Is No News Good News? Reconciling Evidence from ARCH and Stochastic Volatility Models

1. Is No News Good News?Reconciling Evidence from ARCH and Stochastic Volatility Models By Jun Yu Singapore Management University

2. Outline of Talk • Motivation and literature • Our approach • Estimation and inferential methods • Empirical results • Sensitivity check • Conclusions

3. Log-volatility at t+1 News Impact Function Return News at t

4. Motivation and Literature • How volatility responds to return news? • This is an important question. • Volatility forecasting, asset allocation, risk management, asset pricing (option pricing) • Asymmetry

5. Motivation and Literature • Wisdom 1: volatility feedback effect of (Campbell and Hentschel, 1992). • arrival of a large piece of news is evidence of high current volatility. • volatility tends to cluster. • Higher future volatility should amplify the negative impact of bad news but dampen the positive impact of good news. So no news is good news.

6. Motivation and Literature • Wisdom 2: leverage effect of Black (1976). • Bad news decreases the value of a firm’s equity and increases its leverage. Hence volatility should go up. • Good news increases the value of a firm’s equity and decreases its leverage. Hence volatility should go down. So good news is better than no news.

7. Motivation and Literature • Difficulty: Volatility is not directly observable. • Two empirical methods: • Autoregressive Conditional Heteroskedasticity (ARCH) models • Stochastic Volatility (SV) models

8. Motivation and Literature • ARCH models: • First generation: ARCH of Engle (1982), GARCH of Bollerslev (1986) • Second generation: EGARCH of Nelson (1991), GJR-GARCH of Glosten, Jagannathan, and Runkle (1993)

9. Motivation and Literature • ARCH(1)

10. Motivation and Literature • Engle and Ng (1993) introduced the News Impact Function (NIF) to examine the relationship between holding other lagged variables constant • Let

11. Motivation and Literature • The NIF for the ARCH(1) model is

12. Motivation and Literature • NIF is symmetric. Volatility responds to the good news in the same way as to the bad news of the same size. • The larger the piece of news, the higher the volatility next period. This holds for each single period! • As far as the volatility is concerned, no news is good news because volatility is the smallest when no news arrives.

13. Motivation and Literature • GARCH(1,1) • Fix at its long run mean. The NIF of GARCH(1,1) is the same as that of ARCH(1).

14. Motivation and Literature • EGARCH(1,1) • NIF for EGARCH(1,1) is

15. Motivation and Literature • Three special cases: • NIF is V-shaped if • NIF is asymmetrically V-shaped if, in addition, • NIF slopes downwards if • Only case 3 is consistent with the leverage hypothesis. The first 2 cases imply no news is good news while the last case implies good news is better.

16. Motivation and Literature Case 1 Case 2 Case 3

17. Motivation and Literature • Case 2 is often found empirically since empirical estimates of EGARCH(1,1) often suggest that and • Some commercial software, such as Splus, unnecessarily requires . As a result, a downward-sloping NIF is ruled out.

18. Motivation and Literature • GJR-GARCH(1,1) • NIF for GJR is

19. Motivation and Literature • Three special cases: • NIF is V-shaped if • NIF is asymmetrically V-shaped if, in addition, • NIF slopes downwards if • Only case 3 is consistent with the leverage hypothesis. Only case 2 is empirically relevant and implies that no news is good news.

20. Motivation and Literature • Summary of features of ARCH models • ARCH models allow for volatility clustering • ARCH models allow for excess kurtosis • The same set of parameters generates volatility clustering and excess kurtosis • Volatility is a deterministic function of a few lagged variables • NIF for EGARCH and GJR-GARCH has flexible shapes • The empirical NIF is often asymmetrically V-shaped

21. Motivation and Literature • SV models • Log-normal SV model:

22. Motivation and Literature • Summary of features of SV models • SV allows both volatility clustering and excess kurtosis. • Volatility clustering is captured by phi. • Excess kurtosis is captured by sigma. • Log-volatility is AR(1) and hence stochastic. • SV only postulates a relationship between volatility and return news on average. • NIF of Engle and Ng (1993) is not applicable to SV.

23. Motivation and Literature • We have to extend the definition of NIF so that it is applicable to SV models. We define NIF to be the relationship between holding other lagged variables constant • Let

24. Motivation and Literature • To derive the NIF for the log-normal SV model, we rewrite it as • So the NIF is

25. Motivation and Literature • The NIF for the log-normal SV model is linear and slopes downwards. It is consistent with the leverage hypothesis. • It is consistent with Bollerslev and Zhou (2005) where the integrated volatility is found to be negatively related to the standardized shock in the context of the continuous time square root SV model.

26. Motivation and Literature

27. Motivation and Literature • SV models is more flexible than ARCH models. • It is known that the ARCH effect becomes insignificant in the context of the basic SV model (Fridman and Harris 1998 and Danielsson 1994).

28. Motivation and Literature • We design two Monte Carlo experiments. We simulate 1000 samples, each with 2000 observations from the following lognormal-SV model. • The true NIF slopes downwards (slope=-0.04) • Fit both EGARCH and GJR-GARCH to each simulated sequence using maximum likelihood (ML) and then examine the estimated NIF

29. Motivation and Literature

30. Motivation and Literature • Observations: EGARCH and GJR lead to a V-shaped NIF when the data are simulated from a SV model with monotonic NIF.

31. Our Approach • Concern 1: The lognormal SV model is not flexible enough in terms of possible shapes of NIF. • Concern 2: EGARCH has more flexible shapes for NIF but specification of volatility dynamics may be too restricted.

32. Our Approach • Unified approach: Combine lognormal SV and EGARCH

33. Our Approach • Three special cases: • If , it becomes a lognormal-SV model • If , it becomes an EGARCH model • If, in addition, , it is an EGARCH model without asymmetry. • NIF of the proposed model

34. Our Approach • Merits of the proposed model: • Nests both lognormal-SV and EGARCH and hence provides a simple way to test two specifications. • Being a SV model, allow more flexible NIF than lognormal-SV model.

35. Our Approach • Theorem 1: {y} and {h} are covariance stationary, strictly stationary and ergodic if and only if . {y} possesses finite moments of arbitrary order and the expression for the moments is given in the paper.

36. Estimation Methods • It is known SV models are more difficult to estimate than ARCH models since calculation of likelihood is non-trivial. • Likelihood based estimation and inference can now be done via a Bayesian method called Markov Chain Monte Carlo (MCMC). • Instead of searching for the analytic expression of a posterior density, MCMC methods aim to provide a general mechanism to sample the parameter vector (including latent volatility) from its posterior density.

37. Estimation Methods • In particular, MCMC methods set up a Markov chain for each variate whose invariant distribution is the same as the posterior density. When the Markov chain converges, the simulated values may be regarded as a (correlated) sample obtained from the posterior and hence can be used as the basis for making statistical inferences.

38. Estimation Methods • To estimate EGARCH, we use the MCMC method of Vrontos, et al (2000) • To estimate lognormal SV and proposed model, we use the MCMC method of Yu (2005). • Prior distributions: most of them are not informative. • 110,000 samples are simulated with first 10,000 samples discarded.

39. Estimation Methods • Monte Carlo results when data are simulated from: • 2500 observations are simulated. The replication is 500.

40. Methods for Model Comparison • We compare three models: the proposed model, the EGARCH, and the lognormal SV model • Examine the significance of relevant estimated coefficients in the proposed model • Evaluate posterior odds – marginal likelihood method of Chib (1985) • Deviance information criterion (DIC) – generalization of AIC

41. Empirical Results • Data: S&P500 index continuously compounded returns from July 1, 1993 to December 31, 2003. Excluding weekend and holidays, there are 2,645 daily observations. The sample mean was subtracted from the data.

42. Empirical Results

43. Empirical Results

44. Empirical Results

45. Empirical Results • Summary of empirical results • EGARCH suggests a V-shaped NIF • EGARCH is rejected • Lognormal SV cannot be rejected • NIF slopes downwards although more flexible shapes are allowed. Good news is better than no news

46. Sensitivity Check • In ARCH both volatility clustering and excess kurtosis are generated by the same set of parameters. In practice, serving the dual roles is difficult. • One way to break the tension is to use a heavy-tailed error distribution, that is,

47. Sensitivity Check • Another way to break the tension is to introduce jumps into the return process, that is,

48. Sensitivity Check • Estimate these two models using the MCMC method of Berg, Meyer and Yu (2004). • 310,000 samples are simulated with first 60,000 samples discarded.

49. Sensitivity Check

50. Sensitivity Check • Results: • Both models suggest downward-sloping NIF. Good news is better than no news. • Reject t-EGARCH. • Reject EGARCH+jumps. • LN-SV+Jumps model performs the best, followed by LN-SV+t error distribution, and then LN-SV.