An Adaptive, Rate-Optimal Test of a Parametric Mean-Regression Model Against a Nonparametric Alternative

Abstract

We develop a new test of a parametric model of a conditional mean function against a nonparametric alternative. The test adapts to the unknown smoothness of the alternative model and is uniformly consistent against alternatives whose distance from the parametric model converges to zero at the fastest possible rate. This rate is slower than n y1 r2 . Some existing tests have nontrivial power against restricted classes of alternatives whose distance from the parametric model decreases at the rate n y1 r2 . There are, however, sequences of alternatives against which these tests are inconsistent and ours is consistent. As a consequence, there are alternative models for which the finite-sample power of our test greatly exceeds that of existing tests. This conclusion is illustrated by the results of some Monte Carlo experiments.

Nonparametric Cointegrating Regression With Endogeneity and Long Memory

“…Local alternatives of the form (3.5) are commonly used in the theory of nonparametric inference involving stationary data. See, for instance, Horowitz and Spokoiny (2001). We impose the following smoothness conditions on m(x) and the consistency rate condition onθ n under H 1 to aid the asymptotic development here.…”

“…Condition (iii) excludes functions m (x) which lie in the span of g (x,θ 0 ) and its derivativesġ (θ 0 ) , thereby ensuring that there is discriminatory power in the direction of m (x) . A related condition is used in Horowitz and Spokoiny (2001).…”

Testing parametric assumptions of trends of a nonstationary time series

“…Hypothesis testing in nonparametric regression under independence has been discussed by Härdle & Mammen (1993), Hart (1997), Fan et al (2001) and Van Keilegom et al (2008). Other contributions can be found in Azzalini et al (1989), Eubank & LaRiccia (1992), Aerts et al (1999), Eubank (1999), Horowitz & Spokoiny (2001) and Fan & Jiang (2007). In this paper we adopt the following for-25 mulation: suppose we observe…”

Nonparametric Tests of Moment Condition Stability

“…In this paper we follow Zheng (1996) and Li (1999) in that we explore several choices of bandwidth to examine the sensitivity of size and power for the tests. An alternative strategy was suggested by Horowitz and Spokoiny (2001). They propose calculating the supremum of nonparametric tests over a range of possible bandwidth parameters as a test statistic.…”