What is Vector autoregression

Understanding Vector Autoregression in Time Series Analysis

Vector autoregression (VAR) is a statistical model for analyzing the relationship between multiple time series variables. It is commonly used in econometrics, finance, and other fields that deal with time series data. VAR models are versatile and can be used to forecast future values, estimate causal relationships between variables, and conduct model diagnostics.

This article aims to provide an overview of VAR models, including how they work, when to use them, and the different variations of the model.

How does VAR work?

VAR is a type of regression model, where each variable in the system is regressed on its own past values and the past values of all the other variables in the system. In other words, it assumes that each variable is influenced by its own past behavior as well as the behavior of the other variables in the system. This makes it a useful tool for studying how variables interact over time.

The basic VAR(p) model is:

• Y_t = C + A_1Y_(t-1) + A_2Y_(t-2) + ... + A_pY_(t-p) + u_t

In this equation, Y_t is a k-dimensional vector of variables at time t, C is a constant term, A_1, A_2, ..., A_p are parameter matrices of lagged values up to p, and u_t is a k-dimensional vector of error terms. The p parameter is known as the order of the model, and determines how many lagged values are included in the model.

To estimate the parameters of a VAR model, maximum likelihood estimation or least squares estimation can be used.

When to use VAR?

VAR can be used to analyze any system of time series variables that are believed to influence each other. For example, it can be applied to macroeconomic data like gross domestic product (GDP), inflation rates, and interest rates, to study their interrelationships. It can also be used in finance to examine the relationships between stock prices, exchange rates, and other financial variables that impact each other over time.

VAR models can be used to forecast future values of the variables in the system. Depending on the data and the assumptions made by the model, forecasts can be accurate and useful for decision-making. Additionally, VAR can be used to estimate the causal relationships between variables, which can be helpful in understanding how different variables interact over time.

Variations of VAR

There are several variations of VAR that are commonly used in different fields:

• Vector Autoregressive Moving Average (VARMA): A combination of VAR and moving average models that adds a moving average component to the lagged variables. This model can account for short-term shocks, whereas VAR only considers the long-term relationships between variables.
• VAR Integrated Moving Average (VARIMA): A variation of VARMA that also includes differencing to make data stationary. This is commonly used in time series data that exhibit trends or seasonality.
• Vector Error Correction Model (VECM): This model is a type of VAR that accounts for cointegration between variables, where a linear combination of non-stationary variables creates a stationary variable. This is commonly used in finance, where many variables are believed to have a long-run equilibrium relationship.
• Structural VAR (SVAR): A variation of VAR that includes a structural form of the model, which specifies the underlying causal relationships between variables. This can be useful in studying the impact of one variable on the others in the system.
Model Diagnostics

It is important to test the validity of VAR models before using them for forecasting or analysis. A common diagnostic tool is the residual autocorrelation function (ACF), which helps to check for the presence of omitted variables or lagged effects. If the ACF shows significant autocorrelation at a certain lag, it may indicate that the model is misspecified or that additional variables need to be included.

Other diagnostic tools include the impulse response function (IRF), which shows the impact of one variable on the others in the system, and the forecast error variance decomposition (FEVD), which shows the contribution of each variable to the forecast error.

Conclusion

Vector autoregression is a powerful tool for analyzing the relationship between multiple time series variables. It is widely used in economics, finance, and other fields that deal with time series data. With its ability to forecast future values and estimate causal relationships between variables, VAR is a valuable tool for decision-making and modeling. Additionally, the different variations of the model offer flexibility and adaptability for different types of data and research questions.