A Practical Introduction to Econometric Methods Classical by Patrick Kent Watson, Sonja Sabita Teelucksingh

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3 Consider the following “true” model: y t = β1x1t + β 2 x 2 t + u t The economist, however, wrongly specifies that: y t = β1x1t + v t and this misspecified model is fitted by OLS to yield the following: −1 βˆ 1 = (X ′X) X ′y where ⎡ ⎢ ⎢ (X′X)−1 = ⎢(x11 ⎢ ⎢ ⎢ ⎣ −1 x12 ⎛ x11 ⎞ ⎤ ⎜ x12 ⎟ ⎥ ⎛ ⎜ ⎟⎥ … … x1 n ) ⎜ % ⎟ ⎥ = ⎜ ⎝ ⎜ % ⎟⎥ ⎜ ⎟⎥ ⎝ x1n ⎠ ⎥⎦ n ∑ t =1 ⎞ x12t ⎟ ⎠ −1 = 1 n ∑x 2 1t t =1 and (X′y) = (x11 x12 ⎛ y1 ⎞ ⎜ y2 ⎟ ⎜ ⎟ … … x1 n ) ⎜ % ⎟ = ⎜ %⎟ ⎜ ⎟ ⎝ yn ⎠ n ∑ (x y ) 1t y t =1 Therefore: n −1 βˆ 1 = (X ′X) X ′y = ∑ (x y ) 1t t t =1 n ∑x 2 1t t =1 3 In Part II of this book, tests for this and other kinds of model misspecification will be studied.

Highly significant. 3 OLS Regression fit of Mt = β1 + β3 pmt/pdt + β4 Cgt + ut ============================================================== LS // Dependent Variable is IMPORTS Date: 08/12/95 Time: 18:44 Sample: 1967 1991 Included observations: 25 ============================================================== Variable Coefficient Std. Error T-Statistic Prob. D. E. 71113 Sum squared resid 6517534. 83 on imports). Furthermore, the R2 and F-statistics leave us in no doubt that this is a good fit.

The null is rejected if the test statistic exceeds tα/2 and not rejected in the event that it does not. 6993. 074 and so the null hypothesis is resoundingly rejected at the 5% level (and therefore at all higher levels, including 10%). 819 and so the null hypothesis is rejected at this level as well. The test of the nullity of a given regression coefficient is so popular that standard econometric packages like EViews provide the value of its test statistic automatically. 1 in the previous chapter under the column headed “T-Statistic”.

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