A vector error correction model (VECM) can lead to a better understanding of the nature of any nonstationarity among the different component series and can also improve longer term forecasting over an unconstrained model.
The VECM(p) form with the cointegration rank is written as
where is the differencing operator, such that ; , where and are matrices; is a matrix.
It has an equivalent VAR(p) representation as described in the preceding section.
where is a identity matrix.
An example of the second-order nonstationary vector autoregressive model is
with
This process can be given the following VECM(2) representation with the cointegration rank one:
The following PROC IML statements generate simulated data for the VECM(2) form specified above and plot the data as shown in Figure 35.12:
proc iml; sig = 100*i(2); phi = {-0.2 0.1, 0.5 0.2, 0.8 0.7, -0.4 0.6}; call varmasim(y,phi) sigma=sig n=100 initial=0 seed=45876; cn = {'y1' 'y2'}; create simul2 from y[colname=cn]; append from y; quit; data simul2; set simul2; date = intnx( 'year', '01jan1900'd, _n_-1 ); format date year4. ; run;
proc timeseries data=simul2 vectorplot=series; id date interval=year; var y1 y2; run;
The following statements use the Johansen cointegration rank test. The COINTTEST=(JOHANSEN) option does the Johansen trace test and is equivalent to specifying COINTTEST with no additional options or the COINTTEST=(JOHANSEN=(TYPE=TRACE)) option.
/*--- Cointegration Test ---*/ proc varmax data=simul2; model y1 y2 / p=2 noint dftest cointtest=(johansen); run;
Figure 35.13 shows the output for Dickey-Fuller tests for the nonstationarity of each series and Johansen cointegration rank test between series.
Figure 35.13: Dickey-Fuller Tests and Cointegration Rank Test
Unit Root Test | |||||
---|---|---|---|---|---|
Variable | Type | Rho | Pr < Rho | Tau | Pr < Tau |
y1 | Zero Mean | 1.47 | 0.9628 | 1.65 | 0.9755 |
Single Mean | -0.80 | 0.9016 | -0.47 | 0.8916 | |
Trend | -10.88 | 0.3573 | -2.20 | 0.4815 | |
y2 | Zero Mean | -0.05 | 0.6692 | -0.03 | 0.6707 |
Single Mean | -6.03 | 0.3358 | -1.72 | 0.4204 | |
Trend | -50.49 | 0.0003 | -4.92 | 0.0006 |
Cointegration Rank Test Using Trace | ||||||
---|---|---|---|---|---|---|
H0: Rank=r |
H1: Rank>r |
Eigenvalue | Trace | Pr > Trace | Drift in ECM | Drift in Process |
0 | 0 | 0.5086 | 70.7279 | <.0001 | NOINT | Constant |
1 | 1 | 0.0111 | 1.0921 | 0.3441 |
In Dickey-Fuller tests, the second column specifies three types of models, which are zero mean, single mean, or trend. The third column ( Rho ) and the fifth column ( Tau ) are the test statistics for unit root testing. Other columns are their p-values. You can see that both series have unit roots. For a description of Dickey-Fuller tests, see the section PROBDF Function for Dickey-Fuller Tests in Chapter 5: SAS Macros and Functions.
In the cointegration rank test, the last two columns explain the drift in the model or process. Since the NOINT option is specified, the model is
The column Drift in ECM indicates that there is no separate drift in the error correction model, and the column Drift in Process indicates that the process has a constant drift before differencing.
H0 is the null hypothesis, and H1 is the alternative hypothesis. The first row tests against ; the second row tests against . The trace test statistics in the fourth column are computed by , where is the available number of observations and is the eigenvalue in the third column. The p-values for these statistics are output in the fifth column. If you compare the p-value in each row to the significance level of interest, such as 5%, the null hypothesis that there is no cointegrated process (H0: ) is rejected, whereas the null hypothesis that there is at most one cointegrated process (H0: ) cannot be rejected.
The following statements fit a VECM(2) form to the simulated data. From the result in Figure 35.13, the time series are cointegrated with rank = 1. You specify the ECM= option together with the RANK=1 option. For normalizing the value of the cointegrated vector, you specify the normalized variable by using the NORMALIZE= option. The PRINT=(IARR) option provides the VAR(2) representation. The VARMAX procedure output is shown in Figure 35.14 through Figure 35.16.
/*--- Vector Error-Correction Model ---*/ proc varmax data=simul2; model y1 y2 / p=2 noint lagmax=3 ecm=(rank=1 normalize=y1) print=(iarr estimates); run;
The ECM= option produces the estimates of the long-run parameter, , and the adjustment coefficient, . In Figure 35.14, "1" indicates the first column of the and matrices. Since the cointegration rank is 1 in the bivariate system, and are two-dimensional vectors. The estimated cointegrating vector is . Therefore, the long-run relationship between and is . The first element of is 1 since is specified as the normalized variable.
Figure 35.15 shows the parameter estimates in terms of lag one coefficients, , and lag one first differenced coefficients, , and their significance. "Alpha * Beta" indicates the coefficients of and is obtained by multiplying the "Alpha" and "Beta" estimates in Figure 35.14. The parameter AR1 corresponds to the elements in the "Alpha * Beta" matrix. The t values and p-values corresponding to the parameters AR1 are missing since the parameters AR1 have non-Gaussian distributions. The parameter AR2 corresponds to the elements in the differenced lagged AR coefficient matrix. The "D_" prefixed to a variable name in Figure 35.15 implies differencing.
Figure 35.15: Parameter Estimates for the VECM(2) Form
Parameter Alpha * Beta' Estimates | ||
---|---|---|
Variable | y1 | y2 |
y1 | -0.46680 | 0.91295 |
y2 | 0.10667 | -0.20862 |
AR Coefficients of Differenced Lag | |||
---|---|---|---|
DIF Lag | Variable | y1 | y2 |
1 | y1 | -0.74332 | -0.74621 |
y2 | 0.40493 | -0.57157 |
Model Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Equation | Parameter | Estimate | Standard Error |
t Value | Pr > |t| | Variable |
D_y1 | AR1_1_1 | -0.46680 | 0.04786 | y1(t-1) | ||
AR1_1_2 | 0.91295 | 0.09359 | y2(t-1) | |||
AR2_1_1 | -0.74332 | 0.04526 | -16.42 | 0.0001 | D_y1(t-1) | |
AR2_1_2 | -0.74621 | 0.04769 | -15.65 | 0.0001 | D_y2(t-1) | |
D_y2 | AR1_2_1 | 0.10667 | 0.05146 | y1(t-1) | ||
AR1_2_2 | -0.20862 | 0.10064 | y2(t-1) | |||
AR2_2_1 | 0.40493 | 0.04867 | 8.32 | 0.0001 | D_y1(t-1) | |
AR2_2_2 | -0.57157 | 0.05128 | -11.15 | 0.0001 | D_y2(t-1) |
The fitted model is given as
The PRINT=(IARR) option in the previous SAS statements prints the reparameterized coefficient estimates. For the LAGMAX=3 in the SAS statements, the coefficient matrix of lag 3 is zero.
The VECM(2) form in Figure 35.16 can be rewritten as the following second-order vector autoregressive model: