saving. The strength of this precautionary saving effect is proportional to the square of risk aversion, y2.

Given the historical average short-term real interest rate of 1.8%, the historical average consumption growth rate of 1.8%, and the historical averagestandard deviation of consumption growth of 3.3% shown in Table 8.1, а у of 19 implies a discount factor 5 of 1.12; this is greater than one, corresponding to a negative rate of time preference. Weil (1989) calls this the riskfree rale puzzle. Intuitively, the puzzle is that if investors arc extremely risk-averse (y is large), then with power utility they must also be extremely unwilling to substitute intcrtcmporally (i/r is small). Given positive average consumption growth, a low riskless interest rate and a positive rate of lime preference, such investors would have a strong desire to borrow from ihe future. A low riskless interest rate is possible in equilibrium only if investors have a negative rale of time preference that reduces their desire to borrow.

Of course, these calculations depend on the exact moments given in Table 8.1. In some data sets an even larger coefficient of relative risk aversion is needed to fit the equity premium: Kandel and Stambaugh (1991), for example, consider a risk-aversion coefficient of 29. With risk aversion ibis large, the precautionary savings term -y2a2/2 in equation (8.2.8) reduces the equilibrium riskfree rate and so Kandel and Stambaugh do not need a negative rate of time preference to lit the riskfree rate. A visual impression t)f this effect is given in Figure 8.3, which shows the mean stochastic discount factor first decreasing and then increasing as у increases with a fixed \S. Since the riskless interest rate is the reciprocal of the mean stochastic jdiscount factor, this implies that the riskless interest rale first increases and thcn decreases with y. The behavior of the riskless interest rate is always a .problem for models with high y, however, as the interest rale is extremely sensitive to the parameters g and a2 and reasonable values of the interest rate are achieved only as a knife-edge case when the effects of g and a2 almost exactly offset each other.

Is the Equity Premium Puzzle a Robust Phenomenon?

\nother response to the equity premium puzzle is to argue that it is an irtcfact of the particular data set on US slock returns. While we have not eported standard errors for risk-aversion estimates, careful empirical research by Cecchctti, I .am, and Mark (1994), Kochcrlakola (199(i), and others shows that the data can reject risk-aversion coefficients lower than about JO using standard statistical methods. However, the validity of these tests depends on the characteristics of the data set in which they arc used. Rielz (1988) has argued that there may be a peso problem in these data. A peso problem arises when there is a small positive probability of an important event, and investors lake this probability into account when selling market prices. If the event docs not occur in a particular sample period, investors

may appear to be irrational in the sample. While it may seem implausible that this could be an important problem in 100 years of data, Rietz (1988) argues that an economic catastrophe that destroys almost all stock-market value can be extremely unlikely and yel have a major depressing elfect on slock prices.

A related point has been made by Brown, Goetzniann, and Ross (1995). These authors argue thai financial economists concentrate on die US stock market precisely because it has survived and grown to become the worlds largest market. In some other markets, such as those of Russia, investors have had all their wealth expropriated during the last 100 years and so there is no continuous record of market prices; in others, such as the Argentine market, returns have been so poor that today these markets are regarded as comparatively less important emerging markets. If this survivorship effect is important, estimates of average US stock returns are biased upwards.

Although these points have some validity, they are unlikely to be the whole explanation for the equity premium puzzle. The difficulty with the Riclz (1988) argument is lhat it requires not only an economic catastrophe, but one which affects slock market investors more seriously than investors in short-term debt instruments. The Russian example suggests lhat a catastrophe causes very low returns on debt as well as on equity, in which case the peso problem affects estimates of the average levels of returns but not estimates of the equity premium. Also, there seems to be a surprisingly large equity premium not only in the last 100 years of US data, but also in US data from earlier in die 19lh Century as shown by Siegel (1994) and in data from other countries as shown by Campbell (1990b).

Time-Variation in Expected Asset Returns and Consumption Growth Equation (8.2.5) gives a relation between rational expectations of asset returns and rational expectations of consumption growth. It implies lhat expected asset returns are perfectly correlated with expected consumption growth, but the standard deviation of expected asset returns is у times as large as the standard deviation of expected consumption growth. Equiva-lently, the standard deviation of expected consumption growth hip = 1/y limes as large as the standard deviation of expected asset returns.

This suggests an alternative way to estimate у or ф. I lansen and Singleton (1983), followed by Hall (1988) and others, have proposed instrumental variables (IV) regression as a way to approach the problem. If wc define an error term ,.,+ , = rll+1 - E,[r,.,+ ] - y(Ao+i - E,l Д<м 11). then we can rewrite equation (8.2.5) as a regression equation,

>li+\ = lit + уД,+ + (8.2.9)

In general the error term ; ,+ ! will be correlated with realized consumption growth so OI.S is not an appropriate estimation method. However );,.,f. is

Table 8.2. h,M,mm;,l l variables .egressions for returns and consumption growth.

I .og u-al t oiisttnipitim gi imth i .lies .mil .issrl i rliи us .ire liicaslnril in alillti.il PS dala. I KM9 In 100 I. Ilie 4 iiliiiuus liiatU-il l-ii si-stage u-giessions renin I the Л- statistics autt juint significance levels til Он- explanatory v.u ialtli-s in I rgl cssious ill red It us anil I i.iisuillpliiill growth on lite instruments. Ilie miliums lu-.uleil ) anil t/ report Iwo-slagc least squares insli umcillal-vatial.les (IV) estimates ol die parameters у anil i/i in regressions (H.2.9) anil (H.2.10) respec-livelv. I lie ciilimiiis l.eaileil Test (N.2.9) anil Тем (К.2.10) герои lite II- statistics and juint significance levels nl the explanatory vai tables in rcgrevsions of IV regression residuals (H.J !l) anil (H.2.10) i.n the instruments. Tin- instruments inclutle either i.ne lag (in rows marked I), i.t пне and two lags (in mux market! 1 and 2) of the real commercial paper rale, the teal rousiuupliun growth rale, anil the lug dividend-price ratio.

uncoiiclalcd willi any variables in ibe information sel al lime /. 1 lence any lagged variables correlated with asset relurns can be used as inslrumcnts in an l\ regression to estimate die coefficient of relative risk aversion y.

Table 8.2 illustrates two-stage least squares estimation ol (N.2.9). In this table the asset relurns are die real commercial paper rale and real slock return from fable K. I, and coiistmiplion growlh is the annual growth rale of real nondurables and services consumption. The instruments are either one lag, or one and iwo lags, of die real euiuuicrciitl paper rale, the real consumption growth rale, and the log dividend-price ratio.

for each asset and sel of instruments, the table first reports the Л- slalislics and significance levels for fust-stage regressions of the asset return and ronsuinplion growlh rale onto the instruments. The table then shows the IV esliuiale of у with its standard error, and-in the column headed

in postwar quarterly data there is stronger evidence of predictable variation in consumpj lion growih. Campbell and Mankiw (1990) show that this variation is associated with predictable, income growth.

Test (8.2.9) -the R2 statistic for a regression of the residual on the in- struments together with the associated significance level of a test of the over-identifying restrictions of the model. This test is discussed at the end of Section A.l of the Appendix.

Table 8.2 shows strong evidence that the real commercial paper rate is forecastable, and weaker evidence that the real stock return is forecastable. There is very little evidence that consumption growth is forecastable.9 The IV estimates of у are negative rather than positive as implied by the underlying theory, but they are not significantly different from zero. The overiden-tifying restrictions of the model are strongly rejected when the commercial paper rate is used as the asset.

One problem with IV estimation of (8.2.9) is that the instruments are only very weakly correlated with the regressor because consumption growth is hard to forecast in this data set. Nelson and Startz (1990) have shown that in this situation asymptotic theory can be a poor guide to inference in finite samples; the asymptotic standard error of the coefficient tends to be too small and the overidentifying restrictions of the model may be rejected even when it is true. To circumvent this problem, one can reverse the regression (8.2.9) and estimate

Дс+, = r,- + tK,+i+?,.,+ (8.2.10)

If the orthogonality conditions hold, then as we have already discussed the estimate of \p in (8.2.10) will asymptotically be the reciprocal of the estimate of у in (8.2.9). In a finite sample, however, if у is large and \p is small then IV estimates of (8.2.10) will be better behaved than IV estimates of (8.2.9).

In Table 8.2 \b is estimated to be negative, like y, but is small and insignificantly different from zero. The overidentifying restrictions of the model ( Test (8.2.10) ) are not rejected when only 1 lag of the instruments is used, and they are rejected at the 10% level when 2 lags of the instruments are used. Table 8.2 also shows that the residual from the IV regression is only marginally less forecastable than consumption growth itself. These results are not particularly encouraging for the consumption model, but equally they do not provide strong evidence against the view that investors have power utility with a very high у (which would explain the equity premium puzzle) and a correspondingly small \U (which would explain the unpredictability of consumption growth in the face of predictable asset returns).

8.2.2 Power Utility and Generalized Method of Moments

So far wc have worked with a restrictive loglinear specification and have discussed cross-sectional and time-series aspects of the data separately. The Generalized Method of Moments (GMM) of Hansen (1982), applied to the consumption CAPM by Hansen and Singleton (1982), allows us to estimate and test the power utility model without making distributional assumptions and without ignoring either dimension of the data. Section A.2 of the Appendix summarizes the GMM approach, and explains its relation to linear instrumental variables.

When GMM is used to estimate the consumption CAPM with power utility, using the same asset returns and instruments as in Table 8.2 and assuming white noise errors, the overidentifying restrictions of the model are strongly , rejected whenever stocks and commercial paper are included together in the system. The weak evidence against the model in Table 8.2 becomes much stronger. This occurs because there is predictable variation in excess returns on stocks over commercial paper.1 Such predictable variation is ruled out j by the loglinear homoskedastic model (8.2.5) but could in principle be ex-j plained by a heteroskedastic model in which conditional covariances of asset returns with consumption arc correlated with the forecasting variables. The GMM system allows for this possibility, without linearizing the model or imposing distributional assumptions, so the GMM rejection is powerful evidence against the standard consumption CAPM with power utility.

Faced with this evidence, economists have explored two main directions for research. A first possibility is that market frictions invalidate the standard consumption CAPM. The measured returns used to test the model may not actually be available to investors, who may face transactions costs and constraints on their ability to borrow or shortsell assets. Market frictions may also make aggregate consumption an inadequate proxy Tor the consumption of stock market investors. A second possibility is lhat investors have more complicated preferences than the simple power specification. We explore each of these possibilities in the next two sections.

8.3 Market Frictions

We now consider various market frictions that may be relevant for asscl pricing. If investors face transactions costs or limits on their ability to borrow

Recall 111 al in (Chapter 7 we presented evidence thai the dividend-price ratio forecasts excess stock returns. The dividend-price ratio is one of the instruments used here.

One can understand this by considering a heieroskedastic version of the lineari/ed model (8.2.5) in which the variances have time subscripts. Campbell (1У87) and Harvey (I.18.)) apply GMM to models of ibis type which impose the restriction that asset retlims conditional means are linear functions of their conditional second moments. We discuss this work further in Chapter 12.

or sell assets short, then they may have only a limited ability to exploit the empirical patterns in returns. In Section 8.3.1 we show how this can alter the basic Hansen and Jagannathan (1991) analysis of the volatility of the stochastic discount factor.

The same sorts of frictions may make aggregate consumption an inadequate proxy for the consumption of stock market investors. In Section 8.3.2 we discuss some of the evidence on this point, and then follow Campbell (1993a, 1996) in developing a representative-agent asset pricing theory in which the consumption of live representative investor need not be observed. The theory uses a generalization of power utility, due to F.pslcin and /in (1989, 1991) and Weil (1989), that.breaks the link between risk aversion and intertemporal substitution. The resulting model, in the spirit of Mer-ton (1973a), is a multifactor model with restrictions on the risk prices of the factors; hence it can be tested using the econometric methods discussed in Chapter О.

8.3.1 Market Erictions and Ilansen-Jagannalhan Bounds

The volatility bounds of Hansen ami Jagannathan (1991), discussed in Section 8.1.1, assume that investors can freely trade in all assets without incurring transactions costs and without limitations on borrowing or short sales. These assumptions arc obviously rather extreme, but they have been relaxed by lie and Modest (1995) and I.utlmer (1994). To understand die approach, note that if asset < cannot be sold short, then the standard equality restriction F.[( I + /\ )М,] = I must be replaced by an inequality restriction

li[(l -t-/f/,)M,l < I. (8.3.1)

If the inequality is strict, then an investor would like lo sell the asset but is prevented from doing so by the shortsalcs constraint. Instead, the investor holds a zero position in the asset.

Shortsalcs constraints may apply to some assets but not others; if they apply to all assets, then they can be interpreted as a solvency constraint, in that they ensure that an investor cannot make investments today that deliver negative wealth tomorrow. Assuming limited liability for all risky assets, so that the minimum value ofeach asset tomorrow is zero, a portfolio with nonneg-alive weights in every asset also has a minimum value of zero tomorrow.

Investors may also face borrowing constraints that limit their ability to sell assets lo finance consumption today. Such constraints deliver inequality restrictions of the form (8.3.1) for all raw asscl returns, but the standard equality constraint holds for excess returns since the investor is free to short one asset in order lo take a long position in another asset.

Shortsalcs constraints can also be used lo model proportional transactions costs of the type that might result from a bid-ask spread lhat does not