When the error termerror term independence premise is

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Focus on the fourth step. When the error term/error term independence premise is satisfied, that is, when the error terms are independent, we can ignore the covariance terms when calculating the variance of a sum of variables. 1 1 2 2 3 3 2 2 2 2 1 2 3 1 Var ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) [ ] x x x x x x x x x x x x = + + + + e e e Error Term/Error Term Independence Premise The error terms are independent: Var[ x + y ] = Var[ x ] + Var[ y ] 1 1 2 2 3 3 2 2 2 2 1 2 3 1 Var[( ) ] Var[( ) ] Var[( ) ] ( ) ( ) ( ) [ ] [ ] x x x x x x x x x x x x = + + + + e e e When autocorrelation is present, however, the error terms are not independent and the covariance terms cannot be ignored. Therefore, when autocorrelation is present the fourth step is invalid: 1 1 2 2 3 3 2 2 2 2 1 2 3 1 Var[ ] Var[( ) ] Var[( ) ] Var[( ) ] ( ) ( ) ( ) [ ] [ ] x x x x x x x x x x x x = + + + + x b e e e Consequently, in the presence of autocorrelation, the equation we used to describe the relationship between the variances of the probability distribution for the error term and the probability distribution coefficient estimate is no longer valid: 2 1 Var[ ] Var[ ] ( ) T t t x x = = x e b The procedure used by the ordinary least squares (OLS) to estimate the variance of the coefficient estimate’s probability distribution is flawed. 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 = = x e b é ã 2 1 EstVar[ ] EstVar[ ] ( ) T t t x x = = x e b
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16 The equation that the ordinary least squares (OLS) estimation procedure uses to estimate the variance of the coefficient estimate’s probability distribution is flawed when autocorrelation is present. Consequently, how can we have faith in the variance estimate? Our Suspicions Let us summarize. After reviewing the algebra we suspect that when autocorrelation is present the ordinary least squares (OLS) estimation procedure for the coefficient value will still be unbiased. variance of the coefficient estimate’s probability distribution may be biased. Confirming Our Suspicions We shall use a simulation to confirm our suspicions. Econometrics Lab 17.2: The Ordinary Least Squares (GLS) Estimation Procedure and Autocorrelation [Link to MIT-Lab 17.2 goes here.] Is OLS estimation Is OLS estimation procedure procedure for the for the variance of the coefficient’s value of the coefficient estimate’s unbiased? probability distribution unbiased? ã é ã é Actual Estimate of Variance of the Estimate of the variance coefficient coefficient estimated coefficient for coefficient estimate’s value value values probability distribution Sample Size 30 Mean (Average) Variance of the Average of Actual of the Estimated Estimated Coefficient Estimated Variances, Estim Value Values, b x , from Values, b x , from EstVar[ b x ], from Rho Proc of β x All Repetitions All Repetitions All Repetitions 0 OLS 2.0 2.0 .22 .22 .6 OLS 2.0 2.0 1.11 .28 Table 17.1: Autocorrelation Simulation Results
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