29 Structural VAR
29.1 The Structural Framework
Now we reconsider the problem of estimating structural shocks. We have introduced the structural shock framework in Chapter 15 the underlying econometric framework of understanding our economy. To briefly recap, we envision our economy as an MA process in which multiple structural shocks are the fundamental driving forces:
\[ y_t = \Theta(L) \epsilon_t \tag{29.1}\]
Structural shocks must be distinguished from residuals in a regression. Residuals are prediction errors based on past observations. Residuals can be cross-sectionally or serially correlated. Structural shocks are attached with specific economic meaning. They are also assumed to be unforeseeable and uncorrelated. We do not usually observe structural shocks but they are the conceptualized driving forces in the background.
If \(\Theta(L)\) is invertible, we would have
\[ \Theta^{-1}(L) y_t = \epsilon_t \]
which is an infinite order AR process. This motivates us to estimate structural shocks via vector autoregressive processes. Suppose we have a VAR process:
\[ A(L) y_t = u_t \tag{29.2}\]
where \(u_t = y_t - \text{Proj}(y_t | y_{t-1},y_{t-2},\dots)\) are the projection residuals. The question is, to what extend, or under what conditions, can we identify the structural shocks from this VAR specification?
The answer is easier than you might have thought. We only need to identify
\[ u_t = \Theta_0 \epsilon_t \tag{29.3}\]
where \(\Theta_0\) is the first coefficient matrix in the lag polynomial. That is, the condition for identification is that we can find a linear transformation to decompose \(u_t\) into \(\epsilon_t\).
29.2 Invertibility
The structural MA process is said to be invertible if \(\epsilon_t\) can be linearly determined from current and lagged values of \(y_t\):
\[ \epsilon_t = \text{Proj}(\epsilon_t| y_t, y_{t-1},\dots). \]
This means there is no “omitted variable” in the observable space, in the sense that the space spanned by \(\{\epsilon_t, \epsilon_{t-1},...\}\) is fully covered by \(\{y_t, y_{t-1},…\}\).
\[ \text{span}\{\epsilon_t,\epsilon_{t-1},...\} = \text{span}\{y_t, y_{t-1},...\} = \text{span}\{u_t, u_{t-1},...\} \]
This is a strong assumption. Under invertibility, the knowledge of the past true shocks would not even improve the the VAR forecast. But it does not require our VAR system being exhaustive, including everything observable variables in our economy. In a particular application, we would only be interested in a few structural shocks. The invertibility condition requires the observables fully cover the space spanned by the structural shocks of particular interests.
With the above condition satisfied, we can show that the identification problem is reduced to identify \(\Theta_0\). Given Equation 29.1 and Equation 29.2, we have
\[ u_t = A(L)y_t = A(L)\Theta(L)\epsilon_t \overset{?}= \Theta_0 \epsilon_t \]
By definition,
\[ \begin{aligned} u_t &= y_t - \text{Proj}[y_t | y_{t-1},y_{t-2},...] \\[1em] &=\Theta(L)\epsilon_t - \text{Proj}[\Theta(L)\epsilon_t | y_{t-1},y_{t-2},...]\\[1em] &=\Theta_0\epsilon_t + \Theta_1\epsilon_{t-1} +\cdots+ \text{Proj}[\Theta_0\epsilon_t + \Theta_1\epsilon_{t-1} + \cdots | y_{t-1},y_{t-2},...] \\ &= \Theta_0\epsilon_t - \Theta_0\underbrace{\text{Proj}[\epsilon_t|y_{t-1},...]}_{=0\ \text{by definition}} + \sum_{j=1}^{\infty}\Theta_j\{\epsilon_{t-j} - \underbrace{\text{Proj}[\epsilon_{t-j}|y_{t-1},...]}_{=\epsilon_{t-j}\ \text{by invertibility}}\} \\ &= \Theta_0\epsilon_t. \end{aligned} \]
Proposition 29.1 (Assumptions of Structural VAR)
- All variables are stationary;
- The space spanned by the innovations and the structural shocks coincide such that \(u_t = \Theta_0\epsilon_t\);
- The structural process \(y_t = \Theta(L)\epsilon_t\) is invertible.
Under the assumptions, identifying \(\Theta_0\) is equivalent to identify the structural shocks \(\epsilon_t = \Theta_0^{-1} u_t\).
In essence, structural identification is equivalent to sorting out the contemporaneously correlated residuals into uncorrelated shocks that can be attached to certain economic meanings. As we will see, the decomposition is largely subjective, according to researchers’ understanding of how structural shocks are correlated contemporaneously.
If the invertibility assumption fails, that means there exists no mapping from VAR residuals to the structural shocks. Non-invertibility arises when the observed variables fail to span the space of the state variables (structural shocks). If this is the case, we can include more variables to expand the space; or we may choose to simply ignore it if we believe the wedge between the spaces spanned by VAR residuals and structural shocks are small.
29.3 Identification
With invertibility, the essential task of SVAR is to decompose Equation 29.3 to recover the structural shocks. The key is to estimate \(\Theta_0\). Consider the second-order moments of Equation 29.3:
\[ \Theta_0\mathbb{E}(\epsilon_t\epsilon_t')\Theta_0' = \mathbb{E}(u_tu_t') \]
Estimating the VAR system Equation 29.2 by OLS gives \(\hat\Omega = \mathbb{\hat E}(u_tu_t')\). By definition, elements of \(\epsilon_t\) are orthogonal to each other, so \(D=\mathbb E (\epsilon_t\epsilon_t')\) is diagonal. Estimated \(\hat\Omega\) gives \(n(n+1)/2\) distinct values. Identification of \(D\) requires \(n\) values. So no more than \(n(n-1)/2\) parameters in \(\Theta_0\) can be identified. That means, we cannot identify the full \(n\times n\) matrix \(\Theta_0\) without restrictions.
29.3.1 Recursive restriction
One common way to impose restrictions on \(\Theta_0\) is to require it being lower triangular. Thus eliminating \(n(n-1)/2\) entries. We also assume the structural shocks have the same magnitude as the residuals, so the diagonal entries are \(1\)s. For example, in the three variable Keynesian system, we may assume
\[ \begin{bmatrix} u_t^\pi \\ u_t^y \\ u_t^m \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ * & 1 & 0 \\ * & * & 1 \end{bmatrix} \begin{bmatrix} \epsilon_t^{S} \\ \epsilon_t^{IS} \\ \epsilon_t^{MP} \end{bmatrix} \]
The recursive structure is equivalent to imposing restrictions on the contemporaneous relationships between variable, or imposing different reaction speed to the structural shocks. In the above example, we assume the observed monetary policy (interest rate) responds to IS shock, supply shock and monetary policy shock contemporaneously; but inflation and output respond to monetary policy shock with a lag (sluggish response). Output responds to IS shock and supply shock contemporaneously, but inflation responds to IS shock with a lag. Of course, one can question the validity of these assumptions, or even the validity of the conceptualization of the three structural shocks. But this is a structural question, not an econometric one. Economists have always been debating what are the proper structures to describe the economy.
In the recursive identification scheme, the ordering of the variables is the vital decision to make. Typically, the slow-moving variables are ordered first, and the fast-moving variables last, provided the \(\Theta_0\) is upper triangular. In the literature involving monetary policy, a “slow-r-fast” scheme is widely adopted. That is, low-moving variables such as real output and price levels are ordered before interest rates; and fast-moving variables such as financial market indexes are ordered after interest rates. Because financial market absorbs information in real time, even ahead of the monetary policy decision. But it take time for real variables to materialize the impact of monetary policy changes.
29.3.2 Non-recursive restriction
We may also impose non-recursive structure based on theories or intuitions. Consider a model constituted of the demand and supply of an agriculture product, and the weather condition that affects the supply of the product. We assume weather does not depend on market behaviors. In addition, the supply but not the demand is influenced by the weather. This results in an identification matrix as follows
\[ \begin{bmatrix} u_t^d \\ u_t^s \\ u_t^w \end{bmatrix} = \begin{bmatrix} 1 & -\beta & 0 \\ 1 & -\gamma & -\delta \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \epsilon_t^{d} \\ \epsilon_t^{s} \\ \epsilon_t^{w} \end{bmatrix} \]
Note that there are only three parameters to be estimated in the matrix. So \(\Theta_0\) in this case can also be identified.
Structural VAR literature has invented lots of identification schemes, such as long-run restrictions, sign restrictions, and so on. These are left for the readers to explore themselves.