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

Spline Garch as a Measure of Unconditional Volatility and its Global Macroeconomic Causes. Robert Engle and Jose Gonzalo Rangel NYU and UCSD. GOALS. ESTIMATE THE DETERMINANTS OF GLOBAL EQUITY VOLATILITY How are long run volatility forecasts affected by macroeconomic conditions?

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

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

  2. GOALS • ESTIMATE THE DETERMINANTS OF GLOBAL EQUITY VOLATILITY • How are long run volatility forecasts affected by macroeconomic conditions? • What volatility can be expected for a newly opened financial market? • MEASURE AND MODEL CHANGING UNCONDITIONAL VOLATILITY

  3. WHAT MOVES ASSET PRICES AND VOLATILITY? • NEWS vs OTHER THINGS • RESEARCH STRATEGIES • VOLATILITY MODELS • e.g.Officer(1973), Schwert(1989) • ANNOUNCEMENT + NEWS MODELS • e.g.Roll(1988), Cutler Poterba and Summers(1990) • In all cases, macro effects appear small

  4. A MODEL • CAMPBELL(1991), CAMPBELL& SHILLER(1988) LOG LINEARIZATION • Decompose into Innovations to the present discounted value of future dividends or expected returns

  5. MULTIPLICATIVE EFFECTS • The impact of a news event may depend upon the macro economy. • Eg. News about a firm will have a bigger effect in a recession or close to bankruptcy

  6. NEWS EVENTS • Return is a function of news times its impact • e = observable news • z = macro or deterministic events • if news is not observable, then there is just an innovation, u

  7. NEWS VARIANCE • The variance of the news also depends upon macro and other deterministic elements both through the intensity and the magnitude of the news.

  8. REALIZED VARIANCE • Realized Variance is the unconditional variance plus an error. Assuming mean zero returns:

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

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

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

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

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

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

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

  16. SPLINE • ASSUME UNCONDITIONAL VARIANCE IS AN EXPONENTIAL QUADRATIC SPLINE OF TIME • For K knots equally spaced

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

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

  19. 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

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

  21. VOLATILITY BY ASSET CLASS

  22. 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

  23. 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:

  24. EXPLANATORY VARIABLES

  25. 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

  26. ONE VARIABLE REGRESSIONS

  27. MULTIPLE REGRESSIONS

  28. CPI VOLATILITY T-STAT

  29. DROP ARGENTINA? • OUTLIER? • HIGHLY INFORMATIVE? • ESTIMATE BOTH WAYS.

  30. PANEL ESTIMATE • RANDOM COUNTRY EFFECTS • AR(1) DYNAMIC COUNTRY EFFECTS • TIME FIXED EFFECTS

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