Stock returns and volatility: empirical evidence from fourteen countries

Abstract

This is a pioneering effort to test in 14 countries the relationship between stock market returns and their forecast volatility derived from the symmetric and asymmetric conditional heteroscedasticity models. Both weekly and monthly returns and their volatility are investigated. An out-of-sample testing methodology is employed using volatility forecasts instead of investigating the relation between stock returns and their in-sample volatility estimates. Expected volatility is derived from the ARCH(p), GARCH(1,1), GJR-GARCH(1,1) and EGARCH(1,1) forecast models. Expected volatility is found to have a significant negative or positive effect on country returns in a few cases. Unexpected volatility has a negative effect on weekly stock returns in six to seven countries and on monthly returns in nine to eleven countries depending on the volatility forecasting model. However, it has a positive effect on weekly and monthly returns in none of the countries investigated. It is concluded that the return variance may not be an appropriate measure of risk.

Notes

 In the data set for each country, there are 522 weekly volatility observations. Of these, the first 261 of the weekly observations (from December 1987 to November 1992) are used for estimation, while the second 261 observations (from December 1992 to December 1997) are used for forecasting purposes. In the case of the monthly analysis, there are 120 monthly volatility observations, split evenly between estimation and forecasting.

 The order of p in ARCH models across countries is not specified to save space.

 A 261-week (60-month) rolling estimation procedure is adapted; i.e., each month's forecast is based on the estimated daily model parameters in the last 261 (60) calendar weeks (months), on the average, 1250 daily observations. All models are estimated using quasi-maximum likelihood (see Bollerslev and Wooldridge, 1992). All the estimated parameters for the ARCH(p), GARCH(1,1) and GJR-GARCH(1,1) models are always consistent with the standard parameter restrictions for stationarity and non-negativitiy of conditional variance series. Since the EGARCH(1,1) model is in logarithmic form, there is no parameter restriction. The standardized residuals (ε _t /h _t ) and their squared values from all models always obey the standard assumptions of no autocorrelation and no heteroscedasticity although the (ε _t /h _t )'s are not normally distributed. The daily forecast variance converges to its unconditional value fast in all models, a common finding of the previous research using time varying volatility models.

The whole procedure requires estimation of conditional volatility models, from which multi-step (n-day) ahead forecasts of daily variances are constructed. We sum up these out-of-sample forecasts of n daily variances to obtain weekly and monthly total variance (see Akgiray, 1989, among others). Dividing the last figure by the number of trading days (n) in each week or month and then taking its square root gives each model's forecast (σ _f , _w or σ _f ,m ) of within-week or within-month standard deviation which is compared to the actual (estimated) standard deviation (σ _a , _w or σ _a , _m ). Unexpected volatility is simply the difference between forecast volatility and observed volatility for each week or month: [σ _u,w  = σ _f , _w  − σ _a , _w ].

 The findings where both expected and unexpected components of volatility enter simultaneously into the empirical tests are not reported herein because the results previously stated remain unchanged. However, all results are available upon request.