Econometrics-I-18

# This can be computed at any vector and scalar 2 you

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zero (one of the assumptions of the linear model). This can be computed at any vector and scalar 2. You can take expected values of the parts of the matrix to get ™  14/47 σ σ 2 i 4 1 -E[ ]= n 2 i i x x 0 H 0

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Part 18: Maximum Likelihood Estimation Asymptotic Variance p The asymptotic variance is {–E[ H ]}-1 i.e., the inverse of the information matrix. p There are several ways to estimate this matrix n Inverse of negative of expected second derivatives n Inverse of negative of actual second derivatives n Inverse of sum of squares of first derivatives n Robust matrix for some special cases ™  15/47 ( 29 - - - σ σ σ σ 1 1 2 2 i 1 4 4 {-E[ ]} = = 2 2 n n i i x x 0 X X 0 H 0 0
Part 18: Maximum Likelihood Estimation Computing the Asymptotic Variance We want to estimate {-E[ H ]}-1 Three ways: (1) Just compute the negative of the actual second derivatives matrix and invert it. (2) Insert the maximum likelihood estimates into the known expected values of the second derivatives matrix. Sometimes (1) and (2) give the same answer (for example, in the linear regression model). (3) Since E[ H ] is the variance of the first derivatives, estimate this with the sample variance (i.e., mean square) of the first derivatives, then invert the result. This will almost always be different from (1) and (2). Since they are estimating the same thing, in large samples, all three will give the same answer. Current practice in econometrics often favors (3). Stata rarely uses (3). Others do. ™  16/47

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Part 18: Maximum Likelihood Estimation Poisson Regression Model ™  17/47 Application of ML Estimation: Poisson Regression for a Count of Events exp( ) Poisson Probability: Prob[y=j]= , 0,1,... ! Regression Model: = E[y|x] = exp( x) Competing estimators: N j j j -λ λ = λ β onlinear Least Squares - Consistent Maximum Likelihood - Consistent and Efficient
Part 18: Maximum Likelihood Estimation Application: Doctor Visits p German Individual Health Care data: n=27,236 p Model for number of visits to the doctor: n Poisson regression (fit by maximum likelihood) n Income, Education, Gender ™  18/47

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Part 18: Maximum Likelihood Estimation Poisson Model ™  19/47 [ ] 1 Density of Observed y exp( ) Prob[y = j | ] = ! Log Likelihood logL( | , ) = log log ! Likelihood Equations = Derivatives of log likelihood exp( ) exp( ) = j i i i i n i i i i i i i i i j y y = λ -λ + λ - ∂λ = = x y X x x x β β β β β β β [ ] [ ] 1 1 1 log = = = i i n i i i i i n n i i i i i i i L y y = = = λ + = - λ ε x x x 0 x x β
Part 18: Maximum Likelihood Estimation Asymptotic Variance of the MLE ™  20/47 Variance of the first derivative vector: Observations are independent. First derivative vector is the sum of n independent terms. The variance is the sum of the variances. The variance of each term is V 1 ar[ ( )] = Var[ ] = Summing terms log Var i i i i i i i i i i n i i i i y y L = - λ - λ λ = λ = ∂β x x x x x x x XΛX

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Part 18: Maximum Likelihood Estimation
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