A very good example of a model with several cointegrating equations has been given by Johansen and Juselius (1990) (1992) (see also Harris (1995)). They considered a single equation approach to combine both Purchasing Power Parity and Uncovered Interest rate Parity condition in one model.

In this model we expect two cointegrating equations between the UK consumer price index P, the US consumer price index P*, USD/GBP exchange rate S and two interest rates I and I * in the domestic and foreign countries respectively. If we denote their log counterparts by the corresponding small letter, the theory suggest that the following two relationships should hold in efficient markets with rational investors: pt — p* = st and Ast+i = it — i*. The data is considered within the range from January 1989 to November 2008 is given in PPPFPl.wfl file.

We create the log counterparts of the variables in the standard ways, like series lcpi_uk=log(cpi_uk) and so on. In order to check for cointegration we can either estimate VECM (open 5 series as VAR model) or create a Group with the variables. Johansen and Juselius (1990) included into the model seasonal dummy variables as well as crude oil prices. We restrict ourself with only seasonal dummy for simplicity. We can create dummy variables by using a command @expand, which allows to create a group of dummy variables by expanding out one or more series into individual categories. For this purposes we need first to create a variable indicating the quarter of the observation. We do it in the following way

series quarter=@quarter(cpi_uk)

The command @quarter returns the quarter of the year in which the current observation begins. The second step is to create the dummy variables:

group dum=@expand(quarter)

EViews will create a new group object dum containing four dummy variables for each of the quarter of the observation.

In both cases, either with VAR or with group objects, one can perform Jo-hansen's test procedure by clicking on View/Cointegration Test....

The dialog window will ask offer to specify the form of the VECM and the cointegrating equation (with or without intercept or trend components). We choose the first option with no trend and intercept to avoid perfect collinearity since we include four dummy variables as exogenous in the model. In the box Exogenous Variables enter the name of the dummy variables group dum.

In the box Lag Intervals for D(Endogenous) we set 14 - we include 4 lags

Johansen Cointegration Test | X |

Figure 6.8: Johansen's Cointegration test dialog window

in the model. This is determined by EViews as optimal according to 3 criteria (first estimate VAR with any of the lag specifications, check the optimality of the lag order in View/Lag Structure/Lag Specification/Lag Length Criteria and then re-estimate the VECM with the optimal lag order).

Figure 6.9: Output for Johansen's Cointegration test

EViews produces results for various hypothesis tested, from no cointegration (r = 0) to increasing number of cointegrating vectors (see Figure ??). The eigenvalues of matrix П is given in the second column. In the third column Xtrace statistic is higher than the corresponding critical value at 5% significance for the first hypothesis. This means that we reject the null hypothesis of no cointegration.

However, we cannot reject the hypothesis that there is at most one cointegrating equation. On the basis of Xmax statistics (the second panel) it is also possible to accept that there is only one cointegrating relationship. The following two panels provide estimates of matrices в and a respectively.

Note the warning on the top of the output window that saying that critical values assume no exogenous series. This means that we have to take into account that the critical values we are using might not be fully correct as we included exogenous dummy variables in the model. This may give as an explanation why we detected only one cointegrating equation instead of two which were expected. Another reason may be that the second relation based on the UIP condition involves changes of exchange rate rather than levels considered in the VAR model.

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