Structural Vector Autoregression (SVAR)

While standard Vector Autoregression (VAR) models excel at capturing temporal dependencies and forecasting multiple time series simultaneously, and Granger causality tests assess predictive influence, they often fall short of identifying structural causal relationships. In a VAR model, $Y_t = c + \sum_{i=1}^p A_i Y_{t-i} + u_t$, the reduced-form errors ($u_t$) are typically contemporaneously correlated ($E[u_t u_t'] = \Sigma_u$, where $\Sigma_u$ is non-diagonal). This correlation obscures the immediate, structural impact one variable has on another within the same time period. Attributing an observed correlation in errors to a specific causal link requires imposing additional structure.

Structural Vector Autoregression (SVAR) extends the VAR framework precisely to address this challenge. It allows us to move from mere forecasting or predictive causality to estimate the contemporaneous causal effects among variables, by using theoretical assumptions or domain knowledge about the underlying data generating process.

From Reduced Form to Structural Form

The core idea of SVAR is to posit a structural model related to the reduced-form VAR. Consider a system with $k$ endogenous variables represented by the vector $Y_t$. The SVAR model can be written as:

$$B_0 Y_t = k + \sum_{i=1}^p B_i Y_{t-i} + \epsilon_t$$

Here:

• $Y_t$ is the $k \times 1$ vector of variables at time $t$.
• $B_0$ is a $k \times k$ matrix capturing the contemporaneous relationships between the variables. The off-diagonal elements of $B_0$ represent the direct causal impact of variable $j$ on variable $i$ at time t.
• $k$ is a vector of intercepts.
• $B_i$ are $k \times k$ matrices of coefficients for lagged variables.
• $\epsilon_t$ is a $k \times 1$ vector of structural shocks (or innovations). These are assumed to be mutually uncorrelated, representing the underlying exogenous economic or system disturbances. Typically, their covariance matrix $E[\epsilon_t \epsilon_t'] = \Sigma_\epsilon$ is assumed to be diagonal, often normalized to the identity matrix $I$ for simplicity.

We can relate the SVAR to the standard VAR by multiplying by $B_0^{-1}$ (assuming $B_0$ is invertible):

$$Y_t = B_0^{-1} k + \sum_{i=1}^p (B_0^{-1} B_i) Y_{t-i} + B_0^{-1} \epsilon_t$$

Comparing this to the reduced-form VAR, $Y_t = c + \sum_{i=1}^p A_i Y_{t-i} + u_t$, we see the mapping:

• $c = B_0^{-1} k$
• $A_i = B_0^{-1} B_i$
• $u_t = B_0^{-1} \epsilon_t$ (These are the reduced-form residuals)

The covariance matrix of the reduced-form residuals $\Sigma_u$ is related to the structural shocks $\epsilon_t$ and the contemporaneous effects matrix $B_0$:

$$\Sigma_u = E[u_t u_t'] = E[(B_0^{-1} \epsilon_t) (\epsilon_t' (B_0^{-1})')] = B_0^{-1} E[\epsilon_t \epsilon_t'] (B_0^{-1})' = B_0^{-1} \Sigma_\epsilon (B_0^{-1})'$$

If we assume $\Sigma_\epsilon = I$ (uncorrelated structural shocks with unit variance), then:

$$\Sigma_u = B_0^{-1} (B_0^{-1})' = (B_0' B_0)^{-1}$$

or equivalently,

$$B_0' B_0 = \Sigma_u^{-1}$$

The Identification Problem

The standard VAR estimation yields estimates for $A_i$ and $\Sigma_u$. The challenge in SVAR is to recover the structural parameters, particularly the contemporaneous effects matrix $B_0$ and the properties of the structural shocks $\epsilon_t$ (often just their variances if assumed diagonal), from the estimated $\Sigma_u$.

The equation $B_0' B_0 = \Sigma_u^{-1}$ (assuming $\Sigma_\epsilon = I$) involves $k^2$ unknown elements in $B_0$. However, since $\Sigma_u^{-1}$ is symmetric, this equation only provides $k(k+1)/2$ unique restrictions. To uniquely determine the $k^2$ elements of $B_0$, we need an additional $k^2 - k(k+1)/2 = k(k-1)/2$ restrictions. These restrictions must come from outside the model, typically based on economic theory, domain knowledge, or specific assumptions about the causal structure. This is known as the identification problem in SVAR.

Common Identification Strategies

Several strategies exist to impose the necessary restrictions to identify $B_0$:

1. Short-Run Restrictions (Recursive Ordering): This is perhaps the most common approach, pioneered by Christopher Sims. It assumes a recursive structure for the contemporaneous effects. Specifically, it imposes zero restrictions on certain elements of the $B_0$ matrix, making it lower (or upper) triangular. For a lower triangular $B_0$, the assumption is that the first variable in the system ($Y_{1t}$) is only affected contemporaneously by its own shock ($\epsilon_{1t}$), the second variable ($Y_{2t}$) is affected by $\epsilon_{1t}$ and $\epsilon_{2t}$, and so on. The last variable ($Y_{kt}$) can be affected by all structural shocks contemporaneously.

   • Mechanism: A lower triangular $B_0$ provides exactly the required $k(k-1)/2$ zero restrictions.
   • Implementation: This structure corresponds directly to the Cholesky decomposition of the estimated variance-covariance matrix $\Sigma_u$. If $P$ is the lower triangular Cholesky factor such that $P P' = \Sigma_u$, then setting $B_0^{-1} = P$ (up to scaling/sign of columns) satisfies $\Sigma_u = B_0^{-1} (B_0^{-1})'$.
   • Caveat: The identification depends crucially on the chosen ordering of variables. Different orderings imply different causal assumptions and can lead to substantially different results. The ordering should ideally be justified by strong theoretical arguments about the speed of responses.

   A recursive (lower triangular $B_0$) structure for $k=3$. Variable $Y_1$ only responds contemporaneously to shock $\epsilon_1$. $Y_2$ responds to $\epsilon_1, \epsilon_2$ and $Y_1$. $Y_3$ responds to all shocks and variables. Dashed lines indicate potential non-zero off-diagonal elements in $B_0$.

2. Long-Run Restrictions (Blanchard-Quah): Instead of restricting immediate effects ($B_0$), this approach imposes restrictions on the long-run effects of certain shocks. The long-run impact matrix is given by $\sum_{i=0}^\infty C_i$, where $C_i$ are the moving average (MA) coefficient matrices representing the impulse responses. Restrictions are placed such that certain shocks (e.g., supply shocks) have a permanent effect on some variables (e.g., output), while others (e.g., demand shocks) only have temporary effects. This requires assumptions about the long-run behavior of the system.

3. Sign Restrictions: This method uses qualitative theoretical knowledge. Instead of setting specific coefficients in $B_0$ or the long-run matrix to zero, it restricts the sign of the impulse responses of certain variables to specific shocks over a given horizon (e.g., a positive monetary policy shock should decrease inflation and output for some periods). This approach doesn't yield a unique $B_0$ but rather a set of possible models consistent with the sign restrictions. Inference is often done by analyzing the distribution of results across this set (e.g., using median responses and confidence bands). Computationally more demanding as it involves searching or sampling over possible rotation matrices.

4. Proxy SVAR / External Instruments: This approach uses external variables (proxies) that are correlated with a specific structural shock of interest but are uncorrelated with other structural shocks and do not directly affect the endogenous variables except through that specific shock. This is analogous to using instrumental variables (as discussed in Chapter 4) in a time-series context. Finding valid and strong proxies is the main challenge, but when available, this method can be effective for identifying specific shocks (e.g., using high-frequency financial data to proxy for monetary policy shocks).

Analyzing SVAR Output: IRFs and FEVD

Once the SVAR model is identified (i.e., $B_0$ is estimated), the primary tools for causal interpretation are:

• Structural Impulse Response Functions (IRFs): These trace out the dynamic effect of a one-unit (or typically, one-standard-deviation) structural shock ($\epsilon_{j,t}$) on the future path of all endogenous variables ($Y_{k, t+h}$ for $h=0, 1, 2, \dots$). Unlike reduced-form IRFs, which show responses to correlated residuals $u_t$, structural IRFs isolate the causal impact of distinct, uncorrelated underlying shocks. This allows for more meaningful causal narratives (e.g., "What is the effect of an unexpected 1% increase in the policy interest rate on inflation and unemployment over the next 24 months?").

  Example impulse responses showing the dynamic effect of a one standard deviation structural shock originating in variable Y2 on both Y1 and Y2 over 10 periods.

• Forecast Error Variance Decomposition (FEVD): FEVD decomposes the variance of the forecast error for each variable ($Y_k$) at different horizons ($h$) into proportions attributable to each of the structural shocks ($\epsilon_j$). It helps quantify the relative importance of different sources of variation or causal influences driving the fluctuations in each variable over time. For example, FEVD might show that 60% of the 12-month forecast error variance in inflation is due to supply shocks, 30% to demand shocks, and 10% to monetary policy shocks.

Practical Notes and Limitations

While powerful, SVAR analysis rests on several assumptions and faces limitations:

• Identification Validity: The credibility of SVAR results depends entirely on the validity of the chosen identification scheme (recursive, long-run, sign, proxy). These restrictions are often non-testable assumptions derived from theory. Sensitivity analysis, exploring alternative plausible identification schemes, is highly recommended.
• Stationarity: Basic SVAR assumes the time series are stationary. If variables are non-stationary and cointegrated, a Structural Vector Error Correction Model (SVECM) is required.
• Linearity: The relationships are assumed to be linear. Non-linear SVAR models exist but are more complex.
• Lag Length: Choosing the appropriate lag length $p$ for the underlying VAR is important. Standard information criteria (AIC, BIC, HQIC) are typically used, but checking residual autocorrelation is also needed.
• Parameter Uncertainty: IRFs and FEVDs are based on estimated parameters and thus have uncertainty. Confidence intervals (often generated via bootstrapping) should always be reported and considered.
• Dimensionality: The number of parameters in a VAR/SVAR ($k^2 p$ autoregressive coefficients plus the covariance matrix elements) grows quadratically with the number of variables $k$. Estimation becomes unreliable for large $k$. Regularization techniques (LASSO VAR) or Bayesian VARs (BVARs) might be necessary precursors in high-dimensional settings before imposing structural identification.

SVAR provides a framework for imposing causal structure onto multivariate time series models, enabling the estimation of contemporaneous effects and the analysis of dynamic causal impacts through structural shocks. Its identification relies heavily on external assumptions, making careful justification and sensitivity analysis essential components of any applied SVAR study. The insights gained, particularly through IRFs, can be valuable for understanding dynamic systems and informing policy or business decisions in contexts where temporal dynamics and feedback are present.