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```					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

First establish that two series are I(1) i.e. difference stationary.

If both Yt and Xt 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 Yt and Xt are I(1), we estimate regression of Yt on a
constant and Xt
Yt = β0 + β1Xt + ut                             (1)
Cointegration in Single Equations

We have both formal and informal methods of establishing cointegration
based on the regression residuals ut from equation

Yt = β0 + β1Xt + ut                                     (1)

(a) Informal:   Plot the time series of the regression residuals ut and
correlogram (i.e. autocorrelation function).
Are they stationary?

(b) Formal:     Test directly if the disequilibrium errors are I(0).
ut =Yt - β0 - β1Xt
If ut 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 Yt = β0 + β1Xt + ut 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 0.38.

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: ut = ut-1 + zt-1
Ha: ut = ρut-1 + zt-1             ρ<1

N.B. Assumes that the disequilibrium errors ut 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

Yt = β0 + β1Xt + ut

We then test regression residuals ut 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.

Δut = φ* ut-1 + θ1Δut-1 + θ2Δut-2 + θ3Δut-3 + θ4Δut-4 + et

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* < 0      =>    cointegration (i.e. TS is less than CV)
Testing for Cointegration

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

- 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).
55
X    Y
50

45

40

35

30

25

20

15

10

0      10       20   30   40    50    60    70       80   90   100

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.
Yt = β0 + β1Xt + ut

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               4.85755      0.1375      35.3    0.000   0.9279
X                      1.00792    0.005081      198.    0.000   0.9975

R^2                   0.997541   F(1,97) =    3.935e+004 [0.000]**
log-likelihood        -82.8864   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.

1.0
residuals

0.5

0.0

-0.5

0    10             20       30       40       50   60       70       80        90        100
1.0
ACF-residuals

0.5

0.0

-0.5

0   1           2        3        4        5    6   7    8        9        10        11     12
Testing for Cointegration

Using CRDF we incorporate lagged dependent variables into our
regression
Δut = φ* ut-1 + θ1Δut-1 + θ2Δut-2 + θ3Δut-3 + θ4Δut-4 + et
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         -74.238306        1.8212       1.7406     1.6859
EQ( 3)         94    4 OLS         -74.793519        1.7847       1.7202     1.6765
EQ( 4)         94    3 OLS         -74.797849        1.7364       1.6881     1.6553
EQ( 5)         94    2 OLS         -74.948145        1.6913       1.6591     1.6372
EQ( 6)         94    1 OLS         -75.845305        1.6621       1.6459     1.6350
Tests of model reduction (please ensure models are   nested for   test validity)
EQ( 2) --> EQ( 6): F(4,89) = 0.77392 [0.5450]
EQ( 3) --> EQ( 6): F(3,90) = 0.67892 [0.5672]            All model reduction tests are
EQ( 4) --> EQ( 6): F(2,91) =    1.0254 [0.3628]          accepted hence move to most
EQ( 5) --> EQ( 6): F(1,92) =    1.7730 [0.1863]          simple model

Consequently we choose                       Δut = φ* ut-1 + et
Testing for Cointegration

The estimated results from our model Δut = φ* ut-1 + et 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_1           -1.16140      0.1024     -11.3     0.000   0.5805

log-likelihood        -75.8453   DW                          1.95

Which means             Δut = -1.161 ut-1 + et
(-11.3)

CRDF test statistic = -11.3 << -3.39 = 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?

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