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

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

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

DEFINITIONS  r t is a mean zero random variable measuring the return on a financial asset  CONDITIONAL VARIANCE  UNCONDITIONAL VARIANCE

GARCH(1,1)  The unconditional variance is then 

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

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

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

SPLINE  ASSUME UNCONDITIONAL VARIANCE IS AN EXPONENTIAL QUADRATIC SPLINE OF TIME

THIS IS EASY TO COMPUTE  For K knots equally spaced, construct new regressors

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

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

RESULTS LogL: SPGARCH Method: Maximum Likelihood (Marquardt) Date: 08/04/04 Time: 16:32 Sample: Included observations: Evaluation order: By observation Convergence achieved after 19 iterations CoefficientStd. Errorz-StatisticProb. C(4) E W(1)-1.89E E W(2)2.71E E W(3)-4.35E E W(4)3.28E E W(5)-3.98E E W(6)6.00E E W(7)-8.04E E C(5) C(1) C(2) Log likelihood Akaike info criterion Avg. log likelihood Schwarz criterion Number of Coefs.11 Hannan-Quinn criter

PATTERNS OF VOLATILITY  ASSET CLASSES –EQUITIES –EQUITY INDICES –CURRENCIES –FUTURES –INTEREST RATES –BONDS

VOLATILITY BY ASSET CLASS

PATTERNS OF EQUITY VOLATILITY  COUNTRIES –DEVELOPED MARKETS –EUROPE –TRANSITION ECONOMIES –LATIN AMERICA –ASIA –EMERGING MARKETS  Calculate Median Annualized Unconditional Volatility using daily data

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:

EXPLANATORY VARIABLES

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

ONE VARIABLE REGRESSIONS

MULTIPLE REGRESSIONS

ANNUAL REALIZED VOLATILITY

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.