Financial Time Series Analysis (FTSA) Lecture 6: Conditional Heteroscedastic Models Few models are capable of generating the type of ARCH one sees in the data.... Most of these studies are best summarized with the adage that to get GARCH you need to begin with GARCH. ~- Adrian Pagan (1996) GARCH Model The GARCH Model was introduced by Bollerslev (1986) primarily to overcome the large number of ARCH parameters that were needed to model the volatility process. For example, it is common knowledge that to adequately model the S&P500 Index one would need almost 9 ARCH parameters. He therefore proposed the following generalized form of the ARCH model: Let: then follows a GARCH(m,s) model if: Where: is a sequence of iid r.v. with and,,, for I,j > 0, and. The last constraint is required to make sure that the unconditional variance of is finite where as its conditional variance evolves over time.
If we use the following re-parameterization and define the squared shocks as: The GARCH model becomes:. un-correlated series. Which is nothing but the ARMA form for the squared series Advantages of a GARCH Model Focusing on a GARCH(1,1) model we have, With, for I,j > 0, and 1. Weak stationarity 2. Volatility clusters are modeled because a large or a large will give rise to a large. 3. The distribution of the series will have heavy tails if: which implies that 4. Simple parametric representation to describe the volatility evolution. Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 2
Dis-advantages of a GARCH Model Symmetric to both positive and negative prior returns Restrictive on and otherwise we will have an infinite fourth moment. Provides no explanation as to what causes the variation in volatility Not sufficiently adaptive in prediction because they react slowly to large isolated shocks. Tail behavior of GARCH models remains too short even with a standardized Student-t innovations. Estimating the Unconditional Variance of of a GARCH(1,1) Forecasting Using the GARCH Model In general, we can write an h-step ahead forecast model for the GARCH(1,1) as: Which is exactly the same as that of an ARMA(1,1) model with AR polynomial equal to Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 3
IGARCH(1,1) Model If the AR polynomial of the GARCH model has a unit root, then we have an IGARCH model. That is, IGARCH model is a unit-root GARCH model. This occurrence is due to the fact that the impact of the past squared shocks is persistent. for i > 0, on For an IGARCH(1,1) model we have, NOTE: The unconditional variance of for an excess returns series. is not defined which makes this model suspect In general, if when IGARCH(1,1) as: we can write an h-step ahead forecast model for the That is, the effect of forecasts form a straight line with slope. on future volatilities is persistent, and the volatility Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 4
GARCH-M Model When the return of a security is dependent on its volatility we can use the GARCH-M model formulation. That is, For a GARCH-M model we have, Where c is the risk-premium, which is expected to be positive. The GARCH-M model implies that: There exists serial correlations in the return series. These serial correlations are introduced by the volatility process due to a riskpremium. NOTE: Upon solving the model, if the risk-premium is not statistically significant do not use the GARCH-M model because the model will reduce to the simple GARCH model specification. Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 5
EGARCH Model Know limitation of the prior GARCH models is that they assume that there is asymmetry of responses to positive and negative returns. Nelson (1991) therefore introduced the EGARCH model which introduced the concept of weighted innovation. An EGARCH(m,s) model showing the evolution of the conditional variance is of specified as: is Where the weighted innovation is specified as: And θ (leverage parameter), and γ (magnitude) are real constants, and are zero-mean iid sequence with continuous distributions and thus E( ) = 0. The leverage parameter shows the effect of the sign of Properties of the EGARCH model The unconditional mean of = which is similar to the GARCH model. However, it is different from the GARCH model because it uses logged conditional variance to relax the positiveness constraint of the model coefficient. The use of allows the model to respond asymmetrically to positive and negative lagged value of the innovation. Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 6
Example: EGARCH(1,1) model Under normality, and the model becomes: Let then we have: Taking anti-log transformation we have: An alternative form for the EGARCH(m,s) that is used both by SAS and S-Plus is as follows: The parameter signifies the leverage effect of. Generally, we expect the leverage effect to be negative. Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 7
TGARCH Model This volatility model is used to handle leverage effects. It is defined as shown in Equation 3.33. CHARMA Model This model uses random coefficients to produce conditional heteroscedasticity. Most importantly, this model is used if one needs to study the interaction effect of previous returns on volatility. However, the model gets very complex very quickly. This model extends easily to include explanatory variables. Bottom Line Hansen and Lunde (2001) implemented the same methodology of Andersen and Bollerslev (1998) that established that volatility models do provide good forecasts of the conditional variance. Their goal was to evaluate whether the evolution of volatility models has led to better forecasts of volatility when compared to the two base models ARCH and GARCH(1,1). To do this, they compared 330 different volatility models using daily exchange data and IBM stock prices. They conclude that although some of these newer models outperform the GARCH(1,1) model the forecasting gains are not significantly better than the GARCH(1,1) model. However, they do note that the ARCH(1) model is clearly outperformed. Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 8
Additional Volatility Models Stochastic Volatility Models These models introduce an innovation to the conditional variance. These models are very similar to the EGARCH models. They also have shown to provide improvements in model fitting. However, their contributions is mixed for out-of-sample volatility forecasts. Long Memory Stochastic Volatility Models Stochastic Volatility models have been extended to allow for long memory in volatility. Recall: A time-series has a long-memory process if its ACF decreases at a hyperbolic rate instead of at an exponential rate as the lag increases. These models are generally estimated using fractional difference. Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 9
Using High Frequency Data for Volatility Modeling It is possible to due to the properties of log returns to use simple averaging methods to compute a monthly volatility from daily or even higher-frequency observations. However some obvious factors must be considered before one can make inference on the parameters. That is the parameters may not be consistent. Some of these factors are: Distribution of the higher frequency returns in not known There may be excess kurtosis in the series There may be serial correlation in the series In essence one needs to be careful in using parametric approaches to modeling when using high frequency data. Using Daily O/H/L/C Volatility can be computed based on some or all of the prices (see page 144) of a stock. One must however be aware that since information is captured only at finite points one could underestimate the volatility. The underestimation is greater for stocks that are not traded as frequently. Kurtosis of GARCH Models To assess the variability of an estimated volatility, one must consider the excess kurtosis of a volatility model. By studying the excess kurtosis we can determine the tail behavior of the innovation. That is, we can determine if the distribution has heavy tails or not; the distribution is really a student-t distribution instead of Gaussian; or the tail behavior is simply standardized noise. Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 10
Executing the Models in SAS filename retin disk 'c:\timeseries\sp500mnthret.txt'; data a; infile retin; input spret; time = _n_; run; Proc autoreg data=a; model spret = / archtest dwprob; proc autoreg data=a; model spret = / nlag=3 garch=(p=1,q=1); * AR(3)+garch(1,1); model spret = / garch=(p=1,q=1); * garch(1,1); model spret = / garch=(p=1,q=1,type=stn); * garch(1,1) stationary; model spret = / garch=(p=1,q=1,mean=linear); * garch-m(1,1); model spret = / garch=(p=1,q=1,type=integrated); * igarch(1,1); model spret = / garch=(p=1,q=1,type=exp); * egarch(1,1); *output out=b cev=spcev r=resid; run; Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 11
AR(3)-GARCH(1,1) Dependent Variable spret Ordinary Least Squares Estimates SSE 2.70318776 DFE 791 MSE 0.00342 Root MSE 0.05846 SBC -2244.3895 AIC -2249.0641 MAE 0.03973876 AICC -2249.059 MAPE 135.870252 Regress R-Square 0.0000 Durbin-Watson 1.8161 Total R-Square 0.0000 Intercept 1 0.006143 0.002077 2.96 0.0032 Estimates of Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 0 0.00341 1.000000 ******************** 1 0.000307 0.089861 ** 2-0.00009-0.026766 * 3-0.00043-0.126933 *** Preliminary MSE 0.00333 Estimates of Autoregressive Parameters Lag Coefficient Error t Value 1-0.088717 0.035356-2.51 2 0.023737 0.035486 0.67 3 0.122425 0.035356 3.46 Algorithm converged. GARCH Estimates SSE 2.68240068 Observations 792 MSE 0.00339 Uncond Var 0.00324551 Log Likelihood 1270.21398 Total R-Square 0.0077 SBC -2493.706 AIC -2526.428 MAE 0.03966542 AICC -2526.2851 MAPE 140.542747 Normality Test 84.3783 Pr > ChiSq <.0001 Intercept 1 0.007474 0.001580 4.73 <.0001 AR1 1-0.0337 0.0385-0.87 0.3819 AR2 1 0.0312 0.0383 0.82 0.4146 AR3 1 0.0101 0.0356 0.28 0.7759 ARCH0 1 0.0000805 0.0000240 3.35 0.0008 ARCH1 1 0.1200 0.0202 5.94 <.0001 GARCH1 1 0.8552 0.0196 43.54 <.0001 Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 12
GARCH (1,1) The SAS System 00:31 Saturday, October 24, 2009 52 The AUTOREG Procedure Dependent Variable spret Ordinary Least Squares Estimates SSE 2.70318776 DFE 791 MSE 0.00342 Root MSE 0.05846 SBC -2244.3895 AIC -2249.0641 MAE 0.03973876 AICC -2249.059 MAPE 135.870252 Regress R-Square 0.0000 Durbin-Watson 1.8161 Total R-Square 0.0000 Intercept 1 0.006143 0.002077 2.96 0.0032 Algorithm converged. GARCH Estimates SSE 2.70454696 Observations 792 MSE 0.00341 Uncond Var 0.00324989 Log Likelihood 1269.46195 Total R-Square. SBC -2512.2257 AIC -2530.9239 MAE 0.03970158 AICC -2530.8731 MAPE 146.032719 Normality Test 95.0051 Pr > ChiSq <.0001 Intercept 1 0.007453 0.001547 4.82 <.0001 ARCH0 1 0.0000818 0.0000238 3.44 0.0006 ARCH1 1 0.1203 0.0197 6.12 <.0001 GARCH1 1 0.8545 0.0189 45.15 <.0001 Model: Unconditional variance = 0.00324 Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 13
GARCH-M(1,1) The SAS System 00:31 Saturday, October 24, 2009 54 The AUTOREG Procedure Dependent Variable spret Ordinary Least Squares Estimates SSE 2.70318776 DFE 791 MSE 0.00342 Root MSE 0.05846 SBC -2244.3895 AIC -2249.0641 MAE 0.03973876 AICC -2249.059 MAPE 135.870252 Regress R-Square 0.0000 Durbin-Watson 1.8161 Total R-Square 0.0000 Intercept 1 0.006143 0.002077 2.96 0.0032 Algorithm converged. GARCH Estimates SSE 2.7049062 Observations 792 MSE 0.00342 Uncond Var. Log Likelihood 1270.11495 Total R-Square. SBC -2506.8571 AIC -2530.2299 MAE 0.03978412 AICC -2530.1536 MAPE 147.943475 Normality Test 98.7311 Pr > ChiSq <.0001 Intercept 1 0.005385 0.002267 2.37 0.0175 ARCH0 1 0.0000844 0.0000246 3.43 0.0006 ARCH1 1 0.1212 0.0203 5.97 <.0001 GARCH1 1 0.8524 0.0195 43.79 <.0001 DELTA 1 1.0292 0.8207 1.25 0.2098 Model: Std. Error of risk premium is 0.8207 Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 14
IGARCH(1,1) The SAS System 00:31 Saturday, October 24, 2009 55 The AUTOREG Procedure Dependent Variable spret Ordinary Least Squares Estimates SSE 2.70318776 DFE 791 MSE 0.00342 Root MSE 0.05846 SBC -2244.3895 AIC -2249.0641 MAE 0.03973876 AICC -2249.059 MAPE 135.870252 Regress R-Square 0.0000 Durbin-Watson 1.8161 Total R-Square 0.0000 Intercept 1 0.006143 0.002077 2.96 0.0032 Algorithm converged. Integrated GARCH Estimates SSE 2.70447825 Observations 792 MSE 0.00341 Uncond Var. Log Likelihood 1268.20368 Total R-Square. SBC -2516.3837 AIC -2530.4074 MAE 0.03970226 AICC -2530.3769 MAPE 145.768595 Normality Test 83.7218 Pr > ChiSq <.0001 Intercept 1 0.007420 0.001545 4.80 <.0001 ARCH0 1 0.0000511 0.0000162 3.16 0.0016 ARCH1 1 0.1429 0.0188 7.62 <.0001 GARCH1 1 0.8571 0.0188 45.71 <.0001 Model: Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 15
EGARCH(1,1) The AUTOREG Procedure Dependent Variable spret Ordinary Least Squares Estimates SSE 2.70318776 DFE 791 MSE 0.00342 Root MSE 0.05846 SBC -2244.3895 AIC -2249.0641 MAE 0.03973876 AICC -2249.059 MAPE 135.870252 Regress R-Square 0.0000 Durbin-Watson 1.8161 Total R-Square 0.0000 Intercept 1 0.006143 0.002077 2.96 0.0032 Algorithm converged. Exponential GARCH Estimates SSE 2.70360869 Observations 792 MSE 0.00341 Uncond Var. Log Likelihood 1271.91981 Total R-Square. SBC -2510.4668 AIC -2533.8396 MAE 0.03971555 AICC -2533.7633 MAPE 141.488491 Normality Test 99.7411 Pr > ChiSq <.0001 Intercept 1 0.006872 0.001604 4.28 <.0001 EARCH0 1-0.1540 0.0599-2.57 0.0101 EARCH1 1 0.2269 0.0342 6.63 <.0001 EGARCH1 1 0.9734 0.009897 98.35 <.0001 THETA 1-0.2573 0.1051-2.45 0.0144 Model: Leverage: -0.2573 ; Therefore a standardized shock of magnitude +2 contributes (0.2269*(1-0.2573)*2)=0.3370 to the log volatility. Similarly a magnitude of -2 contributes 0.5706 to the log volatility. Time Series Analysis Lecture 6 Edited: Oct 23, 2009 Page 16