Chapter Chapter 12 Autoregressive Conditional Heteroscedasticity (ARCH) and Generalized ARCH (GARCH) Models Section Section 12. 1 Introduction ARCH and GARCH Models • ARCH and GARCH models are designed to model heteroscedasticity (unequal variance) of the error term with the use of timeseries data • Objective is to model and forecast volatility Example: Understand the risk of holding an asset; useful in financial situations • ARCH -- Autoregressive Conditional Heteroscedasticity • GARCH -- Generalized ARCH Engle, R. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation,” Econometrica 50 (1982):987-1008.
Engle noticed that in some time series, particularly those involving financial data, large and small residuals tend to come in clusters, suggesting that the variance of an error may depend on the size of the preceding error. 3 ARCH-GARCH Models Recent development in financial econometrics require the use of models and techniques that are able to model the attitude of investors not only towards expected returns, but towards risk (or uncertainty) as well. This fact requires models that are capable of dealing with the volatility (variance) of the series. Such models are the ARCH-family of models. In
In other words, we observe that large changes in stock returns seem to be followed by other large changes and vice versa. This phenomenon is what financial analysts call volatility clustering. In such cases it is clear that the assumption of homoskedasticity (or constant variance) is very limiting, and in such instances it is preferable to examine patterns that allow the variance to depend upon its history. 4 Section Section 12. 2 The ARCH(q) Model The ARCH(q) Model The ARCH(q) model will simultaneously examine the mean and the variance of a series according to the following specification: Yt = a + ? ?X t + u t ut = ht e t e t ~ IN ( 0 ,1) t I ? t ~ iid N ( 0 , ht ) q ht = ? 0 + ? ? j u t2? j j =1 is the information set; that is, the dependent variable, the explanatory variables, the specification of ut and the specification of ht. The estimated coefficients of the ? s should be positive in order to guarantee that the variance is greater than zero. t 6 Before estimating ARCH(q) models, it is important to check for the possible presence of ARCH effects in order to know which models require the ARCH estimation method instead of the use of OLS. This test can be done along the lines of the Breusch-Pagan Lagrange Multiplier (LM) Test, which entails estimation of the mean equation:
Yt = a + ? ?X t + ut by OLS as usual (note that the mean equation can have not only explanatory variables (the xt vector), but also autoregressive terms of the ? error error term); to obtain the residuals ut , subsequently run an auxiliary regression of the squared residuals (ut2 ) upon the lagged squared ? 2 2 terms (ut ? 1 ,... , ut ? q ) and a constant as in: ? ? u = ? 0 +? u ? ? 2 t Test 2 1 t ? 1 + ... + ? u ? 2 q t ? q + wt H 0 : ? 1 = ? 2 = ... = ? q = 0. Rejection of H0 suggests evidence of ARCH(q) effects. 7 2 The LM test statistic follows a ? distribution with q degrees of freedom.
Alternatively, we may compute the test statistic (Q) based on squared residuals to test for ARCH(q) effects (McLeod and Li, 1983). ? r (i : ut2 ) Q ( q ) = N ( N + 2) ? , where i =1 ( N ? i ) q ? ? ? tN=i+1 (ut2 ? ?? 2 )(ut2? i ? ?? 2 ) and 2 ? r (i : ut ) = N (ut2 ? ? 2 ) 2 ? t =1 ? ? 1 ?= ? N 2 N ut2 ?? t =1 Similar to the LM statistic, the Q statistic is computed from squared OLS residuals, assuming that the disturbance term in the mean or structural equation is white noise. The Q statistic 2 also follows a ? distribution with q degrees of freedom. 8 SAS Program for ARCH Models Publix Data roc reg data=PublixFF20; model logunits=week logdisc logprice fsi1 logdisp logad logdist q1 q2 q3 / dw; proc autoreg data=PublixFF20; model logunits=week logdisc logprice fsi1 logdisp logad logdist q1 q2 q3 / nlag=1 normal method=ml; proc autoreg data=PublixFF20; model logunits=week logdisc logprice fsi1 logdisp logad logdist logdist q1 q2 q3 / garch=(q=2) maxit=500 archtest; proc autoreg data=PublixFF20; model logunits=week logdisc logprice fsi1 logdisp logad logdist q1 q2 q3 / garch=(q=2) maxit=500 dist=t archtest; proc autoreg data=PublixFF20; model logunits=week logdisc logprice fsi1 logdisp logad ogdist q1 q2 q3 / nlag=1 garch=(q=2) maxit=500 archtest; 9 proc autoreg data=PublixFF20; model logunits=week logdisc logprice fsi1 logdisp logad logdist q1 q2 q3 / nlag=1 garch=(q=2) maxit=500 dist=t archtest; run; With the archtest command, SAS conducts the LM and Q test statistics for various lags. garch = (q = 2) calls for the ARCH(2) model. nlag = 1 also allows the structural or mean equation to exhibit serial correlation. The lag of the autoregressive process in the disturbance terms is 1. That is, the disturbance terms follow an AR(1) process. 10 Information Criteria AIC, AICC, and SBC
Akaike’s information criterion (AIC), the corrected Akaike’s information criterion (AICC) and Schwarz’s Bayesian information criterion (SBC) are computed as follows: AIC = ? 2 ln( L) + 2k k (k + 1) AICC = AIC + 2 N ? k ? 1 SBC = ? 2 ln( L) + ln( N )k N = sample size k = number of explanatory or right-hand side variables L = value of the likelihood function 11 Section Section 12. 3 SAMPLE PROBLEM: ARCH(q) Model Demand for a Cereal in Publix Supermarkets The AUTOREG Procedure Dependent Variable LOGUNITS Ordinary Least Squares Estimates SSE MSE SBC MAE MAPE Durbin-Watson 13 4. 72154649 0. 03066 -61. 963382 0. 12757372 . 33157044 1. 3796 DFE Root MSE AIC AICC Regress R-Square Total R-Square 154 0. 17510 -96. 128782 -94. 403292 0. 9505 0. 9505 Miscellaneous Statistics Statistic Normal Test Value Prob Label 96. 1683 ChiSq Variable Estimate Intercept WEEK LOGDISC LOGPRICE FSI1 LOGDISP LOGAD LOGDIST Q1 Q2 Q3 14 DF Standard Error 1 1 1 1 1 1 1 1 1 1 1 13. 2451 0. 000840 2. 6339 -2. 8900 0. 001777 1. 0218 1. 2339 13. 2609 -0. 008809 -0. 0754 -0. 1470 0. 7926 0. 000309 0. 1890 0. 5534 0. 0893 0. 1873 0. 1020 3. 7027 0. 0406 0. 0419 0. 0429 t Value Approx Pr > |t| 16. 71 2. 72 13. 94 -5. 22 0. 02 5. 46 12. 10 3. 58 -0. 22 -1. 80 -3. 43