Econometric take home APPS_Part_31

# Econometric take home APPS_Part_31 - t-1 = ut-1)ut-2)ut-3...

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ε t- 1 = u t- 1 + ( ρ - λ ) u t- 2 + ρ ( ρ - λ ) u t -3 + ρ 2 ( ρ - λ ) u t -4 + ... Therefore, the middle term is zero and the third is simply λσ u 2 . Thus, Cov[ ε t , ε t -1 ] = σ u 2 {[ ρ (1 + λ 2 - 2 ρλ )]/(1 - ρ 2 ) - λ ]} = σ u 2 [( ρ - λ )(1 - λρ )/(1 - ρ 2 )] For lags greater than 1, Cov[ ε t , ε t-j ] = ρ Cov[ ε t -1 , ε t-j ] + Cov[ ε t-j , u t ] - λ Cov[ ε t-j , u t -1 ]. Since ε t-j involves only u s up to its current period, ε t-j is uncorrelated with u t and u t -1 if j is greater than 1. Therefore, after the first lag, the autocovariances behave in the familiar fashion, Cov[ ε t , ε t-j ] = ρ Cov[ ε t , ε t - j +1 ] The autocorrelation coefficient of the residuals estimates Cov[ ε t , ε t -1 ]/Var[ ε t ] = ( ρ - λ )(1 - ρλ )/(1 + λ 2 - 2 ρλ ). 3. Since the regression contains a lagged dependent variable, we cannot use the Durbin-Watson statistic directly. The h statistic in (15-34) would be h = (1 - 1.21/2)[21 / (1 - 21(.18 2 )] 1/2 = 3.201. The 95% critical value from the standard normal distribution for this one-tailed test would be 1.645. Therefore, we would reject the hypothesis of no autocorrelation. 4. It is commonly asserted that the Durbin-Watson statistic is only appropriate for testing for first order autoregressive disturbances. What combination of the coefficients of the model is estimated by the Durbin-Watson statistic in each of the following cases: AR(1), AR(2), MA(1)? In each case, assume that the regression model does not contain a lagged dependent variable. Comment on the impact on your results of relaxing this assumption. In each case, plim d = 2 - 2 ρ 1 where ρ 1 = Corr[ ε t , ε t -1 ]. The first order autocorrelations are as follows: AR(1): ρ (see (15-9)) and AR(2): θ 1 /(1 - θ 2 ). For the AR(2), a proof is as follows: First, ε t = θ 1 ε t -1 + θ 2 ε t -2 + u t . Denote Var[ ε t ] as c 0 and Cov[ ε t , ε t -1 ] as c 1 . Then, it follows immediately that c 1 = θ 1 c 0 + θ 2 c 1 since u t is independent of ε t -1 . Therefore ρ 1 = c 1 / c 0 = θ 1 /(1 - θ 2 ). For the MA(1): - λ / (1 + λ 2 ) (See (15-43)). To prove this, write ε t = u t - λ u t -1 . Then, since the u

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