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I am studying ECM alone using a book and some parts are not explained.

First, the book advise to test for a unit and for the order of integration of the series. In eviews options are not pretty clear to me :

  • What is the difference between level, 1st difference and 2nd difference in the ADF Unit root test ?

Running the test (ADF and intercept) I conclude that my series is not stationary :

Null Hypothesis: LTD_P51S_DHFZ7_CH has a unit root              
Exogenous: Constant             
Lag Length: 1 (Automatic - based on SIC, maxlag=13)             

            t-Statistic   Prob.*

Augmented Dickey-Fuller test statistic          -0.874719    0.7939
Test critical values:   1% level        -3.475500   
    5% level        -2.881260   
    10% level       -2.577365   

*MacKinnon (1996) one-sided p-values.

But when using Trend & Intercept it is not very clear that it is not stationnary :

Null Hypothesis: LTD_P51S_DHFZ7_CH has a unit root              
Exogenous: Constant, Linear Trend               
Lag Length: 2 (Automatic - based on SIC, maxlag=13)             

            t-Statistic   Prob.*

Augmented Dickey-Fuller test statistic          -2.811152    0.1957
Test critical values:   1% level        -4.022586   
    5% level        -3.441111   
    10% level       -3.145082   

*MacKinnon (1996) one-sided p-values.

  • Which test should I use ? Intercept ? Trend ? None ?

  • Also how to test it's order of integration ? I(1), I(2) etc.

Cointegration

Also when you have multiple variables (more than two) in you long term static equation, it is possible that some variable are cointegrated two by two and that you need to run a VECM (from what I have understood).

To test for that I read that you can make a Johansen System Cointegration but I really do not understand all the options (Intercept, trend ; in CE, in VAR etc.) and what they imply, I tried multiple but the results differ greatly.

Thank you.

William
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2 Answers2

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I'll answer your questions pertaining to cointegration.

1) If the context of your exercise is the forecasting of a particular dependent variable by using a set of independent variables as opposed to jointly forecasting a set of variables, then you want to explore an ECM not a VECM, with the latter one being potentially overkill.

2) In your process of exploring an ECM, you can test for cointegration between your dependent variable Y and a set of independent variables {X1, X2, X3} by testing that the residual obtained after regressing Y on X1, X2, and X3 is weakly stationary. This is the first step of the so-called Engle-Granger two-step process. If the residual is stationary then the variables are cointegrated. Cointegrated relationships, however, need not be unique. The shortcoming of the Engle-Granger two-step process is that it will give you at most one cointegrated relationshpip. If there is a reason why you'd expect there to be more than one cointegrated relationship between Y on X1, X2, and X3, the Johansen method could be appealing because it is able to give you more than one cointegrated relationship, but at the price of switching from a single regression to a vector-based regression (or set of regressions), which could be overkill, if you are looking to forecast one single variable, rather than forecast an entire set of variables. Another reason for staying away from the Johansen method, especially if you are new to the method, is that it will give you potentially erroneous results if the VAR is not specified correctly.

So, if you are forecasting a single-variable, stick with ECM and the Engle-Granger method over VECM and Johansen method, but remember that the Engle-Granger method will give you at most one cointegrated relationship, which is perfectly fine in many if not most situations.

ColorStatistics
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1) In ADF, the $H_0$ is the presence of unit root and $H_a$ is for stationary (weakly dependent) series. In your both examples, you don't reject $H_0$. To reject $H_0$, you want the "Augmented Dickey-Fuller test statistic" to be farther from zero (more negative) than the critical value at a significance level chosen. To find out the order of integration, you repeat the test on differenced series (see answer 2).

2) Series are generally stationarized by differencing. So if you don't reject $H_0$ on level values, you take first differences and test for unit root again. If the series is stationary after first differences, its a I(1) series (integrated of order one), if 2nd differences are necessary to stationarize, the series is I(2) - hence the setting in EViews: it lets you determine order of integration by testing on levels, 1st diffs, etc.

3) Constant / Trend / None settings depend on your data generating process. Plot your series to see the best setting - alternatively, use all three settings ("Trend" being the most general, it allows for constant and trend in the DGP).

4) For co-integration, you may test residuals from a cointegration equation for stationarity. I would recommend you to read Chapter 18 from Wooldridge, Introductory econometrics before you actually proceed with the ECM modelling.

Tomas
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