Financial Econometrics Jeffrey R. Russell Midterm 2014

Transcription

1 Name: Financial Econometrics Jeffrey R. Russell Midterm 2014 You have 2 hours to complete the exam. Use can use a calculator and one side of an 8.5x11 cheat sheet. Try to fit all your work in the space provided. If you find you need more space continue on the back of the page. Don t panic and budget your time. Students in my class are required to adhere to the standards of conduct in the Booth Honor Code and the Booth Standards of Scholarship. The Booth Honor Code also require students to sign the following Honor pledge, "I pledge my honor that I have not violated the Honor Code during this examination. Discussing any portion of this exam prior to all students taking the exam is a violation of the honor code. Please sign here to acknowledge

2 Unless otherwise stated, t is iid N(0, 2 ) 1. (12 points) Consider the three series y1, y2, y3, and y4. Match each of the series up with the models the model that appears most appropriate below. r r. Let t 0 1 t 1 1 t 1 t a. 0 =1, 1 =.9, =.9, and =5 b. 0 =1, 1 =.9, =.9, and =1 c. 0 =1, 1 =0, =.9, and =1 d. 0 =1, 1 =.9, =0, and =1 2

3 2. (28 points) Consider the AR(1) model yt 0 1yt 1 t a. For what values of 0, 1 and is the mean reverting? 2 where t ~ iid N 0,. Now let 0 =.5, 1 =.9 and =2. To get full credit for these problems be sure to show your work and find a final numeric answer. b. What is the unconditional mean? c. What is the unconditional variance? d. Let y T =4.1. What is the conditional mean E y y T 1 T? e. Let y T =4.1. What is the conditional variance Var y y T 1 T? f. What is the conditional distribution f y y T 1 T 3

4 g. Let y T =4.1. Find E y y t 2 t h. Let y T =4.1. Find E y y t 3 t 4

5 . 3. (28 points) Here are continuously compounded returns for the DJIA from 1990 to 1/26/ Series: RET Sample 1/03/2000 1/25/2012 Observations 3034 Mean 3.83e-05 Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque-Bera Probability For parts a. and b. of this problem, assume that the natural log of the DJIA prices follows a random walk model. The index level (not log level) at the close on 1/26/2012 was 12,756 a. Explain what the skewness number is measuring in intuitive terms (no equations are needed) and interpret the implications of the skewness in the table to the right of the histogram. b. Explain what the kurtosis is measuring in intuitive terms (no equations are needed) and interpret the implications of the kurtosis in the table to the right of the histogram. c. The drift can be estimated from the sample mean and the standard deviation of the error can be estimated by the standard deviation in the above output. Find the k step ahead forecast of the log DJIA index level as a function of k and the initial log index. Use sample estimates where they appear in the forecast equation. 5

6 d. Find the k step ahead forecast error variance associated with part c. as a function of k. Use sample estimates where they appear in the forecast error variance expression. e. Consider the following output for the DJIA continuously compounded returns. Are then predictable? Explain with the use of supporting evidence. 6

7 Next, consider the following models fit for the DJIA returns data. Two MA models are estimated below for the DJIA returns data. The first is a pre financial crisis data set prior to The second is a data set that primarily focuses on the period of the financial crisis, from January 2007 to present. Dependent Variable: RET Method: Least Squares Date: 02/08/10 Time: 22:16 Sample: 1/01/1990 1/26/2007 Included observations: 4436 Convergence achieved after 6 iterations MA Backcast: 12/27/ /29/1989 Variable Coefficient Std. Error t-statistic Prob. C MA(1) MA(2) MA(3) R-squared : Dependent Variable: RET Method: Least Squares Date: 02/08/10 Time: 22:15 Sample (adjusted): 1/01/2007 6/03/2009 Included observations: 633 after adjustments Convergence achieved after 7 iterations MA Backcast: 12/27/ /29/2006 Variable Coefficient Std. Error t-statistic Prob. C MA(1) MA(2) MA(3) R-squared : f. For the crisis sample, find the forecast of the return E r,,,...,,,... t t 1 t 2 t k t t 1 t 2 as a function of, and the estimated parameters above. You should have forecasts for all k>0. 7

8 g. Explain in plain English what the difference is between the pre-crisis and the crisis return dynamics. 4. (20 points) Below is the estimated GARCH model for the DJIA return series in problem 3. Variable Coefficient Std. Error z-statistic Prob. Variance Equation C 1.06E E RESID(-1)^ GARCH(-1) a. The last return in the sample is.0104 and the value of h T associated with the last observation in the sample is Write the one step ahead forecast of the variance. b. Find the expected variance for the cumulative return over the next 5 days. Report the annualized volatility. 8

9 c. What number does the forecast for the variance of the cumulative return get close to (in an annualized basis) when the number of days gets large? Now consider the following TARCH (asymmetric GARCH model). Variable Coefficient Std. Error z-statistic Prob. Variance Equation C 1.44E E RESID(-1)^ RESID(-1)^2*(RESID(-1)<0) GARCH(-1) d. What is the one step ahead conditional variance if the last return in the sample is.0104 and the value of h T associated with the last observation in the sample is

10 Now consider the following output obtained from the symmetric GARCH(1,1) model. e. What do the autocorrelations tell you about whether or not the GARCH(1,1) model appears to fit the volatility dynamics well? Be specific. 10

11 5. (12 points) Let r t denote the continuously compounded returns on an asset and let d t denote the price dividend ratio of the asset. You use the price dividends to forecast future returns over different time horizons. Suppose that rt 1dt 1 t and dt dt 1 t where t and t are iid, mean zero and independent of each other. Both r t and d t are mean reverting and stationary. a. Find E r r d t 1 t t 1 b. Find E r r r d t 2 t 1 t t 1 c. What is E r... r r d t k t 1 t t 1 d. What does the forecast converge to as k gets large? 11