Asymptotic Behaviour of Sums of Powers of Residuals in the Classic Linear Regression Model

Abstract

In this paper, we study the asymptotic properties of sums of powers of residuals, a subject which, apart from some problems in Econometric Theory, has not received much attention in the econometric literature. We intend to show that some powerful results can be achieved. Furthermore, an inequality, not much used in econometrics, will be intensively applied—we refer to Markov’s inequality. The Chebychev inequality, much more used, is a special form of the Markov inequality. The inequality simply states that, if $g(x) \geqslant 0{\text{ }} \Rightarrow {\text{ }}P[g(X) \geqslant \theta ] \leqslant \frac{{E\{ g({X^\alpha })\} }}{{{\theta ^\alpha }}},\alpha > 0$ where E denotes expectation and P denotes probability, see [2, p.74]. Using this inequality implies that certain consistency results can be proven much more easily as we are completely free in the choice of a as long as it is positive.