or a high-frequency variable such as the Treasury-Bill rate. This point
will be examined later using the methodology proposed in Section 4.
For the forecasts of PCE and core inflation (Panels B and C in Appendix Figure G.7), there is more variation
across horizons and the convergence is around ℎ 3, 4, implying the availability of forward signals up to a year ahead,similar to the baseline findings for CPI inflation forecasts. The coefficient estimates for the longer horizons provides ameasure of persistence that is quite steady over the years, since the data is available for these variables only for the last
decade, after the great decline in inflation persistence. Also notice that the persistence in PCE inflation is around 0.6,
which is close to the persistence in CPI inflation during these years, as described in Figure 2. For the core inflation,
persistence is estimated at around 0.8. This higher degree of persistence in core inflation is in line with the notion that
this measure approximates trend inflation, by excluding CPI components with high-frequency volatility. However, themeasured persistence is still lower than 1.
For two variables, there is no clear pattern across horizons in the persistence regressions: The GDP growth(Panel C in Appendix Figure G.6) and the inflation by GDP deflator (Panel A in Appendix Figure G.7). In order tounderstand the reason for this exception, we should recall that the pattern of convergence across horizons depends on
two important factors: First, the structure of information and secondly, the underlying process, specifically, how wellthis process is approximated by AR(1). Considering the first issue, it should be noticed that unlike other variables in the
SPF which are forecasted in terms of the rate of change, the original forecasts of GDP and GDP deflator are reported inlevels. The original forecasts are then transformed to GDP growth and inflation rates, and the estimation is applied to
the rate-of-change forecasts, as it is applied to forecasts of other variables. However, it is not clear if forward signals
were applied by SPF forecasters to predict the level or the rate of change and this may obscure the pattern acrosshorizons. For instance, suppose that the process in growth rates follows AR(1):𝜌𝑥𝜔𝑥where 𝑥Δ𝑧 . Thus, the process in levels follows AR(2):
16

𝜌Δ𝑧
𝜔 ⟺𝑧
1 𝜌 𝑧
𝜌𝑧
𝜔
Δ𝑧
Forecasters receive multiple forward signals on the levels 𝑧(ℎ 0,1, … , 𝐻) and apply the corresponding state-space
representation in terms of 𝑧 . Accordingly, they form Kalman filter level-forecasts 𝑧 | . The appropriate persistence
regression for this case would therefore be:
𝑐 𝜌 𝑧
𝜌 𝑧
𝑒𝑟𝑟𝑜𝑟
𝑧 |
|
|
This specification also relates to the second issue mentioned above. We may not only need to estimate thespecification using forecasts in levels, but also to expand the dynamics beyond AR(1) by including more “lags.”
Appendix Figure G.8 shows the results of applying this specification to the original level forecasts of GDP and GDP
deflator. It describes the persistence estimate as the sum of the coefficients (𝜌𝜌 ) and the R-squared for the various
horizons, where ℎ starts at 2 due to the additional “lag”. As opposed to the results in Appendix Figure G.6 (Panel C)
and Appendix Figure G.7 (Panel A), the results from the modified specification are much closer to a pattern of “layers,”
as was documented for the other variables. The big swings in the 1980s may indicate that more “lags” are still required.
Indeed, the estimation of AR(4) persistence (adding the forecasts 𝑧| and 𝑧| would diminish these swings,
but the pattern of layers cannot be demonstrated with high-order dynamics. It should also be noted that somemeasurement issues are also possible since explicit forecasts of real GDP has been provided only since the beginning of
the 1980s (see documentation at the SPF website). In sum, the results from the persistence regressions broadly confirm
the patterns documented in Section 2. Forward information thus plays a more general role in expectations ofmacroeconomic variables. Our findings also demonstrate how patterns may change when the underlying process is more
complex.
We also apply the direct method proposed in section 4, for testing and estimating the role of forward
information, to SPF forecasts of the above macro variables. First, we test the presence of noise in realization, according
to the standard noisy information framework, by applying specification (26) to deviations of individual backcasts fromthe mean (ℎ 0, the test is not performed for GDP growth and GDP inflation because of the absence of backcasts forthese variables, as noted above). The estimation results are reported in Appendix Table G.5. Recall that perfect
information about realizations would imply zero coefficients, as obtained above for CPI inflation forecasts (Table 2).
The same result is also obtained using forecasts of PCE and core measures of inflation. For the unemployment andinterest rate, some significance is documented for several coefficients, but the size of coefficient estimates is still very
close to zero, as reported in Appendix Table G.5. Thus, information about realizations is close to perfect, even for thesevariables. At the same time, the limited amount of noise found in the backcasts of unemployment and interest rate is inline with the faster pattern of convergence across horizons in the persistence estimates, as documented above for thesame variables.
Second, we estimate specification (26) using the forecasts of each variable for the various available horizons
and then compare it with the restricted version in specification (27), which rules out the availability of forward signals(zero coefficients on non-diagonal elements). Appendix Table G.6. presents the BIC statistics from the two
specifications, for each variable and each forecasting horizon (ℎ 1,2,3, 4). Similar to the findings in section 4, theunrestricted specification (26) outperforms the restricted version. Hence, there is direct evidence for the presence of
forward information, even in forecasts of GDP growth and GDP inflation, for which the indirect approach based on
persistence patterns, provided less conclusive evidence.
Finally. we briefly examine how our approach for quantifying forward information in Section 5 can identify
forward information with predictive power in forecasts of additional variables. We employ SPF forecasts of
unemployment, interest rate and GDP growth and estimate series of 𝐹𝐼 | for each variable using (32). As for inflation,
it involves the estimation of quarter-by-quarter persistence for each variable with the proper horizon, based on the steps
described above (constant term is also estimated). Appendix Table G.7. reports highly significant effects of thoseforward information series in a simple univariate AR. The significant effect obtained for highly persistent variables,such as the unemployment and interest rate also indicate the long-lasting impact of forward information on movements
in macroeconomic fundamentals (We also considered the unit root possibility, by taking the differenced series of
unemployment and interest rates and running them on the forward information series. The coefficient on forward
information was again highly significant in these specifications.).

F.2. Fed Forecasts
The Greenbook forecasts of the Fed staff are available for multiple horizons, thus allowing us to examine if the patternsof persistence across horizons that were documented in the SPF also exists in forecasts of the Fed. Accordingly, we
regress 𝑥 | on 𝑥
0, … ,4 as in Section 2, with the difference that only time-series regressions can be| for ℎapplied (no cross-section). The results are reported in Appendix Table G.3. for CPI inflation and additional variables
with a comparison to the SPF (Panels A and B). Interestingly the same pattern of an increase in the coefficient over ℎfrom Section 2 is documented in the Greenbook forecasts (and similarly in mean-level forecasts of the SPF). The
17

robustness of this pattern in Greenbook forecasts across several variables points to utilization of forward information by
the Fed staff when preparing their periodic projections for the U.S. economy.
We also note that in the Greenbooks, unlike in the SPF, the forecasts of both GDP and GDP deflator are
originally in growth rates. Indeed, the results reported in Appendix Table G.3. demonstrate an increase in persistence
across horizons for both variables, when applying time-series regressions to the Greenbook forecasts (Panel B). PanelA further shows that a similar pattern arises in SPF mean-level forecasts of GDP growth based on a time-series
regression, but not in the GDP inflation forecasts. These findings shed more light on issues that were raised above
regarding the SPF micro-level evidence on these variables.

F.3. ECB SPF inflation forecasts
We also investigate inflation forecasts in the European SPF, managed by the European Central Bank since 1999Q1. The
forecasts also refer to consumer inflation (the Harmonized Index of Consumer Prices across European countries), andlike the US SPF, the survey is quarterly. Unlike the US SPF, though, there are no quarterly forecasts in the ECB SPF.
Instead, all forecasts refer to annual changes. Specifically, participants provide rolling-year forecasts for the inflation ayear from now, and the year afterwards. Additionally, they provide forecasts for calendar years, mainly for the current
year and one year ahead. Thus, we cannot estimate persistence regressions across quarterly horizons as in Section 2.
Rather, we can obtain only two persistence estimates: the persistence in the rolling-year forecasts and the persistence incalendar-year forecasts. Notice, however, that the forward information model implies an important distinction between
the two estimations. If forward information is utilized by forecasters, the calendar-year persistence should vary acrosscalendar quarters, while the rolling-year persistence should not. The reason is that the target years of calendar forecasts
are fixed when advancing from Q1 to Q4 of the year. As a result, calendar forecasts apply more forward informationfrom survey to survey, during the calendar year, thus, leading to more biased estimates of persistence, according to ourabove analysis.
We examine this conjecture by estimating persistence regressions, for each calendar-quarter separately.
Appendix Table G.8. reports the estimates for the two types of annual forecasts in Panels A and B (rolling and calendar
forecasts, respectively). The results support the presence of a forward information effect. For the calendar-year forecasts,
there is a sizable reduction in persistence estimates when moving from Q1 to the other quarters. The persistence for Q1
surveys is 0.429, while the estimates using Q2, Q3 or Q4 surveys are between 0.355 and 0.397. The null of coefficient
equality across calendar quarters is also strongly rejected. By contrast, with the rolling-year forecasts, coefficientestimates are higher and vary little across quarters, between 0.453 and 0.502.
We can also check the variation across calendar quarters in the US SPF, since besides the quarterly forecasts it
further provides forecasts for the calendar year, like the ECB SPF. Panel C in Appendix Table G.8. reports the results,
using CPI inflation forecasts for the same period as for the European survey (1991Q1-2007Q4). Strikingly, we find evenmore clear pattern of reduction in the persistence estimates, across the calendar quarters. Notice also that when using
Q1 forecasts, which should provide the least-biased estimation (longer horizons), the coefficient estimate for the US
SPF is the same as the estimate for the ECB SPF – 0.429, implying a similar degree of inflation persistence in the twoareas during the recent years, when considering a far enough horizon to rule out the forward information bias. Finally,
we note that the regression constant is a little bit higher for the US, in line with a small difference in trend inflation.
Overall, these findings provide additional support for the presence of forward information in both surveys ofprofessionals.

18

Appendix G: Additional figures and tables
Figure G.1: Persistence Patterns with Various AR Orders

1

.2

.2

.4

.4

R-Squared.6

Inflation persistence
.6
.8

.8

1

Panel A: AR(2)1985q1

1990q1

1995q1h=4

2000q1
h=3

2005q1

2010q1

h=2

2015q1

1985q1h=1

1990q1

1995q1h=4

2000q1
h=3

2005q1

2010q1

h=2

2015q1
h=1

1
.2

.4

.4

R-Squared.6

Inflation persistence
.6
.8

.8

1

Panel B: AR(3)1985q1

1990q1

1995q1

2000q1

h=4

2005q1

h=3

2010q1

2015q1

1985q1

1990q1h=2

1995q1

2000q1

h=4

2005q1

h=3

2010q1

2015q1

h=2

.5

.2

.6

.4

R-Squared.6

Inflation persistence
.7
.8

.8

.9

1

1

Panel C: AR(2)1985q1

1990q1

1995q1

2000q1
h=4

2005q1
h=3

2010q1

2015q1

1985q1

1990q1

1995q1

2000q1
h=4

2005q1

2010q1

2015q1h=3Notes: The figure plots smoothed estimates of persistence 𝜌 and R-squared measures, based on
estimating augmented version of specification (4) for different forecast horizons in the SPF survey.
The augmented version includes additional forecasts for previous quarters according to the order of
the autoregression, as specified in each panel. Each quarterly point is based on OLS estimation using
the forecast data from the last 8 quarters for a specific horizon. The smoother is a local mean whichuses an Epanechnikov kernel.

19Figure G.2: Simulation Results – Fixed Noise Across Horizons
Panel A: Persistence Estimates

Panel B: R-Squared of Persistence Regressions

Notes: The two panels in the figure show estimation results of specification (4), applied to a simulated
data of forecasts, according to the model presented in section 3. Coefficient estimate (panel A) and Rsquared (panel B) are averaged across 1000 draws of the simulation. Each simulation applies a different
value of persistence in the state process. The variance of the shock in the state process is standardized toone. Regressions were estimated for seven forecast horizons out of eight horizons for which noisy signals
are available (two consecutive horizons in each regression). The noise-to-signal ratio across all thehorizons is set to 2.

20

Figure G.3: Forward Information about Major Events
Panel A: COVID-19
ℎ

1

3

.5

0

Persistence

Persistence

1

1

2

1.5

ℎ

CPI

GDP

INT

2020Q1
2020Q20-12020Q1
2020Q2
UNEMP

CPI

GDP

INT

UNEMP

Panel B: 2008 Financial Crisis
1

ℎ

3

.5

.5

Persistence

Persistence

1

1

1.5

1.5

ℎ

CPI

GDP

INT

2008Q3
2008Q4002008Q3
2008Q4
UNEMP

CPI

GDP

INT

UNEMP

Panel C: 9/11 Attack
ℎ

1

3

0

.5

Persistence

Persistence
.5
1

1

1.5

2

1.5

ℎ

CPI

GDP

INT

UNEMP

2001Q3
2001Q40-.52001Q3
2001Q4

CPI

GDP

INT

UNEMP

Notes: The figure plots estimates of persistence coefficients for specified quarters around three major
events, based on specification (4). The estimation is applied to cross-sections of SPF forecasts of four
macroeconomic variables in the specified quarters. Each circle presents a point estimate and whiskers
show the 95% confidence interval. Blue and red estimates refer, respectively, to the surveys before
and after the event became known. CPI = CPI inflation, GDP = real GDP growth, INT = 3-month
Treasury bill interest rate, UNEMP = unemployment rate.

21

Figure G.4: Impulse Responses to Inflation News
Panel A: Forward Information Method
Unemployment

-.2

-.5

0

Percentage Points-.10Percentage Points
.5
1

1.5

.1

Inflation

0

5

10
Horizon

15

20

0

5

15

20

15

20

15

20

15

20

Percentage Points
0
.1
-.1
-.2

-.2

-.1

Percentage Points
0
.1

.2

10-Year Interest Rate

.2

3-Month Interest Rate

10
Horizon

0

5

10
Horizon

15

20

0

5

10
HorizonPanel B: Barsky-Sims News-Shock Method
Unemployment

-.5

-.5

Percentage Points
0

Percentage Points
0

.5

.5

Inflation

0

5

10
Horizon

15

20

0

5

10-Year Interest Rate

-1

-.4

-.2

-.5

Percentage Points
0.2Percentage Points
0
.5

.4

.6

1

3-Month Interest Rate

10
Horizon

0

5

10
Horizon

15

20

0

5

10
HorizonNotes: The figure plots impulse responses to inflation news in a VAR system estimated on the
1983Q1-2017Q4 sample. Inflation news is obtained by the forward information method in panel A,
or by the Barsky-Sims news-shock method in panel B. The solid lines are the estimated impulse
responses and the shaded areas show the 95% confidence interval based on bootstrap replications.

22

.2

FI variation over forecast variation
.3
.4
.5
.6

.7

Figure G.5: Contribution of Forward Information over Time

1995q1

2000q1

2005q1

2010q1

2015q1

Notes: The figure plots the ratio between forward information and forecast variation, using∑ 𝐹𝐼 |∑ 𝑥|. The component of forward information 𝐹𝐼 | is evaluated by equation (32).

The ratio is computed with a rolling window of 40 obs.

23

Figure G.6: Persistence Patterns of Additional Macroeconomic Variables

.85

.7

R-Squared.8.9

Unemployment persistence
.9
.95
1

1

1.05

Panel A: Unemployment Rate

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1
h=4

h=3

h=2

h=1

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1

h=0

h=4

h=3

h=2

h=1

h=0

.2

.6

.4

R-Squared.6.8

Interest rate persistence
.8
1
1.2

1

1.4

Panel B: Interest Rate

1985q1

1990q1h=4

1995q1

2000q1

h=3

2005q1

h=2

2010q1

2015q1h=1

1985q1h=0

1990q1h=4

1995q1

2000q1

h=3

2005q1

h=2

2010q1

2015q1

h=1

h=0

-.2

.1

0

.2

R-Squared.3

Growth persistence
.2
.4

.4

.6

.5

.8

Panel C: Real GDP Growth

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1
h=4

h=3

h=2

h=1

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1
h=4

h=3

h=2

h=1

Notes: The figure plots smoothed estimates of persistence 𝜌 and R-squared measures based on
estimating specification (4) for different forecast horizons, and using different macroeconomic
variables from the SPF survey, as specified in each panel. Each quarterly point is based on OLS
estimation using the forecast data from the last 8 quarters for a specific horizon. The smoother is alocal mean which uses an Epanechnikov kernel.

24Figure G.7: Persistence Patterns of Additional Inflation Variables

.5
0

0

.1

R-Squared.2.3

GDP deflator persistence
.2
.4
.6

.4

.8

Panel A: GDP Deflator

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1
h=4

h=3

h=2

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1

h=1

h=4

h=3

h=2

h=1

.6
0

0

.2

R-Squared.4

PCE inflation persistence
.2
.4
.6

.8

Panel B: PCE Inflation

2009q1 2010q1 2011q1 2012q1 2013q1 2014q1 2015q1 2016q1 2017q1
h=4

h=3

h=2

h=1

2009q1 2010q1 2011q1 2012q1 2013q1 2014q1 2015q1 2016q1 2017q1

h=0

h=4

h=3

h=2

h=1

h=0

0

.2

.2

R-Squared.4.6

Core inflation persistence
.4
.6
.8

1

.8

Panel C: Core Inflation

2009q1 2010q1 2011q1 2012q1 2013q1 2014q1 2015q1 2016q1 2017q1
h=4

h=3

h=2

h=1

h=0

2009q1 2010q1 2011q1 2012q1 2013q1 2014q1 2015q1 2016q1 2017q1
h=4

h=3

h=2

h=1

h=0

Notes: The figure plots smoothed estimates of persistence 𝜌 and R-squared measures based on
estimating specification (4) for different forecast horizons, and using forecasts for different inflation
measures from the SPF survey, as specified in each panel. Each point on the lines is based on OLS
estimation using the forecast data from the last 8 quarters. The smoother is a local mean which uses
an Epanechnikov kernel.

25

Figure G.8: Real GDP and GDP Deflator – Levels and Higher Order

.94

.85

.96

GDP persistence.98

R-Squared
.9
.95

1

1

1.02

Panel A: Real GDP

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1
h=4

h=3

h=2

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1

h=1

h=4

h=3

h=2

h=1

.97

.99

.975

R-Squared.98.985

GDP deflator persistence
1
1.01
1.02

.99

.995

1.03

Panel B: GDP Deflator

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1
h=4

h=3

h=2

h=1

1975q1 1980q1 1985q1 1990q1 1995q1 2000q1 2005q1 2010q1 2015q1
h=4

h=3

h=2

h=1

Notes: The figure plots smoothed estimates of persistence 𝜌 and R-squared measures for various
forecast horizons, using level forecasts of GDP and GDP deflator from the SPF survey. The ℎ-stepahead forecast is regressed on ℎ 1 and ℎ 2-step ahead forecasts, as in AR(2), and the persistenceis estimated as the sum of coefficients. Each point on the lines is based on OLS estimation using theforecast data from the last 8 quarters. The smoother is a local mean which uses an Epanechnikov
kernel.

26

TABLE G.1.
Regressions of the deviation from the mean forecast: Simulation – Fixed noise across horizons
Dependent variable:
𝑥|

𝑥

𝑥

𝑥

𝑥

|

|

|

|

𝑥|

𝑥

𝑥

𝑥

𝑥

|

|

|

|

𝑥|

𝑥|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

0.903

-0.010-0.001

0.000

0.000

0.903

-0.011-0.001-0.0010.000-0.0100.892-0.013-0.0020.000-0.0100.892-0.013-0.002-0.001-0.001-0.0130.879-0.016-0.003-0.001-0.0130.879-0.017-0.0030.000-0.002-0.0160.862-0.0220.000-0.002-0.0160.862-0.025

0.000

0.000

-0.003-0.022

0.838

0.000

-0.001-0.005-0.022

0.824

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

-0.004-0.0010.000-0.031-0.007-0.002

𝑥

|

𝑥

|

0.000

0.000

0.000

-0.004-0.0310.804-0.047-0.016

𝑥

|

𝑥

|

0.000

0.000

0.000

-0.001-0.007-0.0470.749-0.084

𝑥

|

𝑥

|

0.000

0.000

0.000

0.000

-0.002-0.016-0.084

0.639|

Notes: The table reports true parameters and simulated coefficient estimates from regressions of the individual deviation from the mean forecast, based on
specification (26). Each column reports in bold the simulated regression results for a specified dependent variable, with truncation at horizon ℎ 4. The
true parameters in each column correspond to separate rows in the matrix 𝐼 𝐺 (right column corresponds to the first row and so on). Forecasts are
simulated with 𝜌 0.5 and noise-to-signal ratio of 2 for all signals.

27

TABLE G.2.
Regressions of the deviation from the mean forecast: Simulation
Dependent variable:
𝑥|

𝑥

𝑥

𝑥

𝑥

|

|

|

|

𝑥|

𝑥

𝑥

𝑥

𝑥

|

|

|

|

𝑥|

𝑥|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

0.513

-0.046-0.007-0.002

0.000

0.512

-0.045-0.008-0.002-0.001-0.0180.618-0.059-0.013-0.004-0.0200.618-0.062-0.016-0.006-0.001-0.0300.686-0.072-0.0220.000-0.0280.689-0.070-0.0200.000-0.003-0.0360.764-0.072-0.001-0.003-0.0380.764-0.0710.000-0.001-0.007-0.048

0.783

0.001

-0.002-0.009-0.055

0.751

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

|

𝑥

0.000

0.000

0.000

-0.002-0.0010.000-0.009-0.004-0.002-0.028-0.014-0.007-0.086-0.043-0.021

𝑥

|

𝑥

|

0.000

0.000

-0.002-0.014-0.0640.776-0.111-0.055

𝑥

|

𝑥

|

0.000

0.000

0.000

0.000

-0.001-0.0040.988-0.006

𝑥

|

𝑥

|

0.000

0.000

0.000

0.000

0.000

0.000

0.000

1.000

|

Notes: The table reports true parameters and simulated coefficient estimates from regressions of the individual deviation from the mean forecast, based on
specification (26). Each column reports in bold the simulated regression results for a specified dependent variable, with truncation at horizon ℎ 4. The
true parameters in each column correspond to separate rows in the matrix 𝐼 𝐺 (right column corresponds to the first row and so on). Forecasts are
simulated according to the details in Section 3, with 𝜌 0.5.

28TABLE G.3.
Persistence across forecast horizons in time-series regression: SPF consensus and Greenbook
Forecast horizon:

ℎ

0

ℎ

1

ℎ

2

ℎ

3

ℎ

4

Panel A: SPF consensus
CPI Inflation

0.395***(0.060)GDP Inflation
PCE Inflation
Core CPI Inflation
Interest Rate
Unemployment Rate
GDP Growth
Industrial Production

0.299***(0.052)0.454***(0.046)
0.986***(0.013)0.971***(0.021)0.606***(0.081)
0.962***(0.180)0.299***(0.043)0.785***(0.056)
0.989***(0.006)
0.974***(0.014)0598***(0.030)0520***(0.032)

0.942***(0.034)1.003***(0.031)0.509***(0.055)0.868***(0.014)0.989***(0.007)0.970***(0.010)0.571***(0.042)0.628***(0.072)

1.002***(0.019)1.009***(0.026)
0.572***(0.055)0.895***(0.017)0.982***(0.008)0.965***(0.006)
0.639***(0.056)
0.693***(0.046)0.999***(0.016)
0.957***(0.027)0.678***(0.052)0.866***(0.045)0.978***(0.008)0.960***(0.004)0.790***(0.069)0.777***(0.078)Panel B: Greenbook
CPI Inflation

0.426***
0.366***
0.870***
0.995***
1.019***(0.061)(0.079)(0.055)(0.025)(0.014)
GDP Inflation
0.555***
0.637***
0.898***
0.956***
0.975***(0.068)(0.077)(0.042)(0.037)(0.034)PCE Inflation
0.361***
0.183***
0.323***
0.842***
0.944***(0.082)(0.060)(0.127)(0.063)(0.064)
Core CPI Inflation
0.588***
0.727***
0.983***
0.967***
0.953***(0.071)(0.092)(0.035)(0.023)(0.022)
Unemployment Rate
0.982***
0.971***
0.968***
0.957***
0.949***(0.026)(0.012)(0.007)(0.007)(0.006)
GDP Growth0.528***
0524***
0.598***
0.717***
0.808***(0.071)(0.052)(0.071)(0.096)(0.112)
Industrial Production
0.553***
0346***
0.491***
0.701***
0.739***(0.056)(0.036)(0.077)(0.130)(0.105)
Notes: The table reports coefficient estimates from regressions of mean forecasts 𝑥 | on 𝑥
| . Each entry reports
an estimate from a different regression. The variable 𝑥 and forecast horizon ℎ that were applied in each regression are
specified in the headers of the table's rows. Panels A and B use forecasts from SPF and Greenbook, respectively. The
sample period is 1983Q1-2015Q4, except for PCE and core CPI inflation, for which the sample begins in 2007Q1.
Newey-West standard errors are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

29

TABLE G.4.
Share of inflation volatility explained by news
Forecast Horizon

Forward information method Barsky-Sims method

1
4
8
12

54.3
53.5
51.8
51.2
50.6

20

2.5
5.2
8.4
9.6
10.0

Notes: The table reports the share of inflation volatility explained by news from a variancedecomposition applied to a VAR system estimated on the 1983Q1-2017Q4 sample. Inflation news is
obtained by the forward information method, or by the Barsky-Sims news-shock method, as indicatedin the columns' headers.

TABLE G.5.
Regressions of the deviation from the mean backcast
Unemployment
Dependent variable: 𝑥 |
𝑥|
𝑥

𝑥

𝑥

𝑥

𝑥|

Interest Rate

PCE Inflation

Core Inflation𝑥 | (backcasts)
0.027***

0.018***-0.0000.001(0.006)(0.003)(0.000)(0.001)-0.012*-0.013***-0.000-0.001(0.007)(0.003)(0.000)(0.001)0.011*

0.012***

0.000

0.000

(0.007)(0.003)(0.000)(0.001)-0.009-0.010***0.000-0.001(0.007)(0.004)(0.000)(0.001)

0.003-0.001-0.0000.001(0.004)(0.002)(0.000)(0.001)0.000-0.000-0.000-0.000(0.001)(0.000)(0.000)(0.000)

Obs.

5,476

3,825

1,288

1,394

𝑅

0.006

0.019

0.001

0.005

𝑥

|

𝑥

|

𝑥

|

𝑥

|

Constant

|

|

|

|

Notes: The table reports coefficient estimates from regressions of the individual deviation from the mean backcast,
based on specification (26) with ℎ 0. Each column reports results, using SPF forecasts for a specified
macroeconomic variable. Driscoll-Kraay standard errors are in parentheses. ***, **, * denote significance at 0.01,
0.05, and 0.10 levels.

30

TABLE G.6.
Forward noisy information vs. the standard model: SPF forecasts of additional variables
Dependent variable

𝑥
𝑥
𝑥
𝑥

𝑥
𝑥
𝑥
𝑥

𝑥
𝑥
𝑥
𝑥

|
|
|
|

|
|
|
|

|
|
|
|

𝑥
𝑥
𝑥
𝑥

𝑥
𝑥
𝑥
𝑥

𝑥
𝑥
𝑥
𝑥

|
|
|
|

|
|
|
|

|
|
|
|

Forwardinformation

Standard
framework

Forwardinformation

Standard
framework

Panel A: Unemployment Rate
-6962
-7247
-3385
-3343
-1085
-835
73
117

Panel D: GDP Deflator
16299
17522
15099
16437
15626
16796
16489
16739

Panel B: Interest Rate
-1568
-1694
1079
1109
2277
2313
3164
3155

Panel E: PCE Inflation
3129
3129
2267
2279
1557
1577
1453
1439

Panel C: Real GDP Growth
19336
20871
18891
20473
19047
20544
20541
20654

Panel F: Core Inflation
1247
1273
1058
1074
1043
1095
818
827

Notes: The table reports BIC statistics associated with specifications (26) and (27), for the forward information and thestandard noisy information framework, respectively. Each panel presents results for a specified macroeconomic variable,using the SPF forecasts. The specifications were estimated for several forecast horizons as indicated in the first column(ℎ 0,1,2,3).

31

TABLE G.7.
Regressions of forward information: Additional variables
Dependent Variable
Unemployment

(𝑥 ):

(1)
𝐹𝐼 |
𝑥
𝑥
𝑥
𝑥

Constant

𝑅(2)

Interest Rate
(3)

(4)

Real GDP Growth
(5)

(6)

1.110***

1.144***

1.146***(0.084)(0.064)(0.133)

1.626***

1.095***

1.507***

0.972***

0.313***

0.259***(0.090)(0.087)(0.094)(0.067)(0.074)(0.057)-0.641***-0.109-0.479***

0.084

0.108

0.062

(0.183)(0.105)(0.145)(0.094)(0.088)(0.076)-0.0080.013-0.033-0.165***

0.027

0.011

(0.168)(0.082)(0.141)(0.046)(0.078)(0.069)-0.017-0.044-0.0190.092**

0.008

0.007

(0.065)(0.046)(0.090)(0.038)(0.090)(0.070)0.256***

0.186***

0.066**

0.116***

1.546***

2.402***(0.078)(0.053)(0.033)(0.032)(0.379)(0.412)

0.975

0.989

0.985

0.996

0.142

0.442

Notes: The table reports coefficient estimates from various regressions. The variable 𝑥 is indicated in the columns'
headers. The forward information variable is computed according to (32). The sample period is 1972Q1-2017Q4 forunemployment and GDP growth, and 1983Q3-2017Q4 for the interest rate. Newey-West standard errors are inparentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

32

TABLE G.8.
Persistence in ECB and US annual forecasts across calendar quarters (1999Q1-2017Q4)
Whole Sample
Panel A: 𝑥|

Constant

𝑐

𝜌𝑥

|

Q1

Q2

Q3

Q4

𝑢 (ECB SPF)

0.956***

0.940***

1.000***

0.927***

0.959***(0.047)(0.060)(0.067)(0.037)(0.086)

0.483***

0.488***

0.457***

0.502***

0.484***(0.027)(0.033)(0.040)(0.023)(0.049)

Obs.

3378

946

836

752

844

𝑅

0.439

0.438

0.406

0.507

0.405

𝐹𝑥

Panel B: 𝑥|

Constant𝑐𝜌𝑥|𝑢 (ECB SPF)

1.031***

0.984***

1.087***

0.990***

1.013***(0.039)(0.068)(0.029)(0.058)(0.063)0.379***

0.429***

0.355***

0.397***

0.370***(0.023)(0.035)(0.026)(0.040)(0.034)

Obs.

4214

1067

1057

991

1099

𝑅

0.560

0.492

0.509

0.631

0.600

𝐹𝑥

Panel C: 𝑥
Constant

|

𝑐

𝜌𝑥

|

𝑢 (US SPF)

1.617***

1.391***

1.521***

1.673***

1.689***(0.091)(0.179)(0.114)(0.103)(0.122)0.269***

0.429***

0.338***

0.236***

0.193***(0.041)(0.082)(0.043)(0.041)(0.042)

Obs.

2753

667

686

656

744

𝑅

0.183

0.361

0.280

0.161

0.091

𝐹𝑥

Notes: The table reports coefficient estimates for the specified equations at the top of each panel. 𝑥 | and 𝑥 | arerolling-year forecasts for one and two years ahead, respectively. 𝑥 | and 𝑥 | are calendar-year forecasts for thecurrent and the next calendar years. Regressions were estimated for the period of 1999Q1-2017Q4 and separately foreach of the calendar quarters in this period as indicated by columns' headers. Standard errors of Driscoll and Kraay(1998) are in parentheses. ***, **, * denote significance at 0.01, 0.05, and 0.10 levels.

33