Lecture2full

# Step 5a if these tests show no evidence of

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Step 5a: If these tests show no evidence of misspecification in any relevant form, go on to conduct statistical inference about the parameters. Step 5b: If these tests show evidence of misspecification in one or more relevant forms, then two possible courses of action seem to be implied: If you are able to establish the precise form in which the model is misspecified, then it may be possible to find an alternative estimator which will is optimal or will have other desirable qualities when the regression model is statistically misspecified in a particular way. Regard statistical misspecification as a symptom of a flawed model. In this case, one should search for an alternative, well-specified regression model, and so return to Step 1. For example, if all of the conditions of the normal classical linear regression model (NCLRM) are satisfied,

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2 then the ordinary least squares estimator is BLUE, and is the optimal estimator. Furthermore, given that estimators of the error variance (and so of coefficient standard errors) will also have desirable properties, then the basis for valid statistical inference exists. The CLASSICAL linear regression model assumes, among other things, that each of the regressor variables is NON-STOCHASTIC. This is very unlikely to be satisfied when we analyse economic time series. We shall proceed by making a much weaker assumption. The regressors may be either stochastic or non-stochastic; but if they are stochastic, they are asymptotically uncorrelated with the regression model disturbances. So, even though one or more of the regressors may be correlated with the equation disturbance in any finite sample, as the sample size becomes indefinitely large this correlation collapses to zero. One other point warrants mention. In these notes, I am assuming that each of the regressor variables is “covariance stationary”. At this point I will not explain what this means. That will be covered in detail later in the course. It will be useful to list the assumptions of the linear regression model (LRM) with stochastic variables. These are listed in Table 4.1. TABLE 4.1 THE ASSUMPTIONS OF THE LINEAR REGRESSION MODEL WITH STOCHASTIC REGRESSORS The k variable regression model is t 1 2 2t k kt t Y = + X + ... + X + u t = 1,...,T β β β (1) The assumptions of the CLRM are: (1) The dependent variable is a linear function of the set of possibly stochastic, covariance stationary regressor variables and a random disturbance term as specified in Equation (1). No variables which influence Y are omitted from the regressor set X (where X is taken here to mean the set of variables X j , j=1,...,k), nor are any variables which do not influence Y included in the regressor set. In other words, the model specification is correct. (2) The set of regressors is not perfectly collinear. This means that no regressor variable can be obtained as an exact linear combination of any subset of the other regressor variables.
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