Nonparametric cointegration analysis

2. The data-generating process

Consider the q-variate unit root process with drift z_t = μ + z_t-1 + u_t, where u_t is a zero
mean stationary process, and μ is a vector of drift parameters. We assume that z_t is
observable for t = 0,1,2,..,n. Due to the Wold decomposition theorem, we can write (under
some mild regularity conditions),
∞

^u_t = Cv C,^vt-_j = C(L ) V^,                                                                       (¹)

J=0

where v_t is a q-variate stationary white noise process, and C(L) is a q×q matrix of lag
polynomials in the lag operator L. For convenience we assume that C(L) is a rational lag
polynomial, and that the v_t ’s are Gaussian white noise, so that u_t is a Gaussian VARMA
process:

Assumption 1. The process u_t can be written as (1), with v_t i.i.d. N_q(0,I_q) and C(L) =
₁(L)^-1C₂(L), where C₁(L) and C₂(L) are finite-order lag polynomials, with all the roots of
det(C₁(L)) lying outside the complex unit circle.

This assumption is more restrictive than necessary, but it will keep the argument below
transparent, and focussed on the main issues. See Phillips and Solo (1992) for weaker
conditions in the case of linear processes. Also, we could assume instead of Assumption 1
that u_t is stationary and ergodic, so that we can write u_t = ε_t + w_t - w_t-1, where ε_t is a
martingale difference process with variance matrix comparable with C(1)C(1)^T. Cf. Hall and
Heyde (1980, p.136), and equation (2) below. Note that we do not restrict the lag polynomial
₂(L), except for the implicit restrictions imposed by Assumption 2 below.

Since by construction the lag polynomial C(L) - C(1) is zero at L = 1, we can write

u_t = C(L)V_t = C(1)V_t ÷ (C(L)-C(1))V_t = C(1)V_t ÷ (1-L)D(L)V_t

C C(¹)^V_t + w_t - wt_ 1

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