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ECON 898: ECONOMETRICS III
PROF. LE WANG
Problem Set #
Do the following empirical exercises in Stata. Turn in your do file, log file, and graphs. Set the seed at 123456 at the start of each problem.
Part I: Analytical Questions
Question 1. We know that ARCH/GARCH models have the following form
A(L)yt = B(L)t t = vt
ht
where A(L) and B(L) are polynomials of orders p and q in the lag operator L, et is the error term, depending on vt ∼ N (0, 1) and ht.
(0.1) ARCH(1) : ht = α 0 + α 1 ^2 t− 1
(0.2) GARCH(1, 1) : ht = α 0 + α 1 ^2 t− 1 + β 1 ht− 1
Verify the following: (1) Verify that 0.1 is essentially an AR(1) model for ^2 t. Notice that the error term in this AR model is time heteroscedastic. Hence the conventional estimation procedure of AR models does not produce optimal results here. Similarly, ARCH(q) is essentially an AR(q) model for ^2 t. This result has an important implication for determination of the order of ARCH(q). (2) Verify that 0.2 is essentially an ARMA(1,1) model for ^2 t. Notice that the error term in this model is time heteroscedastic. 1
2 PROF. LE WANG
(3) Apply backward substitution of 0.2 to show that GARCH model is essentially an ARCH(∞) process. That is, the GARCH term captures all the history of the shocks of t. This is the reason why in practice it is often preferred to ARCH(q) models. (4) Derive the first four moments of the unconditional distribution of t (mean, variance, skewness, and kurtosis). State clearly the assumptions you make. Explain why these models can capture (a) volatility clustering; (b) fat tail distribution with outliers. (5) Show that ht is indeed the conditional variance of t
Part II: Applied Questions
Question 1. Go to the course webpage and download the file entitled ibmln.dta. This file contains the monthly log returns of IBM stocks from Jan 1926 to Dec 1997 for 864 observations. Use the series to perform the following tasks.
(1) Estimate an AR(1) model. After examining the residuals, do you find any evidence of serial correlation? (2) Examine the properties of the squared residuals. Using both informal and formal tests discussed in the class, do you find any evidence of ARCH/GARCH effects? Explain. (3) Use the result from Part I, Q1.1, you can identify an ARCH model.
- Estimate an ARCH([1,2,11])
- Examine the residuals. Is there still conditional heteroskedasticity left? Use both formal and informal tests.
- Is the normality assumption of standardized residuals plausible? If not, perform robust inference. (4) Estimate a GARCH(1,1) model.
- Examine the residuals. Is there still conditional heteroskedasticity left? Use both formal and informal tests.
- Is the normality assumption of standardized residuals plausible? If not, perform robust inference.
- If α 1 +β 1 is close to unity, estimate an Integrated-GARCH model as well. Perform related diagnostic tests of residuals. (5) Estimate a GARCH in Mean model using GARCH(1,1). Is there any evidence of risk-return trade-off? (6) Among the estimated models above, which one is the best model? Explain.
