Econometrics-I-6

N i i i 1 = = β ε ∑

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Unformatted text preview: n i i i 1 = = β + ε ∑ v &#152;&#152;&#152;&#152;™ ™ 30/34 Part 6: Finite Sample Properties of LS Implications of Gauss-Markov p Theorem: Var[ b* | X ] – Var[ b | X ] is nonnegative definite for any other linear and unbiased estimator b* that is not equal to b . Implies: p b k = the kth particular element of b. Var[ b k| X ] = the kth diagonal element of Var[ b | X ] Var[ b k| X ] < Var[ b k*| X ] for each coefficient. p cb = any linear combination of the elements of b. Var[ cb | X ] < Var[ cb *| X ] for any nonzero c and b* that is not equal to b . &#152;&#152;&#152;&#152; ™ 31/34 Part 6: Finite Sample Properties of LS Aspects of the Gauss-Markov Theorem Indirect proof: Any other linear unbiased estimator has a larger covariance matrix. Direct proof: Find the minimum variance linear unbiased estimator Other estimators Biased estimation – a minimum mean squared error estimator. Is there a biased estimator with a smaller ‘dispersion’? Yes, always Normally distributed disturbances – the Rao-Blackwell result. (General observation – for normally distributed disturbances, ‘linear’ is superfluous.) Nonnormal disturbances - Least Absolute Deviations and other nonparametric approaches may be better in small samples &#152;&#152;&#152;&#152; &#152;™ 32/34 Part 6: Finite Sample Properties of LS Distribution =- = + ε ′ ′ = ∑ β n i i i 1 1 i i i Source of the random behavior of ( ) where is row i of . We derived E[ | ] and Var[ | ] earlier. The distribution of | is that of the linear combination of the disturbanc b v v X X x x X b X b X b X-- ε ε σ ′ ′ + σ ′ ′ σ = σ β ε ε β β i 2 i 2 1 2 2 1 es, . If has a normal distribution, denoted ~ N[0, ], then | = where ~ N[0, ] and = ( ) | ~ N[ , ] N[ , ( ) ]. Note how b inherits its stochastic properties from b X A I A X X X . b X A I A X X ε . &#152;&#152;&#152;&#152; &#152;™ 33/34 Part 6: Finite Sample Properties of LS Summary: Finite Sample Properties of b p Unbiased: E[ b ]= p Variance: Var[ b | X ] = 2( XX )-1 p Efficiency: Gauss-Markov Theorem with all implications p Distribution: Under normality, b | X ~ N[ , 2( XX )-1 (Without normality, the distribution is generally unknown.) &#152;&#152;&#152;&#152; &#152; 34/34...
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