timing of the structural change under the alternative hypothesis is estimated endogenously.
Gregory and Hansen suggest three alternative models accommodating changes in parameters of
the cointegration vector under the alternative. A level shift model allows for the change in the
intercept only (C):
y11 = μ + μ2^tτ + a’y2t + et, t = 1,......,n (1)
.
The second model accommodating a trend in data also restricts shift only to the change in
level with a trend (C/T):
y 11 = μ + μ2Ψtτ + βt + a'y21 + et, t = 1,......,n (2)
The most general specification allows for changes both in the intercept and slope of the
cointegration vector (R/S):
y 11 = μ + μ2Ψtτ + a y 11 + a2' y2tψtτ + et, t = 1,......, n
(3)
The dummy variable, which captures the structural change, is represented as:
Ψtτ
0, t ≤ [nτ]
Д, t > [nτ]
(4)
where τ ∈ (0,1) is a relative timing of the change point. The trimming interval is usually
taken to be (0.15n, 0.08n), as recommended in Andrews (1993). The models (1)-(3) are estimated
sequentially with the break point changing over the intervalτ∈ (0.15n,0.85n) . Non-stationarity
of the obtained residuals, expected under the null hypothesis, is checked by ADF test. Setting the
test statistics (denoted as ADF*) to the smallest value of the ADF statistics in the sequence, we
select the value that constitutes the strongest evidence against the null hypothesis of no
cointegration.
5.2. Recursive Cointegration Tests
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