Measuring variability of monetary policy lags : a frequency domain approach

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Economics Department

The Asymptotic Variance of the Estimated Roots in a Cointegrated Vector Autoregressive Model

S0ren Johansen

ECO No. 2001/1

EUI WORKING PAPERS

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European University Institute 3 0001 0 0 3 4 3 8 2 0 9 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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EUROPEAN UNIVERSITY INSTITUTE, FLORENCE

ECONOMICS DEPARTMENT

EUI Working Paper ECO No. 2001/1

The Asymptotic Variance of the Estimated Roots in a Cointegrated Vector Autoregressive Model

S 0 R EN J O H A N S E N

BADIA FIESOLANA, SAN DOMENICO (FI)

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No part of this paper may be reproduced in any form without permission of the author.

European University Institute Badia Fiesolana I - 50016 San Domenico (FI)

Italy © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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The Asymptotic Variance

of the Estimated Roots

in a Cointegrated Vector

Autoregressive Model

S0ren Johansen Department of Economics European University Institute

Badia Fiesolana

50016 San Domenico di Fiesole (FI) Italy

http://w w w .iue.it/P ersonal/Johansen/ January 17, 2001

A bstract

We show that the asymptotic distribution of the estimated stationary roots in a vector autoregressive model is Gaussian. A simple expression for the asymptotic variance in terms of the roots and the eigenvectors of the companion matrix is derived. The results are extended to the cointegrated vector autoregres sive model and we discuss the implementation of the results for complex roots.

Key words: Autoregressive process; estimated eigenvalues; error

correction model; asymptotic distribution JEL classification: C32 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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1 Introduction

In a stationary vector autoregressive model the estimated coefficients are asymptotically Gaussian, see Anderson (1971). In this note we show that the asymptotic distribution of the estimated stationary simple roots is also Gaussian, and derive a simple expression for the asymptotic variance matrix. We thereby extend and simplify results by Wymer (1972).

In order to prove this, we give a necessary and sufficient condition for a root of the characteristic polynomial to be a simple root, that is have multiplicity one, and use that a simple root is a continuous and differentiable function of the coefficients in the model.

The results are shown to hold also for the stationary roots of a cointegrated vector autoregressive model by investigating the compan ion form for the stationary cointegrating relations and the differences. Finally it is shown how the Arcsin transformation helps eliminate a sin gularity of the variance when a root is close to one.

In Section 2 the roots and the companion form are investigated, and in Section 3 we find the asymptotic distribution of the simple eigenvalues. In Section 4 we extend the result to the cointegrated vector autoregres sive model and discuss some issues in relation to the calculation of the variances. We conclude in Section 5 with a few simulations to illustrate the results.

2 The roots of an autoregressive process

Let the n —dimensional process X t be given by the vector autoregressive model

x t =

r iiX f- i +

n.

2 Xt

~2

+ ri

3 .Xt

_3

+ et,

(i)

where et are i.i.d. with mean zero and variance matrix fi, and the initial conditions are fixed. We have taken three lags for notational convenience. We define the characteristic polynomial as

fl(p) = p3In - p2IIi - pU2 - n3, (2)

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with determinant |Il(p)| and derivative

ri(p) = 3p2/ n - 2plh - n 2.

We let p i, . . . , pm denote the roots of

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|n(p)| = o,

and associate with each root p the n —vectors rp and kp, with the property

that

Note that in general the roots and the vectors rp and np may be complex valued. Hence if r = rT +iTit we use the notation r* = r ' — ir[ =

f'p. The vector tp corresponds to the root p. Note also that if p is not a simple root, the vectors tp and np are not unique. For any full rank (complex) matrix a (n x m ),m < n, we define aj_ (n x (n — m)) of full rank, so that a*a± = 0. We give in the next theorem a condition on II(p) to ensure that po is a simple root. The result is a slight reformulation of Theorem 3 from Johansen and Schaumburg (1998).

T h e o rem 1 If po is a root of the equation

then n(po) = ab* where a and b are n x r matrices of full rank for some

r < n. It then holds that po is an (n — r) —double root if and only if

r ; u (p) = 0, n (p)kp = 0. ₍₄₎

ln (p)| = o,

|ain(p0)6x| ^ o. (5)

In this case

|ain (p o )6 x |(l + 0 ( p - p 0)), (6)

and n ( p) 1 has a pole of order 1 at po :

II(p )-1 = — — 6 x (a in (p 0)6 x)-1a*x + 0(1). P - P o (7) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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P ro o f. The matrix polynomial II(p) has the expansion at p = p0

U(p) = n(po) + {p ~ Po)n(p), (8)

where the polynomial fl(p) satisfies fi(po) = n(po)- Because n (p 0) is singular, it has the representation

n(po) = ab*,

where a and b axe n x r matrices of rank r < n. Now multiply (8) by the full rank matrices (a, ax)* and (b, bx ) and we find that

(a ,a x )*II(p)(6,6x) / a*a6*6

V

0

a*n(p0)6x

airi(po)&x ^ + 0 (p - po) (p - p0)/„ -r J

°

Ì

This relation immediately implies the result about the determinant (6) and the expression for

Km (p - po)n(p)-1 = hm (p - p0)(6, bx ) [(a, ax)*n(p)(6,6x)]_1 (a, ai)*, which shows (7). ■

As a special case of Theorem 1 we find th at p is a simple root (r = 1) if and only Il(p) = ab*, with a and b of rank n — 1, and a^n(p)6x ^ 0. In this case we associate to p two vectors rp = a± and kp = bx , which we

normalize by

r;fl(p)/cp = 1, (9)

see (5).

We give a result on the differentiablity of a simple eigenvalue as function of the coefficient matrices, see Magnus and Neudecker (1995, p. 161)

T h e o re m 2 Let IIo(-) be a matrix polynomial with a simple root at po-

I f II( ) is close to IIo(-), then it has a simple root close to p0. This defines a continuous function p(II) in a neighborhood o fU 0. The function is also differentiable with differential

dp = - T ; ( d u ) ( P)Kp = —T*(p2(dn1) + p(dn2) + (dn3))^. (

10

) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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P roof. From the relations (2) and (3) we find, by taking differentials, that

d(n{p)Kp) = n{p)np{dp) + (dii)(p)/cp + n(p)(d/cp) = o.

Multiplying by r* we get, since rptl(p)Kp — 1 and r*II(p) = 0, that

T*il(p)Kp(dp) + r;(dU)(p)Kp + TpU{p){dKp) = dp + r*(dll)(p)/cp = 0.

■

2.1 T h e c o m p a n io n fo rm o f th e v e c to r a u to r e g r e s siv e m o d e l

The companion form of (1) is

Xt — AXt-i + Bet, (11) where ' x t ^ * i i i n 2 1I3^ f In ) Xt-, ,A = In 0 0 , B = 0 \ X‘-2 ) O O l 0 ) It is well-known that IPhn ~A\ = |n(p)|,

see for instance Johansen (1996, p. 16), which implies that the roots of | n ( p ) [ = 0 are the same as the eigenvalues of A. A right eigenvector is

k'p = K'p(p2In, PIn, I „), (12)

where kp satisfies I1(p)kp = 0. Similarly a left eigenvector tp for A satisfies

r; = r*(/„, pin - u u p2i n - pih - n 2), (13)

where tp satisfies rpU(p) — 0, see (4). Further we find from (12) and (13) that t*kp = rp (3p2 - 2p h i - n 2)/cp = Tp tl(p)Kp = 1, (14) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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by the normalization of kp and rp, see (9) and (3). From the equation

A k p = pkp,

which is equivalent to U (p)k p = 0, we find the differential (10) can be written

dp = fp{dA)kp. (15)

In the following we sometimes work with the assumption th at all roots are simple roots. In this case they are denoted pk, k = 1,. .. ,m, and the corresponding vectors are t*, f k, *k- From the relation

t^ Aki = PkT^ki = r^kipi, (16)

it follows that for pk ^ pi, the vectors kk and fi are orthogonal in the sense th at f'fkk = 0.

The relations (14) and (16) can be summarized as ( n , - • • j rm) — (fci, • ■ •, ftm)

3 Asymptotic distributions

We first discuss briefly the asymptotic distribution of the estimated co efficients in the autoregresive model, Anderson (1971), and then apply the differentiability of the roots to derive the asymptotic distribution of the estimated roots. The parameters in (1) are estimated by regression and for stationary processes (|pfc| < 1, k = 1 , . . . , m) it holds that

T i [ ( i i u h 2,i i3) - ^ Nnx3n( o , n ® x ~ l ).

The variance matrix E is defined as

E = Var(Xf) = Var(X;, X't_u X't_2) \ and can be found from

E = AY, A! + (17) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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The asymptotic distribution of

t* ( i - A) = b(

nx

- n u

n

2

- n2l n

3

- n3)

is therefore Gaussian in (3n x 3n) dimensions with mean zero and asymp totic variance matrix

asV ar{T^{A - A)) = BClB' <g> E -1.

An estimate of the variance fi is found from the regression, and E can be estimated from (17).

T h e o rem 3 Let X t be the stationary autoregressive process given by (1)

with characteristic polynomial (2). I f p is a simple root of |Il(p)| = 0

and the vectors rp and kp satisfy rpYl(p) = U.(p)kp= 0 and r*tl(p)K = 1,

then T^ (p — p) has the same limit distribution as T l>f;(A - A )kp = - T W ; ii( p ) Kp, that is, asymptotically Gaussian with mean zero.

If pk and pi are simple real roots the asymptotic covariance is

asCov(T^dpk,T^dpi) = /c^E-1^ - (18)

If pk and pi are simple complex roots the asymptotic covariances are given in terms of the matrices

l kl = UJkl wrr tjk‘_ir ri i xt

H

TrkflTrl T^ClTr, ■ ~kl Irr ~klJri _{) - (} KkX-'Zrl ~kl Hr ~klHi K'fcE _1KrJ ik^Tu t^ T u

The asymptotic variances and covariances of the estimated real and imag inary part of the roots are given by

asC ov(T^ dprk, T3 dpri) asCov(T*dpik, T*dpu) asCcnj(Tkdprk ,T kdpu) (19) (20) - « # £ + u£ tS (21) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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P ro o f. From the continuity of p(II) and the consistency of ft, it follows that also p is consistent. From the differential (10) and (15), it follows that

t Hp - p) = T h ; ( A - A)iip + oP( 1) = - T i r;n(p)Kp + oP(i) (

22

)

which should be compared with Wymer (1972, page 575), where

vec(diag(dpi, . . . ,dpm)) is found in terms of a matrix derived from the

eigenvectors.

We first assume that pk and pi are two simple real roots, in which case we get from (22)

asCov(T?dpk,T^dpi) = f'kBLlB,TiR'k'B~1 Ri,

which proves (18).

If pk and pi are two simple complex roots the formulae are somewhat more complicated. We get from (15) that

dp = (f' — ifl)dA(Rr + iRi)

= r'r.(dA)Rr + f[{dA)Ri + i ( f lr(dA)Ri - T-(dA)Rr),

— dpr idpi,

and hence in terms of u kl and 7fci, we find the results in (19), (20), and (21). ■

If all the roots are simple the variances can be expressed in terms of the quantities rkUri, pk, and pi :

C o ro llary 4 Let X t given by (1) be stationary and assume that all roots

are simple. We define the matrix 0 with elements

Tjjn-r, Wfci — z--- —•

1 ~ PkPi

I f all roots are real we get

asCov(T^dpk,T^dpi) = t^ t,( ^ )kl = r(flr;0 w. (23) 1 PiPj © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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In general, if all roots are possibly complex, we have the expressions (19), (20), and (21) with

k l = i ( e " + e f ©?' - e f \

y

2 \ -ef* -

e f

©*' -

e ?

) ’

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where l is defined by pj = pi.

P roof. From (13) it follows that f k B — rk , and from (17) we get f ’k Efj = rk AHA'fi + TkBQB'fi = pkPif^Uft + T*fb|

since A'fi = pfy, and hence

f:Ef, = Tk£lri

1 ~ PkPi = e

kl-Note that © in general is a complex matrix which is self-adjoint: © = 6*. For r = ( f i , . .. , f m) we get

= /e;T(T*Er)-1r*S| = e'*©- ^ , = ©“ , (25) the kl'th element of © '. This proves (23).

From the relations

Qkl = k*kE = (k'rk - i«'ifc)E l (kTi + iku) = 7r' + 7« + »(7r' ~ 7,>)

0 " = = («rfc - *«tt)E"1(SP| - iKa) = 7rr - 7« - *(7>“ + 7,>)

we find the expression (24) for 7W. ■

A simple example is a bivariate system with one lag and two real roots, where the asymptotic variance of the estimated roots is

( t(Qt! r{ n r2 y t^CIti t'2VIt2 r f O r i Ht-j \ 1- P l 1- P1P2 I r'fin r^n-ri I 1—PiPa 1-Pj / (26) Here the • denotes the element-wise product. For a univariate system with one lag this simplifies to the well-known result that the asymp totic variance is 1 - p2. It is seen from (26) and in general (18) that

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/ \ \ a» 2<V K s

the expression for the asymptotic variance matrix is independent oH i^j normalization of the eigenvectors {t*}.

normalization of the eigenvectors {r*}

9§1,!,\ ov

The matrix 0 1 can be calculated using real matrices as follows. The complex number is represented by the matrix

( Kk <k \ ( n 0 \ ( T« - r u \ \ ~ 7ik <k

A

0 ra Tr, ) ’

and (1 - pkpi) by

( 1 0 \ f Prk ~Pik \ / Prl Pu \ \ 0 1 / \ Pik Prk ) \ -P it Prl )

In this way we obtain an expression for 0 as a 2m x 2m matrix, the inverse of which represents 0 _1 with 2 x 2 blocks of the form

©H 0 « - 0 « \

0

?

0

rw

J ■

4 The cointegrated V A R

o

In this section we show how the above results of Section 3 hold for the stationary roots in a cointegrated 7(1) process. Let X t be an 7(1) process given by the vector autoregressive model (1), but reparametrized as

A X t — a f t X t - 1 + TjAXe-i + r 2A 7ft_2 + £t- (27)

The parameters are related by

Lb = 7„ + a f f + Tj, a/3' = IIj + II2 + II3 — 7n,

n 2 = —r

1

+ r 2) r i = —n 2 —

113

,

1I3 = —r2,

r

2 = —

1I3.

In terms of the new parameters we find

n(p) = - p 2( 1 - p)In ~ P2ctff + p( 1 - p ) I\ + (1 - p)T2. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Note that 11(1) = —a(3', has reduced rank, which causes the nonsta- tionarity of X t. From the companion form (11) we derive the companion form for Yt = {X[p± , X[p, AX 't, A X ; _ J

y£ = K A K ~ lYt- X + K B e t = MYt- i + N e t,

with P 0 0 N

P

- I n 0 ,

P

- I n - I n ) and ( In-r P'±a PI Tr P'J'2 \ ( P I \ M = K A K ~ X = 0 Ir + P'a P'Ti /3T2 , N = K B = P' 0 a Ti r 2 In l 0 0 In o ) \ o /

For the stationary process Z[ = (X'tP, AX't, A X ('_ j)' we have the equa tions Z t = P Z £_ i + Qet with P = ( Ir + P'a a P'T\ Ti P 'r2 ) r 2 , Q = t pr \ I n { 0 I n _{0 )} _{{ 0 /}

It follows from (12) and (13) th at the eigenvectors of M satisfy

K'p{M) = K'p(p2In, p ln, In) K ', K = P' I n

V o

u o - I n In u 0 0 -In ( p±

P±

\ k r;(M ) = r ; ( / n, p in - nllP 2i n - Pu t - n 2)k ~ \

A calculation then shows the following result:

L em m a 5 If X t is an 7(1) process given by (27), then the companion

matrix P for the stationary part of the process and the companion matrix

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M have the same eigenvalues apart from the unit roots of M . If p is a root of M , then a right eigenvector is given by

n'p{M) = ^ ( p 2/3±, p2/3, {p - l)p /„, (p - 1 )/„),

where U(p)kp —0, and a left eigenvector is

r;(M) = r;(01^ - i a ,p -a(pri + r ^ . p - 'r , ) ,

where r*II(p) = 0. The corresponding eigenvectors of P, for p / 1, are

Vp = K'p{p2P,(P - l)pJn, (p - l)/n ),

r ; = Tp (^ riQiP-2(p r1 + r ■2) ,p - 1r 2),

which are normalized on

Tp*P = Tf i { p ) KP =

1-4 .1 A s y m p t o t ic d is tr ib u tio n o f e ig e n v a lu e s a n d im  p r o v e d a p p r o x im a tio n for p c lo se to 1

In a cointegrated VAR we estimate the coefficients a and (3 by reduced rank regression of A X t on X t- i corrected for &XW , AX,_2, see Johansen (1996) or Anderson (1951). Because the estimate of (3 is T _1 consistent, we determine the asymptotic distribution of the other coefficients by regression of A X t on ((3'Xt- \, A X t- \, A X t- i) where (3 is replaced by (3. We find th at T ^ (d — a ,f j — Ti, f 2 — T2) is asymptotically Gaussian with mean zero and variance

fi® 'I'-1,

where-$ denotes the variance of the stationary part of the stacked process * = V a i ( p X t , A X t , A X t - i ) ,

which, as in the stationary case, can be found from the equations

¥ = P V P 1 + QCIQ'. (28) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Prom t * ( P -p) = tIq(&

-

a ,

f

i

- rlf f

2

- r2)

we find T ln ; { P - P )k p = - T b ; Q ( h ( p ) - = -tW ;(

n(P) - n(P))v

As in (15) we find dp = T*(dP)Zpi

and we can summarize the results in

T h e o re m 6 Let X t be the cointegrated 1(1) process given by (27), and

let

|n(p)|

= 0 have n — r unit roots and define

= V a r i p X t .A X ^ A X t - i ) ,

then the conclusions of Theorem 3 and Corollary \ hold with £ replaced by

The above result holds for stationary roots. In order to see what happens if a root P l, say, tends to 1, consider the result for two real roots, see (26). In this case the variance is

( (1 — p\ )t[LIti pr{f2r 2 \ / r( flri i/r(flr2 \

\ pr^flTi (1 — p2)r2Qr2 ) \ tI£It2 J with P = \ /( l - Pi)(! For pi —» 1 it holds th at p converges towards

(o

o

\

(

T(f2n o \_1= /o

o \

[ 0 (1 - p1)t^ t2 I • V 0 t£ It2 ) [ 0 (1 - p \ ) ) ---57 7 ( 1 - P i ) ( l - P i ) ~ « ) , v = -*---- --- — ---. 1 — P\Pi

—> 0 and v —* 0, and the variance matrix

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Thus the variance of px tends to zero for px —* 1, and simula tions show th at we get a very skew and concentrated distribution. It is therefore convenient to transform the roots before confidence interval are found.

For y = sinx, x E] — ^7r, |7r[, we find x = Arcsin(y), y e] — 1 ,1[ and d x /d y = 1 /^ (1 — x2). We introduce the Arcsin transformation of the roots which eliminates the factor (1 — p2) and find

T h e o re m 7 If all roots are real and distinct the asymptotic distribution

of

T^ (Arcsin(pfc) — Arcsin(p*)), k = 1 , . . . , m is Gaussian with mean zero and variance matrix given by

{rlflTk} • {r/ftrjti/ju}- 1 , with vu given by

Vki = --- --- .

1 - pkpl

P ro o f. The asymptotic distribution of (pk — pk) j has covariance matrix with elements given by (23). It follows th at the asymptotic covariance matrix of the vector

| t* (Arcsin(pfc) - Arcsin(pfc) ) | is given by the matrix with elements

riflTi v ' U - P Î X i - P ? / 1 - ^ rlÇÏTj y i = Tk^T l(\ /( l - P i)(l - Pj) 1 - PiPj © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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5 Simulations

In this section we simulate a bivariate system in order to get an impres sion of the distributions and some confidence intervals. We take T = 100 and the number of simulation is 200. We simulate the system

and A is chosen to give suitable roots. Figure 1 shows the estimated roots if we have a process with the roots ±0.5i. It is seen that the real and imaginary part have a joint distribution which is described by the variance matrix given by the relations (19), (20), and (21).

Next consider in Figure 2 the simulation of a system with two real roots at ±0.5. It is seen that all estimated roots are real, and the distri bution is given by the variance in (18). If, however, we consider Figure 3, where the roots are 0.9 and 0.5, we see that the empirical distribution spreads into the complex plane, a phenomenon that is not captured by the asymptotic distribution, which is concentrated on the real axis. Thus the finite sample performance of the estimated real roots needs further

An explanation of this phenomenon is the following. The char acteristic equation is of second degree and the roots are found as the intersection of a parabola with the x axis. If the estimated parabola is sufficiently close to the true one, assumed to have two real roots, the estimated roots are real too. If, however, the random coefficients have perturbed the parabola so much th at it does not intersect the x axis, then the roots become complex.

This is a relevant observation also for a discussion of unit roots, where one sometimes finds th at there are two complex roots with rather

( X u \ _ f -^11 A\2 \ / -Xit-i \ / £it \

\ A2t / \ A2

i

A22 J

\ A2t_1

)

\ £21

)

where et are i.i.d. iV(0,fi), with

study. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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large absolute value. The question then is whether one of them or possi bly both Eire unit roots. Obviously if we have two unit roots we sometimes will observe two complex roots with a high modulus, but even if we have roots at 1.0 and 0.9, say, the random variation could be such th at we observe two large complex roots with a real part of .95.

In the simulation in Figure 4 we have a root at 0.95 and a root at 0.5. It is seen in Figure 5 th at the distribution is highly skewed, and the

Arcsin transformation makes it a lot more symmetric in Figure 6. The

last Figure shows a 95% confidence ellipse for a bivariate time series with

T = 50 observations. The simulations were performed in RATS 4.20.

5.1 R e fe r e n c e s

Anderson, T.W. (1951) Estimating linear restrictions on regression

coefficients for multivariate normal distributions. Annals of Math

ematical Statistics, 22, 327-351.

Anderson, T.W. (1971) Statistical Analysis of Time Series New York:

John Wiley.

Johansen, S. (1996) Likelihood-Based Inference in Cointegrated Vector

Autoregressive Models. Oxford: Oxford University Press.

Johansen, S., and Schaumburg, E. (1998) Likelihood analysis of seasonal cointegration, Journal of Econometrics 88, 301-339. Magnus, J.R., and Neudecker, H. (1995) Matrix Differential Calcu

lus with Applications in Statistics and Econometrics. New York:

John Wiley.

W Y M E R , C.R. (1972) Econometric estimation of stochastic differential

equation systems. Econometrica 40, 565-577.

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T=100, Sim - 50, rho1 « +I/2, rho2 a 4/2 rMi Figure 1: © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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T=100, Sim * 50, rhol » +1/2, rho2 » -1/2 Figure 2: © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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1.00 T=100, Sim = 50, rho1 = 0.9, rho2 = 0.5 0.75 - 0.50 - 0.25 - 0.00 - -0.25 - •0.50 - -0.75 - 1.00 --1.0 -0.5 0.0 0.5 1.0 real D □ o a O o o o Figure 3: © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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T*100, Sim » 50, rho1 = 0.95, rho2 = 0.5 f M l Figure 4: © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Histogram for rhohat T=100, Sim ■ 500, rho1 = 0.95, rhohat Figure 5: © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Histogram for Arcsin(rhohat) T=100, Sim a 500, rho1 = 0.95, Arc«in(rhohat) Figure 6: © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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95% confidence interval with T=50t rho1 =(1+IV2, rho2=(1-iV2 real Figure 7: © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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EUI

WORKING

PAPERS

EUI Working Papers are published and distributed by the European University Institute, Florence

Copies can be obtained free of charge - depending on the availability of stocks - from:

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Publications of the European University Institute

To The Publications Officer

European University Institute Badia Fiesolana

1-50016 San Domenico di Fiesole (FI) - Italy Fax: +39-055-4685 636

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□ Please send me a complete list of EUI Working Papers □ Please send me a complete list of EUI book publications □ Please send me the EUI brochure Academic Year 2001/2002 Please send me the following EUI Working Paper(s):

Dept, n°, author ... Title: ... Dept, n°, author ... Title: ... Dept, n°, author ... Title: ... Dept, n°, author ... Title: ... Date Signature

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Working Papers of the Department of Economics Published since 2000

E C O 2000/1 Marcelo FERNANDES

Central Limit Theorem for Asymmetric Kernel Functionals

E C O 2000/2

Ray BARRELL/Alvaro M. PINA How Important are Automatic Stabilizers in Europe? A Stochastic Simulation Assessment

E C O 2000/3

Karen DURY/ Alvaro M. PINA Fiscal Policy in EMU: Simulating the Operation of the Stability Pact E C O 2000/4

Marcelo FERNANDES/Joachim GRAMMIG

Non-parametric Specification Tests for Conditional Duration Models

E C O 2000/5

Yadira GONZALEZ DE LARA

Risk-sharing, Enforceability, Information and Capital Structure

E C O 2000/6 Karl H. SCHLAG

The Impact of Selling Information on Competition

E C O 2000/7

Anindya BANERJEE/Bill RUSSELL The Relationship Between the Markup and Inflation in the G7 Plus One Economies

E C O 2000/8 Steffen HOERNIG

Existence and Comparative Statics in Heterogeneous Cournot Oligopolies E C O 2000/9

Steffen HOERNIG

Asymmetry and Leapfrogging in a Step-by-Step R&D-race E C O 2000/10 Andreas BEYER/Jurgen A. DOORNIK/David F. HENDRY E C O 2000/11 Norbert WUTHE

Exchange Rate Volatility's Dependence on Different Degrees of Competition under Different Learning Rules - A Market Microstructure Approach E C O 2000/12

Michael EHRMANN

Firm Size and Monetary Policy Transmission - Evidence from German Business Survey Data

E C O 2000/13 Gunther REHME

Economic Growth and (Re-)Distributive Policies: A Comparative Dynamic Analysis

E C O 2000/14 Jarkko TURUNEN Leaving State Jobs in Russia E C O 2000/15

S0ren JOHANSEN

A Small Sample Correction of the Test for Cointegrating Rank in the Vector Autoregressive Model

E C O 2000/16 Martin ZAGLER

Economic Growth, Structural Change, and Search Unemployment

Aggregate Demand, Economic Growth, and Unemployment

Growth Externalities, Unions, and Long term Wage Accords

E C O 2000/19

Joao AMARO DE MATOS/ Marcelo FERNANDES

Market Microstructure Models and the Markov Property © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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ECO 2000/20

Anindya BANERJEE/Massimiliano MARCELLINO/Chiara OSBAT Some Cautions on the Use o f Panel Methods for Integrated Series of Macro- economic Data

ECO 2000/21

Anindya BANERJEE/Bill RUSSELL The Markup and the Business Cycle Reconsidered

ECO 2000/22

Anindya BANERJEE/Bill RUSSELL Industry Structure and the Dynamics of Price Adjustment

ECO 2000/23

Jose Ignacio CONDE-RUIZ/ Vincenzo GALASSO

Positive Arithmetic of the Welfare State ECO 2000/24

Jose Ignacio CONDE-RUIZ/ Vincenzo GALASSO Early Retirement ECO 2000/25 Klaus ADAM

Adaptive Learning and the Cyclical Behavior of Output and Inflation

*** ECO 2001/1 Sdren JOHANSEN

The Asymptotic Variance of the Estimated Roots in a Cointegrated Vector

Autoregressive Model © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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