Tests of polynomial speci\u85cation in semiparametric regression

Spline smoothing is a popular method of nonparametric curve \u85tting. It is known that, after a reordering of the data, the spline can be obtained from the smoothed estimates of a state space representation of a time series model for irregularly spaced observations. The key parameter is the signal-to-noise ratio. When this is zero the underlying model is a polynomial trend plus noise, otherwise it is a high order stochastic trend plus noise. Time-series tests for the presence of stochastic trends in the context of ir-regularly spaced data are derived and used as tests for the null hypothesis of polynomial speci\u85cation against a general nonparametric alternative for cross-sectional data. The distribution of the tests depends on the design matrix of the regressors but critical values can be easily obtained by Monte Carlo techniques. Importantly, the test statistic does not depend on any smoothing parameters. The power properties of the linearity test are evalu-ated for a range of alternative non-linear speci\u85cations and compared with the properties of other tests of the same hypothesis. As an illustration, we test for linearity of Engel curves