Essays on Estimation of a Nonlinear Commodity Price Model without a Closed-form Solution
This thesis is about estimation of classic and modified versions of the rational expectations competitive storage model in the tradition of Gustafson (1958) (the storage model for short), an important economic theory of price determination of storable primary commodities.
The first chapter proposes and evaluates a procedure for approximating the optimal instruments under the context of the classic storage model. This procedure involves calibrating the unknown, true conditional variance function of price disturbance in the optimal instrument using the counterpart of an auxiliary storage model. Monte Carlo simulation suggests that this procedure brings small-sample efficiency gain relative to the benchmark Generalized Method of Moments (GMM) estimator of Deaton and Laroque (1992) and a few other alternatives at the sample size of 100. Its performance is also robust to parameterizations of the auxiliary model within moderate range from the true. This chapter also studies the estimators that require no preliminary estimation, which provide preliminary estimates for the proposed and other infeasible estimators. I find that a well-performing preliminary estimator does not contain instruments of lagged-more-than-one price or increasing transformations of lag prices, and does not use estimated optimal weighting matrix or adopt the continuous-updating approach of Hansen, Heaton and Yaron (1996). Therefore, an instrument of a constant plus reciprocal of lag one price and an identity weighting matrix in general form a good preliminary estimator.
Chapter 2 addresses two concerns about the usefulness of the theory of storage. While commodity speculators can induce serial dependence in price, Deaton and Laroque (1992, 1995 and 1996) argue that speculation explains only a small fraction of the observed autocorrelation in the actual data. Furthermore, the expected rate of return on storage implied by previous econometric estimates is implausibly small. This chapter addresses these two concerns about the validity of the theory of speculative storage by recognizing the downward trend in real price. The existence of a unique non-stationary equilibrium is proved for a rational-expectations competitive-storage model with a trend, and testable implications of the model are also derived. I show that, when a downward price trend in part or all of the sample is ignored, the autocorrelation coefficient in price tends to be overestimated while the expected rate of return tends to be underestimated. Finally, I offer an empirical illustration of the trending storage model using annual corn price over the period from 1961 to 2005.
Chapter 3 discusses the empirical implications of the distributional misspecification of two nonlinear least squares estimators of a modified storage model with unbounded prices. The existence of infrequent, extremely low harvest generates extremely high cutoff price which is difficult to pass in finite periods. Meanwhile, due to the tiny chance of such events, it is easy for the practitioners to ignore them during the estimation and apply a false storage model with relatively low cutoff price. This chapter studies how such misspecification can affect the empirical implications of estimating the storage model. Surprisingly, I find that misspecified econometric models yield better estimates for the real interest rate; and the estimated cutoff price, actually captures the sharp turning point of the equilibrium price function. Therefore, though misspecified, the estimates are practically useful. Nevertheless, it is also emphasized that such interesting property of the two estimators should by no means be understood as a defense of ignoring the infrequent influential event in the asset pricing problems.
Mathematical proofs for general results and further discussions of a few econometric issues can be found in the Appendices. While a few theoretical results have been derived, this thesis relies heavily on Monte Carlo simulation and numerical functional approximation. Numerical methods turn out to be a convenient and many times necessary tools to study small-sample econometric problems when asymptotic results cannot provide an accurate approximation to the exact sampling.