Gordon C.R. Kemp

THE BEHAVIOR OF FORECAST ERRORS FROM A NEARLY INTEGRATED AR(1) MODEL AS BOTH SAMPLE SIZE AND FORECAST HORIZON BECOME LARGE

We develop asymptotic approximations to the distribution of forecast errors from an estimated AR(1) model with no drift when the true process is nearly I(1) and both the forecast horizon and the sample size are allowed to increase at the same rate. We find that the forecast errors are the sums of two components that are asymptotically independent. The first is asymptotically normal whereas the second is asymptotically nonnormal. This throws doubt on the suitability of a normal approximation to the forecast error distribution. We then perform a Monte Carlo study to quantify further the effects on the forecast errors of sampling variability in the parameter estimates as we allow both forecast horizon and sample size to increase.

Scale equivariance and the Box-Cox transformation

The Box-Cox family of transformations has two useful features: first, it includes linear and logarithmic transformations as special cases; and, second, it possesses strong scale equivariance properties, including the property that the transformation parameter is unaffected by the rescaling. Its main disadvantage is that both the domain and the range of the transformation are, in general, bounded. We show that, for a certain class of models, if a model demonstrates these scale equivariance properties for the dependent variable then the transformation on the dependent variable must be a variant of the Box-Cox transformation.

Invariance and the Wald test

Many models and hypotheses of interest in econometrics are invariant to certain types of data transformations such as measurement unit changes. Dagenais and Dufour (1991, 1992) and Dufour and Dagenais (1992) have shown that Wald test are not invariant in general to such data transformations. In this paper, I provide a simple set of sufficient conditions to ensure that a Wald test for a null hypothesis is invariant to such a data transformation. I then use this set of conditions to help account for certain features of the Monte Carlo results from Gregory and Veall (1986) on the properties of a variety of Wald tests for Sargans COMFAC restriction (Sargan 1980) in the first-order autoregressive distributed-lag model.

The Joint Distribution of Forecast Errors in the AR(1) Model

Second-order asymptotic expansion approximations to the joint distributions of dynamic forecast errors and of static forecast errors in the stationary Gaussian pure AR(1) model are derived. The approximation to the dynamic forecast errors distribution can be expressed as a multivariate normal distribution with modified mean vector and covariance matrix, thus generalizing the results of Phillips [12]. However, the approximation to the static forecast errors distribution includes skewness and kurtosis terms. Thus the class of multivariate normal distributions does not provide as good approximations (in terms of error convergence rates) to the distributions of the static forecast errors as to the distributions of the dynamic forecast errors. These results cast some doubt on the appropriateness of model validation procedures, such as Chow tests, which use the static forecast errors and implicitly assume that these have a distribution which is well approximated by a multivariate normal.

When is a proportional hazards model valid for both stock and flow sampled duration data

Abstract It is shown that if the flow-sampled total durations and stock-sampled elapsed durations from stationary renewal processes both exhibit proportional hazard rates and are continuous then they belong to the family of survival distributions with linear mean residual functions introduced by Hall and Wellner (1984) [Hall, W.J., Wellner, J.A., 1984. Mean residual life. In: Csorgo, M., Dawson, D.A., Rao, J.N.K., Saleh, A.K.M.E., (Eds.), Proceedings of the International Symposium on Statistics and Related Topics. North Holland, Amsterdam, pp. 169–184].

On Wald tests for globally and locally quadratic restrictions

In this paper I analyze the properties of Wald tests for quadratic and locally quadratic restrictions. Initially I consider the finite-sample properties of Wald tests for quadratic restrictions in the Classical Linear Regression (CLR) model. Then I extend these results to the asymptotic properties of Wald tests for locally quadratic restrictions in more general models. Finally, I apply these results to the problem of testing for second-order COMFAC restrictions. This analysis indicates that the conventional general-to-specific approach to testing is inappropriate for the sequential testing of COMFAC restrictions.

ON THE CONSTRUCTION OF BOUNDS CONFIDENCE REGIONS

We modify the procedure for constructing exact bounds confidence regions introduced by Dufour (1990, Econometrica 58, 475-494) so as to drop the requirement that the confidence regions for the nuisance parameters are marginal with respect to the parameters of interest, i.e., that they are the same for all values of the parameters of interest. We illustrate this modified procedure with an application to a dependent variable heteroskedasticity model and, using a Monte Carlo study, compare the confidence regions constructed by this procedure with asymptotically justified regions.

The potential for efficiency gains in estimation from the use of additional moment restrictions

It is frequently assumed that in Generalised Method of Moments estimation the explicit utilization of valid additional moment restrictions will lead to asymptotic efficiency gains. In this paper I develop a general framework for examining the possibility of such efficiency gains. Using this framework, I then consider in more detail the potential for efficiency gains from utilizing a constant conditional disturbance variance restriction when estimating a single equation from a nonlinear Simultaneous Equations Model (SEM) by nonlinear Two Stage Least Squares (2SLS). In general such efficiency gains will arise; however, for a linear SEM with constant disturbance covariances and zero third moments no efficiency gains arise for fairly general instrument sets. This generalises, in certain respects, the usual result that 2SLS is asymptotically efficient for estimating a single equation from a linear SEM with independently identically distributed normal disturbances.