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1 File: C:\WINWORD\ECONMET\HYPTEST.DOC HYPOTHESIS TESTING IN THE GENERAL LINEAR MODEL The general linear model can be represented in various ways: or in terms of all n observations on each variable written explicitly Y = X + X + X + u 1 2 2 k k 1 β β β ... + in which we have X 1 = 1 for all observations. In compact matrix notation it is written as Here, Y is a (n × 1) vector of observations on the dependent variable, X is a (n × k) matrix of n observations on k explanatory variables, one of which will usually be an intercept. β is a (k × 1) vector of parameters, and u is a (n × 1) vector of disturbance terms. The following assumptions are being made throughout this set of notes: (1) The dependent variable is a linear function of the set of non-stochastic regressor variables and a random disturbance term as specified in Equation (1). This model specification is taken to be the correct one. (2) The n × k matrix X has full rank, k. (3) The random errors, u i , i=1,. .,n, are mean zero, homoscedastic and independent random variables. That is E( u ) = 0 and Var( u ) = E( uu / ) = σ 2 I n OLS POINT ESTIMATES OF PARAMETERS The OLS estimator of the parameter vector β is given by i 1 2,i 2 k,i k i Y = + X + X + u i = 1,. ..,n β β β ... + Y = X + u β (1) β = (X X) X Y -1 (2)
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2 Substituting for Y in (2) from (1) we obtain ( ) β β β β β = (X X) X (X + u) = (X X) X X + (X X) X u and so = + (X X) X u -1 -1 -1 -1 3 SAMPLING PRECISION OF THE OLS ESTIMATOR OF β Information about the sampling precision of β - and the basis for testing hypotheses - comes from the variance/covariance matrix of the OLS estimator, β Var/Cov ( β ) = E X X [( )( ) ] ( ) β β β β σ - - = - 2 1 (4) In (4), for a given sample X is known but σ 2 is not, so (4) as it stands can not be used directly for statistical inference. To proceed, we require an estimator of the scalar term σ 2 . AN UNBIASED LEAST-SQUARES BASED ESTIMATOR OF σ 2 An unbiased estimator of σ 2 is given by k n u ˆ u ˆ ˆ 2 - = σ where u Y Y Y X = - = - β is the least squares residuals vector. AN ADDITIONAL ASSUMPTION
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This note was uploaded on 03/01/2012 for the course EC 408 taught by Professor Rogerperman during the Fall '07 term at Uni. Strathclyde.

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