Question 3 Suppose a researcher estimated the following model. where the error t

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Question 3 Suppose a researcher estimated the following model. where the error term is denoted tt and th rescarcher estimated this model using 98 observations, and found: e are parameters to be estimated. Suppose the 0.765 1.763 X 0.693 X (0.976) (0.568) (0.398) and also that the ESS = 76.76 and the TSS-749.67. (a) Test whether cach of the coefficients are significant. (b) Calculate the R2 and test the overall significance of the model. Suppose that a critic argued that several additional variables should be added. When the new variables are added, the estimated regression model is = 0.587 + 1.673 Xii + 0.599 + 0.598 X3i + 0.956 (0.371) (0.856) (0.533) (0.278) (0.401) and also that the ESS = 132.66. (c) Test whether the added variables are significant. (d) Calculate the R2 and test the overall significance of the new model. (e) Test whether the additional variables significantly improve the fit of the model.

Explanation / Answer

(a) To test for significance we use the t test which is given by t= B/SE(B) where B is the coefficient and SE is the standard error of the coefficient. Thus in this case for the 3 variables the coefficients are 0.765/0.976=0.783, 1.763/0.568=3.103 and 0.693/0.398=1.741. Thus only the second coefficient is significant at the 5% level as the t value is greater than 2. So only X1i is significant.

(b) The R squared is given by the ESS/TSS = 76.76/749.67=10.23%. We use the F test to test for significance of the overa model. This is F= (0.1023/1)/(0.8977/96) = 0.1023/0.0094 = 10.88. So as the F stat is greater than 2 the model is significant.

(c) Now we have the t stats for the coefficients above computed in the same way as 0.686,3.138,1.614,2.151 and 2.384. Thus as the t stats need to be greater that 2, X1i,X3i and X4i are significant.

(d) The R squared is 132.66/ 749.67= 18% and the F stat = (0.1769/3)/(0.8231/94)=0.0589/0.008= 7.3. So as the F stat > 2 the model is significant.

(e) The model receives a boost in the R squared as the additional variables are added and so they improve the fit of the model.

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