Consider as an example the Forward Premium Puzzle. Due to rational expectation hypothesis, forward rate should be unbiased predictor of future spot exchange rate. This means that in the regression of levels of spot St+ on forward rate Ft the intercept coefficient should be equal to zero and the slope coefficient should be equal to unity.

Consider monthly data of the USG/GBP spot and forward exchange rate for the period from January 1986 to November 2008 (the data is in FPP.wf1 file).

Unit roots are often found in levels of spot and forward exchange rates. Augmented Dickey-Fuller test statistic values are -2.567 and -2.688 which are high enough to fail rejecting the null hypothesis at 5% significance level. Phillips-Perron test produces test statistic which value on the border of the rejection region. Thus, if two series are not cointegrated, there is a danger to obtain spurious results from the OLS regression. However, if we look at plots of the two series we can see that they co-move together very closely, so we can expect existence of cointegrating relation between them.

To perform Engle-Granger test for cointegration let us run OLS regression St+i = [3Ft + ut in EViews and generate residuals from the model.

Is f_spt fwd series resid1=resid

The second step is to test the residuals for stationarity. Augmented Dickey-Fuller test strongly rejects the presence of a unit root in the residual series in the favour of stationarity hypothesis.

Similar results are generated by other testing procedures. Thus, we conclude that future spot and forward exchange rates are cointegrated. Hence, the OLS results are valid for the regression in levels as well. In this case the slope coefficient is equal to 0.957 which is positive and close to unity. However, we reject the null hypothesis H0: в = 1 with the Wald test.

Thus, the forward premium puzzle also exists even for the model in levels for the exchange rates.

Figure 6.4: Plots of forward and future spot USD/GBP exchange rates

Figure 6.5: Results of Augmented Dickey-Fuller test for residuals from the long-run equilibrium relationship

Figure 6.6: Wald test results for testingH0: $ = 1

Another way of estimating cointegrating equation is to estimate a vector error correction model. To do this, open both forward and spot series as VAR system (select both series and in the context menu choose Open/as VAR...). In the VAR type box select Vector Error Correction and in the Cointegration tab click on Intercept (no trend) in CE - no intercept in VAR. EViews' output is given in Figure ??.

As expected, the output shows that the stationary series is approximately St+i — Ft with the mean around zero. Deviations from the long-run equilibrium equation have significant effect on changes of the spot exchange rate. Another highly significant coefficient a^2 indicates a significant impact of ASt on AFt which is not surprising. This underlies the relationships between the spot and forward rate through the Covered Interest rate Parity condition (CIP).

The following subsection introduces an approach of testing for cointegration

Figure 6.7: Output of the vector error correction model

when there exists more than one cointegrating relationship.

6.2.6. Tests for Cointegration: The Johansen's Approach

An alternative approach to test for cointegration was introduced by Johansen (1988). His approach allows to avoid some drawbacks existing in the Engle-Granger's approach and test the number of cointegrating relations directly. The method is based on the VAR model estimation.

Consider the VAR{p) model for the k x 1 vector Yt

where ut ~ IN(0, E).

Since levels of time series Yt might be non-stationary, it is better to transform Equation (6.2.5) into a dynamic form, calling vector error correction model (VECM)

Let us assume that Yt contains non-stationary I(1) time series components. Then in order to get a stationary error term ut, ÏÏYt_i should also be stationary. Therefore, ÏÏYt_i must contain r < k cointegrating relations. If the VAR(p) process has unit roots then П has reduced rank rank(II) = r < k. Effectively, testing for cointegration is equivalent to checking out the rank of the matrix П.

If П has a full rank then all time series in Y are stationary, if the rank of П is zero then there are no cointegrating relationships.

If 0 < rank (П) = r < k. This implies that Yt is I(1) with r linearly independent cointegrating vectors and k — r non-stationary vectors. Since П has rank r it can be written as the product

where a and ¡3 are k x r matrices with rank(a) = rank(3) = r. The matrix 3 is a matrix of long-run coefficients and a represents the speed of adjustment to disequilibrium. The VECM model becomes

Johansen's methodology of obtaining estimates of a and (3 is given below. Johansen's Methodology

Specify and estimate a VAR(p) model (6.2.5) for Yt.

Determine the rank of LT; the maximum likelihood estimate for (3 equals the matrix of eigenvectors corresponding to the r largest Eigen values of a k x k residual matrix (see Hamilton (1994), Lutkepohl (1991), Harris (1995) for more detailed description).

Construct likelihood ratio statistics for the number of cointegrating relationships. Let estimated eigenvalues are Ai > A2 > > Ak of the matrix n. Johansen's likelihood ratio statistic tests the nested hypotheses

The likelihood ratio statistic, called the trace statistic, is given by

It checks whether the smallest k — r0 eigenvalues are statistically different from zero. If rank (n) = r0 then Aro+i k should all be close to zero and LRtrace(r0) should be small. In contrast, if rank (n) > r0 then some of Aro+1 Ak will be nonzero (but less than 1) and LRtrace (r0) should be large.

We can also test H0: r = r0 against H1: r0 = r0 + 1 using so called the maximum eigenvalue statistic

Critical values for the asymptotic distribution of LRtrace(r0) and LRmax(r0) statistics are tabulated in Osterwald-Lenum (1992) for к — r0 = 1 10.

In order to determine the number of cointegrating vectors, first test H0: r0 = 0 against the alternative H: r0 > 0. If this null is not rejected then it is concluded that there are no cointegrating vectors among the к variables in Yt. If Ho: r0 = 0 is rejected then there is at least one cointegrating vector. In this case we should test H0: r0 < 1 against H: r0 > 1. If this null is not rejected then we say that there is only one cointegrating vector. If the null is rejected then there are at least two cointegrating vectors. We test H0: r0 < 2 and so on until the null hypothesis is not rejected.

In a small samples tests are biased if asymptotic critical values are used without a correction. Reinsel and Ahn (1992) and Reimars (1992) suggested small samples bias correction by multiplying the test statistics with T — kp instead of T in the construction of the likelihood ratio tests.

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