The Term Structure of Uncertainty: New Evidence from Survey Expectations

Transcription

1 The Term Structure of Uncertainty: New Evidence from Survey Expectations Carola Binder Tucker S. McElroy Xuguang S. Sheng Abstract We construct measures of individual forecasters subjective uncertainty at horizons ranging from one to five years, incorporating a rich information set from the European Central Bank s Survey of Professional Forecasters. Our measures reflect the subjective expectations of market participants and are ex ante measures that do not require the knowledge of realized outcomes. We find that the uncertainty curve is more linear than the disagreement curve uncertainty at the one-year and two-year horizons can almost perfectly predict uncertainty at the five-year horizon, but not so for disagreement. We document substantial heterogeneity across forecasters in both the level and the term structure of uncertainty, and show that short-run uncertainty is countercyclical, but the difference between long-run and short-run uncertainty is procyclical. We develop a learning model that captures the two channels through which large shocks affect the term structure of uncertainty: wake-up effect the visibly large shocks induce immediate and synchronized updating of information for economic agents, and wait-and-see effect the occurrence of those shocks generates increased uncertainty among agents. Depending on the magnitude of the shock, and whether it is temporary or permanent, the response of uncertainty afterwards displays heterogeneous patterns over different time horizons. Keywords: Density forecasts; Heterogeneity; Inflation; Surveys; Term structure; Uncertainty The views expressed herein are those of the authors and do not necessarily reflect the views of U.S. Census Bureau. Haverford College, cbinder1@haverford.edu. U.S. Census Bureau, tucker.s.mcelroy@census.gov. American University, sheng@american.edu. 1

2 1 Introduction Uncertainty varies over time and with the business cycle (Bloom, 2009; Bachmann and Bayer, 2013). Heightened uncertainty in the Great Recession has prompted renewed efforts to investigate the expectations formation process, the nature of economic agents information problems, and sources and consequences of uncertainty (Hellwig and Venkateswaran, 2009; Baker and Bloom, 2013; Bachmann et al., 2013; Leduc and Liu, 2016). Survey forecasts can provide valuable information about the expectations formation process and the information environment (Mankiw et al., 2004; Coibion and Gorodnichenko, 2012). In this paper, we construct measures of individual forecasters subjective uncertainty at horizons ranging from one to five years, using density (histogram) forecasts from the European Central Bank s Survey of Professional Forecasters (ECB SPF). Uncertainty refers to the spread (e.g. variance) of an individual agent s probability distribution about an outcome. We also construct measures of disagreement, or dispersion of expectations across forecasters, at each horizon. We explore the properties of uncertainty and disagreement over shorter and longer horizons, documenting four stylized facts. First, the term structure of uncertainty is linear that is, uncertainty at the one-year and two-year horizons can almost perfectly predict uncertainty at the five-year horizon. This is true for both aggregate uncertainty and at the individual forecaster level. Barrero et al. (2017) show that implied volatility data is also characterized by linearity, so the entire term structure can be summarized by a level and a slope term. Our second stylized fact is that the term structures of uncertainty and disagreement are qualitatively quite different. The term structure of disagreement is not characterized by high linearity. Average uncertainty nearly always increases with forecast horizon, while disagreement is at times increasing, decreasing, or non-monotonic in forecast horizon. We thus contribute to a literature on the empirical and theoretical distinctions between disagreement and uncertainty, but our distinction is to focus on differences in their term structures (Zarnowitz and Lambros, 1987; Boero et al., 2008; D Amico and Orphanides, 2008; Lahiri 1

3 and Sheng, 2010; Rich and Tracy, 2010; Abel et al., 2016; Kozeniauskas et al., 2018). Third, we document substantial heterogeneity across forecasters in both the level and term structure of uncertainty. While average uncertainty increases with forecast horizon, a sizeable minority of forecasters have higher uncertainty at shorter horizons. This heterogeneity is persistent. That is, particular forecasters tend to have particularly wide or narrow (or even inverted) term structures of uncertainty. We then explore the forecaster characteristics, including ex post forecast accuracy, tendency to disagree, and revision frequency, associated with the slope of the term structure. Fourth, we show that not only the level, but also the slope of the term structure of uncertainty, is time-varying. Other authors have shown that uncertainty is countercyclical (Bloom, 2009, 2014; Baker et al., 2016). We confirm this, but also show that the slope is procyclical. In recessions, short-run uncertainty rises significantly more than long-run uncertainty. Lahiri and Sheng (2008) explore the dynamics of disagreement at varying horizons and estimate the relative importance of each component of disagreement due to differences in (i) the initial prior beliefs, (ii) the weights attached on priors, and (iii) interpreting public information. Similarly, Patton and Timmermann (2010) show that the term structure of disagreement can be used to determine the relative importance of various causes of disagreement differences in priors versus differences in private information. We argue that the term structure of uncertainty is also key to understanding the public and private information in the expectations formation process of economic agents. Guided by our stylized facts, we model forecasters signal extraction process under an information structure with private and public channels of information that are subject to shocks, and derive implications for uncertainty and dispersion across forecast horizons. 2

4 2 Data and Measurement of Uncertainty and Related Variables The European Central Bank (ECB) has conducted the Survey of Professional Forecasters (SPF) since Each quarter, approximately 60 forecasters provide point and density forecasts for inflation (π), GDP growth (g), and unemployment (u) over several horizons. Density forecasts take the form of histograms; the forecaster assigns probabilities, summing to 100%, that the realization will fall in each bin. The density forecasts have a consistent bin width of 0.5 percentage points. This is an advantage over the U.S. Survey of Professional Forecasters, for which the bin width changes over time. The number of bins occasionally changes to try to ensure that large probabilities are not assigned to the upper and lower intervals, which are open-ended. Thus, for example, additional lower bins were added for the inflation and growth density forecasts in We consider forecasts at three horizons that are included on most survey dates: one-year, two-year, and five-year. The one- and two-year forecasts have rolling horizons, where the rolling one-year horizon refers to the month (for inflation and unemployment) or quarter (for growth) one year ahead of the latest available observation at the time of the survey. The rolling two-year horizon refers to the month or quarter two years ahead of the latest available observation. For surveys on the third and fourth quarter of the year, the five-year horizon refers to five calendar years ahead, while on the first and second quarter of the year, the five-year horizon refers to four calendar years ahead. Uncertainty Measurement We use the density forecasts to construct a measure Uith x of ex ante, subjective uncertainty for each forecaster i, variable x, and horizon h in quarter t, defined as the variance of the density forecast. We estimate the variance and the mean of each forecaster s density 1 In the first quarter of 2009, many forecasters assigned high probability sometimes even 100% to the lowest bin for growth, corresponding to growth rates less than -1%. As a result, measured uncertainty in this period is artificially low, so we omit 2009Q1 growth uncertainty from subsequent analysis. In the second quarter, bins for growth rates of less than -6%, -6% to -5.5%,..., -1.5% to -1% were added. 3

5 forecasts both parametrically and non-parametrically. Our parametric estimates come from fitting (via maximum likelihood) a generalized beta distribution to the density forecast, with supports determined by the individual forecast values. Liu and Sheng (2018) show that this distributional setting, which is highly flexible, performs best in terms of goodness of fit in mimicking the empirical histograms in the data, which can be asymmetric or irregular. 2 Alternatively, following D Amico and Orphanides (2008), we compute the variance of each density forecast nonparametrically by assuming that the probability is concentrated at the midpoint of each bin. 3 Our parametric and non-parametric uncertainty estimates are extremely similar, with correlation coefficients above Therefore, in all subsequent analysis, we use the parametric variance estimate as our measure of uncertainty at the individual forecaster level. We also consider aggregate uncertainty Ū th x, the average across forecasters of U x ith. Disagreement Measurement Let F x ith and let µ x ith denote forecaster i s point forecast of variable x for horizon h made at time t, denote the mean of her density forecast. The parametric and non-parametric estimates of µ x ith have correlation coefficients above 0.99, and in the following analysis we use the parametric estimates. Engelberg et al. (2009) show that for the US SPF, point forecasts are usually close to the central tendency of density forecasts. Likewise, we find that the correlation between F x ith and µx ith is around 0.9 with respondents point forecasts. See Appendix Figure A.1 for plots of Fith x against µx ith for each variable and horizon. Disagreement D x th is defined as the cross-sectional variance of forecasts for variable x at horizon h. Note that this could refer to the heterogeneity of point forecasts or of the density means across forecasters. Since both measures are very similar (with correlation coefficients 2 The four distribution settings that Liu and Sheng compare include the normal distribution, as used by Giordani and Soderlind (2003), the generalized beta distribution with no parameter constraint, the generalized beta distribution with supports determined by individual forecast values, and a combination of beta and triangle distributions, as used by Engelberg et al. (2009). 3 D Amico and Orphanides show that estimates of uncertainty computed with the assumption of midpoint probability concentration are very similar if probability is assumed to be uniformly distributed across each bin or if a normal distribution is fitted to the density forecast. Therefore, estimates of uncertainty are not sensitive to this assumption. 4

6 around 0.8 or 0.9 depending on variable and horizon), for consistency with the literature we use the variance of F x ith. We also define an individual-level measure of disagreement, which is the absolute difference of forecaster i s point forecast at time t for variable x at horizon h from the cross-section mean: D x ith = F x ith 1 N t N t j=1 F x jth (1) where N t is the number of forecasts made for that variable and horizon at time t. Revision Frequency and Mean Squared Error We also use the point forecast in defining a respondent s revision frequency and mean squared error (MSE). A forecaster s revision frequency for variable x at horizon h is the share of periods in which she makes a nonzero revision to her point forecast for x at horizon h. Estimating the MSE requires data on the realization of the variables being forecasted, which we obtain from Eurostat. More detailed information on these variables appears in Appendix Tables A.1 and A.2. The forecast error e x ith is the difference between the actual realization and forecast. The squared forecast error is sometimes used as a measure of ex post uncertainty or accuracy. 3 Stylized Facts In this section, we present four stylized facts from our empirical analysis of the term structures of uncertainty and disagreement. (1) The term structure of uncertainty is linear. Figure 1 displays estimates of aggregate uncertainty by horizon for inflation, growth, and unemployment. For all variables, uncertainty increases with forecast horizon. A term structure is linear if the entire curve is well characterized by a level and slope statistic. We regress 5

7 aggregate 5-year uncertainty on 1-year uncertainty (the level) and the difference between 1- and 2-year uncertainty (the slope). Table 1 shows that aggregate uncertainty at the 5-year horizon is largely explained by the level and slope statistic, with R 2 around 0.9 for each variable. Table 2 shows a panel version of this regression, in which the dependent variable is 5-year uncertainty for forecaster i in quarter t. Columns (1) through (3) do not include time or forecaster fixed effects, while (4) through (6) do. Again, for all variables, the term structure of uncertainty is highly linear. In the similar spirit, Barrero et al. (2017) document that volatility curves are linear, or in other words, that a regression of long-run implied volatility on short-run volatility and the difference between short- and medium-run volatility has a high R 2. Table 1: Regression for linearity of the term structure of aggregate uncertainty (1) (2) (3) Inflation Growth Unemp 1-year (0.02) (0.03) (0.04) 2-yr minus 1-yr (0.16) (0.12) (0.27) Constant (0.04) (0.05) (0.08) Observations r Notes: Robust, time-clustered standard errors in parentheses. Dependent variable is log uncertainty at 5-year horizon for indicated variable. *** p< 0.01, ** p< 0.05, * p<

8 Figure 1: Time Series of Aggregate Uncertainty Notes: ECB SPF data. Uncertainty is the variance of a beta distribution fitted to an individual s density forecast. We take the average of log uncertainty across forecasters at time t. 7

9 Table 2: Panel regression for linearity of the term structure of uncertainty (1) (2) (3) (4) (5) (6) Inflation Growth Unemp Inflation Growth Unemp 1-yr (0.04) (0.03) (0.04) (0.02) (0.02) (0.02) 2-yr minus 1-yr (0.04) (0.05) (0.05) (0.03) (0.03) (0.03) Constant (0.05) (0.04) (0.07) (0.07) (0.07) (0.09) Observations r2 o Time FE No No No Yes Yes Yes Forecaster FE No No No Yes Yes Yes Notes: Robust, time-clustered standard errors in parentheses. Dependent variable is log uncertainty for forecaster i at 5-year horizon for indicated variable. *** p< 0.01, ** p< 0.05, * p< (2) Uncertainty is distinct from disagreement. Figure 2 displays the evolution of disagreement for each variable; it is visually quite distinct from Figure 1. Uncertainty increases in the Great Recession, remaining high thereafter. Disagreement also spikes in the Great Recession, but unlike uncertainty, does not remain elevated. For unemployment, disagreement increases monotonically with horizon, but the term structure is often inverted for the other variables. For growth, 1-year disagreement in 2009Q3 spiked to 0.61, while 5-year disagreement was just Most forecasters expected growth to return to around 1.8% in the long-run, though they disagreed. The term structure of disagreement is not as linear as the term structure of uncertainty. In Table 3, we regress 5-year disagreement on 1-year disagreement and the difference between 1- and 2-year disagreement, and the R 2 values are much lower: 0.39, 0.14, and 0.40 for inflation, growth, and unemployment, respectively. Table 4 shows results from an analogous panel regression, where the dependent variable is the log absolute difference of forecaster i s 5-year forecast at time t from the consensus forecast. As in Table 2, columns (4) through (6) include fixed effects. But here, the R 2 values are much closer to zero. Note, too, that the ex ante uncertainty measure is also distinct from mean squared error 8

10 Figure 2: Time Series of Forecast Disagreement Notes: ECB SPF data. Disagreement is the cross-sectional variance of the point forecasts. 9

11 (ex post uncertainty), which also has a less linear term structure (see Appendix Figure A.2 and Table A.3). For growth, for example, the one-year forecast made in 2008Q4 had the largest ex post error. The mean forecast was 1.2%, and the realization was -5.5%. For the longer horizon forecasts, of course, even earlier forecasts were associated with larger errors ex post due to the recession. But ex ante, forecasters were not particularly uncertain in 2008Q4. The mean variance of their density forecasts was 0.3. It was not until the recession was fully underway that ex ante uncertainty rose, and it remained elevated even after ex post uncertainty fell. Table 3: Regression for linearity of the term structure of disagreement (1) (2) (3) Inflation Growth Unemp 1-year (0.14) (0.08) (0.06) 2-yr minus 1-yr (0.17) (0.10) (0.14) Constant (0.40) (0.20) (0.15) Observations r Notes: Robust, time-clustered standard errors in parentheses. Dependent variable is log disagreement at 5-year horizon for indicated variable. *** p< 0.01, ** p< 0.05, * p<

12 Table 4: Panel regression for linearity of the term structure of disagreement (1) (2) (3) (4) (5) (6) Inflation Growth Unemp Inflation Growth Unemp 1-yr (0.03) (0.03) (0.03) (0.03) (0.03) (0.03) 2-yr minus 1-yr (0.02) (0.02) (0.03) (0.02) (0.02) (0.02) Constant (0.08) (0.09) (0.07) (0.19) (0.19) (0.19) Observations r2 o Time FE No No No Yes Yes Yes Forecaster FE No No No Yes Yes Yes Notes: Robust, time-clustered standard errors in parentheses. Dependent variable is log absolute difference of forecaster i s 5-year forecast at time t from the average forecast for the indicated variable. *** p< 0.01, ** p< 0.05, * p< (3) There is substantial heterogeneity across forecasters in both the level and term structure of uncertainty. Individual forecasters differ in both their level and term structure of uncertainty. Figure 3 displays the variability of variance across forecasters over time for inflation. We find that heterogeneity of forecast uncertainty is both substantial and time-varying. See Figures A.4, and A.5 for analogous results for growth and unemployment. Figure 4 displays the share of forecasters for whom longer-horizon uncertainty is greater than shorter-horizon uncertainty for each variable over time. For inflation, about 70% of forecasters have higher uncertainty at the two-year than the one-year horizon, and 72% have higher uncertainty at the five-year than at the one-year horizon. For growth, these figures are 70% and 76%, and for unemployment, 75% and 85%. Figure A.3 displays histograms of the difference between 5-year and 1-year uncertainty for individual forecasters, pooled across all time periods. For each variable, for most respondents, 5-year uncertainty is slightly greater than 1-year uncertainty, but, especially for inflation, there is a sizeable minority for whom the term structure of uncertainty is inverted. Bruin et al. (2011) and Boero et al. (2015) find evidence of persistence in uncertainty 11

13 Figure 3: Distribution of inflation uncertainty across forecasters over time Notes: ECB SPF data. Interior line is the median, bottom and top of boxes are the 25th and 75th percentiles, and bottom and top of lines are 10th and 90th percentiles of inflation uncertainty. 12

14 Figure 4: Share of Forecasters with Higher Uncertainty at Longer Horizon Notes: ECB SPF data. Centered 5-quarter moving average. for consumers and professional forecasters, respectively. We corroborate this finding that an individual forecaster s uncertainty is persistent, and also show that the term structure of uncertainty is persistent. Following Patton and Timmermann (2010), for each quarter, we rank forecasters in order of uncertainty for each variable and horizon, then compute transition matrices for each quartile. If uncertainty is not persistent, all table entries should be approximately 25%, but if uncertainty is persistent, the diagonal entries should be greater than the off-diagonals. Tables 5, 6, and 7 show these transition matrices for inflation, growth, and unemployment, respectively, for each forecast horizon. They also show transition matrices for quartiles for the slope of the term structure of uncertainty. In all cases, the diagonal entries are significantly greater than 25%. Hence, both the level and the slope of the term structure of uncertainty are persistent, though the persistence is greater for the level than for the slope. Also see Appendix Table A.4, which shows panel regressions summarizing the persistence of the level and term structure of uncertainty. To investigate the forecaster characteristics associated with the level and term structure of uncertainty, we limit our sample to the 40 forecasters who make at least 25 density 13

15 Table 5: Quartile transition matrices for inflation uncertainty One-year uncertainty Total Two-year uncertainty Total Five-year uncertainty Total Slope (two-year minus one-year) Total Notes: Rows correspond to respondent s quartile in quarter t and columns in quarter t

16 Table 6: Quartile transition matrices for growth uncertainty One-year uncertainty Total Two-year uncertainty Total Five-year uncertainty Total Slope (two-year minus one-year) Total Notes: Rows correspond to respondent s quartile in quarter t and columns in quarter t

17 Table 7: Quartile transition matrices for unemployment uncertainty One-year uncertainty Total Two-year uncertainty Total Five-year uncertainty Total Slope (two-year minus one-year) Total Notes: Rows correspond to respondent s quartile in quarter t and columns in quarter t

18 forecasts for each variable at each horizon. For each of these forecasters, we compute the mean level of log uncertainty over time at the one-year horizon (ln Ui1) x and the mean slope of her uncertainty term structure (ln Ui5 x ln Ui1) x for each variable x {π, g, u}. First, we note that both the mean level and the mean slope are correlated across variables. That is, a forecaster with relatively high average inflation uncertainty also has relatively high average growth and unemployment uncertainty. And a forecaster with a relatively wide term structure of uncertainty for inflation is also likely to have a relatively wide term structure for growth and unemployment (Appendix Table A.5). We test whether the level and slope of a respondent s term structure of uncertainty vary with the forecaster s mean squared errors, tendency to disagree, and revision frequency. When considering cross-sectional differences in mean squared forecast errors, it is important to control for differences over time in the difficulty of forecasting (Clements, 2016). Following DiAgostino et al. (2012) and Clements (2014), we also construct an adjusted measure of MSE. First we divide the forecast error e x ith by the cross section MSE at time t: ẽ x ith = e x ith 1 Nt. (2) N t j=1 (ex jth )2 Then forecaster i s adjusted MSE for variable x at horizon h is the mean across t of (ẽ x ith )2. The adjusted MSE can differ substantially from the unadjusted MSE, since there is so much time variation in the cross section MSE (see Figure A.2). We consider analogous adjustments for log uncertainty, individual disagreement, and revision frequency. 4 However, for these, the adjusted and unadjusted variables are almost perfectly correlated (see Appendix Table A.6). Thus, for simplicity, we use unadjusted log uncertainty, individual disagreement, and revision frequency. Table 8 uses regression analysis to test the association of the level and slope of uncertainty with each of these characteristics. The independent variables are adjusted mean squared 4 Specifically, for variable z x ith, the adjusted variable is zx ith = z x ith 1 Nt N t j=1 zx jth. 17

19 error, tendency to disagree, and revision frequency, each at the one-year and five-year horizon. We omit the two-year horizon of the independent variables to avoid collinearity issues. For each of the 40 forecasters, we have three observations corresponding to their inflation, growth, and unemployment forecasts. We use a panel regression with between-effects estimator, since we are interested in explaining why different forecasters have different uncertainty term structures. 5 The first three columns show that uncertainty at each horizon decreases with long-run disagreement and revision frequency, but increases with long-run MSE. Thus, those who update more frequently and dissent more at the long-run horizon tend to be more confident at all horizons. And those who face high ex ante uncertainty at varying horizons are most likely to make large ex post forecast errors in the long run (but not in the short run). For MSE and disagreement, these effects are stronger as the forecast horizon increases. Thus, as shown in column 4, the slope of the uncertainty term structure (i.e. the difference between two-year and one-year log uncertainty) decreases with short-run MSE and long-run disagreement. Appendix Table A.7 presents specifications with three alternative dependent variables: the share of a respondent s forecasts for which two-year uncertainty exceeds one-year uncertainty, the share for which five-year exceeds one-year uncertainty, and a dummy variable indicating that the forecaster s five-year uncertainty exceeds one-year uncertainty in at least 90% of periods. Results are qualitatively consistent with column (4) of Table 8. Most notably, short-run disagreement is positively associated and long-run disagreement negatively associated with having higher long-run than short-run uncertainty, implying that disagreement might be a key driver of the term structure of uncertainty. 5 Results are similar if we use OLS with variable fixed effects. Statistical significance of coefficient estimates is somewhat reduced. 18

20 Table 8: Regressions of level and slope of uncertainty on forecaster characteristics (1) (2) (3) (4) lnu 1 lnu 2 lnu 5 Slope Adjusted MSE (1y) (0.78) (0.79) (0.83) (0.14) Adjusted MSE (5y) (0.51) (0.52) (0.54) (0.09) Disagreement (1y) (0.45) (0.46) (0.48) (0.08) Disagreement (5y) (0.44) (0.45) (0.47) (0.08) Revision Freq. (1y) (1.61) (1.64) (1.72) (0.30) Revision Freq. (5y) (0.56) (0.57) (0.60) (0.10) Constant (1.97) (2.00) (2.10) (0.36) N R Notes: Robust, standard errors in parentheses. *** p< 0.01, ** p< 0.05, * p< Sample includes the 40 forecasters who made at least 25 forecasts of each variable and horizon. (4) Short-run uncertainty is countercyclical, but the difference between longrun and short-run uncertainty is procyclical. Both the level and term structure of uncertainty display some notable time variations. Uncertainty is countercyclical, but the slope of the term structure is procyclical. That is, in periods of low growth, uncertainty rises, but moreso at the shorter horizon, so the term structure narrows. The level of uncertainty also rises, and the term structure narrows, when inflation is far from target. Uncertainty rises when there is high economic policy uncertainty (as measured by the Economic Policy Uncertainty Index for Europe from Baker et al. (2016)), 6, and the slope is unchanged or falls. Each of these patterns can be seen in Appendix Table A.8, which summarizes the cor- 6 The EPU is based on newspaper articles regarding policy uncertainty. Data was downloaded in April 2018 from monthly.html. The newspapers include Le Monde and Le Figaro for France, Handelsblatt and Frankfurter Allgemeine Zeitung for Germany, Corriere Della Sera and La Repubblica for Italy, El Mundo and El Pais for Spain, and The Times of London and Financial Times for the United Kingdom. 19

21 relation between aggregate log uncertainty or the slope of the uncertainty term structure with growth, the deviation of inflation from target, and the EPU. The level of uncertainty is most strongly correlated with the EPU. The correlation of the EPU with the slope of uncertainty is only statistically significant for inflation uncertainty. This is consistent with Binder s (2017) finding that the U.S. EPU is more strongly correlated with shorter-run than longer-run inflation uncertainty. For inflation and growth, more than for unemployment, the difference between 5-year and 1-year log uncertainty is strongly procyclical (see Figure 5). In particular, the lowest growth rates in the first two quarters of 2009 correspond to the narrowest term structure of growth uncertainty and the only time when aggregate 1-year growth uncertainty was greater than aggregate 5-year growth uncertainty. These patterns also appear in the panel regressions with forecaster fixed effects in Table 9. For inflation, growth, and unemployment, we regress log uncertainty at the short and long horizon on growth, the deviation of inflation from target, the EPU, and forecaster fixed effects. We also regress the difference between 2-year and 1-year log uncertainty, the difference between 5-year and 1-year log uncertainty, and a dummy variable indicating that 5-year uncertainty is greater than 1-year uncertainty, on the same variables. For all variables, uncertainty declines with growth and increases with π 2, and both effects are greater at the shorter horizon. Uncertainty increases with the EPU, with similar effects at each horizon. Column 4 shows that the slope decreases with π 2, and for inflation and growth forecasts, the slope increases with g. These results clearly show that while the level of term structure of uncertainty is countercyclical, its slope is procyclical. 20

22 Table 9: Regressions of level and slope of uncertainty on macro variables A. Inflation uncertainty (1) (2) (3) (4) (5) 1y 5y 2y 1y 5y 1y 5y>1y g (0.005) (0.006) (0.003) (0.005) (0.005) π (0.015) (0.018) (0.010) (0.015) (0.013) EPU/ (0.017) (0.020) (0.011) (0.017) (0.015) N R B. Growth uncertainty (1) (2) (3) (4) (5) 1y 5y 2y 1y 5y 1y 5y>1y g (0.006) (0.007) (0.004) (0.006) (0.005) π (0.016) (0.019) (0.010) (0.015) (0.013) EPU/ (0.018) (0.021) (0.012) (0.018) (0.015) N R C. Unemployment uncertainty (1) (2) (3) (4) (5) 1y 5y 2y 1y 5y 1y 5y>1y g (0.006) (0.007) (0.004) (0.007) (0.004) π (0.018) (0.021) (0.011) (0.020) (0.012) EPU/ (0.020) (0.024) (0.012) (0.022) (0.014) N R Notes: Robust, standard errors in parentheses. *** p< 0.01, ** p< 0.05, * p< Panel regressions with forecaster fixed effects. In columns 1 and 2, dependent variable is log uncertainty for the 1-year and 5-year horizon, respectively, for inflation, growth, or unemployment, as indicated. In columns 3 and 4, dependent variable is the difference between 2-year and 1-year log uncertainty or the difference between 5-year and 1-year log uncertainty. In column 5, dependent variable is a dummy variable indicating greater uncertainty at 5-year than at 1-year horizon. All regressions include a constant term. EPU is rescaled (divided by 100) for ease of presenting coefficients. 21

23 Figure 5: Economic growth and slope of term structure of uncertainty Notes: The slope of inflation uncertainty is ln U π 5 ln U π 1. The slope of growth uncertainty is ln U g 5 ln U g 1. 4 Information Structure and Expectation Updating Following Baker et al. (2017), we use a model in which N agents forecast an m-dimensional stationary Markov signal process {π t } using noisy public and private signals. The observation process for agent i is y t (i) = A(i) B π t + ν t(i) η t. (3) where {η t } and {ν t (i)} are the public and private noise and the matrix A (i) (B) corresponds to the private (public) manifestation of the signal. We assume that {η t } is serially uncorrelated and to allow for time-varying uncertainty has a stochastic covariance matrix Σ t. The private noises {ν t (i)} are serially uncorrelated with deterministic covariance matrix Σ (i). 22

24 4.1 State Space Representation We first consider the forecasters Kalman filtering process in the case that Σ t, the parameters governing {π t }, and the matrices B and A (i) are known. Later, to introduce first- and secondmoment shocks, we will consider the unknown parameters case. Suppose there exists a matrix G such that π t = G x t, where x t is a Markovian state vector x t with transition equation: x t = Φ x t 1 + ɛ t (4) for t 1, with initial value x 0. We assume the transition matrix Φ has eigenvalues less than one in magnitude and that the signal innovations {ɛ t } are uncorrelated with x 0, so that ɛ t is uncorrelated with x t 1 for t 1. The innovations common covariance matrix is denoted Σ ɛ. Let δ t (i) = ν(i) t η t H(i) = A(i) B G, so that combining (3) with π t = G x t yields the observation equation y t (i) = H(i) x t + δ t (i). (5) Evidently, {δ t (i)} is heteroscedastic white noise, with covariance matrix S t given by S t = Var[δ t (i)] = Σ(i) 0 0 Σ t. (6) Note that the only data available to the ith agent is {y t (i)}, and so estimates of the signal are to be constructed on this basis, without reference to the data available to some other agent j. Together, equations (5) and (4) describe the information structure in state space form. Then x t = [π t, π t 1,..., π t p+1] with G = [I m, 0,...] (and I m is the m-dimensional identity matrix) corresponds to the companion form, yielding a VAR(1) for the state vector, 23

25 writing Φ = Φ 1 Φ 2... Φ p I m I m 0. We define the following quantities: the forecast of the state vector is x t+1 t (i) = E[x t+1 y 1 (i),..., y t (i)], and its mean square error matrix is P t+1 t (i) = Cov[x t+1 x t+1 t (i)]. The residual is the data minus its forecast, namely e t (i) = y t (i) ŷ t t 1 (i), and its mean square error matrix is denoted V t (i). The Kalman gain is by definition K t (i) = Cov[x t+1, e t (i)] Var[e t (i)] 1, and plays a key role in updating a signal extraction estimate given new information. Initialization of the recursive Kalman filter algorithm is given by x 1 0 (i) = 0 and P 1 0 (i) = Var[x 1 ], which are the correct quantities given a stationary state vector. In the case of a VAR(p) signal process, this initial variance can be computed directly from the companion form. Then for 1 t T, we compute e t (i) = y t (i) H(i) x t t 1 (i) (7) V t (i) = H(i) P t t 1 (i) H(i) + S t (8) K t (i) = Φ P t t 1 (i) H(i) V t (i) 1 (9) x t+1 t (i) = Φ x t t 1 (i) + K t (i) e t (i) (10) P t+1 t (i) = (Φ K t (i) H(i)) P t t 1 (i) Φ + G Σ ɛ G. (11) Equation (9) gives a recursive formula for the Kalman gain, and its dependence on the heteroscedastic noise is clearly given through V t (i) in (8). Moreover, equations (10) and (11) tell us how to update our one-step ahead prediction and forecast error variance for the state 24

26 vector. Again, because the Kalman gain depends upon the heteroscedastic variance Σ t, both the state vector forecast and its uncertainty will be impacted. In order to obtain (k +1)-step ahead forecasts (k 0), we compute π t+1 t k (i) = G Φ k x t+1 k t k (i) (12) Var[ π t+1 t k (i) π t+1 ] = G P t+1 t k (i) G, (13) where P t+1 t k (i) = Var[ x t+1 t k (i) x t+1 ] and is further described below. So (12) provides an optimal estimate of the forecasted signal, given the presence of noise with time-varying volatility. We require a flexible and broad specification for the signal and noise processes, and propose the VAR(p) class for the signal, where p is taken sufficiently large to approximate a generic signal. We parametrize this process such that stationarity is guaranteed, as described in Roy, McElroy and Linton (2017); the private noise has variance matrix Σ (i), which can be parameterized as a member of the space of symmetric positive definite matrices. For the dynamics of the public signal, we describe Σ t as a stochastic volatility process. 7 Following the work of Cogley and Sargent (2005), Primiceri (2005), and Neusser (2016), consider the Cholesky decomposition Σ t = B t Ω t B t, where B t is unit lower triangular and Ω t is diagonal. The diagonal entries of Ω t each follow an exponential random walk, and the B t = exp{c t }, where C t is a lower triangular with zeroes on the diagonal. Each lower triangular element of C t follows an independent random walk. This framework of a VAR(p) signal with the exponential random walk model for volatility can be tailored to the user s specification through the parameter settings, which determine the dispersion for the random walk increments in the volatility process. 7 Chiu, Leonard and Tsui (1996) modeled Σ t as the matrix exponential of a symmetric matrix A t (which can take negative values), whose vech was modeled as a VAR process. Uhlig (1997) modeled Σ t via a generalized Cholesky decomposition, wherein the diagonal factor followed a positive random walk model. A Wishart autoregressive process was studied in Gouriéroux, Jasiak and Sufana (2009). 25

27 4.2 Temporary and Permanent Shocks A temporary shock at some time index τ can be generated by scaling the diagonal entries of a single Σ t by some a > 0, but without altering B t or Ω t, so that the effect is transitory: Σ t = B t Ω t B t (1 + a 1 {t=τ} ). (14) This ensures that Σ τ has values multiplied by 1+a, but the corresponding shock η τ will not be large unless the random vector is generated from the right tail of the normal distribution. We proceed by generating m random variables Z independently from the marginal distribution P[Z > x Z > 2] = (1 Φ(x))/(1 Φ(2)), and multiplying the corresponding vector ζ by Σ 1/2 τ to obtain η τ. This modification to η t and Σ t at time t = τ will be designated as temporary first-moment and second-moment shocks. A permanent shock at some time index τ involves dilating Σ t in the same manner as the temporary shock, but with the effect lasting for all times t τ: Σ t = B t Ω t B t (1 + a 1 {t τ} ). (15) This second-moment shock is paired with a first-moment shock obtained by generating a temporary first-moment shock ζ at time t = τ in the manner described above, and for t > τ we draw an independent standard normal random vector υ t, and set η t = Σ 1/2 t (ζ + υ t ). Note that in this construction, ζ corresponds to an initial up-swing at time τ, which is persistent at later times t > τ. 4.3 A Framework for Expectation Formation The key three ingredients of our framework are: (i) multi-step ahead forecasting, (ii) stochastic volatility, and (iii) Kalman filter updating. The ith agent (1 i N) observes both public and private signals and makes the forecast through a signal extraction process. Given 26

28 the data {y t (i)} for 1 t T, for each agent i, we obtain the (k + 1)-step ahead forecast π t+1 t k (i) from (12); calculation of the uncertainty relies upon P t+1 t k (i) in (13), which is a special case of the multi-step ahead error covariance R (ij) k,l (t) = Cov [ x t+1 t k (i) x t+1, x t+1 t l (j) x t+1 ], i.e., the covariance of forecast errors for the ith agent (k + 1 steps ahead) and the jth agent (l + 1 steps ahead). The following result provides a recursive algorithm for computing these covariances, based upon recursions for the one-step ahead error covariances Q (ij) t+1 t = Cov[ x t+1 t(i) x t+1, x t+1 t (j) x t+1 ]. Note that setting j = i and l = k we obtain P t+1 t k (i) = R (ii) k,k (t), and furthermore Q(ii) t+1 t = P t+1 t (i). Proposition 1 The covariance of one-step ahead prediction errors across agents, Q (ij) t+1 t, can be computed recursively by Q (ij) t+1 t = [Φ K t(i) H(i)] Q (ij) t t 1 [Φ K t(j) H(j)] + Σ ɛ K t (i) Σ t K t (j), (16) with the initialization Q (ij) 1 0 = Var[x 1] for all i and j. The covariance of prediction errors 27

29 across forecast horizons and agents, R (ij) k,l (t), can be computed if k l by R (ij) l,l (t) = l 1 Φl Q (ij) t+1 l t l Φ l + Φ n Σ ɛ Φ n n=0 R (ij) l 1,l (t) = l 1 Φl 1 (Φ K t+1 l (i) H(i)) Q (ij) t+1 l t l Φ l + Φ n Σ ɛ Φ n R (ij) k,l (t) = Φk + l 1 (Φ K t n (i) H(i)) Q (ij) n=k k Φ n Σ ɛ Φ n, n=0 n=0 t+1 l t l Φ l + l k l m Φ k m=2 n=k (Φ K t n (i) H(i)) Σ ɛ Φ l m+1 where k l 2 in the last case, and where the matrix products are computed with the lowest index matrix first, and multiplying on the right by matrices of higher index. It is clear that Proposition 1 can be used to compute the mean squared error (MSE) of multi-step ahead forecast errors, and is a new result in the state space literature of interest in its own right. Beyond supplying the covariances needed to compute (13), the proposition also allows us to compute aggregate forecaster MSE. If we have interest in some linear composite of economic agents results, say π t+k+1 t = N w i π t+k+1 t (i) (17) i=1 for given weights w i, then the corresponding target is N i=1 w iπ t+k+1, which equals π t+k+1 when the weights sum to one. The variance of the discrepancy between π t+k+1 t and π t+k+1 is the forecast MSE, given by MSE t+k+1 t = N i,j=1 w i w j G R (ij) k,k (t + k) G. (18) To compute (18) recursively, for fixed k, we only need the first formula in Proposition 1, which in turn requires us to know Q (ij) t+1 t, and is calculated from (16). The disagreement 28

30 across agents is defined as the sample variability of the forecasts across forecasters, i.e., D t+k+1 t = N ) ). w i ( πt+k+1 t (i) π t+k+1 t ( πt+k+1 t (i) π t+k+1 t (19) i=1 This is easily computed from (12) and (17). To see this, we can show that π t+k+h+1 t (i) π t+k+h+1 t = G Φ h ( x t+k+1 t (i) x t+k+1 ), where x t+k+1 = N i=1 w i x t+k+1 t (i). Substituting into (19) yields D t+k+h+1 t = N w i G Φ ( x ) ) h t+k+1 t (i) x t+k+1 t ( xt+k+1 t (i) x t+k+1 t Φ h G, i=1 which expresses the (h+k+1)-step ahead dispersion in terms of (k+1)-step ahead dispersion (of the state vector). Let the aggregate forecast uncertainty be defined as U t+k+1 t = N w i P t+k+1 t (i), i=1 which represents an average (across agents) of the variability in k-step ahead forecasting of the state vector. Then we have the following result. Proposition 2 The covariance of prediction errors across one forecast horizon and multiple agents can be recursively computed via R (ij) k+h,k+h (t + k + h) = Φh R (ij) k,k (t + k) h 1 Φ h + Φ n Σ ɛ Φ n. Hence the aggregate forecast uncertainty satisfies n=0 k 1 U t+k+1 t = Φ k U t+1 t Φ k + Φ n Σ ε Φ n. (20) 29 n=0

31 5 Simulation [TO BE DONE] 6 Conclusion It is well known that uncertainty rises in recessions, and did so especially in the Great Recession. Less known is that the term structure of uncertainty also changes in recessions, becoming more narrow. We have documented the cyclicality and other properties of the term structure of uncertainty for European professional forecasters. We have also built a model with noisy public and private information subject to shocks that can capture the key empirical properties of the term structure of uncertainty. 30

32 References Abel, Joshua, Robert Rich, Joseph Song, and Joseph Tracy (2016) The Measurement and Behavior of Uncertainty: Evidence from the ECB Survey of Professional Forecasters, Journal of Applied Econometrics, Vol. 31, pp Bachmann, Rudiger and Christian Bayer (2013) Wait-and-See Business Cycles? Journal of Monetary Economics, Vol. 60, pp Bachmann, Rudiger, Steffen Elstner, and Eric Sims (2013) Uncertainty and Economic Activity: Evidence from Business Survey Data, American Economic Journal: Macroeconomics, Vol. 5, pp Baker, Scott and Nicholas Bloom (2013) Does Uncertainty Reduce Growth? Using Disasters as Natural Experiments, NBER Working Paper No Baker, Scott, Nicholas Bloom, and Steven Davis (2016) Measuring Economic Policy Uncertainty, Quarterly Journal of Economics, Vol. 131, pp Baker, Scott, Tucker McElroy, and Xuguang Sheng (2017) Expectation Formation Following Large Unexpected Shocks, Mimeo. Barrero, Jose Maria, Nicholas Bloom, and Ian Wright (2017) Short and Long Run Uncertainty, NBER Working Paper No Binder, Carola (2017) Economic Policy Uncertainty and Household Inflation Uncertainty, B.E. Journal of Macroeconomics, Vol. 17, pp Bloom, Nicholas (2009) The Impact of Uncertainty Shocks, Econometrica, Vol. 77, pp pp (2014) Fluctuations in Uncertainty, Journal of Economic Perspectives, Vol. 28, 31

33 Boero, Gianna, Jeremy Smith, and Kenneth Wallis (2008) Uncertainty and disagreement in economic prediction: the Bank of England Survey of External Forecasters, Economic Journal, Vol. 118, pp (2015) The Measurement and Characteristics of Professional Forecasters Uncertainty, Journal of Applied Econometrics, Vol. 30, pp de Bruin, Wandi Bruine, Charles F. Manski, Giorgio Topa, and Wilbert van der Klaauw (2011) Measuring Consumer Uncertainty About Future Inflation, Journal of Applied Econometrics, Vol. 26, pp Clements, Michael (2014) Forecast Uncertainty Ex Ante and Ex Post: U.S. Inflation and Output Growth, Journal of Business and Economic Statistics, Vol. 32, pp (2016) Are Macro-Forecasters Essentially The Same? An Analysis of Disagreement, Accuracy and Efficiency, ICMA Centre Discussion Papers in Finance. Coibion, Olivier and Yuriy Gorodnichenko (2012) What can survey forecasts tell us about information rigidities? Journal of Political Economy, Vol. 120, pp D Amico, Stefania and Athanasios Orphanides (2008) Uncertainty and Disagreement in Economic Forecasting, Finance and Economics Discussion Series, Board of Governors of the Federal Reserve System, Vol. 56, pp DiAgostino, A., K. McQuinn, and K. Whelan (2012) Are some forecasters really better than others? Journal of Money, Credit and Banking, Vol. 44, pp Engelberg, Joseph, Charles Manski, and Jared Williams (2009) Comparing the Point Predictions and Subjective Probability Distributions of Professional Forecasters, Journal of Business and Economic Statistics, Vol. 27, pp Giordani, Paolo and Paul Soderlind (2003) Inflation Forecast Uncertainty, European Economic Review, Vol. 47, pp

34 Hellwig, Christian and Venky Venkateswaran (2009) Setting the right prices for the wrong reasons, Journal of Monetary Economics, Vol. 56, pp Kozeniauskas, Nicholas, Anna Orlik, and Laura Veldkamp (2018) What Are Uncertainty Shocks? Forthcoming in Journal of Monetary Economics. Lahiri, Kajal and Xuguang Sheng (2008) Evolution of Forecast Disagreement in a Bayesian Learning Model, Journal of Econometrics, Vol. 144, pp (2010) Measuring Forecast Uncertainty by Disagreement: The Missing Link, Journal of Applied Econometrics, Vol. 25, pp Leduc, Sylvain and Zheng Liu (2016) Uncertainty Shocks are Aggregate Demand Shocks, Journal of Monetary Economics, Vol. 82, pp Liu, Yang and Xuguang S. Sheng (2018) The Measurement and Transmission of Macroeconomic Uncertainty: Evidence from the U.S. and BRIC Countries, Forthcoming in International Journal of Forecasting. Mankiw, N. Gregory, Ricardo Reis, and Justin Wolfers (2004) Disagreement about Inflation Expectations, in Mark Gertler and Kenneth Rogoff eds. NBER Macroeconomics Annual, Vol. 18: MIT Press. Patton, Andrew and Allan Timmermann (2010) Why do forecasters disagree? Lessons from the term structure of cross-sectional dispersion, Journal of Monetary Economics, Vol. 57, pp Rich, Robert and Joseph Tracy (2010) The relationships among expected inflation, disagreement, and uncertainty: evidence from matched point and density forecasts, Review of Economics and Statistics, Vol. 92, pp Zarnowitz, Victor and Louis A. Lambros (1987) Consensus and Uncertainty in Economic Prediction, Journal of Political Economy, Vol. 95, pp

35 Appendix A Tables and Figures Table A.1: Variables Forecasted by the European Central Bank Survey of Professional Forecasters Name Notation Description Inflation π Year on year percentage change of the Harmonised Index of Consumer Prices (HICP) published by Eurostat Growth g Year on year percentage change of real GDP based on ESA definition Unemployment u Unemployment as percentage of labor force based on Eurostat definition Notes: For more information, see ECB Survey of Professional Forecasters (SPF): Description of SPF Dataset. Table A.2: Survey Return Dates and Available Information Quarter Return Date Last π Obs. Last g Obs. Last u Obs. 1 Late Jan/early Feb Dec Q3 Nov 2 Late April/early May March Q4 Feb 3 Late July/early Aug June Q1 May 4 Late Oct/early Nov Sep Q2 Aug Notes: For more information, see ECB Survey of Professional Forecasters (SPF): Description of SPF Dataset. 34

36 Table A.3: Regression for linearity of the term structure of aggregate mean squared forecast error (1) (2) (3) Inflation Growth Unemp 1-year (0.14) (0.16) (0.84) 2-yr minus 1-yr (0.10) (0.06) (0.41) Constant (0.30) (2.72) (0.90) Observations r Notes: Robust, time-clustered standard errors in parentheses. Dependent variable is mean squared error at 5-year horizon for indicated variable. *** p< 0.01, ** p< 0.05, * p< Table A.4: Persistence of Uncertainty and Term Structure Inflation Growth Unemployment (1) (2) (3) (4) (5) (6) logu IncreasingTerm logu IncreasingTerm logu IncreasingTerm L.logU (0.01) (0.02) (0.02) L.IncreasingTerm (0.04) (0.04) (0.05) Constant (0.03) (0.04) (0.03) (0.04) (0.04) (0.05) Observations r2 o Notes: Robust standard errors in parentheses. *** p< 0.01, ** p< 0.05, * p< Table A.5: Correlation of average level and slope of term structure across variables Levels ln U g i1 ln Ui1 u ln Ui1 π ln U g i Slopes ln U g i5 ln U g i1 ln Ui5 u ln Ui1 u ln Ui5 π ln Ui1 π ln U g i5 ln U g i Notes: Table shows the correlation across variables of the mean level and slope of the term structure of uncertainty for the 40 forecasters who made at least 25 forecasts of each variable and horizon. 35

37 Table A.6: Correlation between adjusted and non-adjusted variables for cross section of 40 forecasters Variable Horizon MSE ln Uncertainty Disagreement Revision Freq. Inflation Inflation Inflation Growth Growth Growth Unemployment Unemployment Unemployment Notes: Sample includes the 40 forecasters who made at least 25 forecasts of each variable and horizon. See Equation 2 and footnote 4 for definition of adjusted variables. Table A.7: Regressions of level and slope of uncertainty on forecaster characteristics (1) (2) (3) Share 2y>1y Share 5y>1y Increasing Adjusted MSE (1y) (0.16) (0.21) (0.40) Adjusted MSE (5y) (0.11) (0.14) (0.27) Disagreement (1y) (0.09) (0.12) (0.23) Disagreement (5y) (0.09) (0.12) (0.23) Revision Freq. (1y) (0.34) (0.44) (0.84) Revision Freq. (5y) (0.12) (0.15) (0.29) Constant (0.42) (0.54) (1.03) N R Notes: Robust, standard errors in parentheses. *** p< 0.01, ** p< 0.05, * p< In column (1), the dependent variable is the share of forecasts for which forecaster i s uncertainty is greater at the 2-year than at the 1-year horizon. In column (2), the dependent variable is the share of forecasts for which forecaster i s uncertainty is greater at the 5-year than at the 1-year horizon. The column (3) dependent variable is a dummy variable indicating that forecaster i has greater uncertainty at the 5-year than at the 1-year horizon in at least 90% of periods. This dummy variable has mean Sample includes the 40 forecasters who made at least 25 forecasts of each variable and horizon. 36

38 Table A.8: Correlation of level and slope of term structure of uncertainty with growth, inflation, and policy uncertainty g π 2 EPU ln U1 π ln U g ln U1 u ln U2 π ln U g ln U2 u ln U5 π ln U g ln U5 u ln U2 π ln U1 π ln U g 2 ln U g ln U2 u ln U1 u ln U5 π ln U1 π ln U g 5 ln U g ln U5 u ln U1 u Notes: Table shows the correlation of level or slope of term structure of aggregate log uncertainty with growth, absolute deviation of inflation from 2%, and economic policy uncertainty index. 37

39 Figure A.1: Comparison of point forecasts and density forecast means Notes: ECB SPF data, pooled across forecast dates. Density mean is estimated by maximum likelihood using generalized beta distribution. 38

40 Figure A.2: Time Series of Mean Squared Error Notes: ECB SPF data. Mean of the squared difference between forecast made at time t and realization in t + 1, t + 2, or t

41 Figure A.3: Difference between 5-year and 1-year log uncertainty for individual forecasters Notes: ECB SPF data. Figure shows histograms of the difference between log uncertainty at the 5-year horizon and log uncertainty at the 1-year horizon for individual forecasters, pooled across all time periods. 40

42 Figure A.4: Distribution of growth uncertainty across forecasters over time Notes: ECB SPF data. Interior line is the median, bottom and top of boxes are the 25th and 75th percentiles, and bottom and top of lines are 10th and 90th percentiles of growth uncertainty. 41