Nonparametric cointegration analysis



the generalized eigenvalue problem det(P-λQ) = 0,

we need to transform one of our matrices such that its limiting matrix becomes a.s.
nonsingular. As will be shown below, choosing
Pn = Âm and Qn = Bm + n2Am 1 yields a
suitable pair (
Pn , Qn ), such that if rank(C(1)C(1)T) = q-r then the q-r largest solutions of
det(
P-λQ) = 0 are a.s. positive and free of nuisance parameters, whereas the r smallest
solutions are zero.

Now choose the functions Fk such that

1 n

(6

(7


—∑Fk( t / n ) ^ o (1),
√n t"1

1
nJn


n

Σ tFk( t /n ) = o (1),
t= 1


and for ij,

ʃʃFi (x ) Fj (y )min(x ,y ) dxdy = 0,                                                         (8)

ʃFj(x)jF(y)dydx o 0,                                                              (9)

0

ʃFj (x)Fj(x) dx o 0.                                                                       (10)

Note that the integrals involved are taken over the unit interval [0,1] if not otherwise
indicated, as will be in the sequel. It is a standard exercise in Wiener measure calculus to
show (see, e.g., Billingsley 1968, Phillips 1987, Bierens 1994, Ch.9) that for each
k,

(                ʌ

Mnz(Fk)/F

M (Fk ) F

V k


Cd)ʃF,(x) W(x )dx
C
(1)(F,(1) W(1) -ʃf,( x ) W(x )dx),


N2q(0, (C(1)C(1)TΣk),


(11


where W is a q-variate standard Wiener process, fk is the derivative of Fk, and

Σk


ʃʃFk(x )Fk(y )min(x ,y ) dxdy

0


ʌ

0

ʃFk(x )2 dx


(12




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