Lecture2full

The disturbance term is actually serially correlated

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incorrectly omitted, there are insufficient lags in the model, or the functional form is incorrect. the disturbance term is actually serially correlated Only in the third case would it be appropriate to respond to a significant serial correlation statistic by reasoning that the disturbance is serially correlated and then using a GLS type estimator in preference to OLS. The position we take in this course is that the most likely cause is model misspecification - the appropriate response is then to respecify the regression model. It is worth noting that a test of common factor restrictions may suffice to establish whether the disturbance term is actually serially correlated. We do not cover this in our course. B:2: CONSTANCY OF DISTURBANCE TERM VARIANCE (HOMOSCEDASTICITY) The LRM assumes that the variance of the equation disturbance term is constant over the whole sample period. That is t 2 2 = for all t, t = 1,...,T σ σ If this assumption is false, the OLS estimator is no longer efficient . Estimating the regression model by OLS when the assumption of heteroscedasticity is not valid has the consequence that the OLS estimator is inefficient (although still unbiased where regressors are non-stochastic). This can be explained intuitively in a similar manner to the case of serial correlation, discussed above. If the error term did in fact exhibit heteroscedasticity, the OLS estimator would be making no use of this information. It is the failure of the estimator to use that information that explains the inefficiency. Another (and probably more serious) consequence of disturbance serial correlation is that the standard errors of the OLS estimators are in general biased (as was the case also with serially correlated disturbances). This means that the use of t and F statistics to test hypotheses is misleading or invalid. Conversely, if the researcher knew the true structure of the heteroscedasticity, then an alternative estimator - the Generalised Least Squares (GLS) estimator - could be used instead of OLS, and would one again yield unbiased and efficient parameter estimates. If the assumption of homoscedasticity is false, then by definition the disturbance terms are heteroscedastic. There are an infinite quantity of ways in which the disturbance term could be heteroscedastic. Each of the following mechanisms involves the variance being related to the value taken by one variable, Z (which may or may not be included as a variable in your regression model):

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13 Additive heteroscedasticity u Z u Z u Z u Z Multiplicative heteroscedasticity u Z Z Discrete switching heteroscedasticity u t T u t T T t t t t t t t t t t t t t t t t t t Var( ) Var( ) Var( ) Var( ) Var( ) exp( ) exp( ) exp( ) Var( ) , ,..., Var( ) , ,..., = = = = = = + = = + = = + = = = = = + σ δ σ δ σ δ δ σ δ δ σ δ δ δ δ σ σ 2 2 2 2 1 2 2 1 2 2 2 1 2 1 2 1 2 1 2 2 1 1 1 Another possibility is that the variance of the equation disturbance term is linearly related to the values taken by a set of p variables, an intercept plus p-1 other variables denoted Z 2 to Z p . Let us consider this case.
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