hyptest - File C\WINWORD\ECONMET\HYPTEST.DOC HYPOTHESIS...

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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 In addition to the three assumptions made previously, we now also assume that the vector of error terms is normally distributed. Thus we may write u N I n ~ ( , ) 0 2 σ This assumption implies that the vector Y is multivariate normally distributed with mean vector as
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3 Y N X I n ~ ( , ) β σ 2
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