Question are these estimation procedures still

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Question: Are these estimation procedures still unbiased when autocorrelation is present? Ordinary Least Squares (OLS) Estimation Procedure for the Coefficient Value Begin by focusing on the coefficient value. Previously, we showed that the estimation procedure for the coefficient value was unbiased by applying the arithmetic of means; and recognizing that the means of the error terms’ probability distributions equal 0 (since the error terms represent random influences). Let us quickly review. First, recall the arithmetic of means: Mean of a constant plus a variable: Mean[c + x ] = c + Mean[ x ] Mean of a constant times a variable: Mean[c x ] = c Mean[ x ] Mean of the sum of two variables: Mean[ x + y ] = Mean[ x ] + Mean[ y ] To keep the algebra straightforward, we focused on a sample size of 3: Equation for Coefficient Estimate: 1 1 1 2 2 3 3 2 2 2 2 1 2 3 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) T t t t x x x T t t x x e x x e x x e x x e b x x x x x x x x β β = = + + = + = + + +
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11 Now, some algebra: 1 1 1 2 2 3 3 2 2 2 1 2 3 ( ) ( ) ( ) Mean[ ] Mean ( ) ( ) ( ) [ ] x x x x x x x x x x x x x x β + + = + + + e e e b Applying Mean[ c + x ] = c + Mean[ x ] 1 1 2 2 3 3 2 2 2 1 2 3 ( ) ( ) ( ) Mean ( ) ( ) ( ) [ ] x x x x x x x x x x x x x β + + = + + + e e e Rewriting the fraction as a product 1 1 2 2 3 3 2 2 2 1 2 3 1 Mean ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) [ ] x x x x x x x x x x x x x β = + + + + + e e e Applying Mean[ c x ] = c Mean[ x ] 1 1 2 2 3 3 2 2 2 1 2 3 1 Mean ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] x x x x x x x x x x x x x β = + + + + + e e e Applying Mean[ x + y ] = Mean[ x ] + Mean[ y ] 1 1 2 2 3 3 2 2 2 1 2 3 1 Mean[( ) ] Mean[( ) ] Mean[( ) ] ( ) ( ) ( ) [ ] x x x x x x x x x x x x x β = + + + + + e e e Applying Mean[ c x ] = c Mean[ x ] 1 1 2 2 3 3 2 2 2 1 2 3 1 ( )Mean[ ] ( )Mean[ ] ( )Mean[ ] ( ) ( ) ( ) [ ] x x x x x x x x x x x x x β = + + + + + e e e Since Mean[ e 1 ] = Mean[ e 2 ] = Mean[ e 3 ] = 0 x β = What is the critical point here? We have not relied on the error term/error term independence premise to show that the estimation procedure for the coefficient value is unbiased. Consequently, we suspect that the estimation procedure for the coefficient value will continue to be unbiased in the presence of autocorrelation.
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12 Ordinary Least Squares (OLS) Estimation Procedure for the Variance of the Coefficient Estimate’s Probability Distribution Next, consider the estimation procedure for the variance of the coefficient estimate’s probability distribution used by the ordinary least squares (OLS) estimation procedure: The strategy involves two steps: First, we used the adjusted variance to estimate the variance of the error term’s probability distribution: EstVar[ ] Degreesof Freedom SSR = e estimates Var[ e ]. Second, we applied the equation relating the variance of the coefficient estimates probability distribution and the variance of the error term’s probability distribution: 2 1 Var[ ] Var[ ] ( ) T t t x x = = x e b Step 1: Estimate the variance of the error term’s Step 2: Apply the relationship between the probability distribution from the available variances of coefficient estimate’s and information – data from the first quiz error term’s probability distributions EstVar[ ] Degreesof Freedom SSR = e 2 1 Var[ ] Var[ ] ( ) T t t x x = =
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