Analysis of Cointegration and Error Correction Models
In-Class Exercise 7
Question 1
This question helps to familiarize with the analysis of cointegration and error correction models. The dataset we are using is forex.csv, which can be downloaded from Moodle.
Part 1: Using Eviews for Analysis
First, load the file into Eviews. To get a sense of the data over the sampling period, highlight and open the two series spotrate and forwardrate. Then choose View -> Graph and click OK.
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Clearly, the two series show some trending behavior, and they seldom cross their own mean values. Therefore, it is believed that they are non-stationary.
Also, the general trend behavior looks quite similar, providing visual (not statistical) evidence that the spot and forward rates are cointegrated.
Next, we have to use the ADF test to check whether the two level series are stationary or not.
For Spot Rate:
(Level Data)
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As the ADF test statistic is greater than the critical value at 5%, it suggests that the series contains a unit root and is non-stationary.
(First-Differenced Series)
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As the ADF test statistic is less than the critical value at 5%, it suggests that the series does not contain a unit root and is stationary.
For Forward Rate:
(Level Data)
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As the ADF test statistic is greater than the critical value at 5%, it suggests that the series contains a unit root and is non-stationary.
(First-Differenced Series)
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As the ADF test statistic is less than the critical value at 5%, it suggests that the series does not contain a unit root and is stationary.
To check the presence of cointegration, we first run the regression of Y = spotrate against X = forwardrate.
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Then, we have to save the residuals of the model and name it as u. This can be done by typing the following statement in the command window:
genr u=resid
Next, we perform the EG test for u:
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Notice that the critical value for testing the stationarity of u is not correct. The correct critical value is provided by Engle and Granger (1987) and is found in slide 8 of the lecture notes.
Since the ADF test statistic is less than the critical value of -3.37 at the 5% level, we conclude it does not contain a unit root and is stationary. In other words, the spot rate and forward rate are cointegrated, and the regression of the spot rate on the forward rate will not be spurious. Here, the coefficient on the forward rate is referred to as the long-run multiplier.
Then, we proceed to the error correction model. ECM can be written as:
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where
SR = Spot Rate
FR = Forward Rate
u = Residual from the regression model of spot rate against forward rate
The ECM can be estimated by typing the following statement in the estimation window of Eviews:
d(spotrate) c d(forwardrate) u(-1)
The output is:
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Consistently, the coefficient on ut-1 is negative. For a positive error, the spot rate will fall next period to restore equilibrium. In particular, an equilibrium error of one unit tends to cause the spot rate to fall by 1.085 units in the next period, ceteris paribus.
If the forward rate remains unchanged for some time (ΔFR = 0), but then suddenly it changes by one unit, the model implies that Y would instantly change by 1.044 units.
Question 2
This question focuses on the Granger causality test. The dataset we employ is stockpab.csv.
Part 1: Using Eviews for Analysis
By the ADF test, we find that the level series are non-stationary, but its first differenced series are stationary.
Since the level series are expressed as the log of the price, we can convert them by taking the first difference. This can be done by typing the following command:
genr da=100*d(countrya)
genr db=100*d(countryb)
Then, we are going to test whether they are cointegrated. We regress the log price of country A on that of country B, and then obtain the residual. The EG test concludes that there is no cointegrating relationship between the two.
Since there is no integration relationship, we simply run the following models for the Granger causality test:
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and
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Here, we type the following commands to estimate the above models:
da c db(-1) db(-2) db(-3) db(-4) da(-1) da(-2) da(-3) da(-4) time
db c db(-1) db(-2) db(-3) db(-4) da(-1) da(-2) da(-3) da(-4) time
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From the first table, it is found that the stock return in country B has strong explanatory power for the stock return in country A. In other words, the stock return in country B Granger causes the stock return in country A.
On the other hand, we find that the stock returns in country A are all insignificant. Hence, the stock return does not seem to Granger cause the stock returns in country B.
Actually, under no cointegrating relationship, Eviews has a built-in function for the Granger causality test:
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Here is the procedure:
Choose Quick -> Group Statistics -> Granger Causality Test
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In the box, we type the name of the differenced variables.
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Choose the number of lags (for example, lag = 4).
The output will be generated:
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For the first test, the p-value is 0.0016, which means that the null hypothesis that the stock return in country B does not Granger cause the stock return in country A is rejected. In other words, the stock return in country B Granger causes the stock return in country A.
For the second test, the p-value is 0.9459, which means that the null hypothesis that the stock return in country A does not Granger cause the stock return in country B cannot be rejected. In other words, the stock return in country A does not Granger cause the stock return in country B.