## (Stata13) VAR Estimation and Discussions : CrunchEconometrix

How can you explain a vector autoregressive (VAR) model?

The word “autoregressive” indicates the presence of the lagged values of the dependent variable on the right-hand side of the equation. The word “vector” implies that the system contains a vector of two or more variables.

A VAR model is constructed only if the variables are integrated of order one. That is, stationary after first difference. If the variables are cointegrated, construct both short-run (VAR) and long-run (VEC) models.

If variables are NOT cointegrated, construct only the short-run (VAR) model. All the variables in a VAR system are endogenous; there are no exogenous variables. The stochastic error terms are often called impulses, or innovations or shocks. All variables in the system have equal lags.

VAR must be specified in levels, hence VAR in differences is a mis-specification!

The VAR model is estimated by ordinary least squares (OLS).

Deciding on the maximum lag length, k (an empirical issue). If you use too many lags, you will lose many degrees of freedom, incur statistically insignificant coefficients and multicollinearity. If there are too few lags, there may be specification errors. So, choose optimal lags using the information criterion: AIC, SC, HQIC etc.

Also, the interpretation of the short-run coefficients is as in any other linear model; they are ceteris-paribus effects and inference can be based on the usual OLS standard errors and test statistics.

What are some of the reasons for estimating VAR model? (1) Because there is no cointegration among the variables in the system; (2) to establish causal relationships; (3) to simulate shocks to the system and trace out the effects of shocks on the endogenous variables; (4) for forecasting (decomposing shocks to the VAR system).

[Watch video clip on VAR estimation and discussions]

## VECM Estimation, Discussion and Diagnostics

So, what do you understand by vector error correction model (VECM)?

You may say any of the following: that it is a system having a vector of two or more variables;

that all the variables in a VECM are endogenous; there are no exogenous variables;

VECM is constructed only if the variables are cointegrated;

cointegration implies evidence of a long-run relationship among the variables;

it is a restricted VAR model with cointegrating restrictions built into the specification;

constructed to examine long- and short-run dynamics of the cointegrated series;

restricts the long-run behaviour of endogenous variables to converge to their cointegrating relationships;

that the cointegrating term is known as the error correction term;

it is a representation of cointegrated VAR (courtesy of Granger’s representation theorem) and that the resulting VAR from VECM representation has more efficient coefficient estimates.

Also, note that VAR specified in differences is a mis-specification while VECM is obtained by differencing a VAR, hence losing a lag.

So, you construct a VECM with a (p-1) lag lengths for all the variables in the system.

These are the basic steps required to estimating a VECM. (1) series must be stationary (integrated of same order); (2) determine optimal lag length for the model; (3) perform Johansen cointegration test; (4) if there is no cointegration, estimate the unrestricted VAR model; (5) but if there is cointegration, then specify the restricted VAR model (i.e. VECM). In this video using Stata13, I show you the rudiments of the VECM specification.

## (Stata13): VAR and 3-Ways Causality Checks (1) : CrunchEconometrix

So how can you explain causality? A statement such as “J causes Q” will have the following meaning in different scenarios and disciplines such as J leads Q, J is the only cause of Q, J is only one of the possible causes of Q, J must always lead to Q (that is, J determines Q), the occurrence of J makes the occurrence of Q  more probable, J is a probabilistic cause of Q, J must occur either before or simultaneously with Q, but not afterwards, past values of J forecasts future values of Q.

But Regression analysis deals with the dependence of one variable on other variables, it does not necessarily imply causation. In other words, the existence of a relationship between variables does not prove causality or the direction of influence. But in regressions involving time series data, the situation may be somewhat different.

Short-run causal effects: through the F-statistics and the statistical significance of the regressors.

Long-run causal effects: through the statistical significance error-correction term (applicable to VECM only).

Joint causal effects: through the F-statistics and the significance of the independent variables and the statistical significance error-correction term (applicable to VECM only).

Unidirectional causality: occurs from J to Q if the set of estimated coefficients of the lagged J are significantly different from zero and the set of estimated coefficients of lagged Q are not significantly different from zero.

Bi-directional causality: occurs from J to Q if the set of estimated coefficients of the lagged J are significantly different from zero and vice-versa.

Independence: occurs from Q (J) to J (Q) if the set of estimated coefficients of the lagged Q (J) are not significantly different from zero. Using EViews10, this video shows you how to perform causality tests in four different ways within a VAR framework and interpret the results.

[Watch video clip shown below]

## (Stata13): VAR and 3-Ways Causality Checks (2) : CrunchEconometrix

So what is causality? A statement such as “J causes Q” will have the following meaning in different scenarios and disciplines such as J leads Q, J is the only cause of Q, J is only one of the possible causes of Q, J must always lead to Q (that is, J determines Q), the occurrence of J makes the occurrence of Q  more probable, J is a probabilistic cause of Q, J must occur either before or simultaneously with Q, but not afterwards, past values of J forecasts future values of Q.

But Regression analysis deals with the dependence of one variable on other variables, it does not necessarily imply causation. In other words, the existence of a relationship between variables does not prove causality or the direction of influence. But in regressions involving time series data, the situation may be somewhat different.

Short-run causal effects: through the F-statistics and the statistical significance of the regressors.

Long-run causal effects: through the statistical significance error-correction term (applicable to VECM only).

Joint causal effects: through the F-statistics and the significance of the independent variables and the statistical significance error-correction term (applicable to VECM only).

Unidirectional causality: occurs from J to Q if the set of estimated coefficients of the lagged J are significantly different from zero and the set of estimated coefficients of lagged Q are not significantly different from zero.

Bi-directional causality: occurs from J to Q if the set of estimated coefficients of the lagged J are significantly different from zero and vice-versa.

Independence: occurs from Q (J) to J (Q) if the set of estimated coefficients of the lagged Q (J) are not significantly different from zero.

Using EViews10, this video shows you how to perform causality tests in four different ways within a VAR framework and interpret the results.