6 GARCH.pdf

Applied Econometrics GARCH Models Lecture handout 6 Autumn 2023 D.Sc. (Econ.) Elias Oikarinen Professor (Associate) of Economics University of Oulu, Oulu Business School 1 This Handout • Univariate time series analysis: modelling & forecasting conditional volatility • GARCH & ARCH models • ARCH-M...

Applied Econometrics GARCH Models Lecture handout 6 Autumn 2023 D.Sc. (Econ.) Elias Oikarinen Professor (Associate) of Economics University of Oulu, Oulu Business School 1 This Handout • Univariate time series analysis: modelling & forecasting conditional volatility • GARCH & ARCH models • ARCH-M model • Some other GARCH models briefly • Brooks, Chapter 9 (for the purposes of this course, until 9.17; especially 9.7-; many interesting volatility related issues after 9.17 as well, though, for an interested student) • Important reading before Eviews exercises: 9.7.4, 9.9.3, 9.16.1, 9.17.1 • Enders, Chapter 3 • Eviews UG II, Chapter 25 2 ARCH / GARCH Models  ARCH = autoregressive conditional heteroscedasticity (Engle, 1982)  GARCH = generalized autoregressive conditional heteroscedasticity (Bollerslev, 1986)  ARCH is particular type of heteroscedasticity  Basic idea: Large shocks are more likely to be followed by large shocks, small shock by small ones (large ”shock” = large error term in absolute value)  Clustered volatility, fat tails in residual distribution  Volatility modelling has raised substantial interest in financial market research since the 1980s  Robert Engle: the winner of the 2003 Nobel Memorial Prize in Economic Sciences, "for methods of analyzing economic time series with time-varying volatility“:  Engle showed that it is possible to estimate simultaneously both the expected value and conditional variance of a time series 3 ARCH / GARCH Models  Of notable importance regarding e.g. risk management and derivatives pricing, among other things  Note: Even if the expected value of series (e.g. stock return) is not predictable, its volatility can exhibit predictability  GARCH is a model for error term variance / standard deviation  We relax the assumption of constant (conditional) variance; Residuals are not white noise, IID   GARCH model is always based on an underlying equation for the mean of time series, e.g. a simple ARMA model (at the very least: yt = c + t) The variance of  at time t depends on the variance in previous period(s)  conditional heteroskedasticity 4 ARCH(q) Model  Let’s first consider a model: yt = c + 1yt-1 + t , where the conditional variance of yt, ht is of the form: ht = var(yt|yt-1) = Et-1[(yt – c – 1yt-1)2] = Et-1[t2]   Box-Jenkings procedure assumes that ht = 2 = constant In ARCH process the conditional variance is not constant, but depends on q lags of squared errors: ht = 0 + 12t-1 + 22t-2 + … + q2t-q  If all parameters 1 … q equal zero, there is not heteroscedasticity 5 Example: ARCH(1)  Equation for conditional variance: ht = 0 + 12t-1 + t  if t-1 is large in absolute value, expected value of t is large in absolute terms t = error term from the mean equation ht = conditional variance t = white noise (IID) ht0.5 = conditional volatility / standard deviation    Error terms are uncorrelated: E[t,t-i] = 0 …but squared error terms are correlated – hence the error terms t are not independent Variance has to be nonnegative  conditions for the model viability, ”positivity conditions”  In ARCH(1) model: 0 > 0, 0 < 1< 1 6 Why do we have squared  in RHS?  Equation for conditional variance:   ht = 0 + 12t-1 + t Equation for conditional volatility (s.d.): ht0.5 = (0 + 12t-1 + t)0.5  Why not   ht0.5 = t = 0 + 1t-1 + t But then t would exhibit predictability and the mean (ARMA) model would be insufficient  we would need to estimate better model for yt so to better capture predictability and remove autocorrelation in t If the model is well-specified, t is not autocorrelated by definition However, 2t can be autocorrelated! While the conditional variance 2 may exhibit predictability, we are not able to predict the sign of the error term (otherwise we would of course revise the prediction to remove the expected error!)   7 GARCH(p,q) Model  Extension to ARCH model (Bollerslev, 1986): t  N(0,ht)  GARCH is an ARMA-type model for residual conditional variance, ht  q is the lag length for ”moving average component” and p for autoregressive component  Advantages of GARCH (vs. ARCH)   Typically requires less parameters Easier to identify and estimate  ARCH terms () are ”short memory”, GARCH terms (h) ”long memory”  NOTE: The sign of error term () is unpredictable  E[t] is still zero 8 GARCH(1,1)  Estimation is usually good to start with the GARCH(1,1) model, which is often the most suitable model: ht = 0 + 12t-1 + 1ht-1  Positivity conditions, to ensure that ht > 0 : 0 > 0, 1 ≥ 0, 1 ≥ 0  Stationarity condition: 1 + 1 < 1  Typically: 1 < 1 , and the sum is close to 1  After estimation, one should check whether the conditions are fulfilled (if parameters have not been restricted in advance) POLL 9 Other GARCH(p,q) Models  An example, GARCH(2,1): 0 ≥ 0, 1 ≥ 0, 1 ≥ 0,  1 + 2 < 1, 12 + 42 ≥ 0  More generally, you are pretty safe with the positivity and stationarity conditions, if: all coefficients i and i are nonnegative (for positivity) & the sum of i < 1 (for stationarity)  However, Nelson & Cao (1992, in Journal of Business & Economic Statistics) show that all the coefficients do not necessarily need to be positive for ht to be strictly positive  GARCH model also can include other explanatory variables  E.g. transaction volume  Dummy variable to capture extreme events, e.g. 11.9.2001: ht = 0 + 12t-1 + 1ht-2 + 1Dt Dt = 1, kun t = 11.9.2001 Dt = 0 kun t ≠ 11.9.2001 if 1 > 0, if 11.9.2001 ”event” increased volatility 10 Testing for Heteroscedasticity 1. Estimate the best possible ARMA model for {yt} (or some other regression model for the mean of {yt}) 2. Square the error terms to form {t2} 3. Inspect the ACF of the squared residuals: Significant autocorrelation indicates heteroscedasticity, and thus ARCH effects in the data; Q-test applies here as well • If no significant autocorrelations are observed: no (G)ARCH effect With Eviews: Residual Diagnostics – Correlogram for squared residuals 11 Engle LM test • Lagrange multiplier (LM) test to formally investigate for ARCH effects • As with the Q-test: 1. as good a model for mean as possible, 2. form series {t2} • Presence of ARCH-effect is investigated with regression (”auxiliary regression”): q = selected lag length • Null hypothesis: residuals do not exhibit ARCH effects  i = 0 for all q ( auxiliary model R2 is low) • Test statistic: TR2  2q, where T = number of obs, q = degrees of freedom • If TR2 > 2 distribution critical value, null hypothesis is rejected 12 Estimating GARCH Model: An Example  OMXH, Small Cap total return index Weekly frequency over 2 Oct 2006 – 6 July 2020 Natural log taken before estimations  Data can be downloaded in Moodle right below GARCH lecture handout   13 Estimating GARCH Model: An Example - Index is I(1) - DGP for return not clear, could be e.g. AR(1), MA(1) or higher-order, ARMA(1,1)… 14 Estimating GARCH Model: An Example  ARIMA(2,1,|3|), i.e. ARMA(2,|3|) for returns, seems like the best option based on the Box-Jenkins procedure  Not normally distributed error term: fat tails (often caused by heteroscedasticity) Adding a Covid19 dummy does not affect much NOTE: the notation for the sole MA component, i.e. the use of |.|   15 Estimating GARCH Model: An Example Both Q-test and ARCH test indicate heteroscedasticity  Makes sense to continue with GARCH modeling  16 Estimating GARCH Model: An Example  AR(1) is now preferred over ARMA(2,|3|) (compare AIC & SC)  This is AR(1)-GARCH(1,1) model Adding a Covid19 dummy does not affect much  17 Standardized Residuals  If the ARCH effects have been successfully modeled, the standardized residuals, Zt should be:  Note: Eviews automatically provides the JB test and histogram for standardized residuals, when a GARCH model is estimated  Also Zt distribution often has fat tails, although is closer to normal than that of t (in this case, GARCH model can partially eliminate the fat tails)  What if Zt is not normally distributed?   Parameter estimates are ok (consistent) ML estimation does not produce correct std. errors for coefficients  Quasi-ML estimation; in Eviews: The Bollerslev and Wooldridge (1992) QML covariances and standard errors are estimated by selecting Options - Coefficient covariance - Bollerslev-Wooldridge 18 Estimating GARCH Model: An Example     Zt should not exhibit autocorrelation, otherwise the model is not wellspecified No autocorrelation in the squared Zt either (otherwise there are still ARCH effects that are not captured by the model) These conditions are fulfilled by the AR(1)-GARCH(1,1) model Homoskedastic residuals  GARCH(1,1) captures the conditional volatility well 19 Standardized Residuals  Zt is not normally distributed: excess kurtosis (flat tails) 20 Estimating GARCH Model: An Example    Non-normal residuals  QML estimation Coefficients estimates do not change, but their std. errors (and hence p-values) change slightly All coefficients still statistically highly significant All stationarity and non-negativity conditions are fulfilled The index returns appear to exhibit surprisingly strong predictability (at least in-sample) 21 Estimating GARCH Model: An Example  Conditional standard deviation ( ℎ𝑡 ) graphed (Eviews: View Garch graph): 22 Estimating GARCH Model: An Example   The estimated model for OMX Helsinki Small Cap index is ARIMA (1,1,0)-GARCH(1,1) For the difference, i.e. for index returns: ARMA(1,0)-GARCH(1,1), shorter notation AR(1)GARCH(1,1): yt = c + 1yt-1 + t ht = 0 + 12t-1 +  1ht-1  In this example: Lehmann dummy takes value 1 during four weeks after the shock  some small changes in parameter estimates, AIC and SBC slightly ”better” We could use dummy variables to test the Lehmann collapse effect (Sep 15, 2008), or Covid19 effect, on conditional volatility POLL 23 Unconditional vs. Conditional Mean  ”Unconditional” refers to the long-run constant mean, while ”conditional” refers to the short-run predictable expectation of yt  AR(1) Example:  Unconditional mean as presented earlier on already:  Conditional mean: Since: (It-1 = information set available in time period t-1, i.e. historical observations) 24 Unconditional vs. Conditional Variance  Unconditional variance = long-term variance of the time series (around which the conditional variance moves)  In GARCH(1,1), the unconditional variance: ht = ht-1 = h E[2t] = h h = 0 + 12t-1 + 1ht-1 = 0 + 1h + 1h h – 1h – 1h = 0 h = 0 / (1 – 1 – 1)  Equilibrium unconditional variance exists, if 1 + 1 < 1 ; if 1 + 1 = 1 cannot be measured; 1 + 1 > 1 would give negative variance = non-stationarity • Conditional variance: E[ 2 |I ] t t = ht = 0 + 12t-1 + 1ht-1 25 Extensions to the Basic GARCH Model 26 GARCH-in-Mean Model  ARCH-M (Engle, Lilien & Robbins, 1987)  Later extended to GARCH-M  Brooks, pp. 445-446; Enders, pp. 143-146  Basic idea: investors should be compensated for bearing higher risk through a higher reward (return)  Particularly suited to the study of asset markets 27 GARCH-in-Mean Model  Engle et al. express this idea by writing the excess return from holding a risky asset as: yt = t + t , where: yt = excess return from holding a long-term asset relative to a risk-free asset (i.e. one-period U.S. treasury bill in their case) 𝜇t = risk premium necessary to induce the risk-averse agent to hold the asset rather than the risk-free bond, i.e., expected excess return over the risk-free rate 𝜀t = unforecastable shock to the excess return on the long-term asset  The expected excess return must equal to the risk premium: Et-1yt = t  Risk premium is an increasing function of the conditional variance of 𝜀t : t =  + ht ,  > 0 , where ht is an ARCH(q)-process: ht = 0 + 12t-1 + 22t-2 + … + q2t-q • The three equations above together form the ARCH-M model 28 Testing for GARCH-M Effect      If conditional variance is constant (1 = 2 =…= q = 0), risk premium is constant GARCH-M effect can be tested similar to ARCH or GARCH effects LM test: Test statistic TR2  2, T = nobs, df = number of parameter restrictions If test statistic > 2 distribution critical value at the selected level of significance, the null hypothesis of no ARCH-M effect is rejected See also: Engle et al. (1987, Econometrica); the article can be downloaded from the Moodle (”Additional course material” section) 29 GARCH-M Example  Excess return of OMXH Small Cap index return (risk-free interest rate measured by the U.S. gvt. 10-year bond yield)  Correlogram: which ARMA model this could be? (POLL) 30 GARCH-M Example  1st step: Box-Jenkins for excess return, if heteroscedasticity, then continue with GARCH modeling 31 GARCH-M Example     Again, AR(1) is enough when GARCH model is included Passes the residual checks, except for normality EViews: ARCH-M box, select either std. dev., variance, or log var No GARCH-M effect observed – maybe would be observed at some other data frequency 32 Selected Other GARCH Models  IGARCH (Integrated GARCH)   In the baseline model: 1 + 1 = 1, non-stationary variance EGARCH (exponential GARCH)  Nelson (1991)  Conditional variance equation includes also (t-i/ h0.5t-i ) terms (in the righthand side.  Brooks, pp. 441-443, including empirical examples 33 Selected Other GARCH Models   TARCH (threshold ARCH, myös GJR-GARCH)  Zakoian (1994); Glosten et al. (1993)  An additional term added to account for possible asymmetries: different coefficient on negative and positive error terms  In Eviews: include treshold order in GARCH/TARCH model APARCH (asymmetric power ARCH)  Ding, Engle & Granger (1993) 34 Recommended Reading! BEFORE TUESDAY’S LECTURE • On forecasting with ARMA & GARCH: • Brooks, pp. 285-298 • On unit root tests • Brooks, Chapter 8, pp. 353-373 BEFORE GOING THROUGH WEEK 4 EXERCISE VIDEOS  On estimating GARCH models with EViews:  Brooks 9.7.4, 9.9.3, 9.16.1, 9.17.1  On Maximum Likelihood estimation:  Corresponding handout provided in Moodle  Brooks, pp. 431-435 35