Modeling and Forecasting Stock Return Volatility Using a Random Level Shift Model

1. Modeling and Forecasting Stock Return Volatility Using a Random Level Shift Model Lu Hongda Yu Lu Meng Shuangshuang

2. Introduction •  Main goal : to forecast volatility proxied by daily squared returns • Our approach: extends Starica and Granger (2005)’s work • Apply log-absolute returns and daily returns rather than intra-daily data

3. Three main parts

4. 1. The model and the estimation method Lu Hongda

5. 1.1 Model • The model we apply to log-absolute returns is given by is a constant is the random level shift components is a short-memory process, included to model the remaining noise. • The level shift component is specified by: Where ,is a binomial variable that takes value 1 with probability α and value 0 with probability (1 − α). If it takes value 1, then a random level shift occurs, specified by ∼ i.i.d. N(0,).

6. 1.2 Assumptions For simplicity, we assume: • Short memory: =, () • The components and are mutually independent • Normality assumption for , and (needed to construct the likelihood function)

7. 1.3 Model Estimation • Specify level shift component as a random walk process with innovations distributed according to a mixture of two normally distributed processes: • where , and is a Bernoulli random variable that takes value one with probability α and value 0 with probability 1 − α. By specifying and

8. Estimation of : The probability of level shifts is very small in all cases considered. Thus… • Given that the shifts occur so infrequently, the noise component accounts for the bulk of total variation.

9. Then… Apply the method of Bai and Perron (2003) to obtain the estimates of the break dates that globally minimize the following sum of squared residuals: • where m is the number of breaks, (i = 1,… , m) are the break dates with and and(i = 1, ..., m + 1) are the means within each regime which can easily be estimated once the break dates are.

10. 1.3 Model Estimation

11. Continue… • We next specify the model in terms of first-differences of the data: where • We then have the following state space form: • Or more generally • Where, in the case of an AR(p) process

12. And… and is a p-dimensional normally distributed random vector with mean zero and covariance matrix

13. 1.4 The estimation method • Similar to Markov regime switching models in estimation methodology • The log likelihood function is: where

14. 1.4 The estimation method where • where 1 represents a (4*1 )vector of ones denotes element-by-element multiplication • with element, • withthe element of

15. 1.4 The estimation method For when We first focus on the evolution of

16. 1.4 The estimation method • Therefore, the evolution of is given by: • Or more compactly by:

17. 1.4 The estimation method • With • The conditional likelihood for is the following Normal density: • : the prediction error : the prediction error variance

18. 1.4 The estimation method • The best forecast for the state variable and its associated variance conditional on past information and are We have the measurement equation

19. 1.4 The estimation method • Where • Hence, the prediction error and variance are: • Applying standard updating formulae, we have (given ),

20. 1.4 The estimation method • To reduce the dimension of the estimation problem, we adopt the re-collapsing procedure suggested by Harrison and Stevens (1976), given by

21. 1.4 The estimation method • By doing so, we make unaffected by the history of states before time t − 1 • If we define we then have four possible states corresponding to • with the transition matrix Π as defined in (4)

22. 1.4 The estimation method • The vector of conditional densities • thus have a more compact representation given by: • Where are as defined in (5)

23. 1.5. Apply model and estimation method • Four major market indices: the S&P 500, AMEX, Dow Jones and NASDAQ • Daily returns: • Offset parameter/small constant: 

24. 1.5. Apply model and estimation method Specification of short-memory component • , ,by Starica and Granger(2005) • , process, as a robustness check

25. 1.5. Apply model and estimation method • Initialize the state vector and its covariance matrix by their unconditional expected values(all components of states are stationary), i.e., and • We obtain estimates by directly maximizing the likelihood function:

26. 2. Empirical results for returns on stock market indices Yu Lu

27. 2.1. Estimation results • The cases in which the short-memory component is specified to be white noise and an AR (1) process • the standard deviation of the original series • : the standard deviation of level shift component, • α: the probability of a shift • :the standard deviation of the stationary component • φ :the autoregressive coefficient when considering the AR (1) specification for .

28. Estimation results? Table 1: Maximum Likelihood Estimates

29. 2.2. Noteworthy features of results: Firstly, for short-memory component • When consideringprocess, estimate of φ is very small and close to zero • Adopting either an AR(1) or white noise specification for yields very similar results Thus… In subsequent sections we shall only consider results based on the white noise specification for the short-memory component. These results are in agreement with those of Starica and Granger (2005)

30. 2.2. Noteworthy features of results: Secondly, for level shift component • The probability of level shifts is very small in all cases considered. Thus… • Given that the shifts occur so infrequently, the noise component accounts for the bulk of total variation.

31. 2.3. The effect of level shifts on long-memory and conditional heteroskedasticity • Investigating whether the shifts can explain: • a) the well-documented feature of long-memory • b) the presence of conditional heteroskedasticity

32. 2.3.1. Effects of level shift on long-memory Whether the level shift can account for the long-memory feature of the series? Plot: the ACF of the original series the ACF of its short-memory component(obtained by subtracting the fitted trend from )

34. the ACF of the original series • The log-absolute returns clearly display an autocorrelation function that resembles that of a long-memory process: it decays very slowly and the values remain important even at lag 300.

36. b) the ACF of its short-memory component • For all practical purposes, we can view the short-memory component as being nearly white noise.

37. 2.3.2 Effects of level shift on conditional heteroskedasticity • GARCH (1, 1) model with Student-t errors given by, for the demeaned returns process • i.i.d.Student-t distributed with mean 0 and variance 1 • the parameters of interest and measure the extent of conditional heteroskedasticity present in the data.

38. CGARCH model • where if tis in regime i, i.e., and 0 otherwise, with (i = 1, ..., m) being the break dates documented in Figure 1 (again and ).The coefficients , which index the magnitude of the shifts, are treated as unknown and are estimated with the remaining parameters, while the number of breaks is obtained from the point estimate of α.

39. Table2. Parameter Estimates

40. 2.3.2 Effects of level shift on conditional heteroskedasticity Use the standard GARCH (1,1) model allow for level shifts(CGARCH) • Both estimates are highly significant for all series. In particular, the value of is quite high. • when we none of the estimates ofare significant.

41. Summary: • The level shifts model with white noise errors appears to provide an accurate description of the data. • The level shift component is an important feature that explains both the long-memory and conditional heteroskedasticity features generally perceived as stylized facts.

42. 3 Forecasting Meng Shuangshuang Performance of the level shift model &GARCH(1,1)in forecasting volatility

43. 3.1 Design of forecasting experiment • Following Starica and Granger(2005) • Start forecasting at observations 2,000 • Re-estimate every 20 days, up to 200 days • Proxy: realized squared returns( )

44. 3.2 Evaluation of forecasting performance • evaluated by the ratio of their MSE: • MSE(p) • where n : no. of forecasting produced

45. 3.3 construction of the forecastsThe level shift model(1) • The level shift model • an appropriate transformation • where C : a positive constant used to bound returns away from zero • yields

46. 3.3 construction of the forecastsThe level shift model(2) • continue • ignore level shifts when forecasting • rare uncertain about timing and magnitudes

47. 3.3 construction of the forecastsThe level shift model(3) • Continue • the K -period ahead forecast of the squared returns is :

48. 3.3 construction of the forecastsThe GARCH model(1) • Student-t GARCH(1,1) model • Transformation:

49. 3.3 construction of the forecasts The GARCH model(2) • assuming < 1, • recursive form for the squared demeaned returns • So for k > 1, the k-period ahead forecast • Where • the forecasts for the squared returns

50. 3.4 Forecasting Comparisons The comparison indicates that with a precise estimate of the mean of log-absolute returns at a given date, we can obtain much better forecasts from the level shift model than the other models. the random level shifts model the GARCH(1,1) • use the fitted means obtained from the full sample to get the mean • re-estimate every 20 observations to get the mean