Prueba de las hipótesis especificas

We first discuss the results from the simplest model which only includes consumption growth. Moreover, we assume that the variance of the innovations and therefore the covariance between the innovations (i.e. between the excess return and consumption growth) are constant over time. Table 5.3 presents the parameter estimates; sandwich based standard errors are given in parentheses. Note that θ^c has been normalized to unity.

The persistence parameter in the estimated autoregressive equation for consump-tion growth is 0.48 which indicates that this series is not very persistent. In sharp

4The equations describing the first three specifications are shown in Appendix 5.A.3.

5.4 Results 111

contrast, the AR(1) parameter for relative risk aversion — conveniently labeled γ₁ to distinguish it from the series generated by the other specifications — is 0.97, close to upper bound of unity for non-explosive series. The bottom panel shows the correlation between the innovations in consumption growth and (1) the innovations in the excess return and (2) the innovations in relative risk aversion. The first is positive and hence indicates that these innovations are positively associated. This implies that when there is a positive shock to consumption growth it is likely that real excess returns are higher as well. The correlation between (the innovations in) consumption growth and relative risk aversion is negative which is in line with our a priori expectation: a positive sur-prise in consumption growth should be associated with a decline in risk aversion. A positive shock to consumption growth is considered to be good news and hence risk aversion should decline. Note that all parameters estimates are statistically significant from zero at the 5% significance level.

Our main variable of interest, the time-varying estimate of log relative risk aver-sion, is shown in Figure 5.1. As indicated by the parameter estimate in Table 5.3, this series is highly persistent. Although the value of 0.97 is not significantly differ-ent from the value of unity which would make the variable of an explosive nature, the figure clearly indicates that over the period from January 1960 until June 2011 it exhibits mean reversion. The shaded regions indicate the NBER recession dates and are the main tool to judge our model. Our sample period contains the following eight recession periods:⁵

I. April 1960 until February 1961 II. December 1969 until November 1970 III. November 1973 until March 1975

IV. January 1980 until July 1980 V. July 1981 until November 1982

5We refer to Labonte and Makinen (2002) and Bordo and Landon-Lane (2010) for a more detailed exposition of these eight periods including likely causes, severity and policy responses.

(c) and the log coefficient of relative risk aversion (labeled γ1). Estimation results refer to equations (5.13)-(5.16), see Appendix 5.A.3. This model only features consumption growth and keeps the inno-vation variances fixed. As a result the covariance between excess returns and consumption growth is also constant. In addition, θ^cis normalized to unity. Correlations (denoted by ρ) are calculated between the innovations of the series in question. Every column pertains to a single series and shows its relevant parameter estimates, with superscript i referring to these series. The sample period runs from January 1960 until June 2011; sandwich based standard errors in parentheses.

exr c γ₁

θⁱ - 1.00

-- (0.36)

-¯i - 0.02 1.32

- (5.3E-3) (0.54)

φⁱ - 0.48 0.97

- (0.11) (0.22)

(σⁱ)² 1.9E-3 1.6E-5 489.45 (4.9E-4) (5.3E-6) (236.76)

ρ_(exr,i) - 0.76

-- (0.21)

-ρ_(i,γ1) - -0.49

-- (0.14)

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VI. July 1990 until March 1991 VII. March 2001 until November 2001 VIII. December 2007 until June 2009

Before we discuss the path of log relative risk aversion in more detail, it is impor-tant to realize the following. As we keep the covariance between the innovations in the real excess return and consumption growth (k^c) constant, one might be tempted to conclude that all variation in γ₁comes from variation in the variance of the real excess return. This is not the case. The reason is the presence of ^exr and ^γ which are both unobserved. In other words, not all variation in the real excess return is reflected in the path of risk aversion. Moreover, we present the smoothed series, i.e. we also apply the Kalman smoothing technique. This implies that all information is used to estimate the most likely path of risk aversion: historical, current but also future realizations of excess returns and consumption growth. This is standard practice when using the state space technique, but one should be aware of its influence.

Although risk aversion rises during the first recession period, it keeps increasing for an extended period of time after the downturn officially ended. The leveling of log risk aversion between 3 and 4 during 1962 until 1964 is caused by a spell of mostly negative real excess returns in this period, but the innovations in γ1 also absorbed part of the variation in the dependent variable. During periods II and III risk aversion rises, but while it keeps on rising after the first recession period in the 1970s, it approxi-mately peaks at the end of the trough for the recession related to the oil crises. The declining path of γ1during recession period IV (also known as the first leg of the ‘Dou-ble Dip’ recession) is due to the predominantly positive returns in this period. During the second leg of the ‘Double Dip’ recession (period V), risk aversion rises and again peaks near the end of the trough.

Risk aversion rises in late 1987 due to the stock market crash but the effect of this event is rather limited. This is partly due to negative innovations in the excess return and some positive innovations in γ1. A likely reason for the somewhat muted impact

on risk aversion is that this crash was unexpected and although its impact was huge, its impact on equity returns was short-lived: the excess return of November is negative but returns are positive for the period thereafter. The path slopes downward in period VI but here negative and positive returns are more balanced so the decline is less steep than during the fourth recession period. During the longest expansion in the history of the U.S. (i.e. in between period VI and VII), γ1 is predominantly rising, especially during the late 1990s. This can be explained by the events that took place during the summer of 1998, i.e. the Russian ruble crises and the subsequent default of Long-Term Capital Management which contributed to a spell of negative excess returns. The fact that these events span a longer time period and have such an impact only supports our view concerning the relatively small response of risk aversion to the 1987 stock market crash.

The decade of the 1990s also exemplifies the impact of the Kalman smoother. As mentioned, through the use of the smoother future realizations are used to get the best estimate of risk aversion at time t. This implies that the level of γ just after the 1990 recession is to some degree influences by the events in 1998 and the recession starting in the first quarter of 2001. In other words, in 1990 it is already known that risk aversion should increase later in the decade as a result of these two events. In combination with the high level of persistence, this produces the upward sloping path of risk aversion during this decade.⁶

Surprisingly, risk aversion peaks just before 2001 recession and is already declining during the recession period. Returns are mostly negative in this timespan so this can only be the result of the unobserved innovations and this is indeed the case. Note that this period also comprises the September 2001 terrorist attacks, for which one would expect to observe rising levels of risk aversion. However, in light of the discussion concerning the 1987 stock market crash, this results is less surprising. The path slopes

6As our interest is to show the time variation in risk aversion, applying the smoother is validated.

If one is interested in forecasting future levels of risk aversion one should of course use the output generated by the Kalman filter.

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upward again in the run-up the ‘Great Recession’, rises during this period and peaks in November 2009, five months after the recession officially ended. The decline in γ1 at the end of the sample period is the result of the good performance of the stock market after the most recent recession and the very low yields on government debt which, combined, produce a large excess return.

With respect to the level of relative risk aversion, it has its minimum at the late 1960s (0.67) and in the period after the 1990s recession (0.74). Based on this specifi-cation, the final recession of the sample period has been accompanied by the highest level of risk aversion (6.81), but the 1975 (6.27) and 2001 (6.23) recessions are a close second and third. The average log relative risk aversion is equal to 3.51 such that the coefficient of relative risk aversion is approximately 98. This value corresponds well to the literature on the equity premium puzzle in which such high values are not uncommon.

Figure 5.1: Smoothed log relative risk aversion coefficient generated by the first specification (γ1) over the sample period running from January 1960 until June 2011. Shaded bars indicate official recession dates. Risk aversion is estimated from equations (5.13)-(5.16), see Appendix 5.A.3. This specification only involves consumption growth and innovations in this series and the excess return are assumed to have a constant variance. As a result the covariance between shocks in the excess return and consump-tion growth innovaconsump-tions is also constant.