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**Cointegration and Error Correction Models**

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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.

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**OLS Regression with I(1) data**

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

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**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

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**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:

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Trend Stationary A series is said to be trend stationary when it is stationary around a trend:

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**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:

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**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.

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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:

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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.

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**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.

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**Augmented Dickey-Fuller Test**

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

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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).

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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.

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**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.

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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

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**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.

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**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.

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**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:

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**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 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.

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**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

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**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.

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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.

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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.

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**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 π matrix (coefficients on the lagged dependent variable).

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**Steps in Testing for Cointegration**

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

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