Econometrics-I-6

Definition b is efficient in this class of estimators

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Definition : b is efficient in this class of estimators. n i i i 1    =    = β + ε v ™    30/34
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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 cb = any linear combination of the elements of b. Var[ cb | X ] < Var[ cb *| X ] for any nonzero c and b* that is not equal to b . ™    31/34
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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 ™    32/34
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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 . ™    33/34
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Part 6: Finite Sample Properties of LS Summary: Finite Sample Properties of b p Unbiased: E[ b ]= p Variance: Var[ b | X ] = 2( XX )-1 p Efficiency: Gauss-Markov Theorem with all implications p Distribution: Under normality, b | X ~ N[ , 2( XX )-1 (Without normality, the distribution is generally unknown.)     34/34
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