Joon Y. Park

Statistical Inference in Regressions with Integrated Processes: Part 1

This paper develops a multivariate regression theory for integrated processes which simplifies and extends much earlier work. Our framework allows for both stochastic and certain deterministic regressors, vector autoregressions, and regressors with drift. The main focus of the paper is statistical inference. The presence of nuisance parameters in the asymptotic distributions of regression F tests is explored and new transformations are introduced to deal with these dependencies. Some specializations of our theory are considered in detail. In models with strictly exogenous regressors, we demonstrate the validity of conventional asymptotic theory for appropriately constructed Wald tests. These tests provide a simple and convenient basis for specification robust inferences in this context. Single equation regression tests are also studied in detail. Here it is shown that the asymptotic distribution of the Wald test is a mixture of the chi square of conventional regression theory and the standard unit-root theory. The new result accommodates both extremes and intermediate cases.

Canonical cointegrating regressions

A new procedure for statistical inference in cointegrating regressions is developed. The author introduces canonical cointegrating regressions (regressions formulated with the transformed data). The required transformations involve simple adjustments of the integrated processes using stationary components in cointegrating models. Canonical cointegrating regressions, therefore, represent the same cointegrating relationships as the original models. They are, however, constructed in such a way that the usual least squares procedure yields asymptotically efficient estimators and chi-square tests. The methodology presented here is applicable to a very wide class of cointegrating models, including models with deterministic and singular, as well as stochastic and regular, cointegrations. Copyright 1992 by The Econometric Society.

Nonlinear Regressions with Integrated Time Series

An asymptotic theory is developed for nonlinear regression with integrated processes. The models allow for nonlinear effects from unit root time series and therefore deal with the case of parametric nonlinear cointegration. The theory covers integrable, asymptotically homogeneous and explosive functions. Sufficient conditions for weak consistency are given and a limit distribution theory is provided. In general, the limit theory is mixed normal with mixing variates that depend on the sojourn time of the limiting Brownian motion of the integrated process. The rates of convergence depend on the properties of the nonlinear regression function, and are shown to be as slow as n^{1/4} for integrable functions, to be generally polynomial in n^{1/2} for homogeneous functions, and to be path dependent in the case of explosive functions.

ASYMPTOTICS FOR NONLINEAR TRANSFORMATIONS OF INTEGRATED TIME SERIES

An asymptotic theory for stochastic processes generated from nonlinear transformations of nonstationary integrated time series is developed. Various nonlinear functions of integrated series such as ARIMA time series are studied, and the asymptotic distributions of sample moments of such functions are obtained and analyzed. The transformations considered in the paper include a variety of functions that are used in practical nonlinear statistical analysis. It is shown that their asymptotic theory is quite different from that of integrated processes and stationary time series. When the transformation function is exponentially explosive, for instance, the convergence rate of sample functions is path-dependent. In particular, the convergence rate depends not only on the size of the sample, but also on the realized sample path. Some brief applications of these asymptotics are given to illustrate the effects of nonlinearly transformed integrated processes on regression. The methods developed in the paper are useful in a project of greater scope concerned with the development of a general theory of nonlinear regression for nonstationary time series.

A COINTEGRATION APPROACH TO ESTIMATING PREFERENCE PARAMETERS

In this paper, we estimate the (long-run) intertemporal elasticity of substitution of non-durable consumption, which has often been estimated with the generalized methods of moments (GMM). The GMM estimator, however, is not consistent in the presence of liquidity constraints, aggregation over heterogeneous consumers, unknown preference shocks, or a general form of time-nonseparability. We use Engle and Grangers cointegration methodology in order to develop an estimator which is consistent even in the presence of these factors. We then form a formal test that compares the estimates obtained using cointegration techniques with those obtained using GMM.

ON THE FORMULATION OF WALD TESTS OF NONLINEAR RESTRICTIONS

This paper utilizes asymptotic expansions of the Edgeworth type to investigate alternative forms of the Wald test of nonlinear restricti ons. Some formulae for the asymptotic expansion of the distribution of the Wald statistic are provided for a general case that should include most econometric applications. When specialized to the simple cases that have been studied recently in the literature, these formulae are found to explain rather well the discrepancies in sampling behavior that have been observed by other authors. It is further shown how the corrections delivered by Edgeworth expansions m ay be used to find transformations of the restrictions that accelerate covergence to the asymptotic distribution. Copyright 1988 by The Econometric Society.

A Sieve Bootstrap for the Test of a Unit Root

In this paper, we consider a sieve bootstrap for the test of a unit root in models driven by general linear processes. The given model is first approximated by a finite autoregressive integrated process of order increasing with the sample size, and then the method of bootstrap is applied for the approximated autoregression to obtain the critical values for the usual unit root tests. The resulting tests, which may simply be viewed as the bootstrapped versions of Augmented Dickey-Fuller (ADF) unit root tests by Said and Dickey (1984), are shown to be consistent under very general conditions. The asymptotic validity of the bootstrap ADF unit root tests is thus established. Our conditions are significantly weaker than those used by Said and Dickey. Simulations show that bootstrap provides substantial improvements on finite sample sizes of the tests. Copyright 2003 Blackwell Publishing Ltd.

ON THE ASYMPTOTICS OF ADF TESTS FOR UNIT ROOTS

ABSTRACT In this paper, we derive the asymptotic distributions of Augmented-Dickey–Fuller (ADF) tests under very mild conditions. The tests were originally proposed and investigated by Said and Dickey (1984) for testing unit roots in finite-order ARMA models with i.i.d. innovations, and are based on a finite AR process of order increasing with the sample size. Our conditions are significantly weaker than theirs. In particular, we allow for general linear processes with martingale difference innovations, possibly having conditional heteroskedasticities. The linear processes driven by ARCH type innovations are thus permitted. The range for the permissible increasing rates for the AR approximation order is also much wider. For the usual t-type test, we only require that it increase at order o(n 1/2) while they assume that it is of order o(n κ) for some κ satisfying 0 < κ ≤ 1/3.

Bootstrap Unit Root Tests

We consider the bootstrap unit root tests based on finite order autoregressive integrated models driven by iid innovations, with or without deterministic time trends. A general methodology is developed to approximate asymptotic distributions for the models driven by integrated time series, and used to obtain asymptotic expansions for the Dickey-Fuller unit root tests. The second-order terms in their expansions are of stochastic orders O(p) (n-super- - 1/4) and O(p) (n-super- - 1/2), and involve functionals of Brownian motions and normal random variates. The asymptotic expansions for the bootstrap tests are also derived and compared with those of the Dickey-Fuller tests. We show in particular that the bootstrap offers asymptotic refinements for the Dickey-Fuller tests, i.e., it corrects their second-order errors. More precisely, it is shown that the critical values obtained by the bootstrap resampling are correct up to the second-order terms, and the errors in rejection probabilities are of order o(n-super- - 1/2) if the tests are based upon the bootstrap critical values. Through simulations, we investigate how effective is the bootstrap correction in small samples. Copyright The Econometric Society 2003.

Nonlinear econometric models with cointegrated and deterministically trending regressors

This paper develops an asymptotic theory for a general class of nonlinear nonstationary regressions, extending earlier work by Phillips and Hansen (1990) on linear cointegrating regressions. The model considered accommodates a linear time trend and stationary regressors, as well as multiple I(1) regressors. We establish consistency and derive the limit distribution of the nonlinear least squares estimator. The estimator is consistent under fairly general conditions but the convergence rate and the limiting distribution are critically dependent upon the type of the regression function. For integrable regression functions, the parameter estimates converge at a reduced n^{1/4} rate and have mixed normal limit distributions. On the other hand, if the regression functions are homogeneous at infinity, the convergence rates are determined by the degree of the asymptotic homogeneity and the limit distributions are non-Gaussian. It is shown that nonlinear least squares generally yields inefficient estimators and invalid tests, just as in linear nonstationary regressions. The paper proposes a methodology to overcome such difficulties. The approach is simple to implement, produces efficient estimates and leads to tests that are asymptotically chi-square. It is implemented in empirical applications in much the same way as the fully modified estimator of Phillips and Hansen.