Macroeconomic Uncertainty Through the Lens of Professional Forecasters

Transcription

1 Macroeconomic Uncertainty Through the Lens of Professional Forecasters Soojin Jo Rodrigo Sekkel This version: January 3, 5 Abstract We use consensus forecasts from the Survey of Professional Forecasters to estimate macroeconomic uncertainty. Using a factor stochastic volatility model, we quantify macroeconomic uncertainty as the volatility of a common factor simultaneously affecting the size of unpredictable changes in different indicators. Our macroeconomic uncertainty has three major spikes aligned with the , 98-8, and 7-9 recessions, while other recessions were mainly characterized by increases in indicatorspecific uncertainties. We show that data revisions have a substantial effect on the estimated uncertainty series. Finally, we compare our index to a dispersion-based uncertainty measure, and revisit the impact of uncertainty shocks. JEL classification: C38, E7, E3 Keywords: Factor stochastic volatility model; Survey forecasts; Uncertainty Soojin Jo (sjo@bankofcanada.ca) and Rodrigo Sekkel (rsekkel@bankofcanada.ca): Canadian Economic Analysis, Bank of Canada. We thank Natsuki Arai, Christiane Baumeister, Marcelle Chauvet, Rafaella Giacomini, Lutz Kilian, James D. Hamilton, Monica Jain, Michael McCracken, Ulrich Mueller and Alexander Überfeldt for useful comments and suggestions, as well as seminar participants at Bureau of Economic Analysis, Hamilton College, University of Alberta, the 4 Workshop on Uncertainty and Economic Forecasting (UCL), 4 Canadian Economic Association, 4 Computing in Economics and Finance, 4 IAAE meetings, and the 4 Society for Economic Measurement meeting. The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada.

2 Introduction The literature on the impacts of uncertainty on real economic activity has recently witnessed a rapid growth following the Great Recession and the seminal contribution of Bloom (9). Building on this line of literature, several studies have aimed at empirically quantifying the effect of uncertainty. Central to these studies is the need for a measure of time-varying uncertainty, as uncertainty is not directly observable. One popular practice in the literature has been to use an observable proxy for macroeconomic uncertainty. For instance, Bloom (9) uses the VIX index, the implied stock market volatility based on the S&P index, and Bloom et al. () use the cross-sectional dispersion of total factor productivity shocks. Another popular proxy for uncertainty is the cross-sectional dispersion of individual forecasts, as in Bachmann et al. (3). A proxy constructed using word searches from newspaper articles is proposed in Alexopoulos and Cohen (9), and Baker et al. (3) s Economic Policy Uncertainty index combines news article counts with the number of federal tax code provisions set to expire as well as the forecast dispersions. However, in estimated dynamic stochastic general equilibrium (DSGE) models, uncertainty is modeled as the time-varying conditional volatility of unpredicted shocks to the economy (see e.g., Fernández-Villaverde et al., Leduc and Liu and Gilchrist et al. 4). In particular, uncertainty is incorporated using stochastic volatility processes that are parsimonious, yet efficiently captures the time-varying volatility of macroeconomic data. We contribute to the literature by proposing a straightforward method to construct subjective and real-time measures of common (macroeconomic) as well as idiosyncratic uncertainties. Using the Survey of Professional Forecasters for various indicators, our framework jointly models both series consistent with the definition of uncertainty from the DSGE literature. We use a factor stochastic volatility (FSV) model as introduced by Pitt and Shephard (999), where we define macroeconomic uncertainty as the conditional time-varying standard deviation of a factor that is common to the forecast errors for various macroeconomic indicators. Hence, our uncertainty index quantifies the uncer- See, for example, Arellano et al. (), Caggiano et al. (4), Aastveit et al. (3), and Mumtaz and Surico (3), among many others.

3 tainty around a common driver simultaneously affecting the size of unpredicted variations in different macroeconomic indicators. By jointly estimating time-varying common as well as indicator-specific uncertainty, our approach provides a clear picture of how much of the total variations in individual series can be attributed to fluctuations in common and idiosyncratic uncertainties. Using survey forecasts provides the following advantages when estimating macroeconomic uncertainty. First, they are not tied to any particular econometric forecasting model and are easily available. Hence, it is not necessary to select and estimate a specific forecasting model in order to obtain forecast errors. In addition, they provide an effective way of removing expected variations in macroeconomic series. As highlighted by Jurado et al. (forthcoming), it is crucial to remove the predictable component of macro series when estimating macroeconomic uncertainty, as not to attribute some of the predictable variability to unpredictable shocks. Subjective forecasts have been shown to be at least as accurate as forecasts from econometric models. 3 Therefore, survey forecasts are good candidates to control for the predictable variations in the economic indicators. Our paper shares similarity with recent studies focusing on estimating macroeconomic uncertainty. Scotti (3) also exploits survey forecasts and creates an uncertainty index as the sum of the squared survey forecast errors for different indicators, weighted by the loadings from a factor model estimated to construct a business cycle conditions index in Aruoba et al. (9). Jurado et al. (forthcoming) fit a factor model to a variety of macro and financial variables to generate forecasts. They assume that the volatilities of individual forecast errors follow a univariate stochastic volatility process, whose average becomes macroeconomic uncertainty. Our approach, on the other hand, postulates a factor structure for a set of survey forecast errors, where both macroeconomic and indicatorspecific uncertainties evolve as independent stochastic volatility processes, but are jointly estimated. Our estimated uncertainty measure shows persistent dynamics. In particular, all major spikes of uncertainty are associated with episodes of economic recessions, i.e., the , 3 Ang et al. (7) and Faust and Wright (3) document the advantage of surveys over forecasting models for inflation. Aiolfi et al. () study the optimal combination of the two types of forecasts for different indicators, and find that combinations always improve over time series models, but still fail to systematically improve on the survey forecasts alone.

4 98-8, and 7-9 recessions, similar to the findings in Jurado et al. (forthcoming). However, other recessions (i.e. the 99 and recessions) are still notable in the dynamics of idiosyncratic uncertainty, but not mainly picked up by the macroeconomic uncertainty series, suggesting that increases in uncertainty during these events were not as broad-based as during other recessions. We further gauge the effect of using different data vintages. To the best of our knowledge, ours is the first paper that examines the effect of data revisions on the estimation of macroeconomic uncertainty. As macroeconomic variables are constantly revised, uncertainty measures based on the most recent data vintage use a different information set than that previously available to professional forecasters. We find large quantitative differences in the uncertainty series with real-time and revised forecast errors. More specifically, while the and 98-8 recessions exhibit the largest jumps in our baseline uncertainty index with real-time forecast errors, uncertainty is at its highest in the Great Recession when the most recent vintage is used, although the two have similar dynamics. Hence, the use of a specific data vintage can be very important especially when the interest lies in estimating the level of macroeconomic uncertainty. We compare our measure to another popular survey-based proxy of uncertainty, namely forecast disagreement. We construct a common disagreement factor using the same set of variables as in our uncertainty index. The most notable difference is that a common disagreement factor is significantly more volatile and less persistent than our uncertainty index. A further investigation of the dynamic relationship between the two shows that while common disagreement reacts strongly to uncertainty shocks, shocks to common disagreement actually lead to a small decreases in uncertainty. In addition, we find that an uncertainty shock has a significant and persistent effect on a variety of real economic variables in the VAR analysis using our measure of macroeconomic uncertainty: investment, non-durable and durable consumptions and GDP retract after an increase in macroeconomic uncertainty. Although the size of effects differs depending on the ordering of the variables, the negative effects remain significant. The rest of the paper is organized as follows. Section introduces the dataset. Section 3 provides an exposition of our econometric model. The next section presents the estimated 3

5 uncertainty series, further examining the impact of data revision and the usefulness of survey forecasts. We also compare our macroeconomic uncertainty measure to forecast disagreement and analyze the effects of innovations to macroeconomic uncertainty on a series of economic activity indicators. Section 5 concludes. Data We use the data from the U.S. Survey of Professional Forecasters (SPF). The survey was initially introduced by the National Bureau of Economic Research and the American Statistical Association in 968, which the Federal Reserve Bank of Philadelphia took over in 99. Professional forecasters are asked to report their quarterly forecasts for total 3 macroeconomic indicators ranging from real-side variables (e.g., Gross Domestic Product (GDP) and residential investment) to financial side (e.g., BAA corporate bond yields); some variables have been surveyed since 968Q4 and others have been added over time starting from 99Q. We use four variables that can span the longest period from 968Q4 to 4Q: real GDP, unemployment rate, industrial production (IP) and housing starts. With the selected variables, we can obtain a long time series of macroeconomic uncertainty. Since forecasters are surveyed on a quarterly basis, the most recent quarter of data in their information set would be the previous quarter. The forecast submission deadline of the survey tends to occur close to the middle of the quarter (after the Bureau of Economic Analysis advance report of the notional income and product accounts (NIPA), which contains the first estimates of the previous quarter s GDP), so for macroeconomic variables released on a monthly basis, forecasters would have access to the first month s realized data for the current forecast horizon, before the survey is submitted. We use their one-step-ahead forecasts, namely their nowcasts, in order to construct the forecast errors. The calculation of forecast errors at any point in time is contingent on the realized value of the series. The NIPA data go through substantial revisions, and these revisions can ultimately affect our measurement of uncertainty. Thus, we use both the first and last data vintages 4 to compute two possible values of forecast errors. When calculating the forecasting errors, we take consensus forecasts, i.e., averages of individual professional 4 Our last data vintage is 4Q. 4

6 forecasters forecasts, to minimize potential influences from individual forecasting biases. The errors are then standardized before the estimation of the FSV model. This prevents one variable from dominating the dynamics of a factor. Table provides summary statistics of the forecasting errors based on the first and last vintages. 3 Factor Stochastic Volatility Model We use the FSV of Pitt and Shephard (999) to estimate macroeconomic as well as idiosyncratic uncertainty indexes. First, we define the forecasting error of a variable i in period t, denoted as ε i,t, as follows: () ε i,t = (x i,t E[x i,t I t ]), where x i,t is the realization of variable i in time t, and E[x i,t I t ] is a conditional mean of forecasts of variable i for the quarter t across different forecasters (i.e., consensus forecast). Again, one of the key differences of our measure from other uncertainty indexes based on a particular forecasting model is that we obtain E[x i,t I t ] from the consensus forecasts, instead of using forecasts from a specific econometric model. The information set (I t ) also has the same time-subscript t, as it contains information obtained until the middle of the quarter t. As discussed in the previous section, for monthly macroeconomic indicators such as IP, the first month s value in the quarter t is included in I t along with the first NIPA release of x i,t. For indicators of quarterly frequency, I t includes the first NIPA estimate of x i,t which is only available in the middle of the quarter t. Next, we postulate that the forecasting errors of the macroeconomic series have a factor structure: () ε t = λf t + u t, where ε t = [ ε,t,..., ε n,t ] is a (n ) vector of standardized forecasting errors, λ = [λ... λ n ] is a vector of factor loadings, f t is a factor and u t = [u,t,..., u n,t ] is a (n ) vector of idiosyncratic errors. Hence, equation () implies that there is a factor that drives the 5

7 common dynamics across the forecasting errors of n economic indicators, and u t captures the remaining indicator-specific components. In addition, we assume that (3) (4) f t N(, h f,t ), u t N(, Σ t ), where Σ t is a n n diagonal matrix of time-varying idiosyncratic volatilities. In other words, h,t h Σ t =,t h n,t Finally, the common as well as indicator-specific volatilities follow independent stochastic volatility processes: log h f,t = log h f,t + σ f η f,t (5) log h i,t = log h i,t + σ i η i,t, where σ f and σ i s are time-invariant parameters determining the variability of the volatilities. In the current setup, our key estimate of interest is the time-varying standard deviations of the factor, i.e., { h f,t } T t= which we define as our measure of the macroeconomic uncertainty index: the time series of macroeconomic uncertainty captures the volatility of a common driver that simultaneously affects the magnitude of forecasting errors across our four variables, by determining the magnitude of common variations across all indicators. Bayesian Estimation Approach Our model is estimated using Bayesian methods, since the model features high dimensionality as well as non-linearity. The Bayesian methods deal with such features by separating 6

8 parameters into several blocks, which greatly simplifies the estimation process. In particular, the Markov Chain Monte Carlo (MCMC) algorithm breaks the parameters into several blocks and repeatedly draws from conditional posterior distributions, in order to simulate the joint posterior distributions. In one of the sub-steps when the volatility states are drawn, we incorporate Metropolis methods within the overall Gibbs sampler, following the algorithm by Jacquier et al. (). We collect 5, draws by storing every th draw in order to avoid potential autocorrelation across draws. Thus, we iterate over the Metropolis-within-Gibbs sampler a total of 6, times, but discard the first, draws of parameters. We follow the common identification scheme of a factor model which sets the first factor loading (of real GDP) equal to unity. However, the resulting baseline index of macroeconomic uncertainty is robust to equalizing other factor loadings to one or changing the ordering of variables. The choice of prior distributions and their parameter values is very similar to Pitt and Shephard (999). A detailed description of the prior distribution setup and the MCMC algorithm is provided in Appendix A. 4 Results We plot our baseline macroeconomic uncertainty series in Figure : the solid line is the median posterior draw of the common stochastic volatility ({ h f,t } T t=), and the shaded area represents the 95% posterior confidence set. For our baseline estimates, we use the first data release to calculate the forecasting errors. In this way our baseline measure reflects an information set available to forecasters at the time they are surveyed. 5 There are three main spikes in macroeconomic uncertainty, all associated with deep recessions. The first spike was observed during the recession, the second during the 98-8 recession, and the last one during the recent Great Recession. It is also clear from the figure that, in general, the level of macroeconomic uncertainty was significantly higher in the period than from 985 onwards, consistent with the findings in Kim and Nelson (999) and McConnell and Perez-Quiros (). The index also increases 5 We assess, in the next section, the effect of data revisions on our uncertainty measure. 7

9 around the 99 and recessions, but these increases are mild in comparison with the three critical spikes. Table reports the median posterior draws of factor loadings. 6 We find that housing starts loads least on the common factor, while the loadings of the other three variables are in a comparable range. Using the median posterior draws, we further examine how much of the total variation in forecasting errors of each indicator is driven by the common versus idiosyncratic components. This is calculated by using the factor structure of our model. In particular, our model implies a total variance of each variable, var( ε i,t ), to be var( ε i,t ) = var(λ i f t + u i,t ) = λ i var(f t ) + var(u i,t ) = λ i h f,t + h i,t, as the factor, and idiosyncratic error terms are assumed to be uncorrelated. We then measure the size of the total common variation driven by macroeconomic uncertainty as λ i var(f t ) = λ i h f,t, incorporating the heterogeneity due to the difference in factor loadings. Figure plots the decomposition of each total variation into components explained by macroeconomic and idiosyncratic uncertainties. The most notable spikes in the total variances are driven to a large extent by macroeconomic uncertainty. More interestingly, recessions that were not accompanied by distinct increases in macroeconomic uncertainty do show up in the total variations of unemployment (the 99 and recessions) and IP (the recession), indicating that these two recessions were not as broad-based as the others. Among the four variables, it is IP whose volatility is explained the most by the macroeconomic uncertainty. The volatilities of real GDP and unemployment are also largely driven by the common uncertainty, but their respective idiosyncratic uncertainties still account for a sizable share of their total variations. In particular, from 985 to 7, when the baseline macroeconomic uncertainty index was relatively subdued, idiosyncratic volatilities of real GDP and unemployment were still high, peaking in different periods. In the case of GDP, idiosyncratic volatility explained more than 5 percent of the total 6 Note again that we set the factor loading of GDP to unity. However, the relative size of the factor loadings are robust to different normalization. 8

10 variation of GDP forecasting errors during this period. Finally, most of the total variation of housing starts is idiosyncratic rather than driven by macroeconomic uncertainty, and it leads the total volatility of the other indicators. It is interesting to note that idiosyncratic uncertainty of housing starts was on an increasing trend from the mid-99s to the Great Recession. 4. The Impact of Data Revisions A large body of literature in forecasting and econometrics highlights the importance of data revisions (see Croushore for a discussion on real-time data analysis). Data revision should also be relevant for the measurement of uncertainty, since it directly affects the magnitude of forecast errors. Our baseline measure is calculated based on forecast errors computed with the first data release; however, we also examine how our macroeconomic uncertainty index differs if we use the final vintage to calculate the forecast errors. To our knowledge, this is the first paper that examines the effect of data revisions for the estimation of macroeconomic volatility. Figure 3 shows the macro uncertainty index estimated with the final 4Q vintage. Though the correlation between the two indexes is high (86%), there are significant quantitative differences between the two. Most notably, for the macroeconomic uncertainty series estimated with the final vintage, the increase in uncertainty associated with the recent Great Recession is the largest since the beginning of the series. In contrast, with the real-time forecast errors, the uncertainty during the recessions in the pre-great Moderation periods, i.e., the and 98-8 recessions, are higher than the level of uncertainty during the last recession. Hence, there are important quantitative differences between the uncertainty estimates depending on which vintage is used to calculate the forecast errors. This further implies that studies estimating macroeconomic volatility using the final-revised data will likely underestimate the actual volatility faced by professional forecasters in the 97s and 98s. 9

11 4. How Important is the Information in the Survey? It is important to remove predictable components from the economic indicators when estimating macroeconomic uncertainty, in order not to attribute the predictable variation to unpredictable shocks. As highlighted by Jurado et al. (forthcoming), the most-commonly used proxies of uncertainty such as stock market volatility (i.e., VIX) and forecast dispersions, do not account for this fact. Several papers have documented that the predictive ability of survey forecasts is a difficult benchmark for econometric forecasting models to beat, especially at short horizons (see Ang et al. 7, Faust and Wright 3 and Aiolfi et al. ). As a consequence, survey forecasts efficiently control for predictable components for the estimation of uncertainty. To demonstrate the effectiveness of survey forecasts in removing predictable variations, here we re-estimate our factor stochastic volatility model using forecasting errors from simple autoregressive (AR) models: (6) y i,t = α i + p β i,j y i,t j + γ yi,t + γ yi,t 3 + γ 3 yi,t + ɛ i,t, j= where p is the lag length chosen based on the Akaike information criterion with a rolling fixed window of 6 observations. For the variables released at a monthly frequency, the monthly values reported, i.e., the first month value in quarter t (y i,t, only if the release date is before the survey deadline) and the third and second month values of quarter t (y 3 i,t and y i,t ), are augmented to the AR model, to closely mimic information available to professional forecaster at the time of the survey. 7 It is worthwhile to note that the real-time data available in each quarter t are used for the AR models to match our baseline index more closely. We then estimate stochastic volatility of a common factor using ˆɛ i,t, the forecast errors from the AR model above, for each of our four variables. The dotted line in Figure 4 plots the estimated macroeconomic volatility with the AR forecast errors. As easily seen, the AR-based macroeconomic uncertainty index is considerably higher throughout the whole 7 The forecast errors from the above AR models are provided at the Federal Reserve Bank of Philadelphia website along with a detailed description regarding the model and real-time available information sets in Stark ().

12 sample period, and more volatile than our baseline index based on survey forecasts. The result implies that a significant share of the forecast errors from the AR models are indeed captured by consensus forecasts, especially during recessions. Hence, simple statistical models such as AR may have substantially larger forecasting errors which are subsequently attributed to higher macroeconomic uncertainty. 8 More importantly, this analysis shows that using a consensus survey forecast is a parsimonious and effective way of eliminating the predictable variations from economic variables, thereby providing a more accurate measure of uncertainty. 4.3 Comparison with Measures of Disagreement A widely-used proxy for uncertainty based on survey forecasts is forecast disagreement, commonly measured as the interquartile range (e.g. Bachmann et al. 3). Underlying this practice is the assumption that predictions of forecasters are more likely to be close to each other when economic uncertainty is low. However, forecast disagreement may just reflect heterogeneous, but not uncertain, beliefs. 9 In this section, we investigate the relationship between our macroeconomic uncertainty measure and an analogous, disagreement-based proxy of uncertainty. We extract a common factor from the forecast disagreement of the same economic indicators used for our baseline index. That is, we estimate the first principal component of the disagreement series for each of the four underlying series. Disagreement is measured as the 75th percentile minus the 5th percentile of individual forecasts. We then compare the estimated common component of disagreement to our measure of macroeconomic uncertainty. Figures 5 and 6 presents the individual disagreement among forecasters for the four economic indicators, and the first principal component of the disagreement series, respectively. The most notable difference is that the common disagreement factor is significantly 8 The Real-Time Data Research Center at the Federal Reserve Bank of Philadelphia provides statistics related SPF forecast errors by comparing the forecast errors from the SPF to those from other simple but widely-used forecasting models: no-change forecast, iterative AR forecast, and horizon-specific AR forecast. The SPF forecast of the four variables we use have consistently lower RMSEs compared to other forecasting errors, and the differences are statistically significant. This further indicates that survey forecasts integrate wider sets of information than a simple statistical mode. 9 See, for example, Mankiw et al. (4), Lahiri and Sheng () and Sill () for a more detailed discussions of measuring uncertainty using forecast disagreement.

13 more volatile than our baseline uncertainty index. Nonetheless, the three major spikes in disagreement coincide with the three main episodes of uncertainty increases in our baseline index. Interestingly, however, the relative size of the increment in disagreement during the Great Recession is substantially smaller than that of our measure of macroeconomic uncertainty. The correlation between the two series is.76. We further investigate the dynamic relationship between our measure of uncertainty and the common component of forecast dispersions by estimating a bivariate VAR(4), recursively identified with the disagreement factor ordered first. Figure 7 shows the impulse response functions to both uncertainty and disagreement factor shocks. A few interesting results emerge. First, it is evident that the response to its own shock is significantly more persistent for our measure of macroeconomic uncertainty, compared to that of common disagreement. Furthermore, we find that while common disagreement reacts strongly positively to uncertainty shocks, the contrary is not true: shocks to common disagreement actually lead to a small decreases in uncertainty. If the dispersion factor was a close proxy of macroeconomic uncertainty, one would expect a significant and positive impact on uncertainty of a dispersion shock. We thus conclude that, even though we find a high unconditional correlation, an increase in disagreement is likely the result of heightened uncertainty, but not vice versa. 4.4 VAR Analysis In this section, we examine the dynamic relationship between our measure of macroeconomic uncertainty and a set of macroeconomic indicators using a standard recursively identified VAR. Previous studies using proxies for uncertainty, like the VIX in Bloom (9), tend to find a significantly negative, but short-lived impact of uncertainty on economic activity. This drop in activity is then followed by an overshoot, as economic activity rebounds. In contrast, studies that estimate macroeconomic uncertainty, as Jurado et al. (forthcoming), find that these shocks have a much more persistent effect on economic activity, and no evidence of a strong rebound and overshooting. Our benchmark specification of a VAR comprises of six variables, with the following order: log(private investment), log(nondurable consumption expenditure), log(durable

14 consumption), log(gdp), log(s&p index), and our macroeconomic uncertainty index. The VAR is estimated in levels and with four lags. A natural choice of the ordering of variables in the VAR is not clear, as our uncertainty measure should react to real activity shocks within a quarter, while it is also possible that other real variables respond to uncertainty during the same quarter. Given the above difficulty, we choose to order our uncertainty measure last in our baseline VAR analysis. Hence, we purge innovations to our uncertainty measure from any contemporaneous and past movements in the real activity variables, as well as the S&P index. This choice of ordering implies, by construction, both a zero contemporaneous impact of uncertainty on economic activity and a more conservative estimate of the impact of uncertainty shocks. Figure 8 presents the estimated dynamic responses of the various economic activity measures to a one standard deviation innovation to macroeconomic uncertainty. All of the economic activity variables show a significant and persistent decline following an uncertainty shock, supporting the findings of long-lived negative effects of uncertainty as in Bachmann et al. (3) and Jurado et al. (forthcoming). Moreover, we find no evidence of overshooting in economic activity, as the uncertainty shock dissipates. There is some heterogeneity in how the different economic activity measures respond to innovations in our measure of macroeconomic uncertainty: uncertainty shocks generate a larger fall of durables consumption and investment compared to the responses of nondurables consumption and GDP. These findings are consistent with theoretical models of business investment and durable consumption where irreversibility plays a significant role; any investment or durable consumption is accompanied by a fixed adjustment cost that makes it difficult to reverse such decisions, and as a result, economic agents will postpone investment or durable consumption when uncertainty is high. To verify the quantitative importance of an uncertainty shock, Table 3 reports the forecast error variance decomposition for the real economic activity variables included in the VAR. As discussed earlier, given our choice of the benchmark VAR with uncertainty ordered last, these estimates should be viewed as a lower bound. The decomposition shows that an uncertainty shock can explain the maximum of 5.58 to 8.8% of the variance of See e.g., Pindyck (99) and Bertola et al. (5). 3

15 various real economic activity measures within 5 years after the shock. These numbers are near the lower end of those reported by Bachmann et al. (3), Jurado et al. (forthcoming) and Caggiano et al. (4). Next, we investigate the robustness of the findings to the ordering assumption of the VAR. We estimate a VAR with the same set of variables, but now our uncertainty measure is ordered second after the S&P index, before the real economic activity indicators. Figure 9 reports the impulse response functions of the economic activity indicators, and Table 4 the resulting forecast error variance decomposition. As one would expect, this change in ordering results in a larger decline of economic activity, following a positive innovation to uncertainty. Its importance in the variance decomposition of economic activity also rises significantly. Under this ordering, uncertainty shocks account for the maximum of.8 to 8.98% of the variance of the real economic activity indicators. Notwithstanding the more pronounced effects, the qualitative evidence is very similar to our baseline ordering choice. We also conduct a comparison of our impulse response functions to the ones obtained with other estimates and proxies for macroeconomic uncertainty. Figure plots responses to innovations in our baseline uncertainty series, as well as Jurado et al. (forthcoming) estimates of macroeconomic uncertainty, the common disagreement estimated in Section 4.3 and the VIX. There is a clear divergence in results. Both innovations to our measures, as well as JLN measure of macroeconomic uncertainty result in large and persistent drops in economic activity. On the other hand, innovations to proxies of macroeconomic uncertainty, as the disagreement and the VIX result in very small and short lived negative impact on economic activity, followed by strong rebounds. 5 Conclusion This paper estimates macroeconomic uncertainty from 968Q4 to 4Q as perceived by professional forecasters. Using a FSV model proposed by Pitt and Shephard (999), we estimate volatilities of a common factor and idiosyncratic components across consensus forecast errors of different economic indicators. We define the time-varying standard deviation of the factor as a measure of macroeconomic uncertainty, and estimate it jointly with indicator-specific uncertainties. 4

16 In general, macroeconomic uncertainty was higher in the period compared to the post-985 period. Our baseline uncertainty measure is relatively smooth and persistent with all major spikes associated with economics recessions (the , 98-8, and 7-9 recessions), consistent with Jurado et al. (forthcoming). Additionally, we find that data revisions have a substantial effect on the estimated macroeconomic uncertainty. We also compare our baseline measure of uncertainty to another survey-based uncertainty proxy, namely forecast disagreement. The first principal component of disagreement is significantly more volatile than our measure throughout the sample period. Further investigation on the dynamic relationship between the two shows that shocks to common disagreement does not have any meaningful impact on uncertainty, while common disagreement reacts strongly positively to uncertainty shocks. Finally, we conduct a VAR analysis to investigate the dynamic relationship between uncertainty and real economic variables. A one-standard deviation increase in our baseline uncertainty index results in a significant and persistent decrease in various measures of economic activity such as investment, durable and non-durable consumptions, in line with the findings in Bachmann et al. (3) and Jurado et al. (forthcoming). However, this evidence is at odds with the short-lived negative impact followed by a strong rebound, as suggested by Bloom (9). Appendix A Bayesian Estimation Method Here we describe in detail the setup of the priors as well as the MCMC posterior distribution simulation process in detail. Our choice of prior distributions and their parameter values is very similar to Pitt and Shephard (999) except for the values of the conditional inverse Gamma prior, σ IG( v, δ ). We set v = and δ =, which makes the conditional prior distribution flatter than the one in Pitt and Shephard (999) and more so than the ones in other recent studies incorporating stochastic volatility (see e.g., Primiceri 5 and Baumeister et al. 3). This is related to the setup of our model. We have four variables from which we extract one common factor. In addition, we set the factor loading of the first 5

17 variable to be for identification, consistent with the conventional identification strategy. Hence, we need to allow for a large time variation for stochastic volatilities a priori, without which the factor would only mimic the dynamics of real GDP rather than capturing the common dynamics of the four indicators. Moreover, compared to the previous studies using time-varying VAR models with stochastic volatility (e.g., Primiceri 5 and Baumeister and Peersman 3), the total number of parameters to estimate is substantially smaller in our case. Thus, we use a more diffuse prior and put a larger weight on data. A. Prior Distributions and Starting Values The prior distribution for factor loadings is the Normal distribution, i.e., λ i N(λ, Λ ) with λ = and Λ = 5, as in Pitt and Shephard (999). The choice of relatively large Λ represents a fair degree of uncertainty around the factor loadings. The initial value of the factor loadings is the OLS estimates of forecasting errors on the first principal component as a proxy of a factor. Since the factor and loadings are not completely identified in a factor model, we set the loading of the first variable (i.e., real GDP) to be equal to, a commonly-used identification strategy of a factor model. Factor loadings of the second variable to the last are only drawn from the posterior distribution introduced below. A diffuse Normal prior is used as the prior distribution for a factor conditional on {h f,t } T t=, consistent with equation (3) (i.e., f t N(θ, Θ ) where θ = and Θ = h f,t ). As mentioned above, we use the first principal component for the initial iteration. The prior for the variability of volatilities is the inverse Gamma, i.e., σ f and σ i IG( v, δ ), where v = and δ =. As discussed above, we choose a larger value of δ compared to a conventional setup to be less informative about the variability of volatilities, allowing for potentially larger time variation of volatilities at the same time. The prior of each time-varying volatility is the log-normal. In particular, for the initial period s stochastic volatility, h, we have log h N(µ h, V h ), where µ h = and V h = to allow a good chance for the data to determine the posterior distribution. 6

18 A. Posterior Distribution Simulation We divide the parameters in the model into four blocks; a) the factor loadings (λ), b) the time series of the factor ({f t } T t=), c) the hyperparameters of volatilities (σ f and σ i for all i), and d) the volatility states ( {h f,t } T t= and {h i,t } T t= for all i). As will be explained in detail, the volatility states are drawn via Metropolis methods within the overall Gibbs sampler. Denoting by z T the time-series of a variable z from t = to T, the sampler algorithm is described below. A.. Factor loadings Conditional on all other parameters, this step is a simple Bayesian regression of forecasting errors on the factor with known heteroskedastic error structures. Moreover, because all correlations are captured by the factor by definition, each i-th loading can be separately drawn from the following distribution: λ i ε T, f T, σ f, σ i, h T f, h T i N(λ, Λ ), where Λ = (Λ + T t= f t /(h i,t )) and λ = Λ (Λ λ + T t= f t ε i,t /(h i,t )). A.. Factor Conditional independence also simplifies this step. Given all other parameter values, this step again becomes a Bayesian regression of n-forecasting errors on factor loadings with known heteroskedasticity for each period t. That is, f t ε T, λ i, σ f, σ i, h T f, h T i N(θ, Θ ), where Θ = {h f,t + n i= λ i (h i,t )} and θ = Θ ( n i= λ i ε i,t /(h f,t )). A..3 Innovation variance of volatilities Since we model each stochastic volatility to follow a unit-root process without a drift, the conditional posterior distribution of σ can be simplified from the posterior inverse Gamma 7

19 distribution in Kim et al. (998). Hence, σ is drawn from σ ε T, f T, h T f, h T i IG( v, δ ) where v = v + T, and δ = δ + T t= (h i,t h i,t ). A..4 Volatility states This step further decomposes to n+ sub-steps of univariate stochastic volatility draws, based on the Markovian property of stochastic volatility. It follows the algorithm by Jacquier et al. () as used in Cogley and Sargent (5). For each volatility series of an idiosyncratic error i or of the factor, the algorithm draws the exponential of volatility (h i,t) one by one for each t, based on f(h i,t h i,t, h i,t+, yi T, λ, f T, σ). Before sampling the states, we first transform forecasting errors to be ε i,t = ε i,t λ i,t f t. Such transformation is unnecessary for the factor. Then, we apply Jacquier et al. () s algorithm for each date, i.e., f(h i,t (h i ) T t, y T i, σ) = f(h i,t h i,t, h i,t+, y T i, σ) f(yi,t h i,t )f(h i,t h i,t )f(h i,t+ h i,t ) ( y = (h i,t ).5 ) ( ) i,t (log hi,t µ t ) exp exp, where µ t and σ c are the conditional mean and variance of log h i,t, respectively. Under the h i,t unit-root specification of this paper, they can be calculated as µ t = (log h i,t + log h i,t+ ), σ c = σ i, for t =,, T. Hence, a trial value of log h i,t is drawn from the Normal distribution with mean µ t and variance σ c. For the beginning and end periods, the following conditional σ c 8

20 mean and variance are used instead: t = : σc = σ iv h σ i +V h, µ = σ c ( µ h V h t = T : σ c = σ i, µ = log h i,t. + log h i,t+ σ i ), After obtaining a draw, the conditional likelihood f(y i,t h i,t ) is evaluated in order to obtain the acceptance probability, completing a Metropolis step (see Cogley and Sargent 5 for a detailed description). We can summarize the estimation procedure as following:. Assign initial values for λ, f T, σ f, σ i for all i, h T f, and ht i for all i.. Draw λ from p(λ ε T, f T, σ f, σ i, h T f, ht i ) 3. Draw f T from p(f T ε T, λ, σ f, σ i, h T f, ht i ) 4. Draw σ f and σ i s from p(σ ε T, f T, h T f, ht i ) 5. Draw h T f, and ht i s from p(h ε T, f T, σ f, σ i ) 6. Go to step. We iterate over the Metropolis-within-Gibbs sampler a total of 6, times, discarding the first, draws of parameters. Then we store every th draw in order to avoid potential autocorrelation across draws, and finally obtain 5, draws from the joint posterior distribution. References Aastveit, K. A., G. J. Natvik, and S. Sola (3): Economic Uncertainty and the Effectiveness of Monetary Policy, Norges Bank Working Paper Series, 7. Aiolfi, M., C. Capistran, and A. Timmermann (): Forecast Combinations, in The Oxford Handbook of Economic Forecasting, ed. by M. Clements and D. Hendry, Oxford University Press, The Oxford Handbook of Economic Forecasting, Alexopoulos, M. and J. Cohen (9): Uncertain Times, Uncertain Measures, University of Toronto Department of Economics Working Paper, 35. Ang, A., G. Bekaert, and M. Wei (7): Do Macro Variables, Asset Markets, or Surveys Forecast Inflation Better? Journal of Monetary Economics, 54, 63. 9

21 Arellano, C., Y. Bai, and P. J. Kehoe (): Financial Frictions and Fluctuations in Volatility, Federal Reserve Bank of Minneapolis Research Department Staff Report, 466. Aruoba, S. B., F. X. Diebold, and C. Scotti (9): Real-Time Measurement of Business Conditions, Journal of Business & Economic Statistics, 7, Bachmann, R., S. Elstner, and E. Sims (3): Uncertainty and Economic Activity: Evidence from Business Survey Data, American Economic Journal: Macroeconomics, 5, Baker, S. R., N. Bloom, and S. J. Davis (3): Measuring Economic Policy Uncertainty, Chicago Booth Research Paper, 3-. Baumeister, C., P. Liu, and H. Mumtaz (3): Changes in the Effects of Monetary Policy on Disaggregate Price Dynamics, Journal of Economic Dynamics and Control, 37, Baumeister, C. and G. Peersman (3): Time-Varying Effects of Oil Supply Shocks on the US Economy, American Economic Journal: Macroeconomics, 5, 8. Bertola, G., L. Guiso, and L. Pistaferri (5): Uncertainty and consumer durables adjustment, The Review of Economic Studies, 7, Bloom, N. (9): The Impact of Uncertainty Shocks, Econometrica, 77, Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. J. Terry (): Really Uncertain Business Cycles, NBER Working Paper, w845. Caggiano, G., E. Castelnuovo, and N. Groshenny (4): Uncertainty Shocks and Unemployment Dynamics: An Analysis of Post-WWII US Recessions, Journal of Monetary Economics, 67, Cogley, T. and T. J. Sargent (5): Drifts and Volatilities: Monetary Policies and Outcomes In the Post-WWII US, Review of Economic Dynamics, 8, 6 3. Croushore, D. (): Frontiers of Real-Time Data Analysis, Journal of Economic Literature, 49, 7. Faust, J. and J. H. Wright (3): Forecasting Inflation, in Handbook of Economic Forecasting, ed. by G. Elliott and A. Timmermann, Elsevier, vol., Part A of Handbook of Economic Forecasting, 56. Fernández-Villaverde, J., P. A. Guerrón-Quintana, K. Kuester, and J. Rubio-Ramírez (): Fiscal Volatility Shocks and Economic Activity, NBER Working Paper Series, 737. Gilchrist, S., J. W. Sim, and E. Zakrajšek (4): Uncertainty, Financial Frictions, and Investment Dynamics, NBER Working Paper, w38.

22 Jacquier, E., N. G. Polson, and P. E. Rossi (): Bayesian Analysis of Stochastic Volatility Models, Journal of Business & Economic Statistics,, Jurado, K., S. Ludvigson, and S. Ng (forthcoming): Measuring Uncertainty, American Economic Review. Kilian, L. (998): Small-Sample Confidence Intervals for Impulse Response Functions, Review of economics and statistics, 8, 8 3. Kim, C.-J. and C. R. Nelson (999): Has the US Economy Become More Stable? A Bayesian Approach Based on a Markov-Switching Model of the Business Cycle, Review of Economics and Statistics, 8, Kim, S., N. Shephard, and S. Chib (998): Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models, Review of Economic Studies, 65, Lahiri, K. and X. Sheng (): Measuring Forecast Uncertainty By Disagreement: The Missing Ink, Journal of Applied Econometrics, 5, Leduc, S. and Z. Liu (): Uncertainty Shocks Are Aggregate Demand Shocks, Federal Reserve Bank of San Francisco Working Paper,. Mankiw, N. G., R. Reis, and J. Wolfers (4): Disagreement About Inflation Expectations, in NBER Macroeconomics Annual 3, Volume 8, The MIT Press, 9 7. McConnell, M. M. and G. Perez-Quiros (): Output Fluctuations in the United States: What Has Changed since the Early 98 s? American Economic Review, 9, Mumtaz, H. and P. Surico (3): Policy Uncertainty and Aggregate Fluctuations, Working Paper, School of Economics and Finance, Queen Mary, University of London. Pindyck, R. S. (99): Irreversibility, Uncertainty, and Investment, Journal of Economic Literature, 48. Pitt, M. and N. Shephard (999): Time-Varying Covariances: A Factor Stochastic Volatility Approach, Bayesian Statistics, 6, Primiceri, G. E. (5): Time-Varying Structural Vector Autoregressions and Monetary Policy, Review of Economic Studies, 7, Scotti, C. (3): Surprise and Uncertainty Indexes: Real-Time Aggregation of Real- Activity Macro Surprises, FRB International Finance Discussion Paper, 93. Sill, K. (): Measuring Economic Uncertainty Using the Survey of Professional Forecasters, Federal Reserve Bank of Philadelphia Business Review, 4, 6 7. Stark, T. (): Realistic Evaluation of Real Time Forecasts in the Survey of Professional Forecasters, Federal Reserve Bank of Philadelphia Special Report.

23 Table : Summary Statics of Forecasting Errors real GDP IP Unemployment Housing starts First Vintage () Mean Std. Dev Last Vintage () Mean Std. Dev Note: This table shows the sample means and standard deviations of four forecasting errors used in the construction of the macroeconomic uncertainty index. The sample means are tested to see whether significantly different from (H : mean= ), and and denote that the mean is significantly different from at 5% and %, respectively. Table : Summary Statics of Posterior Draws of Factor Loadings Real GDP IP Unemployment Housing starts Median Std. Dev Note: This table shows the median and standard deviations calculated from the posterior draws of four factor loadings. Since our identification strategy is to set the loading of real GDP to unity, the standard deviation is not reported for real GDP. Table 3: Variance Decomposition for the Baseline VAR Quarters GDP RPI PCE-ND PCE-D Note: This table shows forecast error variance decomposition for our baseline VAR with the following variables and Cholesky ordering: real private investment (RPI), real personal consumption expenditures on non durables (PCE-ND), real personal consumption expenditures on durables (PCE-D), GDP, S&P index, and our uncertainty measure. The VAR is estimated in levels with 4 lags. All variables, except our uncertainty measure, enter the VAR in logs.

24 Table 4: Variance Decomposition for alternative VAR ordering: Uncertainty Before Real Activity Quarters GDP RPI PCE-ND PCE-D Note: This table shows forecast error variance decomposition for our baseline VAR with the following variables and Cholesky ordering: S&P index, our uncertainty measure, real private investment (RPI), real personal consumption expenditures on non durables (PCE-ND), real personal consumption expenditures on durables (PCE-D), and GDP. The VAR is estimated in levels with 4 lags. All variables, except our uncertainty measure, enter the VAR in logs. Figure : Estimated Macroeconomic Uncertainty Series Note: This figure plots the baseline macroeconomic uncertainty series estimated using real-time data. The series is the time-varying standard deviation of a common factor across forecasting errors of four macroeconomic indicators. The black solid line represents the median posterior draws along with the 95% posterior credible set. 3

25 Figure : Total Variation versus Common Variation Gross Domestic Product Industrial Production.8.6 Total Variation Macro Variation.8 Total Variation Macro Variation Unemployment Housing Starts.8.6 Total Variation Macro Variation.8.6 Total Variation Macro Variation Note: This figure shows how much of total variation of each variable is explained by the macroeconomic uncertainty. The grey line is the total variation of one variable (defined as standard deviation), and the black line is the macroeconomic uncertainty multiplied by a factor loading. All calculations are based on the median posterior draws. 4