Hansen and Jagannathan (1991) proposed a volatility bound for evaluating asset-pricing models that is a restriction on the volatility of a representative agentÌs intertemporal marginal rate of substitution (IMRS). We develop a generalization of their bound that (i) incorporates the serial correlation properties of return data and (ii) allows us to calculate a spectral version of the bound. That is, we develop a bound and then decompose it by frequency; this enables us to judge whether models match important aspects of the data in the long run, at business cycle frequencies, seasonal frequencies, etc. Our generalization is related to the space in which the bounding IMRS lives. Instead of specifying the bounding IMRS to be a linear combination of contemporaneous returns, we let the bounding IMRS live in a linear space of current, past and future returns. We also require the bounding IMRS to satisfy additional restrictions that resemble Euler equations. Our volatility bound not only uses the unconditional first and second moment properties of return data but also the serial correlation properties. Incorporating this additional information results in a tighter bound for two reasons. First, we impose additional orthogonality conditions on our bounding IMRS. Second, our projection is onto a larger space (current, past and future returns). We also show that the spectrum of the model IMRS must exceed the spectrum of our bounding IMRS. Using the serial correlation properties of returns (together with the mean and variance), we are able to derive the spectrum of the bounding IMRS. That is, the lower bound on the spectrum of the model IMRS is completely pinned down by asset return data. This permits a frequency-by-frequency examination of the fundamental component of the model, namely, the Euler equation that links asset returns to the IMRS. In particular, we can identify the frequencies at which an asset-pricing model does not perform well. The researcher can then decide whether or not failures at a particular set of frequencies are troubling. We illustrate our method with four asset pricing models -- time-separable CRRA preferences, state non-separable preferences (Epstein-Zin, 1989, 1991), internal habit formation (Constantinides, 1990), and external habit formation preferences (Campbell and Cochrane, 1999) -- using two data sets, annual data from 1889-1992 and quarterly data spanning 1950:1-1995:4.
