Estimated GJR-GARCH results

For each of the three stock indices, we estimate a base model that excludes sentiment as an explanatory variable in the mean and conditional volatility equations. We estimate GJR-GARCH in the period from August 22, 1996 to December 31, 2007, July 22, 1987 to December 31, 2007, and February 1, 1971 to December 31, 2006 for daily, weekly, and monthly data, respectively. The period of our daily data is approximately ten years and the period of our weekly data which is approximately twenty years is relatively longer. The period of our monthly data which is the longest across the three periods is approximately thirty-five years. The estimated coefficients of the base models for the three stock indices for daily, weekly, and monthly data are reported in Table 5.

< Table 5 is inserted about here >

First, in the base model, the time-invariant portion of excess returns is not significant; and the time-varying portion of excess returns in the base model is not significant with conditional volatility too. The results are not consistent with previous findings of a negative price for time-varying risk (Glosten, Jagannathan, and Runkle, 1993; De Santis and Gerard, 1997; Lee, Jiang, and Indro, 2002).

Second, across the three stock indices, not all of the estimated GARCH coefficients in the base models are significant. We confirm that surprises have an asymmetric effect on conditional volatility and this result is consistent with our forecast, because most of coefficients of the asymmetric effect which is β₂ are significant and positive except the model of monthly NASDAQ return. Negative shocks cause higher upward revisions in volatility.

In addition, as Glosten, Jagannathan, and Runkle (1993) find, volatility is generally greater when inflation rates are projected to be higher in the future. Except the NASDAQ, the

coefficients for the risk-free rate are positive and significant for both the DJIA and S&P500.

This result is the same with Lee, Jiang, and Indro (2002). Despite the coefficients for the risk-free rate of the daily and weekly NASDAQ are not significant and significantly negative, respectively, it is significant and positive for the monthly NASDAQ.

In our base models, the seasonal effects which are January and October effect are not significant that only the dummy variables of October effect of monthly DJIA and S&P500 are significance at 10% level and negative. The season effect is very weak in our data period. The coefficients of the dummy variables for dot-com bubble in the mean equation are negative and are significant in the daily models of S&P500 and NASDAQ and weekly models of DJIA and NASDAQ, especially in NASDAQ (significant at 1% level). But all of dummy variables for dot-com bubble in the monthly base models are not significant. The impact from the crash of dot-com bubble to NASDAQ is the largest and the most obvious, because NASDAQ index is composed of many small high tech companies where many of their investors are small investor.

To the base model in Table 5, we then add measures of noise trader risk associated with shifts in sentiment in the mean and volatility equations. The percentage changes in sentiment for daily, weekly, and, monthly data are utilized in Table 6, 7, and 8, respectively. The major findings are summarized below.

< Table 6 is inserted about here >

< Table 7 is inserted about here >

< Table 8 is inserted about here >

As shown in Table 6, the time-invariant portion of excess returns and the coefficient of GARCH in mean equation are not significant. January and October effect is the same as in its insignificance. The dummy variable for dot-com bubble is not only significant and great, it is

negative only in NASDAQ. The dot-com bubble variable is not significant, except S&P500 of PCO.

The three sentiment indices which are ARMS, the OEX put-call open interest ratio (PCO), and the OEX put-call trading volume ratio (PCV) in our daily data are bearish indicators. In daily model, only the coefficient of lagged shifts in PCO in mean equation for excess returns of all stock indices are very significant (significance at 1% level) and negative, but PCV and ARMS are not. We find that PCO can be used to forecast the excess return of a particular stock index. When the markets are more bearish, PCO goes up. This will affect stock market in that the excess return of stock market will go down in the future.

In the variance equation of the daily model, after adding the shifts of sentiment, the phenomenon that volatility is greater when inflation rates are projected to be higher in the future is no longer clear. In addition, it is the same with base model that surprises have an asymmetric effect on conditional volatility. This is the leverage effect that is different for negative than for positive shocks and the magnitude of the change in market volatility is greater for bad news than for good news.

For the models of ARMS, we find bullish shifts in sentiment in the current period result in statistically significant downward revisions in the volatility of future returns, and the coefficient of bearish shifts in ARMS in variance equation are very small. For PCO and PCV, bearish shifts in sentiment in the current period lead to upward revisions in volatility of future returns. Bullish shifts in PCO and PCV in the current period also lead to upward revisions, but it is less significance (only NASDAQ p-value is less than 5%). We find the three daily sentiment indices are good indicators to forecast the volatility of excess return of stock index.

As reported in Table 7, we use two direct sentiment indices AAII and II as the bullish sentiment indicator in our weekly models. The season effects are very weak. The dummy

variable of dot-com bubble for each indices and sentiment are significant and great negative other than DJIA for AAII.

First for AAII, in mean equation, the time-invariant portion and the time-invariant portion of excess returns of DJIA and NASDAQ are not significant. But the time-invariant portion of excess return of S&P500 is significant and negative (-0.908) and its coefficient of GARCH in mean equation is significant and positive (0.159). We find AAII in the current period can be used to forecast the excess return of S&P500 in the future, because the coefficient of the shift of AAII is significant at 5% level and positive (0.771). When AAII rises, it will affect stock market in that the excess return of stock market will go up in the future. In variance equation, we don’t find the surprises having an asymmetric effect on conditional volatility and it is surprising that the coefficients of the risk-free rate are negative. We find that bullish shifts in AAII sentiment index in the current period result in statistically significant downward revisions in the volatility of future returns of DJIA and S&P500, and the coefficient of bearish shifts in AAII in the variance equation are not significant.

Second in the models of II, in the mean equation, each time-invariant portions and coefficient of GARCH-in-mean are not significant. In the mean equation for DJIA and S&P500, a shift in sentiment has a significant positive impact on excess return and the coefficients of DJIA and S&P500 are -1.576 and -1.559, respectively. It means the II is a good contrary indicator for the excess returns of DJIA and S&P500, especially for S&P500.

In the variance equation, there is the leverage effect that is different for negative than for positive shocks, but II in the current period can’t affect the volatility in the future.

As shown in Table 8, we use IPON and IPORET as bullish sentiment index in our monthly data. In Table 8, we don’t find IPORET in the current period can affect the excess return in the future and each of coefficients of October effect is significant. Although IPORET in the current period can affect the volatility of the excess return of DJIA in the future, the

coefficient is very small. So we can say with confidence that IPORET has poor forecasting power.

In addition, we find that when IPON rises, the excess returns of S&P500 and NASDAQ will go up in the future. However, IPON also has forecast power for the volatility of NASDAQ, bullish shifts in sentiment in the current period result in statistically significant downward revisions in the volatility of future returns.

Overall, we find that investor sentiment is an important factor in explaining equity excess returns and changes in conditional volatility. PCO can be used to forecast the excess returns of all stock indices. As PCO goes up, the excess returns will go down. AAII can be used to forecast the return S&P500 and II can be used to DJIA and S&P500. There is positive correlation between excess returns and shifts of AAII, but there is negative correlation between excess returns and II. When the bullish percentage of II rises up, the excess return will go down. We confirm II is a contrary indicator for excess returns of large capitalization stocks, because II represents the newsletters’ sentiment and they are medium investors. Their opinion will affect other investor especially with the small investors and AAII represents the small investors’ sentiment. Although there is great positive correlation (0.513) between AAII and II in Table 1, their results are very different. We consider that AAII might be close to noise traders’ sentiment. We also find that there is a positive relationship between change in IPON in the current period and excess return of S&P500 or NASDAQ.

Our results show that shifts in sentiment have an asymmetric impact on conditional volatility. As the magnitude of shifts in bullish sentiment increases, there is a downward (upward) revision in the volatility of future returns. First, PCO and PCV can be used to estimate the effect of change of bearish sentiment to excess return of volatility, and ARMS can be used to estimate the effect of change of bullish sentiment. Second, when change of bullish sentiment percentage of AAII goes up in the current period, the volatility of excess

return of DJIA and S&P500 which are large stock will go down in the future. But the effect of the change of bearish sentiment within AAII is not as clear. IPON also has the same effect, but there is some difference in that it is for volatility of NASDAQ.

However, only AAII used as a sentiment indicator to estimate the excess return of S&P500 fits in with our empirical hypothesis which is the noise trader model of De Long et al (1990a), so we use this result to explain the economical reasoning. In the mean equation, a shift in sentiment has a statistically significant positive impact in excess return. The hold-more effect tends to dominate the price-pressure effect and leads to an increase in excess returns when noise traders are more bullish in their sentiments. In particular, when sentiment becomes more bullish, optimism induces noise traders to hold more of the risky assets than fundamentals would indicate, this secures the compensation for bearing the increase in risk associated with sentiment. Nevertheless, the higher risk premium due to increased demand is partially offset by the unfavorable price at which noise traders transact.

If sentiment becomes more bearish, there is a reduction in excess returns. Noise traders choose to hold less of the risky assets when they are more pessimistic, and consequently, are unable to capture the risk premium related to sentiment. Moreover, there is a negative price impact caused by sentiment-induced sale of securities.

In the volatility equation, we also find that bullish shifts in sentiment in the current period result in significant downward revisions in the volatility of future excess returns. And bearish shifts in sentiment in the current period lead to upward revisions in volatility of future excess returns. As the magnitude of shifts in bullish (bearish) sentiment increases, there is a downward (upward) revision in the volatility of future excess returns resulting in lower (higher) future excess returns.

Because of their tendency to trade together, noise traders usually have poor market timing

where they end up buying high and selling low. The Friedman effect implies that asset prices tend to be negatively affected when noise traders’ misperceptions are more severe. But the extent that asset prices are adversely influenced by the Friedman effect depends on the space which noise trading creates. The lower excess return associated with volatility revisions due to bullish sentiment shifts indicates that the positive effect on price of the space created by sentiment-induced noise trading is not large enough to offset the negative effect on price of poor market timing. In contrast, there is a higher excess return associated with volatility revisions due to bearish sentiment shifts. In this case, the positive effect on price associated with noise trader created space is sufficient to offset the negative effect on price associated with poorly timed sales of securities triggered by bearish shifts in sentiment.