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The regression results of equation (3.1.a) and equation (3.1.b) using weighted least squares (WLS) are shown in Table 3.3. The weights are just the inverse of the intraday standard deviation. WLS is adopted to correct heteroskedasticity. Table 3.3 also contains a simple regression where only the expected volatility variable exists in the right side of the equation.

<Table 3.3>

From Table 3.3, I find that the expected volatility is significantly positive at traditional significance levels for 15 stocks in the simple regression. Six stocks show significance at the 1%

level; an additional 4 stocks show significance at the 5% level; and an additional 5 stocks show significance at the 10% level. A total 15 stocks out of 30 support the hypothesis that excess returns are positively related to expected volatility. However, there is one count-evidence in my sample. Stock PG (index=11) shows that there is a significant negative relation between excess returns and expected risk at the 5% level.The rest of the 14 stocks generally show positive signs (except that stock DIS, index=19 and stock MCD, index=21, show a negative sign) but no significance.

In the multiple regression of equation (3.1.a), nineteen out of 30 stocks show significant support for the hypothesis that there is a positive relation. In addition, the rest of the stocks support a zero relation. The stock PG shows a negative but insignificant coefficient because of the inclusion of the control variable: unexpected volatility. Only stock PG (index=11) and stock

MCD (index=21) out of 30 stocks show a negative coefficient of the expected risk variable in the regression model (3.1.a). In the multiple regression of equation (3.1.b), the estimates of the expected risk variable are very similar to those of the regression of equation (3.1.a). I can conclude that I find a positive relation between excess returns and expected risk for big capitalized stocks overall.

Turning to other explanatory variables, I find that excess returns are negatively related to unexpected volatility. Twenty-seven out of 30 stocks show negative significance on the unexpected volatility variable. The rest of the 3 stocks show no significance. That there is a negative relation between excess returns and unpredicted volatility is indirect evidence that supports the risk-return trade-off.²⁹

I see from Table 3.3 that the intraday skewness coefficient variable has a very high t-value.

This is the case for all 30 stocks. This indicates that the excess return is strongly positively related to the intraday skewness coefficient. ³⁰ Comparing the adjusted R² of three regression models, I find that the intraday skewness coefficient variable explains a bigger portion of the variation in excess returns than do other variables. The average adjusted R² of the regression model (3.1.b) among these 30 stocks is 8.53% while the adjusted R² of the model without the intraday skewness variable is 2.54%.

The adjusted R² of the model that contains only the expected risk variable is mere 0.15%, and the maximum among 30 stocks is 0.94%. The small adjusted R² of the regression model tells one

29 When I do an ARMA process for the standard deviation, I find that the standard deviation generally follows a positive coefficient moving average process. When the unexpected standard deviation increases, all predicted standard deviations will be revised upward for all future time periods. If the hypothesis that the risk premium is positively related to the predicted standard deviation is true, then the discount rate for future cash flows increases. If the cash flows are unaffected, the current stock price will be reduced. Thus, I observe a negative relation between excess returns and the unexpected standard deviation (French et al., 1987).

30 I also regress the daily stock return on the daily market return and the stock’s intraday skewness coefficient. The intraday skewness coefficient is still very significant for all 30 stocks. That indicates that the intraday skewness coefficient contains shocks to the individual stock. (Results are not reported here.)

thing: the expected risk really explains very little about excess returns, even though almost two third of the sample stocks show a significant coefficient in the regression equation. This result strengthens the claim that investors consider some other risk measure to be more important than the standard deviation of portfolio returns (Baillie and DeGennaro, 1990). Comparing with the explanatory power of the skewness variable, I conclude that the intraday skewness coefficient works as a good control variable. It reflects overall impact from individual stocks’ news shocks.

Since it explains the variation in excess returns incrementally, the model has smaller standard errors of the regression. Adding such a variable in the regression is where my model is different from the models used in French et al. (1987), Campbell (1987), and Ghysels et al. (2005).

There might be a missing variable problem in equation (3.1.b). Nonetheless, since I already included the ex ante part risk and innovation part risk, I can expect that any missing variables will mainly affect the coefficients of the innovation part risk variables and the coefficient of the expected volatility variable will be affected only marginally. Moreover, any missing variables will affect the intraday skewness coefficient according to the efficient market hypothesis (EMH).

Including the intraday skewness coefficient in the regression partially solves the missing variable problem.

The AR(1) variable is significant for 13 out of the 30 stocks. The signs, however, are mixed.

The residual checks reveal that problems of autocorrelation still exist in the residual series from adopting only one lag dependent variable. However, accurate identification of the ARMA process for all 30 stocks makes the model more complicated and distracts from the focus on the risk-return relation. The existence of the autocorrelation can be understood as the result of a mis-specification problem (Greene, 2003). As mentioned before, the mis-mis-specification problem

should affect the expected volatility variable only marginally, since I include the unexpected volatility variable and the intraday skewness coefficient variable in the regression.

Since I have 30 stocks in the sample, panel data analysis is feasible. However, the Hausman test for random effects and the F Test for fixed effects point out that a pooled regression is best.

On the bottom of Table 3.3, I report the pooled regression results, which indicate that there is a significant positive relation between excess returns and expected volatility. The adjusted R² is 6.54%. Still, the skewness variable explains the most part. The adjusted R² of the model that doesn’t include the skewness variable is 1.33%, while the expected standard deviation alone only contributes 0.08%.

Now, I can conclude that there is evidence to support a positive relationship between excess returns and expected risk for big capitalized stocks at the daily level.³¹ Stock PG shows a negative relation in the whole sample but it becomes insignificant after controlling unexpected risk. The pooled regression also supports a positive relation. The unanswered question is why some stocks support an argument that excess returns statistically have nothing to do with expected risk.

3.4 Quantile Regression on the Relation Between Returns and Risk