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**Robert Engle and Jose Gonzalo Rangel NYU and UCSD**

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

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**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

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**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

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**A MODEL CAMPBELL(1991), CAMPBELL& SHILLER(1988) LOG LINEARIZATION**

Decompose into Innovations to the present discounted value of future dividends or expected returns

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**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

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**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

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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.

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

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**HISTORY OF THE US EQUITY MARKET VOLATILITY: S&P500**

PLOT PRICES AND RETURNS HOW MUCH DO RETURNS FLUCTUATE?

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**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

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

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GARCH(1,1) The unconditional variance is then

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**GARCH(1,1) If omega is slowly varying, then**

This is a complicated expression to interpret

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**SPLINE GARCH Instead, use a multiplicative form**

Tau is a function of time and exogenous variables

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**UNCONDITIONAL VOLATILTIY**

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

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

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**ESTIMATION FOR A GIVEN K, USE GAUSSIAN MLE**

CHOOSE K TO MINIMIZE BIC FOR K LESS THAN OR EQUAL TO 15

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**EXAMPLES FOR US SP500 DAILY DATA FROM 1963 THROUGH 2004**

ESTIMATE WITH 1 TO 15 KNOTS OPTIMAL NUMBER IS 7

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**RESULTS LogL: SPGARCH Method: Maximum Likelihood (Marquardt)**

Date: 08/04/04 Time: 16:32 Sample: Included observations: 12455 Evaluation order: By observation Convergence achieved after 19 iterations Coefficient Std. Error z-Statistic Prob. 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 Hannan-Quinn criter

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**PATTERNS OF VOLATILITY**

ASSET CLASSES EQUITIES EQUITY INDICES CURRENCIES FUTURES INTEREST RATES BONDS PUT TOGETHER AN EVIEWS WORKFILE WITH ALL SIX TYPES OF ASSET CLASSES. FOR THE BONDS USE A LONG BOND YIELD SO THAT PRICE IS COUPON/YIELD. THEN FIGURE OUT RETURN. SIMILARLY FOR SHORT TERM INTEREST RATES – APPROXIMATELY TAKE FIRST DIFFERENCES. SHOULD I THINK ABOUT THE VOLATILITY OF PRICE DIFFERENCES VS LOG DIFFERENCES?

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**VOLATILITY BY ASSET CLASS**

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**PATTERNS OF EQUITY VOLATILITY**

COUNTRIES DEVELOPED MARKETS EUROPE TRANSITION ECONOMIES LATIN AMERICA ASIA EMERGING MARKETS Calculate Median Annualized Unconditional Volatility using daily data

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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:

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**EXPLANATORY VARIABLES**

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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

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**ONE VARIABLE REGRESSIONS**

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MULTIPLE REGRESSIONS

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CPI VOLATILITY T-STAT

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

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**PANEL ESTIMATE RANDOM COUNTRY EFFECTS AR(1) DYNAMIC COUNTRY EFFECTS**

TIME FIXED EFFECTS

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**ANNUAL REALIZED VOLATILITY**

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**CONCLUSIONS AND IMPLICATIONS**

Unconditional volatility changes in systematic ways. Macro volatility and growth are important determinants of financial volatility. Unconditional volatility and realized volatility give similar results but the former fits better. Big swings in financial volatility are common across the globe.

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