Abstract
This paper proposes two simple and new specication tests based on the use of an orthogonal series for a considerable class of cointegrated time series models with endogeneity and nonstationarity. The paper then establishes an asymptotic theory for each of the proposed tests. The rst test is initially proposed for the case where the regression function involved is integrable, which lls a gap in the literature, and the second test is an extended version of the rst test for covering a class of non{integrable functions. Endogeneity in two general forms is allowed in the models to be tested. A potential global departure in the alternative hypothesis, which is being overlooked by the literature, is investigated. The nite sample performance of the proposed tests is examined through using several simulated examples. Meanwhile, the second test is naturally applicable to the case where there is a type of endogeneity inherited in the relationship between the United States aggregate consumers’ consumption expenditure and disposable income over the period of 1960{2009. Our experience generally shows that the proposed tests are easily implementable and also have stable sizes and good power properties even when the ’distance’ between the null hypothesis and a sequence of local alternatives is asymptotically negligible.
Highlights
Econometric model estimation for nonlinear structural cointegrating models is an increasingly active area in recent years
A rigorous specification procedure should be used to test whether the suggested parametric model may be accepted statistically. This introduces the recent literature about nonparametric specification testing for parametric specification of nonlinear and nonstationary time series
Note that Gao et al (2009a) propose a nonparametric kernel–based test for parametric specification of nonstationary nonlinear models where the regressor that is an integrated time series is independent of the equation error term that is a martingale difference sequence; Gao et al (2009b) consider a testing issue on a nonstationary nonlinear autoregression models where strict conditions are imposed on the density of the error term; Hong and Phillips (2010) test linearity of cointegrating relations with an application; and Wang and Phillips (2012a) investigate parametric specification of nonstationary nonlinear regression models where the regression function is imposed to have a certain growth rate when its variable tends to infinity such as polynomials, power functions, etc
Summary
Econometric model estimation for nonlinear structural cointegrating models is an increasingly active area in recent years. This paper aims at proposing a test statistic for the specification of integrable nonstationary regression models, in which two forms of endogeneity are allowed. A large class of functions, such as polynomials and power functions, can be allowed for m(x) These tests clearly cover the existing papers such as Gao et al (2009a) and Wang and Phillips (2012a) as a subclass and fill the gap in the literature. Our experience shows that while each of the tests is of a simple quadratic form, it is not necessarily easy to establish and prove an asymptotic distribution for each of the test statistics This is mainly because existing central limit theorems available for standardised versions of quadratic forms (see, e.g., Theorem A.1 of Gao, 2007) are not applicable in our case where there is no martingale structure involved in model (1.1).
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