# C bt 1 2 it 2 1 it 1 2 1 s it s1

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Unformatted text preview: nt Common Factor Test The presence of the restriction, λ4 = −λ2 λ3 enables us to test the validity of the model speciﬁcation. This test is known as the Common factor test. This test discriminates two cases d statistic is low because the disturbance term is genuinely subject to an AR(1) process. d statistic is low for other reasons. The usual F test of a restriction is not appropriate because the restriction is nonlinear. Common Factor Test procedure 1 Run the two regressions (the ﬁrst in nonlinear and the second is OLS) 2 Grab the RSS of each 3 RSSrestricted T = nlog RSSunrestricted ∼ χ2 restrictions # 55 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent Common Factor Test The presence of the restriction, λ4 = −λ2 λ3 enables us to test the validity of the model speciﬁcation. This test is known as the Common factor test. This test discriminates two cases d statistic is low because the disturbance term is genuinely subject to an AR(1) process. d statistic is low for other reasons. The usual F test of a restriction is not appropriate because the restriction is nonlinear. Common Factor Test procedure 1 Run the two regressions (the ﬁrst in nonlinear and the second is OLS) 2 Grab the RSS of each 3 RSSrestricted T = nlog RSSunrestricted ∼ χ2 restrictions # 56 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent Past Exam Speciﬁcation; AR(1), MA(1) Statistic; Durbin-Watson, Durbin h, Common factor test, F test Removal of autocorrelation Short run vs Long run effect Restricted vs. Unrestricted (ADL(1,1)) Dynamic Models; Adaptive Expectation, Partial Adjustment Lagged Dependent Variables 57 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent 2007 Q7 58 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent 2007 Q7 59 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent (a) Want to get rid of expectation since it is not observable. To begin with, ite 1 = λit + (1 − λ) ite + Then, Bt = β1 + β2 λit + β2 (1 − λ) ite + ut Also, Bt −1 = β1 + β2 ite + ut −1 β2 ite = Bt −1 − β1 − ut −1 Combining the above equations, Bt = β1 λ + β2 λit + (1 − λ) Bt −1 + ut − (1 − λ) ut −1 γ1 = β1 λ, γ2 = β2 λ, γ3 = 1 − λ, vt = ut − (1 − λ) ut −1 60 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent (b),(c) (b) Lagged dependent variables with autocorrelated error ⇒ Inconsistency due to correlation between one of regressors, Bt −1 and error. (c) Bt = β1 + β2 λit + β2 λ (1 − λ) it −1 + ... +β2 λ (1 − λ)s it −s+1 + ut Nonlinear estimation 61 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Assumption C7 Autocorrelation Detection Lagged Dependent (d),(e),(f) (d) OLS can be used due to no correlation between the regressors and error 62...
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