The goal of this paper is to fill a void in the literature. There are, to our knowledge, no head-to-head, statistical (i.e. likelihood based or asymptotically equivalent) comparisons of asset pricing models from macro/finance. This paper fills the void. The asset pricing models considered are the habit persistence model of Campbell and Cochrane (1999), CC hereafter, the long-run risks model of Bansal and Yaron (2004), BY hereafter, and the prospect theory model of Barberis, Huang, and Santos (2001), BHS hereafter. There are two reason for this choice: These three models are arguably the leading contenders and the authors describe their computational methods precisely enough to permit replication of thier results.
The need for a statistical comparison of asset pricing models is underscored by the ongoing debate between advocates of the long-run risks and habit models. Beeler and Campbell (2009) claim that the long-run risks model is rejected by historical data on the basis of the predictability of excess returns, consumption growth, dividend growth, and their respective volatilities by the price to dividend ratio. Bansal, Kiku, and Yaron (2009) argue that the long-run risks model provides adequate predictability results when using a vector auto regression (VAR) based on consumption growth, price to dividend ratio, and the real risk-free rate. Bansal, Kiku and Yaron also argue that the habit model provides counterfactual predictability results for the price to dividend ratio when using lagged consumption growth as a regressor.
We know of only one other study that attempts a head-to-head, statistical comparison of asset pricing models: Bansal, Gallant, and Tauchen (2007), BGT hereafter. BGT compared the habit model to the long-run risks model using frequentist methods. Their methods could not distinguish between the two models because frequentist non-nested model comparison methods require abundant data. Abundant data are not available in macro/finance. The typical sampling frequency used to calibrate and assess macro/finance models is annual and there are only about 80 annual observations available on the U.S. economy. The papers cited above use annual data. BHS insist that annual is the only frequency that is appropriate to their model. Using more abundant higher frequency data to compare models is not an option. They were not designed to explain high frequency data.
Failing to achieve a definitive statistical result, BGT proceeded to compare the models using the more traditional methods of macro/finance, which consist of enumerating some moments, evaluating them both from the data and from a model simulation, and comparing, often without taking sampling variability into account. On the basis of such comparisons, BGT conclude that the long-run risks model is preferred. Of these comparisons, they relied mostly on the fact that the habit model provides counter factual predictability results for the price to dividend ratio when using lagged consumption growth as a regressor.
In addition to the fact that the BGT comparison, in the end, was not statistical, there are other concerns. BGT did not actually compare the models proposed by CC and BY. They modified them to impose cointegration on macro variables that ought not diverge. They also used a general purpose method to solve them; specifically, a Bubnov-Galerkin method (Miranda and Fackler, 2002, p 152–3). Our view is that fairness dictates that one should use the model that was actually proposed by the originator in comparisons, not a modified model, and that one should use the same solution method that was proposed. To state our view succinctly, the model is the simulation algorithm proposed by the originator; it is not the mathematical equations that suggested the algorithm.
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Habit, Long-Run Risks, Prospect? A Statistical Inquiry
