4 Empirical Results
4.1 Data
Here we use the daily risk premium of Taiwan Stock Exchange Capitalization Weighted
Stock Index (TAIEX) compiled by Taiwan Stock Exchange Co., Ltd. (TWSE) in our
empirical test. The period is from January 2006 to December 2010, including 1246 daily
observations. Entire samples are all collected from TEJ (Taiwan Economic Journal). TEJ was
founded in April 1990 to provide quality, in-depth and extensive historical financial data and
information in the major financial markets in Asia. There is a definition about equity risk
premium in this paper: we use the difference, return rates of TSEC weighted index minus
two-year Taiwan treasury-bill rates, as a proxy to be explored, including various frequencies
such as daily, weekly and monthly data form. In the meantime, statistical software E-Views is
applied to analyze and compute some relevant data.
Table 1 shows the descriptive statistics about the sampled equity risk premium. We find
the mean for ERP is negative. That means on average there is no premium investors acquire in
the stock market during this period. Conversely, they even get some losses. Variances are used
in this table because of the relation between risk and return. Specifically, we focus on
connections of average return and conditional variance, not standard deviations
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Table 1: Descriptive Statistics
Descriptive statistics of ERP with different sampling frequencies from 2006 to 2010, included are mean, variance, skewness, kurtosis. The number of samples for each frequencies is also reported in the table.
Mean (%) Variance Skewness Kurtosis
Monthly -0.49 0.0056 -0.3167 3.2388
Weekly -1.09 0.0011 -0.5692 4.0098
daily -1.22 0.0003 -0.3370 4.2252
4.2 MIDAS Estimation
This subsection is integrated from two parts. As we mentioned before, we decide to use
Exponential weight specification and apply the 30 days lags length. For first part, we apply
the suggestion under setting 1 = −0.01 and 2 = 0 (see Ghysels, Snata-Clara, and
Valkanov, 2006b) as a benchmark. Then we compare it with other two cases: 1 = 0 and 2 = 0 (shown as equal weight) , 1 = −1 and 2 = 0 (considered as reasonable pattern).
We plot the weights that the MIDAS estimator places of the first 22 lagged daily squared risk
premiums corresponding to one month in Figure 2. The top panel is case1, the middle panel is
case2 and the bottom panel displays case3.
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Figure 2: MIDAS Weight on Variables Predictors
Weight
Lag (in days)
Weight
Lag (in days)
Weight
Lag (in days)
The figure plots the estimated weights of conditional variance on the lagged daily squared risk premiums corresponding to one month. Three panels are representative of three different declining weight shapes respectively. We then use the weights to estimate related parameters by MIDAS approach.
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Now we jointly estimate the parameters and by nonlinear least squares (NLS). In
Table 2, we show three different weight polynomials and two various types of the volatility
predictors. We also explore the estimation results between MIDAS approach and rolling
window approach.
Table 2: MIDAS Estimation of Equity Risk Premiums
The table shows estimates of ERP with MIDAS estimation using TAIEX form Jan 2006 to Dec 2010. Exponential lag is used and lagged daily (weekly) squared (absolute) risk premiums are respectively used in the construction of conditional variance estimator. The estimated equations are as follows:
ERP = + ∑ ( ; 1, 2)ERP + ε / ERP = + ∑ ( ; 1, 2)|ERP | + ε ,
where ( ; 1, 2) = ∑ ( )
( )
The coefficients and corresponding p-value are shown in the middle columns and the right column is shown as corresponding R-squared value.
MIDAS Estimation
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*indicates the statistics reach 0.05 of the significant level
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This subsection presents the result of MIDAS approach based on the Merton’s ICAPM
model. We find the coefficients and are almost statistically significant. First, we start
from MIDAS estimation. In daily data, the estimated risk aversion coefficient is ranging
between -0.29 and -6.73. There is not a very small gap between the both sides. The risk
aversion absolute seems greater in daily data than in weekly data, and that means the degree
of risk aversion which can be tolerated by investors. In addition, we see that there are just
little differences between the weight 1 and weight 2 polynomials and R-square values
respectively. We also find such t-statistics of the corresponding estimated coefficient are
significant by judging from the p-values. We can conclude that volatility predictors of weight
1 and weight 2 are obviously better than weight 3. Actually, these results with polynomial
weight 3 are not explainable enough.
In weekly data, the estimated risk aversion coefficient is of -0.03 to -3.04 and the
difference is much closer. However, the result of weight 3 case becomes better because its
R-squares value is getting obviously higher, even over 20% extra. While mentioning to
R-square value, it is reports to quantify the explanatory power of the variance estimators in
predictive regressions for sampled premiums. To sum up, the estimation of daily risk premium
performs better than weekly risk premium because the significance of coefficients and
variance explanations level performs more outstanding, up to 46%. Moreover, the result of
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squared daily return volatility is also comparable to the result of absolute daily return
volatility. The risk aversion coefficients of weight 1 and 2 are -1.47and -1.46, and the model
explained variation levels are around 46.15% and 46.08% respectively. Basically both are
almost equivalent, but we still prefer to choose the estimation model under 1 = −0.01 and
2 = 0 with daily frequency. These results point to the importance of having a flexible
functional form for the weights on past daily squared returns. Then we use out-sample to
measure forecasting errors in following subsection to make certain whether the estimation is
appropriate.
However, one thing important needs to be noticed. We all have negative magnitude of
risk aversion coefficients in above cases, no matter whether the squared risk premium or the
absolute risk premium is. It clearly points out that the tradeoff relation in our empirical study
is negative. These “negative” results are obviously corresponding to some previous classical
studies. Actually we think the results may depend on what the estimated method for the
conditional variance of returns is used. Campbell (1987) use generalized method moments
(GMM) to verify the relationship between expected stock returns and the conditional variance
of stock returns. The coefficient estimates of GMM for stock suggest that stocks have a higher
expected return when their conditional variance is low. Correspondingly, Nelson (1991) uses
the GARCH method to estimate a model of the risk premium on the CRSP value-weighted
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market index form 1962 to 1987. The outcome is shown as a statistically significantly
negative relation between both. In recent studies, Glosten et al. (1993) use the CRSP data and
find support of a negative relation between conditional expected monthly return and
conditional variance of monthly return, using the modified GARCH-M model. More related
interpretation we leave in Section 5.
4.3 GARCH-in-Mean Estimation
Before applying GARCH-M estimation, time series data should be processed by a kind of
unit tests and we find out the result shows significant rejections of null hypothesis which
mean the risk premium data is not autocorrelated. Then we directly use the data under
GARCH-M estimation.
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Table 3: GARCH-M Estimation of Equity Risk Premiums
The table shows estimates of ERP with GARCH-M estimation using TAIEX form Jan 2006 to Dec 2010. The estimated equations are as follows:
= + + , where = + + .
The coefficients and corresponding p-value are shown in the middle columns and the right column is shown as corresponding R-squared value.
GARCH-M Estimation
*indicates the statistics reach 0.05 of the significant level
Table 3 shows the empirical results of GARCH(1,1)-M estimation of risk premium data
with different frequencies. The estimated coefficients are obtained by a sort of maximum
likelihood estimations, and we assuming error term is normally distributed. Compared
with other three different frequencies in GARCH-M estimation, the R-squared statistics with
daily frequency data is much better. In addition, the GARCH-M model with daily frequency
shows the statistical significance of mean equation and variance equation, excluding intercept
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term. The risk aversion coefficient γ is around of -0.77 for the mean equation and the p-value
of the corresponding estimated coefficient looks very significantly. Here we notice that the
risk aversion is still negative, consistent with the MIDAS estimation as we mentioned above.
Besides, under the GARCH-M approach the R-squared statistic is around of 22.47%,
lower than in the MIDAS approach which is shown as 46.15%. Take this for example, it is
because the MIDAS approach estimates two parameters rather than three as GARCH-M
model does and employs more observations to forecast market volatility under variance
equation. In generally speaking, traditional GARCH-M estimation outcome in explainable
range is not superior to the MIDAS approach.
4.4 Rolling Window Estimation
We discuss about the rolling window estimation with daily and weekly frequency data.
The results of rolling window approach are shown in Table 4. The estimate of is still
negative (around of -1.4), and the coefficient is very significant because the p-value is far
lower than the significant level. It is shown consistently under this situation with the MIDAS
estimation. Besides, R-square value is 46.08% of rolling window estimation, and almost as
same as the MIDAS estimation, 46.15%. They are so close but obviously we still recognize
that the daily frequency specification is better as a result of the higher R-squared value. The
rolling window approach can be thought as a robust check of the MIDAS estimation because
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it is a simple estimator of conditional variance with no parameters to be estimated. Besides its
simplicity, the use of daily data has some advantages: first, as with MIDAS approach, it can
increase the precision of the variance estimator. Second, the stock market variance is quite
persistent (see Officer, 1973; Schwert, 1989), so the realized variance on a given month ought
to be a good forecast of next month’s variance.
Table 4: Rolling Window Estimation of Equity Risk Premiums
The table shows estimates of ERP with rolling window estimation using TAIEX form Jan 2006 to Dec 2010. The estimated equations are as follows:
= + + , where = ∑
The coefficients and corresponding p-value are shown in the middle columns and corresponding R-squared values are shown in the right column.
Rolling Window Estimation
*indicates the statistics reach 0.05 of the significant level