Cointegration in Single Equations: Lecture 6 Statistical Tests for Cointegration Thomas 15.2 Testing for cointegration between two variables Cointegration is useful since it identifies a long-run relationship between I(1) variables. Nelson and Plosser (1982) argued that many variables in economics are I(1). Cointegration avoids the problem of spurious regressions. We test to see whether there is a cointegrating relationship between our variables.

Cointegration in Single Equations Main Task First establish that two series are I(1) i.e. difference stationary. If both Y t and X t are I(1), there exists the possibility of a cointegrating relationship. If two variables are integrated of different orders, say one is I(2) and other is I(1), there will not be cointegration. Assuming both Y t and X t are I(1), we estimate regression of Y t on a constant and X t Y t = β 0 + β 1 X t + u t (1)

Cointegration in Single Equations Main Task We have both formal and informal methods of establishing cointegration based on the regression residuals u t from equation Y t = β 0 + β 1 X t + u t (1) (a) Informal: Plot the time series of the regression residuals u t and correlogram (i.e. autocorrelation function). Are they stationary? (b) Formal:Test directly if the disequilibrium errors are I(0). u t =Y t - β 0 - β 1 X t If u t is I(0) then an equilibrium relationship exists. And hence we have evidence of cointegration.

Testing for Cointegration (1) Cointegrating Regression Durbin Watson (CRDW) Test Suggested by Engle and Granger (1987). Makes use of Durbin-Watson statistic – similar to Sargan and Bhargava (1983) test for stationarity. Test residuals from regression Y t = β 0 + β 1 X t + u t using DW stat. Low Durbin-Watson statistic indicates no cointegration. Similar to spurious regression result where the Durbin-Watson statistic was low for non-sense regression.

Testing for Cointegration (1) Cointegrating Regression Durbin Watson (CRDW) Test At 0.05 per cent significance level with sample size of 100, the critical value is equal to Ho: DW = 0 => no cointegration (i.e. DW stat. is less than 0.38) Ha: DW > 0 => cointegration (i.e. DW stat. is greater than 0.38) Ho: u t = u t-1 + z t-1 Ha: u t = ρu t-1 + z t-1 ρ < 1 N.B. Assumes that the disequilibrium errors u t can be modelled by a first order AR process. Is this a valid assumption? May require a more complicated model.

Testing for Cointegration (2) Cointegrating Regression Dickey Fuller (CRDF) Test Again based on OLS estimates of static regression Y t = β 0 + β 1 X t + u t We then test regression residuals u t under the null of nonstationarity against the alternative of stationarity using Dickey Fuller type tests. Stationary residuals imply cointegration.

Testing for Cointegration (2) Cointegrating Regression Dickey Fuller (CRDF) Test Use lagged differenced terms to avoid serial correlation. Δu t = φ* u t-1 + θ 1 Δu t-1 + θ 2 Δu t-2 + θ 3 Δu t-3 + θ 4 Δu t-4 + e t Use F-test of model reduction and also minimize Schwarz Information Criteria. Critical Values (CV) are from MacKinnon (1991) Ho: φ* = 0 => no cointegration (i.e. TS is greater than CV) Ha: φ* cointegration (i.e. TS is less than CV)

Testing for Cointegration Advantages of (CRDF) Test Engle and Granger (1987) compared alternative methods for testing for cointegration. (1) Critical values depend on the model used to simulated the data. CRDF was least model sensitive. (2) Also CRDF has greater power (i.e. most likely to reject a false null) compared to the CRDW test.

Testing for Cointegration Disadvantage of (CRDF) Test - Although the test performs well relative to CRDW test there is still evidence that CRDF have absolutely low power. Hence we should show caution in interpreting the results.

Testing for Cointegration Example: Are Y and X cointegrated? First be satisfied that the two time series are I(1). E.g. apply unit root tests to X and Y in turn.

Testing for Cointegration Once we are satisfied X and Y are both I(1), and hence there is the possibility of a cointegrating relationship, we estimate our static regression model. Y t = β 0 + β 1 X t + u t EQ( 1) Modelling Y by OLS (using Lecture 6a.in7) The estimation sample is: 1 to 99 Coefficient Std.Error t-value t-prob Part.R^2 Constant X sigma RSS R^ F(1,97) = 3.935e+004 [0.000]** log-likelihood DW 2.28 no. of observations 99 no. of parameters 2 CRDW test statistic = 2.28 >> 0.38 = 5% critical value. This suggests cointegration - assumes residuals follow AR(1) model.

Testing for Cointegration After estimating the model save residuals from static regression. (In PcGive after running regression click on Test and Store Residuals) Informally consider whether stationary.

Testing for Cointegration Using CRDF we incorporate lagged dependent variables into our regression Δu t = φ* u t-1 + θ 1 Δu t-1 + θ 2 Δu t-2 + θ 3 Δu t-3 + θ 4 Δu t-4 + e t And then assess which lags should be incorporated using model reduction tests and Information Criteria. Progress to date Model T p log-likelihood SC HQ AIC EQ( 2) 94 5 OLS EQ( 3) 94 4 OLS EQ( 4) 94 3 OLS EQ( 5) 94 2 OLS EQ( 6) 94 1 OLS Tests of model reduction (please ensure models are nested for test validity) EQ( 2) --> EQ( 6): F(4,89) = [0.5450] EQ( 3) --> EQ( 6): F(3,90) = [0.5672] EQ( 4) --> EQ( 6): F(2,91) = [0.3628] EQ( 5) --> EQ( 6): F(1,92) = [0.1863] Consequently we choose Δu t = φ* u t-1 + e t All model reduction tests are accepted hence move to most simple model

Testing for Cointegration The estimated results from our model Δu t = φ* u t-1 + e t are as follows EQ( 6) Modelling dresiduals by OLS (using Lecture 6a.in7) The estimation sample is: 6 to 99 Coefficient Std.Error t-value t-prob Part.R^2 residuals_ sigma RSS log-likelihood DW 1.95 Which means Δu t = u t-1 + e t (-11.3) CRDF test statistic = << = 5% Critical Value from MacKinnon. Hence we reject null of no cointegration between X and Y.

Testing for Cointegration: Summary To test whether two I(1) series are cointegrated we examine whether the residuals are I(0). (a) We firstly use informal methods to see if they are stationary (1) plot time series of residuals (2) plot correlogram of residuals (b) Two formal means of testing for cointegration. (1) CRDW - Cointegrating Regression Durbin Watson Test (2) CRDF - Cointegrating Regression Dickey Fuller Test

Next Lecture: Preview In the next lecture we consider - the relationship between cointegration and error correction models. - we illustrate how the disequilibrium errors from a cointegrated regression can be incorporated in a short run dynamic model. - What happens when we have more than two variables. Do you have one cointegrating relationship between say three variables? Do we have more than one cointegrating relationship?