Herman J. Bierens

Consistent model specification tests

In this paper we propose two consistent tests for functional form of nonlinear regression models without employing specified alternative models. The null hypothesis is that the regression function equals the conditional expectation function, which is tested against the alternative hypothesis that the null is false. These tests are based on a Fourier transform characterization of conditional expectations.

Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate

Abstract This paper is concerned with testing the unit root with drift hypothesis against a very general trend stationarity hypothesis, namely the alternative that the time series is stationary about an almost arbitrary deterministic function of time. Our approach employs the fact that any function of time can be approximated arbitrarily close by a linear function of Chebishev polynomials. We propose various tests on the basis of an Augmented Dickey-Fuller auxiliary regression with linear and nonlinear deterministic trends, where the nonlinear deterministic trend is approximated by detrended Chebishev time polynomials. Also, we propose a model-free test. We apply our tests to the GNP deflator, the consumer price index, and the interest rate for the USA, taken from the extended Nelson-Plosser data set. The results indicate that these series are nonlinear trend stationary.

Nonparametric cointegration analysis

In this paper we propose consistent cointegration tests, and estimators of a basis of the space of cointegrating vectors, that do not need specification of the data-generating process, apart from some mild regularity conditions, or estimation of structural and/or nuisance parameters. This nonparametric approach is in the same spirit as Johansen s LR method in that the test statistics involved are obtained from the solutions of a generalized eigenvalue problem, and the hypotheses to be tested are the same, but in our case the two matrices in the generalized eigenvalue problem involved are constructed independently of the data-generating process. We compare our approach empirically as well as by a limited Monte Carlo simulation with Johansen s approach, using the series for ln(wages) and ln(GNP) from the extende

Uniform Consistency of Kernel Estimators of a Regression Function under Generalized Conditions

Abstract In this article we prove uniform consistency of kernel estimators of a multivariate regression function under various assumptions on the distribution of the data. In addition to the usual assumptions that the data are iid and that the distribution of the regressors is absolutely continuous, we consider the cases that some regressors are discrete and the data are either stationary ϕ-mixing themselves or generated by a class of functions of one-sided infinite stationary ϕ-mixing sequences. Moreover, we demonstrate the performance of the kernel estimation method under these generalized conditions by a numerical example.

Nonparametric Nonlinear Cotrending Analysis, With an Application to Interest and Inflation in the United States

Given the assumption that the components of a vector time series are stationary around nonlinear deterministic time trends, nonlinear cotrending is the phenomenon that one or more linear combinations of the time series are stationary around a linear trend or a constant; hence, the series have common nonlinear deterministic time trends. In this article, I develop nonparametric tests for nonlinear cotrending, and I derive nonparametric estimators of the cotrending vectors. I apply this approach to the federal funds rate and the consumer price index inflation rate in the United States, using monthly data, to analyze the price puzzle.

Model specification testing of time series regressions

Abstract Given the specification of the lag length and functional form of a (non)linear time series regression we shall propose a test of the null hypothesis that the expectation of the error conditional on the exogenous variables, all lagged exogenous variables and all lagged dependent variables equals zero with probability 1. In the case that the data-generating process is strictly stationary this test is consistent with respect to the alternative hypothesis that the null is false. The test is also applicable for a particular class of non-stationary time series regressions, although in that case consistency with respect to all possible alternatives is no longer guaranteed. The test involved is a generalization of a test proposed in Bierens (1982b). Moreover, we also present a similar but simpler test of the hypothesis that the errors are martingale differences.

TIME-VARYING COINTEGRATION

In this paper we propose a time-varying vector error correction model in which the cointegrating relationship varies smoothly over time. The Johansen setup is a special case of our model. A likelihood ratio test for time-invariant cointegration is defined and its asymptotic chi-square distribution is derived. We apply our test to the purchasing power parity hypothesis of international prices and nominal exchange rates, and we find evidence of time-varying cointegration.

Testing stationarity and trend stationarity against the unit root hypothesis

In this paper we propose a family of relativel simple nonparametrics tests for a unit root in a univariate time series. Almost all the tests proposed in the literature test the unit root hypothesis against the alternative that the time series involved is stationarity or trend stationary. In this paper we take the (trend) stationarity hypothesis as the null and the unit root hypothesis as the alternative. The order differnce with most of the tests proposed in the literature is that in all four cases the asymptotic null distribution is of a well-known type, namely standard Cauchy. In the first instance we propose four Cauchy tests of the stationarity hypothesis against the unit root hypothesis. Under H1 these four test statistics involved, divided by the sample size n, converge weakly to a non-central Cauchy distribution, to one, and to the product of two normal variates, respectively. Hence, the absolute values of these test statistics converge in probability to infinity 9at order n). The tests involved ar...

Robust Methods and Asymptotic Theory in Nonlinear Econometrics

1 Introduction.- 1.1 Specification and misspecification of the econometric model.- 1.2 The purpose and scope of this study.- 2 Preliminary Mathematics.- 2.1 Random variables, independence, Borel measurable functions and mathematical expectation.- 2.1.1 Measure theoretical foundation of probability theory.- 2.1.2 Independence.- 2.1.3 Borel measurable functions.- 2.1.4 Mathematical expectation.- 2.2 Convergence of random variables and distributions.- 2.2.1 Weak and strong convergence of random variables.- 2.2.2 Convergence of mathematical expectations.- 2.2.3 Convergence of distributions.- 2.2.4 Convergence of distributions and mathematical expectations.- 2.3 Uniform convergence of random functions.- 2.3.1 Random functions. Uniform strong and weak convergence.- 2.3.2 Uniform strong and weak laws of large numbers.- 2.4 Characteristic functions, stable distributions and a central limit theorem.- 2.5 Unimodal distributions.- 3 Nonlinear Regression Models.- 3.1 Nonlinear least-squares estimation.- 3.1.1 Model and estimator.- 3.1.2 Strong consistency.- 3.1.3 Asymptotic normality.- 3.1.4 Weak consistency and asymptotic normality under weaker conditions.- 3.1.5 Asymptotic properties if the error distribution has infinite variance. Symmetric stable error distributions.- 3.2 A class of nonlinear robust M-estimators.- 3.2.1 Introduction.- 3.2.2 Strong consistency.- 3.2.3 Asymptotic normality.- 3.2.4 Properties of the function h(?). Asymptotic efficiency and robustness.- 3.2.5 A uniformly consistent estimator of the function h(?).- 3.2.6 A two-stage robust M-estimator.- 3.2.7 Some weaker results.- 3.3 Weighted nonlinear robust M-estimation.- 3.3.1 Introduction.- 3.3.2 Strong consistency and asymptotic normality.- 3.3.3 A two-stage weighted robust M-estimator.- 3.4 Miscellaneous notes on robust M-estimation.- 3.4.1 Uniform consistency.- 3.4.2 The symmetric unimodality assumption.- 3.4.3 The function ?.- 3.4.4 How to decide to apply robust M-estimation.- 4 Nonlinear Structural Equations.- 4.1 Nonlinear two-stage least squares.- 4.1.1 Introduction.- 4.1.2 Strong consistency.- 4.1.3 Asymptotic normality.- 4.1.4 Weak consistency.- 4.2 Minimum information estimators: introduction.- 4.2.1 Lack of instruments.- 4.2.2 Identification without using instrumental variables.- 4.2.3 Consistent estimation without using instrumental variables.- 4.2.4 Asymptotic normality.- 4.2.5 A problem concerning the nonsingularity assumption.- 4.3 Minimum information estimators: instrumental variable and scaling parameter.- 4.3.1 An instrumental variable.- 4.3.2 An example.- 4.3.3 A scaling parameter and its impact on the asymptotic properties.- 4.3.4 Estimation of the asymptotic variance matrix.- 4.3.5 A two-stage estimator.- 4.3.6 Weak consistency.- 4.4 Miscellaneous notes on minimum information estimation.- 4.4.1 Remarks on the function