 # Understanding and Solving the Structural Vector Autoregressive Identification Problem

### Introduction

The structural vector autoregressive model is a crucial time series model used to understand and predict economic impacts and outcomes. In this blog, we look closely at the identification problem posed by structural vector autoregressive models and its solution. In particular, we cover:

1. What is the structural VAR model and what is the reduced form VAR?
2. What is the relationship between structural VAR and reduced form VAR models?
3. What is the structural VAR identification problem?
4. What are common solutions to the structural VAR identification problem?

## Structural VAR versus reduced form VAR

The first step to understanding restrictions in structural vector autoregressive (SVAR) models is understanding the differences and links between the SVAR model and reduced form VAR model.

The reduced form VAR model considers each variable to be a function of:

• Its own past values.
• The past values of other variables in the model.

Reduced form VAR models:

However, a key issue with reduced form VAR models is that it is usually impossible to disentangle what impact a sudden change in one variable will have on the other variables in the model.

Structural VAR models:

• Allow us to examine the causal relationships between variables.
• Use economic theory to add structural restrictions to the VAR model.
• Can be used to examine the impact individual shocks will have on other variables.

### The Relationship Between SVAR and Reduced Form VAR Models

Let’s take a closer look at the mathematics of the structural VAR and reduced form VAR models. We will do this in a simple bivariate model with two endogenous variables, $Y_1$ and $Y_2$.

Suppose we believe that $Y_1$ and $Y_2$ can both be modeled using:

1. Past observations of $Y_1$ and $Y_2$ going back one period.
2. Random shocks to each variable, $\epsilon_{1,t}$ and $\epsilon_{2,t}$.

Mathematically we can represent this in a two-equation system:

$$y_{1,t} = \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + b_{11}\epsilon_{1,t} + b_{12}\epsilon_{2,t}$$ $$y_{2,t} = \phi_{21} y_{1,t-1} + \phi_{22} y_{2,t-1} + b_{21}\epsilon_{1,t} + b_{22}\epsilon_{2,t}$$

This structural VAR model includes separate contemporaneous shocks to each variable, $\epsilon_{1,t}$ and $\epsilon_{2,t}$. These shocks are:

• Unobservable and zero-mean white noise processes.
• Serially uncorrelated and independent of each other.

In this model the matrix

$$B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$$

captures the structural impacts the shocks $\epsilon_{1,t}$ and $\epsilon_{2,t}$ have on the endogenous variables $Y_1$ and $Y_2$.

While this may look like a straightforward system of equations, remember that $\epsilon_{1,t}$ and $\epsilon_{2,t}$ are unobserved, leaving us unable to estimate $B$.

This is where the reduced form VAR comes into play. To see this, let's combine the "shock" components of each equation such that we define:

$$u_{1,t} = b_{11}\epsilon_{1,t} + b_{12}\epsilon_{2,t}$$ $$u_{2,t} = b_{21}\epsilon_{1,t} + b_{22}\epsilon_{2,t}$$

Now our two-equation system becomes becomes a reduced form VAR model:

$$y_{1,t} = \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + u_{1,t}$$ $$y_{2,t} = \phi_{21} y_{1,t-1} + \phi_{22} y_{2,t-1} + u_{2,t}$$

We can use OLS to estimate our unknown parameters in the reduced form VAR model

$$\Phi= \begin{bmatrix} \phi_{11} & \phi_{12} \\ \phi_{21} & \phi_{22} \end{bmatrix}$$

However, the residuals from these estimates do not allow us to determine the impacts of the shocks $\epsilon_{1,t}$ and $\epsilon_{2,t}$ on $Y_1$ and $Y_2$.

## Why do we need SVAR restrictions?

To back out the impacts of the shocks $\epsilon_{1,t}$ and $\epsilon_{2,t}$ on $Y_1$ and $Y_2$ from our reduced form model, it is natural to begin at the relationship

$$u_{1,t} = b_{11}\epsilon_{1,t} + b_{12}\epsilon_{2,t}$$ $$u_{2,t} = b_{21}\epsilon_{1,t} + b_{22}\epsilon_{2,t}$$

or in matrix form

$$U_t = B \epsilon_t$$

From this relationship we can derive, using linear algebra and a few statistical relationships, the identity that is at the heart of implementing structural VAR relationships:

$$\Sigma_u = BB'$$

where $\Sigma_u$ is the covariance matrix of the reduced form residuals:

$$\Sigma_u = {\mathbb E}\left[ u_t' u_t \right]$$

To see the issue with this, let's again consider what this implies in our two-variable system. First, note that $\Sigma_u$ is the covariance matrix of the residuals from our reduced form model:

$$\Sigma_u = \begin{bmatrix} \sigma^2_{11} & \sigma^2_{12} \\ \sigma^2_{21} & \sigma^2_{22} \end{bmatrix}$$

This means that $\Sigma_u = BB'$ is equivalent to

$$\begin{bmatrix} \sigma^2_{11} & \sigma^2_{12} \\ \sigma^2_{21} & \sigma^2_{22} \end{bmatrix} = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \begin{bmatrix} b_{11} & b_{21} \\ b_{12} & b_{22} \end{bmatrix}$$

which we can further expand to a system of equations

\begin{aligned} \sigma^2_{11} &= b^2_{11} + b^2_{12} \\ \sigma^2_{12} &= b_{11}b_{21} + b_{12}b_{22} \\ \sigma^2_{21} &= b_{11}b_{21} + b_{12}b_{22} \\ \sigma^2_{22} &= b^2_{21} + b^2_{22} \end{aligned}

Note though, that $\sigma^2_{12} = \sigma^2_{21}$. This means we have only 3 unique equations but 4 unknowns - this makes the model under-identified. This is the identification problem posed by structural VAR models.

To solve this problem, we need more equations -- these equations come in the form of restrictions.

## How to pick restrictions

Identifying restrictions can take many forms such as:

• No short-run effects.
• No long-run impacts.
• Sign restrictions.

Picking restrictions to identify SVAR models can seem daunting. However, the guiding factor in determining restrictions should always be the theoretical background.

For example, when modeling the impacts of monetary policy on real GDP, the theory of money neutrality implies that monetary policy has no cumulative long-run impacts on real GDP.

### Example SVAR Studies

Authors Topic Restrictions
Lutz Killian (2009) Oil price determination
1. World demand and oil prices do not have short-run (within one month) impact on production.
2. World demand is affected in the short-run by oil production but not by oil prices.
3. Oil prices respond immediatly to oil production and world demand.
Stock and Watson’s (2001) The effects of monetary policy shocks
1. Inflation depends only on past observations of other variables.
2. Contemporaneous federal funds rate does not effect unemployment but contemporaneous inflation does.
3. Contemporaneous inflation and unemployment both inform the funds rate.
Paresh Kumar Narayan, Seema Narayan, and Arti Prasad (2008) The relationship between electricity consumption and real GDP.
1. No short-run relationship between real GDP and electricity consumption.

## Common identification schemes

Picking restrictions is made easier by the fact that economic theory often implies common identification schemes like:

1. Zero short-run restrictions.
2. Zero long-run restrictions.
3. Sign restrictions.

### Zero short-run restrictions (Cholesky identification)

This identification scheme assumes that some shocks have no contemporaneous effect on one or more of the endogenous variables. For example, we may believe that shocks to monetary policy do not have an immediate impact on aggregate demand.

How do we implement zero short-run restrictions? Recall our previous bivariate structural VAR representation:

$$y_{1,t} = \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + b_{11}\epsilon_{1,t} + b_{12}\epsilon_{2,t}$$ $$y_{2,t} = \phi_{21} y_{1,t-1} + \phi_{22} y_{2,t-1} + b_{21}\epsilon_{1,t} + b_{22}\epsilon_{2,t}$$

If we believe that shocks to $y_2$ have no contemporaneous impacts on $y_1$ this implies that $b_{12}=0$.

Put in the context of our $B$ matrix: $$B = \begin{bmatrix} b_{11} & 0 \\ b_{21} & b_{22} \end{bmatrix}$$

Notice that we can order our variables such that the $B$ matrix is lower triangular. This allows us to use the Cholesky decomposition of $\Sigma_u$ for estimation.

### Zero long-run restrictions (BQ restrictions)

This identification scheme is built on the theory that some shocks have no long-run cumulative effects on one or more of the endogenous variables. Recall my example earlier about, the economic theory of money neutrality and the implication that monetary policy has no long-run effects on output.

How do we implement zero long-run restrictions? To understand how to implement zero long-run restrictions we first need to look at how shocks accumulate over time.

To start, let's rewrite our bivariate model:

$$y_{1,t} = \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + b_{11}\epsilon_{1,t} + b_{12}\epsilon_{2,t}$$ $$y_{2,t} = \phi_{21} y_{1,t-1} + \phi_{22} y_{2,t-1} + b_{21}\epsilon_{1,t} + b_{22}\epsilon_{2,t}$$

in matrix form

$$Y_t = \Phi Y_{t-1} + B \epsilon_t$$

Now let's consider time period $T+1$:

$$Y_{t+1} = \Phi Y_t + B \epsilon_{t+1}$$

Substituting for $Y_t$, this becomes:

\begin{aligned}Y_{t+1} &= \Phi (\Phi Y_{t-1} + B \epsilon_t) + B \epsilon_{t+1} \\&= \Phi^2 Y_{t-1} + \Phi B \epsilon_t + B \epsilon_{t+1}\end{aligned}

We can do the same for time period $T+2$: \begin{aligned}Y_{t+2} &= \Phi Y_{t+1} + B \epsilon_{t+2} \\&= \Phi (\Phi^2 Y_{t-1} + \Phi B \epsilon_t + B \epsilon_{t+1}) + B \epsilon_{t+2} \\&= \Phi^3 Y_{t-1} + \Phi^2 B \epsilon_t + \Phi B \epsilon_{t+1} + B \epsilon_{t+2}\end{aligned}

If we focus just on the impact of $\epsilon_t$ in each time period we find that:

Time period Impact of $\epsilon_t$
$T$ $B \epsilon_t$
$T+1$ $\Phi B \epsilon_t$
$T+2$ $\Phi^2 B \epsilon_t$
$\vdots$ $\vdots$
$T+S$ $\Phi^S B \epsilon_t$
$\vdots$ $\vdots$
$T+\infty$ $\Phi^\infty B \epsilon_t$

The long-run cumulative impact then, is equal to the sum of all these impacts: $$B \epsilon_t + \Phi B \epsilon_t + \Phi^2 B \epsilon_t + \ldots + \Phi^\infty B \epsilon_t = \sum_{i=0}^\infty \Phi^i B \epsilon_t$$

As long as our data is stationary, we can simplify this using the fact that

$$\sum_{i=0}^\infty \Phi^i = (1 - \Phi)^{-1}$$

which implies that our cumulative long-run effects of $\epsilon_t$ are given by

$$C = (1 - \Phi)^{-1} B$$

To implement zero long-run restrictions we set some members of the $C$ matrix to be zero.

### Sign restrictions

The intuition behind sign restriction is very simple -- sign restrictions imply that some shocks cause only increases or decreases in one or more of the endogenous variables.

Sign restrictions are commonly used in oil price modeling. For example, Kilian and Murphy (2012) use an SVAR model to explore the effects of shocks to oil supply, aggregate demand, and oil-specific demand.

They are able to identify the structural impacts using sign restrictions:

Shock Global oil production Real economic activity Real price of oil
Decrease in oil supply $-$ $-$ $+$
Increase in aggregate demand $+$ $+$ $+$
Increase in oil-specific demand $+$ $-$ $+$

How do we implement sign restrictions? Sign restrictions are implemented using an iterative guess-and-check method to identify a rotation matrix, $Q$ that can be used to construct the $B= PQ$ matrix. Note that $P$ is the Cholesky decomposition of $\Sigma_u$ from the previously discussed short-run restrictions.

Acceptable $Q$ solutions can be found following the algorithm:

1. Draw a random $Q$ matrix.
2. Compute $P$ as the Cholesky decomposition of $\Sigma_u$.
3. Compute $B$ and the shock impacts associated with $B$.
4. Check that the impacts meet the chosen sign restrictions. If yes, keep $Q$. If, no, discard $Q$.

### Summary of common SVAR identification regimes

Method Description Implementation
Zero contemporaneous effects. Assumes that some shocks at time $t$ have no impact on one or more of the endogenous variables at time $t$.
• Restrict upper diagonal elements of the $B$ matrix to zero.
• $B$ can be found using $B = cholesky(\Sigma_u)$.
Zero long-run effects. Assumes that some shocks at time $t$ have no long-run cumulative impacts on one or more of the endogenous variables.
• Restrict upper diagonal elements of $C = ((1 - \Phi)^{-1})B$ to zero.
• Define $\Omega = CC'= ((1 - \Phi)^{-1})B ((1 - \Phi)^{-1})B)'$.
• $C$ can be found using $C = cholesky(\Omega)$.
• $B = (1 - \Phi)C$.
Sign restrictions. Assumes that some shocks cause either strictly positive or negative impacts on one or more of the endogenous variables.
• Implemented by finding a rotation matrix $Q$ using an iterative guess and check method.
• Draw a random $Q$ matrix.
• Compute $P$ as the Cholesky decomposition of $\Sigma_u$.
• Compute $B$ and the shock impacts associated with $B$.
• Check that the impacts meet the chosen sign restrictions. If yes, keep $Q$. If, no, discard $Q$.

### Conclusion

In this blog, we've looked in-depth at the identification problem in the structural VAR model. After reading this blog you should have a better understanding of:

1. The difference between the structural VAR model and the reduced form VAR model.
2. The relationship between the structural VAR and reduced form VAR models.
3. The structural VAR identification problem.
4. Common solutions to the structural VAR identification problem.

1. Introduction to the Fundamentals of Time Series Data and Analysis
2. Introduction to the Fundamentals of Vector Autoregressive Models
3. The Intuition Behind Impulse Response Functions and Forecast Error Variance Decomposition
4. Introduction to Granger Causality

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