1 Ka-fu Wong University of Hong Kong Volatility Measurement, Modeling, and Forecasting

2 Importance of volatility Good volatility forecasts are crucial for the implementation and evaluation of asset and derivative pricing theories as well as trading and hedging strategies. Two assets: an risky and a riskless (i.e., volatility = 0) Risky asset generally has a higher expected return than the riskless assets. We would like to invest in a portfolio consisting of the two assets. When the risky asset has a very high volatility, the portfolio will consist of the riskless asset only. When the risky asset has a very low volatility, the portfolio will consist of more risky assets.

3 Importance of volatility The variance of inflation may have impact on various macro and investment decisions. High variance in inflation may also imply welfare loss. Previous studies have tried to measure the time-varying variance of inflation.

4 Clustering of volatility It is a well-established fact, dating back to Mandelbrot (1963) and Fama (1965), that financial returns display pronounced volatility clustering. Therefore, models of volatility should allow such clustering.

5 Example: AR(1) y t = φ y t-1 +  t  t ~ WN(0,  2 ) AR(1): y t = φ (φ y t-2 +  t-1 ) +  t = φ 2 y t-2 + φ  t-1 +  t = φ 2 (φ y t-3 +  t-2 )+ φ  t-1 +  t = φ 3 y t-3 + φ 2  t-2 + φ  t-1 +  t =  t + φ  t-1 + φ 2  t-2 + φ 3  t-3 + φ 4  t-4 + φ 5  t-5 + … … E(  t ) = 0, E(y t ) = 0 Var(  t ) = E[(  t – E(  t )) 2 ] =  2 Var(y t ) = E[(y t – E(y t )) 2 ] =  2 (1+ φ + φ 2 + φ 3 + φ 4 +…) Repeatd substitution:

6 Homoskedasticity vs. Heteroskedasticity So far, innovation are assumed to be i.i.d. It is possible to allow variance to change across observations, i.e., Heteroskedasticity. Information available at time t-1

7 A general linear process Consider a general linear process: Need not be i.i.d.

8 Two examples Consider a general linear process: Need not be i.i.d. y t = φ y t-1 +  t y t =  t + φ  t-1 + φ 2  t-2 + φ 3  t-3 + φ 4  t-4 + … b i = φ i AR(1) MA(2) y t =  t + θ 1  t-1 + θ 2  t-2 b 0 =1, b 1 = θ 1, b 2 = θ 2, b 3 =b 4 =…=0

9 Unconditional means and variances Consider a general linear process: y t = φ y t-1 +  t y t =  t + φ  t-1 + φ 2  t-2 + φ 3  t-3 + φ 4  t-4 + … b i = φ i AR(1) y t =  t + θ 1  t-1 + θ 2  t-2 MA(2) E(y t )= E(  t ) + φE(  t-1 ) + φ 2 E(  t-2 ) + … = 0 V(y t )= V(  t ) + φ 2 V(  t-1 ) + φ 4 V(  t-2 ) + … E(y t )= E(  t ) + θ 1 E(  t-1 ) + θ 2 E(  t-2 ) = 0 V(y t )= V(  t ) + θ 1 2 V(  t-1 ) + θ 2 2 V(  t-2 )

10 Conditional variances change with horizon of forecast but are not time-varying given a horizon. Consider a general linear process: Conditional mean is time-varying : h-step ahead forecast is time-varying: Conditional information b 0 =1, b 1 = θ 1, b 2 = θ 2, b 3 =b 4 =…=0 y t =  t + θ 1  t-1 + θ 2  t-2 MA(2) E(y t |  t-1 )= θ 1  t-1 + θ 2  t-2 E(y t+1 |  t )= θ 1  t + θ 2  t-1 E(y t+2 |  t+1 )= θ 1  t+1 + θ 2  t

11 Conditional variances change with horizon of forecast but are not time-varying given a horizon. Consider a general linear process: Conditional variance is not time-varying: Conditional prediction error variance: Conditional information b 0 =1, b 1 = θ 1, b 2 = θ 2, b 3 =b 4 =…=0 y t =  t + θ 1  t-1 + θ 2  t-2 MA(2) E[(y t -E(y t |  t-1 ) ) 2 |  t-1 ] = E(  t 2 |  t-1 ) =  2 Non-time-varying!

12 ARCH(p) process Examples: (1)  ARCH(1):  t 2 =  +  1  t-1 2 (2) ARCH(2):  t 2 =  +  1  t  2  t-2 2 ARCH(p) AutoRegressive Conditional Heteroskedasticy of order p

13 ARCH(p) process Examples: (1)  ARCH(1):  t 2 =  +  1  t-1 2 (2) ARCH(2):  t 2 =  +  1  t  2  t-2 2 ARCH implies volatility clustering. That is, large changes tend to be followed by large changes and small by small, of either sign.

14 ARCH(p) process Examples: (1)  ARCH(1):  t 2 =  +  1  t-1 2 (2) ARCH(2):  t 2 =  +  1  t  2  t-2 2 (1) Unconditional mean (2) Unconditional variance (3) Conditional variance Some properties

15 ARCH(1)  t 2 =  +  1  t-1 2 Note that E[  t 2 ] = E[ E(  t 2 |  t-1 ) ] = E(  t 2 ) =  2 E[(  t -E(  t )) 2 ] = ? E[  t 2 ] =  +  1 E[  t-1 2 ]  2 =  +  1  2  2 =  / (1-  1 )

16 How to simulation ARCH(1)? Suppose we are interested in generating T observations of  t that has the property of ARCH(1).  t ~ N(0,  t 2 ), where  t 2 =  +  1  t-1 2 (1) Fixed the parameters. Compute the unconditional variance of  t.  2 =  / (1-  1 ) (2) Generate T+1 observations of standard normal random variables, v 0, v 1, …., v T (3) Generate  t recursively For t=0,  t 2 =  2,  t = v t  t For t=1,  t 2 =  +  1  t-1 2, and  t = v t  t For t=2,  t 2 =  +  1  t-1 2, and  t = v t  t

17 The inflation example of Engle (1982) First difference of the log of the quarterly consumer price index log of the quarterly manual wage rates Lagged 4 periods Engle, Robert F. (1982): “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 50(4):

18 The inflation example of Engle (1982) OLS regression Restriction imposed.

19 The inflation example of Engle (1982) ML estimation with ARCH(1) The ARCH model comes closer to truly random residuals after standardizing for their conditional distributions.

20 GARCH(p,q) Backward substitution on  t 2 yields A infinite-order ARCH process with some restriction in the coefficients. (Analogy: An ARMA(p,q) process can be written as MA(∞) process.) GARCH can be viewed as a parsimonious way to approximate a high order ARCH process

21 Important properties of GARCH(p,q) (1) Unconditional variance is fixed but conditional variance is time-varying

22 Important properties of GARCH(p,q) (2) Unconditional distribution of conditionally Gaussian GARCH is symmetric and leptokurtic. Real-world financial asset returns, are often found to symmetrically distributed and have a fatter tail than Gaussian distribution. Ordinary Gaussian distribution does not provide a good approximation of the asset returns, but the Gaussian distribution with GARCH does.

23 Important properties of GARCH(p,q) (3) Conditional prediction error variance varies with conditional information set. unbiased forecast Conditional variance of the prediction error Conditional variance approaches unconditional variance

24 Important properties of GARCH(p,q) (3)  t follows GARCH implies  t 2 follows an ARMA.

25 Extension of ARCH and GARCH Models Threshold GARCH When the lagged return is positive (good news yesterday), D=0, so the effect of the lagged squared return on the current conditional variance is simply . When the lagged return is negative (negative news yesterday), D=1, so the effect of the lagged squared return on the current conditional variance is simply . Allowance for asymmetric response has proved useful for modeling “leverage effects” in stock returns, which occur when  < 0.

26 Extension of ARCH and GARCH Models exponential GARCH Volatility is drive by both the size and sign of shocks (both positive and negative). Hence, the model allows for asymmetric response depending on the sign of news. When the shock is positive, the impact of (  t-1 /  t-1 ) on ln(  t 2 ) is  +  When the shock is negative, the impact of (  t-1 /  t-1 ) on ln(  t 2 ) is  + 

27 Extension of ARCH and GARCH Models GARCH with exogenous variables Financial market volume, for example, often helps to explain market volatility.

28 Extension of ARCH and GARCH Models GARCH-in-Mean (i.e., GARCH-M) High risk, high return. Conditional mean regression

29 Estimating, Forecasting, and Diagnosing GARCH Models Diagnostic: Estimate the model without GARCH in the usual way. Look at the time series properties of the squared residuals. Correlogram, AIC, SIC, etc. ARMA(1,1) in the squared residuals implies GARCH(1,1).

30 Estimating, Forecasting, and Diagnosing GARCH Models Estimation: Usually use maximum likelihood with the assumption of normal distribution. Maximum likelihood estimation finds the parameter values that maximize the likelihood function Forecast: In financial applications, volatility forecasts are often of direct interest. 1-step-ahead conditional variance Better forecast confidence interval vs.

31 Application: Stock Market Volatility Objective: Model and forecast the volatility of daily returns on the New York Stock Exchange Data: Daily returns on the New York Stock Exchange (NYSE) form January 1, 1988, through December 31, Excluding holidays, there are 3531 observations. Estimation: Forecast:

32 Time Series Plot, NYSE Returns

33 Histogram and Related Diagnostic Statistics, NYSE Returns

34 Correlogram, NYSE Returns

35 Time Series Plot, Squared NYSE Returns

36 Correlogram, Squared NYSE Returns

37 AR(5) Model, Squared NYSE Returns

38 ARCH(5) Model, NYSE Returns

39 Correlogram, Squared Standardized ARCH(5) residuals, NYSE Returns

40 GARCH(1,1) Model, NYSE Returns  t-1 2  t-1 2

41 Correlogram, Squared Standardized GARCH(1,1) residuals, NYSE Returns

42 Estimated Conditional Standard Deviation, GARCH(1,1) Model, NYSE Returns

43 Estimated Conditional Standard Deviation, Exponential Smoothing, NYSE Returns

44 Conditional Standard Deviation, History and Forecast, GARCH(1,1) Model

45 Conditional Standard Deviation, Extended History and Extended Forecast, GARCH(1,1) Model

46 Is GARCH(1,1) enough most of the time? 330 GARCH-type models are compared in terms of their ability to forecast the one-day-ahead conditional variance. The models are evaluated out-of-sample using six different loss functions, where the realized variance is substituted for the latent conditional variance. Hansen, Peter R. and Asger Lunde (2005): “A Forecast Comparison Of Volatility Models: Does Anything Beat A GARCH(1,1)?” Journal of Applied Econometrics, 20:

47 Is GARCH(1,1) enough most of the time? Data: DM–$ spot exchange rate data, the estimation sample spans the period from October 1, 1987 through September 30, 1992 (1254 observations) and the out-of-sample evaluation sample spans the period from October 1, 1992 through September 30, 1993 (n = 260). IBM stock returns, the estimation period spans the period from January 2, 1990 through May 28, 1999 (2378 days) and the evaluation period spans the period from June 1, 1999 through May 31, 2000 (n = 254).

48 Specifications of the conditional variance

49 Loss functions for forecast evaluation MSE 2 and R 2 Log are similar to R 2 of the MZ regressions.

50 The test Giving benefits of the doubt to the benchmark, i.e., GARCH(1,1). Loss of GARCH(1,1) Loss of alternatiave GARCH models. The maintained hypothesis is that GARCH(1,1) is better unless there is strong evidence against it.

51 Superior Predictive Ability and Reality Check for data snooping

52 The test results superior predictive ability (SPA)

53 IBM data: superior predictive ability and reality check for data snooping

54 Does Anything Beat A GARCH(1,1)? No. So, use GARCH(1,1) if no other information is available.

55 End