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

Table 4: An identified long-run structure in β
The structure : β = (H₁^₁,..., H_rφ_r) , y²(10) = 9.19[0.51]

	pp	∙S12	∆p2	bɪ	b2	∙⅜1	⅜	^t91.1	t
~l^fΓ	—0.01	0.01	0.00	1.00	— 1.00	0.00	0.00	0.00	0.00
~l^fΓ	[-23.49]	[23.49]	[≡]	[≡]	[≡]	[≡]	[≡]	[≡]	[5.60]
β2	0.00	0.00	0.09	0.00	1.00	0.00	— 1.00	0.00	0.00
β2	[≡]	[6.32]	[10.25]	[≡]	[≡]	[≡]	[≡]	[4.36]	[≡]
β^,₃	—0.01	0.00	1.00	0.00	0.00	—0.74	0.00	—0.00	0.00
β^,₃	[-5.51]	[^A]		[^A]		[-16.48]	[^A]	[-4.83]	[^A]
ʌ	—0.92	0.15	0.03	0.03	0.04	0.04	0.04	0.00	—0.01
δ2	—0.51	—0.17	0.02	0.02	0.02	0.03	0.02	0.00	—0.00
δ₃	1.31	—0.07	—0.04	—0.04	—0.05	—0.06	—0.05	0.00	0.00
^β ±1,1	1.00	0.35	11.71	— 1.62	— 1.62	15.80	—0.61	—0.00	0.00
^β ±1,2	2.86	1.00	33.51	—4.67	—4.69	45.23	— 1.80	—0.01	0.01

9 Estimating the I(2) model subject to identifying
restrictions on the long-run structure

The decomposition τ = (β,β±₁) defines three stationary polynomially cointegrating
relations, β'_ix_t+δ'_i∆x_t, i = 1, 2, 3 and five stationary cointegration relations between the
differenced variables, τ'∆x_t. The difference between β'x_t and β±₁x_t is that the latter
can only become stationary by differencing, whereas the former can become stationary
by polynomial cointegration. How to impose and test over-identifying restrictions on
β was discussed in Section 4, but not on β±₁ or τ as their asymptotic distributions are
not yet worked out.

9.1 The estimated long-run structure

To obtain standard errors of the estimated β coefficients we need to impose identifying
restrictions on each of the polynomially cointegrating relations reported above. The
asymptotic distribution of an identified β is given in Johansen (1997).

When interpreting the β relations below we shall only include the first two el-
ements of δ'∆x_t, corresponding to the inflation rate differentials and the deprecia-
tion∕appreciation rate, as they are likely to be more relevant than the other variables.

The first relation is approximately describing the relationship between long-term
interest rate spreads and ppp which have been found in many other VAR models of
similar data:

β^,₁x_t + δ'₁∆x_t = {(b₁ — 0.92∆p₁) — (b₂ — 0.92∆p₂) + 0.15∆s₁2 — 0.01ppp + trend}. (22)

The second is a relation between the US term spread and US inflation relative to
German inflation. It can be interpreted as expected inflation, measured by the term
spread, as a function of actual inflation rates and the change in the Dιnk∕8 rate:

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

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