Department of Mathematics and Statistics, University of Vaasa, Finland January 14 February 27, 2014 Feb 19, 2014
Part VI Cointegration
1 Cointegration (a) Known ci-relation (b) Unknown ci-relation Error Correction Models
Consider I (1) series x t and y t. In general u t = y t βx t I (1) for any β. However, if there exist a β 0 such that y t βx t I (0), then y t and x t are said to be cointegrated. If x t and y t then β in y t βx t is unique. β is called the cointegration parameter and (1, β) is called the cointgration vector (ci-vector). Remark 6.1: If y t βx t I (0), then x t γy t I (0), where γ = 1/β. Note also that for any a 0, ay t aβx t I (0), which implies that the cointegration parameter β is unigue when a is fixed to unity.
Remark 6.2: If x t I (0) and y t I (0) then for any a, b R, ax t + by t I (0) If x t I (1) and y t I (0) then for any a, b R, a 0, ax t + by t I (1).
Cointegrated series do not depart far away from each others. Example 1 Cointegrated series x t and y t with ci-vector (1, 1), such that u t = y t x t is stationary. Cointegrated Series x(t), y(t) 10 0 10 x(t) y(t) 0 100 200 300 400 500 Time Stationary u(t) = y(t) x(t) u(t) 1 1 3 5 0 100 200 300 400 500 Time
1 Cointegration (a) Known ci-relation (b) Unknown ci-relation Error Correction Models
1 Cointegration (a) Known ci-relation (b) Unknown ci-relation Error Correction Models
If the ci-vector (1, β) is known (i.e., β is known, e.g. β = 1), testing for cointegration means testing for stationarity of u t = y t βx t. (1) Testing can be worked out with the ADF testing. Note that in ADF testing the null hypothesis is that the series is I (1). When applied to ci-testing (with known ci-vector) an ADF test indicates cointegration when the ADF null hypothesis is rejected, where u t = y t βx t. H 0 : u t I (1) (2)
Example 2 Consider 6 months and 3 months U.S. T-bill rates (annualized). 20 16 12 1.6 1.2 0.8 0.4 8 4 0 0.0-0.4-0.8 1980 1985 1990 1995 2000 2005 R3 R6 SP63 Ci-test of the spread: ========================================= t-stat p-val ----------------------------------------- ADF -4.932658 0.0001 =========================================
1 Cointegration (a) Known ci-relation (b) Unknown ci-relation Error Correction Models
Ci-parameter β must be estimated from the data. More general cointegration testing can be run on the ci-regression y t = β 0 + βx t + u t (3) where β 0 and β 1 are estimated from the data. Testing can be done with Johansen ML based testing procedure.
Given that x t I (1) and y t I (1), then in the testing procedure there are six different options: 1) No intercept or trend in CE or VAR series: x t = lags(x t, y t ) + e xt, e xt I (0) (4) y t = lags(x t, y t ) + e yt, e yt I (0) (5) y t = βx t + u t (6) 2) Intercept in CE no intercept in VAR x t = lags(x t, y t ) + e xt, e xt I (0) (7) y t = lags(x t, y t ) + e yt, e yt I (0) (8) y t = β 0 + βx t + u t (9)
3) Intercept in CE and in VAR x t = µ x + lags(x t, y t ) + e xt, e xt I (0) (10) y t = µ y + lags(x t, y t ) + e yt, e yt I (0) (11) y t = β 0 + βx t + u t (12) 4) Intercept and trend in CE only intercept in VAR x t = µ x + lags(x t, y t ) + e xt, e xt I (0) (13) y t = µ y + lags(x t, y t ) + e yt, e yt I (0) (14) y t = β 0 + δt + βx t + u t (15)
5) Intercept and trend in both CE and VAR x t = µ x + δ x t + lags(x t, y t ) + e xt, e xt I (0) (16) y t = µ y + δ y t + lags(x t, y t ) + e yt, e yt I (0) (17) y t = β 0 + δt + βx t + u t (18) The most common is option 3). Remark 6.3: In any of the above CI-specification x t and y t are modeled as general VAR (Vector AutoRegression).
Example 3 General cointegration test of the short 3 months and 6 months rates using option 3) (Eviews). As a default the program suggests to add 4 short run lags (four VAR differences) into the model. Unrestricted Cointegration Rank Test (Trace) ============================================================ Hypothesized Trace 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** ------------------------------------------------------------ None * 0.179202 26.80702 15.49471 0.0007 At most 1 0.027409 3.307149 3.841466 0.0690 ============================================================ Trace test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegration Rank Test (Maximum Eigenvalue) ============================================================ Hypothesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** ------------------------------------------------------------ None * 0.179202 23.49987 14.26460 0.0013 At most 1 0.027409 3.307149 3.841466 0.0690 ============================================================ Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values
The Johansen test produces two test statistics, Trace test and Maximum eigenvalue. Both tests suggest the existence of a cointegration relation. The cointegration parameter estimates to ˆβ = 1.039 in the ci-regression regresson r6 = β 0 + βr3 t + u, i.e., the estimated ci-vector is (1, 1.039), which is close to (1, 1).
Error Correction Models 1 Cointegration (a) Known ci-relation (b) Unknown ci-relation Error Correction Models
Error Correction Models Cointegration analysis allows modeling at the same time the long run relation (ci-relation) and short run relations (error corrections, or equilibrium correction). The latter tells in particular how the system gets back to the long run equilibrium indicated by the ci-relation.
Error Correction Models If x t and y t are I (1) series, then usual analysis deals with the differences, e.g., y t = φ 0 + φ 1 y t 1 + γ x t + u t. (19) If y t and x t are cointegrated such that the model can be enhanced to ci t = y t βx t I (0), y t = φ 0 + αci t 1 + φ 1 y t 1 + γ x t + u t. (20) The system is in long run equilibrium if ci t = y t βx t = 0.
Error Correction Models If, for example, ci t 1 = y t 1 βx t 1 < 0, then y t 1 is below its equilibrium value. Then, if α < 0, next period a positive correction effect is expected in y t. Thus α is the short run correction coefficient towards the equilibrium.
Error Correction Models Example 4 Consider the 6 months adjustment towards the equilibrium (spread) s t = r6 t r3 t. r6 t = φ 0 + αs t 1 + lags( r6, r3) + u t (21) (Demonstrated in the classroom).