Table 4: An identified long-run structure in β
The structure : β = (H1^1,..., Hrφr) , y2(10) = 9.19[0.51]
pp |
∙S12 |
∆p2 |
bɪ |
b2 |
∙⅜1 |
⅜ |
t91.1 |
t | |
~lfΓ |
—0.01 |
0.01 |
0.00 |
1.00 |
— 1.00 |
0.00 |
0.00 |
0.00 |
0.00 |
[-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 |
[≡] |
[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 |
[-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'ixt+δ'i∆xt, i = 1, 2, 3 and five stationary cointegration relations between the
differenced variables, τ'∆xt. The difference between β'xt and β±1xt 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'∆xt, 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,1xt + δ'1∆xt = {(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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