Question 1 carries a weight of 25%; question 2 carries 25%; question 3 carries 20%; and question 4 carries 30%.

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1 UNIVERSITY OF EAST ANGLIA School of Economics Main Series PGT Examination FINANCIAL ECONOMETRIC THEORY ECO-7024A Time allowed: 2 HOURS Answer ALL FOUR questions. Question 1 carries a weight of 25%; question 2 carries 25%; question 3 carries 20%; and question 4 carries 30%. Marks awarded for individual parts are shown in square brackets. A formula sheet, t-tables, and F-tables are attached at the end of the exam paper. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. ECO-7024A Module Contact: Prof Peter Moffatt, ECO Copyright of the University of East Anglia Version 2

2 Page 2 QUESTION 1 [25 Marks] ALL WORKING MUST BE SHOWN IN YOUR ANSWER TO THIS QUESTION Consider the regression model: yy ii = ββ 1 + ββ 2 xx ii + εε ii, εε ii XX~NN(0, σσ 2 ), CCCCCC εε ii, εε jj XX = 0 for ii jj, ii = 1, 2,, nn. This model can be written in matrix notation as, yy = XXββ + εε, εε XX~NN(00, σσ 2 II nn ) where yy is (nn 1), XX is (nn 2), ββ is (2 1), εε is (nn 1), II nn is the (nn nn) identity matrix and σσ 2 is a positive constant. For a particular data set, we are given the following information: XX XX = , XX yy = , yy yy = 576 a) Calculate the OLS estimates bb = bb 1 bb 2 of vector ββ = ββ 1 ββ 2. [4marks] b) Compute and interpret the RR 2. Also compute the adjusted RR 2 (RR 2 ). [6 marks] c) Compute the variance of bb, VVVVVV (bb XX), and consequently the standard errors of bb 1 and bb 2. [5 marks] d) Is the slope coefficient ββ 2 statistically different from 1.5? Is it statistically different from 2? [5 marks] e) Based on your answer in part (d), explain why we prefer stating we don t have evidence to reject HH 0 than we have evidence to accept HH 0. [5marks]

3 Page 3 QUESTION 2 [25 marks] Consider the following linear regression model, yy = XXββ + εε, EE(εε XX) = 00, VVVVVV(εε XX) = σσ 2 II nn where yy is (nn 1), XX is (nn KK), ββ is (KK 1), εε is (nn 1) and II nn is the (nn nn) identity matrix. a) Let bb 0 be an arbitrary (KK 1) vector of estimates of ββ, and ee 0 = yy XXbb 0 be the corresponding arbitrary (nn 1) vector of residuals. Prove that the OLS estimator bb = (XX XX) 11 XX yy is the solution to the unconstraint minimization problem min RRRRRR(bb 0 ), where RRRRRR(bb 0 ) = ee 0 ee 0 (hint: differentiate RRRRRR(bb 0 ) with respect to bb 0 bb and solve the first order condition). [6marks] b) Prove that the OLS estimator is an unbiased estimator of ββ. In addition, show that VVVVVV(bb XX) = σσ 2 (XX XX) 1. [6marks] c) One of the MLR assumptions states that XX must have full rank equal to KK. With reference to your answer to part (a), why is this assumption needed? [3marks] Consider any other estimator that is linear and unbiased, bb = [(XX XX) 11 XX + DD]yy, where DD is a (KK nn) matrix of functions of XX or fixed numbers. d) Prove that bb is unbiased only if DDDD = 00. [3marks] e) Show that VVVVVV bb XX = σσ 2 [(XX XX) 1 + DDDD ]. [3marks] f) Show that VVVVVV bb XX VVVVVV(bb XX) is a positive semidefinite matrix. Why is this an important result? [4marks] TURN OVER

4 Page 4 QUESTION 3 [20 marks] We have data on 53 countries in Let p_locali be the price of a Big Mac (the McDonald s hamburger) in country i in local currency in Let ei be the exchange rate for country i against the US dollar in 2017 (that is, ei is the number of units of local currency that can be exchanged for one US dollar in 2017). (a) Data on three of the 53 countries is shown in the following table. Country Currency p_local e Sri Lanka Rupee South Africa Rand Brazil Real Compute the price of a Big Mac in each of the three countries in US dollars. On this basis, which of the three currencies appears under-valued in 2017, and which appears over-valued? [7marks] The following regression model is estimated using data from all 53 countries in 2017 (p_usa is the price of a Big Mac in the USA in 2017): _ log p locali = β + 1 β2log e + i ui ; i = 1, L,53 (1) p _ usa Following the regression, two tests are performed. The results are as follows:. regress log_p_ratio log_e Source SS df MS Number of obs = F(1, 51) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = log_p_ratio Coef. Std. Err. t P> t [95% Conf. Interval] log_e _cons test (_b[_cons]=0) (_b[log_e]=1) ( 1) _cons = 0 ( 2) log_e = 1 F( 2, 51) = Prob > F = test (_b[log_e]=1) ( 1) log_e = 1 F( 1, 51) = Prob > F =

5 Page 5 (b) (c) Consider the two tests performed following the regression above. The first test is a test of the Law of One Price (LOP). Explain the concept of the LOP. Is it rejected by the 2017 Big Mac data? Which theory is being tested by the second test? Is it rejected? [7marks] Explain the Balassa-Samuelson effect, and explain why you might expect it to hold. How would you extend the regression model (1) in order to carry out a test of the Balassa-Samuelson effect? [6marks] TURN OVER

6 Page 6 QUESTION 4 [30 marks] Two years of daily data on the daily return of Unilever (Food and Beverages) stock are analysed. The daily return variable is named r. A plot of the daily return against time is shown below. r jan jul jan jul jan2018 date (a) Describe the return series. Do you think that it exhibits volatility clustering? [5marks] The following regression results are obtained using STATA:. regress r l.r l2.r l3.r l4.r l5.r Source SS df MS Number of obs = F(5, 512) = 2.76 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = r Coef. Std. Err. t P> t [95% Conf. Interval] r L L L L L _cons (b) Based on the F-test for overall significance, is there evidence to reject the Efficient Markets Hypothesis (EMH)? Explain why this test is appropriate for answering this question. [5marks]

7 Page 7 A set of day-of-week dummies are added to the model, with the results:. regress r l.r l2.r l3.r l4.r l5.r day2-day5 Source SS df MS Number of obs = F(9, 508) = 2.61 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = r Coef. Std. Err. t P> t [95% Conf. Interval] r L L L L L day day day day _cons (c) (d) Explain why only four day-of-week dummies have been added to the model. What would happen if you attempted to include all five? [5marks] By comparing the two models estimated above, carry out an F-test of the importance of day-of-week in determining the return. Does the result provide evidence (or further evidence) against the EMH? [5marks] TURN OVER

8 Page 8 The following further analysis is performed:. arch r l.r l2.r l3.r l4.r l5.r day2-day5, arch(1) garch(1) ARCH family regression Sample: Number of obs = 518 Distribution: Gaussian Wald chi2(9) = Log likelihood = Prob > chi2 = OPG r Coef. Std. Err. z P> z [95% Conf. Interval] r r L L L L L day day day day _cons ARCH arch L garch L _cons e test l.r l2.r l3.r l4.r l5.r ( 1) [r]l.r = 0 ( 2) [r]l2.r = 0 ( 3) [r]l3.r = 0 ( 4) [r]l4.r = 0 ( 5) [r]l5.r = 0 chi2( 5) = Prob > chi2 = test day2 day3 day4 day5 ( 1) [r]day2 = 0 ( 2) [r]day3 = 0 ( 3) [r]day4 = 0 ( 4) [r]day5 = 0 chi2( 4) = Prob > chi2 = (e) Explain why ARCH/GARCH models are useful in this situation. On the basis of the above results, which of ARCH and GARCH do you think is more appropriate in this case? Does your answer confirm your answer to (a)? [5marks]

9 Page 9 (f) Following the last estimation, two tests have been performed. Use these test results to make inferences concerning the validity of the EMH. Do the conclusions agree with those of (b) and (d)? Comment on your answer. [5marks] END OF PAPER

10 Page 10 Financial Econometric Theory Formula Sheet The matrix regression model In the model: yy = XXββ + εε, bb = (XX XX) 11 XX yy RRRRRR = yy yy bb XX yy TTTTTT = yy yy nn yy 2 ss 2 = RRRRRR nn KK Let bb jj be the jj tth element of bb where jj = 1,2,, KK. The estimator of the variancecovariance matrix of bb is given by: VVVVVV (bb XX) = ss 2 (XX XX) 11 The standard error of bb jj is given by the jj tth diagonal element of VVVVVV (bb XX). Testing joint restrictions in the multiple regression model F = 2 2 ( RU RR) / r 2 ( 1 RU ) /( n k) ~ F rn, k

11 Page 11 Table 1: Critical values of the t-distribution df α = 0.10 α = 0.05 α = α = 0.01 α =

12 Page 12 Table 2: Critical values of the F-distribution (α=0.05) df 1= df 2= END OF MATERIALS