VAR

VAR models differ from univariate autoregresive models because they allow feedback to occur between the variables in the model. For example, we could use a VAR model to show how real GDP is a function of policy rate and vice versa.

Steps in VAR Modeling

1. Specifying and estimating a VAR model.
2. Using inferences to check and revise the model (as needed)
3. Forecasting
4. Structural analysis.

Reduced Form

consider each variable to be a function of

• its own past values
• the past values of other variables in the model

cons:

• contemporaneous variables are not related to one another
• the error terms will be correlated across equations: we annot consider what impacts individual shocks will have on the system

Recusrive VAR models

contain all the components of the reduced form model, but also allow some variables to be functions of other concurrent variables.

Structural VAR

include restrictions that allow us to identify causal relationships beyond those that can be identified with reduced form or recursive models.

What makes up a VAR model?

A VAR model is made up of a system of equations that represents the relationships between multiple variables. When referring to VAR models, we often use special language to specify:

How many endogenous variables there are included. How many autoregressive terms are included. For example, if we have two endogenous variables and autoregressive terms, we say the model is a Bivariate VAR(2) model. If we have three endogenous variables and four autoregressive terms, we say the model is a Trivariate VAR(4) model.

Forecasting

One of the most important functions of VAR models is to generate forecasts. Forecasts are generated for VAR models using an iterative forecasting algorithm:

1. Estimate the VAR model using OLS for each equation.
2. Compute the one-period-ahead forecast for all variables.
3. Compute the two-period-ahead forecasts, using the one-period-ahead forecast.
4. Iterate until the h-step ahead forecasts are computed.

Structural analysis

begins with the structural vector autoregression (SVAR) which applies restrictions that allow us to identify the impacts that exogeous shocks have on the variables in the system.

Once the SVAR model is estimated, impulse response functions and forecast error variance decomposition are two of the most important structural analysis tools for examining those impacts.