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Furthermore, since by A1, E(u) = 0, then Ε( βˆ ) = β, which proves the first part of the theorem. Proof of Part (2) of the Theorem By definition, Cov( βˆ ) = E{[ βˆ – Ε( βˆ )][ βˆ – E( βˆ )]′}. 23) above, βˆ – β = (X′X)–1X′u so that: () [ ] [(X′X) X′u]′ = E [(X ′X) X ′uu ′X (X ′X) ] = [(X ′X) X ′E (uu ′) X (X ′X) ] −1 Cov βˆ = E (X ′X) X ′u −1 −1 8 −1 −1 −1 In matrix algebra, a positive semidefinite matrix corresponds, in scalar algebra, to a number that is greater than or equal to zero. In stricter terms, a matrix A is said to be positive semidefinite if there exists a vector x such that the quadratic form x′Ax ≥ 0, for all x.

15 The General Linear Regression Model THE GAUSS–MARKOV THEOREM ON LEAST SQUARES The white noise error term introduced above has the following properties: E (u t ) = 0 E u 2t ) = σ 2 E (u t us ) = 0 ( all t all t s≠t (Property 1) (Property 2) (Property 3) Each one of these properties has a simple yet meaningful interpretation. Property 1 implies that the error term can influence import behaviour in either direction – positive or negative – and is prone neither to be more negative than positive or vice versa.

How does one therefore decide whether it is the multicollinearity or the economic theory underlying the model that is to blame? A possible answer lies, not in an examination of the standard errors and p-values of the model alone, but rather in the study of the R 2 , R 2 , and F-statistics as well. Consider the following scenario. The OLS estimation of a linear regression model yields coefficient values that have associated with them extremely large standard errors and as such, very large p-values.