2.1 Model
We investigate the international spillovers of technology and sentiment shocks using a structural vector autoregression (SVAR) framework. We follow the approach proposed initially by Uhlig (
2004) and applied in the confidence literature by Barsky and Sims (
2011), Levchenko and Pandalai-Nayar (
2020) and Angeletos et al. (
2018) and identify the structural shocks by imposing so-called medium-run restrictions on the impact matrix.
Our basic VAR model includes seven variables for the large economy (euro area): total factor productivity (TFP), real GDP, hours worked, short term nominal interest rate, investments, private consumption and GDP forecasts and GDP for the small open economy (Poland) - in this order. We identify three structural shocks in the model, all stemming from the euro area, which also affect Poland as the economy tightly integrated with the euro area.
The method we apply relies on the sequential identification of the subsequent shocks. We extract the respective shocks conditional on the values of the previous shocks. As a first step, we extract two technology shocks in the euro area in the spirit of Barsky and Sims (
2011). The first one will be called surprise technology shock and corresponds to the reduced form innovation to the TFP equation in the VAR model with the TFP variable ordered first. The second one is a news shock about future TFP which we identify as having no contemporaneous impact on TFP but explaining the maximum of the forecast error variance of the TFP series after accounting for the impact of the surprise technology shock. This approach reflects the assumption that TFP in the euro area is affected by these two shocks only:
$$\begin{aligned} \hbox {TFP}_{t}=\lambda _{1}^{\mathrm{TFP}}\epsilon _{t}^{\mathrm{sur}}+\lambda _{2}^{\mathrm{TFP}}\epsilon _{t-s}^{\mathrm{news}} \end{aligned}$$
(1)
where
\(\hbox {TFP}_{t}\) is TFP in the euro area and
\(\epsilon _{t}^{\mathrm{sur}}\) and
\(\epsilon _{t}^{\mathrm{news}}\) are the surprise and news technology shocks, respectively.
Finally, we identify the sentiment shock in the euro area. Our identification procedure is closely linked to the method proposed by Levchenko and Pandalai-Nayar (
2020) who investigate the spillovers of confidence shocks from US to Canada. Like in this approach we identify sentiment as the shock which maximizes the forecast error variance of the GDP forecasts after accounting for surprise and news technology shocks:
$$\begin{aligned} \hbox {GDP}_{t}^{F,\mathrm{EA}}=\lambda _{1}^{F}\epsilon _{t}^{\mathrm{sur}}+\lambda _{2}^{F}\epsilon _{t-s}^{\mathrm{news}}+\lambda _{3}^{F}\epsilon _{t}^{\mathrm{sent}}+\zeta _{t} \end{aligned}$$
(2)
where
\(\epsilon _{t}^{\mathrm{sent}}\) is the sentiment shock in the euro area, while
\(\zeta _{t}\) is another shock (or combination of structural shocks) affecting the expectations of future economic activity in the euro area
\(\hbox {GDP}_{t}^{F,\mathrm{EA}}\) not related to technology or sentiment. Hence, our approach does not exclude that some other shocks may also affect agents expectations about future economic activity.
In what follows, the identification procedure is described in detail. We start with the reduced form VAR(
p) model:
$$\begin{aligned} A(L)Y_{t}=u_{t} \end{aligned}$$
(3)
where
\(Y_{t}\) is the
\(k\times 1\) vector of observable variables in levels and
\(u_{t}\) is a vector of reduced form disturbances.
The moving average representation of model (
3) is:
$$\begin{aligned} Y_{t}=B(L)u_{t}. \end{aligned}$$
(4)
We assume that the reduced form disturbances
\(u_{t}\) are linear combinations of structural shocks
\(\epsilon _{t}\) with the impact matrix
\(C_{0}\):
$$\begin{aligned} u_{t}=C_{0}\epsilon _{t} \end{aligned}$$
(5)
Therefore the structural representation of the VAR(
p) model is:
$$\begin{aligned} Y_{t}=C(L)\epsilon _{t} \end{aligned}$$
(6)
where
\(C(L)=B(L)\cdot C_{0}\). We assume that the structural shocks
\(\epsilon _{t}\) are orthogonal to each other and have unit variance, which implies that:
$$\begin{aligned} C_{0}C_{0}^{\prime }=\varSigma \end{aligned}$$
(7)
where
\(\Sigma \) is the covariance matrix of reduced form innovations
\(u_{t}\).
As is well-known there is an infinite number of matrices satisfying condition (
7). For example the Cholesky decomposition of
\(\Sigma \) provides a lower triangular matrix which fulfills condition (
7) and this matrix, denoted as
\(\tilde{C}_{0}\) is the starting point for the structural decomposition with medium run restrictions.
As a next step we specify matrix D, which satisfies the restriction \(DD^{\prime }=I\) and which defines the impact matrix \(C_{0}\) as \(C_{0}=\tilde{C}_{0}D\).
The subsequent columns of matrix D correspond to the identified structural shocks. The identification of the respective columns of matrix D is based on the assumption that the structural shocks \(\epsilon _{t}\) explain the maximum variance of the forecast error of selected variables in the VAR(p) model. Below we discuss the subsequent steps of our decomposition.
The h-step ahead forecast error from the VAR(
p) model can be derived as:
$$\begin{aligned} Y_{t+h}-\hat{Y}_{t}(h)=\sum _{\tau =0}^{h}B_{\tau }u_{t+h-\tau }=\sum _{\tau =0}^{h}B_{\tau }C_{0}\epsilon _{t+h-\tau }=\sum _{\tau =0}^{h}B_{\tau }\tilde{C}_{0}D\epsilon _{t+h-\tau } \end{aligned}$$
(8)
where
\(\hat{Y}_{t}(h)\) is the h-step ahead forecast of
\(Y_{t}\) while
\(B_{\tau }\) is the respective coefficient matrix in the moving average representation of VAR(
p).
Accordingly the h-step ahead forecast error of variable
i in vector
\(Y_{t}\) is:
$$\begin{aligned} Y_{i,t+h}-\hat{Y}_{i,t}(h)=\sum _{\tau =0}^{h}B_{i,\tau }\tilde{C}_{0}D\epsilon _{t+h-\tau } \end{aligned}$$
(9)
where
\(B_{i,\tau }\) is the
ith row of matrix
\(B_{\tau }\). Then the forecast error variance of variable
i at horizon
h is:
$$\begin{aligned} \varOmega _{i}(h)=\sum _{\tau =0}^{h}B_{i,\tau }\varSigma B_{i,\tau }^{\prime }. \end{aligned}$$
(10)
Let
\(\varOmega _{i,j}(h)\) denote the contribution of the structural shock
j to the forecast error variance of variable
i at horizon
h:
$$\begin{aligned} \varOmega _{i,j}(h)=\frac{\sum _{\tau =0}^{h}B_{i,\tau }\tilde{C}_{0}d_{j}d_{j}^{\prime }\tilde{C_{0}^{\prime }}B_{i,\tau }^{\prime }}{\sum _{\tau =0}^{h}B_{i,\tau }\varSigma B_{i,\tau }^{\prime }} \end{aligned}$$
(11)
where
\(d_{j}\) is the
jth column of matrix
D.
Without loss of generality let us assume that the first two structural shocks are the euro area surprise and news technology shocks and the third one is the euro area sentiment shock. The baseline of the identification proposed by Barsky and Sims (
2011) and adopted in our paper is the assumption expressed by (
1) that only two technology shocks influence TFP for the euro area. This assumption implies:
$$\begin{aligned} \varOmega _{1,1}(h)+\varOmega _{1,2}(h)=1\;\forall h. \end{aligned}$$
(12)
The surprise technology shock is the reduced form innovation in the TFP equation in model (
3), while the news technology shock is the shock, which maximizes the forecast error variance of TFP over
\(H^{\mathrm{news}}\) horizon after accounting for the impact of the surprise technology shock.
The maximization problem can be written as follows (see Barsky and Sims
2011):
$$\begin{aligned} d_{2}=\mathrm {argmax}\sum _{h=0}^{H^{\mathrm{news}}}\varOmega _{1,2}(h)=\mathrm {argmax}\sum _{h=0}^{H^{\mathrm{news}}}\left( \frac{\sum _{\tau =0}^{h}B_{1,\tau }\tilde{C}_{0}d_{2}d_{2}^{\prime }\tilde{C_{0}^{\prime }}B_{1,\tau }^{\prime }}{\sum _{\tau =0}^{h}B_{1,\tau }\varSigma B_{1,\tau }^{\prime }}\right) \end{aligned}$$
(13)
s.t.
$$\begin{aligned}&\tilde{C}_{0}(1,j)=0\;\forall j\ne 1\\&d_{2}(1)=0\\&D^{\prime }D=I \end{aligned}$$
where
\(d_{2}\) is the second column of
D matrix, which specifies the second structural shocks interpreted here as the news technology shock. Therefore
\(\tilde{C}_{0}d_{2}\) is the impact vector of this shock. The first two restrictions guarantee that the news shock does not have a contemporaneous effect on TFP. The third constraint ensures that vector
\(d_{2}\) is a column of an orthonormal matrix.
Uhlig (
2004) shows that the maximization problem defined by (
13) is equivalent to finding the eigenvector (which is a non-zero portion of
\(d_{2}\)) associated with the largest eigenvalue of the lower
\(\left( k-1\right) \times \left( k-1\right) \) submatrix of matrix
\(\Lambda ^{\mathrm{news}}\), which is a weighted sum of the matrices
\(\left( B_{1,\tau }\tilde{C}_{0}\right) ^{\prime }\left( B_{1,\tau }\tilde{C}_{0}\right) \) over
\(H^{\mathrm{news}}\):
$$\begin{aligned} \Lambda ^{\mathrm{news}}=\sum _{\tau =0}^{H^{\mathrm{news}}}\left( H^{\mathrm{news}}+1-\max \left( 1,\tau \right) \right) \left( B_{1,\tau }\tilde{C}_{0}\right) ^{\prime }\left( B_{1,\tau }\tilde{C}_{0}\right) . \end{aligned}$$
(14)
Next, we identify the euro area sentiment shock, assumed to maximize the remaining forecast error variance of the euro area GDP forecast over
\(H^{\mathrm{sent}}\) horizons after accounting for the contribution of surprise and news technology shocks. The forecast horizon set for the identification of the sentiment shock is assumed to be shorter than the horizon chosen to identify the technology shocks since the impact of the sentiment shock on GDP is supposed to be temporary (this is a demand shock). As already mentioned we assume that the euro area GDP forecast is ordered seventh in the VAR(
p) model while the sentiment shock is the third structural shock. It is worth to note that the identification of the sentiment shock does not alter two technology shocks specified in the previous step. Thus the contribution of these shocks to the forecast error variance of GDP forecast is fixed for all horizons.
To identify the sentiment shock we derive vector
\(d_{3}\) by solving the following equation:
$$\begin{aligned} d_{3}=\mathrm {argmax}\sum _{h=0}^{H^{\mathrm{sent}}}\varOmega _{7,3}(h)=\mathrm {argmax}\sum _{h=0}^{H^{\mathrm{news}}}\left( \frac{\sum _{\tau =0}^{h}B_{7,\tau }\tilde{C}_{0}d_{3}d_{3}^{\prime }\tilde{C_{0}^{\prime }}B_{7,\tau }^{\prime }}{\sum _{\tau =0}^{h}B_{7,\tau }\varSigma B_{7,\tau }^{\prime }}\right) \end{aligned}$$
(15)
s.t.
$$\begin{aligned} \tilde{C}_{0}(1,j)=0\;\forall j\ne 1\\ D\left( :,\,2\right) =\hat{d}_{2}\\ D^{\prime }D=I. \end{aligned}$$
The vector
\(d_{3}\) defining the euro area sentiment shock is thus the third column of matrix
D. We solve Eq. (
15) subject to the constraints that the second column of matrix
D is fixed and equal to the impact vector corresponding to the news shock
\(\hat{d}_{2}\) identified in the previous step. Numerically we find vector
\(d_{3}\) by proceeding as follows:
1.
We form a matrix
\(D^{\mathrm{news}}=\left[ \begin{array}{c@{\quad }c} 1 &{} 0\\ 0 &{} \tilde{D}^{\mathrm{news}} \end{array}\right] \), where the subsequent columns of matrix
\(\tilde{D}^{\mathrm{news}}\) are the eigenvectors associated with the eigenvalues (set in descending order) being the solution to problem (
13).
2.
We derive matrix
\(\Lambda ^{\mathrm{sent}}\), as:
$$\begin{aligned} \Lambda ^{\mathrm{sent}}=\sum _{\tau =0}^{H^{\mathrm{sent}}}\left( H^{\mathrm{sent}}+1-\max \left( 1,\tau \right) \right) \left( B_{7,\tau }\tilde{C}_{0}D^{\mathrm{news}}\right) ^{\prime }\left( B_{7,\tau }\tilde{C}_{0}D^{\mathrm{news}}\right) . \end{aligned}$$
(16)
3.
We calculate the eigenvectors corresponding to the eigenvalues of the lower \(\left( k-2\right) \times \left( k-2\right) \) submatrix of matrix \(\Lambda ^{\mathrm{sent}}\). These eigenvectors are set to be the subsequent columns of \(\left( k-2\right) \times \left( k-2\right) \) matrix \(\tilde{D}^{\mathrm{sent}}\).
4.
We derive a \(k\times k\) matrix \(D^{\mathrm{sent}}=D^{\mathrm{news}}\left[ \begin{array}{c@{\quad }c@{\quad }c} 1 &{} 0 &{} 0\\ 0 &{} 1 &{} 0\\ 0 &{} 0 &{} \tilde{D}^{\mathrm{sent}} \end{array}\right] \).
5.
The vector \(d_{3}\) which corresponds to the euro area sentiment shock is the third column of \(D^{\mathrm{sent}}\) matrix.
Angeletos et al. (
2018) construct and estimate a structural DSGE model with structural sentiment shocks and show that those are closely related to a main business cycle shock extracted from a VAR model. The latter is identified using a sequential identification scheme (by maximizing the variance of output and hours worked). The paper shows that the empirical and structural shocks share the same properties (in terms of impulse responses they generate) and have very similar time series. Given the relationship to theory the properties of the empirical shock evidenced by Angeletos et al. (
2018) are exactly what we expect from our extracted sentiment shock. After our estimation has been done, we carefully check whether our estimated shock shares these properties.