By Patrick Kent Watson, Sonja Sabita Teelucksingh
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Extra info for A Practical Introduction to Econometric Methods Classical and Modern
2 The null hypothesis is rejected in favour of the alternative if this value is less than the corresponding value read from the Student distribution for 22 degrees of freedom at the chosen significance level. 725. The null hypothesis that the marginal propensity to import is 40% cannot be rejected at the 5% level of significance. TESTING THE SIMULTANEOUS NULLITY OF THE SLOPE COEFFICIENTS It is important to test the nullity of a given coefficient because, as economists, we want to know whether the associated variable intervenes in the explanation of the dependent 34 A Practical Introduction to Econometric Methods: Classical and Modern variable.
OLS estimation yields: −1 βˆ 1 = (X1′X1 ) X1′y Replacing y by its value gives: −1 −1 −1 βˆ 1 = (X1′X1 ) X1′ (X1β1 + X 2β 2 + u) = β1 + (X1′X1 ) X1′X 2β 2 + (X1′X1 ) X1′u Since the X matrices are fixed and E(u) = 0, then, clearly: ( ) −1 E βˆ 1 = β1 + (X1′X1 ) X1′X 2β 2 so that βˆ 1 is a biased estimator of β1 unless the omitted and excluded variables are orthogonal to each other, a possibility that we can safely rule out when dealing with economic data. We have considered a specification error due to the absence of relevant explanatory variables, and we have seen that this leads to bias in our coefficient estimates.
The (almost) universal notation R2 is used for the former while R 2 is used for the latter. Let us derive these measures. , n. It can be shown that, if the general linear regression model contains a constant term,1 then n y= ∑ t =1 n n yt = ∑ yˆ t t =1 n That is, the average value of the observed y values ( y) is also the average of the fitted values. Therefore, if the regression fit is a reasonably good one, we can expect that the 1 From now on, unless it is specifically stated otherwise, the general linear regression model will be assumed to contain a constant term which, without loss of generality, may be taken to be the first exogenous variable, xlt.
A Practical Introduction to Econometric Methods Classical and Modern by Patrick Kent Watson, Sonja Sabita Teelucksingh