Appendix: Proof of Equation (25)
The international average consumption asset pricing model is given by:
where, Ri=1+ri. Applying the mathematical lemma, E(a · b)=E(a) · E(b)+COV(a, b), to (A.1) yields:
Let Rs=β · E(g-α) and divide both sides of (A.2) by Rs to get:
Applying Stein’s Lemma, COV(f(y), x)=E(f′(y)) · COV(x, y), to the second term in the right hand side (RHS) of (A.3) yields:
where c+1 is the level of consumption next period, and c is consumption today. Multiplying and dividing the RHS of (A.4) by c, we get:
where Ω+1=β · E(α · g-α-1)/(Rs). Subtracting Rs from (A.5), we get:
Substituting Ri by g (the market return) in (A.6), yields:
where, VAR(g) is the variance of consumption growth. Thus, dividing (A.6) by (A.7) implies:
where βiw is the risk premium of asset i. Finally, solving (A.8) for E(Ri) and setting expected values equal to the perfect foresight values yields the following international version of the capital asset pricing model:
Bayoumi, Tamim, Koujianou, R., “The Effect of Financial Deregulation on Interest Rates” (unpublished; Washington: International Monetary Fund, 1989).
Breeden, D.T., Litzenberger, R. H., “Prices of States Contingent Claims Implicit in Option Prices,” Journal of Business, Vol. 51 (October 1978), pp. 621–52
Brown, David P., Gibbons, Michael R., “A Simple Econometric Approach for Utility-based Asset Pricing Models,” Journal of Finance, Vol. 40 (1985), pp. 359–381.
Ceccheti, Stephen G., Nelson C., Mark, “Evaluating Empirical Tests of Asset Pricing Models: Alternative Interpretations,” AEA Papers and Proceedings, Vol. 80 (1990), pp. 48–51.
Constantinides, George M., “Habit Formation: A Resolution of the Equity Premium Puzzle,” Journal of Political Economy, Vol. 98 (1990), pp. 519–43.
Denison, Edward F., Estimates of Productivity Change by Industry: An Evaluation and an Alternative, ed. by The Brookings Institution (New York, 1989).
Fama, Eugene F., MacBeth, James D., “Risk, Return, and Equilibrium: Empirical Tests,” Journal of Political Economy, Vol. 81 (May 1973), pp. 607–36.
Ferson, Wayne E.; Constantinides, George M., “Habit Persistence and Durability in Aggregate Consumption: Empirical Tests” (Unpublished; Chicago: The University of Chicago, 1989).
Hansen, Lars P.; Richard, Scott F.; Singleton, Kenneth J., “Econometric Implications of the Intertemporal Capital Asset Pricing Model” (Unpublished; Pittsburgh: Carnegie Mellon University, 1981).
Hansen, Lars P., “Large Sample Properties of Generalized Method of Moments Estimators,” Econometrica, Vol. 50 (July 1982), pp. 1029–54.
Hansen, Lars P.; Jagannathan, Ravi, “Implications of Security Market Data for Models of Dynamic Economies,” Journal of Political Economy, Vol. 99 (April 1991), pp. 225–62.
Hansen, Lars P.; Singleton, Kenneth J., “Generalized Instrument Variables Estimation of Non Linear Rational Expectations Models,” Econometrica, Vol. 50 (September 1982), pp. 1269–86.
Hansen, Lars P.; Singleton, Kenneth J., “Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns,” Journal of Political Economy, Vol. 91 (April 1983), pp. 249–65.
Kravis, Irving; Heston, Alan; Summers, Robert, “The Share of Services in Economic Growth,” in Global Econometrics, ed. by Adams and Hickman (Cambridge, 1983), pp. 188–218.
Kravis, Irving; Heston, Alan; Summers, Robert, “International Comparison of Real Product and its Composition: 1950-1977,” Review of Income and Wealth, Vol. 26 (March 1980), pp. 19–66.
Mussa, Michael L., “The Nominal Exchange Rate Regime and the Behavior of Real Exchange Rates,” in Carnegie-Rochester Conferences on Public Policy, Vol. 25 (1986), pp. 117–214.
Newey, Whitney K.; West, Kenneth D., “A Simple, Positive Definite, Heteroscedasticity and Autocorrelation Consistent Covariance Matrix,” Econometrica, Vol. 55 (1987), pp. 703–08.
Working, Holbrook, “Note on the Correlation of First Differences of Averages of A Random Chain,” Econometrica, Vol. 28 (October 1960), pp. 916–18.
Zellner, Arnold, “Notes on the Distribution of the Asset Pricing Restriction Error Term” (Unpublished; Chicago: The University of Chicago, 1987).
The author is particularly grateful to Michael Mussa for valuable discussions and suggestions. The comments of Andrew Atkeson, John Cochrane, David T. Coe, Peter Ireland, Robert Kollman, Robert E. Lucas, and Guillermo Mondino are gratefully acknowledged. Any remaining errors are the author’s responsibility.
The bounds obtained by Hansen and Jagannathan (1990) can be interpreted in this fashion where the model under consideration is the risk neutral model (α-0).
To demonstrate that these results do not hinge on the assumption that υ is log-normally distributed, we abandon this assumption below.
Hansen and Singleton (1982) show that all that the choice of the instrumental variables requires is that they be predetermined as of the current time period. For example, current and lagged values of consumption growth can be chosen.
Kuznets (1957); Kravis, Heston, and Summers (1983); and the Council of Economic Advisers (1988) show that for the past forty years the output of services has risen faster than GNP owing to growing demand.
The countries in this sample are Canada, France, Japan, Sweden, the United Kingdom, and the United States.
Estimation of the fourth order autoregressive process for U.S. consumption growth in 1970.1-88.1 using the Citibank data base is consistent with these findings.
The results for Eurocurrency yields are consistent with these findings and, thus, are not reported.
Newey and West Method is used when the weighing matrix is singular or not positive definite.
Chi-square [χ2] tests of the restrictions are displayed in each table.
Theories of habit persistence in consumption (Constantinides, 1990) suggest that lagged consumption growth should be a significant state variable in asset pricing and the determination of future consumption growth.
First lags are inadmissible instruments because consumption is measured as quarterly average rather than at points in time. According to the permanent income hypothesis, consumption is a random walk. Working (1960) shows that averaging a random walk induces serial correlation between the contemporaneous value and the first lag, but not earlier lags, making first lags invalid instruments.
Restricted risk aversion estimates are consistent with these results and, thus, are not reported.
The basic results remain approximately the same if we consider the covariance term between the marginal rate of substitution of consumption and real interest rates.
For the purpose of comparison with the international measures of traded plus nontraded consumption growth, we also performed restricted estimations with β=0.996 in Eurocurrencies restrictions and β=0.999 in Treasury bills restrictions.
The results of fitting pricing restrictions on Eurocurrencies are comparable to the results for Treasury bills and, thus, are not reported.
The consumption data were obtained from the Federal Reserve Board. Real per capita consumption was calculated dividing by the population figures published by the Bureau of Census. The monthly data used are from 1959.2-88.1.
Restricted estimations were performed with the cross-country average unconditional β estimate 0.996 for Eurocurrencies and 0.999 for Treasury bills pricing restrictions.
The results of tests performed for Eurocurrencies are again consistent with those obtained for Treasury bills and, hence, are not reported.
These results for the closed economy models are also consistent with a model in which traded and nontraded goods are not homothetic (Ogaki, 1988).
The international traded consumption growth model suggests that risk aversion estimates for Japan and Sweden are even larger, 1.4. In addition, the United Kingdom is the only country for which a closed economy measure of consumption provides economically plausible risk aversion estimates.
Bayoumi and Koujianou (1989) find evidence that suggest that the importance of liquidity constraints fell in the 1980s.
The vector of instruments is conformed by consumption growth lagged two and three periods.