Danish Graduate Programme inEconomics

Lecture Notes for the course onCointegration analysis of the vector autoregressive

model for I(2) variables

Søren JohansenEconomics Department, University of Copenhagen

October 2007

Abstract

These notes present a brief overview of the statistical analysis of the cointe-grated VAR model for I(2) variables. The lectures will be based on these and thenotes contain more mathematics than is required for understanding the content.

1

I(2) models 2

Contents

1 Introduction 3

2 Generation of I(2) variables. Some examples 3

3 The I(1) and I(2) models and their solutions 63.1 The I(1) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 The I(1) solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 The I(2) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 The I(2) solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Di¤erent parametrizations of the I(1) and I(2) models 94.1 The MLE parametrization . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 The Paruolo-Rahbek parametrization . . . . . . . . . . . . . . . . . . . 104.3 Normalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 The I(1) model as a nonlinear regression model . . . . . . . . . . . . . 114.5 The I(2) model as a nonlinear regression model . . . . . . . . . . . . . 124.6 Transformation to I(1) space . . . . . . . . . . . . . . . . . . . . . . . . 134.7 Weak exogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Algorithms for estimating the I(2) model 14

6 Deterministic terms 14

7 Hypotheses on the parameters 157.1 Test of the ranks r and s1 . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Hypotheses on the I(2) parameters . . . . . . . . . . . . . . . . . . . . 17

7.2.1 The hypotheses on �; �; and � . . . . . . . . . . . . . . . . . . 177.3 Hypotheses on � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 Asymptotic distributions 188.1 The linear regression model with I(2) variables . . . . . . . . . . . . . . 18

9 The asymptotic distribution of the estimators 19

10 The asymptotic distribution of the test on � 2210.1 A simple example with a simulation . . . . . . . . . . . . . . . . . . . . 23

10.1.1 The hypothesis � 2 = 0 . . . . . . . . . . . . . . . . . . . . . . . 2410.1.2 The hypothesis � 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . 2410.1.3 The test that 3 = 0; . . . . . . . . . . . . . . . . . . . . . . . . 2410.1.4 The test that � = 0; . . . . . . . . . . . . . . . . . . . . . . . . 24

11 References 25

I(2) models 3

1 Introduction

The purpose of these notes is to give an overview of the theory behind the statisticalanalysis of I(2) variables using the vector autoregressive model that generates I(2)variables. Thus we discuss the model, its solution, and the cointegration and polyno-mial cointegration properties of the solution. We need the various parametrizationsthat have turned out to be useful and the analysis, and we discuss which hypotheseswe can formulate and which we can analyze in the model.The statistical analysis involves the calculation of maximum likelihood estimators

and the likelihood ratio tests of the various hypotheses.The asymptotic analysis will not be dealt with in detail, but some insight is neces-

sary to understand what can be done and what can not be done in the model.

2 Generation of I(2) variables. Some examples

Example 1 The simplest way of generating a bivariate I(2) variable is through theequations

�2Xt = "t; t = 1; : : : ; T;

where "t; is i.i.d. (0;):The solution of the equations is

�Xt = �X0 +tXi=1

"i and Xt = X0 + t�X0 +tX

j=1

jXi=1

"i:

Note the linear trends generated by the initial values, and that

tXj=1

jXi=1

"i =tXi=1

(t� i� 1)"i = t"1 + (t� 1)"2 + : : :+ "t

V ar(

tXj=1

jXi=1

"i) =

tXi=1

(t� i+ 1)2 = O(t3);

so that the cumulated random walk is OP (t3=2) and dominates the linear trend. �

Example 2 Another possibility is to generate I(2) variables as

�X1t = "1t; (2.1)

�X2t = X1t�1 + "2t:

The solution is

X1t = X10 +

tXi=1

"1i

X2t = X20 +

t�1Xi=0

X1i +

tXi=1

"2i = X20 +X10t+

t�1Xi=1

iXj=1

"1j +

tXi=1

"2i

I(2) models 4

Thus again we �nd a cumulated random walk and a linear trend. Note that it is notthe shock to the I(2) variable that generates the I(2) trend, but the shock to the I(1)variables, X1t; that, due to the dynamics is cumulated twice.�Example 3 Finally we consider taking linear combinations of the variables and

de�ne�Y1tY2t

�=

�a bc d

��X1t

X2t

�=

�aX1t + bX2t

cX1t + dX2t

�=

a(X10 +

Pti=1 "1i) + b(X20 +X10t+

Pt�1i=1

Pij=1 "1j +

Pti=1 "2i)

c(X10 +Pt

i=1 "1i) + d(X20 +X10t+Pt�1

i=1

Pij=1 "1j +

Pti=1 "2i)

!

=

aPt

i=1 "1i + bPt

i=1 "2i + bPt�1

i=1

Pij=1 "1j

cPt

i=1 "1i + dPt

i=1 "2i + dPt�1

i=1

Pij=1 "1j

!+

�aX10 + b(X20 +X10t)cX10 + d(X20 +X10t)

�Note that there are three trends, two I(1) trends and one I(2) trend and that one ofthe I(1) trends is just the di¤erences I(2) trend. Now both variables are (in general)I(2); but we can eliminate the cumulated random walk because

�dY1t + bY2t = (�da+ bc)(tXi=1

"1i +X10) � I(1)

This variable cointegrates with �Y1t = a"1t + b"2t + b(Pt�1

i=1 "1i +X10); so that

�dY1t + bY2t �(�da+ bc)

b�Y1t = �

(�da+ bc)

b(a"1t + b"2t) � I(0):

This is an example of polynomial cointegration meaning that a linear combinationof levels and di¤erences is stationary. We could have chosen to use �Y2t instead of�Y1t, as they are both I(1) because of the common trend

Pti=1 "1i; or in other words

�Y1t and �Y2t are cointegrated because �dY1t + bY2t is I(1), and is called a medianlong-run relation.�The examples above are examples of VAR processes of the form

�Xt = �Xt�1 + "t:

I(2) models 5

The �rst example has � = 0; and the second has given by 1�0 01 0

�: (2.2)

Note that this is a strange matrix: The eigenvalues are found from

j�I2 � �j = j�

� 0�1 �

�j = �2

which has a double root at � = 0; corresponding to two unit roots of the characteristicpolynomial. An eigenvector, v = (v1; v2); for � = 0 has to solve the equation

(�I2 � �)v = ��v = 0

which gives v1 = 0; and hence v0 = (0; 1) is the only eigenvector, despite there beingtwo roots at � = 0: Usually one can diagonalize a matrix using the eigenvector, but thematrix � has two eigenvalues and only one eigenvector and can not be diagonalized.It is a so called Jordan matrix, ad they are exactly the building blocks of the I(2)variables, see Archontakis (1998).Example 4 The �nal example is more complicated

�2X1t = �12[X1t�1 + �2X2t�1 + (� 1 + �� 2)X3t�1

+2�X1t�1 +�X2t�1 + ( 3 + 2� 1)�X3t�1] + "1t;�2X2t = ��X2t�1 � � 2�X3t�1 + "2t;�2X3t = "3t:

We see that X3t is an I(2) variable and that the second equation is

�X2t = �� 2�X3t�1 + "2t;

with solution

X2t = X21 � � 2X3t�1 + � 2X30 +tXi=2

"i;

1Note that this is a strange matrix: The eigenvalues are found from

j�I2 ��j = j�

� 0�1 �

�j = �2

which has a double root at � = 0; corresponding to two unit roots of the characteristic polynomial.An eigenvector, v = (v1; v2); for � = 0 has to solve the equation

(�I2 ��)v = ��v = 0

which gives v1 = 0; and hence v0 = (0; 1) is the only eigenvector, despite there being two roots at � = 0:Usually one can diagonalize a matrix using the eigenvector, but the matrix � has two eigenvalues andonly one eigenvector and can not be diagonalized. It is a so called Jordan matrix, ad they ar exactlythe building blocks of the I(2) variables, see Archontakis (1999)

I(2) models 6

so that also X2t is I(2): Finally we write the �rst equation as

X1t =12X1t�1 + ut

ut = �12(�2X2t�1 + (� 1 + �� 2)X3t�1 +�X2t�1 + ( 3 + 2� 1)�X3t�1] + "1t;

so that ut is I(2): The equation for X1t has the solution

X1t = (1

2)tX10 +

t�1Xi=0

(1

2)iut�i;

which is also an I(2) variable.

3 The I(1) and I(2) models and their solutions

We parametrize the general VAR(k) model (without deterministic terms) as

�2Xt = �Xt�1 + ��Xt�1 +k�2Xi=1

�i�2Xt�i + "t; t = 1; : : : ; T: (3.1)

Note that we have one term in levels one in di¤erences and the rest in seconddi¤erences. Thus it is a model for the levels even if it is written in error correctionform. Here "t are i.i.d. (0;) and the initial values X0; : : : ; X�k+1 are �xed in thestatistical analysis.

3.1 The I(1) model

The parameter restriction needed to �nd the I(1) model is

� = ��0; (3.2)

where � and � are p � r and the parameters (�;�1; : : : ;�k�1;) are otherwise unre-stricted. We �nd the I(1) model

�2Xt = ��0Xt�1 + ��Xt�1 +

k�2Xi=1

�i�2Xt�i + "t; t = 1; : : : ; T: (3.3)

The generating polynomial is de�ned as

�(z) = (1� z)2Ip � ��0z � �(1� z)z �k�2Xi=1

�i(1� z)2zi:

I(2) models 7

3.2 The I(1) solution

It is then known, see Johansen (1988, 2007), and Rahbek and Mosconi (1999) that iffurther the I(1) condition

j�0?��?j 6= 0 (3.4)

is satis�ed, then the solution of (3.1) is I(1) and can be given the representation

Xt = CtXi=1

"i + C0"t +

1Xi=0

C�i�"t�i + A; (3.5)

where A depends on initial values and satis�es �0A = 0: The coe¢ cients are

C = ��?(�0?��?)�1�0?

C0 = �����0 + C�����0 + ����0�C � C(�����0� + Ip �k�1Xi=1

�i)C

The process Xt is I(1) and �0Xt = �0Yt is I(0): The coe¢ cients can be found from the

identity

(1� z)��1(z) = C + (1� z)C0 + (1� z)21Xi=0

C�i zi:

3.3 The I(2) model

Thus in order to get I(2) variables as solutions, we need the condition

�0?��? = ��0 (3.6)

for some matrices � and � of dimensions (p � r) � s1; s1 < p � r: This gives the I(2)model

�2Xt = ��0Xt�1 + ��Xt�1 +k�2Xi=1

�i�2Xt�i + "t; t = 1; : : : ; T; (3.7)

where the parameters are restricted by (3.6).Thus the I(2) model is a parametric restriction of the I(1) model, which again is a

parametric restriction of the general VAR model.The advantage of the representation (3.7) is that one sees immediately that we

have a non-linear (in the parameters) regression model, and the drawback is that theinterpretation of the parameters becomes di¢ cult and the analysis of the likelihoodwith the purpose of �nding estimators and test statistics is not easy because of theextra reduced rank restriction, which implies that the parameters are not variationindependent.

I(2) models 8

3.4 The I(2) solutions

It can be shown that if further the I(2) condition

j�0?�0?(�����0� + Ip �k�2Xi=1

�i)�?�?j 6= 0;

then Xt can be given the representation

Xt = C2

tXs=1

sXi=1

"i + C1

tXi=1

"i + C0"t +

1Xi=0

C�i�"t�i + A+Bt; (3.8)

see Johansen (1992, 2007), where the matrices C2; C1; C0; and C�i can be expressed interms of the parameters of the model through the identity

(1� z)2��1(z) = C2 + (1� z)C1 + C0(1� z)2 + (1� z)31Xi=0

C�i zi: (3.9)

We de�ne � = �����0� + In �Pk�2

i=1 �i and the directions

�?1 = ��?�; �?2 = �?�?; �?1 = ��?�; �?2 = �?�?; (3.10)

so that (�; �?1; �?2) are orthogonal and span Rp; and of dimensions r; s1; and s2 =p� r � s1 respectively: Similarly for (�; �?1; �?2): Then it holds that

C2 = �?2(�0?2��?2)

�1�0?2; (3.11)

C1 = ���?1��0?1 + [��?1��0?1� � ����0�]C2 + C2[���?1��0?1 � �����0]

+C2[�����0� + �����0�� ���1��

01� � �����0�����0� +

1

6

d3

dz3�(z)jz=1]C2; (3.12)

The matrix C0 satis�es�0C0� = �Ir + ��0�C2���: (3.13)

It follows from (3.8) and (3.11) that (�; �?1)0Xt is I(1) because (�; �?1)

0C2 = 0: Theserelations are called the CI(2; 1) cointegrating relations. Moreover, it holds that �0Xt+��0��Xt is stationary because the nonstationarity part of the process is

�0(C2

tXs=1

sXi=1

"i + C1

tXi=1

"i) + ��0�C2

tXs=1

"t = 0;

as �0C2 = 0; and �0C1 + ��

0�C2 = ���0�C2 + ��0�C2 = 0. These relations are called thepolynomially cointegrating relations.

Example 5 In example (2.1) we considered the model

�Xt = �Xt�1 + "t

I(2) models 9

where � is given by (2.2)

�2Xt = �Xt�1 ��Xt�1 + "t

so that in this case

� =

�01

�; � =

�10

�; � = �I2

Note that � has reduced rank and that

�? =

�10

�; �? =

�01

�; �0?��? = 0

so that � = � = 0; s1 = 0; and �?1 = �?1 = 0; whereas �?2 = �? and �?2 = �?:The I(2) condition is satis�ed because

j�0?1(�����0� + Ip �k�2Xi=1

�i)�?2j

= (1; 0)[

�01

�(0; 1) +

�1 00 1

�]

�01

�= 1:

�

4 Di¤erent parametrizations of the I(1) and I(2)models

4.1 The MLE parametrization

Note that we can construct two models by multiplying (for two lags) by �0? and ��0 =

(�0�1�)�1�0�1 respectively (leaving out more lags)

�0?�2Xt = �0?��Xt�1 + "t = ��0��

0?�Xt�1 + �0?�

���0�Xt�1 + �0?"t

= (�; �0?���)

��1�0

��Xt�1 + �0?"t = �� 0�Xt�1 + �0?"t; (4.1)

��0�2Xt = �0Xt�1 + ��

0��Xt�1 + ��

0"t = �0�Xt�1 + �Xt�1 + ��

0"t: (4.2)

Note that Cov(��0"t; �0?"t) = 0; so that the errors in the equations are inde-

pendent. We have introduced the notation � = (�1; �) and � = (�; �0?���) and

= (�0�1�)�1�0�1� = ��0�; and �nally the parameter � which picks out � = ��from the vectors in � .Then we �nd by combining (4.1) and (4.2) using the identify

Ip = �(�0�1�)�1�0�1 + �?(�0?�?)

�1�0?;

I(2) models 10

that�2Xt = �(�0� 0Xt�1 + 0�Xt�1) + �?(�

0?�?)

�1�0� 0�Xt�1 + "t: (4.3)

One can now prove that instead of the restricted parameters (�; �;�;) with the re-duced rank restriction �0?��? = ��0; we can reparametrize the parameters (�; �; � ; ;)and that these parameters vary unrestricted. This is the so-called maximum like-lihood parametrization and we also need the formula for (�; �;�;) as function of(�; �; � ; ;); so that when the latter are estimated we can calculate the former.One advantage of this parametrization is that all terms entering the equations are

stationary, that is �0� 0Xt�1 + 0�Xt�1; �0�Xt�1; as well as �2Xt�i; i = 1; : : : ; k � 2:

The term �0� 0Xt�1 + 0�Xt�1 is the multicointegration term.The parameter � is the adjustment coe¢ cient to the disequilibrium error given by

�0� 0Xt�1+ 0�Xt�1; and �0 is the adjustment coe¢ cient in the equation for �0?�Xt to

the disequilibrium error � 0�Xt�1:Note that because � 0�Xt�1 is stationary; the relation �0� 0Xt�1 + 0��?�

0?�Xt�1 is

also stationary, which points to yet another parametrization, see Paruolo and Rahbek(1999).

4.2 The Paruolo-Rahbek parametrization

We use the identityIp = ���

0 + ��?�0?

and �nd from (4.3) that

�2Xt = �(�0� 0Xt�1 + 0��?�0?�Xt�1) + (�?(�

0?�?)

�1�+ � 0��)� 0�Xt�1 + "t

or introducing �0 = 0��? and � = �?(�0?�?)�1�+ � 0��

�2Xt = �(�0; �0)

�� 0Xt�1� 0?�Xt�1

�+ �� 0�Xt�1 + "t: (4.4)

Again we can use the parametrization consisting of (�; �; �; � ; �;) which are vari-ation independent instead of the parameters (�; �; � ; ;); and we also need the ex-pressions for (�; �; � ; ;) as function of (�; �; �; � ; �;).

4.3 Normalizations

In the I(1) model it is well known that � and � are not identi�ed because ��0 =�m0(�m�1)0; for any full rank r� r matrix m; so that identifying restriction need to beimposed to get estimators and asymptotic distributions. Similarly, in the I(2) modelthe parameters are not identi�ed. In (4.3) we can discuss this issue as follows: We notethat they enter into the likelihood function only through the products

��0� 0; � 0;�?(�0?�?)

�1�0� 0:

I(2) models 11

This means that for any three full rank matrices a; b; c of suitable dimensions and fullrank, we have that

�; �; ; � ; �; �?; �

determine the same probability measure as

a�1�b; �b0�1; b; �a; a�1�; �?c; �c:

Thus in order to calculate the estimators but also to investigate the asymptotic distri-bution, we need to normalize or just identify the parameters.In the discussion of the maximum likelihood estimator in Johansen (1997) it was

useful to introduce a parametrization such that � = �� and � were normalized by��00� = Ir; and �� 00� = Ir+s: This can always be achieved by de�ning ~� = �(��

00�)�1 and

~� = �(�� 00�)�1:This parametrization is in fact also useful for numerical calculations, even if we

do not know the true values �0 and � 0: We can �nd initial values, from the two stepprocedure, see below, for instance, and use these as normalizations in the subsequentiterations of the MLE algorithm.The main reason for choosing the parametrization involving �0 and � 0; is to dis-

cuss asymptotics, and if we have normalized � in another way, like for instance �c =�(c0�)�1 = ~�(c0~�)�1; we can �nd the asymptotic distribution of �c by the ��method.

4.4 The I(1) model as a nonlinear regression model

We can reformulate the I(1) model (3.3) as a nonlinear regression in the following way.We normalize � on ��0; and de�ne ~� = �(��

00�)�1; and ~� = ��0��

0; so that ��0 = ~�~�

0:

Then ��00~� = Ir: Under the hypothesis of I(1) variable we know that �00Xt and �

00?�Xt

are stationary (although unobserved). Then we �nd (with two lags)

�2Xt = ��0Xt�1 + �1�Xt�1 + "t

= ��0��0�00Xt�1 + ��0��

0?�

00?Xt�1 + �1�Xt�1 + "t

= ~��00Xt�1 + ~�~�0��0?�

00?Xt�1 + �1�Xt�1 + "t

= �00Z0t + �01Z1t + "t;

where we de�ne B1 = ��00?~�

Z 00t = (X 0t�1�

0;�X 0t�1); Z1t = �00Xt�1;

�00 = (~�;�1); �01 = ~�B01:

Here �0 varies freely in Rp�r � Rp�p; and �01 = ~�B1 is a product of two matrices ofdimensions p� r and r � (p� r) respectively which are freely varying.The advantage of this regression formulation is that Z01 is I(0) and Z1t is I(1) so that

the asymptotic analysis of the unrestricted model �00 2 Rp�r �Rp�p and �01 2 Rp�(p�r)is relatively straight forward. The I(1) model is then a parametric restriction on theunrestricted model by the condition �01 = ~�B1: Note that the parameter ~� appearsboth in �0 and in �1:We now do something similar for the I(2) model

I(2) models 12

4.5 The I(2) model as a nonlinear regression model

We normalize the parameters by ��00� = Ir; and �� 00� = Ir+s1 ; and note that we havethat � 00�Xt and �

00Xt + 00�� 0?�00?�Xt is stationary. We de�ne the regressors

Z 00t = (X0t�1�

0 +�X 0t�1

0; �X 0t�1�

0);Z 01t = (�X

0t�1�

0?2;X

0t�1�

0?1);

Z 02t = X 0t�1�

0?2;

of dimensions (2r+s1; p�r; s2) respectively. The regressors are constructed so that theorder of integration of Zit is I(i); i = 0; 1; 2: Corresponding to this choice of regressorswe �nd

�00 = (�;�( � 0)0�� 0 + �?(�0?�?)

�1�0);

�01 = (�( � 0)0��0?2 + �?(�

0?�?)

�1�0� 0��0?2;��

0��0?1);

�02 = ��0��0?2:

This leads to introducing the parameters

A = �( � 0)0�� 0 + �?(�0?�?)

�1�0;

B0 = ��00?2( � 0); B1 = ��

00?1�; B2 = ��

00?2�;

C = ��00?2��?:

(4.5)

so that using � 0 = (���0 + ��?�0?)�

0 we �nd

�00 = (�;A);�01 = (�B

00 + �?(�

0?�?)

�1�0(��B02 + ��?C

0); �B01);

�02 = �B02:

Thus the I(2) model can be formulated as a nonlinear restriction on a linear regressionof an I(0) variable on an I(0), an I(1) and an I(2) variable,

�2Xt = �00Z0t + �01Z1t + �02Z2t + "t;

where the last two variables have the true value equal to zero because B00 = 0; B

01 = 0;

and B02 = 0.

A general formulation of such a model is that �0 = �0 is varying freely and thereare parameters �1 and �2 so that �1 = �1(�0; �1; �2) and �2 = �2(�0; �1; �2):We have studied four di¤erent parametrizations of the I(2) model given in

�2Xt = ��0Xt�1 + �1�Xt�1 + "t; (4.6)

�2Xt = �(�0� 0Xt�1 + 0�Xt�1) + �?(�0?�?)

�1�0� 0�Xt�1 + "t; (4.7)

�2Xt = �(�0; �0)

�� 0Xt�1� 0?�Xt�1

�+ �� 0�Xt�1 + "t; (4.8)

�2Xt = �00Z0t + �01Z1t + �02Z2t + "t (4.9)

each with a suitable parametrization. We shall apply these representations in thefollowing. The �rst (4.6) shows the model as a restricted vector autoregressive modeland the second (4.7) gives a parametrization in terms of unrestricted parameters, asdoes the third (4.8). Finally (4.9) shows that the model can be considered a parametricrestriction in a linear regression model with nonstationary regressors.

I(2) models 13

4.6 Transformation to I(1) space

We can reduce the I(2) model to an I(1) model, Kongsted (1998, 1999), by introducingthe variables

Yt = (�0Xt; �

0?�Xt)

which are both I(1): It follows from (4.4)

�2Xt = �(�0; �0)

�� 0Xt�1� 0?�Xt�1

�+ �� 0�Xt�1 + "t:

by multiplying by (� 0; � 0?) that

�2Y1t = � 0�(�0; �0)Yt�1 + � 0��Y1t�1 + � 0"t

�2Y2t = � 0?�(�0; �0)Yt�1 + � 0?��Y1t�1 ��Y2t�1 + � 0?"t

�2Yt =

�� 0

� 0?

�(�(�0; �0)Yt�1 + �Y2t�1) +

�� 0� 0� 0?� �Is2

��Yt�1 + ~"t:

This shows that in the transformed model, the polynomially cointegrating relationsbecomes the cointegrating relation

��0� 0Xt�1 + �0� 0?�Xt�1 = �(�0; �0)Yt�1 + �Y2t�1;

and that the model is a restricted I(1) model: This can form a basis for �nding theproperties of the I(2) model, see Johansen (1996).Note also how the last term 0���Y1:t�1 appears in �1, and that the last block of

columns in �k�2 is zero. Still one has the possibility, once � has been identi�ed, totransform the model back to I(1); and then analyze a (slightly extended) VAR modelfor I(1) variables. This is clearly not a statement that we can analyze � 0Xt by an I(1)model, we need to include also the di¤erences � 0?�Xt.

4.7 Weak exogeneity

Paruolo and Rahbek (1998) have derived conditions for a0Xt to be weakly exogenousfor the parameters of interest � and �: We �nd the conditional and marginal of

b0�2Xt = !a0�2Xt + (b0 � !a0)�(�0� 0Xt�1 + 0�Xt�1)

+(b0 � !a0)�?(�0?�?)

�1�0� 0�Xt�1 + (b0 � !a0)"t

a0�2Xt = a0�(�0� 0Xt�1 + 0�Xt�1) + a0�?(�0?�?)

�1�0� 0�Xt�1 + a0"t

If the marginal model should not contain the parameters of interest we need the con-ditions

a0(�;�?(�0?�?)

�1�0) = 0;

in which case the marginal model does not contain the parameters of interest, andestimation from the conditional model

b0�2Xt = !a0�2Xt+b0�(�0� 0Xt�1+

0�Xt�1)+b0�?(�

0?�?)

�1�0� 0�Xt�1+(b0�!a0)"t

is e¢ cient.

I(2) models 14

5 Algorithms for estimating the I(2) model

The CATS program for estimating I(2) model is built on a switching algorithm whichcombines reduced rank and regression.Brie�y explained it exploits the following simple observations. If you consider the

parametrization (4.4)

�2Xt = �(�0; �0)

�� 0Xt�1� 0?�Xt�1

�+ �� 0�Xt�1 + "t:

it is clear that if we knew � and �?; we could determine the parameters �; (�0; �0); �;

by reduced rank regression of �2Xt on (X 0t�1� ;�X

0t�1�?)

0 corrected for lagged seconddi¤erences and � 0�Xt�1.On the other hand if we have found these parameters and only need to �nd � ; the

representation (4.3)

�2Xt = �(�0� 0Xt�1 + 0�Xt�1) + �?(�0?�?)

�1�� 0�Xt�1 + "t

shows that the right hand side is a linear function of � ; and hence � can be estimatedby GLS. By combining these two steps the algorithm is constructed. This is not theonly possible algorithm, but one that we have found convenient.A simple estimate is given by �rst estimating the I(2) model without the restriction

�0?��? = ��0; that is, as if it is an I(1) analysis of the data. With the estimated valuesof �; �; �?; and �?; we consider the derived model, see (4.1),

�0?�2Xt = ��0(��

0?�Xt�1) + �0?�

��(�0�Xt�1) + �0?"t;

which directly exhibits the second reduced rank condition and we can estimate � and� by a reduced rank regression of �0?�

2Xt on ��0?�Xt�1 corrected for �

0�Xt�1 andpossible lagged second di¤erences. This is the so-called two step procedure, which isused as a starting point for maximum likelihood estimation.

6 Deterministic terms

Example 6 Let�2Xt = �+ "t

then the solution is

�Xt = �X0 + t�+

tXi=1

"i and Xt = X0 + t�X0 +1

2t(t+ 1)�+

tXj=1

jXi=1

"i;

so that a constant cumulates to a quadratic trend. If instead we consider the e¤ect ofan impulse dummy we �nd

�2Xt = �1ft=t0g + "t

I(2) models 15

with solution

Xt = X0 + t�X0 +

tXj=1

jXi=1

(�1ft=t0g + "i)

= X0 + t�X0 +

tXj=1

jXi=1

"i + 1ft�t0g(t� t0 + 1)�

so that an impulse dummy is cumulated twice to a broken linear trend.�In the I(1) model the constant term generates a linear trend unless it is restricted

to the cointegrating space, and a linear term generates a quadratic trend unless itis restricted to the cointegrating space. This is the justi�cation for modelling thedeterministic terms in the I(2) model as follows:

�2Xt = �(�0� �0�Xt�1t

�+ �0

��Xt�11

�) (6.1)

+�?(�0?�?)

�1�0� �0��Xt�11

�+ "t:

If we de�ne � �0 = (� 0; � l) and �0 = ( 0; c) we �nd

�2Xt = �(�0� 0Xt�1 + 0�Xt�1) + �?(�0?�?)

�1�0� 0�Xt�1 + �t + "t:

�t = ��0� lt+ � c + �?(�0?�?)

�1�0� l;

which shows a linear term in the equation with a complicated parametrization.One can now prove using the expressions for C1 and C2 in (3.8) that �t only generatesa linear trend in the data.

7 Hypotheses on the parameters

The purpose of this section is to �nd out which hypotheses that are of interest in theanalysis of I(2) data. The model is given in the maximum likelihood parametrization(4.7) and the hypotheses are formulated in terms of the parameters � = ��; � ; �0 = 0��? and of course the two ranks indices r and s1:

7.1 Test of the ranks r and s1

The model de�ned by the equations

�2Xt = �(�0� 0Xt�1 + 0�Xt�1) + �?(�0?�?)

�1�0� 0�Xt�1 + "t;

is called Hr;s1 referring to the rank of ��0 and �0?��? = ��0 respectively. Note that the

model Hr;p�r is the model where �0?��? has full rank so there is no I(2) components,that is the I(1) model which we can call Hr: Furthermore the model Hp is the unre-stricted VAR model and �nally H0;p is the VAR in di¤erences. The likelihood ratio

I(2) models 16

The I(2) models, Hr;s1 ; and their relations to the I(1) model Hr

p� r r4 0 H0;0 � H0;0 � H0;0 � H0;0 � H0;4 = H0

3 1 H0;0 � H0;0 � H0;0 � H1;3 = H1

2 2 H0;0 � H0;0 � H2;2 = H2

1 3 H0;0 � H3;1 = H3

0 4 H4;1 = H4

s2 4 3 2 1 0

Table 1:

test of Hr;s1against Hp is easily calculated as

�2 logLR(Hr;s1jHp) =Lmax(� = ��0; �0?��? = ��0)

Lmax(�:� unrestricted)

using the algorithms described in section 5.We would like to give so intuition for this test statistic and show that it is almost

a sum of two trace tests as they are known in the I(1) model. The argument is asfollows. Let

LR(Hr;s1jHp) =Lmax(� = ��0)

Lmax(�:� unrestricted)Lmax(� = ��0; �0?��? = ��0)

Lmax(� = ��0)

The �rst factor is the usual test of reduced rank of the matrix �; that is,

�2 log Lmax(� = ��0)

Lmax(�:� unrestricted)=L(�; �; �; )

L(~�; ~�; ~);

say is the usual trace test, where � is the I1) estimator and ~� is the estimator for theunrestricted VAR.The second factor is

Lmax(� = ��0; �0?��? = ��0)

Lmax(� = ��0)=L(��; �� = ����; ��; � ; �)

L(�; �; �; ); (7.1)

where we use �� for the I(2) estimator.It is a curious fact that although the estimators ��; �� and �; �; are not the same,

the second estimator which is the I(1) estimator of �; �; is very close to ��; �� and havethe same limit distribution as ��; �� ; which is the I(2) maximum likelihood estimator.Thus the maximum likelihood estimator from the I(1) model is in fact asymptoticallye¢ cient in the I(2) model. If we approximate

L(��; �� = ����; ��; � ; �) = max�;� ;�; ;:�=��;��=��

L(�; �; � ; �; ;)

bymax

�;� ;�; ;:�=�;��=�L(�; �; � ; �; ;)

I(2) models 17

where we use the I(1) estimator for � and �; we therefore get almost the same, so that

L(��; �� = ����; ��; � ; �)

L(�; �; �; )�max�;� ;�; ;:�=�;��=� L(�; �; � ; �; ;)

max�;:�=�;�=� L(�; �;�;):

This last test is the test for reduced rank of �0?��? when we assume we know (�; �) =(�; �). The two step procedure analyzes the equation

�0?�2Xt = ��0(��

0?�Xt�1) + �0?�

��(�0�Xt�1) + �0?"t; (7.2)

by reduced rank as if we know �; � from an I(1) analysis. This means that the secondfactor (7.1) is the trace test in (7.2) when we assume we have determined �; � from anI(1) analysis.This argument is the intuition for the proof that the limit distribution of the trace

test for Hr;s1 is the sum of two trace tests, see Nielsen, and Rahbek (2007).

7.2 Hypotheses on the I(2) parameters

We shall discuss the hypotheses that are useful in terms of the parameters of the model

�2Xt = �(�0� 0Xt�1 + 0�Xt�1) + �?(�0?�?)

�1�0� 0�Xt�1 + "t;

and the parameter �0 = 0��? which appears in the parametrization

�2Xt = �(�0; �0)

�� 0Xt�1� 0?�Xt�1

�+ �� 0�Xt�1 + "t:

7.2.1 The hypotheses on �; �; and �

The parameters in � = (�; �?1) determine the cointegrating relations from I(2) to I(1),and the relations � 0Xt are the medium run cointegrating relations. We may want totest homogeneity � = H�; that some � vectors are known � = (b; ) and in generalidentifying restrictions

R0i� i = 0 or � i = Hi�i:

The parameter � = �� picks out those C(2; 1) relations that cointegrate with the I(1)process �Xt because

�0Xt + 0�Xt

or equivalently�0Xt + �0� 0?�Xt = �0Xt + 0��?�

0?�Xt

is I(0):This means that we may be interested in hypotheses on � similar to those on � ;

but also in hypotheses on the polynomial cointegrating coe¢ cient �:Due to the e¢ ciency of the I(1) estimator of �; � we can actually use the tests

on � from the I(1) model, but they will not be likelihood ratio tests and presumablytherefore not e¢ cient. Obviously for the parameters � and �; we need the analysis ofthe I(2) model because these parameters do not appear in the I(1) model.

I(2) models 18

7.3 Hypotheses on �

The polynomial cointegrating relations contain the parameters �; and hypotheses on �can help �nd those linear combinations of Xt that are stationary. Thus for instance� = 0; implies that �0Xt is I(0):The general test that no multicointegration is present, or that �0Xt is stationary,

can be expressed as the hypothesis

H : 0�? = 0; or = ��:

Paruolo (1996) has found the asymptotic distribution of the estimator of this coe¢ cientand it has a di¢ cult limit distribution, which is not mixed Gaussian as far as we know.Thus inference is at best very di¢ cult and may involve nuisance parameters.What we can do here is to circumvent the problem by �rst identifying the space

spanned by � ; or equivalently that spanned by �2; and then testing that is in thesame space as � :

Example 7 Let Xt = ( p1; p2; exch) and assume that Xt is I(2): Assume furtherthat r = 1; s1 = 1; s2 = 1; and that we �nd that �0?2 = (1; 1; 0) indicating that p1tand p2t have the same I(2) trend and that excht is at most I(1): Then � and �?2 arecompletely known:

� 0 =

�1 �1 00 0 1

�and the polynomial cointegrating relation becomes

�1p1t + �2p2t + �3excht + 1�p1t + 2�p2t + 2�excht

Now�excht is stationary since excht is at most I(1): Similarly�p1t��p2t is stationarysince there is price homogeneity in �?2: Thus if we can test the hypothesis

1 = � 2;

then all terms involving the di¤erences are stationary and hence so is �0Xt: This hy-pothesis can also be formulated as � = � 0? = 1 + 2 = 0:�

8 Asymptotic distributions

8.1 The linear regression model with I(2) variables

We �rst discuss the linear regression model with I(0); I(1); and I(2) regressors, to setthe scene for the more complicated I(2) model

�2Xt = �00Z0t + �01Z1t + �02Z2t + "t; (8.1)

I(2) models 19

where "t is independent of Zt = (Z 00t;Z01t;Z

02t)

0: The estimator of �0 = (�00; �01; �

02) is the

regression of �2Xt on Zt :

� = (

TXt=1

ZtZ0t)�1

TXt=1

Zt�2X 0

t

� � � = (TXt=1

ZtZ0t)�1

TXt=1

Zt"0t:

We want the asymptotic distribution of ��� suitably normalized, and for that we needmany limit results from the theory of weak convergence of stochastic processes. Wede�ne (n0; n1; n2) = (12 ; 1; 2) and introduce the product moments

M"j = T�njTXt=1

"tZ0jt; Mij = T�ni�nj

TXt=1

ZitZ0jt; (8.2)

which are normalized to converge. We next assume that the errors and regressorssatisfy

T�12

0@ P[Tu]t=1 "tZ1[Tu]

T�1Z2[Tu]

1A w!

0@ W (u)H1(u)H2(u)

1A ; u 2 [0; 1]; (8.3)

where W (u) is Brownian motion with variance ; and H1 and H2 are de�ned as thelimits of T�1=2Z1[Tu] and T�3=2Z2[Tu] respectively and will be a Brownian motion anda integrated Brownian motion respectively in the application to the I(2) model.Thus the �rst result states that a random walk normalized by T 1=2 converges to a

Brownian motion, and the next result states that an I(1) variable, Z1t; normalized thesame way, also converges to a Brownian motion, H1; and �nally that an I(2) variable,Z2t; need to be normalized by T 3=2 to get convergence.

Theorem 1 Under the assumptions above the unrestricted estimators of the regressionmodel (8.1) has an asymptotic distribution given by

T 1=2(�0 � �0)w! N(0; ��1)�

T (�1 � �1)

T 2(�2 � �2)

�w!�Z 1

0

�H1

H2

�0�H1

H2

�00du

��1 Z 1

0

�H1

H2

�(dW )0

and T 1=2(�0 � �0) and (T (�1 � �1); T2(�2 � �2)) are asymptotically independent.

9 The asymptotic distribution of the estimators

We saw in section ?? that the I(2) model is a submodel of the linear regression modelwith I(2) regressors de�ned as

Z 00t = (X0t�1�

0 +�X 0t�1

0; �X 0t�1�

0; �2Xt�1; : : : ;�2Xt�k+1);

Z 01t = (�X0t�1�

0?2;X

0t�1�

0?1);

Z 02t = X 0t�1�

0?2;

I(2) models 20

of dimensions (2r+s1; p�r; s2) respectively. The regressors are constructed so that theorder of integration of Zit is I(i); i = 0; 1; 2: Corresponding to this choice of regressorswe �nd that the I(2) model is given by the parametric restrictions

�00 = (�;�( � 0)0�� 0 + �?(�0?�?)

�1�0);

�01 = (�( � 0)0��0?2 + �?(�

0?�?)

�1�0� 0��0?2;��

0��0?1);

�02 = ��0��0?2:

These are obviously not freely varying and it is convenient to introduce the freelyvarying parameters

A = �( � 0)0�� 0 + �?(�0?�?)

�1�0;

B0 = ��00?2( � 0); B1 = ��

00?1�; B2 = ��

00?2�;

C = ��00?2��?;

so that

�00 = (�;A);

�01 = (�B00 + �?(�

0?�?)

�1�0� 0��0?2;�B

01) = �01(�0; B1; C;B2);

�02 = �B02 = �02(�0; B2):

(9.1)

Note that at the true value we have B0 = 0; B1 = 0; B2 = 0; C = 0; and hencethat �01 = 0 and �02 = 0: To show that �1 = �1(�0; B0; B1; C;B2) we �nd that � as alinear function of B1 :

� = �� 00�� = �� 00�� = �� 00�

= �� 00(�0��00+ �0?1

��00?1 + �0?2

��00?2)�

= �0��00� + �� 00�0?1

��00?1� + ��

00�0?2��00?2�

� = �0 + �� 00�0?1B1

so that �?(�0?�?)�1�0� 0��

0?2 is a function of �0; B1; B2; and C :

�?(�0?�?)

�1�0� 0��0?2

= �?(�0?�?)

�1�0���0� 0��0?2 + �?(�

0?�?)

�1�0��?�0?�

0��0?2

= �?(�0?�?)

�1�0��(B1)B2 + �?(�0?�?)

�1�0��?(B1)C0:

Thus a rather complicated nonlinear parameter restriction is found. It turns out,see below, that T 1=2(�0 � �0); T (B0; B1; C) and T 2B2 are bounded in probability andwe therefore call �0 and I(0) parameter, B0; B1; and C are the I(1) parameters and�nally B2 is the I(2) parameter.The parametrization (9.1) has the following properties which turn out to be crucial

for the analysis

@�1@�0

j�=�0 = 0;@�2@�0

j�=�0 = 0;@�2@B0

j�=�0 = 0;@�2@B1

j�=�0 = 0;@�2@B2

j�=�0 = 0

I(2) models 21

Thus the parameters �j may depend on the parameters with a lower order, but notvery much close to the true value 0.Close to the true value we also �nd that

@�1@B0

j�=�0 = (�; 0)

@�1@B1

j�=�0 = (0;�)

@�1@Cj�=�0 = (0�0?(�

00?

0�0?)�1�00��0?; 0)

@�2@B2

j�=�0 = �0

so that locally we can recover the I(1) and I(2) parameters from the estimates of �1 and�2: These conditions (??) and (??) are the conditions needed for �nding asymptoticdistribution both of estimators and test statistics, see Johansen (2007).In order to describe the limit distributions and the limits of the regressors, we de�ne

the independent Brownian motions

W1(u) = (�0�1�)�1�0�1W (u);W2(u) = (��0?�(�

0?�?)

�1�0��?)�1��0?�(�

0?�?)

�1�0?W (u):

and the processes H1B; H1C ; and H2 by

T�12Z1[Tu] = T�

12

��0?2�X[Tu]

�0?1X[Tu]

�w!�H1C(u) = �0?2C2W (u)H1B(u) = �0?1C1W (u)

�;

T�32Z2[Tu] = T�

32�0?2X[Tu]

w! H2(u) = �0?2C2

Z u

0

W (s)ds

Based on these we de�ne the mixed Gaussian distribution ofB1 = (B100 ; B10

1 ; B102 )

0 andC1

B1 = [

Z 1

0

HH 0dt]�1Z 1

0

H(dW1)0; (9.2)

where H = (H 01C ; H

01B; H

02)0 and

C1 = [

Z 1

0

H1CH01Cdt]

�1Z 1

0

H1C(dW2)0: (9.3)

With this notation we can �nd the asymptotic distribution of the estimators to be:

Theorem 2 When � is normalized on �� and � is normalized on �� ; the asymptoticdistributions of the matrices ; �, and � are given by

T ��0?2( � )

w! B10 ; (9.4)

T ��0?1(� � �)

w! B11 ; (9.5)

T 2��0?2(� � �)

w! B12 ; (9.6)

T ��0?2(� � �)�?

w! C1: (9.7)

I(2) models 22

Further for�00 = (�;A;�1; � � � ;�k�2) (9.8)

we �ndT

12 (�0 � �0)

w! N(0; ��1); (9.9)

where� = V ar(X 0

t�1� +�X0t�1 ;�X

0t�1� ;�

2X 0t�1; � � � ;�2X 0

t�k+2):

10 The asymptotic distribution of the test on �

We want to discuss test of hypotheses on the parameters of the II(2) model. For thiswe need to formulate these as hypotheses on the parameters A;B;C; and an exampleis given in the next section. We formulate hypotheses in terms of parameters �1 and�2; where the idea is to de�ne them in such a way that T �1 and T

2�2 are convergentin distribution. Roughly speaking we need to be able to recover �2 from B2 and �1from B0; B1; C: Because the limit distribution of (B; C) is quite complicated we furtherneed to split the parameter �1 = (�1B; �1C) so that �1B can be recovered from B0; B1and �1C from C; see condition (10.3) below.We formulate some conditions that are enough to ensure asymptotic �2 distributions

of likelihood ratio test statistics.

Weak dependence :@B2@�1

j�=0 = 0;@2B2

@�21j�=0 = 0; (10.1)

Separation of �1B and �1C :@(B0; B1)

@�1Cj�=0 = 0;

@C

@�1Bj�=0 = 0: (10.2)

and �nally a condition that ensures that we can identify the parameters. We split theparameter �1 = (�1B; �1C) of dimensions q1B and q1C respectively (q1B + q1C = q1); sothat

Identi�cation : rank(@(B0; B1)

@�1Bj�=0) = q1B; rank(

@C

@�1Cj�=0) = q1C ; rank(

@B2@�2

j�=0) = q2

(10.3)Then the we have

Theorem 3 Under the assumption that the parameters B0; B1; B2; C are smoothlyparametrized by the continuously identi�ed parameters �1B;�1C ; and �2; of dimensionq1B; q1C ; and q2 and that conditions (10.1), (10.2), and (10.3) are satis�ed; the like-lihood function is local asymptotic quadratic and the asymptotic distributions of T �1Cand (T �1B; T

2�2) are mixed Gaussian, so that with KC =@Cv

@�v1C0 ; of dimension nC�q1C ;

we have

T �v

1Cw!�K 0C(

Z 1

0

H1CH01Cdt �12 )KC

��1K 0C(

Z 1

0

H1CH01Cdt �12 )(C1)v;

I(2) models 23

and with the (nB � (q1B + q2)) matrix KB de�ned by

KB =

@(Bv00 ;B

v01 )

0

@�v1B0 0

0@Bv2@�v2

0

!

we have

(T �1B; T2�2)

v w!�K 0B

�Z 1

0

HH 0du �11�KB

��1K 0B

�Z 1

0

HH 0du �11�(B1)v:

The asymptotic distribution of the estimates of the remaining parameters

�00 = (�;A;1; � � � ;k�2)

is given in Theorem 2. Hence the asymptotic distribution of the likelihood ratio test forthe hypothesis � = �(�) is asymptotically �2(nB + nC � (q1 + q2)):

This result is taken from Johansen (2007), see also Boswijk (2000).

10.1 A simple example with a simulation

In order to illustrate the �ndings we consider a three variable system

�2X1t = �0:5[X1t�1 + �2X2t�1 + (� 1 + �� 2)X3t�1+2�X1t�1 +�X2t�1 + ( 3 + 2� 1)�X3t�1] + "1t;

�2X2t = ��X2t�1 � � 2�X3t�1 + "2t;�2X3t = "3t:

The example is so constructed that the cointegration parameters are

� =

0@ 1 00 1� 1 � 2

1A ; � =

�1�2

�; � =

0@ 1�2

� 1 + �2� 2

1A ; =

0@ 21

3 + 2� 1

1A ;

and the true value is taken to be � 1 = � 2 = �2 = 3 = 0; so that � and � arenormalized on �� 0 and ��0: We have �xed the parameters � = (�0:5; 0; 0)0;3 = I3; and 0�� 0 = (2; 1); as these parameters do not enter the cointegrating relations. The newparameters (4.5) become

B0 = 3 + 2� 1; B1 = �2; B2 = � 1 + �2� 2; C = ��2� 1 + � 2;

which are variation free. This shows that the model can be tested by an asymptotic�2 test since the conditions of Theorem 3 are satis�ed.We now consider four hypotheses and for each check the conditions behind Theorem

3. At the end of this section we have conducted a small simulation experiment toillustrate what may happen when we cannot prove asymptotic �2 inference.

I(2) models 24

10.1.1 The hypothesis � 2 = 0

The hypothesis � 2 = 0 is equivalent to C = �B1B2: We �nd @C=@B1 = �B2; whichis zero at the true value so that inference is asymptotic �2(1) by Theorem 3. Thehypothesis is a special case of test on a coe¢ cient in � , but it is also a special case oftesting that a given vector is contained in sp(�):

10.1.2 The hypothesis � 1 = 0

The hypothesis � 1 = 0 is equivalent to B2 = B1C: In this case we have @B2=@B1 = Cand @B2=@C = B1; which are both zero at the true value. We also �nd, however,that @2B2=@C@B1 = 1; so that condition (10.1) is not satis�ed. The simulation inTable 10.1 indicates that indeed the asymptotic distribution is not �2(1) as the mean,variance and 95% quantile are all too small.The hypothesis is a special case of test on a coe¢ cient in � ; see section ??, but it is

also a special case of testing that a given vector b is contained in sp(�); see section ??.The example has been constructed so that b = �0; and if this is the case one cannotuse the �2 distribution for inference.

10.1.3 The test that 3 = 0;

The test that 3 = 0; is equivalent to B0 = 2(B2 � CB1)=(1 + B21): In this case we

have to check the derivative @B0=@C = �2B1=(1+B21) which is zero at the true value.

Thus we get asymptotic �2 inference.

10.1.4 The test that � = 0;

Finally we consider the test that � = 0�? = 3 � � 2 = 0: The test is equivalentto B0 = (2(B2 � CB1) + B1B2 + C)=(1 + B2

1): The conditions of Theorem 3 are notsatis�ed as we �nd @B0=@C = (�2B1 + 1)=(1 +B2

1)jB1=0 = 1. Note, however, that thesimulations in Table 1 indicates that we nevertheless get an asymptotic distribution thatis very close to that of a �2(1): Thus the conditions given in Theorem 3 are probablynot necessary. The possibility of asymptotic �2 inference, based on simulations wasalso pointed out by Paruolo, see Paruolo (1995), in connection with the derivation ofthe distribution.

Hypothesis E(�2 logLR) V ar(�2 logLR) 95% quantile� 1 = 0 0.731 1.119 2.84� = 0 0.993 1.926 3.85

Table 2: The simulation has T = 500; and 10; 000 simulations. The table shows theestimated mean, variance, and 95% quantile of the log likelihood ratio test statistic forthe hypotheses � 1 = 0 and � = 0. For the �2(1) we have E(�2(1)) = 1; V ar(�2(1)) = 2;and �2(1)0:95 = 3:84:

I(2) models 25

11 References

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