A Discourse on Modified Likelihood Ratio (LR), Wald and Lagrange Multipliers (LM) Tests for Testing General Linear Hypothesis in Stochastic Linear Regression Model

In this research paper various new advanced inferential tools namely modified likelihood ratio (LR), Ward and Lagrange Multiplier test statistics have been proposed for testing general linear hypothesis in stochastic linear regression model. In this process internally studentized residuals have been used. This research study has brought out some new advance tools for analysing inferential aspects of stochastic linear regression models by using internally studentized residuals. Miguel Fonseca et.al [1] developed statistical inference in linear models dealing with the theory of maximum likelihood estimates and likelihood ratio tests under some linear inequality restrictions on the regression coefficients. Tim Coelli [2] used Monte carlo experimentation to investigate the finite sample properties of maximum likelihood (ML) and correct ordinary least squares (COLS) estimators of the half –normal stochastic frontier production function. In 2011, p. Bala siddamuni et.al [3] have developed advanced tools for mathematical and stochastic modelling.

A Treatise on Testing General Linear Hypothesis in Stochastic Linear Regression Model

The main objective of this research article is to propose test statistics for testing general linear hypothesis about parameters in stochastics linear regression model using studentized residuals, RLS estimates and unrestricted internally studentized residuals. In 1998, M. Celia Rodriguez -Campos et.al [1] introduced a new test statistics to test the hypothesis of a generalized linear model in a regression context with random design. Li Cai et.al [2] provide a new test statistic for testing linear hypothesis in an OLS regression model that not assume homoscedasticity. P. Balasiddamuni et.al [3] proposed some advanced tools for mathematical and stochastical modelling.

Econometric Identification of the Impact of Real Estate Characteristics Based on Predictive and Studentized Residuals

The paper proposes a means of determining the impact of real estate characteristics based on the residuals of an accordingly specified econometric model. The econometric model contains explanatory variables whose values are easily measurable. Then, the hypothesis that the residuals of the econometric model encompass the impact of specific factors indicating that the real estate is atypical is verified, thus supporting real estate market analysis. The work describes various types of residuals (predictive and studentized residuals).

Distribución de las transformaciones lineales de los residuos mínimos cuadrados studentizados internamente = Distribution of linear transformations of internally studentized least squares residuals

<p>Los residuos de regresión por mínimos cuadrados ordinarios tienen una distribución que depende de un parámetro escalar. El término “<em>Studentización</em>” se utiliza comúnmente para describir una cantidad <em>U</em> dependiente de un parámetro de escala dividida por una estimación de escala <em>S</em>, de forma que el ratio resultante,<em> </em><em>U</em>/<em>S</em>, sigue una distribución que no tiene el inconveniente del parámetro de escala desconocido. La <em>Studentización</em> externa hace referencia a un ratio en que el numerador y el denominador son independientes, mientras que la <em>Studentización</em> interna se refiere al ratio en que ambos son dependientes. La ventaja de la <em>Studentización</em> interna es que puede utilizarse cualquier estimador de escala común, mientras que en la <em>Studentización</em> externa, cada residuo es obtenido por un estimador de escala diferente, con el fin de alcanzar la independencia. Con errores de regresión normales, la distribución conjunta de un conjunto arbitrario (linealmente independiente) de residuos <em>Studentizados</em> internamente está bien documentada. Sin embargo, en algunas aplicaciones una combinación lineal de residuos internamente <em>Studentizados</em> puede resultar útil. Sus limitaciones han sido bien documentadas, pero la distribución no parece haberse derivado en la literatura. Este trabajo contribuye a la literatura existente, en el sentido de obtener la distribución conjunta de una transformación arbitraria lineal de residuos de regresión por mínimos cuadrados ordinarios internamente <em>Studentizados</em> con distribución esférica de error. Todas las principales versiones de los residuos de regresión internamente <em>Studentizados</em> que se han utilizado comúnmente en la literatura son casos especiales de la transformación lineal.</p><p>Ordinary least squares regression residuals have a distribution that is dependent on a scale parameter. The term 'Studentization' is commonly used to describe a scale parameter dependent quantity <em>U</em> divided by a scale estimate <em>S</em> such that the resulting ratio, <em>U</em>/<em>S</em>, has a distribution that is free of from the nuisance unknown scale parameter. <em>External</em> Studentization refers to a ratio in which the nominator and denominator are independent, while <em>internal</em> Studentization refers to a ratio in which these are dependent. The advantage of the internal Studentization is that typically one can use a single common scale estimator, while in the external Studentization every single residual is scaled by different scale estimator to gain the independence. With normal regression errors the joint distribution of an arbitrary (linearly independent) subset of internally Studentized residuals is well documented. However, in some applications a linear combination of internally Studentized residuals may be useful. The boundedness of them is well documented, but the distribution seems not be derived in the literature. This paper contributes to the existing literature by deriving the joint distribution of an arbitrary linear transformation of internally Studentized residuals from ordinary least squares regression with spherical error distribution. All major versions of commonly utilized internally Studentized regression residuals in literature are obtained as special cases of the linear transformation</p>

Distribución de las transformaciones lineales de los residuos mínimos cuadrados studentizados internamente = Distribution of linear transformations of internally studentized least squares residuals

<p>Los residuos de regresión por mínimos cuadrados ordinarios tienen una distribución que depende de un parámetro escalar. El término “<em>Studentización</em>” se utiliza comúnmente para describir una cantidad <em>U</em> dependiente de un parámetro de escala dividida por una estimación de escala <em>S</em>, de forma que el ratio resultante,<em> </em><em>U</em>/<em>S</em>, sigue una distribución que no tiene el inconveniente del parámetro de escala desconocido. La <em>Studentización</em> externa hace referencia a un ratio en que el numerador y el denominador son independientes, mientras que la <em>Studentización</em> interna se refiere al ratio en que ambos son dependientes. La ventaja de la <em>Studentización</em> interna es que puede utilizarse cualquier estimador de escala común, mientras que en la <em>Studentización</em> externa, cada residuo es obtenido por un estimador de escala diferente, con el fin de alcanzar la independencia. Con errores de regresión normales, la distribución conjunta de un conjunto arbitrario (linealmente independiente) de residuos <em>Studentizados</em> internamente está bien documentada. Sin embargo, en algunas aplicaciones una combinación lineal de residuos internamente <em>Studentizados</em> puede resultar útil. Sus limitaciones han sido bien documentadas, pero la distribución no parece haberse derivado en la literatura. Este trabajo contribuye a la literatura existente, en el sentido de obtener la distribución conjunta de una transformación arbitraria lineal de residuos de regresión por mínimos cuadrados ordinarios internamente <em>Studentizados</em> con distribución esférica de error. Todas las principales versiones de los residuos de regresión internamente <em>Studentizados</em> que se han utilizado comúnmente en la literatura son casos especiales de la transformación lineal.</p><p>Ordinary least squares regression residuals have a distribution that is dependent on a scale parameter. The term 'Studentization' is commonly used to describe a scale parameter dependent quantity <em>U</em> divided by a scale estimate <em>S</em> such that the resulting ratio, <em>U</em>/<em>S</em>, has a distribution that is free of from the nuisance unknown scale parameter. <em>External</em> Studentization refers to a ratio in which the nominator and denominator are independent, while <em>internal</em> Studentization refers to a ratio in which these are dependent. The advantage of the internal Studentization is that typically one can use a single common scale estimator, while in the external Studentization every single residual is scaled by different scale estimator to gain the independence. With normal regression errors the joint distribution of an arbitrary (linearly independent) subset of internally Studentized residuals is well documented. However, in some applications a linear combination of internally Studentized residuals may be useful. The boundedness of them is well documented, but the distribution seems not be derived in the literature. This paper contributes to the existing literature by deriving the joint distribution of an arbitrary linear transformation of internally Studentized residuals from ordinary least squares regression with spherical error distribution. All major versions of commonly utilized internally Studentized regression residuals in literature are obtained as special cases of the linear transformation</p>