2.1. The dangers of ignoring the persistence properties of shocks; an illustration

It is worth considering the properties of the persistence profile in the univariate case when the variable is driven by shocks with different persistence properties. A simple illustration shows that failure to take account of the persistence of different types of shocks can introduce biases in modeling output movements, mis-measurement of uncertainty and inaccurate claims on the role of uncertainty in business cycle dynamics. To illustrate these points, denote output at $t$ by $y_{t}$ , denote the expectation of $y_{t+h}$ published in a survey at time $t$ by $_{t}y_{t+h}$ and consider the case where output growth is driven by two shocks

(3) \begin{equation} y_{t}-y_{t-1}=\rho (y_{t-1}-y_{t-2})+\epsilon _{t}+\omega _{t}-\omega _{t-1}\text{ .} \end{equation}

Here $\epsilon _{t}\sim N(0,\sigma _{\epsilon }^{2})$ has a permanent effect on $y_{t}$ and $\omega _{t}\sim N(0,\sigma _{\omega }^{2})$ is a shock whose effect on output is known to be short-lived and offset next period, providing a simple example of “correlated news”. The effects of both types of shock are propagated over time through the presence of the lagged growth term. Assuming expectations are formed rationally (purely for the purpose of exposition in this illustration), the survey measure of expected growth in $t+1,$ as reported in $t$ , is

\begin{equation*}_{t}y_{t+1}-y_{t}=\rho (y_{t}-y_{t-1})-\omega _{t} \end{equation*}

and, if $\rho$ is known, the survey measure can be used to obtain a direct measure of the transitory innovation experienced in $t$ .Footnote ⁷ The range of outcomes surrounding output at different horizons in the future depends on the relative size of the $\epsilon _{t}$ and $\omega _{t}$ and is described by the persistence profile of (1) which is given here by

\begin{equation*} P_{t}^{y}(h)=\left ( \sum \nolimits _{k=0}^{h}\rho ^{k}\right ) ^{2}\sigma _{\epsilon }^{2}+\left ( \rho ^{h}\right ) ^{2}\sigma _{\omega }^{2}{,\quad }h=0,1,2,3,{\ldots }..\text {.} \end{equation*}

The persistence profile is dominated by the transitory shocks at the short horizons if $\sigma _{\epsilon }^{2}$ is sufficiently smaller than $\sigma _{\omega }^{2}$ , but tends to $\frac{\sigma _{\epsilon }^{2}}{1-\rho ^{2}}$ at longer horizons if there are permanent effects (i.e., $\sigma _{\epsilon }^{2}\neq 0$ ) and tends to zero if output is subject only to transitory shocks.

In the absence of direct measures of expectations, output growth would be modeled using only actual output data, captured by the ARMA(1,1) process

(4) \begin{equation} y_{t}-y_{t-1}=\rho (y_{t-1}-y_{t-2})+\upsilon _{t}+\theta \upsilon _{t-1} \end{equation}

with $\upsilon _{t}\sim N(0,\sigma _{\upsilon }^{2})$ . Matching the variance/covariances of the characterizations at (3) and (4), we have $\sigma _{\epsilon }^{2}=(1+\theta )^{2}\sigma _{\upsilon }^{2}$ and $\sigma _{\omega }^{2}=-\theta \sigma _{\upsilon }^{2}$ , with $\theta \in \lbrack \!-1,0]$ depending on the relative size of $\sigma _{\epsilon }^{2}$ and $\sigma _{\omega }^{2}$ . The associated persistence profile is

\begin{equation*} \widetilde {P}_{t}^{y}(h)=\left ( \sum \nolimits _{k=0}^{h}\rho ^{k}+\theta \sum \nolimits _{k=0}^{h-1}\rho ^{k}\right ) ^{2}\sigma _{\upsilon }^{2} \end{equation*}

which is readily used to show that

\begin{align*} \widetilde{P}_{t}^{y}(h) &\gt P_{t}^{y}(h) \, \, \text{ for }h=0,1,2,,{\ldots }\text{ } \\[5pt] \text{with} \, \, \, \widetilde{P}_{t}^{y}(h) &\rightarrow P_{t}^{y}(h) \, \, \text{ as } \, h\rightarrow \infty ; \end{align*}

that is the size of the response of $y_{t+h}$ to the news conveyed by actual output alone overstates the true persistence associated with the series at all but the infinite horizon. In the absence of survey data, the shocks with different persistence properties cannot be identified separately, the consequences of known-to-be-short-lived shocks cannot be taken into account and the persistence profile measures based on the univariate ARMA representation therefore overstate the size of the effect of shock at all horizons, including the contemporaneous uncertainty measure.

In practice, the size of the shocks impacting $y_{t}$ might change over time so that $\epsilon _{t}\sim N(0,\sigma _{\epsilon t}^{2})$ and $\omega _{t}\sim N(0,\sigma _{\omega t}^{2})$ with the $t$ -subscript on the variance terms denoting time variation in the size of the underlying innovations. The persistence profiles defined at (1)—updated to include the time variation in the variances—are still calculable if direct measures of expectations are used and still reflect the influence of shocks on output at each future time horizon. But the univariate ARMA(1,1) specification at (4) is now misspecified as the $\theta$ parameter, defined by the relative size of the permanent and transitory shocks, varies over time. The extent of the overstatement of the size of the persistent effect of shocks on output will also change over time therefore, and estimated errors from the ARMA(1,1) specification will be correlated with the $\sigma _{\epsilon t}^{2}$ and $\sigma _{\omega t}^{2}$ . This means that, if these variance measures had been included as additional regressors in a time-invariant ARMA(1,1) specification, a spurious relationship would be found between growth and uncertainty. The illustrative exercise shows that the use of surveys in empirical work will not only allow us to distinguish the effects of shocks with different persistence properties, capturing explicitly the influence of correlated news, but will also deliver measures of the associated time-varying uncertainties the effects of which can be badly misinterpreted in the absence of survey data.

3.1. Identifying the permanent and transitory shocks

The expectations series provided by the surveys are necessary to find an accurate time series representation of the actual and expected output series. But, as noted in the illustration above, they can also be used to obtain observations on the shocks distinguished by the survey respondents according to their persistence properties. Specifically, in the example at (5), we can distinguish three types of shock according to their persistence: a very short-lived transitory shock whose effects influence output one period ahead but not two periods ahead, denoted, $\omega _{1t}$ say; a transitory shock which continues to affect output for two periods and beyond but whose effects die away at the infinite horizon, denoted $\omega _{2t}$ ; and a permanent shock, $\omega _{pt}$ , representing the common stochastic trend driving the three series in the long run.Footnote ¹¹ The structural shocks are related to the reduced form shocks as follows:

(10) \begin{eqnarray} \left ( \begin{array}{ccc} k_{1} & k_{2} & k_{3} \\[5pt] 0 & 1 & 0 \\[5pt] 0 & 0 & 1\end{array}\right ) \left ( \begin{array}{c} \varepsilon _{0t} \\[5pt] \varepsilon _{1t} \\[5pt] \varepsilon _{2t}\end{array}\right ) &=&\left ( \begin{array}{ccc} 1 & 0 & 0 \\[5pt] \ast & 1 & 0 \\[5pt] \ast & \ast & 1\end{array}\right ) \left ( \begin{array}{c} \omega _{pt} \\[5pt] \omega _{2t} \\[5pt] \omega _{1t}\end{array}\right ) \end{eqnarray}

(11) \begin{eqnarray}{\bf Q \boldsymbol{\varepsilon }}_{t} &=&{\bf R}\boldsymbol{\omega } _{t} \end{eqnarray}

\begin{eqnarray*} \text{and } \, \,\,\,\,\,\, \boldsymbol{\omega } _{t} ={\bf W \boldsymbol{\varepsilon } }_{t} \, \,\,\,\,\, \text{ where } \,\,\,\,\, {\bf W}={\bf R}^{-1}{\bf Q} \end{eqnarray*}

and where $\boldsymbol{\omega } _{t}=(\omega _{pt}$ , $\omega _{2t}$ , $\omega _{1t})^{\prime }$ , $\boldsymbol{\omega } _{t}\sim N(0,{\bf \Omega } _{t})$ and ${\bf \Omega } _{t} ={\bf W\Sigma }_{t}{\bf W}^{\prime }$ is a diagonal matrix. The permanent shock is identified by the (known) combination of reduced form shocks, according to the coefficients $k_1, k_2$ , and $k_3$ , that defines the stochastic trend as in ${\bf C}(1)$ ; the shock with the transitory but long-lived effect is identified as being that part of news on two-period-ahead expectations not explained by the permanent shock (with the short-lived shock assumed to have no effect); and the short-lived shock is that part of news on one-period-ahead expectations not explained by the other two shocks.

As mentioned, the $\varepsilon _{it}$ ’s represent all the new information arriving at time t on the actual and expected series. We do not make any assumptions on the source of the news and, indeed, consider this to be a virtue of our approach. Appendix 1 shows how the complexity of the modeling framework rises as the number of sources of information increases so that identification of shocks based on arguments relating to the source of the shocks become difficult to sustain. Our modeling framework and our identification strategy is based on how survey respondents interpret the news they receive and is unrelated to the source.

An implication of this approach is that it does not distinguish between “genuine” shocks and the effects of misperceptions. As noted, the approach accounts for the effects of, for example, sticky information, through restrictions on the dynamics of the VAR, so misperceptions of this sort (whereby agents simply do not update their information sets) are accommodated within the model.Footnote ¹² However, in this case, and more generally, the effects of a genuine short-lived weather shock, say, cannot be distinguished from the noise introduced into survey responses from an incorrect understanding or interpretation of events. This noise can be assumed to be transitory and to have stable structure (in the sense that some elements might usually last only one quarter while some might be more prolonged).Footnote ¹³ But the identified shocks from our approach will be a hybrid of genuine shocks, and the effects of these misperceptions and our approach will not separate the effects by source.

The translation of the reduced form shocks to the structural shocks described in (10) provides a more meaningful interpretation of the model in (7) and (6), but the complexity of the dynamics remains unchanged and, in particular, the implied MA specification for the shocks in the corresponding univariate model for actual output growth will continue to display the “correlated news” property emphasized in Walker and Leeper’s discussion of the news-driven business cycle. The characterization of the time variation in the variance and the persistence profiles is also unchanged. To see this, note that ${\bf \Omega _{t}}={\bf W\Sigma }_{t}{\bf W}^{\prime }$ so that

\begin{equation*}{\bf \Omega } _{t} =\widetilde {{\bf H}}_{0}^{\prime }\widetilde {{\bf H}}_{0}+ \widetilde {{\bf H}}_{1}^{\prime }{\bf \Omega }_{t-1}\widetilde {{\bf H}}_{1}+\widetilde {{\bf H}}_{2}^{\prime }{\bf \boldsymbol {\omega } }_{t-1}{\bf \boldsymbol {\omega } }_{t-1}^{\prime }\widetilde {{\bf H}}_{2} \end{equation*}

where $\widetilde{{\bf H}}_{i}^{\prime }={\bf WH}_{i}^{\prime }{\bf W}^{-1}$ for $i=0,1,2$ and the GARCH structure is maintained (noting that the $\widetilde{{\bf H}}$ ’s are not necessarily diagonal so each of the time-varying elements of $\boldsymbol{\Omega } _{t}$ will respond to all of the ${\bf \boldsymbol{\omega } }_{t-1}$ shocks). Similarly, the moving average representation in (8) can be written in terms of the economically meaningful shocks

\begin{equation*} {\bf \Delta Y}_{t}={\bf C}(L){\bf W}^{-1}{\bf \boldsymbol {\omega } }_{t}={\bf D}(L){\bf \boldsymbol {\omega } }_{t} \end{equation*}

where ${\bf D}(L)={\bf C}(L){\bf W}^{-1}$ and the persistence profiles are given by

\begin{eqnarray*}{\bf P}_{t}(h) &=&\widetilde{{\bf C}}_{h}\text{ }{\bf \Sigma }_{t}\text{ }\widetilde{{\bf C}}_{h}^{\prime } \\[5pt] &=&\widetilde{{\bf D}}_{h}{\bf W}\text{ }{\bf \Sigma }_{t}\text{ }{\bf W}^{\prime }\widetilde{{\bf D}}_{h}^{\prime }=\widetilde{{\bf D}}_{h}\text{ }\boldsymbol{\Omega } _{t}\text{ }\widetilde{{\bf D}}_{h}^{\prime }. \end{eqnarray*}

Written in this way, the persistence profiles now highlight the contribution of the shocks of different persistence to the profiles at different future horizons. Specifically, with $\boldsymbol{\Omega } _{t}$ being diagonal, we can decompose the persistence profile by writing

(12) \begin{equation}{\bf P}_{t}(h)=\widetilde{{\bf D}}_{h}\boldsymbol{\Omega } _{t}^{p}\widetilde{{\bf D}}_{h}^{\prime }+\widetilde{{\bf D}}_{h}\boldsymbol{\Omega } _{t}^{2}\widetilde{{\bf D}}_{h}^{\prime }+\widetilde{{\bf D}}_{h}\boldsymbol{\Omega } _{t}^{1}\widetilde{{\bf D}}_{h}^{\prime } \end{equation}

where $\boldsymbol{\Omega } _{t}=$ $\boldsymbol{\Omega } _{t}^{p}$ $+$ $\boldsymbol{\Omega } _{t}^{2}+\boldsymbol{\Omega } _{t}^{1}$ separates the variances into the elements relating to the permanent shock $\omega _{pt}$ and the transitory shocks $\omega _{2t}$ and $\omega _{1t}$ . Here, the persistence profile relating to the permanent shock $\widetilde{{\bf D}}_{h}$ ${\bf \Omega }_{t}^{p}$ $\widetilde{{\bf D}}_{h}^{\prime }$ will converge to the single value of ${\bf P}_{t}(\infty )$ as the time horizon grows while the profiles relating to the transitory shocks $\widetilde{{\bf D}}_{h}$ $\boldsymbol{\Omega } _{t}^{2}$ $\widetilde{{\bf D}}_{h}^{\prime }$ and $\widetilde{{\bf D}}_{h}$ $\boldsymbol{\Omega } _{t}^{1}$ $\widetilde{{\bf D}}_{h}^{\prime }$ will tend to zero.

3.2. News, persistence and uncertainty in a multivariate GARCH-in-mean model

The model we consider in our empirical work has the following form:

(13) \begin{eqnarray}{\bf \Delta Y}_{t} &=&{\bf B}_{0}+{\bf B}_{1}{\bf \Delta Y}_{t-1}+{\bf \Pi }\text{ }{\bf Y}_{t-1}+{\bf G}\text{ }{\bf \Theta }_{t}+{\boldsymbol \varepsilon }_{t} \end{eqnarray}

(14) \begin{eqnarray} \text{where} \, \, \, \, \, \, {\boldsymbol \varepsilon }_{t} &\sim &N(0,{\bf \Sigma }_{t}), \, \, \, \, \, \, {\boldsymbol \varepsilon }_{t}^{-}=\min ({\boldsymbol \varepsilon }_{t},0), \nonumber \\[5pt] {\bf \Sigma }_{t} &=&{\bf H}_{0}^{\prime }{\bf H}_{0}+{\bf H}_{1}^{\prime }{\bf \Sigma }_{t-1}{\bf H}_{1}+{\bf H}_{2}^{\prime }{\boldsymbol \varepsilon }_{t-1}{\boldsymbol \varepsilon }_{t-1}^{\prime }{\bf H}_{2}+{\bf H}_{3}^{\prime }{\boldsymbol \varepsilon }_{t-1}^{-}{\boldsymbol \varepsilon }_{t-1}^{-\prime }{\bf H}_{3}\text{, } \end{eqnarray}

(15) \begin{eqnarray} \text{ and } \, \, \, \, \, \, \, \, {\bf \Theta }_{t} &=&diag({\bf \Sigma }_{t})\text{, } \end{eqnarray}

with the $\bf B$ ’s, $\bf \Pi$ , $\bf G$ , and $\bf H$ ’s representing model parameters. This model extends the model described in the previous subsection in a number of ways explained below.

3.2.2. The inclusion of other forward-looking influences on output

The model of (5) and (6) can be readily extended to include other variables and in the empirical work below we consider the vector of seven variables ${\bf Y}_{t}=(y_{t}$ , $_{t}y_{t+1}$ , $_{t}y_{t+4}$ , $r_{t}$ , $_{t}r_{t+1}$ , $_{t}r_{t+4}$ , $s_{t})^{\prime }$ adding actual and expected future values of the interest rate $r_{t}$ and current stock prices $s_{t}$ to the output variables.Footnote ¹⁴ The inclusion of the extra variables provides a broader view of macroeconomic dynamics, but it also improves the modeling approach even if the focus of interest is just on output movements. The interpretation of the shocks as “news,” and of their size as uncertainty, relies on having a relatively complete characterization of the information set dated at time $t-1$ . For the news on output, it can be argued that the inclusion of the lagged expectations series as a regressor means the analysis already uses the best summary measure of the available information since, if expectations are formed rationally and with full information, then the variable $_{t-1}y_{t}-$ $y_{t-1}$ provides a complete description of the information relating to $y_{t}-y_{t-1}$ available at time $t-1$ . But, in the possible absence of full information rationality, it is useful to broaden the information set. The inclusion of lagged values of financial variables such as stock prices and expected future interest rates aims to accommodate the relatively rapid reaction and forward-looking nature of financial markets hoping to capture any part of the news available at $t-1$ and relevant to output determination that is not adequately captured by the lagged actual and expected output series. Of course, the inclusion of the additional variables also provides an important connection to the empirical work in the news-driven business cycle literature.

With the inclusion of the extra variables, the model explains stock price growth and actual and expected change in interest rate, with $s_{t}$ and $r_{t}$ assumed integrated of order one and expectational errors on interest rates assumed stationary. Similar to output, the latter assumption implies that actual and expected interest rate are cointegrated and consequently share a single common stochastic trend. The stock price, however, is assumed not to be cointegrated with either output or interest rate. All in all, among the seven variables, there are four cointegrating equations (two between the output variables and two between the interest rate variables) and three stochastic trends consisting of the output trend, the interest rate trend, and the stock price trend.

The model also includes the contemporaneous uncertainties relating to all the additional variables as explanatory variables. The structure of the identification of (10) is then elaborated to give

(16) \begin{equation} \left ( \begin{array}{ccccccc} k_{11} & k_{12} & k_{13} & k_{14} & k_{15} & k_{16} & k_{17} \\[5pt] k_{21} & k_{22} & k_{23} & k_{24} & k_{25} & k_{26} & k_{27} \\[5pt] k_{31} & k_{32} & k_{33} & k_{34} & k_{35} & k_{36} & k_{37} \\[5pt] 0 & 0 & 1 & 0 & 0 & 0 & 0 \\[5pt] 0 & 0 & 0 & 0 & 0 & 1 & 0 \\[5pt] 0 & 1 & 0 & 0 & 0 & 0 & 0 \\[5pt] 0 & 0 & 0 & 0 & 1 & 0 & 0\end{array}\right ) \left ( \begin{array}{c} \varepsilon _{0t}^{y} \\[5pt] \varepsilon _{1t}^{y} \\[5pt] \varepsilon _{4t}^{y} \\[5pt] \varepsilon _{0t}^{r} \\[5pt] \varepsilon _{1t}^{r} \\[5pt] \varepsilon _{4t}^{r} \\[5pt] \varepsilon _{0t}^{s}\end{array}\right ) =\left ( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\[5pt] \ast & 1 & 0 & 0 & 0 & 0 & 0 \\[5pt] \ast & \ast & 1 & 0 & 0 & 0 & 0 \\[5pt] \ast & \ast & \ast & 1 & 0 & 0 & 0 \\[5pt] \ast & \ast & \ast & \ast & 1 & 0 & 0 \\[5pt] \ast & \ast & \ast & \ast & \ast & 1 & 0 \\[5pt] \ast & \ast & \ast & \ast & \ast & \ast & 1\end{array}\right ) \left ( \begin{array}{c} \omega _{pt}^{y} \\[5pt] \omega _{pt}^{r} \\[5pt] \omega _{pt}^{s} \\[5pt] \omega _{4t}^{y} \\[5pt] \omega _{4t}^{r} \\[5pt] \omega _{1t}^{y} \\[5pt] \omega _{1t}^{r}\end{array}\right ) . \end{equation}

The arrangement in (16) relates shocks with different persistence properties to the reduced form errors obtained from the equations explaining the actual and expected series with different forecast horizons. The emphasis of the identification remains on the persistence of the shocks with three of the seven structural shocks identified by long-run restrictions and the remainder identified by short-run restrictions. In terms of the long-run restrictions, the three stochastic trends, $\omega _{pt}^{y}$ , $\omega _{pt}^{r}$ , and $\omega _{pt}^{s}$ , are identified by their permanent effects—as described by the ${\bf C}_{t}(1)$ —assuming that $\omega _{pt}^{y}$ affects all variables while the $\omega _{pt}^{r}$ are shocks that permanently affect interest rates beyond the effect of the $\omega _{pt}^{y}$ , and the $\omega _{pt}^{s}$ are shocks that permanently affect stock prices beyond the effects of the $\omega _{pt}^{y}$ and $\omega _{pt}^{r}.$ In what follows, we call these “permanent output,” “permanent interest rate” and “permanent stock price” shocks. For the short-run restrictions, we assume there are two transitory shocks that last for four periods and beyond, $\omega _{4t}^{y}$ and $\omega _{4t}^{r}$ , and two transitory shocks that last for at least one-quarter but not four quarters, $\omega _{1t}^{y}$ and $\omega _{1t}^{r}$ ; we call these “long-lived” output and interest rate shocks and “short-lived” output and interest rate shocks, respectively.

An important feature of this paper’s approach to identification is the idea that agents can distinguish between news that will have a permanent effect on the macroeconomy from news that has only transitory effects. Examples of shocks that might have permanent effects might include technological advances, wars, demand shocks that are sufficiently severe that, via hysteresis, they have long-term consequences, and so on. Examples of shocks that could permanently affect stock prices and interest rates might include changes in policy regimes, changes in the perception of risk, a permanent shift in the composition of demand, and so on. Such news might be relatively rare and might be dominated in most periods by the news on transitory disequilibrium phenomenon. But importantly, it is the survey respondents who judge which news will have permanent effects and which news will have transitory effects and, as a modeller, we simply identify the different types of news from the reported expectations of outcomes at different forecast horizons.Footnote ¹⁵

The inclusion of the extra variables allows for still greater complexity in the dynamics of the individual series, which will all be subject to correlated news shocks of the form highlighted by Walker and Leeper. These authors, following Beaudry and Portier [BP] (Reference Beaudry and Portier2014) and Barsky and Sims [BS] (Reference Barsky and Sims2011) for example, note the potential role for correlated shocks in the context of a “speculation-driven” model where business cycles are generated primarily by firms anticipating future opportunities. Here stock prices react instantly to the news, but productivity and output react only with a lag. The inclusion of the extra variables in (13) renders the model more comparable to those in BP and BS, and the extended model is equally able to capture such dynamics, although the identification of shocks comes from insights on the timing of effects expressed by survey respondents rather than the sequencing of events imposed by BP and BS (e.g., does not rely on there being instantaneous effects on productivity as in BS). On the other hand, our identification does not consider the source of the shock of the mechanisms by which the effects are propagated over time so is equally consistent with any model involving correlated news, not just “speculation-driven” variants.

4.1. The dynamic effects of permanent and transitory shocks

Figure 2 illustrates the dynamics of the estimated model by showing the average response of actual and the expected output series to two system-wide shocks: one that results in a one-standard deviation increase in the actual output series and one that results in a respective one standard deviation decrease.Footnote ¹⁸ Focusing on actual output first, the figure shows that the dynamics in response to both positive and negative shocks are complex and prolonged: for the positive shock, the initial rise of 0.5 percentage points results in an overshoot at two years converging after around four years to a level in which the effect of the initial shock is doubled, with output around one percentage point higher than in the absence of the shock; for the negative shock, the convergence is achieved after six years and the long run level, at −0.5 percentage points, is roughly equal to the initial shock.

Figure 2.

Response of actual and expected output growth to a positive or negative output shock. The shock is a one standard deviation positive or negative output shock. Top Panel: Positive output shock. Bottom panel: Negative output shock. Solid lines correspond to the average responses across history. Dashed lines correspond to the 95% confidence intervals of the responses of actual output.

The figure also shows that the actual and expected output series converge to the same level eventually (by construction). However, the interactions are not straightforward and the three series are certainly not simple horizontal displacements of each other (with $_{t}y_{t+1}=y_{t+1}$ and $_{t}y_{t+4}=y_{t+4}$ for all forecast horizons $t$ ) as they would if full information rational expectations (FIRE) holds. This shows that information rigidities and/or some form of sentiment play an important role in the macro dynamics and these features are best captured through the inclusion of the survey data.Footnote ¹⁹

It is worth exploring the nature of the prolonged dynamic response,Footnote ²⁰ and Figures 3( $y$ ) and 4( $y$ ) provide some important insights with regard to the time-varying impact of different shocks by their relative persistence. In more detail, Figure 3( $y$ ) plots the (square root of the) persistence profile measures for actual output at one quarter ahead, at one year ahead, and at three years ahead over the sample period; that is, showing $P_{t}^{y}(h)$ for $h=0,$ $3,$ $11$ .Footnote ²¹ The figures illustrate the varying importance of the different types of shock at different times. Figure 3( $y$ ) shows that, at all times over the sample, the persistent effect of shocks to output grows as the future horizon increases—from the black $P_{t}^{y}(0)$ line to the blue $P_{t}^{y}(3)$ line and to the red $P_{t}^{y}(11)$ line—with considerable change occurring over one year and continuing to show for the subsequent two years. The changing balance in the relative importance of the permanent, long-lived, and short-lived shocks to the persistence profiles at different future horizons is illustrated in Figure 4( $y$ ).Footnote ²² Averaging across the whole sample, Figure 4( $y$ ) shows that, for output, the long-lived transitory shocks explain 60% of the variance on impact and continue to have a noticeable impact over the subsequent two to three years, after which the dominance of the permanent shocks is established. Short-lived shocks play a very small role at any time horizon. The prolonged impact of the identified known-to-be-transitory shocks is a key finding of the paper.

Figure 3.

Output and stock price persistence profiles. Persistence profiles from the baseline model that features both survey data and uncertainty feedback.

Figure 4.

Average decomposition of output and stock price persistence profiles over different forecast horizons. The contribution of each type of shock to output and stock price persistence profiles is averaged across the sample periods. The persistence profiles are constructed from the baseline model that features both survey data and uncertainty feedback.

Figure 5.

Contribution of different shocks to output persistence profiles in recessions and expansions over different forecast horizons. The contribution of each type of shock to output persistence profiles in recessions vs the historical distributions of their respective contributions in expansions. The dotted lines provide the 68% historical bands over expansionary periods. The persistence profiles are constructed from the baseline model that features both survey data and uncertainty feedback.

The difference between the response of output and the response of stock prices to shocks is shown by comparison with the plots in Figures 3( $s$ ) and 4( $s$ ). Here, for stock prices, more than 80% of the variance on impact is explained by the permanent shocks, and these permanent shocks explain nearly all the variance within a year. This corresponds well with the news-driven business cycle literature where news about future, permanent output outcomes is reflected very quickly in stock prices but only gradually translates into actual output growth. The additional and novel contribution here, compared to that delivered elsewhere in the literature, is that the permanent shocks and their effects are identified explicitly and separately from the effects of transitory shocks—where the identification structure is an intuitive outcome from the survey responses of expectations.

An important part of the time variation arises through the asymmetric effect of positive and negative shocks. Figure 5 presents the equivalent figures to Figure 4( $y$ ) relating to output growth but showing the decompositions when averaged over expansions/normal times, that is, periods of positive output growth and when averaged over recessions, that is, periods of negative output growth. Output growth was positive over most of the sample, so the decomposition during expansions is similar to the averaged across the whole sample. But Figure 5 shows that the permanent shocks have a much larger impact effect on output during recessionary times than expansionary times (with 65% of the persistence profile on impact relating to permanent shocks compared to 40% in expansionary times) and the long-run effects (or at least 95% of them) are achieved much more quickly, taking around two rather than four years.Footnote ²³ As we will see below, usually, the largest part of the contemporaneous uncertainty surrounding output and interest rates arises out of transitory shocks. But it is recognized that their effects will dissipate over the coming two to three years and the uncertainties arising from this source do not impact significantly on output growth. In contrast, the uncertainties arising surrounding permanent shocks to output and interest rates are influential, and during recessions, a larger part of the contemporaneous uncertainty surrounding output relates to permanent shocks; uncertainty therefore plays a particularly important role in determining output levels in recessionary times. Again, as illustrated in Figure 4, the scaling in Figure 5 reveal that short-lived shocks play a very small role.

In general, the importance of uncertainties surrounding permanent shocks to output and interest rates can also be seen from Table 1. The Table reports the contemporaneous impact of a one standard deviation increase in the uncertainties arising from the various shocks with different persistence properties on the actual and expected outputs.Footnote ²⁴ $^{,}$ Footnote ²⁵ The coefficients show that changes in the uncertainty arising from the permanent shocks to output and interest rates have the most influence on output; a one standard deviation increase in the uncertainty arising from permanent output shocks increases actual output by 0.17 percentage points, while a one standard deviation increase in the uncertainty arising from permanent interest rate shocks decreases actual output by 0.28 percentage points. The finding that it is uncertainty surrounding permanent shocks that impacts on the macroeconomy, and that changes in the uncertainties arising from transitory shocks have little or no impact on output, is another key result from this empirical work.Footnote ²⁶

Table 1.

Impact of uncertainty: structural shocks on actual and expected output

Impact of a one standard deviation increase in uncertainty on actual and expected output growth. Computation steps are provided in the Appendix.

Figure 6 provides a more detailed description of the output persistence profiles at $h=0$ at different points in the sample and decomposing the profiles to see the contribution of the different types of shock. As seen earlier from Figure 4, Figure 6 shows that, for output, usually, the largest part of the contemporaneous uncertainty surrounding output arises out of transitory shocks which we know have little impact in the long run. But there are clear episodic peaks in uncertainty arising from the permanent output shocks at the start of the 1990 Gulf War recession and at the 2008 Global Financial Crisis (GFC). Relatively large values for the uncertainty arising from permanent interest rate shocks are found at these times and the two types of uncertainty also show at the time of the 1987 Stock Market crash and around the early 2000’s dot-com recession. These dates are perhaps obvious candidates for special mention, but it is worth noting—like JLN/LMN before us—that the number of episodes in which these measures of uncertainty rise above 1.65 standard deviations from the mean is relatively small when compared to the very volatile VIX measures for example. As it turns out, and looking back to the profiles at $h=11$ in Figure 3( $y$ ), it is the shocks—and associated uncertainties—of the 1987 Stock Market crash, 1990 Gulf War recession and the 2008 Global Financial Crisis (GFC) which translated into particularly extraordinary permanent output effects.

Figure 6.

Decomposition of output contemporaneous persistence profile. Decomposition of 1-quarter-ahead output persistence profile at each sample period. Transitory shocks are defined as shocks that do not have a permanent effect on all three of output, interest rate, and stock price trends.

Figure 7.

Output persistence profiles from models without survey data or uncertainty feedback. The model without survey data is a tri-variate model of only actual variables but features the GARCH-in-mean component. The model without uncertainty feedback uses both actual and expected variables but does not feature the GARCH-in-mean component.

Figure 8.

Average decomposition of output persistence profiles from models without survey data or uncertainty feedback. G3U: Tri-variate model without survey data that uses only actual variables but features the GARCH-in-mean component. G7: Model without uncertainty feedback that uses both actual and expected variables but does not feature the GARCH-in-mean component.

4.2. The contribution of survey expectations and the feedback of uncertainty

The central role played by the survey data and uncertainty feedback in generating the model’s dynamic properties described above is demonstrated by comparing the output profiles of our model with the corresponding profiles from two alternative models: (i) “G3U” which denotes a trivariate GARCH-in-mean model employing only the actual output, actual interest rate, and stock price data but allowing for an uncertainty feedback; and (ii) “G7” which denotes the same 7-variable model as our baseline GARCH-in-Mean (G7U) model with survey data on expectations but now without the uncertainty feedback.Footnote ²⁷ The profiles from the alternative models are given in the sets of plots in Figures 7(G3U), 7(G7) and 8.

Focusing on the role of the survey data and the importance of explicitly identifying permanent and known-to-be-transitory shocks, in contrast to the baseline model in Figure 3( $y$ ), the output persistence profiles in Figure 7(G3U) show that there is very little gap between $P_{t}^{y}(3)$ and $P_{t}^{y}(11)$ . This indicates that a large part of the response to a permanent shock takes place within the year therefore failing to capture the prolonged dynamic response over three years and beyond exhibited in the baseline model.Footnote ²⁸ Second, in the decomposition of the persistence profiles for a given time horizon, a comparison of Figure 4( $y$ ) to Figure 8, shows that there is also a marked overstatement in the importance of the permanent output shock relative to the baseline model. It is worth noting that the extra dynamic sophistication is achieved in our baseline model not simply through the inclusion of additional variables but through the inclusion of variables capable of distinguishing and capturing the effect of known-to-be-transitory shocks.

Turning to the role of uncertainty, the plots relating to the G7 model differ in important ways from those of the baseline G7U model. First, similar to the G3U model, the G7 model also overstates the importance of permanent shocks in the short-run. In contrast to the G3U model, however, the G7 model features a similar prolonged response to shocks to our baseline model, with a similar gap between $P_{t}^{y}(0)$ , $P_{t}^{y}(3)$ , and $P_{t}^{y}(11)$ in Figure 7(G7) and with the long-lived transitory shocks showing for around two years in Figure 8. However, the presence of significant feedbacks from uncertainty introduces nonlinearities into the model and time variation in the effects of shocks. This time variation is particularly reflected in Figure 4( $y)$ where there are substantial differences between the profiles at the three horizons at different points in the sample; the correlations between the output profiles $P_{t}^{y}(h)$ at $h=0$ and the profiles at $h=3$ and $h=11$ , calculated across the sample, are 0.71 and 0.57, respectively. Of course, some part of the time variation in the profiles is due to the changing size of the shocks captured by the GARCH effects. But the corresponding correlations in the persistence profiles for the G7 model are 0.95 and 0.94 showing that in the absence of the uncertainty feedback, the effects of the shocks simply accumulate over time and in a similar way at all points in the sample. Further, the correlation between the output profiles at $h=0,3$ and $11$ for the G7U GARCH-in-Mean model and the G7 GARCH model—as illustrated in Figures 3( $y$ ) and 7(G7)—are 0.916 at $h=0$ but just 0.662 and 0.578 at $h=3$ and $11$ , respectively, showing the important role played by the uncertainty feedbacks in the preferred G7U model.Footnote ²⁹

The discussions above have elaborated how our proposed G7U model is able to capture important features of the data that are omitted in the absence of expectations data and/or its treatment of uncertainty. While the model is complex, it is useful to summarize the key findings as follows:

• • The inclusion of actual and expected measures of the series allows prolonged and complex dynamics to be captured. The ability to distinguish the effects of shocks with different persistence properties allows the model to capture (i) the slow accumulation of permanent effects and (ii) the prolonged effects of some transitory shocks which are obscured in the absence of survey data. The complexity—including hump-shaped output response to shocks—is partially down to interaction of actual and expected series which is more sophisticated than would be suggested in simple FIRE models (Figure 2, Figures 3(y) and 4(y) vs. Figure 7(G3U) and Figure 8).

• • The results correspond well with the news-driven business cycle literature where news about future, permanent output outcomes is reflected very quickly in stock prices but only gradually translates into actual output growth (Figures 3(y), 3(s), 4(y) and 4(s)).

• • In the absence of explicitly identified permanent and known-to-be-transitory shocks, the impact of permanent shocks on output is fully realized within a year relative to more than three years and is markedly overstated on impact (Figures 3(y) and 4(y) vs. Figures 7(G3U) and 8).

• • The uncertainty feedbacks introduce considerable time-variation in the output response to shocks; the uncertainty surrounding permanent shocks plays a particularly important role in determining output levels in recessionary times (Figure 5, Figure 7(G7)).

• • Uncertainty plays an important role in macrodynamics episodically. The uncertainty surrounding permanent output shocks and permanent interest rates shocks was at its greatest at the time of the 1987 Stock Market crash, at the start of the 1990 Gulf War recession, around the early 2000’s dot-com recession and at the 2008 Global Financial Crisis (GFC). The shocks—and associated uncertainties—of the 1987 Stock Market crash, 1990 Gulf War recession and the 2008 Global Financial Crisis (GFC) translated into particularly extraordinary permanent output effects (Figure 6).

• • Uncertainty arising from the permanent shocks to output and interest rates have significant influence on output, positively in the case of permanent output uncertainty and negatively for permanent interest rate uncertainty. Changes in the uncertainties arising from transitory shocks have little or no impact on output (Table 1).

These baseline results are obtained using 2020:Q2 vintage data. However, real-time data arguably more closely reflects the information held by the forecasters when they submit their responses. While stock price and interest rate are already real-time in nature, real GDP data are often revised. To check the robustness of our main results to the use of real-time real GDP data, we estimate our baseline multivariate GARCH-in-mean model using the first-release real GDP growth series as our measure of actual GDP. The expected GDP data are constructed accordingly from the first-release actual real GDP series.

Overall, qualitatively, our main conclusion does not change—permanent shocks have a higher contribution to short-term output variations in recessions (around 40% for 1-quarter-ahead output uncertainty) than in expansions/normal times (around 20% for 1-quarter-ahead output uncertainty). Relative to the distribution of the contribution of permanent shocks to output in expansions, this difference is statistically significant for 1- to 10-quarter-ahead output uncertainty, which is a longer period than that found with revised GDP data. The mean estimates are indeed lower compared to the ones obtained with revised GDP data and the reduction can be attributed to the higher contribution of long-lived transitory shocks. The contribution of short-lived shocks on output is still small, with only a maximum of around 6% for 1-quarter-ahead output uncertainty on average. In comparison, and consistent with the results using final vintage data, stock price dynamics are primarily affected by permanent shocks in all horizons and in both expansions and recessions. Replications of Figures 4 and 5 using real time data are provided in the Appendix illustrating the robustness of the results in the paper to the use of real time data.