Vector Autoregression (VAR) Model

Abstract

In the regression analysis endogenous and exogenous variables are used alongside each other. An endogenous variable is a variable (generated by a statistical model), which is explained by the relationships between functions within the model. For example, the equilibrium price of a good in a supply and demand model is endogenous because it is set by a producer in response to consumer demand. If the general movement of one variable can be expected to produce a particular result in the other, though not necessarily in the same direction, as long as the change is correlating, it will be considered endogenous. In contrast to endogenous variables, exogenous variables are considered independent. This means one variable within the formula does not directly correlate, to a change in the other, such as personal income and colour preference, or rainfall and gas prices.

Appendix 10.1: The Wald Test

This test is available in EViews and is used for testing restrictions on parameters, especially those derived from regression models. For example, suppose you are conducting a regression analysis where the dependent variable is the price of tea. Further, suppose you believe that the price of tea at time t depends on the prices of coffee at times t − 1 and t − 2, then the model to be examined would be:

$$ \mathrm{TEA}\ {\mathrm{PRICE}}_{\mathrm{t}}=\upalpha +{\beta}_1{\mathrm{COFFEE}\ \mathrm{PRICE}}_{\mathrm{t}-1}+{\beta}_2{\mathrm{COFFEE}\ \mathrm{PRICE}}_{\mathrm{t}-2}. $$

If you believe that the coffee prices at times t − 1 and t − 2 had an equal impact on the price of tea at time t, then you would want to test the parameter restriction H₀ : β₁ = β₂ or equivalently that H₀ : β₁ − β₂ = 0. Parameter restrictions can take a variety of forms and there may be more than one of them. For example you could test the restrictions that α = 0 and that β₁ − β₂ = 0 in the above regression. This would be your null hypothesis. In this case, it is called a composite hypothesis, since it consists of more than one part. Testing the above is performed via the Wald test and its associated F statistic.

I mention this test because it has an application that might be of interest to you in the combination forecasting part of your thesis. Of course, most authors use some measure(s) of forecasting adequacy when using regression, ARIMA etc.—measures such as MAPE and RMSE. However, not so many authors test their forecasts for unbiassedness which is a property that in my view is just as important as a low MAPE.

If a statistic is biased then it is not estimating a population parameter efficiently. There will be consistent error involved in the estimation process. It may be shown that the sample mean is an unbiased estimator of the population mean; the sample regression gradient is an unbiased estimator of the population regression gradient. However, such “logical” rules do not always apply. For example, the sample variance s² is a biased estimator of the population variance σ ² because it consistently underestimates the latter’s value. In fact an unbiased estimator of the population variance is ${\widehat{\sigma}}^2=\frac{ns^2}{n-1}$ .

Now consider your forecasts ${\widehat{Y}}_t$ and your observed values Y_t recorded over the hold back period. Suppose you regress Y_t against ${\widehat{Y}}_t:$

$$ {\mathrm{Y}}_{\mathrm{t}}=\upalpha +\upbeta {\widehat{\mathrm{Y}}}_t. $$

The composite hypothesis H₀: α = 0 and β = 1 is a sufficient for ${\widehat{Y}}_t$ to be an unbiased estimator of Y_t. You perform this regression in EViews via:

Quick

• Estimate Equation

and the form of the regression would be Y C YHAT. Having run this, EViews will refer to the coefficient α as C(1) and β as C(2). Once the regression results have been generated, click the ‘View’ button, then:

Coefficient tests

• Wald − Coefficient Restrictions

and type in C(1) = 0, C(2) = 1 in the box provided. (Note: a comma must separate each restriction). You will be given the value of the F statistic associated with the Wald test and its significance level. You reject H₀ if the significance is less than 0.05, since this is a one tailed test. Acceptance of the null indicates that the forecasts in question are unbiased estimators of Y_t. (Note: EViews also generates a chi-square statistic associated with the Wald test. Should this latter statistic contradict the F statistic, the user must opt for F since it is more sensitive to the sample size in that one of its two degrees of freedom depends on the size of the sample).

Of course, one could apply the Wald test to the forecasts derived from the individual models and/or the combined models (average method and variance-covariance method).