Stock returns, asymmetric volatility, risk aversion, and business cycle: some new evidence.

I. INTRODUCTION

Most asset pricing models, starting with Intertemporal Capital Asset Pricing Model of Merton (1973), suggest a positive relation between risk and return for the aggregate stock market. There is an extensive empirical literature that has tried to establish the existence of such a trade-off between risk and return for stock market indexes. Unfortunately, the results have been inconclusive. Often the relation between risk and return has been found insignificant and, sometimes, even negative. Recently, it has been debated whether risk aversion is state dependent and whether it is procyclical or countercyclical.

In this article, we explore the role of the business cycle in the relation between risk and return for the aggregate stock market. Specifically, by employing asymmetric generalized autoregressive conditional heteroskedasticity in mean models (AGARCH-M), Markov switching models, and a simple theoretical equilibrium framework, we explore how these three related issues--excess stock returns, volatility, and risk aversion--are affected by business cycles. As noted by Fama (1990), of many possible forces that drive the stock market, real economic activities, represented by business cycles, could be an important one since they are highly related to major stock pricing factors; market volatility; and the psychology of investors, particularly investors' attitude toward risk.

An interesting observation about excess stock return is that it is time varying over business cycles and significantly higher in boom periods. (1) According to our estimates for the sample period 1926-2001 presented in Table 1, the mean excess stock return is 0.8963%/mo in boom periods and -0.3247%/mo in recession periods. The excess stock return tends to be very high in the boom period because stock returns remain relatively high with increasing dividend payments. Another interesting observation about the stock market is that the volatility of excess stock return is also time varying over business cycles and significantly lower in boom periods. Our estimates of standard deviation of monthly stock excess return for the sample period 1926-2001 are 4.6% in the boom period and 8.2% in the recession period. These estimates imply that the excess return-risk relation is time varying over the business cycle. Since the coefficient of variance or standard deviation in the conditional mean equation is usually interpreted as being closely related to the coefficient of relative risk aversion as argued by Merton (1980), our finding suggests a potential time-varying risk aversion. (2)

There is a general agreement that investors, within a given time period, require a larger expected return from a security that is riskier. However, there seems to be no relationship between risk and return over time. As such, whether or not investors require a larger risk premium, for investing in a security during periods when the security is more risky relatively remains inconclusive. At first glance, it may appear that rational, risk-averse investors would require a relatively larger risk premium during times when the payoff from the security is more risky. A larger risk premium may not be required, however, because times that are relatively more risky could coincide with times in which investors are better able to bear particular types of risk. Further, a larger risk premium may not be required because investors may want to save relatively more during periods when the future is more risky. (3) Hence, a positive as well as a negative sign for the correlation between the conditional mean and the conditional variance of the excess return on stocks would be consistent with existing theories. Since there are conflicting predictions about the intertemporal trade-off between risk and return, it is important to empirically characterize the nature of this relation.

According to Merton (1980), the coefficient of variance or the standard deviation in the conditional mean equation, which is called volatility feedback, is usually interpreted as the coefficient of relative risk aversion. If so, the changes in excess return and volatility of these magnitudes over business cycles may have important implications for investors' risk aversion. As such, time-varying volatility feedback (or investors' attitudes toward the risk) over business cycles emerges as an appealing idea. The "volatility feedback" effect has been studied by several financial economists including Pindyck (1984); French, Schwert, and Stambaugh (1987); and Campbell and Hentschel (1992), but they have not explored the role of business cycles in the relation explicitly.

In estimating the relation between excess return and volatility, we use an asymmetric generalized autoregressive conditional hetero-skedasticity (AGARCH) model. The model is analytically tractable and captures the asymmetric volatility movement. (4) Further, in an attempt to avoid using the hindsight National Bureau of Economic Research (NBER) information about business cycles, we estimate a regime switching model, where a Markov switching model is estimated simultaneously with a generalized autoregressive conditional heteroskedasticity (GARCH) model. To investigate and confirm potential state-dependent risk aversion, we employ a simple equilibrium asset pricing model with its calibration.

For our sample period 1926-2001, we fail to find a significant relation between risk and excess return in the simple GARCH models without allowing for the business cycle effect. However, once we allow for a business cycle factor (boom and recession), the risk premium coefficient becomes significantly positive for boom periods, whereas it remains insignificantly negative for recession periods. Since the coefficient of volatility in the excess return equation is usually characterized as measuring the time-varying risk aversion parameter as in Merton (1980), this suggests that our finding is in favor of increased risk aversion in the boom period (i.e., procyclical risk aversion).

Using a simple equilibrium asset pricing model with its calibration, we confirm that risk aversion is state dependent and procyclical. We also find that asymmetric volatility movement is weakened during boom periods. Our finding of procyclical risk aversion helps us understand not only the observed larger risk premium for a given risk in the boom periods but also the observed weakened asymmetric volatility during the boom periods.

The remainder of the article is organized as follows. In Section II, we briefly review related literature. In Section III, we describe our data and classification of business cycles. In Section IV, we introduce empirical models of the relation between excess return and volatility--generalized autoregressive conditional heteroskedasticity in mean (GARCH-M) model and Markov switching model--and discuss estimation results. In Section V, we further discuss the implications of the estimation results regarding state-dependent risk aversion over business cycles. In Section VI, we provide several robustness checks. We conclude in Section VII.

II. RELATED LITERATURE

The intertemporal relation between risk and return has been examined extensively; however, empirical evidence is mixed. For example, French, Schwert, and Stambaugh (1987) and Campbell and Hentschel (1992) found that the data are consistent with a positive relation between conditional expected excess stock return and conditional variance, whereas Fama and Schwert (1977); Campbell (1987); Breen, Glosten, and Jagannathan (1989); Turner, Startz, and Nelson (1989); Pagan and Hong (1991); Nelson (1991); Glosten, Jagannathan, and Runkle (1993); and Whitelaw (2000) found a negative relation. Harvey (1989) provided empirical evidence suggesting that there may be some time variation in the relation between risk and return.

Regarding the effect of real economic activity (or business cycle) on market volatility, Schwert (1990) and Fama (1990) argued that future production growth rates explain a large fraction of the variation in stock returns over 1889-1988. McQueen and Roley (1993) found that after allowing for different stages of the business cycle, a stronger relationship between stock prices and economic news becomes evident. They found that when the economy is strong, the stock market responds negatively to news about higher real economic activity. Using a two-variable Markov chain model, Hamilton and Lin (1996) found that economic recessions are a primary factor that drive fluctuations in the volatilities of stock returns.

Whitelaw (1994) found a weak, unconditional, contemporaneous relation between the two moments of stock market returns but a strong, noncontemporaneous relation between the conditional volatility and the conditional expected return. Specifically, he found that volatility appears to lead expected returns over the course of the business cycle. Using a simulated method of moments in a two-state economy, Wu (2001) found that both leverage effect and volatility feedback effect are important to asymmetric volatility movements. Using daily return data and multivariate GARCH models, DeGoeij and Marquering (2002) explained the asymmetric volatility movement of the Treasury bond market with macroeconomic news announcement shocks. By extending these studies and allowing for a business cycle factor (boom and recession) in models of GARCH-M and Markov regime switching, we find that the risk premium coefficient becomes significantly positive for boom periods, whereas it remains insignificantly negative for recession periods.

A number of authors have explored investors' attitudes toward risk by studying the volatility feedback effect. Brown, Harlow, and Tinic (1988) showed that stock price reactions to unfavorable news events tend to be larger than those to favorable events. They attribute this finding to volatility feedback. Poterba and Summers (1986), on the other hand, argued that volatility feedback cannot be important because changes in volatility are too short-lived to have a major effect on stock prices. French, Schwert, and Stambaugh (1987) regressed stock returns on innovations in volatility and found a negative coefficient, which they attribute to volatility feedback. Haugen, Talmor, and Walter (1991) reported a similar result.

An interesting feature of return volatility is its asymmetric movement. Volatility is typically higher after the stock market falls than after it rises, so stock returns are negatively correlated with future volatility. Black (1976), who argued that this could be due to the increase in leverage that occurs when the market value of a firm declines, first discussed this correlation, or predictive asymmetry. However, subsequent studies tended to show that the leverage effect alone is too small to fully account for this phenomenon, according to Christie (1982) and Schwert (1989b).

With regards to the empirical models, early research used moving average measures of volatility, but recent work tends to use the GARCH model of Engle (1982) and Bollerslev (1986). GARCH estimates of stock market variance are typically more persistent than moving average estimates. Attanasio and Wadhwani (1989) and Chou (1988) presented some Monte Carlo evidence that GARCH estimates of persistence in variance are superior to moving average estimates in finite samples.

Since the basic GARCH model assumes a constant mean stock return, it does not fully capture the mechanism underlying volatility feedback. The GARCH-M model of Engle, Lilien, and Robins (1987) allows the conditional mean stock return to depend on the conditional variance or the standard deviation of the returns. French, Schwert, and Stambaugh (1987) estimated a GARCH-M model with conditionally normal innovations and found a significant positive relation between the conditional mean and the variance of stock returns. They argued that it would be desirable to take account of negative skewness from volatility feedback, but they do not pursue this. Chou (1988) also combined an informal discussion of the negative effect of volatility on prices with a formal GARCH-M model, but he does not accommodate this effect.

Recent GARCH models allow returns to be correlated with future volatility and allow for asymmetric effects between positive and negative asset returns, notably the AGARCH model. Using an AGARCH-M, Campbell and Hentschel (1992) argued that volatility feedback can explain asymmetric volatility movement even if the underlying shocks to the market are conditionally normally distributed. Other examples include AGARCH of Engle and Ng (1993); GARCH of Glosten, Jagannathan, and Runkle (1993) (hereafter GJR-GARCH), and "exponential GARCH" of Nelson (1991) model.

Hamilton (1989) extended Goldfeld and Quandt (1973) to the Markov switching model, which is widely adopted by subsequent studies on the unobservable influence of economic events on aggregate output, exchange rate, and unemployment rate. Turner, Startz, and Nelson (1989); Kim, Morley, and Nelson (2004); and Mayfield (2004) explored volatility feedback in the context of Markov regime switching volatility model.

Despite voluminous research on these topics, the role of the business cycle remains unclear in understanding the intertemporal risk-return relation (i.e., volatility feedback), asymmetric volatility movement, and attitude toward risk. By incorporating business cycle factors in AGARCH-M and Markov regime switching models, we find evidence for a state-dependent (i.e., time varying) relation between excess stock returns and conditional volatility. In addition, we find evidence for weakened asymmetric volatility movement during boom periods.

III. DATA AND CLASSIFICATION OF BUSINESS CYCLES

Monthly excess stock returns, [r.sub.t], are computed by subtracting 1-mo U.S. Treasury bill returns from the monthly value weighted with dividend index returns of the NYSE, AMEX, and NASDAQ. The 1-mo U.S. Treasury bill returns are from Ibbotson Associates. For the value-weighted returns of the NYSE, AMEX, and NASDAQ, we use the Center for Research in Security Prices (CRSP) data.

In identifying different states of economic activities such as boom and recession, we use NBER's business cycle classification. According to NBER's business cycle dating procedure, the following factors are considered: employment, personal income less transfer payments, sales of manufacturing and wholesale retail sectors, and industrial production. Previous studies tend to use industrial production as a proxy for the business cycle. NBER's business cycle takes into account more comprehensive information regarding economic activities by considering various economic variables including industrial production. Using NBER's business cycle, we can easily classify economic status into boom and recession periods. More detailed business cycle classifications during the sample period are provided in Appendix A. (5)

In the Markov regime switching model, we use industrial production to identify the two regimes, boom and recession. In Section VI, we extend the model using four variables--employment, personal income less transfer payments, sales of manufacturing and wholesale retail sectors, and industrial production--for identifying the regimes of the economy.

Table 1 presents the summary statistics for the excess stock returns for the sample period January 1926 to December 2001. The average excess stock return is 0.6526%/mo, and the standard deviation 5.5152%/mo. In annual terms, the mean is 8.1185% and the standard deviation 19.1052%. The monthly excess return series exhibits a strong non-normality, with a high skewness (0.2242), a long right tail, and excess kurtosis (10.69686) commonly observed in stock return series.

For boom and recession periods, the excess return series provides quite different pictures. In the boom period, the mean of monthly excess stock return is 0.8963% and its standard deviation 4.5839%. In the recession period, its mean is -0.3247% and its standard deviation 8.2016%. Annualized estimates of mean excess stock returns are 11.302% and -3.9667%/yr in boom and recession periods, respectively. It is evident that the average excess stock return is substantially higher in boom periods. Annualized standard deviations are 15.8791% and 28.4111%/yr in the boom and recession periods, respectively. As expected, the market is more volatile in recession periods.

In Table 2, some statistics of real CRSP stock index returns and real 1 -mo Treasury bill returns are presented. (7) The real stock return shows almost the same statistics as those of excess stock return shown in Table 1. This implies that the statistical properties of the excess stock return are dominated by those of stock return. For example, the mean of real stock return is 0.7117%, and that of the real bond return is 0.0591% over the whole sample period, whereas the mean excess stock return is 0.6525%. For volatility, the standard deviation of real stock return is 5.5289% and that of real bond return is 0.5496%, whereas that of excess return is 5.5152%. In addition, both skewness and kurtosis of real stock returns are very similar to those of excess return. A similar relation holds for both boom and recession period returns.

IV. EMPIRICAL MODELS WITH BUSINESS CYCLES

A. GARCH-M Model

In an attempt to reconcile diverse findings about the relation between expected returns and conditional variance, we consider a more general specification of the GARCH-M model in examining the time-varying relation between excess return and volatility over business cycles. Specifically, we allow for (1) business cycle factors in the GARCH-M model and (2) asymmetries in the conditional variance equation. (8) Among several AGARCH models, we choose GJR-GARCH model following the finding by Engle and Ng (1993) that the GJR-GARCH model is one of the best parametric GARCH models. (9)

Specifically, autoregression (AR)(1)-GJR-GARCH-M is specified based on our preliminary specification tests as follows: (10)

(1) [r.sub.t] = c + [phi][r.sub.t-1] + [delta][square root of[h.sub.t]] + [[epsilon].sub.t] [[epsilon].sub.t] = [square root of [h.sub.t]][v.sub.t]] and [v.sub.t] ~t[degrees of freedom].

(2) [h.sub.t] = [omega] + [[beta]h.sub.t-1] + [[alpha][[epsilon].sup.2.sub.t-1] + [[gamma] [D.sup.-.sub.t-1][[epsilon].sup.2.sub.t-1],

where [r.sub.t] is excess stock return over the l-mo Treasury bill rate; [h.sub.t] is the conditional variance; and [D.sup.-.sub.t] = 1 for [[epsilon].sub.t] < 0 and [D.sub.t] = 0, otherwise. In this GJR-GARCH-M model, a positive coefficient [delta] in Equation (1) implies a positive compensation for volatility or risk over time in the market. The GJR-GARCH model introduces a parameter [gamma] into the volatility Equation (2), which is absent from the simple GARCH model. In the conditional volatility Equation (2), a positive value of 7 implies an asymmetric movement of volatility and a higher volatility for a negative shock ([epsilon.sub.t] < 0) to the market, which is compatible with the leverage effect interpretation. For the distribution of the conditional variance, we employ a t-distribution for the error of the conditional mean equation because the t-distribution helps explain stock returns' fat tail (kurtosis) problem, relative to the Normal distribution. (11)

Now, we introduce two different business cycle states, boom and recession, as a proxy for real economic activities to the above-mentioned AR(1)-GJR-GARCH-M using a business cycle dummy variable d. We allow for dummy variables in both the conditional mean and the volatility equations. In the conditional mean equation, the dummy variable is introduced to the intercept term and to the standard deviation term (or risk term, [square root of [h.sub.t]]). Similarly, we introduce the dummy variable to both the intercept term and the asymmetry term ([D.sup.-.sub.t-1][[epsilon].sup.2.sub.t-1]) in the conditional volatility equation.

With the dummy variable, the GJR-GARCH-M is given by:

(3) [r.sub.t] = [c.sub.1] + [c.sub.2]d + [phi][r.sub.t-1] + [[delta].sub.1][square root of [h.sub.t]] + [[delta].sub.2]d [square root of [h.sub.t]] + [[epsilon].sub.t] [[epsilon].sub.t] = [[square root of [h.sub.t][V.sub.t]]~t[degrees of freedom].

(4) [h.sub.t] = [[omega].sub.1] + [[omega].sub.2]d + [[beta][h.sub.t-1] + [[alpha][[epsilon].sup.2.sub.t-1] + [[gamma].sub.1[D.sup.-.sub.t-1][[epsilon].sup.2.sub.t-1] + [[gamma].sub.2]d[D.sup.-.sub.t-1][[epsilon] .sup.2.sub.t-1],

where [D.sup.-.sub.t-1] = 1 for [[epsilon].sub.t] < 0 and [D.sup.-.sub.t-1] = 0, otherwise, and d denotes a dummy variable, with d = 1 for boom periods and d = 0 for recession periods.

In the conditional mean Equation (3), we have dummy variables for a constant term ([c.sub.2]d) and for the volatility term ([[delta].sub.2]d[square root of [h.sub.t]]). The former detects the change in a constant term and the latter detects the change in the return-risk relation (volatility feedback) in the boom period. The conditional volatility Equation (4) includes dummy variables for a constant term ([[omega].sub.2]d) and the asymmetric volatility term [[gamma].sub.2]d[D.sup.-.sub.t-1][[epsilon].sup.2.sub.t-1]. The coefficient for the asymmetric volatility term (72) measures the intensity of the asymmetric volatility movement over different business cycles, given a positive [[gamma].sub.1] that represents asymmetric volatility movements.

We report the estimation results of the AR(1)-GJR-GARCH-M model in Table 3. (12) First, in the simple GJR-GARCH-M without any business cycle dummy variable, we do not detect a clear and significant relation between risk ([square root of [h.sub.t]]) and excess return ([r.sub.t]). The coefficient of risk premium is positive, [delta] = 0.1376 but not significant. In the conditional volatility equation, we observe asymmetric movements of stock volatility (or leverage effect) with a positive and significant coefficient, [gamma] = 0.1600.

However, when we introduce the business cycle dummy variable d, we observe significant changes in regression coefficients. Over the boom periods (d = 1), the conditional mean equation shows a structural break shifting down in a significant way, [c.sub.2] = -2.9414. In addition, the risk premium coefficient increases substantially over the boom periods ([[delta].sub.2] = 0.7452 and significant). (13) As such, the intertemporal excess return-risk relation is positive and significant in boom periods, whereas the relation is negative but insignificant in recession periods. Among other things, this implies that there is substantially stronger compensation for a risk during boom periods. Without allowing for the difference between boom and recession periods, previous studies might have found a weak and often mixed intertemporal excess return-risk relation. In the conditional volatility Equation (4), the asymmetric volatility relation is significantly different between boom and recession periods. The asymmetric movement of volatility is mostly mitigated over the boom periods ([gamma].sub.2] = -0.2193 and significant).

Overall, we find a significant effect of business cycles on both the risk premium (or volatility feedback) and the intensity of asymmetric volatility movements. During the boom periods (d = 1), we find a significant increase in the risk premium (or volatility feedback) and a weakened asymmetric volatility. As such, we observe a significant positive intertemporal relation between risk and excess return in boom periods but an insignificant negative relation in recession periods. As a result, we observe an insignificant positive relation for the whole sample period in the absence of the differentiation based on business cycles. These findings are quite robust for different specifications of AGARCH models.

B. Markov Switching Model

In general, GARCH models are used to predict next period's (say, t + 1) volatility conditional upon information that is available at time t. In our model, we have taken into account the business cycle information as it is provided by the NBER. However, the NBER releases this information at a time after the state has occurred. That is, the abovementioned models use the hindsight NBER information about the state of the economy. A better approach would be to use only the available information that is known about the economic state at period t, estimating probabilities whether the economy is in a boom or in a recession. For this purpose, we estimate a regime switching model, in which a Markov switching model is estimated simultaneously with a GARCH model.

Specifically, we estimate the following standard AR(1) Markov switching model with GARCH(1,1) in mean:

(5) [r.sub.t] = c + [[phi][r.sub.t-1] + [[delta].sub.1[S.sub.t] + [[delta].sub.0] (1-[S.sub.t)] [square root of [h.sub.t]] + [[epsilon].sub.t], [S.sub.t] [member of] {0, 1}, [for all]t,

where [r.sub.t] is the excess stock return over the risk-free 1-mo Treasury bill rate and St is an unobservable state variable for business cycle varying between boom ([S.sub.t] = 1) and recession ([S.sub.t] = 0), which follows a first-order Markov chain process.

We define the time-varying transition probability matrix of [S.sub.t] between the two states as

(6) [p.sub.ij,t] = P[[S.subt] = j|[[S.sub.t-1] = i] or [p.sub.ii,t] = P[[S.sub.t] = i|[S.sub.t-1] = i] = CDFN([a.sub.i] + [b.sub.i][X.sub.t-1]),

where i, j = 0, 1; [X.sub.t] is the industrial production growth rate; and CDFN is a cumulative density function of Normal distribution. The transition probability matrix P is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The conditional volatility equation or GARCH(1,1) model is specified as:

(7) [[epsilon].sub.t] = [square root of [h.sub.t][v.sub.t]], [v.sub.t]~t[degrees of freedom].

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[??].sub.t] = {[S.sub.t], [S.sub.t-1], ... , [S.sub.1]}.

Our model includes a time-varying transition probability, which is determined by economic conditions measured by the growth rate of industrial production ([X.sub.t]). This implies that the risk aversion parameter, [delta], is determined by the regime shift, which is an endogenous variable rather than an exogenous dummy variable. (14)

In the abovementioned model, we maximize the following log-likelihood function with respect to the parameter set ([theta]) = {c, [phi], [[delta].sub.0], [[delta].sub.1], [a.sub.0], [a.sub.1], [b.sub.0], [b.sub.1], [omega], [beta], [alpha]}:

(9) ln L([theta]) = [T.summation over(t=1)]ln f([r.sub.t]|[r.sub.t-1], [r.sub.t-2], ...).

Table 4 presents estimation results of the Markov switching AR(1)-GARCH(1,1)-M model. Our main interest is on the relation between business cycle ([X.sub.t]) and the risk aversion parameter ([delta]). The risk aversion parameter in State 1 (or in a boom period), [[delta].sub.1], is significantly positive along with the coefficient [b.sub.1], which measures the transition probability from State 1 to the same State 1. More specifically, given State 1 (or boom periods), the effect of the conditional standard deviation, [square root of [h.sub.t]], on excess return ([r.sub.t]) is significantly positive. Also, State 1 shifts more likely to the same State 1. On the other hand, in State 0 (or recession periods), the risk aversion parameter is insignificant along with an insignificant coefficient [b.sub.1]. This provides more evidence that the risk aversion is economic state dependent and procyclical. This procyclical empirical result of the Markov switching model is consistent with our model using the NBER business cycle dummy variable.

V. FURTHER DISCUSSIONS

A. State-Dependent Attitude toward Risk (or Risk Aversion)

In our empirical analysis, we find significant effects of business cycles on both volatility feedback and intensity of asymmetric volatility movements. Since the coefficient of volatility in the excess mean return equation is usually interpreted as measuring the risk aversion parameter as known from Merton (1980), our finding may provide a new insight into time-varying (or state dependent) risk aversion, which is recently debated in the literature. Specifically, we find a significant increase in volatility feedback during the boom periods (d = 1). Then, we may infer that our finding is in favor of increased risk aversion in the boom period or procyclical movement of risk aversion.

Recently, there has been some debate on investors' attitude toward risk under an exogenously given environment. Results from the experimental psychology and economics literature provide support for the hypothesis that risk aversion is state dependent. Isen and Geva (1987); Isen and Patrick (1983); and Nygren et al. (1996) presented evidence suggesting that those who have received a consumption increase are much less willing to gamble than control groups. Isen (2000) interpreted these results as suggesting that persons in a "good mood" are more reluctant to gamble because losing might undermine their good mood. Bosch-Domenech and Silvestre (1999) reported the results of an experiment in which the subjects were given title to a random payout of money and were asked if they wished to insure against a 20% chance of having their personal monetary realization taken from them. Half of the subjects choose to insure but only if their income realization fell within the high-level category, a response that associates greater risk aversion with higher income levels. These experiments are consistent with the hypothesis of associating higher risk aversion with greater consumption growth and higher consumption levels. Broadly speaking, this is the perspective that risk aversion is procyclical, rising during booms and falling during recessions.

Strong empirical evidence to the contrary is provided by Gordon and St-Amour (2002) who postulated a model with time-varying risk aversion. They estimated the implied process on risk aversion arising from per capita consumption and financial return data. Their basic finding is that risk aversion is strongly countercyclical, rising during recessions and falling during expansions. In addition, the constant relative risk aversion estimate by Gordon and St-Amour (2002) moves opposite of the University of Michigan index of consumer confidence, a fact that is also broadly consistent with their finding of countercyclical risk aversion.

B. Simple Equilibrium Model and Its Calibration

To better understand and confirm our interpretation of time-varying (or state dependent) risk aversion, we introduce a simple asset pricing model of Barsky (1989) that allows us to examine the relation between excess stock return and risk aversion. Barsky (1989) explored the possible role of changes in risk and productivity growth for the behavior of bond and stock prices in a simple general equilibrium model (for an extension, see Abel [1988]). His model is the stochastic version of the neoclassical theory of an endowment economy of the type studied by Lucas (1978) and Campbell (1986). It is a two-period, two-asset (equity and risk-free bond) model of the general equilibrium asset pricing model. Our discussion draws heavily from Barsky (1989) to illustrate the relation between excess returns (or equity premium) and risk and then reinterpret it as the relation between excess returns and risk aversion under different economic states.

Agents maximize, subject to the standard intertemporal budget constraints, time-additive, concave expected utility functions of the form:

U([C.sub.1]) + [beta]E[U([[??].sub.2])],

where [C.sub.1] is first-period consumption and [[??].sub.2] is the random second-period consumption. The first-order conditions are:

(10) [[beta]E[U'([[??].sub.2][??]] = U' ([C.sub.1]),

(11) [R.sub.f][beta]E[U' ([[??].sub.2]] = U' ([C.sub.1]),

where [??] is one plus the random rate of return on equity and [R.sub.f] is one plus the real risk-free rate. Since outputs are nonstorable by assumption in this exchange economy, markets clear when agents consume all outputs each period:

(12) [C.sub.1] = [Y.sub.1] and [[??].sub.2] = [[??].sub.2],

where [Y.sub.1] is the first-period output and [[??].sub.2] is the stochastic second-period output. From the first-order conditions and market clearing condition, we obtain returns on risk-free bond and equity as,

(13) [R.sub.f] = [1/[P.sup.b]][U'([Y.sub.1])/[beta]E[U'([[??].sub.2])]],

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [p.sup.b] is the price of a riskless bond and [p.sup.eq] is the price of an equity.

We can derive explicit solutions for riskless interest rates and equity returns by assuming a state-dependent relative risk aversion utility (U(C) = [C.sup.1-[gamma](t)]/t)/(1-[gamma](t))) and a lognormal random variable [[??].sub.2], where logE[[??].sub.2]] = E[log([[??].sub.2])] + (1/2)Var[log([[??].sub.2])] (see also Campbell [1986]). Given the equilibrium condition with a state-dependent relative risk aversion utility function, we interpret the equity premium (Z) with two different states (boom and recession) and risk aversion parameters, [[gamma].sub.1] and [[gamma].sub.2], corresponding to each state:

(15) log([R.sub.f]) + -log([beta]) + E[[[gamma].sub.2]log([[??].sub.2]) -[[gamma].sub.1]log([Y.sub.1])] -0.5[gamma].sup.2.sub.2] Var[log([??].sub.2])],

(16) log [E([??])] = -log([beta]) + E[[gamma].sub.2]log([??].sub.2]) -[[gamma].sub.1]log([Y.sub.1])] + ([[gamma].sub.2] -0.5[gamma].sup.2.sub.2])Var[log([[??].sub.2])],

(17) Z = log[E([??])] - log([R.sub.f]) = [[gamma].sub.2]Var[log([[??].sub.2])],

where Z is the equity premium. Equation (17) shows that equity premium (Z) depends on risk aversion parameter, [[gamma].sub.2] and Var[log([[??].sub.2])].

One way to estimate the value of [gamma] in boom and recession periods is to fit the theoretical relation in Equation (17) into the data provided in Table 5. For the sample period January 1959 to December 2001, the excess return of stock is 0.5554%/mo on average in the boom period and 0.0354%/mo on average in recession period. Variance of real disposable personal income is 2,194,467 in boom periods and 2,140,408 in recession periods. Thus, to explain a higher equity premium (Z) in the boom period by Equation (17), the risk aversion parameter, [gamma], should be higher in the boom period, implying a procyclical risk aversion parameter.

A more specific relation between the economic state and the risk aversion can be discussed using the data provided in Table 5. We rewrite Equation (17) for the risk aversion parameter, [gamma],

(17a) [gamma] = Z/Var[log(Y)].

In other words, the risk aversion parameter of the model is the ratio of excess return to the variance of the logarithm of income Y. Using the data provided in Table 5, we obtain estimates of the risk aversion parameter, [gamma], in boom and recession periods: 3.2289 in the boom period and 0.2212 in the recession period. (15) This calibration result of the risk aversion parameter, [gamma], confirms a procyclical risk aversion.

VI. EXTENSION AND ROBUSTNESS OF THE MARKOV SWITCHING MODEL

A. Markov Switching Model with Four Business Cycle Factors

In the second subsection of Section IV, we have estimated the Markov switching model using only industrial production as a factor for business cycle. In this section, we reestimate the model with multifactors that the NBER considers in their decision of business cycle. These factors are real income, sales, industrial production, and employment ratio. (16) All the data are measured in real terms and seasonally adjusted, and we estimate the model over the period January 1959 to December 2001, given the availability of the data.

Specifically, the AR(1)-GARCH(1,1)-M Markov switching model with the four factors is specified as follows:

(18) [r.sub.t] = c + [phi][r.sub.t-1] + [[delta].sub.1][S.sub.t] + [delta].sub.0](1-[S.sub.t])][square root of [h.sub.t]] + [[epsilon].sub.t], [S.sub.t] [member of] {0, 1}, [for all]t,

(19) [p.sub.ii,t] = Pr[[S.sub.t] = i|[S.sub.t-1] = i] = CDFN([a.sub.i] + [b.sub.1i][X.sub.1,t-1] + [b.sub.2i][X.sub.2,5-1] + [b.sub.3i][x.sub.3,t-1] + [B.sub.4i][X.sub.4,t-1]),

(20) [[epsilon].sub.t] = [square root of [h.sub.t][v.sub.t]], [v.sub.t]~t[degrees of freedom],

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

* CDFN: cumulative density function of normal distribution

* t[degrees of freedom]: t-distribution with degrees of freedom

* [X.sub.1,t]: real disposable personal income

* [X.sub.2,t]: real retail sales

* [X.sub.3,t]: industrial production

* [X.sub.4,t]: civilian employment population ratio.

We present the estimation result in Table 6. We find that the risk aversion parameter in boom period ([delta].sub.1]) is significantly positive at the 5% level. Among the four factors, real income ([b.sub.11]) turns out to be a factor with a significantly positive probability from boom (Regime 1) to boom (Regime 1). With the four factors, we still find, among other things, a procyclical movement of risk aversion under the Markov switching framework. (17)

B. Markov Switching Model with Asymmetric Volatility

In the second subsection of Section IV, we have considered a Markov switching model in the absence of asymmetric volatility. Here, we extend the model by introducing asymmetric volatility movements based on the AR(1)ARCH(1,1)-M Markov switching model with four business cycle variables:

(22) [r.sub.t] = c + [phi][r.sub.t-1]+[[delta].sub.1][S.sub.t] + [[delta].sub.0](1-[S.sub.t])[square root of [h.sub.t]] + [[epsilon].sub.t], [S.sub.t][member of] {0, 1}, [for all]t,

(23) [p.sub.ii,t] = Pr[[S.sub.t] = i|[S.sub.t]= i] = CDFN([alpha].sub.i] + [b.sub.1i][X.sub.1, t-1] + [b.sub.2i][X.sub.2,t-1] +[b.sub.3i][X.sub.3,t-1] + [b.sub.4i][X.sub.4,t-1],

(24) [[epsilon].sub.t] = [square root of [h.sub.t]][v.sub.t]], [v.sub.t]~t[degrees of freedom],

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [D.sup.-.sub.t] = 1 for [[epsilon].sub.t] < 0 and [D.sup.-.sub.t] = 0, otherwise.

We present the estimation results in Table 7. We find that the coefficient for asymmetric volatility ([gamma]) is significant, which indicates asymmetric volatility movements in the Markov switching AR(1)-GARCH(1,1)-M model. In addition, we also find a significant risk aversion in boom period (i.e., a significantly positive [[delta].sub.1]) and a significant transition probability ([b.sub.1l]) from boom to boom for real income's coefficient.

C. Out-of-Sample Forecast Performance with a Business Cycle Dummy

Our major goal in this article was to explore the role of the business cycle in the relation between risk and return for the aggregate stock market. As such, it would be interesting to compare the out-of-sample forecasts between the model including boom/recession dummy and the one without the dummy, to see whether the inclusion of the business cycle dummy improves the model's out-of-sample performance.

As a measure of the forecast performance, we calculate the root mean squared errors (RMSE) from a series of 1-mo-ahead out-of-sample forecasts using the rolling estimation procedure. Specifically, we first estimate the two models (with and without d) using data up to December 1995, and then we use the estimates to generate 1-mo-ahead forecasts of excess returns. We repeat this recursive procedure by adding one more observation at a time up to the last estimation period, November 2001. Hence, we generate forecasts for January 1996 to December 2001. This period coincides with a recent booming period of the expansion of the U.S. economy. Therefore, it provides quite a strenuous test for the AR(1)-GARCH(1,1)-M model with the boom and recession dummy variable (d).

We then calculate the RMSE ratio, which is defined as RMSE for the model with d divided by RMSE for the model without d. A ratio less than 1 indicates a better forecasting performance of the model with the business cycle dummy d. Our estimate of the RMSE ratio based on the GJR-GARCH model is 0.9586, which implies that the model with the boom and recession dummy d produces more accurate forecasts than a simple model without d. (18)

D. Markov Switching Model with a Constant Probabilio'

We have estimated in the second subsection of Section IV the Markov switching model using time-varying probability. In this section, we reestimate the switching model with a constant transition probability to see whether the time-varying transition probability yields a more realistic result. The constant transition probability model is specified as follows:

(26) [r.sub.t] = c + [phi][r.sub.t-1] + [[delta].sub.1][S.sub.t] + [[delta].sub.0](1 - [S.sub.t])][square root of [h.sub.t]] + [[epsilon].sub.t], [S.sub.t][member of] {0, 1], [for all]t,

(27) [p.sub.ij,t] = Pr[[S.sub.t] + j|[S.sub.t-1] = i] = [p.sub.ij], [for all]t,

(28) [[epsilon].sub.t] = [square root of [h.sub.t]][v.sub.t]], [v.sub.t]~t[degrees of freedom],

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, [p.sub.ij,t] = Pr[[S.sub.t] = j|[S.sub.t-1] = i] = [p.sub.ij] indicates that [p.sub.ij] is constant over time

We present the estimation results in Table 8. We find that [p.sub.11] is very large at 0.9705 (i.e., [p.sub.10] is 0.0295) and [p.sub.00] is relatively small at 0.2942 (i.e., [p.sub.01] is 0.7058), and both are significant. The estimates indicate that the probability of the transition from boom to boom is close to 1, whereas the probability of recession-to-recession transition is relatively small.

According to the NBER's classification for the period 1959-2001, about 85% of the whole sample period is in boom periods. Since the sample period is dominated by the boom period, the constant probability switching model seems to simply reflect the dominant boom period for the sample period rather than the two regime (boom and recession) switching economy. As such, the U.S. economy's regime switching property seems poorly captured by the constant probability switching model, which prompts us to extend this model to a time-varying probability model. (19)

E. Conditional Probability of Each Month Being in Boom in Comparison with NBER Business Cycles: Illustration

As an informal evaluation of the Markov switching model's performance, we illustrate in Figure 1 the conditional probability of the boom (or State 1) over the estimation period (January 1959 to April 2001), which is calculated based on the Markov switching AR(1)-GARCH(1,1)-M model with four business cycle factors, and compare that with the NBER classification. We find that the conditional probability of State 1 (boom) tends to be very high during the NBER boom period compared with the NBER recession period.

[FIGURE 1 OMITTED]

VII. CONCLUDING REMARKS

While there is a general agreement about the trade-off between risk and expected return for cross-sectional securities within a given time period, there seems to be no such agreement about the relation between risk and return over time. There is an extensive empirical literature that has tried to establish the existence of such an intertemporal trade-off between risk and return for stock market indexes. Unfortunately, the results have been inconclusive. Recently, it has been debated whether risk aversion is state dependent and whether it is procyclical or countercyclical.

In this article, we have explored these issues using both empirical models and a simple equilibrium framework. We have employed a representative AGARCH-M model allowing for potential business cycle effects to examine a time-varying intertemporal relation between excess return and risk using a broad market index return. To avoid using the hindsight NBER information about business cycles, we estimate a regime switching model, where a Markov switching model is estimated simultaneously with a GARCH-M model. Given our finding of a time-varying risk-return relation over business cycles, we have attempted to infer its implication for time-varying risk aversion over the business cycles.

Our findings can be summarized as follows. First, in the simple GARCH model without allowing for the business cycle effect, the risk premium coefficient does not show any clear and significant relation between risk and excess return. However, once we allow for a business cycle factor (boom and recession), the risk premium coefficient becomes significantly positive for boom periods, whereas it remains insignificantly negative for recession periods.

Second, using a Markov regime switching model in the AR(1)-GARCH(1,1)-M framework, we find that the risk aversion parameter is significantly positive in the boom periods. This procyclical risk aversion is confirmed in extensions of the model. This finding suggests that previous studies may have failed to find a significant relation between risk and return over time in part because they did not take into account the business cycle factor. Since the coefficient of volatility in the excess mean return equation is usually characterized as measuring the time-varying risk aversion parameter (e.g., Merton [1980]), our finding suggests increased risk aversion in boom periods or a procyclical movement of risk aversion.

Third, to further investigate and confirm potential time-varying (or state dependent) risk aversion, we employ a simple equilibrium asset pricing model that allows us to examine the relation between excess return and risk aversion. Using a simple calibration of the model, we confirm that risk aversion is state dependent and procyclical. A stronger risk aversion in the boom period helps explain higher excess stock return and compensation for risk in the boom periods.

Fourth, we find that the asymmetric movement of volatility is also state dependent, and its intensity changes over business cycles. Specifically, we find that asymmetric volatility is weakened significantly in boom periods.

Our finding of state-dependent and procyclical risk aversion helps us understand not only the larger risk premium for a given risk in boom periods but also the weakened asymmetric volatility during boom periods, in particular by extending the argument of Campbell and Hentschel (1992), both of which are observed based on GARCH-M models with business cycle dummies. Regarding the asymmetric volatility movement, Black (1976) argued that it could be due to an increase in leverage that occurs when the market value of a firm declines. However, we find that investors are strongly risk-averse during boom periods. As such, investors become more sensitive to the leverage effect, and the leverage effect hypothesis anticipates that asymmetric volatility will get stronger in boom periods. This prediction is not easily compatible with our finding of weakened asymmetric volatility during boom periods.

Subsequent studies tend to find that the leverage effect alone is too small to fully account for this phenomenon (e.g., Christie 1982; Schwert 1989a). Campbell and Hentschel (1992) partially explained it with the "news effect." They argued that if there is good news about future dividends, then that good news tends to be followed by more good news (i.e., volatility is persistent). Therefore, this piece of good news increases future expected volatility, which in turn increases the required rate of return on stocks and lowers stock prices, dampening the positive impact of the dividend news. This is what usually happens in boom periods. Now consider investors being more risk-averse in boom periods than in recession periods as observed above. Investors will require a higher excess return (and required rate of return) in boom periods than in recession periods. This will result in weakened asymmetric volatility in boom periods. To the extent that this is an important feature in boom periods regarding the volatility of stock returns, our weakened asymmetric volatility can be easily compatible with the news effect hypothesis combined with the state-dependent asset pricing model. As such, our finding provides new insights into these two explanations.

ABBREVIATIONS

AGARCH: Asymmetric Generalized Autoregressive Conditional Heteroskedasticity

AGARCH-M: Asymmetric Generalized Autoregressive Conditional Heteroskedasticity in Mean Models

AMEX: American Stock Exchange

AR: Autoregression

CPI: Consumer Price Index

CRSP: Center for Research in Security Prices

GARCH: Generalized Autoregressive Conditional Heteroskedasticity

GARCH-M: Generalized Autoregressive Conditional Heteroskedasticity in Mean Models

GJR-GARCH: GARCH of Glosten, Jagannathan, and Runkle

NASDAQ: National Association of Securities Dealers Automated Quotation (System)

NBER: National Bureau of Economic Research

NYSE: New York Stock Exchange 

RMSE: Root Mean Squared Errors

APPENDIX A

NBER Business Cycles Reference Dates for January 1926
to December 2001

Trough                       Peak

November 1927 (IV)    October 1926 (III)
March 1933 (I)        August 1929 (III)
June 1938 (II)        May 1937 (II)
October 1945 (IV)     February 1945 (I)
October 1949 (IV)     November 1948 (IV)
May 1954 (II)         July 1953 (II)
April 1958 (II)       August 1957 (III)
February 1961 (I)     April 1960 (II)
November 1970 (IV)    December 1969 (IV)
March 1975 (I)        November 1973 (IV)
July 1980 (III)       January 1980 (I)
November 1982 (IV)    July 1981 (III)
March 1991 (I)        July 1990 (III)
                      March 2001 (I)

Notes: I = first quarter; II = second quarter;
III = third quarter; and IV = fourth quarter.

APPENDIX B

Excess Returns(%) over Business Cycles from January 1926 to December 2001

[GRAPHIC OMITTED]

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SEI-WAN KIM and BONG-SOO LEE *

* We would like to thank the editor and two anonymous referees for their numerous insightful comments that help improve the article.

Kim: Assistant Professor, Department of Economics, Ewha Womans University, 120-750, Seoul, Korea, and Department of Economics, California State University--Fullerton, Fullerton, CA 92834-6848. Phone [phone], Fax [phone], E-mail [e-mail]

Lee: Professor, Department of Finance, College of Business, Florida State University, Tallahassee, FL 32306-1110. Phone [phone], Fax [phone], E-mail [e-mail]

(1.) In this article, excess stock return is measured as the difference between return on CRSP market value-weighted index with dividend and return on 1-mo Treasury bill return. Boom and recession are defined by NBER's classification. In Section III, we discuss the details of the data.

(2.) According to Model 1 of Merton (1980), the coefficient of variance in the expected excess return on the market is equal to a representative investor's relative risk aversion. Even when we use standard deviation in lieu of variance, the time-varying coefficient remains valid.

(3.) If all the productive assets available for transferring income to the future carry risk and no risk-free investment opportunities are available, then the price of the risky asset may be bid up considerably, thereby reducing the risk premium. Abel (1988), Backus and Gregory (1993), Gennotte and Marsh (1993), and Glosten and Jagannathan (1987) have shown that the risk premium on the market portfolio of all assets could, in equilibrium, be lower during relatively riskier times.

(4.) However, the model with conditionally normal innovations does not provide a good fit for a negative skewness or excess kurtosis of returns. We therefore consider the case in which the innovations follow a Student's t-distribution.

(5.) For a detailed explanation on NBER's business cycle decision, see their Web site at www.nber.org. For a graph of excess returns over business cycles for the period January 1926 to December 2001, see Appendix B.

(6.) The skewness of a symmetric distribution, such as the normal distribution, is 0. A positive skewness means that the distribution has a long right tail, and a negative skewness implies that the distribution has a long left tail. Kurtosis measures the flatness of the distribution of the series. The kurtosis of the normal distribution is 3. If the kurtosis exceeds 3, the distribution is peaked (leptokurtic) relative to the normal; if the kurtosis is less than 3, the distribution is flat relative to the normal. The Jarque-Bera statistic is a test statistic for testing whether the series is normally distributed using skewness and peakedness together. It follows [chi square](2) distribution.

(7.) Real returns are calculated by subtracting monthly Consumer Price Index (CPI) growth rate from nominal returns. The CPI data were retrieved from the Bureau of Labor Statistics data archive.

(8.) See Bollerslev, Chou, and Kroner (1992) for an extensive survey of GARCH and GARCH-M models in finance. Campbell and Hentschel (1992) showed that the relationship between excess stock return and volatility feedback can be studied using AGARCH model. They used a quadratic GARCH model, which is very similar to the AGARCH model.

(9.) In a test of volatility models, Engle and Ng (1993) found strong support for both the AGARCH model and the GJR-GARCH model. They found that both models are the best parametric GARCH specifications through diagnostic tests based on news impact curves. Kim and Kon (1994) also found strong support for the GJRGARCH model. As such, among various GARCH models, we have chosen the GJR-GARCH (1993) model. The empirical results based on the AGARCH-M model are also available from the authors upon request.

(10.) The choice of the standard deviation, not variance, in the excess return equation represents the assumption that changes in variance are reflected less than proportionally in the mean, which is consistent with the original ARCH-M model proposed by Engle, Lilien, and Robins (1987). For the selection of the number of lags in AR (lag 1) and GARCH (lag 1 ) in this model, we have also gone through model specification tests, Johansen's test, Information Criteria, and the Partial Autocorrelation Function.

(11.) With a t-distribution, we obtain higher log-likelihood values than Normal distribution. In addition, all degrees of freedom estimates lie between 5 and 8, which explains kurtosis of excess return distribution. For Normal distribution GARCH models, the results are available from the authors upon request.

(12.) Major empirical results are quite consistent and robust over the two GARCH models: GJR-GARCH and AGARCH. To save space, we present only the results of the GJR-GARCH model estimation. The estimation results of the AGARCH model are available from the authors upon request.

(13.) We interpret the insignificant negative relation between excess return and risk in recessions as the absence of a significant relation between excess return and risk, rather than as evidence of risk-neutral or risk-loving in recessions.

(14.) Below, in Section VI, we discuss the model with a constant transition probability matrix and the model with four business cycle factors rather than a single factor, industrial production.

(15.) With consumption data in lieu of income data, the calibrated risk aversion parameter is 5.4990 and 0.3766 for boom and recession periods, respectively.

(16.) All data are retrieved from the St. Louis Fed's data archive. The four variables are defined as follows. [X.sub.1]: real disposable personal income (seasonally adjusted), [X.sub.2]: real retail sales (seasonally adjusted), [X.sub.3]: industrial production (seasonally adjusted), and [X.sub.4]: civilian employment-to population ratio (seasonally adjusted).

(17.) Here, we estimate only [p.sub.ii]. This is because [p.sub.ij] is given by [p.sub.ij] = 1.0-[p.sub.ii]. So we report only [p.sub .00] and [p.sub.11] in Table 6. Therefore, according to our estimates,

[p.sub.11.t] = Pr[[S.sub.t-1] = 1] = CDFN(0.2088 + 0.4826 x [X.sub.1, t-1] - 0.2349[X.sub.2.t-1] -0.0447[X.sub.3,t-1] - 0.6563[X.sub.4,t-1]).

(18.) Our estimate of the RMSE ratio based on the AGARCH model is 0.9508, which also implies that the model with the boom and recession dummy (d) produces more accurate forecasts than a simple model without d., that is, we find that the superior forecasting performance is not sensitive to the GARCH models we employ.

(19.) Another reason for this result may be due to a missing variable problem. Note that the constant probability model would be a constrained version of a more general time-varying transition probability model in the sense that [b.sub.1i] = [b.sub.2i] = [b.sub.3i] = [b.sub.4i] = [b.sub.1i] = 0 is imposed for i = 0 and 1 in the following four-variable transition probability specification:

[p.sub.ii,t] = P[[S.sub.t] = i|[S.sub.t-1] = i] = CDFN ([a.sub.i] + [b.sub.1i][X.sub.1,t-1] + [b.sub.2i][X.sub.2,t-1] + [b.sub.3i][X.sub.3,t-1] + [b.sub.4i][x.sub.4,t-1].

Therefore. the constant transition probability model may not have a sufficient source of regime switching. Although there are two regimes switching over time, the constant probability model with missing variables may produce a bias in the estimation and the regimes may not be fully detected.

TABLE 1
Descriptive Statistics for Monthly Excess Returns ([r.sub.t])

                           Whole Periods             Boom

Observations               912                  730
Mean (%)                     0.6526               0.8963
Standard deviation (%)       5.5152               4.5839
Skewness                     0.2242               0.3347
Kurtosis                    10.6968              10.9062
Jarque-Bera (p value)    2,258.802 (0.0000)   1,914.958 (0.0000)

                             Recession

Observations             182
Mean (%)                  -0.3247
Standard deviation (%)     8.2016
Skewness                   0.3557
Kurtosis                   6.8029
Jarque-Bera (p value)    113.514 (0.0000)

Notes: The monthly excess return is the difference between the
value-weighted market index portfolio return on the NYSE, AMEX,
and NASDAQ (retrieved from CRSP tapes) and 1-mo Treasury bill
rate from January 1926 to December 2001.

TABLE 2
Descriptive Statistics for Monthly Real Stock
and Bond Returns

                                     Real Stock Return
                            Whole
                           Period         Boom        Recession

Observations               912           730         182
Mean (%)                     0.7117        0.8952     -0.0237
Standard deviation (%)       5.5289        4.6218      8.1941
Skewness                     0.2714        0.2901      0.3949
Kurtosis                    10.5506       10.6198      6.7793
Jarque-Berg (p value)    2,177.659     1,776.288     113.047
                            (0.0000)      (0.0000)    (0.0000)

                                     Real Bond Return
                            Whole
                           Period         Boom        Recession

Observations                912           730        182
Mean (%)                      0.0591       -0.0011     0.3009
Standard deviation (%)        0.5496        0.5221     0.5908
Skewness                     -1.9635       -3.1770     0.5444
Kurtosis                     20.6135       27.9552     3.3429
Jarque-Berg (p value)    12,375.08     20,170.46       9.8832
                             (0.0000)      (0.0000)   (0.0000)

Notes: The monthly return series is the value-weighted market
index portfolio return retrieved from CRSP tapes on the NYSE,
AMEX, and NASDAQ from January 1926 to December 2001.

TABLE 3
Estimation of GJR-GARCH-M Model with a Boom/
Recession Dummy Variable d

                                          GJR-GARCH-M
                                      (No Dummy Variable)

Coefficient (Variable)             Estimate      t-Statistic

[c.sub.1]                             0.1833        0.3742
[c.sub.2](d)                          --             --
[phi]([R.sub.t-1]                     0.0634 *      1.7202
[[delta].sub.1]([square
  root of [h.sub.t]]                  0.1376        1.2230
[[delta].sub.2](d [square
  root of [h.sub.t]]                  --             --
[[omega].sub.1]                       1.7689 *      3.6168
[[omega].sub.1](d)                    --             --
[beta]([h.sub.t-1])                   0.8072 *     19.7173
[[alpha].sub.1] ([[epsilon]
  .sup.2.sub.t-1]                     0.0359        0.9234
[[gamma].sub.1] ([[epsilon]
  .sup.2.sub.t-1] [D.sub.t-1]         0.1600 *      2.9259
[[gamma].sub.2] ([[epsilon]
  .sup.2.sub.t-1] d[D.sub.t-1]        --             --
Degrees of freedom                    7.9268 *      4.8882
Log likelihood                   -2,688.209          --

                                          GJR-GARCH-M
                                 (Boom and Recession dummy, d)

Coefficient (Variable)           Estimate        t-Statistic

[c.sub.1]                             1.3109        1.2266
[c.sub.2](d)                         -2.9414 *     -2.1558
[phi]([R.sub.t-1]                     0.0483        1.3040
[[delta].sub.1]([square
  root of [h.sub.t]]                 -0.1448       -0.7688
[[delta].sub.2](d [square
  root of [h.sub.t]]                  0.7452 *      2.6480
[[omega].sub.1]                       2.5481 *      1.7112
[[omega].sub.1](d)                    0.0062        0.0042
[beta]([h.sub.t-1])                   0.7859 *     16.6296
[[alpha].sub.1] ([[epsilon]
  .sup.2.sub.t-1]                     0.0265        0.7053
[[gamma].sub.1] ([[epsilon]
  .sup.2.sub.t-1] [D.sub.t-1]         0.2961 *      2.6709
[[gamma].sub.2] ([[epsilon]
  .sup.2.sub.t-1] d[D.sub.t-1]       -0.2193 *     -2.1387
Degrees of freedom                    7.9124 *      5.0919
Log likelihood                   -2,681.402          --

Notes: * indicates significant at 10% level.

TABLE 4
Estimation of Markov Switching AR(1)-GARCH-M Model

                    Estimates     t-Statistic

c                   0.6216 *       5.3048
[[delta].sub.0]    -0.0701        -0.4465
[[delta].sub.1]     0.3629 *       2.3871
[[alpha].sub.0]     3.4188         1.8664
[b.sub.0]           4.0569         1.5885
[a.sub.1]           1.0199 *       2.1306
[b.sub.1]           1.3584 *       2.1885
[omega]             0.0254 *       3.1110
[alpha]             0.1140 *       5.8655
[beta]              0.8544 *      39.4722
[phi]               0.0458         1.2516

Notes: * indicates significant at

TABLE 5
Calibrating Risk Aversion Parameter (y) in the Model
by Barsky (1989)

                                        Real Consumption

                         Whole Period       Boom        Recession

Observations                  516            440             76
Mean                        1,204.70       1,212.70       1,158.60
Variance                  139,159.25     138,726.26     141,026.99

                                     Log Real Consumption

                         Whole Period       Boom        Recession

Observations              516            440             76
Mean                        7.04           7.05           7.00
Variance                    0.100          0.101          0.094

                                        Real Income

                         Whole Period       Boom        Recession

Observations                   516            440             76
Mean                         3,834.33       3,865.57       3,627.62
Variance                 2,184,565.24   2,194,466.85   2,140,407.84

                                       Log Real Income

                         Whole Period       Boom        Recession

Observations              516            440             76
Mean                        8.17           8.18           8.12
Variance                    0.170          0.172          0.160

                                        Equity Premium

                         Whole Period       Boom        Recession

Observations              516            440             76
Mean (%)                    0.4788         0.5554         0.0354
Variance (%)                0.1943         0.1651         0.3661

                                  [gamma]

                                Whole Period

[gamma] = Z/Var[log(Y)]    0.4788/0.170 = 2.8164
[gamma] = Z/Var[log(C)]    0.4788/0.100 = 4.7880

                                  [gamma]

                                    Boom

[gamma] = Z/Var[log(Y)]    0.5554/0.172 = 3.2289
[gamma] = Z/Var[log(C)]    0.5554/0.101 = 5.4990

                                  [gamma]

                                 Recession

[gamma] = Z/Var[log(Y)]    0.0354/0.160 = 0.2212
[gamma] = Z/Var[log(C)]    0.0354/0.094 = 0.3766

Notes: For consumption and income, we use monthly real personal
consumption expenditures for nondurable goods (seasonally
adjusted) and monthly real disposable personal income
(seasonally adjusted) from January 1959 to December 2001. Both
data series are retrieved from U.S. Department of Commerce:
Bureau of Economic Analysis data archive. The monthly excess
return is the difference between the value-weighted market
index portfolio return on the NYSE, AMEX, and NASDAQ (retrieved
from CRSP tapes) and 1-mo Treasury bill rate (retrieved from
Ibbotson Associates) over January 1926 to December 2001.

TABLE 6
Estimation of Markov Switching AR(1)-
GARCH-M Model with Four Business Cycle
Factors Sample period: January 1959 to
April 2001

                   Estimates      t Value

[[delta].sub.0]    -0.2210       -0.6717
[[delta].sub.1]     0.9993 *      2.6034
[a.sub.0]           0.0000        0.0000
[b.sub.10]         -0.1232       -0.6531
[b.sub.20]         -0.1555       -0.8059
[b.sub.30]          0.0872        0.4951
[b.sub.40]         -0.224        -1.2738
[a.sub.1]           0.2088        0.6572
[b.sub.11]          0.4826 *      2.1517
[b.sub.21]         -0.2349       -1.2317
[b.sub.31]         -0.0447       -0.2621
[b.sub.41]         -0.6563       -1.8693
[omega]             0.0598        1.7744
[beta]              0.0686        2.2266
[alpha]             0.8217 *     15.3059
[phi]              -0.0145       -0.3270
c                   0.1757        0.5545

Notes: * indicates significant at 5% level.

TABLE 7
The AR(1)-GARCH(1,1)-M Markov
Switching Model with Four Business Cycle
Factors

                    Estimates      t Value

[[delta].sub.0]     -0.4633       -1.4490
[[delta].sub.1]      0.7828 *      2.4884
[a.sub.0]            0.0000        0.0000
[b.sub.10]          -0.0681       -1.4262
[b.sub.20]          -0.1135       -0.4813
[b.sub.30]           0.0752        0.5559
[b.sub.40]          -0.2768       -1.3977
[a.sub.1]            0.3389        1.0292
[b.sub.11]           0.4747 *      2.1082
[b.sub.21]          -0.2219       -1.1813
[b.sub.31]          -0.0579       -0.2731
[b.sub.41]          -0.6357       -1.8519
[omega]              0.0679 *      2.3260
[beta]               0.0000        0.0068
[alpha]              0.8110 *     15.8049
[phi]               -0.0083       -0.1615
c                    0.3228        1.2332
[gamma]              0.1152 *      2.6147

Notes: * indicates significant at 5%.

TABLE 8
Estimation of AR(1)-GARCH-M with
a Constant Probability

                    Estimates      t Value

[[delta].sub.0]     -2.0985 *     -4.5171
[[delta].sub.1]      0.2402        0.5734
[p.sub.00]           0.2942 *      2.4015
[p.sub.11]           0.9705 *     75.337
[omega]              0.7992 *      2.8064
[beta]               0.0888 *      3.7143
[alpha]              0.8435 *     27.5098
[phi]               -0.0161       -0.3861
c                    0.1838        0.1035

Notes: * indicates significant at 5% level.