Fin 3618 Financial Econometrics Exercise Sheet 5 Solutions PDF

FIN 3618 Financial Econometrics – Exercise sheet 5: Further development and analysis of the CLRM – 1 Self-Assessment 1. [TRUE/FALSE] The residual sum of squares (RSS) for a multiple linear regression model can be represented by the following expression, where û is the vector...

FIN 3618 Financial Econometrics – Exercise sheet 5: Further development and analysis of the CLRM – 1 Self-Assessment 1. [TRUE/FALSE] The residual sum of squares (RSS) for a multiple linear regression model can be represented by the following expression, where û is the vector of regression residuals: û′ û. TRUE. 2. [TRUE/FALSE] In ordinary least squares (OLS), we select the values of the parameters, β̂1 ,..., βˆk to minimize the residual sum of squares. TRUE. 3. [TRUE/FALSE] The sample regression function contains disturbance terms, but the popu- lation regression function does not. FALSE. 4. [TRUE/FALSE] The product of the matrix (X ′ X)−1 and (X ′ X) is the identity matrix, I. TRUE. 5. [TRUE/FALSE] In a regression with k − 1 regressors and an intercept term, we must account for the loss of k degrees of freedom when computing the variance of the disturbance terms. TRUE. 6. [TRUE/FALSE] The transpose of X β̂ is β̂ ′ X ′. TRUE. 7. [TRUE/FALSE] In a multiple linear regression, the expression for the optimal value of the parameters is given as (X ′ X)−1 X ′ y. TRUE. 8. [TRUE/FALSE] In a multiple linear regression, the variance of the parameter estimates is given by the diagonal elements in the matrix s2 (X ′ X)−1 , where s2 is the variance of the residuals. TRUE. 9. [TRUE/FALSE] Computing the critical values for an F -test requires us to supply the number of restrictions, the number of degrees of freedom, and the significance level. TRUE. 10. [TRUE/FALSE] If we fix the RRSS, URSS, and number of degrees of freedom, the F statistic is increasing in the number of restrictions. FALSE. 1 11. [TRUE/FALSE] The F statistic allows us to test whether the product of two parameters is equal to zero. FALSE. 12. [TRUE/FALSE] In the context of an F -test, the restricted sum of squares may be lower than the unrestricted sum of squares if the restrictions are good. FALSE. 13. [TRUE/FALSE] Adjusted R2 accounts for the loss of degrees of freedom associated with adding more variables to a regression. TRUE. 14. [TRUE/FALSE] R2 is can be calculated by dividing the explained sum of squares (ESS) by the total sum of squares (TSS). TRUE. 2 Exercises 1. You estimate a regression of the form given by Equation (1) below in order to evaluate the effect of various firm-specific factors on the returns of a sample of firms. You run a cross-sectional regression with 200 firms ri = β0 + β1 Si + β2 MBi + β3 PEi + β4 BETAi + ui (1) where ri is the percentage annual return for the stock, Si denotes the size of firm i measured in terms of sales revenue, M Bi refers to the market to book ratio of the firm, P Ei stands for the price/earnings (P/E) ratio of the firm and BET Ai is the stock’s CAPM beta coefficient. You obtain the following results (with standard errors in parentheses) r̂i = 0.080 + 0.801 Si + 0.321 MBi + 0.164 PEi − 0.084 BETAi (0.064) (0.147) (0.136) (0.420) (0.120) Calculate the t-stats. What do you conclude about the effect of each variable on the returns of the security? On the basis of your results, what variables would you consider deleting from the regression? If a stock’s beta increased from 1 to 1.2, what would be the expected effect on the stock’s return? Is the sign on beta as you would have expected? Explain your answer in each case. t-ratios: r̂i = 0.080 + 0.801 Si + 0.321 M Bi + 0.164 P Ei − 0.084 BET Ai (1.25) (5.45) (2.36) (0.39) (−0.70) The critical value from the t-tables for a 2-sided test at the 5% significance level with T − k = 195 degrees of freedom is tcrit = 1.97. Only firm size and market to book value are statistically significantly different from zero, i.e., consider deleting P E and BET A from regression. If the beta increases from 1 to 1.2, then we expect return to FALL by 1.68%. This is at odds with CAPM. 2 2. Why is it desirable to remove insignificant variables from a regression? By definition, variables having associated parameters that are not significantly different from zero are, from a statistical perspective, not helping to explain variations in the dependent variable about its mean value. If the number of degrees of freedom is relatively small, then saving a couple by deleting two variables with insignificant parameters could be useful. 3 3. For the following questions, assume that the econometric model is of the form yt = β1 + β2 x2,t + β3 x3,t + β4 x4,t + β5 x5,t + ut (2) (a) Which of the following hypotheses about the coefficients can be tested using a t-test? Which of them can be tested using an F -test? In each case, state the number of restrictions. i. H0 : β3 = 2 F- or a t- test ii. H0 : β3 + β4 = 1 F-test iii. H0 : β3 + β4 = 1 and β5 = 1 There are two restrictions. Since we are testing more than one hypothesis simultaneously, we would use an F-test. iv. H0 : β2 = 0 and β3 = 0 and β4 = 0 and β5 = 0 We have 4 restrictions. As for (c), we are testing multiple hypotheses so we cannot use a t-test. v. H0 : β2 β3 = 1 Although there is only one restriction, it is a multiplicative restriction. We therefore cannot use a t-test or an F-test to test it. In fact we cannot test it at all using the methodology that has been examined in this chapter. (b) Which of the above null hypotheses constitutes ‘THE’ regression F -statistic in the context of Equation (2)? Why is this null hypothesis always of interest whatever the regression under study? What exactly would constitute the alternative hypothesis in this case? Hypothesis iv. We are always interested in testing this hypothesis since it tests whether all of the coeffi- cients in the regression (except the constant) are jointly insignificant. H1 : β2 ̸= 0 or β3 ̸= 0 or β4 ̸= 0 or β5 ̸= 0 (c) Which would you expect to be bigger - the unrestricted residual sum of squares or the restricted residual sum of squares, and why? RRSS ≥ URSS OLS is selecting parameters optimally to minimize the RSS. It is not possible to reduce it further by picking different values. (d) You decide to investigate the relation given in the null hypothesis of exercise part (a).iii. What would constitute the restricted regression? The regressions are carried out on a sample of 96 quarterly observations, and the residual sums of squares for the restricted and unrestricted regressions are 102.98 and 91.41, respectively. Perform the test. What is your conclusion? The null hypothesis is: H0 : β3 + β4 = 1 and β5 = 1 The first step is to impose this on the regression model yt = β1 + β2 x2,t + β3 x3,t + β4 x4,t + β5 x5,t + ut subject to β3 + β4 = 1 and β5 = 1 We can rewrite the first part of the restriction as β4 = 1 − β3 4  Then rewrite the regression with the restriction imposed yt = β1 + β2 x2,t + β3 x3,t + (1 − β3 ) x4,t + x5,t + ut yt = β1 + β2 x2,t + β3 (x3,t − x4,t ) + x4,t + x5,t + ut yt − x4,t − x5,t = β1 + β2 x2,t + β3 (x3,t − x4,t ) + ut pt = β1 + β2 x2,t + β3 qt + ut This constitutes the restricted regression model The test statistic is calculated as RRSS − URSS T − K test stat = URSS m In this case, m = 2, T = 96, k = 5, so test stat = 5.704. Compare this to an F -distribution with (m, T − k) = (2, 91) degrees of freedom, which is approximately 3.10. Hence we reject the null hypothesis that the restrictions are valid. We cannot impose these restrictions on the data without a substantial increase in the residual sum of squares. 5 4. A researcher estimates the following econometric models including a lagged dependent variable yt = β1 + β2 x2,t + β3 x3,t + β4 yt−1 + ut ∆yt = γ1 + γ2 x2,t + γ3 x3,t + γ4 yt−1 + vt where u and v are independent and identically distributed (i.i.d.) disturbances. Will these models have the same value of... (a) the residual sum of squares (RSS), if β̂4 = γ̂ + 1 and β̂i = γ̂i (for i = 1, 2, 3), residuals will coincide and thus also RSS will coincide (b) R2 , No, because TSS differs. (c) Adjusted R2 ? No, because R2 differs. Explain your answers in each case 6 5. A researcher estimates the following two econometric models yt = β1 + β2 x2,t + β3 x3,t + ut yt = β1 + β2 x2,t + β3 x3,t + β4 x4,t + vt where ut and vt are independent and identically distributed (i.i.d.) disturbances and x4,t is an irrelevant variable which does not enter the data generating process for yt. Will the value of... (a) R2 R22 ≥ R12 (b) Adjusted R2 No clear answer. It depends. be higher for the second model than the first? Explain your answers. 7