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# Vector Autoregression

Updated on January 8, 2024
Article byRutan Bhattacharyya
Edited byAshish Kumar Srivastav
Reviewed byDheeraj Vaidya, CFA, FRM

## What Is Vector Autoregression (VAR)?

The VAR or vector autoregressive model refers to a type of stochastic model that relates a variable’s current observations with the past observations of itself and other variables within the system. It captures the inter-dependencies and evolution between different time series.

For eg:
Source: Vector Autoregression (wallstreetmojo.com)

This model is used in econometrics and finance as it offers a framework for fulfilling crucial modeling objectives, including forecasting, structural inference, policy analysis, and data description. Moreover, it helps in time series research by allowing one to inspect the dynamic relationships between the variable interacting with each other. This model is of three types — structural, recursive, and reduced form.

### Key Takeaways

• Vector autoregression refers to a stochastic model that can forecast multiple time series models utilizing a single model.
• Individuals can utilize recursive, reduced form, and structural VAR models to make forecasts and meet any other modeling objective.
• A noteworthy difference between VAR and ARIMA (Autoregressive Integrated Moving Average) is that the former is bidirectional while the latter is unidirectional.
• There are various vector autoregression model benefits. For example, it allows one to capture complex real-world information in a flexible manner. Moreover, such models do not impose any restrictive assumption on data structure or distribution.

### Vector Autoregression Explained

The vector autoregression refers to a forecasting algorithm that individuals can utilize when at least two-time series influence one another, i.e., a bidirectional relationship exists between the time series. It is one of the most easy-to-use and flexible models for analyzing multivariate time series. This model helps explain the dynamic behavior of financial and economic time series. Moreover, it offers forecasts superior to the univariate time series models’ predictions.

This model is autoregressive as every variable is modeled as the previous values’ function, which means that the predictors are lags of the series.

The projections of a VAR model are quite flexible, as it is possible to make them contingent on the probable future paths of particular variables within the model. In the case of structural analysis, the imposition of specific assumptions regarding the causal structure of the information or data being investigated occurs. The resulting causal effects of the unanticipated shocks or changes to particular variables within the model are encapsulated. Usually, the casual impacts are summarized with FEVD or forecast error variance decomposition and impulse response functions.

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### Types

Let us look at the different types of VAR models in detail.

#### #1 – Structural

Structural vector autoregression models include certain restrictions that enable individuals to spot causal relationships beyond the ones that people can identify with recursive or reduced-form models. The causal relationships can help forecast and model the effects of individual shocks, for example, policy decisions.

#### #2 – Reduced Form

Such models consider every variable a function of its previous values, the previous values of every other variable under consideration, and a sequentially uncorrelated error term. One must note that the error terms are correlated across equations within such models. This means individuals cannot consider the extent of the impact of individual shocks on the system.

#### #3 – Recursive VAR Models

This type of VAR model contains every component of the reduced form VAR model. They also enable a few variables to be other concurrent variables’ functions. The recursive model enables individuals to model structural shocks by imposing short-run relationships. This VAR model considers that error terms in every regression equation do not correlate with previous equations’ errors.

### Assumptions

The assumptions of a VAR are as follows:

• The error term’s conditional mean is zero.
• The possibility of large outliers is low.
• Variables within such a model are stationary.
• Perfect multicollinearity does not exist.

### Examples

Let us look at a few vector autoregression examples to understand the concept better.

#### Example #1

During 2022, the following three types of shocks caused the interest rates in the U.S. to go up.

• Inflation shocks
• Reaction shocks
• Real shocks

Among these, reaction shocks had the most impact as the federal government pivoted more towards aggressive action to curb inflation. The following image shows the drivers of 2-year interest rate yields in the U.S. in 2022.

Source

One must note that the shocks are projected using a sign-restricted Bayesian VAR model with stochastic volatility. Real shocks increase equity prices, interest rates, and inflation expectations. That said, inflation shocks decrease equity prices, while reaction shocks lower equity prices and inflation expectations. Finally, one must note that both raise the U.S. interest rates.

#### Example #2

According to the State Bank of India or SBI, in the week of Diwali 2022, a decrease in currency circulation was witnessed for the first time since 2002, assuming the marginal decrease of 2009 was because of the economic slowdown only. This drop in currency circulation was due to the growing popularity of digital transactions. After all, interoperable payment systems, such as wallets and UPI or Unified Payments Interfaces, made digital funds transfer cheaper.

SBI used an SVAR or structural vector autoregression model to determine the effect of PPI and UPI on monetary base (M0), CIC (currency in circulation), bank deposits, MM or money multiplier, and broad money or M3, individually with short-run constraints.

According to SBI’s expectations, the results disclosed that the rise in PPI or prepaid payment instruments negatively impacted M0 and CIC. Moreover, a surge in PPI positively impacted M3. That said, a rise in UPI had a negative impact on M3 and M0 but had no significant effect on CIC. The model also revealed that PPI and UPI did not significantly impact the money multiplier, although the coefficients were negative.

Let us look at the benefits and limitations of VAR.

• A vector autoregression model offers a systematic yet flexible way of capturing complicated real-world behavior.
• It can capture time series data’s intertwined dynamics.
• Such a model provides individuals with improved forecasting performance.
• VAR models do not impose restrictive assumptions on the distribution or structure of the data.
• Another key autoregression model benefit is that one can use such a model to forecast cointegrated or stationary time series variables.

• VAR models require substantial data and a cautious lag length selection for forecasting.
• As noted above, the error terms are correlated across equations, meaning one cannot determine the individual shocks’ impact on the system.
• Contemporaneous variables do not have any relation with each other.

### Vector Autoregression vs ARIMA

Anyone new to the world of data can find it difficult to fully understand the concepts of VAR and ARIMA or Autoregressive Integrated Moving Average. However, knowing the key differences between the two is essential to understand their meaning and purpose clearly. So, let us dive into their distinct features.

1. What is a Bayesian vector autoregression model?

This model involves utilizing techniques to estimate a VAR model. It assumes a preliminary probability on the innovations covariance matrix and every model coefficient (model constant and linear time trend vector, AR coefficient matrices, etc.).

2. What are AR and vector autoregression?

VAR refers to simply a generalization of an autoregressive or AR model for more than one-time series, spotting the linear relationship existing between them. One can view AR as a specific case of vector autoregression for a single series only.

3. What to do if vector autoregression is not stable?

If a VAR model is unstable, its variables are not stationary. Therefore, individuals must make the variables stationary to apply the model. Alternatively, they can choose to apply the Vector Error Correction Model or VECM if cointegration exists.

4. What are the lags in a VAR model?

In VAR models, a lag refers to a variable’s value in a prior period.

This article has been a guide to what is Vector Autoregression. We explain its examples, assumptions, types, comparison with ARIMA, advantages, and disadvantages. You may also find some useful articles here –