Minimum distance lack-of-fit tests under long memory errors
This paper discusses some tests of lack-of-fit of a parametric regression model when errors form a long memory moving average process with the long memory parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$0>d>1/2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>0</mn> <mo>></mo> <mi>d</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation>, and when design is non-random and uniform on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$[0,1]$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math> </EquationSource> </InlineEquation>. These tests are based on certain minimized distances between a nonparametric regression function estimator and the parametric model being fitted. The paper investigates the asymptotic null distribution of the proposed test statistics and of the corresponding minimum distance estimators under minimal conditions on the model being fitted. The limiting distribution of these statistics are Gaussian for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$0>d>1/4$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>0</mn> <mo>></mo> <mi>d</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </math> </EquationSource> </InlineEquation> and non-Gaussian for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$1/4>d>1/2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> <mo>></mo> <mi>d</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation>. We also discuss the consistency of these tests against a fixed alternative. A simulation study is included to assess the finite sample behavior of the proposed test. Copyright Springer-Verlag Berlin Heidelberg 2015
