Can we estimate the model some other way to begin

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Unformatted text preview: t independent. Can we use the Durbin Watson test to detect autocorrelation? What do we do instead? 39 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent Lagged Dependent Variables The presence of lagged dependent variables makes OLS estimates biased in finite samples. But they are still consistent. But if there is autocorrelation in the errors then we have inconsistency as well. Yt = β1 + β2 Xt + β3 Yt −1 + ut ut = ρut −1 + εt ε is iid. What is the source of inconsistency? ⇒ Yt −1 and ut are not independent. Can we use the Durbin Watson test to detect autocorrelation? What do we do instead? 40 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent Lagged Dependent Variables OLS won’t work. Can we estimate the model some other way? To begin with, let’s look at a conventional way of removing autocorrelation with the following simple model without a lagged dependent variable. Yt = β1 + β2 Xt + ρut −1 + εt Yt −1 = β1 + β2 Xt −1 + ut −1 Subtract ρ times the second equation from the first and we get after rearranging: Yt = β1 − ρβ1 + ρYt −1 + β2 Xt − ρβ2 Xt −1 + εt 41 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent Lagged Dependent Variables OLS won’t work. Can we estimate the model some other way? To begin with, let’s look at a conventional way of removing autocorrelation with the following simple model without a lagged dependent variable. Yt = β1 + β2 Xt + ρut −1 + εt Yt −1 = β1 + β2 Xt −1 + ut −1 Subtract ρ times the second equation from the first and we get after rearranging: Yt = β1 − ρβ1 + ρYt −1 + β2 Xt − ρβ2 Xt −1 + εt 42 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent Lagged Dependent Variables OLS won’t work. Can we estimate the model some other way? To begin with, let’s look at a conventional way of removing autocorrelation with the following simple model without a lagged dependent variable. Yt = β1 + β2 Xt + ρut −1 + εt Yt −1 = β1 + β2 Xt −1 + ut −1 Subtract ρ times the second equation from the first and we get after rearranging: Yt = β1 − ρβ1 + ρYt −1 + β2 Xt − ρβ2 Xt −1 + εt 43 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent Lagged Dependent Variables OLS won’t work. Can we estimate the model some other way? To begin with, let’s look at a conventional way of removing autocorrelation with the following simple model without a lagged dependent variable. Yt = β1 + β2 Xt + ρut −1 + εt Yt −1 = β1 + β2 Xt −1 + ut −1 Subtract ρ times the second equation from the first and we get after rearranging: Yt = β1 − ρβ1 + ρYt −1 + β2 Xt...
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