2.1. The model

We use a modified version of the method recently proposed by Forni et al. (Reference Forni, Gambetti, Maffei-Faccioli and Sala2022). The method aims at estimating a nonlinear moving average representation of the economy where a shock of interest and a nonlinear function of it drive economic variables. The model can be estimated using a two-step procedure where (i) the shock of interest is identified in an informationally sufficient VAR,Footnote ³ and (ii) the estimated shock is used, together with some nonlinear function of it, as an exogenous variable in a VARX which includes a set of variables of interest. By combining the (linear) impulse response functions of the VARX, nonlinearities and asymmetries of the shock of interest can be estimated.

Let $Y_t$ be a vector of $m$ variables of interest and $s_t$ the shock of interest admitting the following structural representation

(1) \begin{equation} Y_t=\mu +\alpha (L)s_t+\beta (L)s_t^2+B(L)u_t \end{equation}

where $s_t$ is the news shock, $B(L)=(I+B_1B_0^{-1}L+B_2B_0^{-1}L^2+\ldots )B_0$ is a $m \times m$ matrix of polynomials in the lag operator $L$ , $\alpha (L)$ and $\beta (L)$ are $m \times 1$ vectors of polynomial in $L$ and $u_t$ is a vector of structural shocks. The vector $\varepsilon _t$ is a vector of shocks orthogonal to $s_t$ and $s_t^2$ . The terms $\alpha (L)$ and $\beta (L)$ represent the impulse response functions of the linear and the nonlinear term on $Y_t$ . The total effect of a positive shock $s_t=\bar s$ is

\begin{equation*} IR(\bar s)=\alpha (L)\bar s+\beta (L)\bar s^2 \end{equation*}

and the effect of a negative shock $s_t=-\bar s$ is

\begin{equation*} IR(\!-\!\bar s)=-\alpha (L)\bar s+\beta (L)\bar s^2. \end{equation*}

Assuming that the term $B(L)u_t$ is an invertible vector moving average, we can rewrite the above model as a VARX for $Y_t$ where $s_t$ and $s_t^2$ and its lags are two exogenous variables

(2) \begin{equation} A(L)Y_t=c+\tilde \alpha (L) s_t+\tilde \beta (L) s_t^2+\varepsilon _t \end{equation}

where $A(L)=(I+B_1B_0^{-1}L+B_2B_0^{-1}L^2+\ldots )^{-1}$ , $\varepsilon _t=B_0u_t$ , $\tilde \alpha (L)=A(L)\alpha (L)$ and $\tilde \beta (L)=A(L)\beta$ are $m \times 1$ vectors of polynomial in $L$ . The impulse response functions to a news shock of size $\bar s$ can be obtained as $A(L)^{-1}(\tilde \alpha (L)\bar s+\tilde \beta (L)\bar s^2)$ for a positive shock and $A(L)^{-1}(-\tilde \alpha (L)\bar s+\tilde \beta (L)\bar s^2)$ for a negative shock.

In order to estimate equation (2), an estimate of $s_t$ is required. We assume that $X_t$ is a vector of variables, possibly different from $Y_t$ , which is informationally sufficient for $s_t$ , that is, $s_t$ can be obtained as a linear combination of current and past values of $X_t$ . We include in the vector $X_t$ the following variables: (log) TFP,Footnote ⁴ (log) stock prices, the Michigan Survey confidence index component concerning business conditions for the next 5 years, (log) real consumption of non-durables and services, the 10-year government bond, the spread between the 3-month Treasury Bill and the 10-year bond, the Moody’s Aaa interest rate (AAA), the spread Aaa-Baa and the CPI inflation.Footnote ⁵ We then estimate the VAR and identify the news shock along the lines of Beaudry and Portier (Reference Beaudry and Portier2014) and Forni et al. (Reference Forni, Gambetti and Sala2014). Precisely, we impose the following restrictions: (i) the news shock has no effects on TFP contemporaneously and (ii) has a maximal effect on TFP in the long run (48 quarters). Condition (ii) is equivalent to impose that there are just two shocks affecting TFP in the long run: the innovation of TFP (the so-called “surprise” shock) and the news shock.Footnote ⁶ This identification scheme has become relatively standard in the news shock literature and is very similar to the one used in Barsky and Sims (Reference Barsky and Sims2011).Footnote ⁷ Having an estimate of the news shock, we estimate model (2) and the related impulse response functions.

Notice that there could be a misspecification problem in the SVAR for $X_t$ . If none of the variables in this SVAR is affected by the square term, then the model will be well specified and the shock well estimated (provided that the variables included are informationally sufficient for the shock). If on the contrary some variables are affected by the square term, the SVAR is misspecified. Despite this, the shock could be correctly estimated, as discussed in Debortoli et al. (Reference Debortoli, Forni, Gambetti and Sala2022). In the empirical section, we perform a simple exercise to verify whether this is the case.

Table 1. Orthogonality test

Note: $P$ -values of the $F$ -test of the null that the coefficients of the lags of the variables are zero in a regression of the estimated shock onto the lagged variables.

3.1. The news shock

The news shock and its square exhibit very large values (more than two standard deviations larger than average) in seven quarters. In Figure 1, we focus on the squared news shock. Five of the seven quarters correspond to periods associated with negative shocks and two are periods associated with positive shocks. The squared news shock is therefore left skewed, with skewness of −0.36. The seven quarters are the following (in parenthesis the sign of the shock and the corresponding event): 1974:Q ( $-$ , Stock Market Oil Embargo Crisis); 1982:Q1 ( $-$ , loan crisis); 1982:Q4 ( $+$ , end of early 80s recession); 1987:Q1 ( $+$ , oil price collapse); 2002:Q3 ( $-$ , WorldCom bankruptcy); 2008:Q3 ( $-$ , Lehman Brothers bankruptcy); 2008:Q4 ( $-$ , stock market crash). Most of these dates correspond to well-identified historical events and/or cycle phases.

Figure 1. Squared news shock. There are seven quarters with peaks corresponding to the following events (in parenthesis the sign of the shock): 1974:Q ( $-$ , Stock Market Oil Embargo Crisis); 1982:Q1 ( $-$ , loan crisis); 1982:Q4 ( $+$ , end of early 80s recession); 1987:Q1 ( $+$ , oil price collapse); 2002:Q3 ( $-$ , WorldCom bankruptcy); 2008:Q3 ( $-$ , Lehman Brothers bankruptcy); 2008:Q4 ( $-$ , stock market crash).

Figure 2 shows the effects of the news shock on the variables in $X_t$ . The impulse response function of TFP exhibits the typical S-shape which is usually found in the literature. Stock prices, E5Y, and the news variable jump on impact, as expected, while consumption increases more gradually. All interest rates reduce on impact, albeit the effect is barely significant. All in all, the effects of the news shock are qualitatively very similar to those found in the literature.

Figure 2. Impulse response functions to the news shock (SVAR). Solid line: point estimate. Light gray area: 90% credible intervals. Dark gray area: 68% credible intervals.

As discussed in the previous section, some of the variables appearing in the vector $X_t$ could be affected by the squared term. In this case, the preliminary SVAR would be misspecified and the news shock potentially poorly estimated. To understand whether this is the case, we estimate the VARX using the same variables as in the SVAR.Footnote ¹⁰ If we find responses to the news shock which differ from those obtained in the SVAR, then the preliminary SVAR is misspecified. Figure 3 displays the results. The black solid lines represent the responses in the VARX, and the red dashed lines represent the responses in the SVAR. The responses are very similar, confirming the validity of the preliminary SVAR.

Figure 3. Impulse response functions to the news shock in the VARX. Solid line: point estimate. Light gray area: 90% credible intervals. Dark gray area: 68% credible intervals. Red lines are the responses obtained in the SVAR.

3.2. The effects on macroeconomic variables

The VARX we employ to study the effects of news on the economy includes (log) real GDP, (log) real consumption of non-durables and services, (log) real investment plus consumption of durables, (log) hours worked, CPI inflation, and the ISM new orders index. The estimated news shock and the squared news shock are used as exogenous variables.

We organize the discussion as follows. First, we present the VARX results relative to the estimated impulse response functions to $s_t$ and $s_t^2$ for $\bar s=1$ . Then, we focus on nonlinearities. Results are reported in Figure 4. The numbers on the vertical axis are percentage variations. The news shock, Figure 4 (left column), has a large, permanent, positive effect on real activity, with maximal effect after about 2 years. The results are in line with the findings of the literature.Footnote ¹¹ The squared news shock (Figure 4, right column) has a significant negative effect on all variables on impact. The maximal effect on GDP is reached after four quarters and is around −1%. Afterward, the effect reduces and vanishes after about 2−3 years. The effects of the square term are also sizable and significant for investment and hours, while the effects on consumption are somewhat milder and not significant. By inspecting the response of inflation, it is clear that square effects are demand-type effects, since both GDP and inflation significantly fall. Since, for $\bar s=1$ , the response to the squared term represents the asymmetry of the responses to positive and negative shocks, whenever the square term response is significant, the asymmetry is significant.

Figure 4. Impulse response functions to the news shock (left column) and the squared news shock (right column) obtained with the VARX. Solid line: point estimate. Light gray area: 90% credible intervals. Dark gray area: 68% credible intervals.

Figure 5 plots the total response of economic variables to the news shock. Recall that the total responses are $IR(\bar s)$ for a positive shock and $IR(- \bar s)$ for negative shocks. We plot the responses to shocks of size $\bar s=1$ , that is, one standard deviation (first column), $\bar s=0.5$ (second column) and $\bar s=2$ (third column). The solid line represents the mean response to a positive news shocks, and the gray areas are the 68% credible intervals. The dashed red line represents the effects of a negative news shock with reversed sign (multiplied by −1), in order to ease the comparison in terms of magnitude between good and bad news.

Figure 5. Nonlinear impulse response functions estimated from the VARX with equation (2). Left column: shock of size $1$ ; middle column: shock of size $0.5$ ; right column: shock of size $2$ . Black solid lines: point estimates. Light gray area: 90% credible intervals of a positive news shock. Dashed red lines are the responses to a negative shock with reversed sign.

A positive news shock permanently increases real economic activity variables: GDP, consumption, investment, and hours worked. The responses however are quite sluggish. Indeed, except for consumption, the impact effects are zero. Inflation significantly falls and new orders increase. By inspecting the two lines, a clear asymmetry emerges. A bad news shock has higher short-run effects than a good news shock on real economic activity variables. Summing up, the impact effects are higher for bad news than for good news. Indeed, for negative shocks the effects of the square term enhance those of news. The contrary holds for positive shocks: the square term mitigates the expansionary effects of news. Interestingly, the result is different for inflation since good news have larger effects than bad news.

The asymmetry is amplified in the case of a large shock $\bar s=2$ (third column) and dampened in the case of a small shock $\bar s=0.5$ (second column). The larger is the shock, the larger is the asymmetry since the square term becomes more important. Notice that in the series of squared news, there are realizations that are as high as four standard deviations; in that case, the importance of the nonlinear component would be extremely high.

Table 2 reports the variance decomposition. In particular, it reports the proportion of variance of the variables attributable to news shocks. This includes both the linear and the quadratic term. The shock has important effects in the medium and long run for GDP, consumption, investment, and hours. For these variables, the shock explains between 40% and 60% of the variance at horizons longer than 1 year.

Table 2. Variance decomposition for macroeconomic variables

Note: Percentage of variance attributable to the news shock, the squared shock, and the sum of the two.

Notice that, in principle, the asymmetry could simply arise because of a different response of TFP to news. To make sure that this is not the case, we add TFP in the VARX and check the response. It turns out that the effect of the nonlinear term is essentially not significant, see the robustness Section and Figure 9. This rules out the possibility that the effects are attributable to a different propagation of news on TFP.

3.3. The effects on financial variables

In order to analyze the effects of news on financial variables and uncertainty, we estimate an additional VARX including stock prices, the 3M T-Bill bond yield, the spread between Baa and Aaa corporate bonds, which may be regarded as a measure of the risk premium, the stock of commercial and industrial loans, and three indices of uncertainty, namely the extended VXO index of implied volatility in option prices (see Bloom (Reference Bloom2009)), the macroeconomic uncertainty index 12-month ahead (denoted as JLN12), developed by Jurado et al. (Reference Jurado, Ludvigson and Ng2015), and the Ludvigson et al. (Reference Ludvigson, Ma and Ng2021) real uncertainty index 12 months ahead (denoted as LMN R12).

Results are reported in Figures 6 and 7. In Figure 6, the left column reports the effects of $s_t$ and the right column the effects of $s_t^2$ . Figure 7 reports the total effects for different magnitudes of the shock.

Figure 6. Impulse response functions to the news shock (left column) and the squared news shock (right column) obtained with the VARX. Solid line: point estimate. Light gray area: 90% credible intervals. Dark gray area: 68% credible intervals.

Figure 7. Nonlinear impulse response functions of financial variables estimated from the VARX with equation (2). Left column: shock of size $1$ ; middle column: shock of size $0.5$ ; right column: shock of size $2$ . Black solid lines: point estimates. Light gray area: 90% credible intervals of a positive news shock. Dashed red lines are the responses to a negative shock with reversed sign.

We start by analyzing the effects of the linear and quadratic components in Figure 6. The linear news shock increases permanently stock prices and reduces uncertainty, the risk premium, and the T-bill. The squared term is, as for macroeconomic variables, contractionary. It is interesting to notice that a positive shock to the squared term has a significant positive effects on the three uncertainty indices, VXO, JLN12, and LMN R12. There is a close link between squared news shock and uncertainty measures. We will come back to this result later on.

Moving to the total effect reported in Figure 7, good news have a large, positive, and persistent effect on stock prices and significantly reduce the risk premium and the uncertainty indices. Bad news have the opposite effects (notice again that we report the response to a negative shock multiplied by −1): persistent and significant reduction of stock prices and increase in the risk premium and uncertainty. Again, a substantial asymmetry arises. Stock prices and the VXO react much more to bad news than to good news and the risk premium reacts faster. The response of the T-Bill is different. This variable displays smaller effects for bad news than good news.

From the variance decomposition in Table 3, it can be seen that the shock is very important for stock price fluctuations, explaining around 40−50% of the variance. On the contrary, the shock plays a smaller role for the other variables. It is interesting to notice that more than 30% of the variance of the VXO is accounted for by the shock while the percentage is a bit less for the JLN12 measure (around 10%) and LMN R12 (around 15%). A sizable part of the existing measure of uncertainty is explained by news. Of course, this leaves the door open for the existence of an exogenous component of uncertainty that has nothing to do with news.

Table 3. Variance decomposition for financial variables

Note: Percentage of variance attributable to the news shock, the squared shock, and the sum of the two.

3.4. Robustness checks

We make three robustness checks. First, we check if there is any additional information coming from uncertainty useful for the identification of $s_t$ , since Cascaldi-Garcia and Galvao (Reference Cascaldi-Garcia and Galvao2021) show that news shocks are correlated with uncertainty shocks. Therefore, we repeat the analysis adding an uncertainty measure in the initial SVAR. The results, displayed in Figure 8, are very similar to those obtained in the baseline model.

Figure 8. Nonlinear impulse response functions of macroeconomic variables using a measure of uncertainty in the first SVAR. Black solid lines: point estimates. Light gray area: 90% credible intervals of a positive news shock.

Figure 9. Impulse response functions to the news shock (left column) and the squared news shock (right column) obtained with the VARX. Solid line: point estimate. Light gray area: 90% credible intervals. Dark gray area: 68% credible intervals.

As a second check, we add TFP in the VARX. Impulse responses are displayed in Figure 9. The nonlinear term on TFP is virtually insignificant, and the responses of the other variables are almost identical to those in Figure 4.

Thirdly, we use the absolute value of the news shock rather than the square as nonlinear function. Figure 10 reports the results. The responses obtained with the absolute value are remarkably similar to those obtained with the square.

Figure 10. Nonlinear impulse response functions of macroeconomic variables using the absolute value as nonlinear function. Black solid lines: point estimates. Light gray area: 90% credible intervals of a positive news shock.

4.1. Evidence

To explore the relation between squared news and uncertainty, we first compute the correlation of the squared shock and a smoothed version of it with a number of uncertainty measures used in the literature, namely (i) the extended VXO index of implied volatility in option prices (Bloom (Reference Bloom2009)); (ii) the Jurado et al. (Reference Jurado, Ludvigson and Ng2015) macroeconomic uncertainty index 1, 3, and 12 months ahead (denoted, respectively, JLN1, JLN3, and JLN12 henceforth); (iii) the Ludvigson et al. (Reference Ludvigson, Ma and Ng2021), financial and real uncertainty indexes 1, 3, and 12 months ahead (denoted, respectively, LMN F and LMN R and the number referring to the month).

Table 4 reports the correlations. The first column refers to the squared shock while the second column to a centered 5-quarter moving average of the squared shock. For the squared shocks, correlations range from 0.24 (VXO) to 0.40 (JLN3 and JLN12) while for the moving average correlations range from 0.36 (VXO) to 0.67 (JLN3 and JLN12). The squared news shock is positively correlated with all measures of uncertainty, the correlation being particularly high for the JLN measures. Figure 11 plots the (standardized) 5-quarter moving average of the news shock (red solid line) together with the (standardized) JLN12 index (blue dotted line), top panel, with the (standardized) LMN R12 measure (blue dotted line), middle panel and with the (standardized) LMN F12 measure (blue dotted line), bottom panel. The result is striking, and the variables closely track each other and display several coincident peaks. Notice that the estimation of the news shock is completely independent of uncertainty since no uncertainty measure is included in the first estimation step. This is in line with the results of the VARX for financial variables, Figure 6. There, we saw that a positive squared news shock generates a positive conditional comovement among uncertainty measures. Here, we see that the positive comovement between squared news and uncertainty also arises unconditionally.

To support the interpretation that the effects of shocks to the squared news capture the effects of shocks to uncertainty, we perform the following exercise. We regress the squared term on the current value, one lag and one lead of the uncertainty measures discussed above. We then repeat the VARX estimation replacing the squared shock with the residual obtained in this regression. If our claim is correct, when the component related to uncertainty is removed from the squared term, its effects should disappear. Figure 12 displays the results. The left column of the figure reports the effects obtained using the residual term together with the responses in the baseline model (red dashed lines). The effects obtained with the residuals are much smaller than those obtained in the baseline model and are not significant. This provides support to the view that the effects of the squared term are closely related to uncertainty.

As a final check, we identify an uncertainty shock as the first Cholesky shock in a VAR including, in order, the VXO index, GDP, Consumption, Investment, Hours worked, CPI inflation, and new orders and compute the related impulse responses.Footnote ¹² We then repeat the estimation with the same specification, but adding the news shock and the squared news shock as the first and the second variable in the VAR, respectively. The uncertainty shock becomes now the third shock in the Cholesky decomposition. If the standard uncertainty shock has nothing to do with news and squared news, the impulse response functions in the two model models, with and without $s_t$ and $s_t^2$ , should be very similar. It turns out that the impulse responses are significantly different (see Figure 13). When the uncertainty shock is cleaned from the effects of news, its effects basically vanish. We interpret this as meaning that a large part of the uncertainty shock is associated with news and squared news. We repeat the same exercise replacing the VXO with LMNR12. Results, see Figure 14, point to the same conclusion. When news and squared news are included, the effects of the uncertainty shocks are substantially mitigated, meaning that to some extent, uncertainty endogenously depends on news.

The evidence suggests the existence of a close link between news shocks and uncertainty and supports the view that the effects associated with the squared term can be interpreted as effects generated by an increase in uncertainty arising from news.Footnote ¹³

In the next section, we discuss a simple framework which allows to interpret the effects associated with $s_t^2$ discussed in the empirical section as effects due to shifts in uncertainty triggered by news.

4.2. A simple informational framework

Here we discuss a simple illustrative framework of limited information flows which can help in understanding why uncertainty can arise from news. By no means this is intended to be an economic model since we do not model how agents react to news and the channels through which uncertainty affect agents’ decisions. Still, we believe, it can be useful since it establishes a potential “uncertainty channel” of news, and this channel can be at the root of the nonlinearities documented in the empirical part.

Table 4. Correlation with JLN and LMN uncertainty

Note: First column: square of the raw shock. Second column: 5-quarter moving average of the shocks.

Let us assume that TFP, $a_t$ , follows

(3) \begin{equation} \Delta a_t = \varepsilon _{t-1} \end{equation}

where $\varepsilon _t\sim N(0,\sigma ^2_{\varepsilon })$ is an economic shock with delayed effects.Footnote ¹⁴ At time $t$ , agents have imperfect information and cannot observe $\varepsilon _t$ , but rather have access to news that report the events underlying the shock, for instance, natural disasters, scientific and technological advances, institutional changes, and political events.Footnote ¹⁵ At each point in time, agents form an expectation, $s_t=E_t\varepsilon _t$ of the true shock.Footnote ¹⁶ The shock and the expectation however, because information is imperfect, do not coincide. We assume that there is a random factor $v_t$ that creates a wedge between the two:

\begin{equation*} \varepsilon _t=s_t v_t. \end{equation*}

The shock $v_t$ has the following properties: the conditional mean is $E_tv_t=1$ , so to satisfy $E_t\varepsilon _t=s_t$ , and the conditional variance is $E_t(v_t-1)^2=\sigma _v^2$ , that is, constant. The above equation can be rewritten as $\varepsilon _t= s_t+ s_t(v_t-1)$ , so that $\varepsilon _t$ is made up by the sum of two components: the observed component $s_t$ and an unobserved component which is proportional to $s_t$ .

This multiplicative noise structure, while common in engineering and control system, to our knowledge has not been employed before in the literature of limited information. Typically, an additive structure is used, mainly for the purpose of analytical tractability. However, we find it particularly attractive since it can describe several relevant economic situations. A few examples can provide a better intuition. Suppose that a diplomatic crisis takes place at time $t$ and is reported by the media. The crisis can lead to a war ( $\varepsilon _t=-1$ ) or not ( $\varepsilon _t=0$ ) with equal probabilities depending on the president’s decision. The decision is taken in $t$ but, for national security reasons, made public only in $t+1$ . So the expected shock is $s_t=-0.5$ . The noise, which captures the uncertainty surrounding the president’s decision, will be $v_t=2$ in case of war and $v_t=0$ otherwise, with equal probabilities. As a second example, suppose the agents observe that a big bank goes bankrupt. The value of the shock, however, is unknown because with some probability, say 0.5, there will be a domino effect and other banks will go bankrupt ( $\varepsilon _t=-3$ ), but with probability 0.5 the government will intervene to rescue them ( $\varepsilon _t=-1$ ). The government’s decision is taken in $t$ but agents do not know it, so the expected shock is $s_t=-2$ and $v_t$ can be either $1.5$ or $0.5$ with equal probabilities.

Figure 11. Each panel displays the 5-quarter moving average of the news shock (red solid) with (a) the (standardized) JLN12 uncertainty measure (blue dotted), top panel; (b) the (standardized) LMN R12 index (blue dotted line), middle panel; (c) the (standardized) LMN F12 measure (blue dotted line), bottom panel.

Figure 12. Impulse response functions to the squared news. Right column, baseline results. Left column, using the residual of the regression of the squared term on the current value, a lag and a lead of uncertainty measures. Black solid line: point estimate. Light gray area: 90% credible intervals. Dark gray area: 68% credible intervals. Red dashed lines on the left column report the baseline estimates.

Figure 13. Impulse response functions to an uncertainty shock identified as the first shock in a Cholesky decomposition with the VXO ordered first. Black solid line: point estimate. Light gray area: 90% credible intervals. Dark gray area: 68% credible intervals. Blue dashed lines are the impulse response functions of the uncertainty shock identified as the third shock in a Cholesky decomposition with the VXO ordered third and news and squared news ordered first and second, respectively.

Figure 14. Impulse response functions to an uncertainty shock identified as the first shock in a Cholesky decomposition with the LMN12 ordered first. Black solid line: point estimate. Light gray area: 90% credible intervals. Dark gray area: 68% credible intervals. Blue dashed lines are the impulse response functions of the uncertainty shock identified as the third shock in a Cholesky decomposition with the LMN12 ordered third and news and squared news ordered first and second, respectively.

In this simple informational framework, the TFP forecast error is

\begin{eqnarray*} u_{t+1}&=&\Delta a_{t+1}-E(\Delta a_{t+1}|s_t)\\[5pt] &=&\epsilon _t-E(\varepsilon _t|s_t)\\[5pt] &=&(v_t-1)s_t. \end{eqnarray*}

Following Jurado et al. (Reference Jurado, Ludvigson and Ng2015), we define uncertainty as the conditional variance of the forecast error, which is

\begin{equation*} E((v_t-1)^2|s_t)s_t^2=\sigma _v^2s_t^2. \end{equation*}

The conditional variance, or uncertainty, depends on the squared expected shock.Footnote ¹⁷ Going back to the previous examples, the bigger the war in case of going to war, or the larger the consequences of the domino effect if government does not intervene, that is in both cases the larger the value of $\varepsilon _t$ in absolute value, the larger is uncertainty since the larger is $s_t$ .

Through the lenses of this interpretative framework therefore, the effects of $s_t^2$ on the economy, not modeled here, can indeed be interpreted as attributable to uncertainty. Can the interpretation be extended to our empirical findings? Can we interpret the asymmetries as arising from the uncertainty generated by news? The answer, essentially, depends on whether the news shock identified in Section 2 can be interpreted as $s_t$ . It is easy to see that this is the case. The model representation of $\Delta a_t$ and $s_t$ is

(4) \begin{equation} \begin{pmatrix} \Delta a_t\\[5pt] s_t\end{pmatrix}=\begin{pmatrix} 1\;\;\;\;& L \\[5pt] 0\;\;\;\; & 1 \end{pmatrix} \begin{pmatrix} u_t \\[5pt] s_{t}\end{pmatrix}. \end{equation}

Notice that (i) the shock $s_t$ satisfies the identifying restrictions used in the empirical model: positive long-run effect and zero impact effect on $a_t$ ; and (ii) the representation above is invertible, that is can be estimated with a SVAR. This means that under this informational assumption $s_t$ is exactly the news shock identified in the SVAR of Section 2. As a result, the effects of the squared term in our empirical findings can be interpreted as effects attributable to uncertainty arising from news.

Summing up, our story unfolds as follows. Agents receive news about economic events and act on the basis of the value of the expected shock (first-order moment effect). However events, due to the fact that they are not seen with certainty, generate uncertainty. The larger is the event, the larger is the uncertainty. Uncertainty generates a contractionary demand-type effect possibly induced by a more cautionary behavior of the agents (second-order moment effect). The two effects combined yield an asymmetry in the effects of news shocks since uncertainty enhances the effects of bad news and mitigate the effects of good news.

It is important to stress that there might be other explanations for why bad news have larger effects. For instance, several works have pointed out that that agents tend to react more to bad news than good news, see Soroka (Reference Soroka2006) or simply bad news have a larger media coverage, see Soroka (Reference Soroka2012). These explanations and ours are of course not mutually exclusive.

Our results also have important implications for DSGE modeling. Second-moment effects, related to changes in conditional volatility, appear in higher order terms of the approximation of DSGE models, see Fernández-Villaverde et al. (Reference Fernàndez-Villaverde, Guerròn-Quintana, Kuester and Rubio-Ramírez2015a, Reference Fernàndez-Villaverde, Guerròn-Quintana and Rubio-Ramírez2015b). Here we show that, at least for the case of the news shock, these terms are important from an empirical point of view, stressing the importance of going beyond linearization to correctly describe fluctuations in macroeconomic variables.

4.3. Simulations

Now that we have a simple model in which uncertainty is generated by the squared news shocks, we come back to our econometric methodology. In this section, we ask the following question: is our method able to detect the first- and second-order effects of the news shock, as generated by the model? We use two simulations to assess our econometric approach.

The first simulation is designed as follows. Consider the simple model of Section 4.2. Assume that $[v_t \;\; s_t]^{\prime }\sim N(0,I)$ .Footnote ¹⁸ Under the assumption $\Delta a_t=\varepsilon _{t-1}$ , and recalling that $s_t=E_ta_{t+1}$ and that $u_t=s_{t-1}v_{t-1}$ is the forecast error, the invertible representation for $\Delta a_t$ is $\Delta a_t = s_{t-1} + u_t$ . We assume that there are two variables, $z_t=[z_{1t}\;z_{2t}]^{\prime }$ , following an MA process, which are affected by $s_t$ and $s_t^2$ . By putting together the fundamental representation for $\Delta a_t$ and the processes for $z_t$ , the data generating process is given by the following MA:

(5) \begin{equation} \begin{pmatrix} \Delta a_t \\[5pt] z_{1t} \\[5pt] z_{2t} \end{pmatrix} = \begin{pmatrix} 1\;\;\;\;\; & L\;\;\;\;\; & 0\\[5pt] 1+m_1L\;\;\;\;\; & 1+n_1L\;\;\;\;\; & 0\\[5pt] 1+m_2L\;\;\;\;\; & 1+n_2L\;\;\;\;\; & 1+p_2L\\[5pt] \end{pmatrix} \begin{pmatrix} u_{t} \\[5pt] s_{t} \\[5pt] w_t\end{pmatrix} \end{equation}

where $w_t=\frac{s^2_{t}-1}{\sigma _{s^2}}$ .

Figure 15. Impulse response functions of the two simulations. Left column: simulation 1. Right column: simulation 2. Solid line: point estimate. Gray area: 90% credible intervals. Red dashed line: true theoretical responses.

Simple MA(1) impulse response functions are chosen for the sake of tractability, but more complicated processes can be also considered. Using the following values $m_1=0.8,\;m_2=1,$ $n_1=0.6,\;n_2=-0.6,\;p_1=0.2,\;p_2=0.4$ , and drawing $[v_t\;\;s_t]$ , we generate 2000 artificial series of length $T=200$ . For each set of series, we estimate a VAR for $[\Delta a_t\;z_{1t}\;z_{2t} ]^{\prime }$ and identify $s_t$ as the second shock of the Cholesky representation. We define $\hat s_t$ as the estimate of $s_t$ obtained from the VAR. In a second step, using the same $2000$ realizations of $[u_t\; s_t\;s_t^2]^{\prime }$ , we generate another variable $\Delta y_t$ (which in the simulation plays the role of one of the variables of interest in the vector $Y_t$ ) asFootnote ¹⁹

\begin{equation*} \Delta y_t = u_t + [L + (1-L)(1+g_1 L)] s_{t} - (1-L)(1+f_1 L) w_t, \end{equation*}

where $g_1=0.7$ and $f_1=1.4$ . We estimate a VARX for $\Delta y_t$ using $\hat s_t^2$ and $\hat s_t$ as exogenous variables.

The second simulation is similar to the first, the only difference being that $w_t$ is an exogenous shock which does not depend on $s_t$ , which implies that the squared news shock has no effects on $z_t$ and $\Delta y_t$ . The values of the parameters are the same as before and $[v_t\;\;s_t\;\;w_t]^{\prime }\sim N(0,I)$ . We then estimate a VARX for $\Delta y_t$ using $\hat s_t^2$ and $\hat s_t$ as exogenous variables.

The results of simulation 1 are reported in the left column of Figure 15, while those of simulation 2 on the right column. The solid line is the mean of the 2000 responses, the gray area represents the 68% credible intervals, while the dashed red lines are the true theoretical responses. In both simulations, and in all cases, our approach succeeds in correctly estimating the true effects of news and uncertainty shock, the theoretical responses essentially overlapping with the mean estimated effects. When none of the variables is driven by uncertainty, our procedure consistently estimates a zero effect.