Testing Hypotheses in an I(2) Model with Applications to the Persistent Long Swings in the Dmk/$ Rate

Table 1: Estimated outlier coefficients

Dummy variables in the model

	Dtax	79:10	82:08	84:01	85:02	86:02	88:08	89:02	97:07
∆∆pp	0.01 [9.77]	*	*	*	*	*	*	*	0.01 [3.49]
∆∆s12	*	*	*	0.06 [2.67]	*	—0.07 [-2.35]	*	*	*
∆∆π	*	*	—0.00 [-2.75]	*	*	—0.00 [-3.33]	*	*	*
∆∆5₁	*	*	*	*	0.00 [2.36]	*	*	*	*
∆∆62	*	0.00 [2.31]	—0.00 [-2.64]	*	0.00 [4.08]	—0.00 [-3.96]	*	*	*
∆∆s₁	*	*	*	*	0.00 [2.79]	*	0.00 [5.55]	0.00 [4.40]	*
∆∆s₂	*	0.00 [5.18]	—0.00 [-12.67]	—0.00 [-13.41]	0.00 [3.27]	*	*	*	*

t-values in brackets, * indicates a t-value < 2.0

robust to moderate ARCH and excess kurtosis.

7 Determining the two reduced rank indices

The number of stationary polynomial cointegrating relations, r, and the number of
I(1) trends, s₁, among the common stochastic trends, p — r, are determined by the
ML trace test procedure in Johansen (1997). Table 3 reports the tests of the joint
hypothesis (r, s₁ , s₂) for all values of r,s₁ and s₂∙ The test procedure starts with the
most restricted model (r = 0, s₁ = 0, s₂ = 5) in the upper left hand corner, continues
to the end of the first row (r = 0, s₁ = 5, s₂ = 0), and proceeds similarly row-wise from
left to right until the first acceptance. Based on the tests, the first acceptance is at
(r = 2, s₁ = 4, s₂ = 1), whereas the next acceptance is at (r = 3, s₁ = 2, s₂ = 2), which
is at a much higher p-values.

Since our model has a broken linear trend restricted to be in the cointegration
relations, and a shift dummy restricted to the differences, the asymptotic trace test
distribution provided by CATS should be shifted to the right, i.e., the test is likely to
be somewhat undersized. The trace tests suggest the possibility of either r = 2 or 3.
Thus, it is useful to perform a sensitivity check before the final choice of r, s₁, and s₂.

The characteristic roots assuming no 1(2) trends show that the choice of (r = 2, s₁ =
5) leaves a large unrestricted root (0.90) in the model, whereas (r = 3,s₁ = 4) leaves
two (0.93 and 0.90). Both cases seem to suggest a total of approximately six (near) unit
roots in the model, consistent with both (r = 2, s₁ = 3, s₂ = 2) and (r = 3, s₁ = 2, s₂ =
2). Thus, the final decision seems to be between two (three) polynomial cointegration
relations (^'x_t + J∆^_t) and three (two) medium-run relations in differences (^^z_l1∆^⅛).
Checking the t—values of a₃ shows five highly significant coefficients (with t-values
in the range of 15.4 to 3.4), which suggests that the third polynomial cointegration

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