Nonparametric cointegration analysis

n -2(R TAmR

r . A _z

Op (n 2) Op (n 1) (O

„ ⇒

O (n 1) n 2Л O

^ p^v ' ^m J ∖

(A.30)

V ¹r,m

where the latter result follows from (15). Q.E.D.

Proof of Lemma 5: By Chebishev inequality:

P(^λU ≤ n∣Kqκm) ≥ ¹

E (λ1,m )

Kκ

у α, q-r, m

E [^trace( Г 1, m^)]

(A.31)

α, q-r, m

Moreover, it follows easily from (23), by first conditioning on the X^*_k’s, that

E( V,,m ) = ∑ γ ²Jr

k= 1

j-1

A-1

= ∑γ I

к-1

= Σ γ²I

к- 1

∑YjEtrace (1/m)£ X^X^^τ W? I,

j= 1

I^k⁼¹

(A.32)

q~,∖∙^ 2_τ

^γ-Σ_t7 k¹r ^,m k= 1

where the second equality follows from fact that the X^*_k’s are i.i.d., hence it follows from (22)
that

(

E [trace( V,₊1,_m )] = 1
∖

ɪɪɪ A
mJ
J

£y2 trace(RrTD(1)D(1)TR_r+J.

<^k=¹ √

(A.33)

Q.E.D.

Proof of Lemma 6: It follows from Fourier analysis that we can write without loss of generality:

Fk⁽^x ) =

У2 c_jkexp(2iπjx), where c_jk = Jexp(2iπjx)F_k(x)dx.

(A.34)

-∞<z' <∞