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 + n²A_m ¹ yields a
suitable pair (P_n , Q_n ), 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 F_k such that

₁ ⁿ

—∑F_k^{( t / n} ) ^ ^{o (1}),
√n ^t"¹

and for i ≠ j,

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

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

0

ʃF_j (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,

where W is a q-variate standard Wiener process, f_k is the derivative of F_k, and

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