Module Contact: Dr P Moffatt, ECO Copyright of the University of East Anglia Version 2

Transcription

1 UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PG Examnaton FINANCIAL ECONOMETRICS ECO-M017 Tme allowed: 2 hours Answer ALL FOUR questons. Queston 1 carres a weght of 25%; Queston 2 carres 30%; Queston 3 carres 25%; Queston 4 carres 20%. Marks awarded for ndvdual parts are shown n square brackets. A formula sheet and t-tables are attached to the examnaton paper. Notes are not permtted n ths examnaton. Do not turn over untl you are told to do so by the Invglator. ECO-M017 Module Contact: Dr P Moffatt, ECO Copyrght of the Unversty of East Angla Verson 2

2 Ths page s delberately left blank. Page 2

3 Queston 1. Page 3 ALL WORKING SHOULD BE SHOWN IN YOUR ANSWER TO THIS QUESTION. The share prce of Standard Lfe (Insurance) was followed for a perod of sx months. The percentage monthly change n the stock market ndex (X), and the percentage monthly return on Standard Lfe stock (Y) are presented n the followng table: Month Market(X) Standard Lfe(Y) January -3-4 February -1 0 March 2 1 Aprl 6 5 May 3 2 June 5 8 (a) Obtan estmates of and n the smple regresson model: Y X u t 1,,6 t t t Var u 2 t Report the beta coeffcent for Standard Lfe 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 Standard Lfe s an aggressve stock, a defensve stock, or nether? Is ths what you would expect for Standard Lfe? [8] TURN OVER

4 rbar Queston 2. Page 4 For each of the 100 companes n the AIM100 ndex, the followng nformaton s found (usng daly data from an unspecfed perod): beta: rbar: beta coeffcent mean (daly) return A scatter plot of the mean return aganst the beta, wth a lowess smoother supermposed, s shown below. Lowess smoother beta bandwdth =.8 (a) (b) Whch features of the plot (f any) are consstent wth the Captal Asset Prcng Model (CAPM)? Whch (f any) are nconsstent wth CAPM? Do you antcpate the problem of heteroscedastcty? Explan your answer. [5] A smple regresson of mean return on beta yelds the followng results:. regress rbar beta Source SS df MS Number of obs = F( 1, 98) = Model e e-07 Prob > F = Resdual e e-08 R-squared = Adj R-squared = Total e e-08 Root MSE =.0001 rbar Coef. Std. Err. t P> t [95% Conf. Interval] beta _cons Interpret the estmate of the ntercept parameter and use t to deduce the annual rsk-free rate (assume there are 260 tradng days n a year). Use a smlar approach to nterpret the slope estmate. [5] (c) The regresson s extended to ntroduce as a second explanatory varable the square of beta (named beta2). The results are:

5 Page 5. gen beta2=beta^2. regress rbar beta beta2 Source SS df MS Number of obs = F( 2, 97) = Model e e-07 Prob > F = Resdual e e-08 R-squared = Adj R-squared = Total e e-08 Root MSE =.0001 rbar Coef. Std. Err. t P> t [95% Conf. Interval] beta beta e-06 _cons hettest Breusch-Pagan / Cook-Wesberg test for heteroskedastcty Ho: Constant varance Varables: ftted values of rbar ch2(1) = Prob > ch2 = Usng a 2-taled t-test, test the sgnfcance of the effect of the varable beta2. Does the result amount to a rejecton of CAPM? Explan your answer.[5] (d) (e) Explan why the problem of heteroscedastcty s antcpated on a theoretcal bass n ths model. Does the test followng the estmaton above confrm that heteroscedastcty s ndeed a problem? [5] The same regresson as n (c) s performed wth the robust opton, wth the followng results:. regress rbar beta beta2, robust Lnear regresson Number of obs = 100 F( 2, 97) = Prob > F = R-squared = Root MSE =.0001 Robust rbar Coef. Std. Err. t P> t [95% Conf. Interval] beta beta _cons In what sense does ths set of results overcome the problem of heteroscedastcty. Whch numbers n the table have changed as a result of usng the robust opton? [5] (f) Conduct a test of CAPM usng the results of (e). In what way does the concluson dffer from that of (c)? [5] TURN OVER

6 gold Queston 3. Page 6 Ths queston s concerned wth the prce of the commodty gold. Daly data on the gold prce was obtaned for the 5 year perod 8 Jan 2008 to 8 Jan 2013, and s represented n what follows by the varable gold. A tme-seres plot of the varable s shown below. 01jan jan jan jan jan jan2013 date (a) (b) Wth reference to the tme seres plot, descrbe the evoluton of the gold prce over the 5 year perod. Does t have the characterstcs of a random walk? [5] The followng STATA results were obtaned:. regress gold l.gold l2.gold l3.gold Source SS df MS Number of obs = F( 3, 1299) =. Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = gold Coef. Std. Err. t P> t [95% Conf. Interval] gold L L L _cons test (l1.gold=1) ( 1) L.gold = 1 F( 1, 1299) = 1.03 Prob > F = test l2.gold l3.gold ( 1) L2.gold = 0 ( 2) L3.gold = 0 F( 2, 1299) = 0.59

7 Page 7 Prob > F = durbna Durbn's alternatve test for autocorrelaton lags(p) ch2 df Prob > ch H0: no seral correlaton Followng the regresson results above are three tests of the (weak-form) Effcent Market Hypothess (EMH). Explan brefly why each of these tests amounts to a test of EMH. In each of the three cases, s EMH accepted or rejected? [5] (c) A set of day-of-week dummes were added to the model of (b) (day1=monday; day5=frday), wth the results:. regress gold l.gold l2.gold l3.gold day2-day5 Source SS df MS Number of obs = F( 7, 1295) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = gold Coef. Std. Err. t P> t [95% Conf. Interval] gold L L L day day day day _cons test day2 day3 day4 day5 ( 1) day2 = 0 ( 2) day3 = 0 ( 3) day4 = 0 ( 4) day5 = 0 F( 4, 1295) = 0.94 Prob > F = Explan why day1 has been excluded from the model. What s the nterpretaton of the coeffcents of the ncluded day dummes? Is there a weekend effect n the market for gold? [5] TURN OVER

8 (d) Page 8 Three lags of the copper prce were added to the model of (b), wth the results:. regress gold l.gold l2.gold l3.gold l.copper l2.copper l3.copper Source SS df MS Number of obs = F( 6, 1296) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = gold Coef. Std. Err. t P> t [95% Conf. Interval] gold L L L copper L L L _cons test l.copper l2.copper l3.copper ( 1) L.copper = 0 ( 2) L2.copper = 0 ( 3) L3.copper = 0 F( 3, 1296) = 4.81 Prob > F = Explan why the F-test conducted followng estmaton amounts to a test of the sem-strong form EMH. Is sem-strong form EMH accepted or rejected? [5] (e) A VAR(3) model of the gold prce and the copper prce s estmated, and a Granger test s performed. The results are as follows:. var gold copper, lags(1 2 3) Vector autoregresson : :. vargranger Granger causalty Wald tests Equaton Excluded ch2 df Prob > ch gold copper gold ALL copper gold copper ALL Explan how the results of the Granger test relate to your answer to (d). What addtonal nformaton does the Granger test convey about the relatonshp between the gold prce and the copper prce? [5]

9 r Queston 4. Page 9 Ths queston s concerned wth a European call opton wrtten on the FTSE250 share ndex, on 8 January The opton has a strke of 8300 and has 60 days to expry. The FTSE250 ndex on 8 January 2013 s (a) Draw a pay-off dagram for ths opton, showng the pay-off at expry aganst the underlyng ndex at expry. Is ths opton n the money, or out of the money? Explan your answer. [5] In order to derve the value of the opton of (a), we requre a measure of the volatlty of the underlyng ndex. We therefore consder the return (r) on the FTSE250 ndex. Fve years of daly returns are plotted below. 01jan jan jan jan jan jan2013 date Summary statstcs for the daly return are:. summ r Varable Obs Mean Std. Dev. Mn Max r (b) Assumng that there are 260 tradng days n the year, deduce an estmate of the annual volatlty of the FTSE250 ndex. [5] TURN OVER

10 (c) Page 10 The Black-Scholes formula s used to compute the value of the opton at varous dfferent values for volatlty. The results are: volatlty call value From the nformaton n the table, deduce as accurately as possble the value of ths call opton. Explan your answer. [5] (d) It s often suggested that the ARCH/GARCH framework s superor to the method used n (b) as a means of computng the volatlty of the prce or ndex underlyng an opton. The followng results are obtaned from a GARCH(1,1) model:. arch r, arch(1) garch(1) : : Sample: Number of obs = 1305 Dstrbuton: Gaussan Wald ch2(.) =. Log lkelhood = Prob > ch2 =. OPG r Coef. Std. Err. z P> z [95% Conf. Interval] r _cons ARCH arch L garch L _cons 1.72e e e e-06 Interpret the results. Do the results provde confrmaton that GARCH s ndeed a superor approach? Explan your answer. [5] END OF PAPER

11 Page 11 The smple regresson model Fnancal Econometrcs Formula Sheet Consder the model: Y X u 1,..., n. The ordnary least squares estmators of and are: ˆ ( X X ) Y ( X X) 2 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: ˆ uˆ2 n 2 The estmated standard errors of ˆ and ˆ are gven by: se( ˆ ) ˆ 1 ( X X) 2 se( ˆ ) ˆ 2 1 X n ( X X) 2 Testng jont restrctons n the multple regresson model F 2 2 RU RR / r 2 1 RU / n k ~ F r, nk

12 Table 1: Crtcal values of the t-dstrbuton Page 12 df = 0.10 = 0.05 = = 0.01 =