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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Journal of Monetary Economics, Volume 54, Issue 8, November 2007, Pages 2269–2290
We develop a generalization of the Hansen–Jagannathan (1991) volatility bound that (i) incorporates the serial correlation properties of return data and (ii) allows us to calculate a spectral version of the bound. This generalization enables us to judge whether models match important aspects of the data in the long run, at business cycle frequencies, seasonal frequencies, etc. Our bound permits evaluation of models without requiring their explicit solution in a way that respects the dynamic implications of the fundamental component of the models, namely, the Euler equation that links asset returns to the intertemporal marginal rate of substitution.
Empirical evaluation of dynamic structural models has a long history in economics. In dynamic general equilibrium frameworks with linear–quadratic preferences, Hansen and Sargent (1980) computed linear decision rules explicitly and linked theory with measurement using the likelihood function. As they showed, a frequency domain approximation to the likelihood may be used in model assessment, in which case the fit of a model is judged by how well its spectral density matches the corresponding spectral density computed from the data. The procedure, unfortunately, rejects practically all models. In circumstances where closed-form expressions for decision rules are not available, Hansen (1982) and Hansen and Singleton (1983) used the Generalized Method of Moments (GMM) to formally estimate and evaluate dynamic models using a subset of the model implications for the data. Tests based on the procedure are less demanding than those based on the full likelihood, though few models pass them. Hansen and Jagannathan (1991) proposed a still less restrictive test that generalizes the variance bounds developed by LeRoy and Porter (1981) and Shiller (1981). They showed how to use asset return data to derive a lower bound on the volatility of a representative consumer's intertemporal marginal rate of substitution (IMRS). A model is said to be consistent with the data if the volatility of the IMRS implied by the model is greater than the Hansen–Jagannathan (HJ) volatility bound. While the HJ test dismisses many models for violating the volatility bound, many others do satisfy the bound. Cochrane and Hansen (1992), for instance, argued that for reasonable parameterizations of time non-separable preferences and of state non-separable preferences, the consumption-based asset-pricing models do satisfy the HJ bound. Tallarini (2000) modified the preferences in the standard business cycle model to allow for non-separabilities across states and showed that the model is consistent with asset return data using the HJ bound. The HJ bound depends on three types of asset return moments: means, variances and contemporaneous correlations. In this paper, we develop a generalized volatility bound that (i) incorporates the serial correlation properties of return data and (ii) allows us to calculate a spectral decomposition of the bound. This enables us to judge whether models fail to match important aspects of the data in the long run, at business cycle frequencies, seasonal frequencies, high frequencies, etc. Our evaluation of models is also based solely on the Euler equation that links the asset returns to the IMRS. This Euler equation governs intertemporal decisions, and hence the propagation of economic fluctuations—so a spectral (i.e., temporal) bound is especially useful in evaluating the model. Specifically, we can identify the frequencies at which a model violates the necessary conditions. Technically, to derive the bound, Hansen and Jagannathan projected the model IMRS onto the space of contemporaneous asset returns and utilized only a necessary condition associated with dynamic models, namely the intertemporal Euler equation. Our generalization involves projecting the model IMRS onto the space of current, past, and future returns. As in Hansen and Jagannathan (1991), the projection involves a covariance between the IMRS and returns, a covariance which is given by Euler equation of the model and which is the sole implication used in the derivation of our bound. We show that the variance of the model IMRS must exceed the variance of the projection. Because of the nature of our projection we also show that the spectrum of the model IMRS must exceed the spectrum of the projection. Using the serial correlation properties of returns (together with the mean and variance), we derive a lower bound on the spectrum of the model IMRS. Similar to GMM and the HJ bound, our procedure does not require solving the entire model. Because our bound involves projection of the model IMRS onto a larger space, it must be at least as tight as the HJ bound. Bounds tighter than the HJ bound are of course, straightforward to find. Adding observable variables to the right-hand side of the implicit HJ regression of the IMRS on returns permits construction of a tighter bound. The virtue of our list of additional variables (past and future returns) is that the residual in the projection must be orthogonal to returns at all leads and lags; this is what enables us to compute the frequency-by-frequency additive decomposition of our volatility bound. The issue that must be confronted in computing our projection is that there are many new covariances between the IMRS and returns—those at leads and lags—that the model does not supply. To fill in these missing covariances, we make use of sample returns and the “sample” IMRS calculated from data (typically just consumption data) for specific model parameters. What is gained by this approach is that when returns and the model IMRS have serial correlation and comovement, the model-implied contemporaneous covariance carries implications for the sort of IMRS volatility that is permissible over short and long horizons. That is, whereas the HJ bound reflects the fact that the variance of the IMRS must exceed a particular minimum given the variance of returns and the Euler equation, our bound reflects the fact that the stochastic process for the IMRS must be sufficiently variable at each frequency given the stochastic process for returns together with the Euler equation. An additional attractive feature of our bound is that bound violations may be localized—at high frequencies, low frequencies, or business cycle frequencies—and this may suggest modifications of the model. To summarize, our bound can be used to provide (i) a more restrictive test of a model because it employs more information than the HJ bound and (ii) a less restrictive test because the additive decomposition enables its application at specific frequencies of interest where a model might satisfy our bound despite violating the HJ bound. We evaluate four asset-pricing models using our bound. In our applications, we also calculate a volatility bound over specific frequency bands (e.g., business cycle frequencies) and compare it to the volatility of the IMRS over the same band. The time-separable model turns out to satisfy the bound at business cycle and lower frequencies, but requires an extraordinary amount of risk aversion to do so. The state non-separable model of Epstein and Zin (1991) fares somewhat better, satisfying the bound at frequencies associated with yearly and longer cycles. We find that the habit formation model of Constantinides (1990) satisfies the bound at high frequencies while violating the bound at business cycle frequencies. The richer habit formation model of Campbell and Cochrane (1999) fares better at business cycle frequencies. In Section 2 of the paper, we review the derivation of the HJ bound and develop the generalized bound. Section 3 applies the spectral bound to four popular asset-pricing models. Section 4 provides a spectral generalization of the He and Modest (1995) and Luttmer (1996) bound for economies with frictions. Section 5 provides confidence bands for our spectral bound. Section 6 concludes.
نتیجه گیری انگلیسی
In this paper, we propose a volatility bound for the evaluation of asset-pricing and business cycle models. The bound, which is a generalization of the volatility bound proposed by Hansen and Jagannathan (1991), incorporates serial correlation properties of returns. The spectral bound that we develop allows for model evaluation by frequency; it thus allows the researcher to determine the frequencies at which the model is not doing well. A business cycle model that violates the bound at business cycle frequencies would be cause for concern, though one might not care that a consumption-based asset-pricing model is inconsistent with the movement of asset prices and consumption at the higher frequencies. Indeed, the higher frequencies present the main difficulties, though the internal habit formation model does reasonably well there. For business cycle and lower frequencies, there are parameterizations—perhaps unreasonable ones—of the workhorse time-separable model, the state non-separable model, and the external habit model that are all consistent with consumption and return data. The bounds we develop in this paper can be easily applied to any dynamic general equilibrium model with virtually no computational cost. It is common in the literature to solve complicated models for economic quantities and then evaluate the fit along the asset returns dimension using the HJ bound. This procedure is used because calculating asset prices in the environments typically used is computationally difficult. Our procedure allows for a more elaborate evaluation of dynamic models along the asset returns dimension without a need to calculate asset prices.