### Regression Analysis and Marks

BRUNEL UNIVERSITY Master of Science Degree examination Specimen Exam Paper 2005-2006 EC5002: Modelling Financial Decisions and Markets EC5030: Introduction to Quantitative Methods Time allowed: 1. 5 hours Answer all of question 1 and at least two other questions 1. COMPULSORY Provide brief answers to all the following: (a) A sample of 20 observations corresponding to the model: Y = + X + u, gave the P P P following data: (X X)2 = 215:4, (Y Y )2 = 86:9, and (X X)(Y Y ) = 106:04. Estimate . 5 marks) (b) Prove that r2 = byx bxy , where byx is the least-squares (LS) slope in the regression of Y on X , bxy is the LS slope in the regression of X on Y , and r is the coe? cient of correlation between X and Y . (5 marks) (c) Present four alternative in‡ ation/unemployment regressions. (5 marks) (d) Give one reason for autocorrelated disturbances. (5 marks) (e) Explain how we might use the Breusch-Godfrey statistic to test estimated residuals for serial correlation. (5 marks) (f) The following regression equation is estimated as a production function for Q: lnQ = 1:37 + 0:632 lnK + 0:452 lnL, cov(bk ; bl ) = 0:055; 0:257) (0:219) where the standard errors are given in parentheses. Test the hypothesis that capital (K ) and labor (L) elasticities of output are identical. (5 marks) Continued (Turn over) 1 ANSWER TWO QUESTIONS FROM THE FOLLOWING: 2. (a) Economic theory supplies the economic interpretation for the predicted relationships between nominal (in‡ ation) uncertainty, real (output growth) uncertainty, output growth, and in‡ ation. Discuss …ve testable hypotheses regarding bidirectional causality among these four variables. (25 marks) + yt b) An investigator estimates a linear relation for German output growth (yt ): yt = 1 + ut , t = 1850; : : : ; 1999. The values of …ve test statistics are shown in Table 1: Discuss the results. Is the above equation correctly speci…ed? (10 marks) 3. (a) i) Show how various examples of typical hypotheses …t into a general linear framework: Rb = r, where R is a (q k) matrix of known constants, with q < k, b is the (k 1) least-squares vector, and r is a q -vector of known constants. ii) Show how the least-squares estimator (b) of about . an be used to test various hypotheses iii) “The test procedure is then to reject the hypothesis Rb = r if the computed F value exceeds a preselected critical value” Discuss. (20 marks) (b) The results of least-squares estimation (based on 30 quarterly observations) of the regression of the actual on predicted interest rates (three-month U. S. Treasury Bills) were as follows: rt = 0:24 + 0:94 rt + et ; RSS = 28:56; (0:86) (0:14) where rt is the observed interest rate, and rt is the average expectation of rt held at the end of the preceding quarter.

FiguresX parentheses are estimated standard errors. in X (rt r )2 = 52. The sample data on r give rt =30 = 10, According to the rational expectations hypothesis expectations are unbiased, that is, the average prediction is equal to the observed realization of the variable under investigation. Test this claim by reference to announced predictions and to actual values of the rate of interest on three-month U. S. Treasury Bills. (Note: In the above equation all the assumptions of the classical linear regression model are satis…ed). 15 marks) Continued (Turn over) 2 4. (a) What are the assumptions of the classical linear regression model? (10 marks) (b) Prove that the variance-covariance matrix of the (k 1) least-squares vector b is: var(b) = 2 (X 0 X) 1 , where 2 is the variance of the disturbances and X is the (n k) matrix of the regressors. (15 marks) b (c) In the two-variable equation: Yi = a+bXi , i = 1; : : : ; n show that cov(a; b) = 2 X= X)2 . (10 marks) X (X 5. (a) Explain how we might use White statistic to test for the presence of heteroscedasticity in the estimated residuals. 10 marks) (b) A speci…ed equation is Y = X +u, with E(u) = 0 and E(uu0 ) = ; where =diagf 2 ; : : : ; 1 Derive White’ correct estimates of the standard errors of the OLS coe? cients. s (15 marks) (c) Explain how we might test for ARCH e¤ects? (10 marks) 2 2g . 3 Table 1. Test statistic Value of the test p-value White heteroscedasticity test 50. 72 0. 00 Box-Pierce Statistic on 82. 263 0. 00 Squared Residuals Jarque-Bera statistic 341. 754 0. 00 ARCH test 65. 42 0. 00 Ramsey test statistic 39. 74 0. 00 4