Specification Testing Driven by Orthogonal Series in Nonstationary Time Series Models

This paper establishes two simple and new specification tests based on the use of an orthogonal series. The paper then establishes an asymptotic theory for each of the proposed tests. The first test is initially proposed for the case where the regression function involved is integrable and the second test is an extended version of the first test for covering a class of non-integrable functions. The finite sample performance of the proposed tests is examined through using several simulated examples. Meanwhile, an application of the second test shows that a second-order polynomial model is more appropriate than a commonly used linear model for modelling the relationship between the United States consumers' consumption expenditure and disposable income over the time 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.