Unit Root and Cointegration

Transcription

1 Unit Root and Cointegration Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign Oct 7th, 016 C. Hurtado (UIUC - Economics) Applied Econometrics

2 On the Agenda 1 Announcements Unit Root 3 Testing for Unit Root 4 Dickey-Fuller in Practice 5 Cointegration C. Hurtado (UIUC - Economics) Applied Econometrics

3 Announcements On the Agenda 1 Announcements Unit Root 3 Testing for Unit Root 4 Dickey-Fuller in Practice 5 Cointegration C. Hurtado (UIUC - Economics) Applied Econometrics

4 Announcements Announcements Problem Set 3: due on Monday C. Hurtado (UIUC - Economics) Applied Econometrics 1 / 16

5 Unit Root On the Agenda 1 Announcements Unit Root 3 Testing for Unit Root 4 Dickey-Fuller in Practice 5 Cointegration C. Hurtado (UIUC - Economics) Applied Econometrics

6 Unit Root Unit Root Let us consider the simplest model y t = ρy t 1 + u t where u t N(0, σ ), starting from y 0 = 0. If ρ = 1, by repeated substitution we can write Then, the variance of y t is t y t = y 0 + u j j=1 t Var(y t ) = σ = tσ j=1 The variance of y t is diverging to infinity with t. C. Hurtado (UIUC - Economics) Applied Econometrics / 16

7 Testing for Unit Root On the Agenda 1 Announcements Unit Root 3 Testing for Unit Root 4 Dickey-Fuller in Practice 5 Cointegration C. Hurtado (UIUC - Economics) Applied Econometrics

8 Testing for Unit Root Testing for Unit Root Let us consider the simplest model where u t N(0, σ ) y t = ρy t 1 + u t The first idea would be to use OLS to test unit root. Denote by ˆρ the OLS estimator of the AR(1) model. We know that n(ˆρ ρ) d N ( 0, σ ( 1 n y t 1 ) 1 ) C. Hurtado (UIUC - Economics) Applied Econometrics 3 / 16

9 Testing for Unit Root Testing for Unit Root Notice that y t = ρy t 1 + u t = ρ y t + ρu t 1 + u t. = ρ j u t j j=0 If ρ < 1, then Var(y t ) = σ 1 ρ C. Hurtado (UIUC - Economics) Applied Econometrics 4 / 16

10 Testing for Unit Root Testing for Unit Root Hence, if ρ < 1, 1 y p σ n t 1 1 ρ Therefore, n(ˆρ ρ) d N ( 0, σ ( 1 n y t 1 ) 1 ) n(ˆρ ρ) d N(0, 1 ρ ) If ρ = 1 we can not use traditional t-test for OLS estimator. C. Hurtado (UIUC - Economics) Applied Econometrics 5 / 16

11 Testing for Unit Root Dickey-Fuller Asymptopia The OLS estimator for ρ is ˆρ = arg min (yt ρy t 1 ) = ρ arg min y t ρ y t y t 1 + ρ yt 1 ρ = yt y t 1 y t 1 If ρ = 1 then, ˆρ 1 = yt y t 1 y t 1 1 = = yt y t 1 yt 1 y t 1 yt 1 (y t y t 1 ) yt 1 u y = t t 1 y t 1 C. Hurtado (UIUC - Economics) Applied Econometrics 6 / 16

12 Testing for Unit Root Dickey-Fuller Asymptopia Let us look to numerator and denominator separately. Numerator: - y t = y t 1 + y t 1u t + u t implies that yt 1 u t = 1 ( yt ) ut - We know that y T N(0, σ T ) and Var( u t ) = σ T, then Where X χ 1 1 d 1 yt 1 σ u t (X 1) T C. Hurtado (UIUC - Economics) Applied Econometrics 7 / 16

13 Testing for Unit Root Dickey-Fuller Asymptopia Let us look to numerator and denominator separately. Denominator: - We know that y t 1 N(0, σ (t 1)), hence E yt 1 = σ T (T 1) (t 1) = σ - Then, the order of y t 1 is T. C. Hurtado (UIUC - Economics) Applied Econometrics 8 / 16

14 Testing for Unit Root Dickey-Fuller Asymptopia Combine the two pieces: T (ˆρ 1) = Where ˇχ 1 is a rescaled and recentered χ 1 1 T yt 1 u t 1 T y t 1 d ˇχ 1 Hence, the τ-statistic τ = ˆρ 1 (ˆσ y t 1) 1/ has distribution that is not the same as that of an ordinary t-statistic, even asymptotically. C. Hurtado (UIUC - Economics) Applied Econometrics 9 / 16

15 Dickey-Fuller in Practice On the Agenda 1 Announcements Unit Root 3 Testing for Unit Root 4 Dickey-Fuller in Practice 5 Cointegration C. Hurtado (UIUC - Economics) Applied Econometrics

16 Dickey-Fuller in Practice Dickey-Fuller in Practice Let us consider a simple AR(1) model y t = ρy t 1 + u t where u t N(0, σ ) If represents the first difference operator, y t = (ρ 1)y t 1 + u t = δy t 1 + u t Testing for a unit root is equivalent to testing δ = 0. From previous slides we know that we need to adjust when testing for unit root. Dickey-Fuller adjusted for three versions: 1. No intercept nor time trend: y t = δy t 1 + u t. Intercept but no time trend: y t = a 0 + δy t 1 + u t 3. Intercept and time trend: y t = a 0 + a 1 t + δy t 1 + u t C. Hurtado (UIUC - Economics) Applied Econometrics 10 / 16

17 Dickey-Fuller in Practice Dickey-Fuller in Practice C. Hurtado (UIUC - Economics) Applied Econometrics 11 / 16

18 Dickey-Fuller in Practice Augmented Dickey-Fuller Test What to do if we have more complicated error process? For example, suppose u t is ARMA(1, 1). i.e. u t = c + Φu t 1 + θε t 1 + ε t with ε iid The testing procedure for the ADF test is the same as for the Dickey-Fuller, but it is applied to the model y t = a 0 + a 1 t + δ 0 y t 1 + δ 1 y t δ k 1 y t k+1 + ε t where a 0 is a constant, a 1 is the coefficient on a time trend and k is the lag order of the autoregressive process. remarkably, the τ-test statistic in this regression has the same asymptotic distribution as in the simple case. C. Hurtado (UIUC - Economics) Applied Econometrics 1 / 16

19 Dickey-Fuller in Practice Augmented Dickey-Fuller: Final Remarks Unit root tests are very sensitive to the number of included lags and/or constant and trends. Very likely, some of the results will indicate the presence of unit root while others will not. How to make a general conclusion on the test results with so many models available? Some authors recommend the use AIC or SIC in the model selection. It is simple to calculate information criteria in ADF tests. Each output of ADF corresponds to a linear regression on the lags, constant, and/or trend of the series. From OLS regression, you recover the sample size, the RSS, and the number of parameters requested to calculate SIC or AIC, plus the original ADF statistic. But remember to use the Dickey-Fuller critical values. C. Hurtado (UIUC - Economics) Applied Econometrics 13 / 16

20 Cointegration On the Agenda 1 Announcements Unit Root 3 Testing for Unit Root 4 Dickey-Fuller in Practice 5 Cointegration C. Hurtado (UIUC - Economics) Applied Econometrics

21 Cointegration Cointegration A time series is integrated of order d if (1 L) d x t is a stationary process, where L is the lag operator and (1 L) is the first difference. Many economic variables are, or at least appear to be, I(1). We know that I(1) variables tend to diverge as T. It might seem that two or more of such variables could never be expected to obey any sort of long-run relationship. However, variables that are all individually I(1), and hence divergent, can in a certain sense diverge together. Formally, it is possible for some linear combinations of a set of I(1) variables to be I(0). If that is the case, the variables are said to be cointegrated. C. Hurtado (UIUC - Economics) Applied Econometrics 14 / 16

22 Cointegration Testing for Cointegration Engle-Granger two-step method: - To test the cointegration of two series {x t, y t } define z t = a + bx t + cy t. - To if there is cointegration it must be the case that b 0 and c 0. - Without loss of generality, divide everything by c and rename z t = α βx t + y t. - But we don t know α nor β. - First step: Estimate α and β using OLS and compute the residuals. - Second step: Proceed with a unit root test on the residuals, i.e. test whether the residuals are I(0) using an ADF test. The critical values to be used here are no longer the same provided by Dickey-Fuller, but instead provided by Engle and Yoo (1987). C. Hurtado (UIUC - Economics) Applied Econometrics 15 / 16

23 Cointegration Testing for Cointegration C. Hurtado (UIUC - Economics) Applied Econometrics 16 / 16