This paper take a new look at Merton’s ICAPM, focus on the trade-off between
conditional variance and conditional mean of the stock market return. We show the existence
of a time-varying risk premium in Taiwan stock market by introducing mixed data sampling
model estimation. Our results are more conclusive because MIDAS estimation confirms the
weighted polynomial with different sampling frequencies performs pretty good. Not the same
as with previous studies, added power obtained from the new MIDAS estimator actually
makes risk premium estimation more flexible.
According to the previous empirical results, conclusions of this study are as follows:
1) The tradeoff between risk and return has long been an important topic in asset valuation
research. Most of this research examine the tradeoff among different securities within a
given time period. We find the common evidence of a negative relation between risk and
return in Taiwan stock market within these years. In fact, we think that what types of
model are used to assume conditional variance of returns as a research framework is
highly relevant to the issue of risk-return relation regardless of positive relation or
negative relation. However, sometimes the models we used cannot completely capture
volatility persistence or reflect positive and negative shocks.
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Black (1976) and Christic (1982) propose financial leverage effect for an
examination of the risk-return tradeoff with asymmetric variance effect. After that,
Campbell and Hentschel (1992) propose volatility feedback effect to explain the same
situations. Most empirical studies show that negative relation between risk and return
might be attributed to asymmetric effects in the conditional variance. Moreover, the type
of relevance is mostly confined by model assumptions which indeed affect these empirical
results.
In addition to asymmetric effect, many different approaches for setting risk as a
proxy variable could also affect the empirical results, especially for risk-return tradeoff.
Moreover, the financial tsunami brings about some potential phenomenon such as
increasing difficulty in predicting expected returns and conditional variances. Meanwhile,
it also indeed related to the sampling period we selected. Although we acquire negative
relation about risk-return tradeoff, which is opposed to some previous research, we still
show some advantages in MIDAS estimation as follows.
2) Comparing with the rolling window and GARCH-M estimation, we conclude that MIDAS
estimation is better and more suitable. As the model explained variation power, 46.15% of
MIDAS is larger than 46.07% of rolling window, also greater than 22.47% of GARCH-M.
The rolling window approach can be thought as a robust check of the MIDAS estimation
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because it is a simple estimator of conditional variance with no parameters to be estimated.
Except for explained variation power, these estimation coefficients are very statistically
significant because of p-values are below significance level. That means the MIDAS
estimation is indeed a well-performed model.
3) By using MIDAS approach, this estimator is behalf of a weight average of past daily
squared returns with flexible functions. MIDAS estimator is not only the superior
estimator because it can be appropriately explained by past risk premiums, but also a
better forecaster in the stock market than rolling window estimators. Last but not the least,
after experiencing investigations of the MIDAS specifications for various volatility
predictors, we obtain that higher frequency predictor such as daily squared return provides
greater results.
The empirical results are statistically significant, at the same time, the forecasting
performance of MIDAS is also reasonable. We still have interests to use MIDAS to process
how these different and jointly estimated weights of volatility predictor work. Next, we
explore the parameters 1 and 2 more deeply. Our purpose is to directly and jointly
estimate the parameters 1 , 2 , μ , γ of Eq. (6) and (7) by nonlinear least error approach.
Owing to the smaller the conditional variance is, the smaller the estimated forecasting error is.
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Therefore we apply MAE and RMSE forecasting error approaches, and then make certain that
the values of MAE and RMSE are both minimum. Our estimated algorithm is as follows,
which is based on rules of minimum error. Take MAE for example, our main purpose is to
minimize the value of ∑ | | , which is under restrains of = ̂ +
( ) ,
and = ∑ ( )
∑ ( ) .
After jointly nonlinear least error calculation with the same period, our results are shown
in Table 7.
Table 7: Errors of Jointly Nonlinear Least Calculation
Forecasting Estimation Minimum Error (%)
0.0096
(RMSE) -0.3163 -0.0503 1.2346 -18.3336 0.0020
0.0564
(MAE) 0.0105 -0.0002 1.2346 -18.3336 ~0.000
*indicates the statistics reach 0.05 of the significant level All forecasting error unit is of percentage (%)
We find that the outcome is not good enough while comparing with previous results
under setting the specific parameter . The both coefficients μ and γ are not statistically
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significant because the p-values are not below the significance level 0.05. Moreover, the
explanatory power is also abnormally low so that we cannot verify this case to be well. Here
is a trivial implication that the suggestion of setting and as some specific values (see
Ghysels et al., 2006) improves the outcomes of MIDAS estimation better. As for the reasons
why the effects of estimated weight polynomial parameter such as are not relatively
outstanding, we leave these issues for future research.
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