I. Introduction

This paper introduces an international version of the consumption asset pricing model that accounts for the low as well as economically implausible risk aversion estimates obtained throughout the finance literature for fixed income assets and closed economy measures of consumption growth. The standard model of intertemporal utility maximization implies stochastic asset pricing restrictions that can be written as:

where g is one plus the rate of consumption growth between any two periods, and r is the real return on any asset between these dates. The utility function used in (1) is given by constant relative risk aversion, where α is the coefficient of relative risk aversion, and β is the subjective time preference discount factor. The risk aversion coefficient measures an investor’s attitude towards risk.

Neoclassical models with complete markets and without distortions usually imply that (1) should be satisfied for a representative closed economy measure of consumption growth and for any measure of asset returns. Hansen and Singleton (1983), Harvey (1987), and Ferson and Constantinides (1989), among others, have found that models with closed economy measures of consumption fit best with low risk aversion (α). Estimates derived using Treasury bills as asset returns usually suggest that α is between 0.1 and 0.2 and, in some cases, produce estimates of α that are not consistent with risk averse behavior. In particular, Hansen and Singleton (1983) find that α is between 0.1 and 0.16 for Treasury bill restrictions, while Ferson and Constantinides (1989) find that the range of α estimates is between -2.2 and 0.2. These results are robust to differences over the frequency of the data and time periods.

This paper provides a statistical and an economic interpretation of these anomalies. For a statistical interpretation, we use a specification errors argument to demonstrate that there is a serious downward bias in parameter estimates derived from the consumption asset pricing model when there is measurement error in consumption and in real rates of return. The analysis suggests that an international measure of consumption growth, and not a national measure, should diminish the effect of specification errors in any given economy. Specifically, we show that international (average) consumption growth across economies reduces the bias introduced by economy specific measurement errors. Thus, the international asset pricing model produces higher risk aversion (α) estimates than national asset pricing models. For an economic interpretation, it is shown that the international model would estimate more precisely the degree of risk aversion only if there is indeed a common consumption component across economies.

To demonstrate the potential importance of measurement errors in empirical estimates of the parameters of (1), we present Monte Carlo simulations that allow for different distributions of random measurement errors in consumption growth and real returns. Numerical simulations are constructed in Section 2 as follows. We add a random disturbance to actual real rates of return and construct a time series of consumption growth so that (1) holds exactly at each point in time for a given initial choice of α and β. To introduce measurement errors, consumption growth and real returns series are then constructed by adding random errors to the measure of consumption growth and real returns for which (1) holds exactly. We perform these simulations for various distributions and correlations of the measurement errors in consumption growth and real returns. Estimates of α and β derived using the simulated series with measurement error are able to replicate the low and economically implausible risk aversion coefficient estimates obtained throughout the literature.

Section 3 demonstrates that asset pricing restrictions may be estimated more accurately if we do not restrict ourselves to a single measure of consumption growth and of rates of return. We examine several measures of consumption growth that may reduce the bias introduced by measurement errors. An important feature of this approach is that these measures are differentially subject to specification errors. Two arguments that characterize a priori the properties and the extent of specification error in various measures of consumption are presented.

First, a signal extraction problem is used to show that a measure of international (average) consumption growth across economies reduces the noise introduced by idiosyncratic measurement errors in national measures and, hence, may reveal a common consumption growth signal across countries that is concealed by economy-specific errors. We show that by reducing the effect of measurement errors, the risk aversion estimates derived from an international measure of consumption growth are superior to those of national measures of consumption growth.

Second, it is shown that there are important differences in the way in which the various measures of consumption growth are affected by specification errors owing to difficulties in measuring accurately changes in the quality of nontraded goods. Thus, we break down national and international measures of consumption growth into their traded and nontraded consumption components and demonstrate that the nontraded consumption growth measures are subject to more significant measurement error in quality changes than the traded measures. In addition to alternative measures of consumption growth, we examine alternative measures of real returns that may also be subject to specification error.

Risk aversion estimates can then be derived from tests of asset pricing restrictions constructed for these various measures. Among all the measures examined, international traded consumption growth is expected to be the least subject to measurement error and thus provide the most reliable risk aversion estimates, while the international measure of nontraded consumption growth is expected to yield the least reliable risk aversion estimates. The reason is that an international nontraded measure is unlikely to contain a common signal of consumption growth across countries.

Section 4 shows that there are important differences in the time series properties of traded and nontraded consumption growth across economies that support the distinction between these measures in characterizing risk aversion. This chapter presents some empirical regularities that describe the behavior of the various measures of consumption growth and of real returns. An important empirical regularity is that traded and international consumption growth exhibit a higher degree of autocorrelation than either nontraded or national consumption growth. Yet, traded consumption growth exhibits lower average growth than total national consumption across economies. Traded and international consumption growth lagged one through four periods are indeed good predictors of future consumption growth, while lags of either nontraded or national consumption growth poorly foresee next period consumption growth.

Section 5 presents a brief discussion of the method of moments estimation and the econometric methodology used to estimate the discount factor (β) and the risk aversion coefficient (α). The method of moments estimation is used in two distinct ways to obtain parameters estimates. First, β and α were estimated using generalized method of moments estimation conditional on information available in the current period (Hansen and Singleton, 1982). Second, for an initial choice of α, we estimate β so that each asset pricing restriction holds as estimated by its sample mean. The international average of the estimated βs was then used in the [restricted] conditional generalized method of moments estimation of the risk aversion coefficient. An important feature of this procedure is that variations in relative risk aversion (a) estimates across countries do not hinge on differences in the choice of the discount factor (β).

The empirical method is then used to estimate the risk aversion coefficient and the subjective discount factor in the class of national and international asset pricing models implied by equation (1) for various measures of consumption growth and asset returns across eight industrialized economies. Section 6 presents tests of the models implied by (1) for international and national measures of traded and of nontraded consumption growth. Tests of asset pricing conditions with NYSE returns are also presented. To evaluate the effect of financial deregulation on asset trading, section 7 derives an international point estimate of the degree of risk aversion in the 1970s and in the 1980s.

The results of these tests suggest that relative risk aversion estimates derived from international and traded consumption growth asset pricing models are significantly higher than those obtained for nationwide and nontraded consumption growth. In particular, the international asset pricing model is consistent with a risk aversion coefficient close to unity. More importantly, the international model implies a common degree of risk aversion across economies. Section 8 presents concluding remarks.

II. Asset Prices and the Specification of Random Disturbances

Rewriting the Euler equation (1), we may define the quantity h as the “utility adjusted return” on a given asset:

We can think of h as the unconditional expectation of the utility adjusted profit from investing one dollar in a given asset. In equilibrium we expect this quantitity to be zero, otherwise an individual with a marginal rate of substitution given by β · g^-α could gain utility by giving up a unit of consumption and investing it, for example, in this asset. An estimator of h is given by the sample average of β · g^-α’(1+r)-1:

To obtain empirical estimates of the risk aversion coefficient (α) and the discount factor (β) using (3), the standard procedure is to choose α and β to minimize h′_T · w_T · h_T, where w_T is a nonlinear instrumental variables weighting matrix. This estimation strategy avoids the theoretical requirement of an explicit representation of the stochastic equilibrium, yet permits estimation and identification of the asset pricing parameters, α and β. Consequently, the estimation can be performed when only a subset of the economic environment is specified a priori. Moreover, Hansen and Singleton (1982) show that the estimators derived from this procedure are consistent and have limiting normal distributions.

Hansen and Jagannathan (1991) suggest that a major drawback of empirical estimates of asset pricing equations using this standard procedure is that they abstract from the presence of measurement and approximation errors. However, the procedure can be extended to take account of measurement errors in consumption and asset returns and specification errors as follows. If we consider the utility adjusted returns as including an unobserved specification error (s) and a forecast error (υ), we can write a modified version of (2) as:

Comparing equation (4) with (2), it is easy enough to see that h_T can be regarded as an estimate of the mean specification error, h=E(s).

We next consider a different kind of specification error where the error (κ) is in the return:

We take unconditional expectations of (5) and assuming that κ and g^-α are uncorrelated, we rewrite the mean specification error (h) in (2) as, β·E(g^-α)·E(κ).

Another interesting case is to consider measurement errors in the marginal rate of substitution together with an error in the return. For the time separable, constant relative risk aversion specification, the corresponding version of (2) that incorporates this type of measurement error is given by:

where ω is an unobserved specification error in consumption growth. ^1/ The nonlinear nature of condition (6) poses severe difficulties in deriving estimates of the bias introduced by ω and κ in the estimation of the relative risk aversion coefficient and the discount factor. We circumvent the analytical difficulty posed by the nonlinear nature of the model by assuming that the error term (υ) is log-normally distributed.

Returning first to the case of no measurement error, Hansen and Singleton (1982) and Zellner (1987) show that if υ is a log-normal error, then (1) can be rewritten as:

where ϵ is a random error with ϵ~N(0, σ²_ϵ)) in the interval -∞<ϵ<∞. We take logs of (7) to get:

We apply the law of iterated expectations to condition (8), and then subtract the resulting restriction from (8) to get:

where ln(g)* = ln(g)-E(ln(g)) and ln(1+r)* = ln(1+r)-E(ln(1+r)). For any z that is an element of the information set at period t, we multiply (9) by z*=(z-E(z)) to get:

We sum (10) across a large sample of time periods, divide by the total number of observations (T), and set average ϵ · z* equal to zero to obtain the unbiased instrumental variables estimator of a:

Next, consider the effect of specification errors in consumption growth and in the rate of return on estimates of α from a linear version of (6). Again, we assume that υ is log-normally distributed and that measurement errors affect consumption growth and returns in a multiplicative fashion, (g)(1+ω) and (1+r)/(1+κ). We substitute this specification of the errors ω and κ in equation (8):

In the same fashion we derived (11), the instrumental variables estimator of α in (12) is given by:

where α^Δ is the risk aversion coefficient estimate implied by (11). Equation (13) suggests that consumption growth and real return errors may introduce a downward bias in the estimation of the relative risk aversion coefficient (a) if consumption growth errors (ω) and real returns shocks (κ) exhibit a positive correlation with z. Moreover, equation (13) implies that the smaller α^Δ is, the more sensitive α^b is to specification errors in consumption growth and in the rate of return. Thus, measurement errors may introduce a serious downward bias in risk aversion estimates. ^2/

These theoretical results for the special case of log-normally distributed random errors suggest that measurement errors may potentially account for the low and often implausible risk aversion estimates derived from fitting the asset pricing model. The following numerical simulations make explicit the extent to which specification errors may contribute to explain the implausible risk aversion (α) estimates obtained throughout the literature, as well as illustrate the sensitivity of relative risk aversion estimates to different distributions of unexpected shocks (ω,κ).

Let g^- be such that Euler equations (1) are satisfied for some initial parameters values and real returns, g^-=[β · (1+r^-)]^1/α, where r^-=r_a + ϵ and r_a is the actual Eurocurrency real return for the United States and ϵ is an independent and identically distributed random error. Parameter values were set at α=-0.9347 and β=0.9976, and correspond to estimates obtained in Section 6. By construction these series for consumption growth and interest rates satisfy (1) exactly for the set values of α and β. To simulate the effects of measurement errors, we next construct consumption growth and real returns series by adding independent and indentically distributed random errors to g^- and r^-, as follows. Let ln(g)-ln(g^-)+ω and r=r^-+κ, where ω and κ are the stochastic shocks previously described.

Table 1 presents Monte Carlo results of generalized method of moments estimations of asset pricing restrictions (1) constructed under different distributions of the unexpected consumption growth shock (ω) and the unexpected realization of cpi inflation (κ), as well as distinct correlations between ω and κ. Each ω and κ was drawn from the random number generator in Gauss. Random disturbances are characterized by their means and standard deviations. Each experiment was performed 100 times. The vector of instruments (z) consists of a constant and consumption growth lagged two and three periods. To analyze the sensitivity of parameters estimates to the choice of the instrumental variables, an additional test is performed with a constant and consumption growth lagged two through seven periods as the vector of instruments. ^3/ Distributions of parameters estimates and standard errors of parameters estimates in table 1 are described by their means and standard deviations.

Table 1.

Monte Carlo Simulations

Initial parameters: α=0.9347 and β=0.9976; ϵ∽(0,0.0100). Distributions were derived from Monte Carlo simulations and are characterized in terms of means (μ) and standard deviations (σ). Each experiment was performed 100 times.

^1/Instruments used in this estimation: constant and consumption growth lagged two through seven periods.

Table 1.

Monte Carlo Simulations

Shocks Distribution				corr(ω,κ)	Parameters Estimates			Std. Errors of Par Estimates
μω	σω	μκ	σκ		β		-α	Std. error β	Std. error α
(0.000,0.014)		(0.000,0.008)		0.010	(0.995,0.008)		(-0.743,1.677)	(0.008,0.018)	(1.484,4.017)
(0.000,0.017)		(0.000,0.015)		-0.300	(0.994,0.007)		(-0.672,1.525)	(0.018,0.019)	(2.958,3.673)
(0.000,0.014)		(0.000,0.018)		-0.498	(0.994,0.017)		(-0.539,2.413)	(0.021,0.028)	(3.051,5.272)
(0.000,0.017)		(0.000,0.019)		-0.576	(0.992,0.010)		(-0.395,1.767)	(0.012,0.016)	(2.304,3.847)
(0.000,0.019)		(0.000,0.017)		-0.810	(0.989,0.007)		(+0.166,1.099)	(0.016,0.051)	(2.342,6.785)
(0.000,0.019)		(0.000,0.017)		-0.820	(0.990,0.003)	1/	(+0.123,0.408)	(0.003,0.001)	(0.351,0.150)

Initial parameters: α=0.9347 and β=0.9976; ϵ∽(0,0.0100). Distributions were derived from Monte Carlo simulations and are characterized in terms of means (μ) and standard deviations (σ). Each experiment was performed 100 times.

^1/Instruments used in this estimation: constant and consumption growth lagged two through seven periods.

View Table

Table 1.

Monte Carlo Simulations

Shocks Distribution				corr(ω,κ)	Parameters Estimates			Std. Errors of Par Estimates
μω	σω	μκ	σκ		β		-α	Std. error β	Std. error α
(0.000,0.014)		(0.000,0.008)		0.010	(0.995,0.008)		(-0.743,1.677)	(0.008,0.018)	(1.484,4.017)
(0.000,0.017)		(0.000,0.015)		-0.300	(0.994,0.007)		(-0.672,1.525)	(0.018,0.019)	(2.958,3.673)
(0.000,0.014)		(0.000,0.018)		-0.498	(0.994,0.017)		(-0.539,2.413)	(0.021,0.028)	(3.051,5.272)
(0.000,0.017)		(0.000,0.019)		-0.576	(0.992,0.010)		(-0.395,1.767)	(0.012,0.016)	(2.304,3.847)
(0.000,0.019)		(0.000,0.017)		-0.810	(0.989,0.007)		(+0.166,1.099)	(0.016,0.051)	(2.342,6.785)
(0.000,0.019)		(0.000,0.017)		-0.820	(0.990,0.003)	1/	(+0.123,0.408)	(0.003,0.001)	(0.351,0.150)

Initial parameters: α=0.9347 and β=0.9976; ϵ∽(0,0.0100). Distributions were derived from Monte Carlo simulations and are characterized in terms of means (μ) and standard deviations (σ). Each experiment was performed 100 times.

^1/Instruments used in this estimation: constant and consumption growth lagged two through seven periods.

The estimates of α and β based on these simulations of measurement and specification error suggest the following conclusions. First, the mean values of the discount factor (β) estimates are below unity and, hence, are economically plausible. Second, the average values of β estimates range from 0.989 to 0.995 and thus are generally close to the true (set) values of β. The standard deviations of standard errors of β estimates range from 0.001 to 0.051. Third, the average values of risk aversion (α) estimates are considerably below the true (set) value and in several simulations not economically plausible. In particular, the average values of α range from -0.17 to 0.74. Fourth, the standard deviations of risk aversion (α) estimates and computed standard errors of α estimates are considerable. For instance, the standard deviations of standard errors of α estimates range from 0.15 to 6.79.

Fifth, risk aversion estimates appear to decline as the standard deviations of measurement errors in consumption growth and real returns increase. Sixth, if additional lags of consumption growth are included in the vector of instruments, the means of parameters estimates do not change significantly, although their standard deviations and computed standard errors were found to decline notably. Finally, as a general conclusion, these simulations indicate that measurement or specification errors in consumption growth or real returns provide a relevant explanation of the implausible estimates of risk aversion in empirical studies based on the asset pricing restriction (1). In particular, numerical simulations in which unexpected consumption growth and real returns fluctuations exhibit a non-zero degree of correlation are able to replicate the low and economically implausible risk aversion estimates obtained by Harvey (1987) and Ferson and Constantinides (1989), among others.

The linear approximation to the risk aversion (a) estimate in (13) provides some intuition of why α estimates are considerably more affected by specification errors than β estimates. Indeed, (13) suggests that a estimates are fairly sensitive to the comovement between specification errors, while an economically plausible estimate of β, given by the sample average of 1/E(g^-α · (1+r)), is consistent with a wide range of risk aversion estimates. For instance, for a quarterly average consumption growth and real return of ¾ of 1 percent and 1 percent, respectively, the estimate of β is 0.997, if the estimate of α is 0.9. However, if estimated α is only 0.45, the estimate of β is 0.994, while if the estimate of α is -0.2 then the estimate of β is 0.989. Note that in this last example a is not economically plausible, yet β is economically sensible and fairly close to 0.99. This indicates that discount factor (β) estimates are not very sensitive to relatively large variations in the extent of estimated risk aversion (α). In addition, low or economically implausible risk aversion estimates are associated with discount factor estimates close to 0.99.

III. Alternative Measures of Consumption Growth and Asset Returns

This section demonstrates that in the presence of measurement error, the degree of risk aversion may be estimated more accurately if asset pricing tests are not restricted to a single measure of consumption growth and asset returns. We focus on the problem of measurement error in consumption growth and examine several alternative measures that may be differentially subject to this problem. Two arguments that characterize a priori the likely properties and the extent of specification error in the various measures of consumption growth are presented. Risk aversion estimates can then be derived from tests of asset pricing restrictions constructed for these alternative measures. In particular, measures of growth that are subject to sizable errors would yield implausible and low risk aversion estimates compared with those measures with little or no error. Discrepancies in the estimated degree of risk aversion are thus related to differences in specification error in consumption growth.

The measures of consumption growth that we examine are, first, national consumption growth measured not only for a single economy but for several countries. Second, we consider an international measure of average consumption growth, obtained from the individual national consumption series. Third, we break down each measure of national consumption into its traded and nontraded consumption components. Finally, international measures of consumption growth that are broken down into their traded and nontraded components are examined. In addition to alternative measures of consumption growth, various measures of real returns are evaluated. To examine the robustness of asset pricing results, the various measures of consumption growth are used to estimate asset pricing restrictions on eurocurrencies and Treasury bills. Some examples are also presented for portfolios of value-weighted and equally-weighted NYSE stocks.

It is shown that the problem of specification error is more severe for national than for international measures. To the extent that averaging across economies reduces the effect of idiosyncratic errors, international consumption growth measures should be subject to less measurement error and, hence, yield higher risk aversion estimates with lower computed standard errors than national measures. Moreover, nontraded measures are subject to greater error than traded measures due to difficulties in measuring quality changes in nontraded goods. Thus, the international average of traded consumption growth provides, in principle, the most reliable risk aversion and discount factor estimates. By contrast, the international average of nontraded consumption growth yields the least reliable parameters estimates since by definition it does not signal a common consumption component across countries.

Table 2 defines these alternative measures of consumption growth and real asset returns. Each measure of consumption in Table 2 is matched with the corresponding asset return measure in the evaluation of asset pricing restrictions. For instance, measures of traded consumption growth are paired with measures of real traded returns. In addition, we associate nondurables with traded commodities, and follow Kravis, Heston and Summers (1980) in treating services as nontraded goods. This provides a convenient way to break consumption into its traded and nontraded components since tests of asset pricing restrictions are generally performed for nondurables and services.

Table 2.

Definitions of Measures of Consumption Growth and Asset Returns

^1/The data are available on a quarterly and yearly basis for Belgium, Canada, France, Germany, Japan, Sweden, the United Kingdom, and the United States. The data on nominal nondurables and services consumption and the implicit deflator for total consumption are from the OECD Quarterly National Accounts. Population data are from the IFS tape. The sample period is 1970.1-1988.1. All data are seasonally adjusted (consumption data for Japan and nominal consumption for the United Kingdom are seasonally adjusted using the smooth command in RATS).

^2/For each international measure, the real consumption of each country was converted to U.S. dollars of 1985 by using the nominal exchange rate.

^3/Traded consumption data for Belgium and Germany were not available.

^4/Three sets of nominal interest rates are considered: eurocurrencies, treasury bills, and NYSE rates. Treasury bill rates are from the Financial Statistics Tape of the International Monetary Fund, except for Japan and France, where the data do not exist so the government bond yield was used. Eurocurrency rates are from the Harris Bank Tape, the recommended series were used. NYSE returns are from CRSP tapes.

Table 2.

Definitions of Measures of Consumption Growth and Asset Returns

Variable	Definition
gi	Per capita real consumption of traded plus nontraded consumption next period in country i relative to current per capita consumption. 1/
g	One plus international per capita consumption growth of traded plus nontraded goods, where international per capita consumption is defined as the sum across countries of real consumption divided by the total population of countries in the sample. 2/
gNi	Per capita real nontraded consumption next period in country i relative to current per capita consumption.
gN	One plus international per capita nontraded consumption growth.
gTi	Per capita real traded consumption in the following period relative to today’s consumption in country i. 3/
gT	One plus international per capita traded consumption growth.
g(-k)	The (k) period lag of the measure of consumption growth examined.
ri	The real return in country i over the period is defined as the nominal interest rate minus the rate of price inflation of the implicit price deflator for traded plus nontraded consumption. 4/
rNi	The real return in terms of nontraded goods in country i is defined as the nominal interest rate minus the rate of nontraded price inflation.
rTi	The real rate of interest in terms of traded commodities in country i is defined as the nominal interest rate minus the price inflation of traded goods.
ri(-k)	Real return under consideration lagged by (k) periods.

^1/The data are available on a quarterly and yearly basis for Belgium, Canada, France, Germany, Japan, Sweden, the United Kingdom, and the United States. The data on nominal nondurables and services consumption and the implicit deflator for total consumption are from the OECD Quarterly National Accounts. Population data are from the IFS tape. The sample period is 1970.1-1988.1. All data are seasonally adjusted (consumption data for Japan and nominal consumption for the United Kingdom are seasonally adjusted using the smooth command in RATS).

^2/For each international measure, the real consumption of each country was converted to U.S. dollars of 1985 by using the nominal exchange rate.

^3/Traded consumption data for Belgium and Germany were not available.

^4/Three sets of nominal interest rates are considered: eurocurrencies, treasury bills, and NYSE rates. Treasury bill rates are from the Financial Statistics Tape of the International Monetary Fund, except for Japan and France, where the data do not exist so the government bond yield was used. Eurocurrency rates are from the Harris Bank Tape, the recommended series were used. NYSE returns are from CRSP tapes.

View Table

Table 2.

Definitions of Measures of Consumption Growth and Asset Returns

Variable	Definition
gi	Per capita real consumption of traded plus nontraded consumption next period in country i relative to current per capita consumption. 1/
g	One plus international per capita consumption growth of traded plus nontraded goods, where international per capita consumption is defined as the sum across countries of real consumption divided by the total population of countries in the sample. 2/
gNi	Per capita real nontraded consumption next period in country i relative to current per capita consumption.
gN	One plus international per capita nontraded consumption growth.
gTi	Per capita real traded consumption in the following period relative to today’s consumption in country i. 3/
gT	One plus international per capita traded consumption growth.
g(-k)	The (k) period lag of the measure of consumption growth examined.
ri	The real return in country i over the period is defined as the nominal interest rate minus the rate of price inflation of the implicit price deflator for traded plus nontraded consumption. 4/
rNi	The real return in terms of nontraded goods in country i is defined as the nominal interest rate minus the rate of nontraded price inflation.
rTi	The real rate of interest in terms of traded commodities in country i is defined as the nominal interest rate minus the price inflation of traded goods.
ri(-k)	Real return under consideration lagged by (k) periods.

^1/The data are available on a quarterly and yearly basis for Belgium, Canada, France, Germany, Japan, Sweden, the United Kingdom, and the United States. The data on nominal nondurables and services consumption and the implicit deflator for total consumption are from the OECD Quarterly National Accounts. Population data are from the IFS tape. The sample period is 1970.1-1988.1. All data are seasonally adjusted (consumption data for Japan and nominal consumption for the United Kingdom are seasonally adjusted using the smooth command in RATS).

^2/For each international measure, the real consumption of each country was converted to U.S. dollars of 1985 by using the nominal exchange rate.

^3/Traded consumption data for Belgium and Germany were not available.

^4/Three sets of nominal interest rates are considered: eurocurrencies, treasury bills, and NYSE rates. Treasury bill rates are from the Financial Statistics Tape of the International Monetary Fund, except for Japan and France, where the data do not exist so the government bond yield was used. Eurocurrency rates are from the Harris Bank Tape, the recommended series were used. NYSE returns are from CRSP tapes.

2. Measurement errors in traded versus nontraded consumption

It is a common procedure to measure national consumption using traded and nontraded consumption. Measurement errors, however, are likely to be more important in nontraded (services) consumption than in traded (manufacturing) consumption. Denison (1989) suggests that there are significant difficulties in treating differences in the quality of nontraded consumption over time as different kinds of goods. These difficulties stem from a methodology, imposed by inadequate information, that cannot partition national product among services industries in a reliable fashion.

Difficulties in assessing changes in the quality of services consumption may indeed prove to be important in fitting asset pricing restrictions (1) across economies. Murphy and Shleifer (1991) introduce a model in which high income countries both produce and demand high quality goods, whereas low income economies both produce and demand low quality goods. It would be interesting to characterize the dynamics of economic growth that would predict, for example, increasing patterns of international trade across economies. We would expect that as income grows, economic agents will demand goods of higher quality, for example, better and time-saving services. ^4/ Indeed, the average rate of services consumption growth during the 1970s and 1980s was ¾ of 1 percent across six of the largest industrialized economies, whereas average nondurable consumption growth was only ¼ of 1 percent. ^5/ This reflects the failure to distinguish between the different kinds of quality services available to individuals.

Differences in the quality of goods over time imply consistent differences in the time series properties of consumption growth and, hence, in parameters estimates derived to fit asset pricing restrictions. Equation (1) implies that, for given α, an unconditional β estimate varies directly with consumption growth. For example, the higher average consumption growth is and the lower mean returns are, the more likely it is that we will obtain low or economically implausible risk aversion (α) estimates if β is below unity. In the same fashion, for values of β below unity, the lower average consumption growth is and the higher mean returns are, the more likely it is that risk aversion estimates are economically plausible.

A failure to treat differences in the quality of services over time as different kinds of services suggests that measured services consumption growth would be increasing over time, when in fact this growth reflects new added services. Thus, for an economically plausible discount factor (β) estimate, a high average rate of consumption growth may imply risk aversion (α) estimates that are low or economically implausible (α<0). In addition, the high average growth in demand is associated with a high average rate of services inflation that implies low real returns in terms of services consumption and, hence, may imply again low or economically implausible risk aversion estimates. Thus, difficulties in measuring quality changes in services (nontradables) suggest that both nontraded consumption and asset returns are subject to more important measurement error than traded measures.

This measurement error argument and the solution to the signal extraction problem imply that among all the measures of consumption growth examined, the international average of traded consumption growth is the least subject to measurement error and thus provides the most reliable parameter estimates. In particular, the international measure of traded consumption growth provides more reliable parameter estimates than either average traded plus nontraded consumption or average nontraded consumption growth, since the former is not subject to the additional noise introduced by measurement errors in nontraded consumption growth. By contrast, an average measure of nontraded growth yields the least reliable parameter estimates because its calculation involves averaging national measures that are subject to important country-specific measurement error.

Table 3 suggests that the relation between national measures of consumption growth and their international (average) measure is indeed remarkably more significant for traded measures of consumption growth than for nontraded measures. This table indicates that the slope estimates derived from least squares regressions of national measures of traded growth on the international measure are systematically higher, in most cases twice as high, than those obtained for nontraded measures across economies. For example, the estimates for traded measures are ¾, ½ and 1 ⅕ for France, Sweden, and the United States, respectively; while, for nontraded growth, the estimates are 0, ¼, and 1 ¼ for France, Sweden and the United States, respectively.

Table 3.

International Least Squares Regressions for Various Measures of Consumption Growth

g_i: one plus national growth of the measure of consumption under consideration in country i; g: one plus average (international) growth across economies of the measure of consumption under consideration. The measure of consumption considered are traded consumption and nontraded consumption. ξ_i is a random disturbance; ${\hat{\gamma }}_{1,\mathrm{i}}$:is the least squares estimate derived from a least squares regression of g_i on g. A constant term was included in these regressions. ():Standard errors. Sample period: 1970.1-1988.1. Source: OECD Quarterly Data.

Table 3.

International Least Squares Regressions for Various Measures of Consumption Growth

gi = γ0 + γi,1·g+ ξi
	Canada	France	Japan	Sweden	United Kingdom	United States
Traded Consumption Growth
${\hat{\gamma }}_{1,\mathrm{i}}$	0.2620	0.8087	0.3935	0.5786	0.4054	1.2093
	(0.2566)	(0.1932)	(0.1027)	(0.4233)	(0.2492)	(0.0751)
Nontraded Consumption Growth
${\hat{\gamma }}_{1,\mathrm{i}}$	0.2944	-0.0256	0.3419	0.3875	0.7650	1.2651
	(0.1929)	(0.1731)	(0.1624)	(0.2308)	(0.2904)	(0.0817)

g_i: one plus national growth of the measure of consumption under consideration in country i; g: one plus average (international) growth across economies of the measure of consumption under consideration. The measure of consumption considered are traded consumption and nontraded consumption. ξ_i is a random disturbance; ${\hat{\gamma }}_{1,\mathrm{i}}$:is the least squares estimate derived from a least squares regression of g_i on g. A constant term was included in these regressions. ():Standard errors. Sample period: 1970.1-1988.1. Source: OECD Quarterly Data.

View Table

Table 3.

International Least Squares Regressions for Various Measures of Consumption Growth

gi = γ0 + γi,1·g+ ξi
	Canada	France	Japan	Sweden	United Kingdom	United States
Traded Consumption Growth
${\hat{\gamma }}_{1,\mathrm{i}}$	0.2620	0.8087	0.3935	0.5786	0.4054	1.2093
	(0.2566)	(0.1932)	(0.1027)	(0.4233)	(0.2492)	(0.0751)
Nontraded Consumption Growth
${\hat{\gamma }}_{1,\mathrm{i}}$	0.2944	-0.0256	0.3419	0.3875	0.7650	1.2651
	(0.1929)	(0.1731)	(0.1624)	(0.2308)	(0.2904)	(0.0817)

g_i: one plus national growth of the measure of consumption under consideration in country i; g: one plus average (international) growth across economies of the measure of consumption under consideration. The measure of consumption considered are traded consumption and nontraded consumption. ξ_i is a random disturbance; ${\hat{\gamma }}_{1,\mathrm{i}}$:is the least squares estimate derived from a least squares regression of g_i on g. A constant term was included in these regressions. ():Standard errors. Sample period: 1970.1-1988.1. Source: OECD Quarterly Data.

It is also important to notice that if stochastic processes are governed by supply and fiscal shocks, risk aversion and discount factor estimates derived with traded measures of consumption growth may also be more accurate and economically plausible than those obtained with measures of nontraded consumption growth. The reason is that trade reduces the importance of the noise relative to the anticipated variation in asset returns and consumption growth captured by the asset pricing model. For nontraded consumption growth, however, the noise introduced by random disturbances may considerably conceal the expected variation in real returns and consumption growth implied by the data.

IV. Time Series Empirical Regularities in the International Economy

This section presents summary statistics that characterize the key features of consumption growth and asset returns measures. Fama (1984) and Fama and French (1988) document the predictability of bond and stock returns. Ferson and Harvey (1990) find time variation in factor loading and risk premia for portfolios of NYSE stocks and various bond portfolios. Cochrane (1988) finds important serial correlation in consumption. Tables 4 through 7 give the mean, standard deviation, and the estimated fourth order autoregressive process for various measures of consumption growth and asset returns across eight industrialized economies.

The statistics displayed in Table 4 suggest that average quarterly consumption growth over the 1970s and 1980s was ½ of 1 percent for the United States and the United Kingdom, and ¾ of 1 percent for Canada and France. Japan had the largest average growth among these countries with 1 percent per quarter, while Sweden had an average consumption growth of only 1/10 of 1 percent; the international average was ½ of 1 percent. However, Sweden had the largest standard deviation of consumption growth, whereas the United States had the smallest.

Table 4.

Summary Statistics for Traded and Nontraded Consumption Growth Across Economies

Consumption growth measure: Traded plus nontraded consumption. Estimation of each autoregressive process includes a constant. ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

Table 4.

Summary Statistics for Traded and Nontraded Consumption Growth Across Economies

Series:	gi						g
Country:	Canada	France	Japan	Sweden	United Kingdom	United States	International
Mean	1.0077	1.0066	1.0086	1.0031	1.0047	1.0051	1.0062
Std. dev.	0.0062	0.0067	0.0090	0.0132	0.0087	0.0049	0.0044
ρ(1)	0.1385	-0.2166	0.1808	-0.5897	0.1675	0.3419	0.4963
	(0.1205)	(0.1234)	(0.1170)	(0.1209)	(0.1248)	(0.1236)	(0.1175)
ρ(2)	0.0598	-0.0464	0.0218	-0.3312	0.0858	-0.0994	-0.2289
	(0.1152)	(0.1220)	(0.1156)	(0.1383)	(0.1264)	(0.1234)	(0.1164)
ρ(3)	0.2109	0.2606	0.1754	-0.1766	0.0763	0.3678	0.5178
	(0.1155)	(0.1221)	(0.1073)	(0.1392)	(0.1273)	(0.1235)	(0.1141)
ρ(4)	0.1042	-0.1481	-0.3041	0.0189	-0.0650	-0.0971	-0.3271
	(0.1179)	(0.1228)	(0.1011)	(0.1227)	(0.1288)	(0.1241)	(0.115)

Consumption growth measure: Traded plus nontraded consumption. Estimation of each autoregressive process includes a constant. ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

View Table

Table 4.

Summary Statistics for Traded and Nontraded Consumption Growth Across Economies

Series:	gi						g
Country:	Canada	France	Japan	Sweden	United Kingdom	United States	International
Mean	1.0077	1.0066	1.0086	1.0031	1.0047	1.0051	1.0062
Std. dev.	0.0062	0.0067	0.0090	0.0132	0.0087	0.0049	0.0044
ρ(1)	0.1385	-0.2166	0.1808	-0.5897	0.1675	0.3419	0.4963
	(0.1205)	(0.1234)	(0.1170)	(0.1209)	(0.1248)	(0.1236)	(0.1175)
ρ(2)	0.0598	-0.0464	0.0218	-0.3312	0.0858	-0.0994	-0.2289
	(0.1152)	(0.1220)	(0.1156)	(0.1383)	(0.1264)	(0.1234)	(0.1164)
ρ(3)	0.2109	0.2606	0.1754	-0.1766	0.0763	0.3678	0.5178
	(0.1155)	(0.1221)	(0.1073)	(0.1392)	(0.1273)	(0.1235)	(0.1141)
ρ(4)	0.1042	-0.1481	-0.3041	0.0189	-0.0650	-0.0971	-0.3271
	(0.1179)	(0.1228)	(0.1011)	(0.1227)	(0.1288)	(0.1241)	(0.115)

Consumption growth measure: Traded plus nontraded consumption. Estimation of each autoregressive process includes a constant. ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

Tables 5 and 6 show these statistics for traded and nontraded consumption growth. An interesting feature of these data is that the time series of traded consumption appear to (i) have a lower average rate of growth and (ii) follow more closely the business cycle than those of nontraded measures for every country under consideration. For example, in the United States, consumption grew at an average quarterly rate of ½ of 1 percent for the nontraded measure and ¼ of 1 percent for the traded measure, with standard deviations of 0.005 and 0.0075 for nontraded and traded growth measures, respectively. In Japan, average growth was 1 percent for nontraded consumption and ½ of 1 percent for the traded measure, while the standard deviations were 0.0057 and 0.0124 for nontraded and traded consumption, respectively. International consumption grew on average ¾ of 1 percent for the nontraded measure and ¼ of 1 percent for the traded measure, with standard deviations of 0.0038 and 0.0056 for the nontraded and traded measures, respectively.

Table 5.

Summary Statistics for Nontraded Consumption Growth Across Economies

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

Table 5.

Summary Statistics for Nontraded Consumption Growth Across Economies

Series:	gNi						gN
Country:	Canada	France	Japan	Sweden	United Kingdom	United States	International
Mean	1.0081	1.0078	1.0098	1.0044	1.0072	1.0051	1.0073
Std. dev.	0.0063	0.0056	0.0057	0.0076	0.0098	0.0050	0.0038
ρ(1)	0.0985	-0.2851	0.4635	-0.4601	0.0918	0.0306	0.1794
	(0.1205)	(0.1224)	(0.1211)	(0.1217)	(0.1244)	(0.1249)	(0.1247)
ρ(2)	0.0741	0.0226	0.1889	-0.3005	0.1488	-0.0265	0.0315
	(0.1183)	(0.1274)	(0.1306)	(0.1316)	(0.1220)	(0.1183)	(0.1162)
ρ(3)	0.0482	-0.0070	0.2653	-0.1769	0.2684	0.3488	0.4166
	(0.1178)	(0.1273)	(0.1282)	(0.1339)	(0.1266)	(0.1182)	(0.1150)
ρ(4)	0.2565	0.0416	-0.2643	0.1154	0.0993	-0.0063	-0.0714
	(0.1169)	(0.1223)	(0.1182)	(0.1252)	(0.1325)	(0.1251)	(0.1244)

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

View Table

Table 5.

Summary Statistics for Nontraded Consumption Growth Across Economies

Series:	gNi						gN
Country:	Canada	France	Japan	Sweden	United Kingdom	United States	International
Mean	1.0081	1.0078	1.0098	1.0044	1.0072	1.0051	1.0073
Std. dev.	0.0063	0.0056	0.0057	0.0076	0.0098	0.0050	0.0038
ρ(1)	0.0985	-0.2851	0.4635	-0.4601	0.0918	0.0306	0.1794
	(0.1205)	(0.1224)	(0.1211)	(0.1217)	(0.1244)	(0.1249)	(0.1247)
ρ(2)	0.0741	0.0226	0.1889	-0.3005	0.1488	-0.0265	0.0315
	(0.1183)	(0.1274)	(0.1306)	(0.1316)	(0.1220)	(0.1183)	(0.1162)
ρ(3)	0.0482	-0.0070	0.2653	-0.1769	0.2684	0.3488	0.4166
	(0.1178)	(0.1273)	(0.1282)	(0.1339)	(0.1266)	(0.1182)	(0.1150)
ρ(4)	0.2565	0.0416	-0.2643	0.1154	0.0993	-0.0063	-0.0714
	(0.1169)	(0.1223)	(0.1182)	(0.1252)	(0.1325)	(0.1251)	(0.1244)

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

Estimation of the fourth order autoregressive process for traded plus nontraded consumption growth across economies are presented in Table 4. The results suggest that the first and third lags of consumption growth are statistically different than zero. An interesting result is that there is a more important serial correlation in international consumption growth. The estimations imply that all four lags of consumption growth are important predictors of future international consumption growth. In particular, the first and third order autoregressive coefficient estimates are about ½, while the second and fourth are negative, -⅕ and -⅓, respectively. This provides further evidence of the findings of Cochrane (1988) for the United States, as well as extends these findings to the international economy. ^6/

Estimation of autoregressive processes for nontraded consumption growth across economies in Table 5 suggests that, in general, only the third lag of consumption growth is relevant in predicting future consumption. Table 6, however, suggests that the first and second lags of traded consumption growth are relevant state variables for predicting future growth. For the United States and the international measure, each of the four lags of consumption growth are meaningful predictors of future traded consumption growth. By contrast, Table 7 suggests that only asset returns lagged by one period reliably foresee future yields. ^7/

Table 6.

Summary Statistics for Traded Consumption Growth Across Economies

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

Table 6.

Summary Statistics for Traded Consumption Growth Across Economies

Series:	gTi						gT
Country:	Canada	France	Japan	Sweden	United Kingdom	United States	International
Mean	1.0031	1.0048	1.0047	1.0012	1.0019	1.0025	1.0033
Std. dev.	0.0123	0.0123	0.0053	0.0204	0.0121	0.0075	0.0056
ρ(1)	-0.1455	-0.4410	0.5477	-0.5191	-0.0252	0.4858	0.5063
	(0.1201)	(0.1231)	(0.1236)	(0.1198)	(0.1258)	(0.1214)	(0.1185)
ρ(2)	0.0564	-0.3015	0.2262	-0.2469	-0.0638	-0.2815	-0.2375
	(0.1115)	(0.1341)	(0.1414)	(0.1358)	(0.1255)	(0.1214)	(0.1192)
ρ(3)	0.1189	0.1171	0.0151	0.0037	-0.0108	0.4847	0.4925
	(0.1074)	(0.1341)	(0.1439)	(0.1362)	(0.1247)	(0.1215)	(0.1191)
ρ(4)	0.0701	-0.0434	-0.1204	-0.0375	-0.0680	-0.2396	-0.3095
	(0.1063)	(0.1227)	(0.1246)	(0.1205)	(0.1256)	(0.1212)	(0.1183)

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

View Table

Table 6.

Summary Statistics for Traded Consumption Growth Across Economies

Series:	gTi						gT
Country:	Canada	France	Japan	Sweden	United Kingdom	United States	International
Mean	1.0031	1.0048	1.0047	1.0012	1.0019	1.0025	1.0033
Std. dev.	0.0123	0.0123	0.0053	0.0204	0.0121	0.0075	0.0056
ρ(1)	-0.1455	-0.4410	0.5477	-0.5191	-0.0252	0.4858	0.5063
	(0.1201)	(0.1231)	(0.1236)	(0.1198)	(0.1258)	(0.1214)	(0.1185)
ρ(2)	0.0564	-0.3015	0.2262	-0.2469	-0.0638	-0.2815	-0.2375
	(0.1115)	(0.1341)	(0.1414)	(0.1358)	(0.1255)	(0.1214)	(0.1192)
ρ(3)	0.1189	0.1171	0.0151	0.0037	-0.0108	0.4847	0.4925
	(0.1074)	(0.1341)	(0.1439)	(0.1362)	(0.1247)	(0.1215)	(0.1191)
ρ(4)	0.0701	-0.0434	-0.1204	-0.0375	-0.0680	-0.2396	-0.3095
	(0.1063)	(0.1227)	(0.1246)	(0.1205)	(0.1256)	(0.1212)	(0.1183)

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: OECD Quarterly Data.

Table 7.

Summary Statistics for Treasury Bills Returns

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: IMF Quarterly Data.

Table 7.

Summary Statistics for Treasury Bills Returns

Series:	ri
Country:	Canada	France	Japan	Sweden	United Kingdom	United States
Mean	1.0064	1.0056	1.0048	1.0008	1.0009	1.0042
Std. dev.	0.0093	0.0099	0.0124	0.0165	0.0139	0.0083
ρ(1)	0.6868	0.6100	0.5052	-0.0192	0.5818	0.7138
	(0.1250)	(0.1252)	(0.1159)	(0.0947)	(0.1244)	(0.1250)
ρ(2)	0.1066	0.0769	0.1959	0.1137	0.1517	0.0249
	(0.1515)	(0.1453)	(0.1216)	(0.0925)	(0.1437)	(0.1562)
ρ(3)	0.0474	0.1753	0.3830	0.0848	0.0639	0.0297
	(0.1513)	(0.1455)	(0.1202)	(0.0931)	(0.1441)	(0.1560)
ρ(4)	0.0298	0.0052	-0.2820	0.6347	0.0478	0.1111
	(0.1249)	(0.1273)	(0.1167)	(0.0919)	(0.1235)	(0.1255)

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: IMF Quarterly Data.

View Table

Table 7.

Summary Statistics for Treasury Bills Returns

Series:	ri
Country:	Canada	France	Japan	Sweden	United Kingdom	United States
Mean	1.0064	1.0056	1.0048	1.0008	1.0009	1.0042
Std. dev.	0.0093	0.0099	0.0124	0.0165	0.0139	0.0083
ρ(1)	0.6868	0.6100	0.5052	-0.0192	0.5818	0.7138
	(0.1250)	(0.1252)	(0.1159)	(0.0947)	(0.1244)	(0.1250)
ρ(2)	0.1066	0.0769	0.1959	0.1137	0.1517	0.0249
	(0.1515)	(0.1453)	(0.1216)	(0.0925)	(0.1437)	(0.1562)
ρ(3)	0.0474	0.1753	0.3830	0.0848	0.0639	0.0297
	(0.1513)	(0.1455)	(0.1202)	(0.0931)	(0.1441)	(0.1560)
ρ(4)	0.0298	0.0052	-0.2820	0.6347	0.0478	0.1111
	(0.1249)	(0.1273)	(0.1167)	(0.0919)	(0.1235)	(0.1255)

ρ: autoregressive coefficient estimate. Period: 1970.2-1988.2. Source: IMF Quarterly Data.

V. The Method of Moments Estimation and Econometric Methodology

Hansen and Singleton (1982) proposed a method of moment estimator that avoids the theoretical requirement of a closed form solution of the stochastic equilibrium, yet permits identification and estimation of parameters of moments conditions, as well as tests of the over-identifying restrictions implied by the model. The basic idea underlying their estimation method is as follows. The optimization problems of economic agents usually imply a set of stochastic Euler equations that must be satisfied in equilibrium. These Euler equations suggest a set of orthogonality conditions that depend nonlinearly on variables and parameters characterizing the objective function.

Nonlinear instrumental variables estimators can be constructed by making sample versions of the orthogonality conditions close to zero according to a certain metric. An important feature of these estimators is that they are consistent and have limiting normal distributions. Also, the models are said to be over-identified in the sense that there are generally more orthogonality conditions than there are parameters to be estimated. Let ν₊ = β · g^-α · (1+r)-1 and m be the number of assets in consideration, we assume that the m constituents of ν₊₁ have finite second moments. Also, let z denote a q dimensional vector of instrumental variables at date t; and define the function f(x₊₁, z, b)-[β(1+r) · g · ^-α-1]⊗z, where f maps R^k X R^p X R^q into R^r [s=m · q], k is the number of variables in υ₊₁, p[=2] is the number of parameters to be estimated, b is a vector of parameters, and ® is the Kronecker product. For example, in asset pricing estimations with one asset and two instruments, s=2. In the same fashion, if we consider Treasury bills issued by seven countries and two instruments, s=14. In our model, the vector of parameters b is conformed by the risk aversion coefficient (α) and a subjective discount factor (β). Hence, our asset pricing model suggests that p=2.

An implication of Euler restrictions (1) and the definition of f(x₊₁, z, b) is that,

where E₀ is the unconditional expectations operator. Equation (20) represents a set of s orthogonality conditions from which an estimator of b=b₀ can be constructed, provided that s is at least as large as the number of unknown parameters, p. Let k₀(b)=E₀[[β(1+r) · g · ^-α-1]⊗z], where b ∈ R^p and it is assumed that the left hand side does not depend on today’s date. Note that (20) suggests that k₀ has a zero at b=b₀. Thus, the method of moments estimator of the function k₀,

evaluated at b=b₀, k_T(b₀), should be close to zero for large values of T. Thus, assuming that f is continuous in its third argument, we can estimate b₀ by choosing from a parameter space Ω ⊆ R^p that makes k_T in (21) close to zero. We follow Hansen (1982) and choose b_T ∈ Ω to minimize the function J_T(b) given by, J_T(b)=k_T(b) ’ W_Tk_T(b), where W_T is an s by s symmetric, positive definite matrix that can depend on sample information.

Hansen and Singleton (1982) introduce an estimator of W_T with the smallest asymptotic covariance matrix among alternative choices of weighting matrices W_T. More precisely, let d(x₊₁, b₀)=υ₊₁=β · g^-α · (1+r)-1, and assume that d is differentiable such that the matrix D₀=E₀[∂d(x₊₁, b₀)/∂b ⊗ z], has full rank. Also, assume that the weighting matrix W_T converges almost surely to a limiting constant matrix W₀ of full rank. Let S₀=E₀[f(x₊₁, z, b₀) · f(x₊₁, z, b₀)′], assuming that S₀ has full rank, Hansen’s (1982) Theorems 3.1 and 3.2 imply that the smallest asymptotic covariance matrix for an estimator b_T that minimizes J_T(b) is obtained by setting W₀ = S^-1₀. The matrix D₀ can be estimated using

A consistent estimator of W_o is given by W*_T,

Note that we can compute W*_T by initially using a suboptimal choice of W_T in minimizing J_T(b) to obtain b_T. Then this b_T can be employed to calculated W*_T using the above formulas. Once W*_T is calculated, b*_T can be obtained by minimizing J_T with W*_T substituted for W_T. Thus, the optimal estimation procedure requires two steps. ^8/ Initially, W₀ is set equal to the identity matrix. For the choice of the instruments, we require only that the z be predetermined. Sufficient conditions for strong consistency and asymptotic normality of these estimators are provided in Hansen (1982).

Hansen and Singleton (1982) suggest testing the restrictions on asset returns as follows. The estimation procedure sets the p linear combinations of the s orthogonality conditions implied by the minimization of J_T equal to zero in estimating b₀. Thus, when s>p, there are s-p remaining linearly independent orthogonality conditions that should be close to zero if the model restrictions are true. Theorem (4.1) in Hansen (1982) implies that T times the minimized value of J(b) is asymptotically distributed as a chi-square with s-p degrees of freedom. It follows that the minimized value of the second step objective function can be used to test the nonlinear asset pricing model (1).

In this paper, the model’s parameters are estimated using this method of moments estimation in two distinct ways. First, estimates of the discount rate (β) and the relative risk aversion coefficient (α) are obtained using generalized method of moments estimation conditional on the information available at any given date. Second, Grossman and Shiller (1981) noted that the discount rate (β) can be estimated from unconditional means of the rate of return on an asset and consumption growth; thus, for α=1, we estimate β so that each Euler equation holds as estimated by its sample mean. The international average of the estimated βs was then used in [restricted] conditional generalized method of moments estimation of the relative risk aversion coefficient (α). This provides a uniform criteria to evaluate risk aversion estimates across economies, as well as the extent to which Euler restrictions are satisfied.

An important feature of this procedure is that any discrepancies or similarities in relative risk aversion estimates do not hinge on differences across economies in the choice of the discount factor. It is well known that a feature of the method of moments estimation is to allow the conditional variances of interest rates to fluctuate with movements of variables in the information set. Hence, if indeed preferences are described by logarithmic utility, minimization of the square disturbance vector generated by Euler restrictions (1) would yield generalized method of moments estimates of the relative risk aversion coefficient that are close to unity with the restrictions satisfied as follows. Prob χ² (DF) is the probability that a χ² (DF) random variate is less than the computed value of the test statistic under the hypothesis that the restrictions are satisfied. ^9/

The restricted method greatly reduces the number of overidentifying restrictions and, thus, the structure imposed on the model. These restrictions are reduced because the degree of risk aversion (α) is estimated conditional on the vector of instruments, while the discount factor (β) estimate is obtained from unconditional moments. This is particularly relevant for joint tests of asset pricing restrictions across economies since the number of overidentifying restrictions in these tests is fairly large.

Models of asset pricing restrictions are estimated using two different sets of instruments: z1_i={constant, (c_{+1, i}/c_i)(-2), (c_{+1, i}/c_i)(-3), (c_{+1, i}/c_i)(-4), (c_{+1, i}/c_i)(-5)} and z2_i={constant, (c_{+1, i}/c_i)(-2), (c_{+1, i}/c_i)(-3), (c_{+1, i}/c_i)(-4), (c_{+1, i}/c_i)(-5), (c_{+1, i}/c_i)(-6), (c_{+1, i}/c_i)(-7)}. ^10/ Hence, z1_i includes a constant and consumption growth lagged two through five periods, while z2_i is conformed by a constant and consumption growth lagged two through seven periods. ^11/

VI. Empirical Evidence on the Asset Pricing Model with the Alternative Measures of Consumption Growth and Asset Returns

This section presents and interprets the results of generalized method of moments estimations of asset pricing restrictions for the alternative measures of national and international consumption growth. The asset returns used in these estimations are Treasury bill returns and Eurocurrency yields. Additional tests for NYSE returns are also presented.

1. National versus international measures of consumption growth

Table 8 summarizes the results of tests performed for the measures of national (g_i) and international (average) consumption growth (g). The results of these tests suggest that, first, estimates of the discount factor (β) for national measures of consumption growth are below similar estimates for the international average measure of growth. Moreover, discount factor estimates are economically plausible for both measures of consumption growth, except for Sweden.

Table 8.

Asset Pricing Estimation of Treasury Bills Returns for Measures of National and International Consumption Growth with z2i

Period: 1970.1-1988.1; unrestricted β estimates; instruments: z2_i. CG: Consumption growth; g_i is one plus national trade plus nontraded consumption growth in the ith economy; g is one plus international traded plus nontraded consumption growth. The vector of instruments (z2_i) is conformed by a constant and the measure of consumption growth under consideration lagged two through seven periods. Unrestricted estimates correspond to conditional estimates of α and β.

Table 8.

Asset Pricing Estimation of Treasury Bills Returns for Measures of National and International Consumption Growth with z2i

E(β·(1+CG)-α·(1+ri) ⊗z) = 1
Country	CG	$\stackrel{\mathrm{\Lambda }}{\beta }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\beta }$	$\stackrel{\mathrm{\Lambda }}{-\alpha }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\alpha }$	DF	Prob χ2
Canada	gi	0.9827	0.0036	+1.4386	0.5232	5	0.0101
France	gi	0.9954	0.0034	-0.2842	0.5233	5	0.0199
Sweden	gi	1.0010	0.0052	-0.6285	0.3373	5	0.6896
United States	gi	0.9983	0.0028	-0.1884	0.4362	5	0.3456
Canada	g	0.9958	0.0033	-0.3095	0.4947	5	0.4743
France	g	0.9957	0.0034	-0.3379	0.5246	5	0.3299
Sweden	g	1.0073	0.0059	-1.4163	0.9522	5	0.6077
United States	g	0.9990	0.0026	-0.4139	0.3760	5	0.1695

Period: 1970.1-1988.1; unrestricted β estimates; instruments: z2_i. CG: Consumption growth; g_i is one plus national trade plus nontraded consumption growth in the ith economy; g is one plus international traded plus nontraded consumption growth. The vector of instruments (z2_i) is conformed by a constant and the measure of consumption growth under consideration lagged two through seven periods. Unrestricted estimates correspond to conditional estimates of α and β.

View Table

Table 8.

Asset Pricing Estimation of Treasury Bills Returns for Measures of National and International Consumption Growth with z2i

E(β·(1+CG)-α·(1+ri) ⊗z) = 1
Country	CG	$\stackrel{\mathrm{\Lambda }}{\beta }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\beta }$	$\stackrel{\mathrm{\Lambda }}{-\alpha }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\alpha }$	DF	Prob χ2
Canada	gi	0.9827	0.0036	+1.4386	0.5232	5	0.0101
France	gi	0.9954	0.0034	-0.2842	0.5233	5	0.0199
Sweden	gi	1.0010	0.0052	-0.6285	0.3373	5	0.6896
United States	gi	0.9983	0.0028	-0.1884	0.4362	5	0.3456
Canada	g	0.9958	0.0033	-0.3095	0.4947	5	0.4743
France	g	0.9957	0.0034	-0.3379	0.5246	5	0.3299
Sweden	g	1.0073	0.0059	-1.4163	0.9522	5	0.6077
United States	g	0.9990	0.0026	-0.4139	0.3760	5	0.1695

Period: 1970.1-1988.1; unrestricted β estimates; instruments: z2_i. CG: Consumption growth; g_i is one plus national trade plus nontraded consumption growth in the ith economy; g is one plus international traded plus nontraded consumption growth. The vector of instruments (z2_i) is conformed by a constant and the measure of consumption growth under consideration lagged two through seven periods. Unrestricted estimates correspond to conditional estimates of α and β.

Second, estimation of the asset pricing conditions with the international measure of consumption growth yield relative risk aversion (α) estimates that are systematically higher than those obtained with national consumption growth across economies. In particular, the estimation of the United States Treasury bill pricing restriction with the national measure of growth implies a risk aversion estimate that is fairly close to ⅙. This estimate is consistent to that obtained by Hansen and Singleton (1983) and Ferson and Constantinides (1989). By contrast, the international measure yields a risk aversion (α) estimate that is economically plausible and fairly close to ⅓ for the United States T-bill restriction. These results are also consistent with those obtained for other countries. For instance, estimated α is consistent with risk aversion for the international measure, but not for the national measures of Canada and France.

Third, risk aversion (α) estimates for international consumption growth models are closer to each other across economies than those obtained for the national models, particularly for Canada, the United States, and France. Moreover, α estimates derived for national measures of growth diverge significantly across economies. For example, the estimated a for national measures of consumption is -1 ½ in Canada and ⅕ in the United States, respectively. Similar estimates for the international measure and T-bills returns in Canada and the United States are ½ and ⅓, respectively. ^12/

Fourth, computed standard errors of parameters estimates derived with the international measure of consumption are consistently smaller than those obtained for national measures of consumption growth. In this sense, the international consumption model yields more precise estimates of α and β than the national consumption models. For T-bill returns and national consumption in the United States, for example, the calculated standard error is ½ for the estimated a. For international consumption growth and the United States T-bill return, however, the computed standard error is only ¼ for the estimated α. Finally, χ² tests suggest that all asset pricing restrictions are satisfied at a 5 percent level of significance for the international models. However, the asset pricing restrictions with national growth of consumption in Canada and France are not satisfied at the 5 percent significance level.

The results support the notion that average consumption growth across economies reduces the effect of country-specific measurement errors on consumption growth. In particular, the international asset pricing model provides a more accurate characterization of risk aversion across countries than national models. A comparison of the models’ underlying forcing variables provides useful insight about the distinct empirical results obtained for the national and international consumption growth models.

One possible explanation of the systematically smaller risk aversion estimates derived for national consumption growth, when compared with those obtained for international consumption growth, is that the average ex post rate of growth of international consumption was smaller than the average ex post national consumption growth, with the exception of the United States and the United Kingdom. The reason is that asset pricing equations (1) imply that relative risk aversion coefficients will be larger the smaller the rate of consumption growth is. Why this occurs is easy enough to see. Solving (1) for the relative risk aversion coefficient, and neglecting the covariance term, yields:

$\begin{array}{cc}\alpha =\left(ln\beta +ln\mathrm{avg}\left(1+r_i\right)\right)/ln\mathrm{avg}\left(g\right)& \left(24\right)\end{array}$

Thus, (24) suggests that, for given β and r_i, the coefficient of risk aversion falls as the average rate of consumption growth rises. Moreover, β estimates derived for national measures of consumption growth are below those obtained for the international measure and hence, from (24), further reduce the risk aversion estimate.

We can gain some intuition about how some of our previous estimates of the risk aversion coefficient (α) were generated, as well as a rough estimate of α, by substituting the average unconditional β estimate across economies (0.9986), the average consumption growth, and the average real yield on Eurocurrencies in (24). The risk aversion estimate implied by the returns on Eurocurrencies and national consumption growth is ¾ for Canada, 1 for France, and ½ for Japan. These estimates are ⅔ the United Kingdom and 1 ⅓ for the United States. In the same fashion, risk aversion estimates derived with Eurocurrencies returns and international consumption growth are 1 1/10 for Canada and France, ¾ for Japan, and ½ and 1 ¼ for the United Kingdom and the United States, respectively.

Equation (24) provides also useful information about the relation between β and α. For example, for the average U.K. T-bill yield per quarter, (24) suggests that the risk aversion coefficient will be negative if β < 1. Thus, the risk aversion estimate is likely to be economically implausible for low discount rate values and average yields, and for high average consumption growth rates. ^13/ The remaining discrepancies in unrestricted risk aversion estimates are due to the important fourth order serial correlation in international, but not in national, consumption growth measures. For example, high serial correlation in international consumption growth helps identify the degree of risk aversion implied by fluctuations in consumption growth and asset returns.

2. National measures of traded versus nontraded consumption growth

Table 9 presents the results of asset pricing tests for two alternative measures of national consumption growth: (1) traded consumption growth (g_i^T) and (2) nontraded consumption growth (g_i^N). The estimations are performed for Treasury bills and Eurocurrencies. ^14/ In addition, some tests are performed for two portfolios of NYSE stocks.

Table 9.

Asset Pricing Estimation of Treasury Bills Returns for National Measures of Traded and Nontraded Consumption Growth

Period: 1970.2-1988.2; instruments: z1_i. CG: Consumption growth; g^T_i is one plus traded consumption growth in the ith economy, and g^N_i is one plus nontraded consumption growth in the ith economy. The vector of instruments (z1_i) is conformed by a constant and the measure of consumption growth under consideration lagged two through five periods. Unrestricted estimates correspond to conditional estimates of α and β.

Table 9.

Asset Pricing Estimation of Treasury Bills Returns for National Measures of Traded and Nontraded Consumption Growth

E(β·(1+CG)-α·(1+ri)⊗z) = 1
Country	CG	$\stackrel{\mathrm{\Lambda }}{\beta }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\beta }$	$\stackrel{\mathrm{\Lambda }}{-\alpha }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\alpha }$	DF	Prob χ2
Canada	gNi	0.9855	0.0034	+1.0718	0.4048	3	0.5100
France	gNi	0.9675	0.0156	+3.7251	2.0796	3	0.6719
Japan	gNi	0.9983	0.0040	-0.2838	0.3983	3	0.6217
United States	gNi	0.9955	0.0024	+0.3833	0.3780	3	0.3041
Canada	gTi	0.9950	0.0021	+0.9477	0.6199	3	0.1540
France	gTi	0.9909	0.0023	+0.3161	0.3897	3	0.1108
Japan	gTi	0.9855	0.0037	+1.1226	0.8586	3	0.0394
United States	gTi	0.9973	0.0016	-0.6492	0.3609	3	0.1398

Period: 1970.2-1988.2; instruments: z1_i. CG: Consumption growth; g^T_i is one plus traded consumption growth in the ith economy, and g^N_i is one plus nontraded consumption growth in the ith economy. The vector of instruments (z1_i) is conformed by a constant and the measure of consumption growth under consideration lagged two through five periods. Unrestricted estimates correspond to conditional estimates of α and β.

View Table

Table 9.

Asset Pricing Estimation of Treasury Bills Returns for National Measures of Traded and Nontraded Consumption Growth

E(β·(1+CG)-α·(1+ri)⊗z) = 1
Country	CG	$\stackrel{\mathrm{\Lambda }}{\beta }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\beta }$	$\stackrel{\mathrm{\Lambda }}{-\alpha }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\alpha }$	DF	Prob χ2
Canada	gNi	0.9855	0.0034	+1.0718	0.4048	3	0.5100
France	gNi	0.9675	0.0156	+3.7251	2.0796	3	0.6719
Japan	gNi	0.9983	0.0040	-0.2838	0.3983	3	0.6217
United States	gNi	0.9955	0.0024	+0.3833	0.3780	3	0.3041
Canada	gTi	0.9950	0.0021	+0.9477	0.6199	3	0.1540
France	gTi	0.9909	0.0023	+0.3161	0.3897	3	0.1108
Japan	gTi	0.9855	0.0037	+1.1226	0.8586	3	0.0394
United States	gTi	0.9973	0.0016	-0.6492	0.3609	3	0.1398

Period: 1970.2-1988.2; instruments: z1_i. CG: Consumption growth; g^T_i is one plus traded consumption growth in the ith economy, and g^N_i is one plus nontraded consumption growth in the ith economy. The vector of instruments (z1_i) is conformed by a constant and the measure of consumption growth under consideration lagged two through five periods. Unrestricted estimates correspond to conditional estimates of α and β.

The results of these tests suggest, first, that discount factor estimates (β) are below unity for the most part and, thus, are economically plausible. Second, β estimates derived for measures of traded consumption growth are systematically higher than those obtained for nontraded measures, except for the Japanese Treasury bill. Third, unrestricted risk aversion (α) estimates are not economically plausible, except for the United States and Japan. This is due to the poor serial correlation in national measures of consumption growth. ^15/ Fourth, computed standard errors of discount factor estimates are, in general, smaller for asset restrictions with traded consumption growth than for restrictions with nontraded consumption growth. The opposite is true for the estimated standard errors of risk aversion estimates, except for France and the United States. Fifth, χ² tests suggest that the hypothesis that asset pricing restrictions are satisfied is not rejected at a 5 percent significance level, except for the restriction on the Japanese Treasury bill for the traded measure.

These results are comparable with those obtained for national and international measures of traded plus nontraded consumption growth. In particular, the results of the restricted estimations suggest that traded consumption growth provides more accurate estimates of risk aversion (α) than nontraded measures. However, unrestricted tests indicate that risk aversion estimates are only economically significant for assets of Japan, the United Kingdom, and the United States. Both sets of results thus suggest that for the United States there may be a more important specification error in nontraded consumption growth than in the traded measure. We provide further tests of this hypothesis by evaluating asset pricing restrictions with value and equally weighted returns on the NYSE index. For comparison with Hansen and Singleton (1982), these tests are performed using 4 and 6 lags of consumption growth and asset returns. ^16/ Table 10 shows the results.

Table 10.

Asset Pricing Estimation of NYSE Returns for Measures of Traded and Nontraded United States Consumption Growth

Period: 1959.2-1988.1; unrestricted β estimates; instruments: ones, lags of CG and r. EWR stands for equally-weighted return, and VWR is value-weighted yield; CG is consumption growth; g_us is one plus traded and nontraded consumption growth in the United States; g^T_us is one plus traded consumption growth in the United States, and g^N_us is one plus nontraded consumption growth in the United States.

Table 10.

Asset Pricing Estimation of NYSE Returns for Measures of Traded and Nontraded United States Consumption Growth

E(β·(1+CG)-α·(1+ri)) = 1
NYSE	CG	$\stackrel{\mathrm{\Lambda }}{\beta }$	Nlags	Standard Error of $\stackrel{\mathrm{\Lambda }}{\beta }$	$\stackrel{\mathrm{\Lambda }}{-\alpha }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\alpha }$	DF	Prob χ2
EWR	gus	4	0.9901	0.0064	+0.0527	2.3961	7	0.2318
EWR	gTus	4	0.9921	0.0038	-0.6375	1.2549	7	0.4903
EWR	gNus	4	0.9886	0.0078	+0.0842	2.6508	7	0.1689
VWR	gus	4	0.9923	0.0052	+1.0359	1.8872	7	0.8360
VWR	gTus E	4	0.9948	0.0026	-0.3748	0.7860	7	0.8114
VWR	gNus	4	0.9917	0.0067	+0.8990	1.7589	7	0.8950

Period: 1959.2-1988.1; unrestricted β estimates; instruments: ones, lags of CG and r. EWR stands for equally-weighted return, and VWR is value-weighted yield; CG is consumption growth; g_us is one plus traded and nontraded consumption growth in the United States; g^T_us is one plus traded consumption growth in the United States, and g^N_us is one plus nontraded consumption growth in the United States.

View Table

Table 10.

Asset Pricing Estimation of NYSE Returns for Measures of Traded and Nontraded United States Consumption Growth

E(β·(1+CG)-α·(1+ri)) = 1
NYSE	CG	$\stackrel{\mathrm{\Lambda }}{\beta }$	Nlags	Standard Error of $\stackrel{\mathrm{\Lambda }}{\beta }$	$\stackrel{\mathrm{\Lambda }}{-\alpha }$	Standard Error of $\stackrel{\mathrm{\Lambda }}{\alpha }$	DF	Prob χ2
EWR	gus	4	0.9901	0.0064	+0.0527	2.3961	7	0.2318
EWR	gTus	4	0.9921	0.0038	-0.6375	1.2549	7	0.4903
EWR	gNus	4	0.9886	0.0078	+0.0842	2.6508	7	0.1689
VWR	gus	4	0.9923	0.0052	+1.0359	1.8872	7	0.8360
VWR	gTus E	4	0.9948	0.0026	-0.3748	0.7860	7	0.8114
VWR	gNus	4	0.9917	0.0067	+0.8990	1.7589	7	0.8950

Period: 1959.2-1988.1; unrestricted β estimates; instruments: ones, lags of CG and r. EWR stands for equally-weighted return, and VWR is value-weighted yield; CG is consumption growth; g_us is one plus traded and nontraded consumption growth in the United States; g^T_us is one plus traded consumption growth in the United States, and g^N_us is one plus nontraded consumption growth in the United States.

The results of these tests indicate, first, that discount factor estimates are below unity and, hence, are economically sensible. Second, traded consumption growth yields systematically higher risk aversion estimates than either nontraded or traded plus nontraded measures. Third, the results for traded consumption restrictions are consistent with previous findings by Hansen and Singleton (1982). In particular, the a estimates ranged from ½ to 1 for the equally weighted return of NYSE stocks. There is, however, a more important change in risk aversion estimates for the traded plus nontraded measure of consumption growth. For this measure, Hansen and Singleton (1982) found that risk aversion estimates range from ½ to 1 in the 1959-88 period. By contrast, Table 10 suggests an important decline in the estimated degree of risk aversion after 1988, to the extent that α estimates are inconsistent with risk aversion for both portfolios of NYSE stocks. Fourth, the estimated standard errors are consistently lower for traded consumption growth than for the nontraded measure. Finally, the χ² statistics suggest that asset pricing restrictions are not rejected at a 5 percent significance level.