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  r t 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 CoefficientStd. Errorz-StatisticProb. C(4)-0.0003197.52E-05-4.2466430.0000 W(1)-1.89E-082.59E-08-0.7294230.4657 W(2)2.71E-072.88E-089.4285620.0000 W(3)-4.35E-073.87E-08-11.247180.0000 W(4)3.28E-075.42E-086.0582210.0000 W(5)-3.98E-075.40E-08-7.3774870.0000 W(6)6.00E-075.85E-0810.263390.0000 W(7)-8.04E-079.93E-08-8.0922080.0000 C(5)1.1372770.04356326.106660.0000 C(1)0.0894870.00241837.008160.0000 C(2)0.8810050.004612191.02450.0000 Log likelihood-15733.51 Akaike info criterion2.528223 Avg. log likelihood-1.263228 Schwarz criterion2.534785 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.