Asymptotic properties of the least squares estimator in local to unity processes with fractional Gaussian noise

This chapter derives asymptotic properties of the least squares (LS) estimator of the autoregressive (AR) parameter in local to unity processes with errors being fractional Gaussian noise (FGN) with the Hurst parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" altimg="E2-0001.gif" display="inline"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math>. It is shown that the estimator is consistent for all values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" altimg="E2-0002.gif" display="inline"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math>. Moreover, the rate of convergence is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" altimg="E2-0003.gif" display="inline"> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math>when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" altimg="E2-0004.gif" display="inline"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0.5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math>. The rate of convergence is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" altimg="E2-0005.gif" display="inline"> <mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>H</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math>when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" altimg="E2-0006.gif" display="inline"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0.5</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math>. Furthermore, the limiting distribution of the centered LS estimator depends on H . When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" altimg="E2-0007.gif" display="inline"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:math>, the limiting distribution is the same as that obtained in Phillips (1987a) for the local to unity model with errors for which the standard functional central limit theorem is applicable. When H > 0.5 or when H < 0.5, the limiting distributions are new to the literature. The asymptotic properties of the LS estimator with fitted intercept are also derived. Simulation studies are performed to check the reliability of the asymptotic approximation for different values of sample size.