Empirical Dynamic Asset Pricing

Model Specification and Econometric Assessment
By Kenneth J. Singleton

Princeton University Press

Copyright © 2006 Princeton University Press
All right reserved.

ISBN: 0-691-12297-0


Chapter One

Introduction

A dynamic asset pricing model is refutable empirically if it restricts the joint distribution of the observable asset prices or returns under study. A wide variety of economic and statistical assumptions have been imposed to arrive at such testable restrictions, depending in part on the objectives and scope of a modeler's analysis. For instance, if the goal is to price a given cash-flow stream based on agents' optimal consumption and investment decisions, then a modeler typically needs a fully articulated specification of agents' preferences, the available production technologies, and the constraints under which agents optimize. On the other hand, if a modeler is concerned with the derivation of prices as discounted cash flows, subject only to the constraint that there be no "arbitrage" opportunities in the economy, then it may be sufficient to specify how the relevant discount factors depend on the underlying risk factors affecting security prices, along with the joint distribution of these factors.

An alternative, typically less ambitious, modeling objective is that of testing the restrictions implied by a particular "equilibrium" condition arising out of an agent's consumption/investment decision. Such tests can often proceed by specifying only portions of an agent's intertemporal portfolio problem and examining the implied restrictions on moments of subsets of variables in the model. With this narrower scope often comes some "robustness" to potential misspecification of components of the overall economy that are not directly of interest.

Yet a third case is one in which we do not have a well-developed theory for the joint distribution of prices and other variables and are simply attempting to learn about features of their joint behavior. This case arises, for example, when one finds evidence against a theory, is not sure about how to formulate a better-fitting, alternative theory, and, hence, is seeking a better understanding of the historical relations among key economic variables as guidance for future model construction.

As a practical matter, differences in model formulation and the decision to focus on a "preference-based" or "arbitrage-free" pricing model may also be influenced by the availability of data. A convenient feature of financial data is that it is sampled frequently, often daily and increasingly intraday as well. On the other hand, macroeconomic time series and other variables that may be viewed as determinants of asset prices may only be reported monthly or quarterly. For the purpose of studying the relation between asset prices and macroeconomic series, it is therefore necessary to formulate models and adopt econometric methods that accommodate these data limitations. In contrast, those attempting to understand the day-to-day movements in asset prices-traders or risk managers at financial institutions, for example-may wish to design models and select econometric methods that can be implemented with daily or intraday financial data alone.

Another important way in which data availability and model specification often interact is in the selection of the decision interval of economic agents. Though available data are sampled at discrete intervals of time-daily, weekly, and so on-it need not be the case that economic agents make their decisions at the same sampling frequency. Yet it is not uncommon for the available data, including their sampling frequency, to dictate a modeler's assumption about the decision interval of the economic agents in the model. Almost exclusively, two cases are considered: discrete-time models typically match the sampling and decision intervals-monthly sampled data mean monthly decision intervals, and so on-whereas continuous-time models assume that agents make decisions continuously in time and then implications are derived for discretely sampled data. There is often no sound economic justification for either the coincidence of timing in discrete-time models, or the convenience of continuous decision making in continuous-time models. As we will see, analytic tractability is often a driving force behind these timing assumptions.

Both of these considerations (the degree to which a complete economic environment is specified and data limitations), as well as the computational complexity of solving and estimating a model, may affect the choice of estimation strategy and, hence, the econometric properties of the estimator of a dynamic pricing model. When a model provides a full characterization of the joint distribution of its variables, a historical sample is available, and fully exploiting this information in estimation is computationally feasible, then the resulting estimators are "fully efficient" in the sense of exploiting all of the model-implied restrictions on the joint distribution of asset prices. On the other hand, when any one of these conditions is not met, researchers typically resort, by choice or necessity, to making compromises on the degree of model complexity (the richness of the economic environment) or the computational complexity of the estimation strategy (which often means less econometric efficiency in estimation).

With these differences in modelers' objectives, practical constraints on model implementation, and computational considerations in mind, this book: (1) characterizes the nature of the restrictions on the joint distributions of asset returns and other economic variables implied by dynamic asset pricing models (DAPMs); (2) discusses the interplay between model formulation and the choice of econometric estimation strategy and analyzes the large-sample properties of the feasible estimators; and (3) summarizes the existing, and presents some new, empirical evidence on the fit of various DAPMs.

We briefly expand on the interplay between model formulation and econometric analysis to set the stage for the remainder of the book.

1.1. Model Implied Restrictions

Let [P.sub.s] denote the set of "payoffs" at date s that are to be priced at date t, for s > t, by an economic model (e.g., next period's cum-dividend stock price, cash flows on bonds, and so on), and let [[pi].sub.t] : [P.sub.s] [right arrow] R denote the pricing function, where [R.sup.n] denotes the n-dimensional Euclidean space. Most DAPMs maintain the assumption of no arbitrage opportunities on the set of securities being studied: for any [q.sub.t+1] [member of] [P.sub.t+1] for which Pr{[q.sub.t+1] = 0}=1, Pr({[[pi].sub.t]([q.sub.t+1]) = 0} [intersection] {[q.sub.t]+1 > 0}) = 0. In other words, nonnegative payoffs at t + 1 that are positive with positive probability have positive prices at date t. A key insight underlying the construction of DAPMs is that the absence of arbitrage opportunities on a set of payoffs [P.sub.s] is essentially equivalent to the existence of a special payoff, a pricing kernel [q*.sub.s], that is strictly positive (Pr{q*.sub.s] > 0} = 1) and represents the pricing function [[pi].sub.t] as

[[pi].sub.t]([q.sub.s]) = E][q.sub.s][q*.sub.s] | [I.sub.t]], (1.1)

for all [q.sub.s] [member of] [P.sub.s], where [I.sub.t] denotes the information set upon which expectations are conditioned in computing prices.

This result by itself does not imply testable restrictions on the prices of payoffs in [P.sub.t+1], since the theorem does not lead directly to an empirically observable counterpart to the benchmark payoff. Rather, overidentifying restrictions are obtained by restricting the functional form of the pricing kernel [q*.sub.s] or the joint distribution of the elements of the pricing environment ([P.sub.s, [q*.sub.s], [I.sub.t]). It is natural, therefore, to classify DAPMs according to the types of restrictions they impose on the distributions of the elements of ([P.sub.s, [q*.sub.s], [I.sub.t]). We organize our discussions of models and the associated estimation strategies under four headings: preference-based DAPMs, arbitrage-free pricing models, "beta" representations of excess portfolio returns, and linear asset pricing relations. This classification of DAPMs is not mutually exclusive. Therefore, the organization of our subsequent discussions of specific models is also influenced in part by the choice of econometric methods typically used to study these models.

1.1.1. Preference-Based DAPMs

The approach to pricing that is most closely linked to an investor's portfolio problem is that of the preference-based models that directly parameterize an agent's intertemporal consumption and investment decision problem. Specifically, suppose that the economy being studied is comprised of a finite number of infinitely lived agents who have identical endowments, information, and preferences in an uncertain environment. Moreover, suppose that [A.sub.t] represents the agents' information set and that the representative consumer ranks consumption sequences using a von Neumann-Morgenstern utility functional

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

In (1.2), preferences are assumed to be time separable with period utility function U and the subjective discount factor [beta] [member of] (0, 1). If the representative agent can trade the assets with payoffs [P.sub.s] and their asset holdings are interior to the set of admissible portfolios, the prices of these payoffs in equilibrium are given by (Rubinstein, 1976; Lucas, 1978; Breeden, 1979)

[[pi].sub.t]([q.sub.s]) = E][m.sup.s-t.sub.s][q.sub.s] | [A.sub.t], (1.3)

where [m.sup.s-t.sub.s] = [beta]U'([c.sub.s])/U'([c.sub.t]) is the intertemporal marginal rate of substitution of consumption (MRS) between dates t and s. For a given parameterization of the utility function U([c.sub.t]), a preference-based DAPM allows the association of the pricing kernel [q*.sub.s] with [m.sup.s-t.sub.s].

To compute the prices [[pi].sub.t([q.sub.s]) requires a parametric assumption about the agent's utility function U([c.sub.t]) and sufficient economic structure to determine the joint, conditional distribution of [m.sup.s-t.sub.s] and [q.sub.s]. Given that prices are s set as part of the determination of an equilibrium in goods and securities markets, a modeler interested in pricing must specify a variety of features of an economy outside of securities markets in order to undertake preference-based pricing. Furthermore, limitations on available data may be such that some of the theoretical constructs appearing in utility functions or budget constraints do not have readily available, observable counterparts. Indeed, data on individual consumption levels are not generally available, and aggregate consumption data are available only for certain categories of goods and, at best, only at a monthly sampling frequency.

For these reasons, studies of preference-based models have often focused on the more modest goal of attempting to evaluate whether, for a particular choice of utility function U([c.sub.t]), (1.3) does in fact "price" the payoffs in [P.sub.s]. Given observations on a candidate [m.sup.s-t.sub.s] and data on asset returns [R.sub.s] [equivalent to] {[q.sub.s] [member of] [P.sub.s] : [[pi].sub.t] ([q.sub.s]) = 1}, (1.3) implies testable restrictions on the joint distribution of [R.sub.s], [m.sup.s-t.sub.s], and elements of [A.sub.t]. Namely, for each s-period return [r.sub.s], E [m.sup.s-t.sub.s][r.sub.s] - 1|[A.sub.t]] = 0, for any [r.sub.s] [member of] [R.sub.s] (see, e.g., Hansen and Singleton, 1982). An immediate implication of this moment restriction is that E]([m.sup.s-t.sub.s] - 1)[x.sub.t]] = 0, for any [x.sub.t] [member of] [A.sub.t]. These unconditional moment restrictions can be used to construct method-of-moments estimators of the parameters governing [m.sup.s-t.sub.s] and to test whether or not [m.sup.s-t.sub.s] prices the securities with payoffs in [P.sub.s]. This illustrates the use of restrictions on the moments of certain functions of the observed data for estimation and inference, when complete knowledge of the joint distribution of these variables is not available.

An important feature of preference-based models of frictionless markets is that, assuming agents optimize and rationally use their available information [A.sub.t] in computing the expectation (1.3), there will be no arbitrage opportunities in equilibrium. That is, the absence of arbitrage opportunities is a consequence of the equilibrium price-setting process.

1.1.2. Arbitrage-Free Pricing Models

An alternative approach to pricing starts with the presumption of no arbitrage opportunities (i.e., this is not derived from equilibrium behavior). Using the principle of "no arbitrage" to develop pricing relations dates back at least to the key insights of Black and Scholes (1973), Merton (1973), Ross (1978), and Harrison and Kreps (1979). Central to this approach is the observation that, under weak regularity conditions, pricing can proceed "as if" agents are risk neutral. When time is measured continuously and agents can trade a default-free bond that matures an "instant" in the future and pays the (continuously compounded) rate of return [r.sub.t], discounting for risk-neutral pricing is done by the default-free "roll-over" return [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For example, if uncertainty about future prices and yields is generated by a continuous-time Markov process [Y.sub.t] (so, in particular, the conditioning information set [I.sub.t] is generated by [Y.sub.t]), then the price of the payoff [q.sub.s] is given equivalently by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

where [E.sup.Q.sub.t] denotes expectation with regard to the "risk-neutral" conditional distribution of Y. The term risk-neutral is applied because prices in (1.4) are expressed as the expected value of the payoff [q.sub.s] as if agents are neutral toward financial risks.

As we will see more formally in subsequent chapters, the risk attitudes of investors are implicit in the exogenous specification of the pricing kernel q* as a function of the state [Y.sub.t] and, hence, in the change of probability measure underlying the risk-neutral representation (1.4). Leaving preferences and technology in the "background" and proceeding to parameterize the distribution of q* directly facilitates the computation of security prices. The parameterization of ([P.sub.s], [q*.sub.s], [Y.sub.t]) is chosen so that the expectation in (1.4) can be solved, either analytically or through tractable numerical methods, for [[pi].sub.t]([q.sub.s]) as a function of [Y.sub.t] : [[pi].sub.t]([q.sub.s]) = P([Y.sub.t]). This is facilitated by the adoption of continuous time (continuous trading), special structure on the conditional distribution of Y, and constraints on the dependence of q* on Y so that the second expectation in (1.4) is easily computed. However, similarly tractable models are increasingly being developed for economies specified in discrete time and with discrete decision/trading intervals.

Importantly, though knowledge of the risk-neutral distribution of [Y.sub.t] is sufficient for pricing through (1.4), this knowledge is typically not sufficient for econometric estimation. For the purpose of estimation using historical price or return information associated with the payoffs [P.sub.s], we also need information about the distribution of Y under its data-generating or actual measure. What lie between the actual and risk-neutral distributions of Y are adjustments for the "market prices of risk"-terms that capture agents' attitudes toward risk. It follows that, throughout this book, when discussing arbitrage-free pricing models, we typically find it necessary to specify the distributions of the state variables or risk factors under both measures.

(Continues...)



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