UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PG Examnaton 016-17 FINANCIAL ECONOMETRICS ECO-7009A Tme allowed: HOURS Answer ALL FOUR questons. Queston 1 carres a weght of 5%; queston carres 0%; queston 3 carres 5%; and queston 4 carres 30%. Marks awarded for ndvdual parts are shown n square brackets. A formula sheet, t-tables, and F-tables are attached at the end of the exam paper. Notes are not permtted n ths examnaton. Do not turn over untl you are told to do so by the Invglator. ECO-7009A Module Contact: Prof. Peter Moffatt, ECO Copyrght of the Unversty of East Angla Verson 1
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Page 3 QUESTION 1 [5 Marks] ALL WORKING MUST BE SHOWN IN YOUR ANSWER TO THIS QUESTION The share prce of Natonal Grd (Utltes) was followed for a perod of seven months. The percentage monthly return on Natonal Grd stock (Y), and the percentage monthly change n the stock market ndex (X), are presented n the followng table: Month Natonal Grd (Y) Market (X) January 1 February - -3 March -1-1 Aprl 0 May 1 June 0 1 July 1 4 (a) Obtan estmates of and n the smple regresson model: Y X u t 1,,7 t t t Var u t Report the beta coeffcent for Natonal Grd stock. [10] (b) (c) Fnd the resduals from the smple regresson performed n (a). Hence fnd an estmate of the parameter. Call the estmate ˆ. What s the nterpretaton of ˆ n ths context? [7] Fnd a 95% confdence nterval for. Does the confdence nterval ndcate that Natonal Grd s an aggressve stock, a defensve stock, or nether? Is ths what you would expect for Natonal Grd? [8] TURN OVER
Page 4 QUESTION [0 marks] We have data on 53 countres n 016. Let p_local be the prce of a Bg Mac (the McDonald s hamburger) n country n local currency n 016. Let e be the exchange rate for country aganst the US dollar n 016 (that s, e s the number of unts of local currency that can be exchanged for one US dollar n 016). (a) Data on three of the 53 countres s shown n the followng table. Country Currency p_local e South Afrca Rand 8 15.81 Norway Kroner 46.8 8.97 Japan Yen 370 118.65 Compute the prce of a Bg Mac n each of the three countres n US dollars. On ths bass, whch of the three currences appears under-valued n 016, and whch appears over-valued? [7] The followng regresson model s estmated usng data from all 53 countres n 016 (p_usa s the prce of a Bg Mac n the USA n 016): _ log p local 1 log e u ; 1,,53 p_ usa Followng the regresson, two tests are performed. The results are as follows:. regress log_p_rato log_e Source SS df MS Number of obs = 53 -------------+---------------------------------- F(1, 51) = 935.98 Model 31.7459 1 31.7459 Prob > F = 0.0000 Resdual 5.5885653 51.10957971 R-squared = 0.989 -------------+---------------------------------- Adj R-squared = 0.986 Total 37.3184 5 6.9447738 Root MSE =.33103 log_p_rato Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- log_e.938301.017157 54.18 0.000.89868.967391 _cons -.81081.0610954-4.60 0.000 -.4037363 -.158479. test (_b[_cons]=0) (_b[log_e]=1) ( 1) _cons = 0 ( ) log_e = 1 F(, 51) = 54.49 Prob > F = 0.0000. test (_b[log_e]=1) ( 1) log_e = 1 F( 1, 51) = 15. Prob > F = 0.0003
Page 5 (b) Consder the two tests performed followng the regresson above. The frst test s a test of the Law of One Prce (LOP). Explan the concept of LOP. Is t rejected by the 016 Bg Mac data? Whch theory s beng tested by the second test? Is t rejected? [7] A further varable, gdp_rato, s generated, defned as GDP per head n the local country n US dollars dvded by GDP per head n the USA. Ths varable s added to the regresson, wth the results:. regress log_p_rato log_e gdp_rato Source SS df MS Number of obs = 53 -------------+---------------------------------- F(, 50) = 053.75 Model 33.376414 161.68807 Prob > F = 0.0000 Resdual 3.9364105 50.0787805 R-squared = 0.9880 -------------+---------------------------------- Adj R-squared = 0.9875 Total 37.3184 5 6.9447738 Root MSE =.8059 log_p_rato Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- log_e.9796333.0178135 54.99 0.000.9438538 1.015413 gdp_rato.594401.1155731 4.58 0.000.973048.7615754 _cons -.66664.09808-6.76 0.000 -.8596675 -.4656605 (c) Does gdp_rato have a sgnfcant effect on log_p_rato? What s the name of the theory that s beng confrmed by ths test? Does the test result provde an explanaton for the results of the tests carred out n (b)? Explan your answer. [6] TURN OVER
Page 6 QUESTION 3 [5 marks] For the 50 stocks n the FTSE-50 Index, and also the precous metal GOLD, the followng varables are computed usng daly return data from an unspecfed perod: beta: beta: sg: rbar: beta coeffcent (beta coeffcent) squared standard measure of unsystematc rsk mean daly return Mean return s plotted aganst beta, and aganst sg, wth lowess smoothers supermposed n each case. The plots are shown below. (a) (b) One of the beta coeffcents seen n the left-hand graph s negatve. Ths s the beta coeffcent for the commodty GOLD. Explan why Gold s usually found to have a negatve beta coeffcent. [5] Are ether of both of the two plots shown above consstent wth the Captal Asset Prcng Model (CAPM)? Explan your answers n detal. [8] TURN OVER
Page 7 Consder the followng model wth mean daly return as the dependent varable: rbar beta beta sg u (1) 1 3 4 Model (1) s estmated n STATA wth the followng results:. regress rbar beta beta sg, robust Lnear regresson Number of obs = 51 F(3, 47) = 13.06 Prob > F = 0.0000 R-squared = 0.1178 Root MSE = 7.4e-05 Robust rbar Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- beta.0001939.0000369 5.6 0.000.000113.000666 beta -.0000686.0000155-4.43 0.000 -.0000991 -.0000381 sg.011651.0179891 0.68 0.500 -.03665.0475967 _cons.47e-06.0000191 0.13 0.897 -.0000351.00004 (c) (d) Usng the STATA results gven above, carry out two dfferent tests of CAPM. Report all relevant test statstcs and p-values. If you reject CAPM, what s the precse nature of the volaton? Is your answer consstent wth your answer to (b)? [7] Explan why the robust opton has been used wth the regress command. Why do you thnk t s mportant to do ths? [5] TURN OVER
Page 8 QUESTION 4 [30 marks] Fve years of daly data on the share prce of TRAVIS PERKINS (Industral Goods and Servces) are used to estmate two models (MODEL 1 and MODEL ). The varable r s the daly return on TRAVIS PERKINS stock. Results from estmaton of the two models are as follows. * MODEL 1. regress r l.r Source SS df MS Number of obs = 1,304 -------------+---------------------------------- F(1, 130) = 6.5 Model.00197061 1.00197061 Prob > F = 0.0108 Resdual.385097055 1,30.00095773 R-squared = 0.0050 -------------+---------------------------------- Adj R-squared = 0.004 Total.38704116 1,303.0009705 Root MSE =.017 r Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- r L1..0705905.076553.55 0.011.0163367.148444 _cons.000386.0004764 0.81 0.418 -.0005484.001307. durbna Durbn's alternatve test for autocorrelaton --------------------------------------------------------------------------- lags(p) ch df Prob > ch -------------+------------------------------------------------------------- 1 8.39 1 0.0041 --------------------------------------------------------------------------- H0: no seral correlaton * MODEL. regress r l.r l.r l3.r l4.r l5.r Source SS df MS Number of obs = 1,300 -------------+---------------------------------- F(5, 194) = 5.60 Model.00817035 5.00163407 Prob > F = 0.0000 Resdual.37764359 1,94.0009187 R-squared = 0.01 -------------+---------------------------------- Adj R-squared = 0.0174 Total.385794711 1,99.00096994 Root MSE =.01708 r Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- r L1..068511.076841.47 0.013.014014.187 L. -.0809848.07787 -.9 0.004 -.135389 -.065867 L3. -.041497.078701-0.87 0.386 -.078853.030558 L4. -.0394986.077807-1.4 0.155 -.0939987.0150015 L5. -.08969.07787-3. 0.001 -.1436673 -.0348711 _cons.000508.0004744 1.07 0.84 -.00045.0014388
Page 9. durbna Durbn's alternatve test for autocorrelaton --------------------------------------------------------------------------- lags(p) ch df Prob > ch -------------+------------------------------------------------------------- 1 0.557 1 0.4554 --------------------------------------------------------------------------- H0: no seral correlaton (a) Frst consder MODEL. Explan why the F-statstc for overall sgnfcance n MODEL amounts to a test of weak-form EMH. Report the F-statstc and the assocated p-value. How strong s the evdence aganst EMH? [5] (b) Test for seral correlaton n both MODEL 1 and MODEL. Report relevant p- values. If you have seral correlaton n one model but not n the other, can you explan ths? [5] A set of day-of-week dummes (day1=monday; day=tuesday; etc.) s added to MODEL. Ths leads to MODEL 3. The results are: * MODEL 3. regress r l.r l.r l3.r l4.r l5.r day-day5 Source SS df MS Number of obs = 1,300 -------------+---------------------------------- F(9, 190) = 3.5 Model.00854657 9.000949184 Prob > F = 0.0007 Resdual.3775053 1,90.0009443 R-squared = 0.01 -------------+---------------------------------- Adj R-squared = 0.0153 Total.385794711 1,99.00096994 Root MSE =.0171 r Coef. Std. Err. t P> t [95% Conf. Interval] -------------+---------------------------------------------------------------- r L1..068464.0778.47 0.014.014067.1861 L. -.0804135.07776 -.90 0.004 -.1348979 -.0599 L3. -.03673.07918-0.85 0.397 -.0784317.0310871 L4. -.0394871.07843-1.4 0.156 -.0940731.0150988 L5. -.090474.07771-3.5 0.001 -.1447308 -.0357639 day.001098.0015006 0.73 0.464 -.0018457.00404 day3.001579.0015011 1.05 0.93 -.0013659.00454 day4.00191.001501 0.86 0.390 -.0016534.004359 day5.0010703.0015006 0.71 0.476 -.0018737.004014 _cons -.0004996.0010614-0.47 0.638 -.005819.001586 (c) (d) Why have only four day-of-week dummes been ncluded n MODEL 3? What problem would arse f all fve were ncluded? [5] Usng an F-test, test the jont sgnfcance of the four day-of-week dummes. Note that ths requres a test of MODEL as a restrcted verson of MODEL 3. Interpret your result. Do you have (further) evdence aganst EMH? [5] TURN OVER
Page 10 You are consderng purchasng a call opton wrtten on the TRAVIS PERKINS stock prce. In order to value the opton, you need to obtan a measure of the annual volatlty of the stock prce. Descrptve statstcs for the daly returns are obtaned as follows:. summ r Varable Obs Mean Std. Dev. Mn Max -------------+--------------------------------------------------------- r 1,305.0004071.01795 -.183804.0751591 (e) (f) From the measure of daly volatlty obtaned above, deduce a measure of annual volatlty. [5] Suppose that you then nput your measure of annual volatlty nto the Black- Scholes formula, and you obtan an opton value that s consderably hgher than the market prce of the opton. What would you conclude about the opton? Would you purchase t? Explan your answer. [5] END OF PAPER
Page 11 Fnancal Econometrcs Formula Sheet The smple regresson model Consder the model: Y X u 1,..., n. The ordnary least squares estmators of and are: ˆ ( X X ) Y ( X X) ˆ Y ˆ X The ftted values of Y are gven by: Yˆ ˆ ˆ X The resduals are: u Y Yˆ ˆ The standard error of the regresson s gven by: uˆ ˆ n The estmated standard errors of ˆ and ˆ are gven by: se( ˆ ) ˆ 1 ( X X) se( ˆ ) ˆ 1 X n ( X X) Testng jont restrctons n the multple regresson model F RU RR / r 1 RU / n k ~ F r, nk
Page 1 Table 1: Crtcal values of the t-dstrbuton df = 0.10 = 0.05 = 0.05 = 0.01 = 0.005 1 3.08 6.31 1.71 31.8 63.66 1.89.9 4.30 6.97 9.93 3 1.64.35 3.18 4.54 5.84 4 1.53.13.78 3.75 4.60 5 1.48.0.57 3.37 4.03 6 1.44 1.94.45 3.14 3.71 7 1.4 1.90.37 3.00 3.50 8 1.40 1.86.31.90 3.36 9 1.38 1.83.6.8 3.5 10 1.37 1.81.3.76 3.17 11 1.36 1.80.0.7 3.11 1 1.36 1.78.18.68 3.06 13 1.35 1.77.16.65 3.01 14 1.35 1.76.15.6.98 15 1.34 1.75.13.60.95 16 1.34 1.75.1.58.9 17 1.33 1.74.11.57.90 18 1.33 1.73.10.55.88 19 1.33 1.73.09.54.86 0 1.33 1.73.09.53.85 1 1.3 1.7.08.5.83 1.3 1.7.07.51.8 3 1.3 1.71.07.50.81 4 1.3 1.71.06.49.80 5 1.3 1.71.06.49.79 6 1.3 1.70.06.48.78 7 1.31 1.70.05.47.77 8 1.31 1.70.05.47.76 9 1.31 1.70.04.46.76 30 1.31 1.70.04.46.75 40 1.30 1.68.0.4.70 50 1.30 1.68.01.40.68 60 1.30 1.67.00.39.66 70 1.9 1.67 1.99.38.65 80 1.9 1.66 1.99.37.64 90 1.9 1.66 1.99.37.63 100 1.9 1.66 1.98.36.63 15 1.9 1.66 1.98.36.6 150 1.9 1.65 1.98.35.61 00 1.9 1.65 1.97.35.60 1.8 1.64 1.96.33.58
Page 13 Table : Crtcal values of the F-dstrbuton (=0.05) df 1=1 3 4 5 6 7 8 10 15 df =1 161.4 199.5 15.7 4.6 30. 34.0 37.0 38.9 41.9 45.9 18.51 19.00 19.16 19.5 19.30 19.33 19.4 19.37 19.40 19.43 3 10.13 9.55 9.8 9.1 9.01 8.94 8.89 8.85 8.79 8.70 4 7.71 6.94 6.59 6.39 6.6 6.16 6.09 6.04 5.96 5.86 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.8 4.74 4.6 6 5.99 5.14 4.76 4.53 4.39 4.8 4.1 4.15 4.06 3.94 7 5.59 4.74 4.35 4.1 3.97 3.87 3.79 3.73 3.64 3.51 8 5.3 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.35 3. 9 5.1 4.6 3.86 3.63 3.48 3.37 3.9 3.3 3.14 3.01 10 4.96 4.10 3.71 3.48 3.33 3. 3.14 3.07.98.85 11 4.84 3.98 3.59 3.36 3.0 3.09 3.01.95.85.7 1 4.75 3.89 3.49 3.6 3.11 3.00.91.85.75.6 13 4.67 3.81 3.41 3.18 3.03.9.83.77.67.53 14 4.60 3.74 3.34 3.11.96.85.76.70.60.46 15 4.54 3.68 3.9 3.06.90.79.71.64.54.40 16 4.49 3.63 3.4 3.01.85.74.66.59.49.35 17 4.45 3.59 3.0.96.81.70.61.55.45.31 18 4.41 3.55 3.16.93.77.66.58.51.41.7 19 4.38 3.5 3.13.90.74.63.54.48.38.3 0 4.35 3.49 3.10.87.71.60.51.45.35.0 1 4.3 3.47 3.07.84.68.57.49.4.3.18 4.30 3.44 3.05.8.66.55.46.40.30.15 3 4.8 3.4 3.03.80.64.53.44.37.7.13 4 4.6 3.40 3.01.78.6.51.4.36.5.11 5 4.4 3.39.99.76.60.49.40.34.4.09 6 4.3 3.37.98.74.59.47.39.3..07 7 4.1 3.35.96.73.57.46.37.31.0.06 8 4.0 3.34.95.71.56.45.36.9.19.04 9 4.18 3.33.93.70.55.43.35.8.18.03 30 4.17 3.3.9.69.53.4.33.7.16.01 40 4.08 3.3.84.61.45.34.5.18.08 1.9 50 4.03 3.18.79.56.40.9.0.13.03 1.87 60 4.00 3.15.76.53.37.5.17.10 1.99 1.84 70 3.98 3.13.74.50.35.3.14.07 1.97 1.81 80 3.96 3.11.7.49.33.1.13.06 1.95 1.79 90 3.95 3.10.71.47.3.0.11.04 1.94 1.78 100 3.94 3.09.70.46.31.19.10.03 1.93 1.77 15 3.9 3.07.68.44.9.17.09.01 1.91 1.75 150 3.90 3.06.66.43.7.16.08.00 1.89 1.73 00 3.89 3.04.65.4.6.14.06 1.98 1.88 1.7 3.84 3.00.60.37.1.10.01 1.94 1.83 1.67 END OF MATERIALS