Stationarity and nonstationarity

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Chapter 3 Testing for cointegration

In this chapter we discuss the various methods of testing for cointegration which include the two-step estimation procedure developed by Engle and Granger in 1987 the Phillips-Ouliaris residual-based tests the Johansen procedure which builds cointegrated variables directly on maximum likelihood estimation instead of relying on OLS estimates.

The Engle-Granger method

Engle and Granger developed this crucial technique in 1987. This technique entails cointegrated variables which are discussed at length including a proof of Granger’s representation theorem, which connects the moving average, the autoregressive, and the error correcting representation for cointegrated systems.
According to both statisticians, the steps for determining whether two integrated variables cointe-grate of the same order are the following:
pre-test each variable to determine its order of integration, and estimate the error-correction model.
If the integrated variables are found to be integrated to the same order, then it must be tested whether these variables are cointegrated (Johansen, 1988) .

Pre-test each variable

Pre-testing each variable aims at determining the order of integration of each variable. By de ni-tion, cointegration necessitates that two variables be integrated of the same order. This is done using the Augmented Dickey-Fuller (ADF) unit root test to infer the number of unit roots (if any) in each of the variables under investigation. The testing procedure for the ADF unit root test is applied to the following model Xj
yt =  +  t +  yt  1 +yt  j + it=1
where is a constant, the coe cient on a time trend series, the coe cient of yt 1, is the lag order of the autoregressive process, yt = yt yt 1 are rst di erences of yt, yt 1 are lagged values of order one of yt, yt j are changes in lagged values, and it is the white noise.
Once the hypothesis of the unit root test is rejected, we estimate the long-run equilibrium rela-tionship in the form of an OLS regression line, yt01xt + t
If the variables cointegrate, an OLS regression equation 3.1.5 yields a \super-consistent » estimator (Enders, 2004). This means that there is a strong linear relationship between the variables under study. The strong linear relationship can be tested in either of the following ways.
(a) The value of 1 falls between 0.5 and 1.
(b) The plot of yt against xt shows coordinates appearing in an increasing or decreasing direction.
In order to determine if variables cointegrate, we test for unit roots on the residual sequence in the equation 3.1.2 using the ADF test. The residual sequence, denoted by t is a series of estimated values of the deviation from the long-run relationship. They are estimated from t = yt   y^t
where y^t are values from the predicted equation 3.1.5. Testing for unit roots on residuals aims at determining whether these deviations are stationary or not. If they are stationary, then the series cointegrate. If the residuals are not stationary, there is no cointegration. The ADF test is performed on the following model ^t = a1^t  1 + « t
where ^t are the estimated rst di erenced residuals, ^t 1 are the estimated lagged residuals, a1 is the parameter of interest representing the slope of the line and « t are errors obtained in tting both di erenced residuals.
Since the t sequences are residuals from a regression equation, there is no need to include the intercept term in equation 3.1.7. To test the hypothesis on a1 to determine whether the residuals are stationary, we follow the following steps.

Estimate the error correction model

An error correction model is de ned as a dynamic model in which the movement of a variable in any period is related to the previous period’s gap from the long-run equilibrium. Although it may be possible to estimate the long-run or cointegrating relationship, yt = xt + t economic systems are rarely in equilibrium, as they are a ected by institutional and/or structural changes that might be temporary or permanent. For example, extra income in the form of a birthday bonus may raise someone’s expenditure pattern in one or two months and then his/her expenditure gradually goes back to equilibrium. Since equilibrium is rarely observed, the short-run evolution of variables (short-run dynamic adjustment) is important. A simple dynamic model of a short-run adjustment model is given by  yt =     0 0xt1xt  11yt  1 + « t
where yt is the dependent variable, xt is the independent variable, yt  1 and xt  1 are lagged values of yt and xt respectively, 0, 1, 0, 1 are parameters, and « t is the error term assumed to be « t iN(0; 2).
The problems associated with the use of the short-run model are the following:
Multicollinearity: This is a situation in which two or more independent variables in a multiple regression model are highly correlated.
Spurious correlation: This is a situation in which two variables have no causal connection, yet it may be inferred that they do as a result of a certain third unseen factor.
These problems are solved by estimating the rst di erences of equation 3.1.11 to obtain
yt00  xt  11  xt  11  yt  1 + « t:
This, however, introduces problems of loss of information about the long-run equilibrium the economic theory is di erenced away.
The possible solution is to adopt the error-correction mechanism (ECM) formulation of the dy-namic structure. We set up the ECM as follows:
yt00xt1xt  11yt  1 + « t:
Subtracting the term yt on both sides leads to yt00xt1xt  1   (1   1)yt  1 + « t:
the error-correction mechanism which measures the distance of the system away from equilibrium
The coe  cient of « t  1 should be negative in sign in order for the system to converge to equilibrium.
The size of the coe  cient  is an indication of the speed of adjustment towards equilibrium in that small values of tending to  1, indicate that economic agents remove a large percentage of disequilibrium in each period larger values, tending to 0, indicate that adjustment is slow extremely small values, less than 2, indicate an overshooting of economic equilibrium positive values would imply that the system diverges from the long-run equilibrium path.
The ECM satis es the assumptions of classical normal linear regression model (CNLRM). These assumptions include:

  • A linear regression model.
  • Residuals are normally distributed.
  • There is no serial correlation among residuals.
  • The number of observations must not exceed the number of parameters to be estimated. There is no perfect multicollinearity.

This means that diagnostic tests have to be conducted on the error-correction mechanism in order to determine whether any of these assumptions have not been violated. These tests include a normality test a heteroscedasticity test a serial correlation test a misspeci cation test.

 Normality test

We use the Jacque-Bera test to determine whether the ECM is normally distributed. This test measures the di erence in kurtosis and skewness of a variable compared to those of the normal distribution (Jarque and Bera, 1980).

Serial correlation test

Serial correlation is cross-correlation of a signal (white noise) with itself. It may be caused by

  • nonstationarity of dependent and explanatory variable
  • data manipulation (averaging, interpolation and extrapolation)
  • incorrect functional form.

Ljung and Box (1978) suggested the use of Ljung-Box test to test the assumption that the residuals contain no autocorrelation up to any order k. The test procedure is as follows: H0: There is no autocorrelation up to order k.
H1: Autocorrelation exists up to order k.
where T is the number of observations, k is the highest order of autocorrelation for which to test, and rj2 is the jth autocorrelation.
We reject the null hypothesis if the p-value level of signi cance and conclude that autocorrelation exists up to order k.
The major drawback of this test is deciding which lag order (k) to use. Ljung and Box (1978) suggested the maximum number of lags to use should be T 3  where T is the number of observations.

Misspeci cation test

Misspeci cation is as the result of

  • incorrect functional form
  • inclusion of irrelevant variables
  • exclusion of relevant variables.

The consequences of a misspeci ed regression are

  • the residuals are not normally distributed
  • there is serial correlation
  • regression is inconsistent with actual working of the economy- the parameter estimates are not robust to samples used.

To test for misspeci cation, Ramsey (1969) suggested the use of Ramsey’s reset test. This is a like-lihood test that compares the likelihood function of original regression to an augmented regression.

 Limitations of the Engle-Granger method

Although the Engle and Granger procedure is easily implemented, it has several defects:
The estimation of the long-run equilibrium regression requires that the researcher place one variable on the right-hand side as the dependent variable and use the other variable on the left-hand side as the independent variable. For example, in the case of two variables, it is possible to run the Engle-Granger method for cointegration by using the residuals from either of the following two \equilibrium » regression equation.

Table of Contents 
1 Introduction 
2 A theoretical overview 
2.1 Stationarity and nonstationarity
2.2 Unit root
2.3 The augmented Dickey-Fuller (ADF) test
2.4 Cointegration tests
3 Testing for cointegration 
3.1 The Engle-Granger method
3.2 Phillips-Ouliaris methods
3.3 Johansen’s procedure
4 Testing Cointegration Results for the Raotbl3 data set 
4.1 Introduction
4.2 Descriptive summary
4.3 Test for normality
4.4 Augmented Dickey-Fuller (ADF) test for unit roots
4.5 Augmented Dickey-Fuller(ADF) test for the order of integration
4.6 Tests for cointegration
4.7 Concluding remarks
5 Testing for Cointegration in the World Economic Indicators 
5.1 Introduction
5.2 Descriptive summary
5.3 Test for normality
5.4 Augmented Dickey-Fuller (ADF) test for unit roots
5.5 Augmented Dickey-Fuller (ADF) test for order of integration
5.6 Tests for cointegration
5.7 Concluding remarks
6 Testing Cointegration Results for the UKpppuip Data Set 
6.1 Introduction
6.2 Descriptive summary
6.3 Test for normality
6.4 Augmented Dickey-Fuller (ADF) test for unit roots
6.5 Augmented Dickey-Fuller (ADF) test for order of integration
6.6 Tests for cointegration
6.7 Concluding remarks
7 Conclusions and recommendations 
8 Appendix

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