FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE - MODULE 2 General Exam - June 2012

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1 FINANCIAL ECONOMETRICS AND EMPIRICAL FINANCE - MODULE 2 General Exam - June 2012 Time Allowed: 105 Minutes Family Name (Surname). First Name. Student Number (Matr.) Please answer all the questions by choosing the most appropriate alternative(s) or by writing your answers in the spaces provided. There might be more than one correct answer(s) for each of the multiple choice questions. Each selected alternative that is a correct answer will be awarded one point. Wrong answers will be penalized with minus 0.5 point. Correct answers not selected and questions that have been left blank will receive zero points. Only answers explicitly reported in the appropriate box will be considered. In the multiple choice case, report your selection by writing one or more of the letters A, B, C, D, E, and F, in BLOCK CAPITAL LETTERS. No other answers on the exam paper or indication pointing to potential answers will be taken into consideration. Section 1 Question 1.1 What does the following graph report? expected rho hat sample size (A) The result of a Monte-Carlo experiment. (B) The bias of the OLS estimate of the parameter associated with the lagged dependent variable in a first order autoregressive process. (C) The results of a recursive estimation exercise. (D) The bias of the OLS estimate of the constant in a first order autoregressive process. (E) The result of a bootstrap exercise. (F) The empirical cumulative density function of the estimate of the autoregressive parameter in a first order autoregressive process. 1

2 Question 1.2 Indicate which of the following statements is/are correct: (A) In a standard mean-variance asset allocation problem the weights of the tangency portfolio do not depend on the coefficient of risk aversion. (B) The Black and Litterman approach delivers weights that are more volatile than those obtained by solving a standard mean-variance asset allocation problem. (C) The Black and Litterman approach will lead the majority of investors to hold the market portfolio. (D) At some investment horizons bonds can be riskier than stocks. (E) In the Black and Litterman approach an investor who is extremely confident in his views on the return of an asset would also be extremely long in that asset. (F) The Black and Litterman approach might lead some investor to concentrate all their portfolio in a single asset. Question 1.3 Consider a model in which portfolio returns follow the simple time homogeneous, IID process: = + IID () where is some distribution left unspecified for our purposes but parameterized by. Fromasample of data on portfolio returns, we estimate the following moments: mean = 0563 variance = 5230 skewness = 0607 kurtosis = 7038 Indicate which of the following statements is/are correct: (A) The estimated model = (0 1) is the most likely model to have generated the data according to the sample moments presented above. (B) We could use the method of moments and hence we will not need to assume any specific distribution for to perform estimation of this model. (C) Given the information we have, adding some structure on (), it is likely that we can use method of moments estimation to estimate the unknown parameters of the model. (D) The estimated model = (5486) is the most likely model (among the choices A, D, and E) to have generated the data according to the sample moments presented above. (E) The estimated model = (4853) is the most likely model (among the choices A, D, and E) to have generated the data according to the sample moments presented above. 2

3 Question 1.4 Consider the Cornish-Fisher approximation for the inverse CDF function in correspondence to a critical value 1 : 1 = Φ (Φ 1 ) (Φ 1 ) 3 3Φ (Φ 1 ) 3 5Φ 1, where 1 indicates the sample skewness coefficient, 2 is the sample excess kurtosis, and Φ 1 is the inverse Gaussian CDF in correspondence to a critical value. With reference to a time series of S&P 500 returns for which at time the forecast of time + 1 volatility is +1 =235%, skewness is -0.68, and the time the forecast of time +1meanis +1 =013%, a colleague of yours has stated the she has just computed a Cornish-Fisher 1% VaR of 9.14%. However, your colleague has forgotten to state what the kurtosis of S&P 500 returns is in her data. Note that Φ = Please indicate which of the following statements is/are correct. (A) The Cornish-Fisher 1% VaR estimate as well as the information provided on the properties of S&P 500 returns imply that their excess kurtosis is approximately (B) The Cornish-Fisher 1% VaR estimate as well as the information provided on the properties of S&P 500 returns imply that their excess kurtosis is approximately (C) The Cornish-Fisher 1% VaR estimate as well as the information provided on the properties of S&P 500 returns imply that their kurtosis is approximately (D) The Cornish-Fisher 1% VaR estimate as well as the information provided on the properties of S&P 500 returns imply that their kurtosis is approximately (E) None of the above are correct. 3

4 Question 1.5 By using the quasi maximum likelihood (QML) method, you have estimated the following univariate time series model for S&P 500 daily returns: +1 = ˆ IID (0 1) ˆ 2 +1 = ( ) (ˆ 2 ) +1 = ( ˆ 2 ) ( 2 ) where (0 1) is some unknown (and therefore not specified) distribution with zero mean and unit variance. However, subsequent tests have shown that the sample estimate [ d ] 6= 0,where ˆ, and that this sample estimate of [ ] has a p-value of using a standard Ljiung-Box test. Moreover, a simple Jarque-Bera test applied on the standardized residuals has revealed that the null hypothesis of normality can be rejected with a p-value of Indicate which of the following statements is/are correct. (A) The QML estimates are unlikely to be valid (i.e., to possess the stated properties of consistency and asymptotic normality) because there is evidence that the standardized residuals resulting from this model are not IID; because the lack of the IID property points to the fact that either the conditional mean function or the conditional variance function have been misspecified, then QML will not valid for this problem. (B) The QML estimates are unlikely to be valid (i.e., to possess the stated properties of consistency and asymptotic normality) because there is evidence that the standardized residuals resulting from this model are not normally distributed, while the QML method assumes normality. (C) In this case, the QML estimates will coincide with the classical ML estimates because this always occurs under the standard distribution assumed in this exercise. (D) The QML estimates will be valid (i.e., they will possess the stated properties of consistency and asymptotic normality) because the component GARCH(1,1) process that has been estimated is stationary in both its components. (E) None of the above. 4

5 Question 1.6 You have estimated the following first-order, three-state Markov switching model for S&P 500 monthly percentage returns obtaining the following point estimates: +1 = (0 1) 2004 if +1 = if +1 = = 0389 if +1 =2 +1 = 5004 if +1 =2 P = if +1 = if +1 = Please indicate which of the following statements is/are correct: (A) Because of the reducible nature of the underlying Markov chain, the ergodic probability of regime 1 is approximately 0.18, the ergodic probability of regime 3 is approximately 0.92, while the ergodic probability of regime 2 is zero. (B) Formally, this is a MSIH(3) model in which the first regime is a bear state in which volatility is high and the third regime is a bull state in which volatility is low; as far as the second regime is concerned, unfortunately, the underlying three-state Markov chain turns out to be reducible to a singlestate chain, in the sense that when the process enters in state 2, it remains in this state for the rest of the time, without any chance to re-enter either state 1 or state 3. (C) The first regime has an average duration of approximately 4.2 months; the third regime has an average duration of approximately 18.9 months; the second regime has an average duration of 11.2 months in spite of its zero ergodic probability. (D) Formally, this is a MSIH(3) model in which the firstregimeisabearstateinwhichvolatility is high and the third regime is a bull state in which volatility is low; as far as the second regime is concerned, it has intermediate properties between the bull and bear regimes but the underlying threestate Markov chain turns out to be reducible to a two-state chain, in the sense that when the process enters either state 1 or 3, it remains in this sub-set of states for the rest of the time, without any chance to re-enter state 2. (E) The estimated structure for the transition matrix P implies that the underlying three-state Markov chain is irreducible. 5

6 Question 1.7 You have estimated a full BEKK GARCH(1,1) with VAR(1) conditional mean function model for Italian and US stock returns, R +1 [+1 +1 ]0, R +1 = μ + ΦR + z +1 z +1 IID (0 Ω +1 ) Ω +1 = CC 0 + AR R 0 A + BΩ +1 B 0, with A, B, andc non-negative and symmetric, that is characterized by 9 parameters in the BEKK GARCH(1,1) part. You know that the model has been estimated by maximum likelihood on a total of 240 observations (i.e., 10 years of monthly data for both Italian and US stock returns) and the resulting Hannan-Quinn information criterion equals Indicate which of the following statements is/are correct: (A) Because the bivariate VAR(1) conditional mean implies a need to estimate 5 parameters, the formula for the Hannan-Quinn information criterion implies that the maximized log-likelihood function must have been , while the saturation ratio for this model is (B) Because the bivariate VAR(1) conditional mean implies a need to estimate 6 parameters, the formula for the Hannan-Quinn information criterion implies that the maximized log-likelihood function must have been , while the saturation ratio for this model is 16. (C) Because estimation has been performed by maximum likelihood, we know that the Cramer- Rao lower bound for the covariance matrix will be reached, and this implies that by construction the maximized log-likelihood function must have been zero, while the saturation ratio for this model is 16. (D) The formula for the Hannan-Quinn information criterion implies that the maximized loglikelihood function must have been , while the saturation ratio for this model is (E) None of the above. 6

7 Question 1.8 Your boss has assigned to you a simple tasks that aims at estimating the mean and variance of S&P 500 (continuously compounded) returns. Assume that S&P 500 returns come from a simple identically and independently distributed process. Your boss cares for you producing estimates of both mean and variance that exploit as much as possible of all the available information on the time series behavior of the S&P 500. Because the necessary data need to be purchased, your boss has offered four different alternatives but has left you free to select only one of four data sets: (i) 10 years of monthly data (ii) 8 months of data sampled at a 1-hour frequency, expressed by highly liquid markets (iii) 130 years of annual data (iv) 5 years of daily data Assume that in a trading day there are approximately 9 hours and therefore 8 returns per day; that in any month there are on average 22 trading days; that in a year there are on average 252 trading days. Indicate which of the following is/are correct. (A) If your boss cares more for using the richest possible information set when estimating the mean than when estimating the variance, then data set (iii) should be purchased. (B) If your boss cares more for using the richest possible information set when estimating the variance than when estimating the mean, then data set (iv) should be purchased. (C) If your boss cares more for using the richest possible information set when estimating the variance than when estimating the mean, then data set (iii) should be purchased. (D) If your boss cares more for using the richest possible information set when estimating the variance than when estimating the mean, then data set (ii) should be purchased. (E) For both the mean and the variance it is best to purchase the data set with the highest number of usable observations and this is data set (ii). 7

8 Question 1.9 Consider the RiskMetrics style model of conditional covariance introduced in the lectures: +1 =(1 ) + (0 1). Please indicate which of the following statements is/are correct: (A) Because of the link between the coefficients (1 ) on the cross return products and on the lagged covariance, the RiskMetrics model implies an explosive covariance process, which will diverge trending up or down in a deterministic way. (B) When compared to a GARCH(1,1)-type model for conditional covariance, +1 = + +, because it is possible to show that +1 = (1 )+ P =0,then the GARCH process +1 simplifies to the RiskMetrics process if and only if =0, =(1 ) 1, and = ; because the persistence of a GARCH(1,1) process is measured by the sum +, itisalso clear that =(1 ) 1 and = implies that + =(1 ) 1 + 1which means that an exponentially smoothed process for conditional covariance implies that covariance is stationary. (C) The RiskMetrics model can be re-written as +1 =(1 P ) 1 =0 1. (D) When compared to a GARCH(1,1)-type model for conditional covariance, +1 = + +, because it is possible to show that +1 = (1 )+ P =0, then the GARCH process +1 simplifies to the RiskMetrics process if and only if =0, =1, and = ; because the persistence of a GARCH(1,1) process is measured by the sum +, itisalso clear that =1 and = implies that + =1 + = 1 which means that an exponentially smoothed process for conditional covariance implies that covariance is non-stationary. (E) The RiskMetrics model can be re-written as +1 =(1 ) P =0. Section 2 Question 2.1 Consider the following model for the daily returns of a mutual fund, +1 and of its benchmark, +1 : +1 = = = ³ IID (0 1) +1 IID (0 1) 2.1(a) Derive the conditional ([+1 ]) and the unconditional mean of the daily mutual fund returns. 8

9 . 2.1(b) Are daily mutual fund returns homoskedastic? Explain your answer. 2.1(c) Are the daily mutual fund returns in excess of the benchmark (+1 +1 )homoskedastic? Explain your answer. 9

10 2.1(d) What is the long-run variance of daily fund returns? Question 2.2 Consider the linear regression model: = + + (Model 1) Youthenaddatimeseries to both and.thus: = + and = + where stands for new. Note that ( ) 6= ( ) 6= 0and( )=0. Consider now the new regression model: = + + (Model 2) Briefly discuss the following statements, showing your work to prove whether these are right or wrong. If needed, please state your assumptions. 2.2(a) The OLS estimates of the parameters and will be the same. 10

11 2.2(b) The 2 from the two regressions will be the same. Question 2.3 Consider the following bivariate dynamic process for two series in logarithm: = (notice is on the right hand side) (1) " 1 2 = (2) # "Ã! Ã!# (3) Answer the following questions justifying your answers. 2.3(a) Is it true that and are conditionally uncorrelated? 11

12 2.3(b) Is it true that and are cointegrated? 2.3(c) Is it true that and will grow (in continuously compounded terms, i.e., when is the rate of growth of any variable ) at the same rate in the long-run? Question 2.4 Consider an ARCH(1) RiskMetrics Dynamic Conditional Correlations model for the case of two assets/securities, " # q " # q = q q = 2 1 = q q =(1 ) , q where ES stands for exponential smoothing, 0 1, (0 1), and 1 = (a) Show that and particular, show that you cannot find any value of for which the two models become identical. Use ( =12) define different processes that can never be identical: in this finding to discuss the fact that in a DCC we define [ q 11 q 22 ] in spite of the fact 12

13 that 1 [ 1 2 ]= (b) Show that conditioning on knowledge of the ARCH process followed by returns, the process for the auxiliary variable 12 can be re-written as: 12 (1 ) = [Hint: Start by re-writing the process for 12 from there] X = as an infinite moving average of past shocks and work 13

14 Question 2.5 Consider the following estimated MSIVARH(2,1) model for the bivariate vector of US and Japanese stock percentage index returns, R +1 [+1 +1 ]0 where +1 = 1 2isafirst-order, two-state irreducible and ergodic Markov chain that characterizes returns in both markets: " # +1 = ˆμ ˆΦ +1 R + ˆΣ +1 ² +1 = " # " #" q ( +1 # + ) 2 +1 " where ˆΣ +1 is the Choleski factor of the (estimated) covariance matrix of the vectors of shocks ² +1, #, ² +1 (0 I 2 ), and " ˆP = # +1 = +1 = ( ( ( 133 if +1 =1 095 if 045 if +1 =2 +1 =1 045 if +1 = 064 if +1 =2 +1 =1 +1 = 0 if +1 =2 ( ( 085 if +1 = if 0 if +1 =2 +1 =1 +1 = 2484 if +1 =2 2.5(a) Write the model in extensive form, i.e., without using vector and matrices and simply by developing one statistical model (equation) for +1 and one statistical model (equation) for +1 from the estimates provided above. Make sure to disentangle what statistical model (equation) applies in regime 1 and which one applies in regime 2, this for each of the two markets under consideration (i.e., the total is therefore 4, 2 markets 2 regimes). 14

15 . 2.5(b) Is there any dynamic linear contagion pattern from the US to Japan in state 2? Is there any simultaneous contagion pattern from the US to Japan in state 2 that goes through the structure of the shocks? What about linear contagion patterns from Japan to the US in state 1? Compute the unconditional mean of Japanese monthly returns in state 2, conditioning on remaining in this regime forever (i.e., only use the estimates that apply to regime 2). 2.5(c) Suppose now that a second round of estimations triggered by the availability of better data or better estimation methods, has given the following estimated model: " # +1 = ˆμ ˆΣ +1 ² +1 i.e., the estimation has revealed that in fact the Markov switching VAR(1) matrix is not different from a matrix of zeros, Φ +1 = O in all regimes. The other estimates ˆμ +1 and ˆΣ +1 are the same as the ones reported above. Can you conclude that there is potential of contagion from the US to Japan and vice-versa? Make sure to carefully explain how such a contagion would take place, i.e., whether dynamically or not. 15

16 Section 3 Question 3.1 Consider the following lines of MATLAB code: reps=100; obs=200; init=15; x=nan(obs,reps); beta1=nan(obs-init,reps); rho=0.9; alpha=0; vol=1; x(1,:)=alpha/(1-rho); u1=vol.*normrnd(0,1,obs,reps); for j=1:reps for i=2:obs x(i,j)=rho*x(i-1,j)+alpha*(1-rho)+u1(i,j)*sqrt((1-rhoˆ2)*vol); end for k=init+1:obs X=NaN(k-1,2); X(:,1)=ones; X(:,2)=x(1:k-1,j); reg1=ols(x(2:k,j), X(1:k-1,:)); beta1(k-init,j)=reg1.beta(2); end end dim=16:200; av beta1 hat(:,1)=mean(beta1,2); State which of the following statement(s) is/are correct: (A) The code will run 100 regressions. (B) The code will run over regressions. (C) The code will run no regressions as no data are imported. (D) The code performs a Monte-Carlo experiment to evaluate the consistency of parameters estimated in a MA model. (E) The unconditional variance of all the series generated in the code is the same and it is 1. (F) The unconditional mean of all the series generated in the code is the same and it is alpha. 16

17 Question 3.2 In one of your lab sessions, Quasi-Maximum Likelihood estimation of a simple GARCH(1,1) DCC(1,1) model has produced the following output printed at the screen along with the following pictures: Number of variables: 2 Functions Objective: dcc mvgarch likelihood Gradient: finite-differencing Hessian: finite-differencing (or Quasi-Newton) Constraints Nonlinear constraints: do not exist Number of linear inequality constraints: 1 Number of linear equality constraints: 0 Number of lower bound constraints: 2 Number of upper bound constraints: 0 Algorithm selected medium-scale: SQP, Quasi-Newton, line-search Max Line search Directional First-order Iter F-count f(x) constraint steplength derivative optimality e Local minimum possible. Constraints satisfied. fmincon stopped because the size of the current search direction is less than twice the default value of the step size tolerance and constraints are satisfied to within the selected value of the constraint tolerance. ============================================ GARCH(1,1)-DCC(1,1) PARAMETERS omega US returns with robust S.E t-stat alpha US returns with robust S.E t-stat beta US returns with robust S.E t-stat omega UK returns with robust S.E t-stat alpha UK returns with robust S.E t-stat beta UK returns with robust S.E t-stat omega German returns with robust S.E t-stat alpha German returns with robust S.E t-stat

18 beta German returns with robust S.E t-stat alpha DCC equation with robust S.E t-stat beta DCC equation with robust S.E t-stat MaxLikelihood: ============================================ 3.2(a) Describe the steps of estimation of a GARCH(1,1) DCC(1,1) making sure that you write down the structure of the model for this particular case. What does it mean that estimation has been performed by quasi maximum likelihood? Why has this method been chosen over maximum likelihood? Was maximum likelihood estimation possible in this case? 18

19 . 3.2(b) In the output screen reported above, what is ()? Is the optimization algorithm minimizing or maximizing such a ()? Why so? Has the optimization been performed under constraints? If so, which constraints? 3.2(c) Why are the predicted monthly mean returns functions completely flat over time? Are the variance processes for Germany, US, and UK stationary? If so, compute their unconditional variances. [Hint: please do NOT set to zero coefficients with t-stats less than 2] 19

20 3.2(d) Why is the DCC process not featuring any estimated constant terms? Is the DCC process for the a stationary one? If not, what other type of model could have been estimated? 20

21 Question 3.3 A series of estimation exercises concerning unconditional variances for a number of portfolios (under theassumptionofnormality) haveproduced5different estimates of unconditional 5% Value-at-Risk (VaR). These are reported in the bar plot below. In particular, the left-most bar that measures Unc. Ptf. VaR has been obtained in a passive fashion by first computing realized portfolio returns over a sample of daily data and then estimating the unconditional variance from such returns; the right-most bar that measures Weighted Avg. is instead obtained as a weighted average of the three separate unconditional VaR measures for Germany, the US, and the UK, estimated over the same sample and each carrying a weight equal to their weight on the portfolio (these are the three bars labeled Unc. Ger VaR, Unc. US VaR, and Unc. UK VaR ). The unconditional correlations for pairs of countries were for the pair US/Germany, for the pair US/UK, and for the pair Germany/UK. Indicate which of the following statements is/are correct: (A) Unc. Ptf. VaR exceeds Weighted Avg. because by computing a simple weighted sum of VaRs, Unc. Ptf. VaR ends up incorrectly assuming that all pairs of time series returns are characterized by unit correlation; therefore Weighted Avg. is preferable to Unc. Ptf. VaR. (B) Weighted Avg. exceeds Unc. Ptf. VaR because by computing a simple weighted sum of VaRs, Weighted Avg. incorrectly assumes that all pairs of return series are characterized by unit correlation; therefore, even if it has passive nature, Unc. Ptf. VaR is preferable to Weighted Avg.. (C) The difference between Weighted Avg. and the passive measure Unc. Ptf. VaR is modest because even though Weighted Avg. incorrectly assumes that all pairs of time series returns are characterized by unit correlation, in fact such correlations are all positive and relatively high. (D) The difference between Weighted Avg. and the passive risk management measure Unc. Ptf. VaR is modest because even though Weighted Avg. incorrectly assumes that all pairs of time series returns are characterized by zero correlation, in fact such correlations are all positive. (E) None of the above. 21

22 Question 3.4 Maximum likelihood (ML) estimation of a simple Gaussian N(A)GARCH(1,1) model +1 & = IID (0 1) 2 +1 = + ( ) on daily S&P 500 returns has produced the following Q-Q (on the left) and kernel density (on the right) plots concerning the standardized returns obtained from the model: Indicate which of the following statements is/are correct: (A) The model cannot be rejected because the standardized returns have an empirical distribution that is approximately normal just around their mean, where most of the total probability mass is. (B) The plots offer powerful evidence that in this case ML is unlikely to have produced consistent and asymptotically normal estimators; if this were the case, then all the crosses on the left Q-Q plot would fall on the 45-degree line and the red kernel density estimate on the right would fall exactly on the blue standard normal density. (C) The model cannot be rejected because the standardized returns show an empirical distribution that is leptokurtic to the left and platykurtic to the right, which is exactly what we expect to happen with daily S&P 500 returns. (D) The model should be rejected because the standardized returns show a left tail (and, to some minor extent, also a right tail) that are considerably thinner than under a standard normal distribution; however, the kernel density plot reveals no important differences between the empirical density and a standard normal around their means. (E) The model should be rejected because the standardized returns show a left tail (and, to some minor extent, also a right tail) that are considerably thicker than under a standard normal distribution; moreover, the kernel density plot reveals important differences between the empirical density and a standard normal around their means. 22

23 Financial Econometrics Rules of conduct during exams or other tests. During exams, students must remain quiet and may not use any external support aids, whether paper or digital (e.g. manuals, lecture notes, personal papers, books, publications, cell phones, handheld computers or other electronic devices), if not expressly authorized by the teacher in class. In addition, students may not copy or look at other students exam paper or contact or attempt to contact other people in any way. Students must remain in the classroom for the whole of the time and only for the time needed to finish his or her exam, unless teachers in class give other orders. Students who have questions for the teacher must raise their hand and wait for the examiner to come to them. At the end of the exam, students must return the exam script and the exam paper to the examining faculty member and leave the room. Any breach of these regulations or any other orders given by the faculty member present at the time of the exam will result in the test being cancelled and an official report sent to the Disciplinary Board in all cases. All disciplinary sanctions will be recorded in the student s academic career. Sanctions greater than a warning will result in forfeiture of benefits for the right to study (scholarships, housing etc.). The Honor Code and detailed regulations for taking exams and other tests are published on the University website Name and Surname (CAPITAL LETTERS) Personal ID Signature: I hereby undertake to respect the regulations described above and undersign my presence at the exam. 23