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]

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:

19

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