Cointegration and Error Correction Models

1
Cointegration and Error Correction Models
2
Introduction

• Assess the importance of stationary variables
  when running OLS regressions.
• Describe the Dickey-Fuller test for stationarity
• Explain the concept of Cointegration with a
  bi-variate model
• Discuss the importance of error correction models
  and their relationship to cointegration.
• Describe how to test for a set theory using
  cointegration.

3
OLS Regression with I(1) data

• The following results were produced when output
  was regressed against stock prices

4
OLS Regression with I(1) data

• In the previous slide, the results can not be
  interpreted as there is clear evidence of
  autocorrelation.
• However the explanatory power is very high
  suggesting a very good result.
• In this case the drift in both variables is
  related, but not explicitly modelled, causing
  autocorrelation. But as the drifts in the two
  variables is related, the explanatory power is
  high
• This produces the case where the R-squared
  statistic is larger than the DW statistic, often
  referred to as an indirect test for cointegration

5
Difference Stationary and Trend Stationary

• The main method for inducing stationarity is to
  difference the data. For instance the random walk
  becomes stationary on differencing

6
Trend Stationary

• A series is said to be trend stationary when it
  is stationary around a trend

7
Differenced Variables

• If in a bi-variate model, both variables are
  difference-stationary, then one way around the
  problem is to run a model with differenced
  variables instead of level variables

8
Differenced Variables

• However this option may not be acceptable as
• - The variables in this form may not be in
  accordance with the original theory
• - This model could be omitting important
  long-run information, differenced variables
  are usually thought of as representing the
  short-run.
• - This model may not have the correct
  functional form.

9
Stationary data

• One of the most important tests for stationarity
  is the Dickey-Fuller Test or Augmented
  Dickey-Fuller Test (ADF).
• The test is based on a random walk and the fact
  that a random walk has a unit root.
• If the variable in question follows a random
  walk, it is therefore not stationary.
• This is why when testing to determine if a
  variable is stationary, it is said to be testing
  for a unit root.

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Dickey-Fuller Test for Stationarity

• The test is based on the following regression.
  The coefficient on the lagged level variable is
  then used to test if it equals zero, in the same
  way as a t-test

11
Dickey-Fuller Test

• This test assumes that the error term (u) follows
  the Gauss-Markov assumptions.
• The test statistic does not follow the
  t-distribution, the critical values have been
  produced specifically for this test.
• A constant and trend could also be included in
  this test, the test statistic would still be the
  test for whether the coefficient on the lagged
  level variable equals zero
• In this case the test is for a unit root against
  no unit root, i.e. the variable needs to be
  differenced once to induce stationarity.

12
Augmented Dickey-Fuller Test (ADF)

• The error term in the Dickey-Fuller test usually
  has autocorrelation, which needs to be removed if
  the result is to be valid. The main way is to add
  lagged dependent variables until the
  autocorrelation has been mopped up.
• The test is the same as before in that it is the
  coefficient on the lagged dependent variable that
  is tested.

13
Augmented Dickey-Fuller Test

• The test is as follows, where the number of
  lagged dependent variables is determined by an
  information criteria

14
I(2) Variables

• When a variable contains two unit roots, it is
  said to be I(2) and needs to be differenced twice
  to induce stationarity.
• When using the ADF test, the data is first tested
  to determine if it contains a unit root, i.e. it
  is I(1) and not I(0)
• If it is not I(0), it could be I(1), I(2) or have
  a higher order of unit roots
• In this case the ADF test needs to be conducted
  on the differenced variable to determine if it is
  I(1) or I(2). (It is very rare to find I(3) or
  higher orders).

15
Dickey-Fuller Test

• Most tests using the Dickey-Fuller (DF) and
  Augmented Dickey-Fuller (ADF) technique are
  considered to have low power. (Accept the null of
  a unit root more often than should). The power
  depends on
• The time span of the data rather than the number
  of observations.
• If ? is roughly equal to one, but not exactly,
  the ADF test may indicate a non-stationary
  process
• These tests assume a single unit root, but many
  time series are I (2) or higher
• The tests fail to account for structural breaks
  in the time series.

16
Engle-Granger Approach to Cointegration

• This is essentially a bi-variate approach and is
  based on the Augmented Dickey-Fuller test for
  stationarity.
• If we have two non-stationary variables
  containing a unit root (i.e. I(1) variables),
  then we describe them as being cointegrated if
  the error term is stationary (i.e. I(0)).
• We test for the stationarity of the error term
  using the ADF test in the same way as the
  individual variables.

17
Cointegration

• When we have an I(0) error term, with two I(1)
  variables, in effect the drift process in the
  I(1) variables have cancelled each other out to
  produce an error term with no drift.
• If there is evidence of cointegration between X
  and Y, we say that there is a long-run
  equilibrium relationship between X and Y

18
Granger Representation Theorem

• According to Granger, if there is evidence of
  cointegration between two or more variables, then
  a valid error correction model should also exist
  between the two variables.
• The error correction model is then a
  representation of the short-run dynamic
  relationship between X and Y, in which the error
  correction term incorporates the long-run
  information about X and Y into our model.
• This implies that the error correction term will
  be significant, if cointegration exists.

19
Engle-Granger Two-Step Method

• The method involves firstly estimating the
  cointegrating relationship and test for
  cointegration.
• The second stage involves forming the error
  correction model, where the error correction term
  is the residual from the cointegrating
  relationship, lagged once.

20
Cointegration Example

• The following cointegrating relationship was run,
  the residual was then tested to determine if it
  was stationary and the error correction model
  (ECM) formed

21
Cointegration Example

• In the previous slide, to determine if the
  variables are cointegrated, the ADF test has been
  conducted on the residual, giving a test
  statistic of (-0.78/0.24) -3.25, this is more
  negative than the -2.89 critical value so we
  reject the null hypothesis of no cointegration.
• The ECM is then formed using the residual lagged
  one time period as the error correction term.

22
Error Correction Models

• An error correction model includes only I(0)
  variables.
• This requires all our non-stationary variables to
  be first-differenced, to produce stationary
  variables
• The error correction term is the residual from
  the cointegrating relationship, lagged one time
  period, this too will be I(0) if the variables
  are cointegrated
• The error correction model can include a number
  of lags on both variables

23
Error Correction Models

• The ECM models the short-run dynamics of the
  model.
• As with short-run models including lags, it can
  be used for forecasting.
• The coefficient on the error correction term can
  be used as a further test for cointegration. It
  is called the Bannerjee ECM test and requires a
  separate set of critical values to determine if
  cointegration has occurred.

24
Error Correction Term

• The error correction term tells us the speed with
  which our model returns to equilibrium following
  an exogenous shock.
• It should be negatively signed, indicating a move
  back towards equilibrium, a positive sign
  indicates movement away from equilibrium
• The coefficient should lie between 0 and 1, 0
  suggesting no adjustment one time period later, 1
  indicates full adjustment
• The error correction term can be either the
  difference between the dependent and explanatory
  variable (lagged once) or the error term (lagged
  once), they are in effect the same thing.

25
Example of ECM

• The following ECM was formed, using 60
  observations

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Example of an ECM

• The error correction term has a t-statistic of 4,
  which is highly significant supporting the
  cointegration result.
• The coefficient on the error correction term is
  negative, so the model is stable.
• The coefficient of -0.32, suggests 32 movement
  back towards equilibrium following a shock to the
  model, one time period later.

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Potential Problems with Cointegration

• The ADF test often indicates acceptance of the
  null hypothesis (no cointegration), when in fact
  cointegration is present
• The ADF test is best when we have a long time
  span of data, rather than large amounts of
  observations over a short time span. This can be
  a problem with financial data which tends to
  cover a couple of years, but with high frequency
  data (i.e. daily data)
• It is only really used for bi-variate
  cointegration tests, although it can be used for
  multivariate models, a different set of critical
  values is required.

28
Multivariate Approach to Cointegration

• A different approach to testing for cointegration
  is generally required when we have more then 2
  variables in the model
• If we assume all the variables are endogenous, we
  can construct a VAR and then test for
  cointegration
• One of the most common approaches to multivariate
  cointegration is the Johansen Maximum Likelihood
  (ML) test.
• This test involves testing the characteristic
  roots or eigenvalues of the p matrix
  (coefficients on the lagged dependent variable).

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Steps in Testing for Cointegration

1. Test all the variables to determine if they are
   I(0), I(1) or I(2) using the ADF test.
2. If both variables are I(1), then carry out the
   test for cointegration
3. If there is evidence of cointegration, use the
   residual to form the error correction term in the
   corresponding ECM
4. Add in a number of lags of both explanatory and
   dependent variables to the ECM
5. Omit those lags that are insignificant to form a
   parsimonious model
6. Use the ECM for dynamic forecasting of the
   dependent variable and assess the accuracy of the
   forecasts.

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Conclusion

• The Dickey-Fuller or Augmented Dickey-Fuller
  tests test for stationarity, based on the test
  for a random walk.
• The Engle-Granger approach to cointegration in a
  bi-variate model, involves testing for
  stationarity of the residual using the ADF test.
• According to the Granger representation theorem,
  if there is cointegration between our two
  variables, we should be able to form the
  appropriate error correction model.