This paper develops a set of diagnostic tests which can shed light on why a particular model is failing and indicate what steps might be taken to make the model consistent with asset returns. Theoretical bounds on the moments of a stochastic discount factor are derived as a function of the moments of observed asset returns. Particular attention is paid to restrictions on moments other than the variance. These bounds can also be used to measure the information about the distribution of the discount factor contained in the moments of various asset returns. As an application of this methodology, bounds on the discount factor are estimated using size‐based portfolios, and the results are used to analyze the small firm effect. Empirical results indicate, for the period 1926–1975, that moments of the returns of small firms contain information about the discount factor that is not contained in the moments of the returns of large firms and/or a proxy of the aggregate wealth portfolio. However, this difference disappears when more recent data is included.

This paper develops a set of diagnostic tests which can shed light on why a particular model is failing and indicate what steps might be taken to make the model consistent with asset returns. Theoretical bounds on the moments of a stochastic discount factor are derived as a function of the moments of observed asset -returns. Particular attention is paid to restrictions on moments other than the variance. These bounds can also be used to measure the information about the distribution of the discount factor contained in the moments of various asset returns. As an application of this methodology, bounds on the discount factor are estimated using size-based portfolios, and the results are used to analyze the small firm effect. Empirical results indicate, for the period 1926-1975, that moments of the returns of small firms contain information about the discount factor that is not contained in the moments of the returns of large firms and/or a proxy of the aggregate wealth portfolio. However, this difference disappears when more recent data is included. ONE OF THE FUNDAMENTAL problems in finance is determininghow future revenue streams are discounted to yield the value of an asset. In the absence of arbitrage opportunities, there is a single stochastic discount factor which can be used to value future revenue. This can be seen in the relationship between the price of a security, qt, and the payoff of that security at some future period, xt+,, which can be represented by E[mt+7rxt+7] = E[ qt] (or E[mt+,xt+,] = 1 if the x is viewed as a gross return) where the variable m represents some positive, stochastic discount factor. This paper extends a methodology proposed by Hansen and Jagannathan (1991a) to extract information about the stochastic discount factor from various moments of observed asset returns and to create bounds on corresponding moments of m. These bounds can be used as diagnostic tests to compare different models in the same framework. Specifically, I apply this methodology to show that the returns of small firms have different implications about the stochastic discount factor than do the returns of large firms. Since different asset pricing models specify different measures of m, we can index models by the stochastic discount factor they imply (see Hansen and Richard (1987)). In the finance literature, m is often represented by the * School of Business, University of North Carolina at Chapel Hill. This paper is based on my dissertation at The University of Chicago. I would like to thank George Constantinides, Wayne Ferson, Peter Knez, and especially Lars Hansen and John Cochrane for their helpful comments and suggestions. 955 956 The Journal of Finance return on some portfolio. For example, the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) implies that m is the linear combination of the returns on the aggregate wealth portfolio and some riskless return. For the Intertemporal Capital Asset Pricing Model (ICAPM), Brown and Gibbons (1985) and Rubinstein (1976) show that, under certain...

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