THE UNIVERSITY OF CHICAGO Graduate School of Business Business 41202, Spring Quarter 2003, Mr. Ruey S. Tsay

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1 THE UNIVERSITY OF CHICAGO Graduate School of Business Business 41202, Spring Quarter 2003, Mr. Ruey S. Tsay Solutions to Homework Assignment #4 May 9, 2003 Each HW problem is 10 points throughout this quarter. The final grade for HW is the average (times 10) over the total number of HW problems.. 1. The file m-mrk4602.dat contains monthly simple returns of Merck stock from June 1946 to December (a) Is there evidence of ARCH effects in the data? Use Ljung-Box statistics for the squared returns with 5 and 10 lags. The approach that we have been using so far to detect ARCH (GARCH) effects is to analyze the squared residuals using the tools we have learned through the course; namely, ACF, PACF, Q-stat, etc. If the (conditional) mean equation of the series is constant, it is exactly equivalent to analyze the squared residuals themselves. If that is not the case, to analyze the squared residuals is a valid explanatory approach and may give very similar results. It is also important to mention that to check the adequacy of a model, we always use the standardized (and squared standardized) residuals. We originally have simple returns for the Merck series. Even though it is not required, I will transform the simple returns to log returns, since people usually work with them, and they seem to have better properties. [* You may use simple returns. The models might be different. But they should be similar. A possible model is an AR(1) model for mean and an ARCH(3) with lag-3 only for the volatility. *] For the squared log returns, Q(5) = with a p-value of ; while Q(10) = with a p-value of This gives evidence of ARCH effects. (b) Use the PACF of the squared returns to identify an ARCH model and fit the identified model. Write down the fitted model. Is the model adequate? Figure 1 shows the PACF of the squared log returns of Merck stock. As we can see from there, the first for lags are significantly different from zero, implying that an ARCH(4) model might be adequate. A more detailed analysis of the series shows that an AR(4) model is needed for the mean equation. This can be seen directly from an exploratory analysis of the log returns, or by fitting an ARCH(4) model with constant mean equation and realizing that the standardized residuals are not white noise (as the Lung-Box statistic suggests.) Let r t be the log return of Merck stock. The model fitted to the data is r t = φ 0 + φ 1 r t 1 + φ 4 r t 4 + a t σt 2 = α 0 + α 2 a 2 t 2 + α 3a 2 t 3 + α 4a 2 t 4 (1) where a t = σ t ε t, and ε t are i.i.d. N(0,1). The commands and output produce by S-Plus, are shown bellow 1

2 Merk squared log returns Partial ACF Lag Figure 1: Partial autocorrelation function of Merck squared log returns. m2.mrk = garch(mrk~ar(4),~garch(4,0)) m2.mrk.rev = m2.mrk$model m2.mrk.rev$ar$which=c(t,f,f,t) m2.mrk.rev$ar$value=c(0.1,0,0,0.1) m2.mrk.rev$arch$which=c(f,t,t,t) m2.mrk.rev$arch$value=c(0,0.1,0.1,0.1) #$ m2.mrk = garch(model = m2.mrk.rev,series=mrk) > summary(m2.mrk) Call: garch(series = mrk, model = m2.mrk.rev) Mean Equation: mrk ~ ar(4) Conditional Variance Equation: ~ garch(4, 0) Conditional Distribution: gaussian C e-007 AR(1) e-002 AR(4) e-002 A e+000 ARCH(2) e-002 ARCH(3) e-003 ARCH(4) e-002 Ljung-Box test for standardized residuals: Statistic P-value Chi^2-d.f Ljung-Box test for squared standardized residuals: Statistic P-value Chi^2-d.f Form the results we can see that the model seems adequate. 2

3 The same model can be estimated using RATS. The results are pretty similar, and the Q-statistics show adequacy of the model. 2. The file m-spln2601.dat contains the monthly log returns of S&P 500 index from 1926 to The file has only one column. Build a GARCH model with normal conditional distributions for the series. What is the fitted model? Is the model adequate? Used the fitted model to compute 1- and 2-step ahead forecasts of the return and volatility. Note that you should also include a mean equation if necessary. The series was analyzed using S-Plus.Similar results can be found using RATS. The model is r t = φ 0 + a t σt 2 = α 0 + α 1 a 2 t 1 + β 1σt 1 2 (2) where a t = σ t ε t, and ε t are i.i.d. N(0,1). The estimates are garch(formula.mean = sp ~ 1, formula.var = ~ garch(1, 1)) Mean Equation: sp ~ 1 Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: gaussian C e-006 A e-004 ARCH(1) e-012 GARCH(1) e+000 The Q(12) statistic for standardized and squared standardized residuals is and 7.588, respectively. These values correspond to p-values of 0.40 and 0.82, which do not give evidence against the white noise assumption. Nevertheless, the normality test fails. The Jarque-Bera statistic is giving a p-value of zero. This suggests that although the residuals are not serial correlated, they do not seem to come from a Normal distribution. Using the RATS output, we see that they are negatively skewed (-0.73) and have large excess kurtosis (2.18). The 1- and 2-step ahead forecasts for the volatility and returns are Again, consider the monthly log returns or S&P 500 index from 1926 to Answer the same questions as problem 2 but building a GARCH model with Student-t conditional distributions. For this part, the model used has the same structure as (2) with the difference that a t = σ t ε t with ε t being i.i.d. t ν ; where ν are the number of degrees of freedom. The estimates are 3

4 garch(formula.mean = sp ~ 1, formula.var = ~ garch(1, 1)) Mean Equation: sp ~ 1 Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: t with estimated parameter and standard error C e-010 A e-003 ARCH(1) e-005 GARCH(1) e+000 The estimates for α 1 and β 1 are very similar to the ones found in the previous question. The estimates for α 0 and φ 0 are quite different. In particular the mean log return for the model using the t distribution is higher. Conversely, the mean of the volatility is smaller. The latter is supported by the fact that the t distribution has heavier tails. Note that the estimate of ν is around 6 (in RATS, this corresponds to V.) It is also the case that the estimate of the unconditional variance is smaller for this model. To see this, simply recall that E(a 2 α 0 t ) = 1 α 1 β 1 and notice that the main difference between the estimates in question 2 and question 3 is driven mainly by α 0. The standardized and square standardized residuals do not seem to be serially uncorrelated. The Q(12) statistics are and 7.83; with p-values of 0.44 and 0.80, respectively. The residuals are still negatively skewed (-0.77), even when the heavy tails (excess kurtosis of 2.5) is explained by the t-distribution. The 1- and 2-step ahead forecasts for the volatility and returns are Again, consider the monthly log returns of S&P 500 index from 1926 to Answer the same questions as problem 2 but using a GARCH(1,1)-M model with Student-t conditional distributions. The model for the log returns of the S&P 500 is r t = φ 0 + cσt 2 + a t σt 2 = α 0 + α 1 a 2 t 1 + β 1σt 1 2 (3) where a t = σ t ε t, and ε t are i.i.d. t ν. To compute a GARCH-M model in S-Plus, one must include the term var.in.mean in the mean equation. The estimates are 4

5 garch(formula.mean = sp ~ 1 + var.in.mean, formula.var = ~ garch(1, 1), cond.dist = "t") Mean Equation: sp ~ 1 + var.in.mean Conditional Variance Equation: ~ garch(1, 1) Conditional Distribution: t with estimated parameter and standard error C ARCH-IN-MEAN A ARCH(1) GARCH(1) We can see that the estimate for c is not significant. Thus, the GARCH-M model does not seem to add any additional information to the model found in the previous question. The statistics for the residuals are very similar to the ones discussed previously, so we will not state them again. The inclusion of the variance in the mean equation does have an impact on the forecasts of the volatility and returns Again, consider the monthly log returns of S&P 500 index from 1926 to Answer the same questions as problem 2 but using an EGARCH(1,1) model with conditional normal distribution. Discuss the implied asymmetric effects of the model. [Note: If you use RATS program, the model is EGARCH(1,0).] The model we really want to use is an EGARCH(1,0) as described in pages of the book. Nevertheless, due to a different parametrization of the model used in S-Plus, the command that must be used to estimate it is actually EGARCH(1,1). The model, as described in the book, is where ε t are i.i.d. N(0,1). Thus, (4b) becomes r t = φ 0 + σ t ε t (4a) (1 αb) log(σ 2 t ) = (1 α)α 0 + g(ε t 1 ), (4b) (1 αb) log(σ 2 t ) = { α + (γ + θ)ε t 1, if ε t 1 0, α + (γ θ)( ε t 1 ), if ε t 1 < 0, (4c) where α = (1 α)α 0 2/π γ. Theta is commonly known as the leverage. The equivalent for (4b) in S-Plus is h t = α 0 + α 1 a t 1 + γ 1 a t 1 σ t 1 + β 1 h t 1 (5) 5

6 where h t = log(σ 2 t ). Thus, we can re-arrange the terms in (5) to arrive to a similar parametrization as before. For simplicity, I will use the results from RATS, but equivalent results can be obtained using S-Plus. Variable Coeff Std Error T-Stat Signif ******************************************************************************* 1. P TH GA A A This implies a negative impact when a negative shock occurs. The residuals show no serial correlation. The Q(10) statistics are and 5.98, with p-values of 0.27 and 0.82 respectively. Finally, the 1- and 2-step ahead forecasts are