Spline Garch as a Measure of Unconditional Volatility and its Global Macroeconomic Causes

1. Spline Garch as a Measure of Unconditional Volatility and its Global Macroeconomic Causes Robert Engle and Jose Gonzalo Rangel NYU and UCSD

2. HISTORY OF THE US EQUITY MARKET VOLATILITY: S&P500 PLOT PRICES AND RETURNS HOW MUCH DO RETURNS FLUCTUATE?

7. MEAN REVERSION QUOTES • “Volatility is Mean Reverting” • no controversy • “The long run level of volatility is constant” • very controversial • “Volatility is systematically higher now than it has been in years” • Very controversial. Cannot be answered by simple GARCH

8. DEFINITIONS • rt is a mean zero random variable measuring the return on a financial asset • CONDITIONAL VARIANCE • UNCONDITIONAL VARIANCE

9. GARCH(1,1) • The unconditional variance is then

10. GARCH(1,1) • If omega is slowly varying, then • This is a complicated expression to interpret

11. SPLINE GARCH • Instead, use a multiplicative form • Tau is a function of time and exogenous variables

12. UNCONDITIONAL VOLATILTIY • Taking unconditional expectations • Thus we can interpret tau as the unconditional variance.

13. SPLINE • ASSUME UNCONDITIONAL VARIANCE IS AN EXPONENTIAL QUADRATIC SPLINE OF TIME

14. THIS IS EASY TO COMPUTE • For K knots equally spaced, construct new regressors

15. ESTIMATION • FOR A GIVEN K, USE GAUSSIAN MLE • CHOOSE K TO MINIMIZE BIC FOR K LESS THAN OR EQUAL TO 15

16. EXAMPLES FOR US SP500 • DAILY DATA FROM 1963 THROUGH 2004 • ESTIMATE WITH 1 TO 15 KNOTS • OPTIMAL NUMBER IS 7

17. RESULTS LogL: SPGARCH Method: Maximum Likelihood (Marquardt) Date: 08/04/04 Time: 16:32 Sample: 1 12455 Included observations: 12455 Evaluation order: By observation Convergence achieved after 19 iterations Coefficient Std. Error z-Statistic Prob. C(4) -0.000319 7.52E-05 -4.246643 0.0000 W(1) -1.89E-08 2.59E-08 -0.729423 0.4657 W(2) 2.71E-07 2.88E-08 9.428562 0.0000 W(3) -4.35E-07 3.87E-08 -11.24718 0.0000 W(4) 3.28E-07 5.42E-08 6.058221 0.0000 W(5) -3.98E-07 5.40E-08 -7.377487 0.0000 W(6) 6.00E-07 5.85E-08 10.26339 0.0000 W(7) -8.04E-07 9.93E-08 -8.092208 0.0000 C(5) 1.137277 0.043563 26.10666 0.0000 C(1) 0.089487 0.002418 37.00816 0.0000 C(2) 0.881005 0.004612 191.0245 0.0000 Log likelihood -15733.51 Akaike info criterion 2.528223 Avg. log likelihood -1.263228 Schwarz criterion 2.534785 Number of Coefs. 11 Hannan-Quinn criter. 2.530420

25. PATTERNS OF VOLATILITY • ASSET CLASSES • EQUITIES • EQUITY INDICES • CURRENCIES • FUTURES • INTEREST RATES • BONDS

26. VOLATILITY BY ASSET CLASS

28. PATTERNS OF EQUITY VOLATILITY • COUNTRIES • DEVELOPED MARKETS • EUROPE • TRANSITION ECONOMIES • LATIN AMERICA • ASIA • EMERGING MARKETS • Calculate Median Annualized Unconditional Volatility 1997-2003 using daily data

32. MACRO VOLATILITY • Macro volatility variables measure the size of the surprises in macroeconomic aggregates over the year. • If y is the variable (cpi, gdp,…), then:

36. EXPLANATORY VARIABLES

37. ESTIMATION • Volatility is regressed against explanatory variables with observations for countries and years. • Within a country residuals are auto-correlated due to spline smoothing. Hence use SUR. • Volatility responds to global news so there is a time dummy for each year. • Unbalanced panel

38. ONE VARIABLE REGRESSIONS

39. MULTIPLE REGRESSIONS

41. ANNUAL REALIZED VOLATILITY

42. CONCLUSIONS AND IMPLICATIONS • Unconditional volatility changes in systematic ways. • Macro volatility is an important determinant of financial volatility • Potential justification for inflation targeting monetary policy as well as stabilization. • Big swings in global financial volatility are associated with US volatility.