Non-Stationary Time Series, Cointegration, and Spurious Regression

Transcription

1 Econometrics II Non-Stationary Time Series, Cointegration, and Spurious Regression Econometrics II

2 Course Outline: Non-Stationary Time Series, Cointegration and Spurious Regression 1 Regression with Non-Stationarity Example of Spurious Regression Simulation Example 2 Definitions and Concepts Cointegration Defined Cointegration and Economic Equilibrium Cointegration and Error Correction 3 Engle-Granger Two-Step Cointegration Analysis Static Regression with Cointegrated Series Super-Consistency Residual-Based Tests for No-Cointegration Engle-Granger Two-Step Analysis 4 Empirical Example: Danish Interest Rates 5 Cointegration Analysis Based on Dynamic Models Estimation in the Unrestricted ADL/ECM Model PcGive Test for No-Cointegration Econometrics II Cointegration and Spurious Regression Slide 2

3 Learning Outcomes After completing this topic, you should be able to: 1 Explain the concept of spurious regression. 2 Give a precise definition and interpretation of the concepts cointegration and error correction. 3 Give an account of statistical models based on cointegration and error correction. 4 Construct statistical tests for cointegration and error correction in economic time series. Econometrics II Cointegration and Spurious Regression Slide 3

4 Course Outline: Non-Stationary Time Series, Cointegration and Spurious Regression 1 Regression with Non-Stationarity Example of Spurious Regression Simulation Example 2 Definitions and Concepts Cointegration Defined Cointegration and Economic Equilibrium Cointegration and Error Correction 3 Engle-Granger Two-Step Cointegration Analysis Static Regression with Cointegrated Series Super-Consistency Residual-Based Tests for No-Cointegration Engle-Granger Two-Step Analysis 4 Empirical Example: Danish Interest Rates 5 Cointegration Analysis Based on Dynamic Models Estimation in the Unrestricted ADL/ECM Model PcGive Test for No-Cointegration Econometrics II Cointegration and Spurious Regression Slide 4

5 Motivation: Regression with Non-Stationarity What happens to the properties of OLS if variables are non-stationary? Consider two presumably unrelated variables: CONS BIRD Danish private consumption in 1995 prices. Number of breeding cormorants (skarv) in Denmark. And consider a static regression model log(cons t) = µ + β 2 log(bird t) + u t. We would expect (or hope) to get β2 0 and R 2 0. Applying OLS to yearly data gives the result: with R 2 = log(cons t) = (80.90) log(bird t) + u t, (6.30) It looks like a reasonable model. But it is complete nonsense: spurious regression. Econometrics II Cointegration and Spurious Regression Slide 5

6 The variables are non-stationary. The residual, u t, is non-stationary and The standard variables results are non-stationary. for OLS do not hold. The In residual, general, regression, is non-stationary models for andnon-stationary standard results variables for OLSgive do not spurious hold. results. In general, regression models for non-stationary variables give spurious results. Only exception is if the model eliminates the stochastic trends to produce Only exception is if the model eliminates the stochastic trends to produce stationary stationary residuals: Cointegration. residuals: Cointegration. For non-stationary variables we should always think in terms of For cointegration. non-stationaryonly variables lookwe at should regression always output thinkifin the terms variables of cointegration. are Only stationary look regression or cointegrate. output if the variables are stationary or cointegrate Consumption and breeding birds, logs Consumption Number of breeding birds 1 Regression residuals (cons on birds) of31 Econometrics II Cointegration and Spurious Regression Slide 6

7 Socrative Question 1 We simulate M = 10, 000 replications from the two data-generating processes: 1 Two independent IID variables: y t = ɛ yt x t = ɛ xt where 2 Two independent random walks y t = y t 1 + ɛ yt x t = x t 1 + ɛ xt where ( ɛyt ) N ɛ xt ( ɛyt ) N ɛ xt (( ) 0, 0 (( ) 0, 0 For each replication, we estimate the static regression model, y t = µ + β 2x t + u t, for t = 1, 2,..., T, ( )) 1 0. (1) 0 1 ( )) 1 0. (2) 0 1 and plot the distribution of β2 and the t-ratio for β2 for an increasing T. What happens to the distribution of the t-ratio in Case 2 as T increases? (A) Nothing. (B) The distribution collapses around zero. (C) The distribution converges to a standard normal distribution. (D) The distribution diverges as the variance goes to infinity. (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 7

8

9 Simulation: Stationary Case Simulation: Stationary Case Consider first two independent IID variables: ( ) (( ) ( )) Consider first y t = two ɛ independent IID variables: yt ɛyt where µ N µµ µ,, x t = ɛ= xt ɛ xt where = for T = 50, 100, 500. for = Here, we get standard results for the regression model Here, we get standard results for the regression model y t = µ + β 2x t + u t. = IID Case, estimates Note the convergence to 2 =0 as T diverges IID Case, t-ratios N(0,1) Looks like a N(0,1) for all T. Standard testing of31 Econometrics II Cointegration and Spurious Regression Slide 8

10 Simulation: I(1) Spurious Regression Simulation: I(1) Spurious Regression Now consider two independent random walks ( ) (( ) ( )) y t = y t 1 + ɛ yt ɛyt Now consider where N, x t = xtwo t 1 independent, + ɛ xt random walks ɛ xt µ µµ µ for T = 50, = 100, where Again, we = estimate 1 + the regression model for = y t = µ + β 2x t + u t. Under the null hypothesis, Under the null hypothesis, 2 =0, the residual is I(1). The condition for consistency β 2 = 0, the residual is I(1). The condition for is not consistency fulfilled. is not fulfilled I(1) case, estimates Looks unbiased, but NO convergence I(1) case, t-ratios The distribution gets increasingly dispersed Note the scale as compared to a N(0,1) Econometrics II Cointegration and Spurious Regression 5of31 Slide 9

11 Course Outline: Non-Stationary Time Series, Cointegration and Spurious Regression 1 Regression with Non-Stationarity Example of Spurious Regression Simulation Example 2 Definitions and Concepts Cointegration Defined Cointegration and Economic Equilibrium Cointegration and Error Correction 3 Engle-Granger Two-Step Cointegration Analysis Static Regression with Cointegrated Series Super-Consistency Residual-Based Tests for No-Cointegration Engle-Granger Two-Step Analysis 4 Empirical Example: Danish Interest Rates 5 Cointegration Analysis Based on Dynamic Models Estimation in the Unrestricted ADL/ECM Model PcGive Test for No-Cointegration Econometrics II Cointegration and Spurious Regression Slide 10

12

13

14

15

16

17

18 Socrative Question 2 Assume that the linear relation, z t = c t b 2y t b 3w t, (3) is a cointegration relation between the unit root processes c t, y t, and w t. Which of the following is NOT a valid cointegration relation? (A) β = (2, 2 b 2, 2 b 3). (B) β = (1, b 2, b 3). (C) β = ( 1 b 2, 1, b 3 b 2 ). (D) β = ( 1 b 3, b 2 b 3, 1). (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 11

19 Socrative Question 3 Consider the error correction models for c t, y t, and w t, c t = δ c + α c ( c t 1 β 2y t 1 β 3w t 1 )+ɛ ct y t = δ y + α y ( c t 1 β 2y t 1 β 3w t 1 )+ɛ yt w t = δ w + α w ( c t 1 β 2y t 1 β 3w t 1 )+ɛ wt. What signs are needed for α c, α y, and α w for c t, y t, and w t, respectively, to error correct? (A) α c < 0, α y < 0, α w < 0. (B) α c > 0, α y < 0, α w < 0. (C) α c < 0, α y > 0, α w > 0. (D) α c > 0, α y > 0, α w > 0. (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 12

20 Cointegration Defined Let x t = ( x 1t x 2t ) be two I(1) variables: x 1t = τ 1t + initial value + stationary process x 2t = τ 2t + initial value + stationary process. where τ 1t and τ 2t are stochastic trends. In general, a linear combination of x 1t and x 2t will also have a random walk. Define β = ( 1 β 2 ) and consider the linear combination: z t = β x t = ( ) ( ) x 1 β 1t 2 = x x 1t β 2x 2t 2t = τ 1t β 2τ 2t + initial value + stationary process. Definition of Cointegration Let x t = ( x 1t x 2t ) be two I(1) variables. If there exists a vector β such that z t = β x t is stationary (i.e., z t I(0)), then x 1t and x 2t are said to co-integrate with cointegration vector β. Econometrics II Cointegration and Spurious Regression Slide 13

21 Remarks 1 Cointegration occurs if the stochastic trends in x 1t and x 2t are the same, so that they cancel out: τ 1t β 2τ 2t = 0. This is called a common stochastic trend. 2 You can think of an equation eliminating the random walks in x 1t and x 2t: x 1t = µ + β 2x 2t + u t. ( ) If u t is I(0) (mean zero) then β = ( 1 β 2 ) is a cointegrating vector. Econometrics II Cointegration and Spurious Regression Slide 14

22 Remarks (Continued) 3 The cointegrating vector is only unique up to a constant factor: If β x t I(0), then so is β x t = bβ x t for b 0. We can choose a normalization β = ( 1 β 2 ) or β = ( β1 1 This corresponds to different variables on the left hand side of ( ) ). 4 Cointegration is easily extended to more variables. The variables in x t = ( x 1t x 2t x pt ) cointegrate if z t = β x t = x 1t β 2 x 2t... β p x pt I(0). Econometrics II Cointegration and Spurious Regression Slide 15

23 Cointegration and Economic Equilibrium Consider a regression model for two I(1) variables, x 1t and x 2t, given by x 1t = µ + β 2x 2t + u t. ( ) The term, u t, is interpretable as the deviation from the relation in ( ). If x 1t and x 2t cointegrate, then the deviation u t = x 1t µ β 2x 2t is a stationary process with mean zero. Shocks to x 1t and x 2t have permanent effects. x 1t and x 2t co-vary and u t I(0). We can think of ( ) as defining an equilibrium between x 1t and x 2t. If x 1t and x 2t do not cointegrate, then the deviation u t is I(1). There is no natural interpretation of ( ) as an equilibrium relation. Econometrics II Cointegration and Spurious Regression Slide 16

24 Empirical Example: Consumption and Income Empirical Example: Consumption and Income Time series for log consumption, c t, and log income, y t, are likely to be integrated of order 1, I(1). Time series for log consumption, Define a vector x t = ( c t y,andlogincome, t ). Consumption and,arelikelytobei(1). income are Define cointegrated a vector with =( cointegration ) 0. vector β = ( 1 1 ) if the (log-) consumption-income ratio, Consumption and income are cointegrated with cointegration vector =(1 1 ) 0 ( ) if the (log-) consumption-income ratio, z t = β ct x t = ( 1 1 ) µ = c y t y t, t = 0 =(1 1 ) = is a stationary process. The consumption-income ratio is an equilibrium is arelation. stationary process. The consumption-income ratio is an equilibrium relation. Consumption and income, logs Income Consumption 0.00 Income ratio, log(c t ) log(y t ) of 31 Econometrics II Cointegration and Spurious Regression Slide 17

25 How is the Equilibrium Sustained? There must be forces pulling x 1t or x 2t towards the equilibrium. Engle and Granger s theorem (1987) x 1t and x 2t cointegrate if and only if there exists an error correction model for either x 1t, x 2t, or both. Econometrics II Cointegration and Spurious Regression Slide 18

26 More on Error-Correction Cointegration is a system property. Both variables could error correct, e.g.: x 1t = δ 1 + Γ 11 x 1t 1 + Γ 12 x 2t 1 + α 1 (x 1t 1 β 2x 2t 1) + ɛ 1t x 2t = δ 2 + Γ 21 x 1t 1 + Γ 22 x 2t 1 + α 2 (x 1t 1 β 2x 2t 1) + ɛ 2t, We may write the model as the so-called vector error correction model, ) ( ) ( ) ( ) ( ) ( ) δ1 Γ11 Γ = + 12 x1t 1 α1 ɛ1t + (x x 2t δ 2 Γ 21 Γ 22 x 2t 1 α 1t 1 β 2 x 2t 1 ) +, 2 ɛ 2t ( x1t or simply x t = δ + Γ x t 1 + αβ x t 1 + ɛ t. Note that β x t 1 = x 1t 1 β 2x 2t 1 appears in both equations. For x 1t to error correct we need α 1 < 0. For x 2t to error correct we need α 2 > 0. Econometrics II Cointegration and Spurious Regression Slide 19

27 Consider Considera asimple simple model model for for two two cointegrated cointegrated variables: variables: ( µ ) ( µ ) x1t = = ( (x x 2t 0.1 1t 1 x 2t 1) )+ µ ( 1 ɛ1t ). 2 ɛ 2t (A) Two cointegrated variables (B) Deviation from equilibrium 0 x 1t x 2t 2.5 'x t =x 1t x 2t (C) Speed of adjustment (D) Cross-plot x 1t x 2t 'x t x of 31 Econometrics II Cointegration and Spurious Regression Slide 20

28 Course Outline: Non-Stationary Time Series, Cointegration and Spurious Regression 1 Regression with Non-Stationarity Example of Spurious Regression Simulation Example 2 Definitions and Concepts Cointegration Defined Cointegration and Economic Equilibrium Cointegration and Error Correction 3 Engle-Granger Two-Step Cointegration Analysis Static Regression with Cointegrated Series Super-Consistency Residual-Based Tests for No-Cointegration Engle-Granger Two-Step Analysis 4 Empirical Example: Danish Interest Rates 5 Cointegration Analysis Based on Dynamic Models Estimation in the Unrestricted ADL/ECM Model PcGive Test for No-Cointegration Econometrics II Cointegration and Spurious Regression Slide 21

29 Socrative Question 4 Consider M = 10, 000 simulated replications from the data-generating process, y t = bx t + ɛ yt ɛ yt N(0, 1) x t = θx t 1 + ɛ xt ɛ xt N(0, 1), for t = 1, 2,..., T. We simulate the two cases: 1 Case 1: θ = 0, so x t and y t are I(0) (IID). 2 Case 2: θ = 1, so x t and y t are I(1) and cointegrated. For each replication, we estimate the static regression, y t = µ + β 2x t + u t. What are the properties of the OLS estimator β2? (A) It is inconsistent and not asymptotically normal. (B) It is consistent, but not asymptotically normal. (C) It is super-consistent, but not asymptotically normal. (D) It is super-consistent and asymptotically normal. (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 22

30

31 Static Regression with Cointegrated Series In the cointegration case there exists a β 2 so that the error term, u t, in x 1t = µ + β 2x 2t + u t. ( ) is stationary. OLS applied to ( ) gives consistent results, so that β2 β 2 for T. Note that consistency is obtained even if potential dynamic terms are neglected. This is because the stochastic trends in x 1t and x 2t dominate. We can even get consistent estimates in the reverse regression x 2t = δ + γ 1x 1t + v t. Unfortunately, it turns out that β2 is not asymptotically normal in general. The normal inferential procedures do not apply to β2! We can use ( ) for estimation but not for testing. Econometrics II Cointegration and Spurious Regression Slide 23

32 Super-Consistency Super-Consistency For stationary For stationary series, series, the variance the variance of b of β2 declines at a rate of T 1 2 declines at a rate of 1.. For cointegrated I(1) series, the variance of β2 declines at a faster rate of For cointegrated T 2 I(1) series, the variance of b. 2 declines at a faster rate of 2. Intuition: Intuition: If b 2 = If β2 2 = then β 2 then is ustationary. t is If b If 2 6= β2 2 βthentheerrorisi(1)andwill 2 error have aand large will variance. have a large The variance. information The information on the parameter on thegrows parameter very grows fast. very fast. 30 Stationary, T=50 30 Stationary, T= Stationary, T= Non-Stationary, T= Non-Stationary, T= Non-Stationary, T= Econometrics II Cointegration and Spurious Regression Slide 24 15

33 Simulation Example Extended Consider M = 10, 000 simulated replications from the data-generating process, y t = bx t + b 1x t 1 + ɛ yt ɛ yt N(0, 1) x t = x t 1 + ɛ xt ɛ xt N(0, 1), for t = 1, 2,..., T and with b = 0.2 and b 1 = 0.8. The variables are cointegrated with cointegration vector β = (1, 1). For each replication, we estimate: 1 The static regression, y t = µ + β 2x t + u t, which is misspecied due the omitted dynamic term, b 1x t 1. 2 The ADL(1,1) model, y t = µ + θy t 1 + φ 0x t + φ 1x t 1 + ɛ t, which nests the DGP. We plot the distribution of β from the static regression and β2 = φ 0+ φ 1 1 θ the ADL model. from Econometrics II Cointegration and Spurious Regression Slide 25

34

35 Test for No-Cointegration, Known β 2 Suppose that x 1t and x 2t are I(1). Also assume that β = ( 1 β 2 ) is known. The series cointegrate if is stationary. z t = x 1t β 2x 2t This can be tested using an ADF unit root test, e.g. the test for H 0 : π = 0 in k z t = δ + c i z t i + πz t 1 + η t. i=1 The usual DF critical values apply to t π=0. Note, that the null, H 0 : π = 0, is a unit root, i.e. no cointegration. Econometrics II Cointegration and Spurious Regression Slide 26

36 Test for No-Cointegration, Estimated β 2 Engle-Granger (1987) two-step procedure. If β = ( 1 β 2 ) is unknown, it can be (super-consistently) estimated in x 1t = µ + β 2x 2t + u t. ( ) β is a cointegration vector if û t = x 1t µ β2x 2t is stationary. This can be tested using a DF unit root test, e.g. the test for H 0 : π = 0 in Remarks: û t = k c i û t i + πû t 1 + η t. i=1 1 The residual û t has mean zero. No deterministic terms in DF regression. 2 The critical value for t π=0 still depends on the deterministic regressors in ( ). 3 The fact that β2 is estimated also changes the critical values. OLS minimizes the variance of û t. Look as stationary as possible. Critical value depends on the number of regressors. Econometrics II Cointegration and Spurious Regression Slide 27

37 Dickey-Fuller distribution for test for no-cointegration Estimated parameters in cointegrating regression (with constant in the regression (***)) DF(constant) 0.4 N(0,1) Econometrics II Cointegration and Spurious Regression 18 Slide of 31 28

38 Critical values for the Dickey-Fuller test for no-cointegration Table 1: A constant term in ( ) Number of estimated Significance level parameters 1% 5% 10% Table 2: A constant and a trend in ( ) Number of estimated Significance level parameters 1% 5% 10% Econometrics II Cointegration and Spurious Regression Slide 29

39 Socrative Question 5 Consider the estimated static regression for consumption, income, and wealth, c t = 0.89 [15.5] [11.2] y t w t + û t, (4) [10.7] over the sample 1973(1)-2010(4) with t-ratios in square brackets, and the model for the estimated residuals from (5), û t, û t = 0.26 û t [ 3.31] [ 3.87] û t 1 + η t. What do you conclude regarding cointegration between c t, y t, and w t? (A) They are not cointegrated. (B) They are cointegrated with cointegration vector β = (0.89, 0.44, 0.31). (C) They are cointegrated with cointegration vector β = ( 0.89, 0.44, 0.31). (D) They are only cointegrated if c t is error-correcting. (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 30

40 Socrative Question 5 Consider the estimated static regression for consumption, income, and wealth, c t = 0.89 [15.5] [11.2] y t w t + û t, (4) [10.7] over the sample 1973(1)-2010(4) with t-ratios in square brackets, and the model for the estimated residuals from (5), û t, û t = 0.26 û t [ 3.31] [ 3.87] û t 1 + η t. What do you conclude regarding cointegration between c t, y t, and w t? (A) They are not cointegrated. (B) They are cointegrated with cointegration vector β = (0.89, 0.44, 0.31). (C) They are cointegrated with cointegration vector β = ( 0.89, 0.44, 0.31). (D) They are only cointegrated if c t is error-correcting. (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 30

41 Models for the Dynamic Adjustment Given the estimated cointegrating vector we can define the error correction term ecm t = û t = x 1t µ β2x 2t, which is, per definition, a stationary stochastic variable. Since β2 converges to β 2 very fast we can treat it as a fixed regressor and formulate an error correction model conditional on ecm t 1, i.e. x 1t = δ + λ 1 x 1t 1 + κ 0 x 2t + κ 1 x 2t 1 + α ecm t 1 + ɛ t, where α < 0 is consistent with error-correction. Given cointegration, all terms are stationary, and normal inference applies to δ, λ 1, κ 0, κ 1, and α. We could estimate ECM models for all variables. Econometrics II Cointegration and Spurious Regression Slide 31

42 Socrative Question 6 Let the estimated residuals, û t, from the static regression, c t = 0.89 [15.5] [11.2] y t w t + û t, (5) [10.7] be stationary, so that c t, y t, and w t are cointegrated. Consider the error correction models (lagged first-differences have been omitted, t-ratios in square brackets), c t = 0.24 û t ɛ ct [ 3.85] y t = 0.05 û t ɛ yt [0.52] w t = 0.05 û t ɛ wt [0.78] Which of the three variables are significantly error-correcting? (A) None of the variables are error-correcting. (B) Only c t are error-correcting. (C) y t and w t are error-correcting. (D) All variables are error-correcting. (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 32

43 Outline of an Engle-Granger Analysis 1 Test individual variables, e.g. x 1t and x 2t, for unit roots. 2 Run the static cointegrating regression x 1t = µ + β 2x 2t + u t. Note that the t ratios cannot be used for inference. 3 Test for no-cointegration by testing for a unit root in the residuals, û t. 4 If cointegration is not rejected, estimate a dynamic (ECM) model like x 1t = δ + λ 1 x 1t 1 + κ 0 x 2t + κ 1 x 2t 1 + αû t 1 + ɛ t. All terms are stationary. Remaining inference is standard. Econometrics II Cointegration and Spurious Regression Slide 33

44 Course Outline: Non-Stationary Time Series, Cointegration and Spurious Regression 1 Regression with Non-Stationarity Example of Spurious Regression Simulation Example 2 Definitions and Concepts Cointegration Defined Cointegration and Economic Equilibrium Cointegration and Error Correction 3 Engle-Granger Two-Step Cointegration Analysis Static Regression with Cointegrated Series Super-Consistency Residual-Based Tests for No-Cointegration Engle-Granger Two-Step Analysis 4 Empirical Example: Danish Interest Rates 5 Cointegration Analysis Based on Dynamic Models Estimation in the Unrestricted ADL/ECM Model PcGive Test for No-Cointegration Econometrics II Cointegration and Spurious Regression Slide 34

45 Empirical Example: Danish Interest Rates 0.2 Bond Yield and money market interest rate Bond yield Money market interest rate 0.05 Interest rate spread Residuals from r t = b t t Impulse responses from b t to r t Econometrics II Cointegration and Spurious Regression Slide 35

46 Empirical Example: Danish Interest Rates Consider two Danish interest rates: r t : b t : Money market interest rate (IMM) Bond Yield (IBZ) for the period t = 1972 : : 2. Test for unit roots in r t and b t (5% critical value is 2.86): r t = (1.35) ( 2.39) r t ( 2.70) r t r t 1 ( 1.80) b t = (0.658) (4.67) b t b t 1 ( 0.909) We cannot reject unit roots. Test if s t = r t b t is I(1) (5% critical value is 2.86): s t = ( 3.71) (2.56) s t s t 1. ( 5.35) It is easily rejected that b t and r t are not cointegrating. Econometrics II Cointegration and Spurious Regression Slide 36

47 Empirical Example: Danish Interest Rates Instead of assuming β 2 = 1 we could estimate the coefficient Modelling IMM by OLS The estimation sample is: 1974 (3) to 2003 (2) Coefficient Std.Error t-value t-prob Part.Rˆ2 Constant IBZ sigma RSS Rˆ F(1,114) = [0.000]** log-likelihood DW 0.82 no. of observations 116 no. of parameters 2 mean(imm) var(imm) Econometrics II Cointegration and Spurious Regression Slide 37

48 Empirical Example: Danish Interest Rates We could test for a unit root in the residuals (5% critical value is 3.34): ɛ t = (2.95) Again we reject no-cointegration. ɛ t ɛ t 1. ( 6.77) Finally we could estimate the error correction models based on the spread: r t = ( 3.23) (4.55) b t (r t 1 b t 1) (5.22) b t = ( 2.11) (4.16) b t r t ( 2.01) (r t 1 b t 1) (2.22) Note that the short-rate, r t, error corrects, while the bond-yield, b t, does not. Econometrics II Cointegration and Spurious Regression Slide 38

49 Course Outline: Non-Stationary Time Series, Cointegration and Spurious Regression 1 Regression with Non-Stationarity Example of Spurious Regression Simulation Example 2 Definitions and Concepts Cointegration Defined Cointegration and Economic Equilibrium Cointegration and Error Correction 3 Engle-Granger Two-Step Cointegration Analysis Static Regression with Cointegrated Series Super-Consistency Residual-Based Tests for No-Cointegration Engle-Granger Two-Step Analysis 4 Empirical Example: Danish Interest Rates 5 Cointegration Analysis Based on Dynamic Models Estimation in the Unrestricted ADL/ECM Model PcGive Test for No-Cointegration Econometrics II Cointegration and Spurious Regression Slide 39

50 Estimation of β in the ADL/ECM Model The estimator of β 2 from a static regression is super-consistent....but it is often biased (due to ignored dynamics) and hypotheses cannot be tested. An alternative estimator is based on an unrestricted ADL model, e.g. x 1t = δ + θ 1x 1t 1 + θ 2x 1t 2 + φ 0x 2t + φ 1x 2t 1 + φ 2x 2t 2 + ɛ t, where ɛ t is IID. This is equivalent to a linear regression error correction model: x 1t = δ + λ 1 x 1t 1 + κ 0 x 2t + κ 1 x 2t 1 + γ 1x 1t 1 + γ 2x 2t 1 + ɛ t, or the non-linear regression model x 1t = λ 1 x 1t 1 + κ 0 x 2t + κ 1 x 2t 1 + α (x 1t 1 µ β 2x 2t 1) + ɛ t. Econometrics II Cointegration and Spurious Regression Slide 40

51 Estimation of β in the ADL/ECM Model (Continued) An estimate of β 2 can be found from the long-run solutions: β 2 = γ2 γ 1 = φ 0 + φ1 + φ2, and µ = δ δ =. 1 θ1 θ2 γ 1 1 θ1 θ2 The main advantage is that the analysis is undertaken in a well-specified model. Inference on β2 is possible. The approach is optimal if only x 1t error corrects. Econometrics II Cointegration and Spurious Regression Slide 41

52 Testing for No-Cointegration Due to Engle and Granger s theorem, the null hypothesis of no-cointegration corresponds to the null of no-error-correction. Several tests have been designed in this spirit. The most convenient is the so-called PcGive test for no-cointegration. Consider the unrestricted ADL or ECM: x 1t = δ + λ 1 x 1t 1 + κ 0 x 2t + κ 1 x 2t 1 + γ 1x 1t 1 + γ 2x 2t 1 + ɛ t. (#) Test the hypothesis H 0 : γ 1 = 0 against the cointegration alternative, γ 1 < 0. This is basically a unit root test (not a N(0, 1)). The distribution of the t ratio, t γ1 =0 = γ1 se( γ, 1) depends on the deterministic terms and the number of regressors in (#). Econometrics II Cointegration and Spurious Regression Slide 42

53 Critical values for the PcGive test for no-cointegration are given by: Table 3: A constant term in (#) Number of variables Significance level in x t 1% 5% 10% Table 4: A constant and a trend in (#) Number of variables Significance level in x t 1% 5% 10% Econometrics II Cointegration and Spurious Regression Slide 43

54 Socrative Question 7 Consider the ADL(2,2) model, where ɛ t is IID. x 1t = δ + θ 1x 1t 1 + θ 2x 1t 2 + φ 0x 2t + φ 1x 2t 1 + φ 2x 2t 2 + ɛ t, Q. Which of the following statements is true for t-ratio for the null hypothesis, H 0 : φ 2 = 0? (A) The t-ratio is asymptotically N(0,1). (B) The t-ratio is asymptotically N(0,1) only if x 1t and x 2t are stationary. (C) The t-ratio is asymptotically N(0,1) only if x 1t and x 2t are cointegrated. (D) The t-ratio follows a non-standard distribution. (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 44

55

56 Socrative Question 8 Consider the estimated linear error correction model for c t, EQ( 2) Modelling dc by OLS The dataset is: ConsumptionData.in7 The estimation sample is: 1973(3) (2) Coefficient Std.Error t-value t-prob Part.Rˆ2 dc_ Constant dy dy_ dw dw_ c_ y_ w_ Q. What is your conclusion to the PcGive test for no cointegration? (A) We reject the null, so c t, y t, and w t are cointegrated. (B) We reject the null, so c t, y t, and w t are not cointegrated. (C) We cannot reject the null, so c t, y t, and w t are cointegrated. (D) We cannot reject the null, so c t, y t, and w t are not cointegrated. (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 45

57 Outline of a (One-Step) Cointegration Analysis 1 Test individual variables, e.g. x 1t and x 2t, for unit roots. 2 Estimate an ADL or ECM model x 1t = δ + θ 1x 1t 1 + θ 2x 1t 2 + φ 0x 2t + φ 1x 2t 1 + φ 2x 2t 2 + ɛ t x 1t = δ + λ 1 x 1t 1 + κ 0 x 2t + κ 1 x 2t 1 + γ 1x 1t 1 + γ 2x 2t 1 + ɛ t. 3 Test for no-cointegration with t γ1 =0. If cointegration is found, the cointegrating relation is the long-run solution. 4 Derive the long-run solution x 1t = µ + β2x 2t. Inference on β 2 is standard (under some conditions). 5 Rest of the analysis is standard: All N(0, 1). Look at dynamics. Econometrics II Cointegration and Spurious Regression Slide 46

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76 Empirical Example 1: Consumption, Income, and Wealth Estimation based on a ADL model with two lags. EQ( 1) Modelling c by OLS The dataset is: ConsumptionData.in7 The estimation sample is: 1973(3) (2) Coefficient Std.Error t-value t-prob Part.Rˆ2 c_ c_ Constant y y_ y_ w w_ w_ sigma RSS Rˆ F(8,111) = [0.000]** Adj.Rˆ log-likelihood no. of observations 120 no. of parameters 9 mean(c) se(c) AR 1-5 test: F(5,106) = [0.4769] ARCH 1-4 test: F(4,112) = [0.2554] Normality test: Chiˆ2(2) = [0.0009]** Hetero test: F(16,103) = [0.0232]* Econometrics II Cointegration and Spurious Regression Slide 47

77 Empirical Example 1: Consumption, Income, and Wealth The PcGive test for no-cointegration is given by (5% critical value -3.51): PcGive Unit-root t-test: The long-run solution is given by: Solved static long-run equation for c Coefficient Std.Error t-value t-prob Constant y w Long-run sigma = ECM = c *y *w; WALD test: Chiˆ2(2) = [0.0000] ** Econometrics II Cointegration and Spurious Regression Slide 48

78 Empirical Example 1: Consumption, Income, and Wealth Plot of the estimated cointegration relation: 0.05 ECM ECM = c *y *w; Econometrics II Cointegration and Spurious Regression Slide 49

79 Empirical Example 1: Consumption, Income, and Wealth Plot of the dynamic multipliers, c t+k y t for k 0, and the accumulated dynamic multipliers. Note, PcGive scales the multipliers by the long-run multipliers given by the cointegration coefficients. y 1.0 y(cum) w w(cum) Econometrics II Cointegration and Spurious Regression Slide 50

80 Empirical Example 1: Consumption, Income, and Wealth Estimation based on a linear ECM model. EQ( 2) Modelling dc by OLS The dataset is: ConsumptionData.in7 The estimation sample is: 1973(3) (2) Coefficient Std.Error t-value t-prob Part.Rˆ2 dc_ Constant dy dy_ dw dw_ c_ y_ w_ sigma RSS Rˆ F(8,111) = [0.000]** Adj.Rˆ log-likelihood no. of observations 120 no. of parameters 9 mean(dc) se(dc) AR 1-5 test: F(5,106) = [0.4769] ARCH 1-4 test: F(4,112) = [0.2554] Normality test: Chiˆ2(2) = [0.0009]** Hetero test: F(16,103) = [0.0023]** Econometrics II Cointegration and Spurious Regression Slide 51

81 Empirical Example 2: Interest Rates Revisited Estimation based on a ADL model. The significant terms are: Modelling IMM by OLS (using PR0312.in7) The estimation sample is: 1973 (4) to 2003 (2) Coefficient Std.Error t-value t-prob Part.Rˆ2 IMM_ Constant IBZ IBZ_ sigma RSS Rˆ F(3,115) = [0.000]** log-likelihood DW 2.16 no. of observations 119 no. of parameters 4 mean(imm) var(imm) Econometrics II Cointegration and Spurious Regression Slide 52

82 Empirical Example 2: Interest Rates Revisited The test for no-cointegration is given by (5% critical value 3.21): PcGive Unit-root t-test: The long-run solution is given in PcGive as Solved static long-run equation for IMM Coefficient Std.Error t-value t-prob Constant IBZ Long-run sigma = Here the t values can be used for testing! β 2 is not significantly different from unity. The dynamic multipliers, x 1t/ x 2t, x 1t/ x 2t 1,... and the cumulated x1t/ x 2t i can be graphed. Econometrics II Cointegration and Spurious Regression Slide 53

83 Review Question 1 Consider the estimated cointegrated relation from a static regression, β EG = c t y t 0.31 w t, (6) and the estimated cointegration relation from an ADL(2,2) model, β ADL = c t y t 0.41 w t. (7) Q. Why do we get different results? And which results would you prefer? Please discuss for three minutes. Econometrics II Cointegration and Spurious Regression Slide 54

84 Review Question 2 Consider the AR(1) process with a unit root, and assume that ɛ t NIID(0, σ 2 ). y t = δ + y t 1 + ɛ t, t = 1, 2,..., T, Q. What is the distribution of the forecast of y T +2 conditional on the information set, I T? (A) y T +2 T N(0, σ 2 ). (B) y T +2 T N(y 0 + (T + 2)δ, (T + 2)σ 2 ). (C) y T +2 T N(y T + δ, σ 2 ). (D) y T +2 T N(y T + 2δ, 2σ 2 ). (E) Don t know. Econometrics II Cointegration and Spurious Regression Slide 55

85 Review Question 3 Assume that there are two cointegration relations between consumption, income, and wealth, such as, c t b 2y t I(0) c t b 3w t I(0). Q. Can the two individual cointegration relations be identified and estimated using the Engle-Granger two-step approach or the ADL/ECM approach? (A) No, we can only identify and estimate one cointegration relation with these approaches. (B) We can identify and estimate the two cointegration relations with the Engle-Granger two-step approach only. (C) We can identify and estimate the two cointegration relations with the ADL/ECM approach only. (D) Yes, we can identify and estimate the two cointegration relations with both of these approaches. (E) Don t know. (More generally, think of at least one advantage and limitation of each approach.) Econometrics II Cointegration and Spurious Regression Slide 56

86

87