Name Financial Economerics Jeffrey R. Russell Miderm Winer 2011 You have 2 hours o complee he exam. Use can use a calculaor. Try o fi all your work in he space provided. If you find you need more space coninue on he back of he page. The las page conains a se of formulas ha migh be useful on he exam. No oher noes or exs are permied. Sudens in my class are required o adhere o he sandards of conduc in he GSB Honor Code and he GSB Sandards of Scholarship. The GSB Honor Code also require sudens o sign he following GSB Honor pledge, "I pledge my honor ha I have no violaed he Honor Code during his examinaion. Please sign here o acknowledge 1
1. (10 poins) Idenify which series is an MA(1) model and which is an AR(1) model. Series 1 Series 2 For series 1 and 2 below, idenify which ACF and PACF corresponds o an MA model and which is an AR model. Series 1 Series 2 2
2. (18 poins) For he AR(1) model y +1 =1+.95y +ε +1 where ~ iid N (0,2) find he following. a. Wha is he uncondiional mean of y? b. Wha is he uncondiional variance of y? c. Wha is he mean of y +1 given ha y =9? d. Wha is he variance of y +1 given ha y =9? e. Find he k-sep ahead forecas of y +k given y =9. f. Find he k-sep ahead forecas error variance as a funcion of k. 3
3. (12 poins) Consider he MA(2) model y.5.9 1.22 where 2 ~ iid N 0,.2. a. Wha is he uncondiional mean of y? b. Wha is he uncondiional variance of y? c. Find he 1, 2, 3, and 4 sep ahead forecas of y +k, given =.8 and -1 =.7. d. Wha would happened if you aemped o fi an AR(p) model o daa generaed by his MA(2) model above? Jus give me a general idea. 4
4. (8 poins) Consider he Augmened Dickey-Fuller es for a uni roo associaed wih he Dollar/Euro foreign exchange rae. Wha is he null hypohesis being esed here? Wha do you conclude? Be specific. 5
5. (12 poins) Consider a random walk model for he log of he S&P500 index. The. monhly reurns have a mean of.008 and a sandard deviaion of.0577. The index level (no log level) a he close on he las in sample day was 1331.29. a. Find he k-sep ahead forecas of he log S&P500 index level as a funcion of k and he iniial log index. b. Find he k-sep ahead forecas error variance associaed wih par a. as a funcion of k. c. Find he expeced reurn over he nex k days. You should be able o wrie he forecas a horizon k as a funcion of k. d. Find a 95% CI for he reurn over he nex k days. You should be able o wrie his inerval as a funcion of k. 6
6. (15 poins) Consider he following oupu from a hreshold (asymmeric) GARCH(1,1) model esimaed on daily reurns daa. The las observed reurn in he sample is r T =-.035 and he volailiy (sandard deviaion) on he las day in he sample h is.024 where T denoes he las observaion in he T sample. a. Find he one sep ahead ou of sample forecas. You should ge a number here. b. Recall ha for a sandard normal, Pr Z 2.33.01. Find he one day ahead 1% Value a Risk 7
c. For a disribuion wih 5.73 degrees of freedom we have Pr.01 3.36. The variance of a disribuion is v v 2. If he GARCH model above used a disribuion wih 5.73 degrees of freedom, wha would he one day ahead 1% Value a Risk be? d. Now, le z r and we find ha he empirical fracion of z s smaller han 2.8% is h 1%. Wha is he boosrapped one day ahead 1% Value a Risk e. If you used he Value a Risk from par b, would your risk be oversaed or undersaed. Discuss his in a couple of senence. 8
7. (13 poins) Consider he reurns of wo asses ha each follow a GARCH process: r h z and r2, h2, z2, for asses 1 and 2 respecively. z 1 and z 2 are iid 1, 1, 1, mean zero and variance 1. Addiionally, 1, 2, correlaed in differen ime periods i.e 1, 2, s cov z, z.5 bu z 1 and z 2 are no cov z, z 0 for s. a. Find he condiional covariance beween reurns for asse 1 and asse 2, i.e. find r1, r2, Er1, r2, F 1 cov,,. b. Wha is he condiional correlaion beween reurns on asse 1 and reurns on asse Er1,, r2, F 1 2 where corr 1,, r2, F 1? 2 2 E r F E r F 1, 1 2, 1 c. Suppose ha you build a porfolio ha pus.5 weigh in asse 1 and.5 weigh in asse 2. The reurns on he porfolio in ime will be p.5 r1,.5r2,. Find he 2 condiional variance of he porfolio 1 E p F. 9
o 8. (12 poins) Consider he model for observed financial prices p p where p is he rue price (he fair marke value) and is a muliplier ha moves he observed price up or down a lile bi away from he fair marke value due o marke microsrucure effecs. The fair marke value follows a random walk 2 model: ln pln p1. N 0,. ln follows an AR(1) is iid 2 where ~ iid N 0, model 1 wih 1., and are independen. a. Wha is he uncondiional mean of he observed coninuously compounded reurns? b. Wha is he uncondiional variance of he observed coninuously compounded reurns? 10
c. Wha are he firs 4 auocorrelaions of he observed coninuously compounded reurns? d. Wha do he auocorrelaions ell you abou which ype of ARMA model would be appropriae? 11
Forecass Saionary 1 <1 Y Mean rever k k k 1 1 = Y 1 Non-saionary 1 =1 Trend up or down depending on sign of 0 k Y Y k0 Forecas errors Iniially increase wih he forecas horizon. Var e k 1 1 2k 1 2 2 1 Increases wih he forecas horizon. Var e k 2 k 12