Linear regression diagnostics

Linear regression is based on five assumptions: 1. $E(\epsilon_t) = 0$ 2. $var(\epsilon_t) = \sigma^2 < \infty$ 3. $cov(\epsilon_i, \epsilon_j) = 0$ 4. $cov(\epsilon_t, x_t) = 0$ 5. $\epsilon_t \backsim N(0, \sigma^2)$

The first assumption $E(\epsilon_t) = 0$ states that the mean of the residuals, or the error term, is zero. This is always the case if a constant term (as a variable) is included in the regression.

The second assumption $var(\epsilon_t) = \sigma^2 < \infty$ states that the variance must be constant, thus equal to σ². Such a variance is called homoscedastic. If variance changes over time, or depending on one variable, it is said to be heteroscedastic and the method of Ordinary Least Squares (OLS) may not be applicable anymore. We describe two test to determine homoscedasticity: Goldfield-Quandt test and White’s test.

The third assumption $cov(\epsilon_i, \epsilon_j) = 0$ ensures that error terms are unrelated to each other, thus not serially correlated nor autocorrelated. A good test for autocorrelation is the Durbin-Watson test.

The fourth assumption $cov(\epsilon_t, x_t) = 0$ ensures that the explanatory variables are not correlated with the error term. Otherwise the OLS estimator is not consistent. Test: ?

The fifth assumption $\epsilon_t \backsim N(0, \sigma^2)$ is the basis for hypothesis testing. To test whether the error term follows a normal distribution we use the Jarque-Bera test.