Fin 642

Chapter 6: Multiple regression

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Chapter 6: Multiple regression

6.1 (a) df for numerator = k and for denominator = n − k − 1 where n = number of observations and k = number of explanatory variables. Here, k = 16 so that n − 16 − 1 = 30. Hence, n = 30 + 16 + 1 = 47. (b)

n−1 ¯ R2 = 1 − (1 − R2 ) n−k−1 = 1 − (1 − 0.943)

47 − 1 = 0.913. 48 − 16 − 1

(c) F = 31.04 on (17,30) df. From Table C in Appendix III, the P -value is much smaller than 0.01. So the regression is highly significant. (d) The coefficients should be compared with a t 30 distribution. From Table B in Appendix III, any value greater than 2.04 in absolute value will be significant at the 5% level. So the constant and variables 4, 8, 12, 13, 14, 15 and 17 are significant in the presence of other explanatory variables. Note that the significance level of 5% is arbitrary. There is no reason why some other significance level (e.g. 2%) could not be used. (e) The next stage would be to reduce the number of variables in the model by removing some of the least significant variables and re-fitting the model. ˆ 6.2 (a) The fitted model is C = 273.93−5.68P +0.034P 2 . For this model, R2 = 0.8315. [Recall: in exercise 5.6, model 1 had R 2 = 0.721 and model 2 had R2 = 0.859.] ¯ So the R2 values for each model are: Model 1 Model 2 Model 3 46 n−1 ¯ = 0.715. R2 = 1 − (1 − 0.721) n−k−1 = 1 − (1 − 0.721) 45 46 n−1 ¯ R2 = 1 − (1 − 0.859) n−k−1 = 1 − (1 − 0.859) = 0.849. 43 46 n−1 ¯ R2 = 1 − (1 − 0.832) n−k−1 = 1 − (1 − 0.832) = 0.824. 44

These values show that model 2 is the best model, followed by model 3. The t values for the coefficients are: Model 1 Model 2 Model 3 α : t = 10.22 α1 : t = 10.33 β0 : t = 8.83 β : t = −5.47 β1 : t = −6.61 β1 : t = −5.62 α2 : t = 4.11 β2 : t = 4.57 β2 : t = −1.99

Of these, only β2 from model 2 is not significantly different from zero. This suggests that a better model would be to allow the second part of model 2 to be a constant rather than a linear function. (b) From the computer...