A. Ronald Gallant** and George Tauchen


We describe a method of nonlinear time series analysis suitable for nonlinear, stationary, multivariate processes whose one-step-ahead conditional density depends on a finite number of lags. Such a density can be represented as a Hermite expansion. Certain parameters of the expansion can be set to imply sharp restrictions on the process such as a pure VAR, a pure ARCH, a nonlinear process with homogeneous innovations, etc. The model is fitted using maximum likelihood procedures on a truncated expansion together with a model selection strategy that determines the truncation point. The estimator is consistent for the true density with respect to a strong norm. The norm is strong enough to imply consistency of evaluation functionals and moments of the conditional density. We describe a method of simulating from the density. Simulation can be used for a great variety of applications. In this paper, we give special attention to using simulations to set sup-norm confidence bands. Fortran code is available via ftp anonymous at ccvrl.cc.ncsu.edu ( in directory pub/arg/snp; alternatively, it is available from the authors in the form of a DOS formatted diskette. The code is provided at no charge for research purposes without warranty.

*Presented at the Institute for Mathematics and its Applications Summer Program, "New Directions in Time Series Analysis," Minneapolis, July 1-29, 1990. To appear in IMA Volumes in Mathematics and its Applications, Springer-Verlag. Research supported by National Science Foundation Grants SES-8808015 and SES-8810357, North Carolina Agricultural Experiment Station Project NCO-6134, and the PAMS Foundation.

**Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203 USA.

Department of Economics, Duke University, Durham, NC 27706 USA.