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This paper studies the intertemporal relation between the conditional mean and the conditional variance of the aggregate stock market return. We introduce a new estimator that forecasts monthly variance with past daily squared returns, the mixed data sampling (or MIDAS) approach. Using MIDAS, we find a significantly positive relation between risk and return in the stock market. This finding is robust in subsamples, to asymmetric specifications of the variance process and to controlling for variables associated with the business cycle. We compare the MIDAS results with tests of the intertemporal capital asset pricing model based on alternative conditional variance specifications and explain the conflicting results in the literature. Finally, we offer new insights about the dynamics of conditional variance.

The Merton (1973) intertemporal capital asset pricing model (ICAPM) suggests that the conditional expected excess return on the stock market should vary positively with the market's conditional variance: equation(1) View the MathML sourceEt[Rt+1]=μ+γVart[Rt+1], Turn MathJax on where γγ is the coefficient of relative risk aversion of the representative agent and, according to the model, μμ should be equal to zero. The expectation and the variance of the market excess return are conditional on the information available at the beginning of the return period, time t. This risk-return trade-off is so fundamental in financial economics that it could be described as the “first fundamental law of finance.” 1 Unfortunately, the trade-off has been hard to find in the data. Previous estimates of the relation between risk and return often have been insignificant and sometimes even negative. Baillie and DeGennaro (1990), French et al. (1987), and Campbell and Hentschel (1992) do find a positive albeit mostly insignificant relation between the conditional variance and the conditional expected return. In contrast, Campbell (1987) and Nelson (1991) find a significantly negative relation. Glosten et al. (1993), Harvey (2001), and Turner et al. (1989) find both a positive and a negative relation depending on the method used.2 The main difficulty in testing the ICAPM relation is that the conditional variance of the market is not observable and must be filtered from past returns.3 The conflicting findings of the above studies are mostly the result of differences in the approach to modeling the conditional variance. In this paper, we take a new look at the risk-return trade-off by introducing a new estimator of the conditional variance. Our mixed data sampling, or MIDAS, estimator forecasts the monthly variance with a weighted average of lagged daily squared returns. We use a flexible functional form to parameterize the weight given to each lagged daily squared return and show that a parsimonious weighting scheme with only two parameters works well. We estimate the coefficients of the conditional variance process jointly with μμ and γγ from the expected return Eq. (1) with quasi-maximum likelihood. Using monthly and daily market return data from 1928 to 2000 and with MIDAS as a model of the conditional variance, we find a positive and statistically significant relation between risk and return. The estimate of γγ is 2.6, which lines up well with economic intuition about a reasonable level of risk aversion. The MIDAS estimator explains about 40% of the variation of realized variance in the subsequent month and its explanatory power compares favorably to that of other models of conditional variance such as the generalized autoregressive conditional heteroskedasticity (GARCH). The estimated weights on the lagged daily squared returns decay slowly, thus capturing the persistence in the conditional variance process. More impressive still is that, in the ICAPM risk-return relation, the MIDAS estimator of conditional variance explains about 2% of the variation of next month's stock market returns (and 5% in the period since 1964). This is substantial given previous results about forecasting the stock market return. For instance, the forecasting power of the dividend yield for the market return does not exceed 1.5%. (see Campbell et al., 1997, and references therein) Finally, the above results are qualitatively similar when the sample is split into two subsamples of approximately equal sizes, 1928–1963 and 1964–2000. To better understand MIDAS and its success in testing the ICAPM risk-return trade-off, we compare our approach to previously used models of conditional variance. French et al. (1987) propose a simple and intuitive rolling window estimator of the monthly variance. They forecast monthly variance by the sum of daily squared returns in the previous month. Their method is similar to ours in that it uses daily returns to forecast monthly variance. However, when French et al. use that method to test the ICAPM, they find an insignificant (and sometimes negative) γγ coefficient. We replicate their results but also find something interesting and new. When the length of the rolling window is increased from one month to three or four months, the magnitude of the estimated γγ increases and the coefficient becomes statistically significant. This result nicely illustrates the point that the window length plays a crucial role in forecasting variances and detecting the trade-off between risk and return. By optimally choosing the weights on lagged squared returns, MIDAS implicitly selects the optimal window size to estimate the variance, and that in turn allows us to find a significant risk-return trade-off. The ICAPM risk-return relation has also been tested using several variations of GARCH-in-mean models. However, the evidence from that literature is inconclusive and sometimes conflicting. Using simple GARCH models, we confirm the finding of French et al. (1987) and Glosten et al. (1993), among others, of a positive but insignificant γγ coefficient in the risk-return trade-off. The absence of statistical significance comes from both GARCH's use of monthly return data in estimating the conditional variance process and the inflexibility of the parameterization. The use of daily data and the flexibility of the MIDAS estimator provide the power needed to find statistical significance in the risk-return trade-off. A comparison of the time series of conditional variance estimated according to MIDAS, GARCH, and rolling windows reveals that while the three estimators are correlated, some differences affect their ability to forecast returns in the ICAPM relation. We find that the MIDAS variance process is more highly correlated with both the GARCH and the rolling windows estimates than these last two are with each other. This suggests that MIDAS combines some of the unique information contained in the other two estimators. We also find that MIDAS is particularly successful at forecasting realized variance both in high and low volatility regimes. These features explain the superior performance of MIDAS in finding a positive and significant risk-return relation. It has long been recognized that volatility tends to react more to negative returns than to positive returns. Nelson (1991) and Engle and Ng (1993) show that GARCH models that incorporate this asymmetry perform better in forecasting the market variance. However, Glosten et al. (1993) show that when such asymmetric GARCH models are used in testing the risk-return trade-off, the γγ coefficient is estimated to be negative (sometimes significantly so). This stands in sharp contrast with the positive and insignificant γγ obtained with symmetric GARCH models and remains a puzzle in empirical finance. To investigate this issue, we extend the MIDAS approach to capture asymmetries in the dynamics of conditional variance by allowing lagged positive and negative daily squared returns to have different weights in the estimator. Contrary to the asymmetric GARCH results, we still find a large positive estimate of γγ that is statistically significant. This discrepancy between the asymmetric MIDAS and asymmetric GARCH tests of the ICAPM turns out to be interesting. We find that what matters for the tests of the risk-return trade-off is not so much the asymmetry in the conditional variance process but its persistence. In this respect, asymmetric GARCH and asymmetric MIDAS models prove to be very different. Consistent with the GARCH literature, negative shocks have a larger immediate impact on the MIDAS conditional variance estimator than do positive shocks. However, we find that the impact of negative returns on variance is only temporary and lasts no more than one month. Positive returns have an extremely persistent impact on the variance process. In other words, while short-term fluctuations in the conditional variance are mostly the result of negative shocks, the persistence of the variance process is primarily driven by positive shocks. This is an intriguing finding about the dynamics of the variance process. Although asymmetric GARCH models allow for a different response of the conditional variance to positive and negative shocks, they constrain the persistence of both types of shocks to be the same. Because the asymmetric GARCH models load heavily on negative shocks and these have little persistence, the estimated conditional variance process shows little to no persistence. The only exception is the two-component GARCH model of Engle and Lee, 1999, who report findings similar to our asymmetric MIDAS model. They obtain persistent estimates of conditional variance while still capturing an asymmetric reaction of the conditional variance to positive and negative shocks. In contrast, by allowing positive and negative shocks to have different persistence, the asymmetric MIDAS model still obtains high persistence for the overall conditional variance process. Since only persistent variables can capture variation in expected returns, the difference in persistence between the asymmetric MIDAS and the asymmetric GARCH conditional variances explains their success and lack thereof in finding a risk-return trade-off. Campbell (1987) and Scruggs (1998) point out that the difficulty in measuring a positive risk-return relation could stem from misspecification of Eq. (1). Following Merton (1973), they argue that if changes in the investment opportunity set are captured by state variables in addition to the conditional variance itself, then those variables must be included in the equation of expected returns. In parallel, an extensive literature on the predictability of the stock market finds that variables that capture business cycle fluctuations are also good forecasters of market returns (see Campbell, 1991; Campbell and Shiller, 1988; Fama, 1990; Fama and French, 1988 and Fama and French, 1989; Ferson and Harvey, 1991; Keim and Stambaugh, 1986, among many others). We include business cycle variables together with both the symmetric and asymmetric MIDAS estimators of conditional variance in the ICAPM equation and find that the trade-off between risk and return is virtually unchanged. The explanatory power of the conditional variance for expected returns is orthogonal to the other predictive variables. We conclude that the ICAPM is alive and well. The rest of the paper is structured as follows. Section 2 explains the MIDAS model and details the main results. Section 3 offers a comparison of MIDAS with rolling window and GARCH models of conditional variance. In Section 4, we discuss the asymmetric MIDAS model and use it to test the ICAPM. In Section 5, we include several often-used predictive variables as controls in the risk-return relation. Section 6 concludes.

This paper takes a new look at Merton's ICAPM, focusing on the trade-off between conditional variance and conditional mean of the stock market return. In support of the ICAPM, we find a positive and significant relation between risk and return. This relation is robust in subsamples, does not change when the conditional variance is allowed to react asymmetrically to positive and negative returns, and is not affected by the inclusion of other predictive variables. Our results are more conclusive than those from previous studies because of the added power obtained from the new MIDAS estimator of conditional variance. This estimator is a weighted average of past daily squared returns, and the weights are parameterized with a flexible functional form. We find that the MIDAS estimator is a better forecaster of the stock market variance than rolling window or GARCH estimators, which is the reason that our tests can robustly find the ICAPM's risk-return trade-off. We obtain new results about the asymmetric reaction of volatility to positive and negative return shocks. We find that, compared with negative shocks, positive shocks have a bigger impact overall on the conditional mean of returns, are slower to be incorporated into the conditional variance, and are much more persistent and account for the persistent nature of the conditional variance process. Surprisingly, negative shocks have a large initial, but temporary, effect on the variance of returns. The MIDAS estimator offers a powerful and flexible way of estimating economic models by taking advantage of data sampled at various frequencies. While the advantages of the MIDAS approach have been shown in the estimation of the ICAPM and conditional volatility, the method itself is general in nature and can be used to tackle several other important questions.