This paper derives the exact finite sample distribution of the two-stage generalized least squares (GLS) estimator in a multivariate linear model with general linear parameter restrictions. This includes the seemingly unrelated regression (SUR) model as a special case and generalizes presently known exact results for the latter system. The usual classical assumptions are made concerning nonrandom exogenous variables and normally distributed errors. The theoretical results of this paper are made possible by the author's development of a matrix fractional calculus. This operator calculus is the main theoretical tool of the paper and may be used to solve a wide range of other unsolved problems in econometric distribution theory. IN THE EARLY 1960's Zellner [10] developed a two-stage GLS estimator for the coefficients in a linear multivariate system that is now popularly known as the SUR model. This two-stage procedure has since been used in many empirical applications. GLS also forms the basis of other commonly used estimators both in linear models with heteroscedastic or autocorrelated errors and in simultaneous equation systems where it leads to three stage least squares (3SLS). In spite of extensive research and perhaps surprisingly in view of the popularity of GLS methods in empirical work, the exact finite sample distribution of the SUR estimator is known only in highly specialized cases. These cases effectively restrict attention to two equation systems and models with orthogonal regressors [2]. Existing distribution theory is even more limited in the case of other commonly used GLS estimators, such as the two-stage estimator in linear models with heteroscedastic errors. Here, only low order moment formulae are known and then only in the simplest two sample setting. The research underlying the present paper is motivated by the deficiencies outlined above. Our initial object of study was the exact distribution of the SUR estimator in the general case. But the methods we have developed open the way to an exact distribution theory for econometric estimators in a much wider setting than the SUR model. The present paper will derive the exact finite sample distribution of the two-stage GLS estimator in the multivariate linear model subject to general linear parameter restrictions. This generalizes all presently known distribution theory for the SUR model itself. Two important specializations of our results will be illustrated in detail: the unrestricted multivariate linear model; and the Zellner model with pairwise orthogonal regressors. The analytical results reported here are made possible by the introduction of a fractional matrix calculus. This calculus is developed in terms of the action of

[The rational expectations hypothesis in a life cycle context asserts that agents seek to keep the marginal utility of discounted expenditure constant across time. In this paper we present estimates of a demand system that takes explicit account of this hypothesis. We show how the restrictions from demand theory enable us to identify the model despite the presence of an unobservable variable that can be interpreted as revisions to the marginal utility of discounted expenditure. This allows us to take explicit account of the different effects of anticipated and unanticipated price changes. An important check on our model is the derivation of expenditure elasticities despite the fact that the demand system does not have expenditure on the right hand side. Our estimates are "sensible" and we find that we can reject the hypothesis that revisions to the marginal utility of discounted money are orthogonal to the past values of our variables. We interpret this to be a rejection of the rational expectations hypothesis.]

This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing. 1. INTRODUCTION THE TERM STRUCTURE of interest rates measures the relationship among the yields on default-free securities that differ only in their term to maturity. The determinants of this relationship have long been a topic of concern for economists. By offering a complete schedule of interest rates across time, the term structure embodies the market's anticipations of future events. An explanation of the term structure gives us a way to extract this information and to predict how changes in the underlying variables will affect the yield curve. In a world of certainty, equilibrium forward rates must coincide with future spot rates, but when uncertainty about future rates is introduced the analysis becomes much more complex. By and large, previous theories of the term structure have taken the certainty model as their starting point and have proceeded by examining stochastic generalizations of the certainty equilibrium relationships. The literature in the area is voluminous, and a comprehensive survey would warrant a paper in itself. It is common, however, to identify much of the previous work in the area as belonging to one of four strands of thought. First, there are various versions of the expectations hypothesis. These place predominant emphasis on the expected values of future spot rates or holdingperiod returns. In its simplest form, the expectations hypothesis postulates that bonds are priced so that the implied forward rates are equal to the expected spot rates. Generally, this approach is characterized by the following propositions: (a) the return on holding a long-term bond to maturity is equal to the expected return on repeated investment in a series of the short-term bonds, or (b) the expected rate of return over the next holding period is the same for bonds of all maturities. The liquidity preference hypothesis, advanced by Hicks [16], concurs with the importance of expected future spot rates, but places more weight on the effects of the risk preferences of market participants. It asserts that risk aversion will cause forward rates to be systematically greater than expected spot rates, usually

This paper develops a continuous time general equilibrium model of a simple but complete economy and uses it to examine the behavior of asset prices. In this model, asset prices and their stochastic properties are determined endogenously. One principal result is a partial differential equation which asset prices must satisfy. The solution of this equation gives the equilibrium price of any asset in terms of the underlying real variables in the economy. IN THIS PAPER, we develop a general equilibrium asset pricing model for use in applied research. An important feature of the model is its integration of real and financial markets. Among other things, the model endogenously determines the stochastic process followed by the equilibrium price of any financial asset and shows how this process depends on the underlying real variables. The model is fully consistent with rational expectations and maximizing behavior on the part of all agents. Our framework is general enough to include many of the fundamental forces affecting asset markets, yet it is tractable enough to be specialized easily to produce specific testable results. Furthermore, the model can be extended in a number of straightforward ways. Consequently, it is well suited to a wide variety of applications. For example, in a companion paper, Cox, Ingersoll, and Ross [7], we use the model to develop a theory of the term structure of interest rates. Many studies have been concerned with various aspects of asset pricing under uncertainty. The most relevant to our work are the important papers on intertemporal asset pricing by Merton [19] and Lucas [16]. Working in a continuous time framework, Merton derives a relationship among the equilibrium expected rates of return on assets. He shows that when investment opportunities are changing randomly over time this relationship will include effects which have no analogue in a static one period model. Lucas considers an economy with homogeneous individuals and a single consumption good which is produced by a number of processes. The random output of these processes is exogenously determined and perishable. Assets are defined as claims to all or a part of the output of a process, and the equilibrium determines the asset prices. Our theory draws on some elements of both of these papers. Like Merton, we formulate our model in continuous time and make full use of the analytical tractability that this affords. The economic structure of our model is somewhat similar to that of Lucas. However, we include both endogenous production and