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IMPACT OF EXOGENEITY AND

MEASUREMENT ERRORS ON

VECTOR ERROR CORRECTION MODEL

by

Hanwoom Hong

Dissertation

Submitted in fulfillment of the requirement for the degree

of

Doctor of Philosophy

in Statistics

The Department of Statistics

College of Natural Sciences

Seoul National University

Febrary, 2014

i

ABSTRACT

IMPACT OF EXOGENEITY AND

MEASUREMENT ERRORS ON VECTOR

ERROR CORRECTION MODEL

Hanwoom Hong

In this thesis we consider two different topics of vector error correction model in

cointegration analysis.

In chapter 3, we consider the impact of the exogeneity on the vector error

correction model. We develop a VECM when vector processes contain exogenous

processes. Our model generalize the condition of Hunter (1990)’s cointegrating

exogeneity, weakly exogeneity of Johansen (1992), Harbo et al. (1998), and Pesaran

et al. (2000), and the non-causality of Mosconi and Gianini(1992) and Pradel and

Rault (2003). We also develop least square estimator and maximum likelihood

estimator of vector error correction model with exogenous processes. Asymptotic

properties of estimators are derived and finite sample properties of the estimators

are examined through a Monte Carlo simulation.

In chapter 4, we investigate the impact of the measurement errors to vector error

correction model with general cointegration rank. We study the impact of

ii

measurement errors to estimators and likelihood ratio test statistic for cointegration.

This is the extension of Hassler and Kuzin (2009)’s study which derived asymptotic

distribution of LR test statistic for the null hypothesis of no cointegration. We also

investigate the impact of measurement errors on the test through a Monte Carlo

simulation study.

Key words: cointegration, error correction model,exogeneity, Granger causality,

Measurement Error

Student Number: 2008-20273

iii

Contents

ABSTRACT i

List of Tables v

List of Figures vi

Chapter 1 Introduction 1

Chapter 2 Preliminary Concepts 6

2.1 Cointegration Analysis and Vector Error Correction Model……….6

2.1.1 Definitions of Cointegration…………………….….…..6

2.1.2 Estimations of Cointegration Analysis………...………..9

2.2 Cointegration Analysis with Exogenous Variables……...…………16

2.3 Cointegration Analysis with Measurement Errors……………...….21

Chapter 3 Impact of Exogeneity on the Vector Error Correction Model 30

3.1 Introduction…………………………………...……………………31

3.2 Parameterization and Least Square Estimation of the Model………32

iv

3.3 Maximum Likelihood Estimation and Its Asymptotic Properties….40

3.4 Numerical Example…………………………………...……………49

3.5 Monte Carlo Results ……….……………………………….....…...54

3.6 Appendix of Asymptotics of the Maximum Likelihood Estomator..59

Chapter 4 Impact of Measurement Errors on the Vector Error Correction

Model 66

4.1 Introduction………………………………………...………………67

4.2 Impact of the Measurement Errors to Least Square Estimator……..69

4.3 Impact of the Measurement Errors to Maximum Likelihood

Estimation……………………………………………….…………77

4.4 Impact of the Measurement Errors to Likelihood Ratio Test.…...…85

4.5 Monte Carlo Results……………………………………..…………90

Chapter 5 Concluding Remarks 103

References 106

국 문 초 록 112

v

List of Tables

3.1 Means and mean squared errors (MSE) of the LSE and MLE for various

sample sizes…………………………………………………..………..…...43

3.2 Average MSEs of and ……….………………………….....….……..…44

4.1 Bias and MSE ( ) for LSE and MLE of cointegration vector

based on 10000 replications for Each Sample Siz.………...…73

4.2. Bias and MSE ( ) for LSE and MLE of adjust matrix

and based on 10000 replications for Each Sample

Size…………………………………………………………………………..75

4.3 Empirical rejection numbers of the likelihood ratio test for the conitegrating

rank r base on 1000 replications and the significance level of 0.05.…........…..78

4.4. Numbers of identified ranks by the likelihood ratio test for the conitegrating

rank r in based on 1000 replications and the level of significance level of

0.05…………………………………………………………………………....79

vi

List of Figures

3.1 Grain and Meat prices during the period January 1980 through December

2008…………………………………………………………………...………....49

3.2 Cointegrating combination of all the variables.…….………..…………...….53

3.3 Cointegrating combination of the exogenous variables…………...……...….53

1

Chapter 1

Introduction

A time series is a collection of time indexed random variables defined on a sample

space. Stationarity is the fundamental concept in time series analysis. However,

many economic time series show nonstationary behavior unlike those encountered

in natural science. For example, stock prices, exchange rates, price index, and sales

volumes either exhibit the stochastic trend or nonhomogenous variance hence do

not satisfy the stationary assumption. Differencing and variance stabilizing

transformations are useful tools to connect the nonstationary and stationary time

series.

By taking a difference we can transform nonstationary processes with stochastic

trend to stationary processes. Unfortunately, differencing processes leads to

2

information loss. In multivariate nonstationary time series, the loss of information is

larger than the univariate case. When there are long run relationships among

components of multivariate nonstationary processes, we lose a lot of information

though differencing, since differenced processes only consider a short-run dynamics

of the processes. Engle and Granger (1987) have solved this problem using the

concept of cointegration and suggested the vector error correction model (VECM).

VECM is a powerful model to analyze nonstationary processes without loss of

information. Engle and Granger (1987) also developed estimation procedure and

test for the existence of cointegration relationship among the multivariate

nonstationary processes. Johansen (1988) and Stock and Watson (1988) developed

tests for cointegration rank. Phillips and Durlauf (1986), Sims et al. (1990), Tsay

and Tiao (1990), and Ahn and Reinsel (1990) studied the asymptotic properties of

least square estimator (LSE) for multivariate nonstationary processes. Johansen

(1988) and Ahn and Reinsel (1990) also developed the maximum likelihood

estimation (MLE) and its asymptotic properties. Their estimation procedures are

remarkable in that they took account of the reduced rank structure of the error

correction term of VECM. Reinsel and Ahn (1992) developed the likelihood ratio

test for cointegration rank which is applicable to reduced rank coefficient matrices.

‘Exogeneity’ is fundamental to most empirical modelling, Engle et al. (1983).

Engle et al. (1983) gave a mathematical definition of the weak and strong

exogeneity. Modelling with consideration of exogeneity could be useful when

3

multivariate process contain exogenous variables. VECM containing exogenous

variables have been studied by many authors. Hunter (1990) defined ‘cointegrating

exogeneity’ and applied his test of cointegrating exogeneity to Purchasing Power

Parity (PPP) and uncovered interest rate parity in the United Kingdom, Hunter

(1992). Johansen (1992), Harbo et al. (1998), and Pesaran et al. (2000) studied

cointegration analysis with exogenous variables. Their approaches used sufficient,

but not necessary condition of weakly exogenous assumptionz. Before Engle et al.

(1983) the concept of exogeneity was used in confusion with ‘causality’. Mosconi

and Gianini (1992) developed estimating method and testing procedure for non-

causality. Pradel and Rault (2003) studied the case of strongly exogenous.

Some time series cannot be observed directly, rather observed with measurement

errors. It is well known that the measurement errors affect the results of statistical

analysis. Fuller (1987) compiled the regression analysis with measurement errors

when time series is stationary. Cook and Stefanski (1994) provided the estimating

procedure, called SIMEX (simulation extrapolation), which adjust the effect of the

measurement errors. Their procedure is applicable to the nonparametric regression.

In the case of nonstationary process which was observed with measurement errors,

Phillips and Durlauf (1986) showed that measurement errors do not affect the

consistency of the LSE. Haldrup et al. (2005). Hassler and Kuzin (2009), however,

showed that the asymptotic distribution is affected by measurement errors. Hassler

and Kuzin (2009) also derived the asymptotic distribution of likelihood ratio (LR)

4

test statistic for no cointegration relationship among the components of multivariate

nonstationary processes.

In the thesis, we investigate the impact of exogeneity and measurement errors on

the VECM. It will be a generalization of the previous studies. This thesis is

organized as follows:

Chapter 2 contains the preliminary concepts which are helpful to understand the

impact of exogeneity and measurement errors on the cointegration analysis. We

introduce definitions of cointegration, the form of the VECM, estimation processes.

Previous studies of cointegration analysis with exogenous variables, and with

measurement errors are reviewed.

In chapter 3, we consider the impact of the exogeneity on the VECM. We develop

a VECM when vector processes contain exogenous processes. Our model generalize

the condition of Hunter (1990)’s cointegrating exogeneity, weak exogeneity of

Johansen (1992), Harbo et al. (1998), and Pesaran et al. (2000), and the non-

causality of Mosconi and Gianini(1992) and Pradel and Rault (2003). We also

develop LSE and MLE of VECM with exogenous processes. Asymptotic properties

of estimators are derived and finite sample properties of the estimators are

examined through a Monte Carlo simulation. A relatively simple numerical example

to illustrate the methods in chapter 3 is also given.

Chapter 4 investigates the impact of the measurement errors to VECM with

general cointegration rank. Previous studies of cointegration analysis with

5

measurement errors by Phillips and Durlauf (1986), Haldrup et al. (2005), and

Hassler and Kuzin (2009) are not applicable to the general VECM model. These can

be considered of the special case of VECM with cointegration rank 1. We study the

impact of measurement errors to LR test statistic for cointegration rank as well as to

estimates of the parameters. This is the extension of Hassler and Kuzin (2009)’s

study which derived asymptotic distribution of LR test statistic for the null

hypothesis of no cointegration. We also investigate the impact of measurement

errors on the test through a Monte Carlo simulation study

Chapter 5 contains the concluding remarks.

6

Chapter 2

Preliminary Concepts

In this chapter, we review some basic concepts which are helpful to understand

the impact of exogeneity and measurement errors on the cointegration analysis.

2.1 COINTEGRATION ANALYSIS AND VECTOR ERROR

CORRECTION MODEL

2.1.1 Definitions of Cointegration

Consider an 𝑚-dimensional nonstationary vector process 𝑋𝑡 with integrated order

𝑘, 𝐼(𝑘), linear combination of X𝑡 will also be 𝐼(𝑘) in general. However, if there

exists a vector 𝛽 such that 𝛽′𝑋𝑡~𝐼(𝑘 − 𝑏) with 𝑘 ≥ 𝑏 > 0, Engle and Granger

(1987) defined that X𝑡 is cointegrated of order (𝑘, 𝑏), denoted by 𝑋𝑡~𝐶𝐼(𝑘, 𝑏)

7

with cointegrating vector 𝛽. The cointegrating rank 𝑟 is defined as the maximal

value 𝑟 = rank(𝐵) for 𝑚 × 𝑟 matrix 𝐵′ = [𝛽1, … , 𝛽𝑟] . In other words, the

cointegrating rank 𝑟 is the number of linearly independent cointegrating vectors.

Note that 𝐵𝑋𝑡 is 𝑟-dimensional vector process of integrated order 𝑘 − 𝑏. If 𝑟 = 0,

no cointegrating vector exists. If 𝑟 = 𝑚, the matrix 𝐵 has a full rank and so that

𝑋𝑡 = 𝐵′−1𝐵′𝑋𝑡~𝐼(𝑘 − 𝑏) is in contradiction to 𝑋𝑡~𝐼(𝑘). Therefore, it can be

assumed that 𝑟 satisfy 0 < 𝑟 < 𝑚 for cointegration analysis. In particular, when

𝑋𝑡 is a 𝐼(1) process and there exists full rank 𝑚 × 𝑟 matrix 𝐵 such that 𝐵′𝑋𝑡 is

𝐼(0) , that is, stationary process, then 𝑋𝑡 is said to be cointegrated with

cointegrating rank 𝑟.

Now, we consider an 𝑚-dimensional vector process X𝑡 of integrated order one,

𝐼(1) generated by an autoregressive process of order 𝑝, VAR(𝑝) given by

Φ(𝐿)𝑋𝑡 = 𝑋𝑡 − ∑ Φ𝑗𝑋𝑡−𝑗 = 휀𝑡

𝑝

𝑗=1

, (2.1)

where 𝐿 is an lag operater such that 𝐿𝑋𝑡 = 𝑋𝑡−1 and the error 휀𝑡 are independent

𝑚-dimensional vector process with mean zero and nonsingular covariance matrix

Ω𝜀. We assume that det{Φ(𝐿)} = 0 has 𝑑 (0 < 𝑑 < 𝑚) unit roots and remaining

roots are outside the unit circle. These assumptions imply that 𝐼𝑚 − Φ(1) =

∑ Φ𝑗𝑝𝑗=1 has 𝑑 linearly independent eigenvectors associated with eigenvalue 1.

The model (2.1) can be written as

8

Φ∗(𝐿)(1 − 𝐿)𝑋𝑡 = −Φ(1)𝑋𝑡−1 + 휀𝑡 , (2.2)

where Φ∗(𝐿) = 𝐼𝑚 − ∑ Φ𝑗∗𝐿𝑗𝑝−1

𝑗=1 with Φ𝑗∗ = − ∑ Φ𝑘

𝑝𝑘=𝑗+1 and Φ(1) = 𝐼𝑚 −

∑ Φ𝑗𝑝𝑗=1 . Let 𝑟 = 𝑚 − 𝑑 > 0. Since det{Φ(𝐿)} = 0 has 𝑑 unit roots, there are

𝑚 × 𝑚 matrices 𝑃 and 𝑄 = 𝑃−1 such that 𝑄(∑ Φ𝑗𝑝𝑗=1 )𝑃 = 𝐽 , where 𝐽 =

diag(𝐼𝑑, 𝛬𝑟) is the Jordan canonical form of ∑ Φ𝑗𝑝𝑗=1 . We can easily see that

Φ(1) = 𝐼𝑚 − ∑ Φ𝑗𝑝𝑗=1 = 𝑃(𝐼𝑚 − 𝐽)𝑄 = 𝑃2(𝐼𝑟 − Λ𝑟)𝑄2

′ when we partition

𝑄′ = [𝑄1, 𝑄2], 𝑄1′ = [𝑄11

′ , 𝑄12′ ], 𝑄2

′ = [𝑄21′ , 𝑄22

′ ], 𝑃 = [𝑃1, 𝑃2], 𝑃1 = [𝑃11, 𝑃21]′,

𝑃2 = [𝑃12, 𝑃22]′ such that 𝑄1 and 𝑃1 are 𝑚 × 𝑑 , 𝑄2 and 𝑃2 are 𝑚 × 𝑟 , 𝑄11′

and 𝑃22 are 𝑟 × 𝑑, 𝑄12′ and 𝑃21 are 𝑑 × 𝑑, 𝑄21

′ and 𝑃12 are 𝑟 × 𝑟, and 𝑄11′

and 𝑃22 are 𝑑 × 𝑟 matrices. The rank of the 𝑚 × 𝑚 matrix Φ(1) is reduced

rank 𝑟. Let 𝐶 = −Φ(1) = 𝐴𝐵, then (2.2) can be written as

Φ∗(𝐿)(1 − 𝐿)𝑋𝑡 = 𝐶𝑋𝑡−1 + 휀𝑡

= 𝐴𝐵𝑋𝑡−1 + 휀𝑡 . (2.3)

The representation of the model (2.3) is a form of the vector error correction

model (VECM) studied by Engle and Granger (1987). 𝐶𝑋𝑡−1 is called the error

correction term which adjust for over-differencing. 𝐶 = 𝐴𝐵 is a full-rank

factorization of 𝐶 since 𝑚 × 𝑟 matrices 𝐴 and 𝐵 have full rank 𝑟 by the

9

defition of 𝐴 and 𝐵. In the VECM, 𝐵𝑋𝑡−1 is assumed to be stationary. Ahn and

Reinsel(1990) showed that 𝐵𝑋𝑡−1 is stationary indeed when we define 𝐴 =

𝑃2(𝐼𝑟 − Λ𝑟)𝑄21′ and 𝐵 = 𝑄21

′−1𝑄2 . Since 𝐵𝑋𝑡−1 is a r-dimensional stationary

process, 𝑋𝑡 is cointegrated with cointegrating rank 𝑟. The matrix 𝐵 is called the

cointegrating matrix, and 𝐴 is called the adjust matrix, or loading matrix. Note that

𝐵 is the coefficient matrix of notstationary process 𝑋𝑡−1 and 𝐴 can be treated as

a coefficient matrix of stationary process 𝐵𝑋𝑡−1.

2.1.2 Estimations of Cointegation Analysis

We rewrite the VECM (2.3) as

Δ𝑋𝑡 = 𝐶𝑋𝑡−1 + ∑ Φ𝑗∗Δ𝑋𝑡−𝑗

𝑝−1

𝑗=1

+ 휀𝑡

= 𝐴𝐵𝑋𝑡−1 + ∑ Φ𝑗∗Δ𝑋𝑡−𝑗

𝑝−1

𝑗=1

+ 휀𝑡 , (2.4)

where Δ is the difference operator and Φ𝑗∗ is as defined following (2.2).

In (2.4), identification problem occurs since we cannot distinguish 𝐶 = 𝐴𝐵 and

𝐶 = 𝐴∗𝐵∗ = 𝐴𝑀𝑀−1𝐵 for some 𝑚 × 𝑚 invertible matrix 𝑀 . Johansen(1988)

developed a maximum likelihood estimation procedure based on the so-called

reduced rank regression with identifying condition 𝐵′Ω𝜀𝐵 = 𝐼. This method is

motivated by canonical correlation analysis and so has a closed form solution of

10

maximum likelihood equation. He also provided the testing procedure of linear

hypothesis about the cointegrating vectors using Chi-square test.

On the other hand, Ahn and Reinsel (1990) used normalized 𝐵 = [𝐼𝑟, 𝐵0] for

the identifying condition. This condition is motivated by the regression analysis

since 𝐵′𝑋𝑡−1 = 𝑋1𝑡−1 + 𝐵0′ 𝑋2𝑡−1 with partitioning 𝑋𝑡 = [𝑋1𝑡

′ , 𝑋2𝑡′ ] where 𝑋1𝑡

is 𝑑 × 1 and 𝑋2𝑡 is 𝑟 × 1. For this normalization, it is assumed that 𝑋𝑡 are

arranged so that 𝑋2𝑡 is purely nonstationary, that is, not cointegrated. Classically,

Engle and Ganger (1987) proposed the two-step regression procedure given by

��0∗ = − (∑ 𝑋1𝑡

𝑇

𝑡=1

𝑋2𝑡′ ) (∑ 𝑋2𝑡

𝑇

𝑡=1

𝑋2𝑡′ )

−1

(2.5)

and

Θ∗ = (∑ Δ𝑋𝑡

𝑇

𝑡=𝑝

��𝑡−1∗′ ) (∑ Δ𝑋𝑡

𝑇

𝑡=𝑝

��𝑡−1∗′ ) (2.6)

where Θ = [𝐴, Φ1∗ , … , Φ𝑝−1

∗ ] and ��𝑡−1∗ = [(��∗𝑋𝑡−1)

′, Δ𝑋𝑡−1

′ , … , Δ𝑋𝑡−𝑝+1′ ]′ .

Instead of Engle and Granger’s procedure, Ahn and Reinsel (1990) developed the

estimation procedure using least square estimator (LSE) and maximum likelihood

estimator (MLE). They showed their LSE and MLE have smaller mean square

errors (MSE) than Engle and Granger (1987)’s classical estimation. Ahn and

Reinsel (1990)’s LSE is given by

11

�� = (∑ Δ𝑋𝑡

𝑇

𝑡=1

𝑈𝑡−1′ ) (∑ 𝑈𝑡−1

𝑇

𝑡=1

𝑈𝑡−1′ )

−1

(2.7)

where 𝐹 = [𝐶, Φ1∗ , … , Φ𝑝−1

∗′ ] and 𝑈𝑡−1 = [𝑋𝑡−1′ , Δ𝑋𝑡−1

′ , … , Δ𝑋𝑡−𝑝+1′ ]′ . Ahn and

Reinsel (1990) also obtained MLE using the Newton-Raphson algorithm as follows;

��(𝑖+1) = ��(𝑖) − (∑𝜕휀𝑡

′

𝜕𝜂Ω𝜀

−1𝜕휀𝑡

𝜕𝜂′)��(𝑖)

−1

(∑𝜕휀𝑡

′

𝜕𝜂Ω𝜀

−1휀𝑡)��(𝑖)

(2.8)

where 𝜂 = (𝛽′, 𝜃′)′, 𝛽 = 𝑉𝑒𝑐{𝐵0′ }, 𝜃 = 𝑉𝑒𝑐{[𝐴, Φ1

∗ , … , Φ𝑝−1∗ ] } . The gradient

vector 𝜕휀𝑡′/𝜕𝜂 is driven as

𝜕휀𝑡

′

𝜕𝜂= − [

𝑋2𝑡 ⊗ 𝐴′

��𝑡−1 ⊗ 𝐼𝑚] (2.9)

where ��𝑡−1 = [(𝐵𝑋𝑡−1)′, Δ𝑋𝑡−1′ , … , Δ𝑋𝑡−𝑝+1

′ ]′. The initial estimator ��(0) can be

driven using LSE ��. The initial estimator of 𝐴 and 𝐵 are obtained as

�� = ��1 (2.10)

and

��0′ = (��′Ω𝜀

−1��)−1

��′Ω𝜀−1��2 (2.11)

12

by partitioning �� = [��1, ��2] where ��1 and ��2 are respectively 𝑚 × 𝑟 and

𝑚 × 𝑑, and Ω𝜀−1 is a sample variance of the regression residuals using (2.7). Ahn

and Reinsel (1990) also derived asymptotic distributions of the above estimators.

The key in the derivation of the asymptotic distribution of the estimators is the

Lemma 1. of Ahn and Reinsel (1990).

Lemma 1. (Ahn and Reinsel, 1990) Let 𝑍𝑡 = 𝑄𝑋𝑡 = [𝑍1𝑡′ , 𝑍2𝑡

′ ]′ and 𝑎𝑡 =

[𝑎1𝑡′ , 𝑎2𝑡

′ ]′ = 𝑄휀𝑡 such that 𝑍1𝑡 and 𝑎1𝑡 are 𝑑 × 1 and 𝑍2𝑡 and 𝑎2𝑡 are 𝑟 × 1,

with Ω𝑎 = 𝑄Ω𝜀𝑄′ and Ω𝑎1= [𝐼𝑑 , 0] Ω𝑎[𝐼𝑑 , 0]′ . In addition, define Ψ11 =

[𝐼𝑑 , 0] Ψ[𝐼𝑑 , 0]′ with Ψ = ∑ Ψ𝑘∞𝑘=0 and Ψ𝑘 are the infinity moving average

coefficients in the representation 𝑢𝑡 = ∑ Ψ𝑘∞𝑘=0 𝑎𝑡−𝑘 for the stationary process

𝑢𝑡 = 𝑄(Δ𝑋𝑡 − 𝐶𝑋𝑡−1) = 𝑍𝑡 − diag(𝐼𝑑 , Λ𝑟)𝑍𝑡−1 , and let 𝐵𝑚(𝑢) and 𝐵𝑑(𝑢) =

Ω𝑎1

−1/2[𝐼𝑑, 0]Ω𝑎1

1/2𝐵𝑚(𝑢) be standard Brownian motions of dimensions 𝑚 and 𝑑

respectively. Under the assumptions of model (2.4), the following distributional

results the hold:

(i) 𝑇−2 ∑ 𝑍1𝑡−1𝑍1𝑡−1′𝑇

𝑡=1

𝑑→ Ψ11Ω𝑎1

1/2∫ 𝐵𝑑(𝑢)𝐵𝑑(𝑢)′𝑑𝑢

1

0Ω𝑎1

1/2Ψ11

′ =: ℬ𝑍𝑍

13

(ii) 𝑇−1 ∑ 𝑎𝑡𝑍1𝑡−1′𝑇

𝑡=1

𝑑→ Ω𝑎

1/2{∫ 𝐵𝑑(𝑢)𝑑𝐵𝑚(𝑢)

1

0

′}

′

Ω𝑎1

1/2Ψ11

′ =: ℬ𝑎𝑍

(iii) 𝑇−3/2 ∑ 𝑈𝑡−1𝑍1𝑡−1′𝑇

𝑡=1 𝑝→ 0

(iv) 𝑇−1 ∑ 𝑈𝑡−1𝑈𝑡−1′𝑇

𝑡=1 𝑝→ Γ𝑈 = Cov(𝑈𝑡)

(v) 𝑇−1/2 ∑ vec(𝑎𝑡𝑈𝑡−1′ )𝑇

𝑡=1 𝑑→ 𝑁(0, Γ𝑈 ⊗ Ω𝑎)

where the operator ⊗ is a Kronecker product.

The asymptotic distribution of LSE �� and MLE ��0 , 𝜃 are summarized at

Theorem 1 and Theorem 2 of Ahn and Reinsel (1990).

14

Theorem 1. (Ahn and Reinsel, 1990) Let �� be the lease square estimator of 𝐹

in (2.7). Then

(�� − 𝐹)𝑃𝐷𝑑→ [𝑀, 𝑁]

where 𝑀 = 𝑃ℬ𝑎𝑍ℬ𝑍𝑍−1 and 𝑁 is such that vec(𝑁) is distributed as 𝑁(0, Γ𝑈

−1 ⊗

Ω𝜀).

Theorem 2. (Ahn and Reinsel, 1990) Let �� denote the MLE obtained from (2.8)

using the initial consistent estimator. Then

𝑇(��0 − 𝐵0) 𝑑→ (𝐴′Ω𝜀

−1𝐴)−1𝐴′Ω𝜀−1𝑃𝑀𝑃21

−1

where the distribution of 𝑀 is specified in Theorem 1 of Ahn and Reinsel (1990),

and

√𝑇(𝜃 − 𝜃) 𝑑→ 𝑁(0, Γ��

−1 ⊗ Ω𝜀)

as 𝑇 → ∞.

15

Before estimating the parameters of VECM (2.4), we should specify the

cointegration rank 𝑟 first. Johansen (1988) and Reinsel and Ahn (1992) obtained

the likelihood ratio (LR) test statistic and its asymptotic distribution for the null

hypothesis of cointegration rank 𝑟, that is, H0: rank(𝐶) is reduced rank 𝑟. This

null hypothesis is equivalent to H0: det{Φ(𝐿)} = 0 has 𝑑 unit roots. The

alternative hypothesis H1 is that rank(𝐶) is full rank 𝑚, or, there are not unit

root in VECM (2.4). The LR test statistic 𝜆 is given as

𝜆 =|𝑆|

|𝑆0| (2.12)

when 𝑆0 is the estimate of the error variance Ω𝜀 under the reduced model and 𝑆

is the estimate of the error variance Ω𝜀 under the full model. Johansen (1988) and

Reinsel and Ahn (1992) have been shown that the LR test statistic 𝜆 is

asymptotically distributed as

−𝑇log𝜆 𝑑→

tr {(∫ 𝐵𝑑(𝑢)𝑑𝐵𝑑(𝑢)′1

0

)

′

(∫ 𝐵𝑑(𝑢)𝐵𝑑(𝑢)′𝑑𝑢1

0

)

−1

(∫ 𝐵𝑑(𝑢)𝑑𝐵𝑑(𝑢)′1

0

)} (2.13)

where 𝐵𝑑(𝑢) is a 𝑑-dimensional standard Brownian motion. Johansen showed

that the choice of the identifying condition of the cointegration matrix 𝐵 does not

16

affect the asymptotic distribution (2.13). Johansen (1988) and Reinsel and Ahn

(1992) also provided the quantiles of the distribution (2.13) for various values of

𝑑 = 1,2,3,4,5 through simulation.

2.2 COINTEGRATION ANALYSIS WITH EXOGENOUS

VARIABLES

Consider the case that the vector process 𝑋𝑡 contains exogenous variables. Assume

that 𝑋𝑡 = (𝑌𝑡′, 𝑍𝑡

′)′, where 𝑌𝑡 is an 𝑚𝑦 -dimensional vector process of

endogenous variables and 𝑍𝑡 is an 𝑚𝑧-dimensional vector process of exogenous

variables with 𝑚𝑦 + 𝑚𝑧 = 𝑚. ‘𝑍𝑡 is exogenous’ means that 𝑍𝑡 is not affected by

𝑌𝑡 while 𝑌𝑡 is affected by 𝑍𝑡 . Although exogeneity is fundamental to most

empirical modelling (Engle et al., 1983), the concept of ‘exogeneity’ was

ambiguous before Engle et al. (1983). Engle et al. (1983) gave the mathematical

definition of the “exogeneity” and “causality”. He defined that 𝑍𝑡 is weakly

exogenous when the conditional probability density function (pdf) of 𝑋𝑡 given past

information (𝑋𝑡−1, … , 𝑋1) can be represented as

𝑓(𝑋𝑡|𝑋𝑡−1, … , 𝑋1; 𝜃) = 𝑓(𝑌𝑡|𝑍𝑡 , 𝑋𝑡−1, . . , 𝑋1; 𝜃1)𝑓(𝑍𝑡|𝑋𝑡−1, … , 𝑋1; 𝜃2) (2.14)

17

where 𝜃1 and 𝜃2 are variation free. In this case, we say (∆𝑌𝑡′, ∆𝑍𝑡

′)′ operates a

sequential cut on the distribution of ∆𝑋𝑡 . 𝑓(𝑌𝑡|𝑍𝑡 , 𝑋𝑡−1, . . , 𝑋1; 𝜃1) is a pdf of

conditional model and 𝑓(𝑍𝑡|𝑋𝑡−1, … , 𝑋1; 𝜃2) is a pdf of marginal model. For

example, consider the model

𝑦𝑡 = 𝛽′𝑧𝑡 + 휀1𝑡 (2.15)

𝑧𝑡 = 𝛿𝑧𝑡−1 + 휀2𝑡 (2.16)

where

[휀1𝑡

휀2𝑡] ~𝑁 (0, [

𝜎11 𝜎12

𝜎12 𝜎22]).

With the conditional model (2.15) and the marginal model (2.16), 𝑧𝑡 is weakly

exogenous only if 𝜎12 = 0 since the parameters of conditional model and marginal

model are not variation free unless 𝜎12 = 0.

Engle et al. (1983) also defined that 𝑍𝑡 is strongly exogenous when (i) 𝑍𝑡 is

weakly exogenous and (ii) 𝑍𝑡 Granger cause 𝑌𝑡 while 𝑌𝑡 does not Granger

cause 𝑍𝑡 . ‘𝑍𝑡 Granger cause 𝑌𝑡 ’ means that the past values of (𝑍𝑡−1, … , 𝑍1)

affect the current value of 𝑌𝑡. Therefore, If 𝑍𝑡 is strongly exogenous, (2.11) can be

represented as

𝑓(𝑋𝑡|𝑋𝑡−1, … , 𝑋1; 𝜃) = 𝑓(𝑌𝑡|𝑍𝑡 , 𝑋𝑡−1, . . , 𝑋1; 𝜃1)𝑓(𝑍𝑡|𝑍𝑡−1, … , 𝑍1; 𝜃2) (2.17)

18

where 𝜃1 and 𝜃2 are variation free. We can observe that 𝑧𝑡 is strongly exogenous

if 𝜎12 = 0 at the example of the model (2.15) and (2.16).

In the cointegration analysis, the 𝑋𝑡 = (𝑌𝑡′, 𝑍𝑡

′)′ does not have a pdf since 𝑋𝑡 is

assumed as 𝐼(1) process. So (2.14) should be modified as

𝑓(Δ𝑋𝑡|𝑋𝑡−1, Δ𝑋𝑡−1, … , Δ𝑋1; 𝜃)

= 𝑓(Δ𝑌𝑡|𝑌𝑡−1, 𝑍𝑡−1, Δ𝑋𝑡−1, . . , Δ𝑋1; 𝜃1)𝑓(Δ𝑍𝑡|𝑍𝑡−1, Δ𝑋𝑡−1, … , Δ𝑋1; 𝜃2). (2.18)

Now, consider the VECM (2.4) contains exogenous variable 𝑍𝑡. Partition the

coefficient matrix 𝐶, 𝛷𝑗∗, 휀𝑡 and Ω𝜀 conformable to 𝑋𝑡 = (𝑌𝑡

′, 𝑍𝑡′)′ as

𝐶 = (𝐶𝑦𝑦 𝐶𝑦𝑧

𝑂 𝐶𝑧𝑧), Φ𝑗

∗ = (Φ𝑗

∗𝑦𝑦Φ𝑗

∗𝑦𝑧

Φ𝑗∗𝑧𝑦

Φ𝑗∗𝑧𝑧) ,

휀𝑡 = (휀𝑦𝑡′ , 휀𝑧𝑡

′ )′ and Ω𝜀 = (

Ω𝑦𝑦 Ω𝑦𝑧

Ω𝑧𝑦 Ω𝑧𝑧) .

The term 𝐶𝑧𝑦 is zero matrix since 𝑍𝑡 is exogenous. We left-multiply

(𝐼 −𝛺𝑦𝑧𝛺𝑧𝑧

−1

𝑂 𝐼)

on both side of (2.4) and obtain

(Δ𝑌𝑡

Δ𝑍𝑡) = (

𝐶𝑦𝑦 𝐶𝑦𝑧 − 𝛺𝑦𝑧𝛺𝑧𝑧−1𝛱𝑧𝑧

𝑂 𝛱𝑧𝑧) (

𝑌𝑡−1

𝑍𝑡−1) + (

𝛺𝑦𝑧𝛺𝑧𝑧−1

𝑂) 𝛥𝑍𝑡

19

+ ∑ (𝛷𝑗

∗𝑦𝑦𝛷𝑗

∗𝑦𝑧− 𝛺𝑦𝑧𝛺𝑧𝑧

−1𝛷𝑗∗𝑧𝑧

𝛷𝑗∗𝑧𝑦

𝛷𝑗∗𝑧𝑧 )

𝑝−1

𝑗=1

(Δ𝑌𝑡−𝑗

Δ𝑍𝑡−𝑗)

+ (휀𝑦𝑡 − 𝛺𝑦𝑧𝛺𝑧𝑧

−1휀𝑧𝑡

휀𝑧𝑡) (2.19)

.

From this, we have conditional model of Δ𝑌𝑡 and marginal model of Δ𝑍𝑡 as

follows

Δ𝑌𝑡 = (𝐶𝑦𝑦 𝐶𝑦𝑧)𝑋𝑡−1 – 𝛺𝑦𝑧𝛺𝑧𝑧−1𝐶𝑧𝑧 𝑍𝑡−1 + 𝛺𝑦𝑧𝛺𝑧𝑧

−1𝛥𝑍𝑡

+ ∑ 𝐻𝑗Δ𝑋𝑡−𝑗

𝑝−1

𝑗=1

+ 𝑒𝑦𝑡 (2.20)

and

Δ𝑍𝑡 = 𝐶𝑧𝑧𝑍𝑡−1 + ∑ Φ𝑗∗𝑧.Δ𝑋𝑡−𝑗 + 휀𝑧𝑡 , (2.21)

𝑝−1

𝑗=1

where 𝐻𝑗 = (𝛷𝑗∗𝑦𝑦

, 𝛷𝑗∗𝑦𝑧

− 𝛺𝑦𝑧𝛺𝑧𝑧−1𝛷𝑗

∗𝑧𝑧 ) , 𝛷𝑗∗𝑧. = (𝛷𝑗

∗𝑧𝑦 , 𝛷𝑗

∗𝑧𝑧 ) and

𝑒𝑦𝑡 = 휀𝑦𝑡 − 𝛺𝑦𝑧𝛺𝑧𝑧−1휀𝑧𝑡, which is uncorrelated with 휀𝑧𝑡.

If 𝐶𝑧𝑧 = 0 , the model (2.20) and (2.21) are variation free and so 𝑍𝑡 is

weakly exogenous. This assumption implies that the exogenous variables do not

cointegrate among themselves. The marginal model (2.21) becomes

Δ𝑍𝑡 = ∑ Φ𝑗∗𝑧.Δ𝑋𝑡−𝑗 + 휀𝑧𝑡 . (2.22)

𝑝−1

𝑗=1

20

Johansen(1992), Harbo et al. (1998) and Pesaran et al. (2000) studied estimation of

parameters and cointegrating rank test procedure based on the models (2.20) and

(2.22) with this assumption. We will show that 𝐶𝑧𝑧 = 0 is not a necessary

condition for weakly exogeneity of 𝑍𝑡 in Chapter 3.

If Φ𝑗∗𝑧𝑦

= 0, 𝑌𝑡 does not Granger cause 𝑍𝑡. With this non-causality assumption,

the marginal model (2.18) becomes

Δ𝑍𝑡 = 𝐶𝑧𝑧𝑍𝑡−1 + ∑ 𝛷𝑗∗𝑧𝑧Δ𝑍𝑡−𝑗 + 휀𝑧𝑡 , (2.23)

𝑝−1

𝑗=1

that is, 𝑍𝑡 has a cointegration structure among themselves when rank{𝐶𝑧𝑧} =

𝑟𝑧 (0 < 𝑟𝑧 < 𝑟). Mosconi and Gianini(1992) developed estimating method and

testing procedure for non-causality based on the model (2.20) and (2.23). If Φ𝑗∗𝑧𝑦

and 𝐶𝑧𝑧 are both zero, 𝑍𝑡 is strongly exogenous process. Pradel and Rault (2003)

studied the case of stongly exogenous 𝑍𝑡.

In Chapter 3, we will re-parameterize and develop the estimation procedure based

on models (2.20) and (2.21) without weakly exogenous and non-causality

assumption. It will be a generalization of the previous results of Johansen(1992),

Harbo et al. (1998), Pesaran et al. (2000), Mosconi and Gianini(1992) and Pradel

and Rault (2003).

21

2.3 COINTEGRATION ANALYSIS WITH

MEASUREMENT ERRORS

Some data are observed with measurement errors. Even when data are observed

without measurement errors, estimated factors from a large set of variables are often

used with a few key variables of interest in dynamic factor modeling (Banerjee and

Marcellino, 2009). And these estimated factors are considered as variables with

measurement errors. It is well known that the measurement errors affect the results

of statistical analysis. For example, the model of simple regression analysis

containing measurement error is

𝑦𝑡 = 𝛽0 + 𝛽1𝑧𝑡 + 휀𝑡 , 휀𝑡~𝑁(0, 𝜎𝜀2) (2.24)

and independent variable 𝑧𝑡 is observed as 𝑧𝑡∗ = 𝑧𝑡 + 𝜂𝑡 where 𝑦𝑡 and 𝑧𝑡 are

both stationary processes. The measurement error 𝜂𝑡, of course, is also stationary.

Then asymptotic distribution of LSE estimate of 𝛽 is

��1

𝑝→

𝐶𝑜𝑣(𝑧𝑡 , 𝑦𝑡)

𝑉𝑎𝑟(𝑧𝑡∗)

=𝛽1𝜎𝑧

2

𝜎𝑧2 + 𝜎𝜂

2 =𝛽1

1 + 𝜎𝜂2/𝜎𝑧

2. (2.25)

where 𝜎𝑧2 and 𝜎𝜂

2 are the variance of 𝑧𝑡 and 𝜂𝑡 .The measurement errors,

therefore, cause a nonzero bias in LSE. This bias does not vanish asymptotically

resulting in the inconsistency of the LSE. More generally, consider the multivariate

regression analysis,

22

𝑌𝑡 = 𝐵0 + 𝐵1𝑍𝑡 + 휀𝑡 , 휀𝑡~𝑁(0, Ω𝜀) (2.26)

where 𝑌𝑡 is an 𝑚-dimensional dependent vector process, 𝑍𝑡 is an 𝑟-dimensional

independent vector process, 휀𝑡 is a regression error, 𝐵0 is an 𝑚-dimensional

vector of intercept, and 𝐵1 is an 𝑚 × 𝑟 matrix of coefficients. The independent

vector process 𝑍𝑡 is observed as 𝑍𝑡∗ = 𝑍𝑡 + 𝜂𝑡, 𝜂𝑡~𝑁(0, Ω𝜂). Then asymptotic

distribution of LSE estimate of 𝐵1 is

��1

𝑝→ 𝐶𝑜𝑣(𝑍𝑡 , 𝑌𝑡 )[𝑉𝑎𝑟(𝑍𝑡

∗)]−1 = 𝐵1Γ𝑧(Γ𝑧 + Ω𝜂)−1

(2.27)

where Γ𝑧 is the variance of the true independent process 𝑍𝑡. Fuller (1987) and

Cook and Stefanski (1994) give different estimating method which reduce (reducing)

the bias caused by measurement errors when the variance of measurement errors

Ω𝜂 is known. Fuller (1987) suggested that

��1 = −��𝑘𝑟��𝑟𝑟−1 (2.28)

where ��.𝑖 are the characteristic vectors of 𝐌𝑍𝑍: = 𝑇−1 ∑ 𝑍𝑡∗𝑍𝑡

∗′ in the metric Ω𝜂

associate with the roots 𝜆𝑚 ≤ ⋯ ≤ 𝜆1 of det{ 𝐌𝑍𝑍 − 𝜆Ω𝜂 }=0 , ��.𝑖 , 𝑖 =

1,2, … , 𝑚 , are the columns of �� = (��𝑟𝑟, ��𝑘𝑟) . Cook and Stefanski (1994)

23

suggested the method called SIMEX (simulation extrapolation). In SIMEX

procedure, we first generate new random variables

��𝑡,𝑖 = 𝑍𝑡,𝑖∗ + √𝜆 Ω𝜂

1/2𝑈𝑡,𝑖 , 𝑈𝑡,𝑖~𝑁(0,1), 𝑖 = 1,2, … , 𝑀 (2.29)

where 𝜆 is scalar and 𝑀 is large number. Then ��1,𝑖(𝜆) obtained by regressing

the 𝑌𝑡 on ��𝑡,𝑖 is a consistent estimate of

𝐵1Γ𝑧(Γ𝑧 + (1 + 𝜆)Ω𝜂)−1

. (2.30)

Define ��1(𝜆) = 𝑀−1 ∑ ��1,𝑖(𝜆𝑀𝑖=1 ) then ��1(−1) is consistent estimate of 𝐵1 .

Therefore, we can expect the estimate of ��1(−1) could reduce the bias caused by

measurement errors. Cook and Stefanski (1994) suggested estimate ��1(−1) as

computing ��1(𝜆) for the various values of 𝜆 such as 0,0.5,1.0,1.5,2.0 and

extrapolating the curve ��1(𝜆) to 𝜆 = −1 using quadratic regression.

Next, consider the case that integrated processes are used as predictor variables in

(time series) regression models.

𝑦𝑡 = 𝛽0 + 𝛽1′𝑍𝑡 + 휀𝑡 (2.31)

where the dependent process 𝑦𝑡 is a univariate 𝐼(1) process and the independent

process 𝑍𝑡 is a 𝑚 − 1 -dimensional non-cointegrated 𝐼(1) process. Equation

24

(2.23) is a special case of ECM of (2.4) where 𝑋𝑡 = (𝑦𝑡 , 𝑍𝑡′)′, 𝑝 = 1, 𝑟 = 1,

𝐴 = (−1, 0𝑚−1′ )′, 𝐵′ = (1, −𝛽′) and 0𝑑 is a 𝑑-dimensional zero vector.

Taking measurement errors at 𝑍𝑡 into (2.23) gives following observed model

gives

𝑦𝑡 = 𝛽0 + 𝛽1′𝑍𝑡

∗ + 휀𝑡∗ (2.32)

where 𝑍𝑡∗ = 𝑍𝑡 + 𝜂𝑡 and 휀𝑡

∗ = 휀𝑡 − 𝛽′𝜂𝑡.

LSE �� from (2.23) is a consistent estimator. Phillips and Durlauf (1986) showed

that LSE ��∗ from (2.24) is still a consistent estimator in spite of the presence of

measurement errors. This is intuitively true since the stationary 𝜂𝑡 does not affect

the asymptotic property of the nonstationary process 𝑍𝑡.

On the other hand, Haldrup et al. (2005) and Hassler and Kuzin (2009) showed

that the asymptotic distribution of 𝑇(��∗ − 𝛽) is affected by measurement errors as

follows. The asymptotic distribution of LSE �� from (2.23) is

𝑇(�� − 𝛽) 𝑑→ ℬ𝑍𝑍

−1ℬ𝑍𝜀 (2.33)

where ℬ𝑉𝑉 and ℬ𝑉𝑣 are stochastic integrals of Brownian motions which denote

the asymptotic distribution of the 𝑇−2 ∑ 𝑉𝑡𝑉𝑡′𝑇

𝑡=1 and 𝑇−1 ∑ 𝑉𝑡𝑣𝑡′𝑇

𝑡=1 respectively,

25

and 𝑉𝑡 is a 𝐼(1) process and 𝑣𝑡 is a 𝐼(0) process. With independent assumption

of the regressor 𝑍𝑡 and the regression error 휀𝑡, (2.25) can be represendented by

(∑ 𝑍𝑡𝑍𝑡′

𝑇

𝑡=1

)

1/2

(�� − 𝛽) 𝑑→ 𝑁(0, Ω𝜀𝐼𝑘). (2.34)

The asymptotic distribution of LSE ��∗ from (2.24) is given by

𝑇(��∗ − 𝛽) 𝑑→ ℬ𝑍𝑍

−1(ℬ𝑍𝜀 − (ℬ𝑍𝜂 + Ω𝜂)𝛽). (2.35)

and

(∑ 𝑍𝑡∗𝑍𝑡

∗′

𝑇

𝑡=1

)

1/2

(��∗ − 𝛽)

is not asymptotically normal since 𝑍𝑡∗ and 휀𝑡

∗ are correlated.

When integrated processes are cointegrated and measurement errors are

stationary, the observed series are also cointegrated (Granger, 1986). In Chapter 4,

we investigate asymptotic properties of estimators of cointegrating vectors in the

ECM of (2.4) when vector process 𝑋𝑡 observed with measurement errors. This will

provide the effect of measurement errors in cointegration analysis. Our study can be

considered as an extension of the model in Phillips and Durlauf (1986), Haldrup et

26

al. (2005) and Hassler and Kuzin (2009) in that our analysis incorporates when the

cointegrating rank is 𝑟 and more general parameters of the loading matrix 𝐴 and

cointegrating matrix 𝐵.

References

Ahn, S. K., and Reinsel, G. C. (1990), “Estimation for Partially Nonstationary

Multivariate Autoregressive Models,” Journal of American Statistical Association,

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27

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on Cointegrating Rank in Partial Systems,” Journal of the American Statistical

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28

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29

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30

Chapter 3

Impact of Exogeneity on the Vector Error

Correction Model

We consider a cointegrated vector autoregressive process of integrated order 1,

where the process consists of endogenous variables and exogenous variables.

Johansen (1992), Harbo et al. (1998), and Pesaran et al. (2000), considered

inference of such processes assuming that the nonstationary exogenous variables are

not cointegrated, and thus they are weakly exogenous. We consider the case where

exogenous variables are cointegrated. Parameterization and estimation of the

model is considered, and the asymptotic properties of the estimators are presented.

The method in this paper is also applicable to the models considered in Hunter

(1990), Mosconi and Giannini (1992), and Pradel and Rault (2003). A real data

example is provided to illustrate the methods. Finite sample properties of the

estimators are also examined through a Monte Carlo simulation.

31

3.1 INTRODUCTION

In this chapter we investigate the estimation of the parameters for models in which

the exogenous variables are cointegrated among themselves as well as with the

endogenous variables. To this end, we employ partial models in (2.17) and (2.18)

that we have introduced in Section 2.2. The partial model provides more convenient

ways to parameterize the cointegration structure considered in this chapter and

implements the zero constraints in the lower block of the coefficient matrices.

Furthermore, the partial model is useful, especially when the dimension of 𝑋𝑡 is

large, see Gonzalo & Pitarakis (2000).

VECM containing exogenous variables has been studied by many authors.

Johansen (1992), Harbo et al. (1998), and Pesaran et al. (2000), studied

cointegration analysis with exogenous variable. Their approach used sufficient (but

not necessary) condition of weakly exogenous assumption. Mosconi and

Gianini(1992) developed estimating method and testing procedure for non-causality.

Pradel and Rault (2003) studied the case of strongly exogenous 𝑍𝑡.

In this chapter we extend above methods. This chapter is organized as follows. The

parameterization and estimation for this model are considered in Section 3.2. MLE

and its asymptotic properties are presented in Section 3.3. A real data example is

provided to illustrate the methods in Section 3.4. Finite sample properties of the

estimators are examined through a Monte Carlo simulation in Section 3.5 and

Section 3.6 contains the proofs.

32

3.2 PARAMETERIZATION AND LEAST SQUARE

ESTIMATION OF THE MODEL

Let 𝑋𝑡 be an 𝑚-dimensional 𝐼(1) cointegrated vector process with cointegration

rank 𝑟. The VECM for 𝑋𝑡 is

Δ𝑋𝑡 = 𝐶𝑋𝑡−1 +∑𝛷𝑗∗ Δ𝑋𝑡−𝑗 + 𝜀𝑡

p−1

j=1

, (3.1)

Now assume that 𝑋𝑡 = (𝑌𝑡′, 𝑍𝑡

′)′, where 𝑌𝑡 is an 𝑚𝑦-dimensional vector process

of endogenous variables and 𝑍𝑡 is an 𝑚𝑧 -dimensional vector process of

exogenous variables with 𝑚𝑦 + 𝑚𝑧 = 𝑚 . If 𝑍𝑡 is sub-cointegrated with

cointegration rank 𝑟𝑥 < 𝑟, we observed that VECM (3.1) can be represented as

following conditional model of Δ𝑌𝑡 and marginal model of Δ𝑍𝑡 in Section 2.2.

Δ𝑌𝑡 = (𝐶𝑦𝑦 𝐶𝑦𝑧)𝑋𝑡−1 –Ω𝑦𝑧Ω𝑧𝑧−1𝐶𝑧𝑧 𝑍𝑡−1 +Ω𝑦𝑧Ω𝑧𝑧

−1𝛥𝑍𝑡

+∑𝐻𝑗Δ𝑋𝑡−𝑗

𝑝−1

𝑗=1

+ 𝑒𝑦𝑡 (3.2)

Δ𝑍𝑡 = 𝐶𝑧𝑧𝑍𝑡−1 + ∑Φ𝑗∗𝑧.Δ𝑋𝑡−𝑗 + 𝜀𝑧𝑡 .

𝑝−1

𝑗=1

(3.3)

33

The error 𝑒𝑦𝑡 of the conditional model is uncorrelated with the error 𝜀𝑧𝑡 of the

marginal model.

Since the entire process 𝑋𝑡 and exogenous process 𝑍𝑡 have a 𝑑 = 𝑚 − 𝑟 and

𝑑𝑧 = 𝑚𝑧 − 𝑟𝑧 unit roots respectively, we have the Jordan canonical form of

∑ 𝛷𝑗𝑝𝑗=1 as 𝑄(∑ 𝛷𝑗

𝑝𝑗=1 )𝑃 = 𝐽, where 𝐽 = diag (𝐼𝑚𝑦−𝑟𝑥 , 𝛬𝑟𝑥 , 𝐼𝑑𝑧 , 𝛬𝑟𝑧) , 𝑟𝑥 = 𝑟 −

𝑟𝑧, 𝛬𝑟𝑥 and 𝛬𝑟𝑧 are diagonal (Jordan block) matrices whose elements are the

stationary roots of the autoregressive operator. Since 𝐶 is an upper block triangular

matrix, we can represent 𝑃 and 𝑄 as upper block triangular matrices

𝑃 = (𝑃𝑦 𝑃𝑦𝑧𝑂 𝑃𝑧

) ,

𝑄 = 𝑃−1 = (𝑃𝑦−1 −𝑃𝑦

−1𝑃𝑦𝑧𝑃𝑧−1

𝑂 𝑃𝑧−1

) ≡ (𝑄𝑦 𝑄𝑦𝑧𝑂 𝑄𝑧

) =(𝑄𝑦 −𝑄𝑦𝑃𝑦𝑧𝑄𝑧𝑂 𝑄𝑧

)

and

Φ(1) = (Φ𝑦𝑦(1) Φ𝑦𝑧(1)

𝑂 Φ𝑧𝑧(1)) .

Conformably partition 𝑄𝑦′ = (𝑄𝑦1, 𝑄𝑦2), 𝑃𝑦 = (𝑃𝑦1, 𝑃𝑦2) , 𝑄𝑦𝑧

′ = (𝑄𝑦𝑧1,

𝑄𝑦𝑧2) = −(𝑄𝑥′𝑃𝑦𝑧′ 𝑄𝑦1 , 𝑄𝑧

′𝑃𝑦𝑧′ 𝑄𝑦2) , 𝑃𝑦𝑧 = (𝑃𝑦𝑧1, 𝑃𝑦𝑧2) , and 𝑄𝑧′ = (𝑄𝑧1, 𝑄𝑧2) ,

𝑃𝑧 = (𝑃𝑧1, 𝑃𝑧2) such that 𝑄𝑦1, 𝑃𝑦1 are 𝑚𝑦 × 𝑑𝑧 𝑄𝑦2, 𝑃𝑦2 are 𝑚𝑦 ×

𝑟𝑧, 𝑄𝑦𝑧1, 𝑃𝑦𝑧1 are 𝑚𝑦 × 𝑑𝑧 , 𝑄𝑦𝑧2, 𝑃𝑦𝑧2 are 𝑚𝑦 × 𝑟𝑧 , and 𝑄𝑧1, 𝑃𝑧1 are

𝑚𝑧 × 𝑑𝑧, 𝑄𝑦2, 𝑃𝑦2 are 𝑚𝑧 × 𝑟𝑧. We get that

34

𝛷(1) = 𝑃(𝐼𝑚 − 𝐽)𝑄

= (𝑃𝑦2 𝑃𝑦𝑧2𝑂 𝑃𝑧2

) (𝐼𝑟𝑥 − 𝛬𝑟𝑥 𝑂

𝑂 𝐼𝑟𝑧 − 𝛬𝑟𝑧) (𝑄𝑦2 ′ 𝑄𝑦𝑧2

′

𝑂 𝑄𝑧2 ′ )

= (𝑃𝑦2(𝐼𝑟𝑥 − 𝛬𝑟𝑥)𝑄𝑦2

′ 𝑃𝑦2(𝐼𝑟𝑥 − 𝛬𝑟𝑥)𝑄𝑦𝑧2′ + 𝑃𝑦𝑧2(𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2′

𝑂 𝑃𝑧2(𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2′

)

= (𝑃𝑦2(𝐼𝑟𝑥 − 𝛬𝑟𝑥)𝑄𝑦2

′ 𝑃𝑦2(𝐼𝑟𝑥 − 𝛬𝑟𝑥)𝑄𝑦2′ 𝑃𝑦𝑧𝑄𝑧 + 𝑃𝑦𝑧2(𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2′

𝑂 𝑃𝑧2(𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2′

)

(3.4)

Therefore, (3.2) and (3.3) can be written as

Δ𝑌𝑡 = −𝑃𝑦2(𝐼𝑟𝑥 − 𝛬𝑟𝑥)𝑄𝑦2′ 𝑌𝑡−1

+{𝑃𝑦2(𝐼𝑟𝑥 − 𝛬𝑟𝑥)𝑄𝑦2′ 𝑃𝑦𝑧𝑄𝑧 + 𝑃𝑦𝑧2(𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2

′ }𝑍𝑡−1

−𝐷𝑃𝑧2 (𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2′ 𝑍𝑡−1 + 𝐷 𝛥𝑍𝑡 + ∑𝐻𝑗𝛥𝑋𝑡−𝑗

𝑝−1

𝑗=1

+ 𝑒𝑦𝑡

= −𝑃𝑦2(𝐼𝑟𝑥 − 𝛬𝑟𝑥)𝑄𝑦2′ (𝐼𝑚𝑦 , −𝑃𝑦𝑧𝑄𝑧 ) 𝑋𝑡−1 − 𝑃𝑦𝑧2(𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2

′ 𝑍𝑡−1

+𝐷(𝛥𝑍𝑡 − 𝑃𝑧2(𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2′ 𝑍𝑡−1) + ∑𝐻𝑗𝛥𝑋𝑡−𝑗

𝑝−1

𝑗=1

+ 𝑒𝑦𝑡 (3.5)

Δ𝑍𝑡 = −𝑃𝑧2(𝐼𝑟𝑧 − 𝛬𝑟𝑧)𝑄𝑧2′ 𝑍𝑡−1 + ∑Φ𝑗

∗𝑧.𝛥𝑋𝑡−𝑗 + 𝜀𝑧𝑡 ,

𝑝−1

𝑗=1

(3.6)

35

where 𝐷 = 𝛺𝑦𝑧𝛺𝑧𝑧−1 . We can easily see that 𝑄𝑦2

′ (𝐼𝑚𝑦 , −𝑃𝑦𝑧𝑄𝑧 ) 𝑋𝑡−1 and

𝑄𝑧2′ 𝑍𝑡−1 are stationary by a similar argument in Ahn and Reinsel (1990)

We let 𝐴 = −𝑃𝑦2(𝐼𝑟𝑥 − 𝛬𝑟𝑥)𝑄𝑦21′ , 𝐵 = 𝑄𝑦21

′−1𝑄𝑦2′ (𝐼𝑚𝑦 , −𝑃𝑦𝑧𝑄𝑧 ) , 𝐴2𝑧 =

−𝑃𝑦𝑧2(𝐼𝑚𝑧 − 𝛬𝑟𝑧)𝑄𝑧21′ , 𝐴𝑧 = −𝑃𝑧2(𝐼 − 𝛬𝑟𝑧)𝑄𝑧21

′ , 𝐵𝑧 = 𝑄𝑧21′−1𝑄2𝑧

′ where

𝑄𝑦2′ = (𝑄𝑦21

′ , 𝑄𝑦22′ ) and 𝑄𝑧2

′ = (𝑄𝑧21′ , 𝑄𝑧22

′ ) such that 𝐴 is 𝑚𝑦 × 𝑟𝑥 , 𝐵 is

𝑚 × 𝑟𝑥 , 𝐴2𝑧 is 𝑚𝑦 × 𝑟𝑧 , 𝐴𝑧 and 𝐵𝑧 are 𝑚𝑧 × 𝑟𝑧 , 𝑄𝑦21′ is 𝑟𝑥 × 𝑟𝑥 , 𝑄𝑧21

′ is

𝑟𝑧 × 𝑟𝑧. Then the model (3.5) and (3.6) can be written as

𝛥𝑌𝑡 = 𝐴𝐵𝑋𝑡−1 + 𝐴2𝑧𝐵𝑧𝑍𝑡−1 + 𝐷(𝛥𝑍𝑡 − 𝐴𝑧𝐵𝑧𝑍𝑡−1)

+ ∑𝐻𝑗𝛥𝑋𝑡−𝑗

𝑝−1

𝑗=1

+ 𝑒𝑦𝑡 (3.7)

𝛥𝑍𝑡 = 𝐴𝑧𝐵𝑧𝑍𝑡−1 +∑Φ𝑗∗𝑧.Δ𝑋𝑡−𝑗 + 𝜀𝑧𝑡 .

𝑝−1

𝑗=1

(3.8)

where 𝐷 = 𝛺𝑦𝑧𝛺𝑧𝑧−1 . Note that 𝐴 and 𝐴𝑧 are coefficient matrix of stationary

process 𝐵𝑋𝑡−1 and 𝐵𝑧𝑍𝑡−1 . 𝐴2𝑧 is a coefficient matrix which represents the

effect of long run variation of exogenous 𝑍𝑡 to conditional model. On the other

hand, 𝐷 is a coefficient matrix which represents the effect of purely short run

variation of exogenous 𝑍𝑡 to conditional model. We note that if 𝐴2𝑧 − 𝐷𝐴𝑧 = 0,

the parameters in the conditional model and those in the marginal models are

variation free. That is, (∆𝑌𝑡′, ∆𝑍𝑡

′)′ operates a sequential cut on the distribution of

36

∆𝑋𝑡. Therefore, 𝐴2𝑧 − 𝐷𝐴𝑧 = 0 is also sufficient condition of weakly exogenous

𝑍𝑡 as well as 𝐶𝑧𝑧 = 0, or 𝐴𝑧 = 0. If we suppose 𝐴2𝑧 − 𝐷𝐴𝑧 = 0, the conditional

model (3.4) be a

𝛥𝑌𝑡 = 𝐴𝐵𝑋𝑡−1 + 𝐷𝛥𝑍𝑡 + ∑𝐻𝑗𝛥𝑋𝑡−𝑗

𝑝−1

𝑗=1

+ 𝑒𝑦𝑡 . (3.9)

We may replace conditional model (3.7) to (3.9) with assumption of weakly

exogenous 𝑍𝑡 . In (3.9), the effect of short-run and long-run effect of 𝑍𝑡 to

conditional model are same and these effects are represented by one coefficient

matrix 𝐷. Johansen (1992), Harbo et al. (1998), and Pesaran et al. (2000)

abandoned VECM structure of 𝑍𝑡 for weakly exogenous assumption, but 𝑍𝑡 is

still weakly exogenous just assuming same long-run and short-run effects of

exogenous variable. Although we may develop estimation procedure using the

conditional model (3.9) and the marginal (3.8) for weakly exogenous of 𝑍𝑡, we

use (3.7) and (3.8) in this thesis for more general situation.

Now, we reparameterize 𝐵 and 𝐵𝑧 using the same parameterization as in Ahn and

Reinsel (1990) which assumes that the last 𝑑 = 𝑚 − 𝑟 components do not

cointegrate. To this end, we consider 𝑋𝑡∗′ = (𝑌1𝑡

′ , 𝑍1𝑡′ , 𝑌2𝑡

′ , 𝑍2𝑡′ ) , which is a

rearrangement of 𝑋𝑡 such that 𝑋2𝑡 ≔ (𝑌2𝑡′ , 𝑍2𝑡

′ )′ is the 𝑑 -dimensional purely

nonstationary components of 𝑋𝑡 and 𝑍2𝑡 is (𝑚𝑧 − 𝑟𝑧)-dimensional. Then, we

have the matrix of the cointegrating vector of the form 𝐵∗ = (𝐼𝑟, 𝐵0∗) associated

37

with 𝑋𝑡 as in Ahn and Reinsel (1990). We partition 𝐵0∗ conformable with 𝑌2𝑡

and 𝑍2𝑡 . Then, since 𝑍𝑡 is exogenous, the lower left block of 𝐵0∗ , which is the

coefficients matrix associated with 𝑌2𝑡 in the equation for 𝑍2𝑡 , is a zero matrix.

As we rearrange the components of 𝑋𝑡∗ to form 𝑋𝑡, we rearrange the column

matrices corresponding to 𝑍1𝑡 and 𝑌2𝑡 , and get 𝐵′ = (𝐼𝑟𝑥 , 𝐵10′ , 𝑂𝑟𝑥×𝑟𝑧 , 𝐵20

′ ) and

𝐵𝑧′ = (𝐼𝑟𝑧 , 𝐵𝑧0′), where 𝐵10 is (𝑚𝑦 − 𝑟𝑥) × 𝑟𝑥, 𝑟𝑥 = 𝑟 − 𝑟𝑧, 𝐵20 is (𝑚𝑧 − 𝑟𝑧) ×

𝑟𝑥 , and 𝐵𝑧0 is (𝑚𝑧 − 𝑟𝑧) × 𝑟𝑧 matrices of parameters. That is, the matrix of

cointegrating vectors is

(𝐼𝑟𝑥 𝐵10

′

𝑂 𝑂

𝑂 𝐵20′

𝐼𝑟𝑧 𝐵𝑧0′ ). (3.10)

We note that 𝐵𝑧′𝑍𝑡−1 = 𝑍1𝑡−1 + 𝐵𝑧0

′ 𝑍2𝑡−1 is the cointegrating combinations of the

exogenous variables only and 𝐵′𝑋𝑡−1 = 𝑌1𝑡−1 +𝐵10′ 𝑌2𝑡−1 +𝐵20

′ 𝑍2𝑡−1 is

cointegrating combinations of both endogenous 𝑌𝑡 and exogenous 𝑍𝑡. If some of

the rows of 𝐵20′ are zero or linearly dependent, then the components corresponding

to these rows are cointegrating combinations of 𝑌𝑡 only.

We consider the estimator of the model in (3.7) and (3.8). For simplicity of

exposition, we consider the case with 𝑝 = 1.

𝛥𝑌𝑡 = 𝐴𝐵′𝑋𝑡−1 + 𝐴2𝑧𝐵𝑧

′𝑍𝑡−1 + 𝐷(𝛥𝑍𝑡 − 𝐴𝑧𝐵𝑧′𝑍𝑡−1) + 𝑒𝑦𝑡 (3.11)

𝛥𝑍𝑡 = 𝐴𝑧𝐵𝑧′𝑍𝑡−1 + 𝜀𝑧𝑡 (3.12)

38

For 𝑝 > 1, we can adjust ∆𝑋𝑡 and 𝑋𝑡−1 for the lagged ∆𝑋𝑡’s in order to reduce

to the AR(1) form. Estimation of 𝐻𝑗 and Φ𝑗∗𝑧. for 𝑝 > 1 can be carried out by

regression of ∆𝑋𝑡 − ��𝑋𝑡−1 on the lagged ∆𝑋𝑡 where �� is the estimator of 𝐶.

We derive the least squares based estimation of (3.11) and (3.12) based on 𝑇

observations, 𝑋1, 𝑋2,⋯ , 𝑋𝑇 . First, we estimate the parameters in (3.12). To this

end, we let 𝐹𝑧 = 𝐴𝑧𝐵𝑧 , and minimize ∑(𝛥𝑍𝑡 − 𝐹𝑧𝑍𝑡−1)′(∆𝑍𝑡 − 𝐹𝑧𝑍𝑡−1). The

least square estimator 𝐹�� of 𝐹𝑧 is

𝐹�� = (∑𝛥𝑍𝑡 𝑍𝑡−1 ′ ) (∑𝑍𝑡−1 𝑍𝑡−1

′ )−1

.

We partition 𝐹�� = (��𝑧1, ��𝑧2) where ��𝑧1 is 𝑚𝑧 × 𝑟𝑧 and ��𝑧2 is 𝑚𝑧 × 𝑑𝑧. Since

𝐹𝑧 = 𝐴𝑧𝐵𝑧′ = 𝐴𝑧(𝐼𝑚𝑧 , 𝐵𝑧0)

= (𝐴𝑧, 𝐴𝑧𝐵𝑧0), we have

��𝑧 = ��𝑧1 (3.13)

��𝑧0 = ( ��𝑧′ ��𝜀

−1��𝑧)−1��𝑧′ ��𝜀

−1��𝑧2, (3.14)

where ��𝜀 is sample variance of residuals in (3.12). Since 𝐹�� is consistent, ��𝑧 is

also consistent. Moreover, Ahn and Reinsel (1990) showed that ��𝑧0 is consistent.

We note that we obtain the generalized least square estimator of 𝐵𝑧0 as in Ahn and

Reinsel (1990) because the LSE ��𝑧2 yields the consistent estimator of 𝐹𝑧2 (which

39

consists of 𝑚𝑧 × 𝑑𝑧 parameters), but not separately of 𝐵𝑧0 (which consists of

only 𝑟𝑧 × 𝑑𝑧 parameters).

For the estimation of the parameters in (3.11), we let 𝐹 = [𝐴𝐵, 𝐴2𝑧, 𝐷] and

𝑈𝑡−1 = [(𝑌𝑡′, 𝑋2𝑡

′ ), (𝐵𝑧𝑍𝑡−1)′, (𝛥𝑍𝑡 − 𝐴𝑧𝐵𝑧𝑍𝑡−1)′]′, where 𝐵𝑧 = [𝐼, 𝐵𝑧𝑜]. If 𝐴𝑧

and 𝐵𝑧 are given, the LSE of 𝐹 is obtained by minimizing

∑(𝛥𝑌𝑡 − 𝐹𝑈𝑡−1)′(∆𝑌𝑡 − 𝐹𝑈𝑡−1) , and is �� = (∑𝛥𝑌𝑡𝑈𝑡−1

′ )(∑𝑈𝑡−1 𝑈𝑡−1 ′ )−1 .

Since 𝐹 = (𝐴, 𝐴𝐵0, 𝐴2𝑧, 𝐷), where 𝐵0 = (𝐵10, 𝐵20), we partition �� = (��1, ��2, ��3,

��4) with ��1, ��2, ��3, ��4 being 𝑚𝑦 × 𝑟𝑥 , 𝑚𝑦 × (𝑚 − 𝑟), 𝑚𝑦 × 𝑟𝑧 , 𝑚𝑦 × 𝑚𝑧 ,

respectively, and obtain

�� = ��4 (3.15)

��2𝑧 = ��3 (3.16)

�� = ��1 (3.17)

��0′ = (��′��𝑒

−1��)−1��′ ��𝑒

−1��2 (3.18)

where Ω𝑒 is the sample variance of the residuals in (3.11). By the similar

arguments that lead to the consistency of (3.13) and (3.14), we can easily show that

��, ��0, ��2𝑧 and �� are consistent. Since 𝐴𝑧 and 𝐵𝑧 are unknown, we replace

these in (3.15)-(3.18) with their consistent estimators ��𝑧 and ��𝑧 = (𝐼, ��𝑧0) in

(3.13) and (3.14).

40

3.3 MAXIMUM LIKELIHOOD ESTIMATION AND ITS

ASYMPTOTIC PROPERTIES

We note that the parameters in the conditional model (3.11) and those in the

marginal model (3.12) are not variation free because 𝐴𝑧 and 𝐵𝑧 are in both

models. Therefore, for the process where the exogenous variables are cointegrated

we need to consider both partial and marginal model simultaneously. Although the

cointegrating relations in the marginal model rely on 𝑍𝑡 alone, efficient estimation

follows from estimating both conditional and marginal models simultaneously as

those in the marginal model appear in both the dynamic equations.

We define 𝑎𝑡 = (𝑒𝑦𝑡′ , 𝜀𝑧𝑡

′ )′ . Then Ω𝑎 = 𝐶𝑜𝑣(𝑎𝑡) = 𝑑𝑖𝑎𝑔(Ω𝑒 , Ω𝜀) where

Ω𝑒 = Ω𝑦𝑦 − 𝐷Ω𝑧𝑦 and 𝛺𝜀 = 𝛺𝑧𝑧 . We further define 𝜃𝑦 = 𝑉𝑒𝑐{(𝐴, 𝐴2𝑧, 𝐷)′ } ,

𝛽 = 𝑉𝑒𝑐{𝐵0} , 𝛼𝑧 = 𝑉𝑒𝑐{𝐴𝑧′} , 𝛽𝑧 = 𝑉𝑒𝑐{𝐵𝑧0} , and 𝜂 = (𝜃𝑦′ , 𝛽′, 𝛼𝑧

′ , 𝛽𝑧′)′

where 𝑉𝑒𝑐{𝐴} is the vector formed by the stacked columns of the matrix 𝐴. With

𝑇 observations 𝑋1, 𝑋2, ⋯ , 𝑋𝑇, the MLE �� maximizing

𝑙(𝜂, 𝛺𝑎) = −𝑇

2𝑙𝑜𝑔|𝛺𝑎| −

1

2∑𝑎𝑡

′

𝑇

𝑡=1

𝛺𝑎−1𝑎𝑡 (3.19)

can be obtained by the iterative approximate Newton-Raphson relations

��(𝑖+1) = ��(𝑖) − (∑𝜕𝑎𝑡

′

𝜕𝜂

𝑇

𝑡=1

𝛺𝑎−1𝜕𝑎𝑡𝜕𝜂′)

��(𝑖)

−1

(∑𝜕𝑎𝑡

′

𝜕𝜂

𝑇

𝑡=1

𝛺𝑎−1𝑎𝑡)

��(𝑖)

(3.20) ,

41

where ��(𝑖) is the estimate at the i-th iteration. For this algorithm, we need to

calculate

𝜕𝑎𝑡

′

𝜕𝜂=

(

𝜕𝑒𝑦𝑡′

𝜕𝜃𝑦𝑂

𝜕𝑒𝑦𝑡′

𝜕𝛽𝑂

𝜕𝑒𝑦𝑡′

𝜕𝛼𝑧

𝜕𝜀𝑧𝑡′

𝜕𝛼𝑧𝜕𝑒𝑦𝑡

′

𝜕𝛽𝑧

𝜕𝜀𝑧𝑡′

𝜕𝛽𝑧)

, (3.21)

where

𝜕𝑒𝑦𝑡′

𝜕𝜃𝑦= −𝐼𝑚𝑦⊗ [(𝐵

′𝑋𝑡−1)′, (𝐵𝑧

′𝑍𝑡−1 )′, (𝛥𝑍𝑡 − 𝐴𝑧𝐵𝑧

′𝑍𝑡−1)′ ]′

𝜕𝑒𝑦𝑡′

𝜕𝛽= −𝐴′⊗𝑋2𝑡−1

𝜕𝑒𝑦𝑡′

𝜕𝛼𝑧= 𝐷′⊗𝐵𝑧

′𝑍𝑡−1

𝜕𝑒𝑦𝑡′

𝜕𝛽𝑧= −(𝐴2𝑧 − 𝐷𝐴𝑧)

′⊗𝑍2𝑡−1

𝜕𝜀𝑧𝑡′

𝜕𝛼𝑧= −𝐼𝑚𝑦⊗𝐵𝑧

′𝑍𝑡−1

𝜕𝜀𝑧𝑡′

𝜕𝛽𝑧= −𝐴𝑧

′ ⊗𝑍2𝑡−1 (3.22)

42

and ⊗ is Kronecker product. As initial values of the iteration in (3.20), we use

the least squares estimators discussed in the previous section because they are

consistent.

To study asymptotic properties of MLE, we define the matrix

𝜏 = 𝑑𝑖𝑎𝑔(√𝑇 𝐼𝑚𝑦(𝑟+𝑚𝑧), 𝑇 𝐼𝑟𝑦(𝑚−𝑟−𝑟𝑧), √𝑇 𝐼𝑚𝑧𝑟𝑧 , 𝑇 𝐼𝑟𝑧(𝑚𝑧−𝑟𝑧)). First-order

Taylor expansion of 𝜕𝑙

𝜕𝜂= ∑

𝜕𝑎𝑡′

𝜕𝜂𝑇𝑡=1 𝛺𝑎

−1𝑎𝑡 at �� may be used to obtain the

asymptotic representation

𝜏(�� − 𝜂) = −(𝜏−1∑𝜕𝑎𝑡

′

𝜕𝜂

𝑇

𝑡=1

𝛺𝑎−1𝜕𝑎𝑡𝜕𝜂′

𝜏−1)

−1

(𝜏−1∑𝜕𝑎𝑡

′

𝜕𝜂

𝑇

𝑡=1

𝛺𝑎−1𝑎𝑡)

+𝑜𝑝(1) . (3.23)

It follows from the discussion in Section 3.6, for �� = (𝜃𝑦′ , ��′, ��𝑧

′ , ��𝑧′)′ in

(3.24) − (3.27), we have

√𝑇 (𝜃𝑦 − 𝜃𝑦)

≈1

√𝑇∑��1

−1{[𝛺𝑒−1⊗ ��𝑡−1]𝑒𝑦𝑡 − ��1[(−𝐷

′, 𝐼𝑚𝑧)𝛺𝑎−1⊗𝐵𝑧

′𝑍𝑡−1]𝑎𝑡}

𝑡

= ��1−1 [

1

√𝑇∑ [𝛺𝑒

−1⊗ ��𝑡−1]𝑒𝑦𝑡𝑡

− ��11

√𝑇∑{[−𝐷′𝛺𝑒

−1⊗𝐵𝑧′𝑍𝑡−1]𝑒𝑦𝑡 + [𝛺𝜀

−1⊗𝐵𝑧′𝑍𝑡−1]𝜀𝑧𝑡}

𝑡

] (3.24)

43

where ��𝑡−1 = [(𝐵′𝑍𝑡−1)

′, (𝐵𝑧′𝑍𝑡−1 )

′, (𝛥𝑋𝑡 − 𝐴𝑧𝐵𝑧′𝑍𝑡−1)′ ]′

𝑇(�� − 𝛽)

≈1

𝑇∑��2

−1{[𝐴′𝛺𝑒−1⊗𝑋2𝑡−1]𝑒𝑦𝑡 − ��2[((𝐴2𝑧 − 𝐷𝐴𝑧)

′, 𝐴𝑧′ )𝛺𝑎

−1⊗𝑍2𝑡−1]𝑎𝑡}

𝑡

= ��2−1 [𝑣𝑒𝑐 {(

1

𝑇∑𝑋2𝑡−1𝑒𝑦𝑡

′

𝑡

)𝛺𝑒−1𝐴}

− ��2𝑣𝑒𝑐 {(1

𝑇∑𝑍2𝑡−1𝑡

𝑒𝑦𝑡′ )𝛺𝑒

−1(𝐴2𝑧 −𝐷𝐴𝑧)

+ (1


𝜀𝑧𝑡′ )𝛺𝜀

−1𝐴𝑧}] (3.25)

√𝑇 (��𝑧 − 𝛼𝑧)

≈1

√𝑇∑��1

−1{[(−𝐷′, 𝐼𝑚𝑧)𝛺𝑎−1⊗𝐵𝑧

′𝑍𝑡−1]𝑎𝑡 − ��1[𝛺𝑒−1⊗ ��𝑡−1]𝑒𝑦𝑡}

𝑡

= ��1−1 [

1

√𝑇∑{[−𝐷′𝛺𝑒

−1⊗𝐵𝑧′𝑍𝑡−1]𝑒𝑦𝑡 + [𝛺𝜀

−1⊗𝐵𝑧′𝑍𝑡−1]𝜀𝑧𝑡}

𝑡

−��11

√𝑇∑[𝛺𝑒

−1⊗ ��𝑡−1]𝑒𝑦𝑡𝑡

] (3.26)

𝑇(��𝑧 − 𝛽𝑧) ≈1

𝑇∑��2

−1{[((𝐴2𝑧 − 𝐷𝐴𝑧)′, 𝐴𝑧

′ )𝛺𝑎−1⊗𝑍2𝑡−1]𝑎𝑡 − ��2[𝐴

′𝛺𝑒−1

𝑡

⊗𝑋2𝑡−1]𝑒𝑦𝑡}

44

= ��2−1 [𝑣𝑒𝑐 {(

1


𝑒𝑦𝑡′ )𝛺𝑒

−1(𝐴2𝑧 − 𝐷𝐴𝑧)

+ (1


𝜀𝑧𝑡′ )𝛺𝜀

−1𝐴𝑧}

− ��2𝑣𝑒𝑐 {(1


′

𝑡

)𝛺𝑒−1𝐴} ] (3.27)

where

��1 = 𝛺𝑒−1⊗(

1

𝑇∑��𝑡−1��𝑡−1

′

𝑇

𝑡=1

) − 𝛺𝑒−1𝐷(𝐷′𝛺𝑒

−1𝐷 + 𝛺𝜀−1)−1𝐷′𝛺𝑒

−1⊗

(1

𝑇∑��𝑡−1𝑍𝑡−1

′ 𝐵𝑧

𝑇

𝑡=1

)(1

𝑇∑𝐵𝑧

′𝑍𝑡−1𝑍𝑡−1′ 𝐵𝑧

𝑇

𝑡=1

)

−1

(1

𝑇∑𝐵𝑧

′𝑍𝑡−1��𝑡−1′

𝑇

𝑡=1

)

��2 = (𝐴′𝛺𝑒−1𝐴)⊗ (

1

𝑇2∑𝑋2𝑡−1𝑋2𝑡−1

′

𝑇

𝑡=1

)

−𝐴′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧)

× [(𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) + 𝐴𝑧

′𝛺𝜀−1𝐴𝑧]

−1

× (𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1 𝐴

⊗ (1

𝑇2∑𝑋2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

)(1

𝑇2∑𝑍2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

)

−1

(1

𝑇2∑𝑍2𝑡−1𝑋2𝑡−1

′

𝑇

𝑡=1

)

��1 = (𝐷′𝛺𝑒−1𝐷 + 𝛺𝜀

−1)⊗ (1

𝑇∑𝐵𝑧


𝑇

𝑡=1

)

−𝐷′𝛺𝑒−1𝐷⊗ (

1

𝑇∑𝐵𝑧

′𝑍𝑡−1��𝑡−1′

𝑇

𝑡=1

) (1

𝑇∑��𝑡−1��𝑡−1

′

𝑇

𝑡=1

)

−1

(1

𝑇∑��𝑡−1𝑍𝑡−1

′ 𝐵𝑧

𝑇

𝑡=1

)

45

��2 = [(𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) + 𝐴𝑧

′𝛺𝜀−1𝐴𝑧] ⊗ (

1

𝑇2∑𝑍2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

)

−(𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1𝐴(𝐴′𝛺𝑒

−1𝐴)−1𝐴′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧)

⊗ (1

𝑇2∑𝑍2𝑡−1𝑋2𝑡−1

′

𝑇

𝑡=1

)(1

𝑇2∑𝑋2𝑡−1𝑋2𝑡−1

′

𝑇

𝑡=1

)

−1

(1

𝑇2∑𝑋2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

)

��1 = −𝛺𝑒−1𝐷(𝐷′𝛺𝑒

−1𝐷 + 𝛺𝜀−1)−1

⊗ (1

𝑇∑��𝑡−1𝑍𝑡−1

′ 𝐵𝑧

𝑇

𝑡=1

)(1

𝑇∑𝐵𝑧


𝑇

𝑡=1

)

−1

��2 = 𝐴′𝛺𝑒−1(𝐴2𝑧 −𝐷𝐴𝑧)[(𝐴2𝑧 − 𝐷𝐴𝑧)

′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) + 𝐴𝑧


−1

⊗ (1

𝑇2∑𝑋2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

)(1

𝑇2∑𝑍2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

)

−1

��1 = −𝐷′⊗(

1

𝑇∑𝐵𝑧

′𝑍𝑡−1��𝑡−1′

𝑇

𝑡=1

) (1

𝑇∑��𝑡−1��𝑡−1

′

𝑇

𝑡=1

)

−1

��2 = (𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1𝐴(𝐴′𝛺𝑒

−1𝐴)−1

⊗ (1

𝑇2∑𝑍2𝑡−1𝑋2𝑡−1

′

𝑇

𝑡=1

)(1

𝑇2∑𝑋2𝑡−1𝑋2𝑡−1

′

𝑇

𝑡=1

)

−1

We use Lemma 1 of Ahn and Reinsel (1990) and get the following asymptotic

results:

��1𝑝→𝛺𝑒

−1⊗ 𝛤�� − 𝛺𝑒−1𝐷(𝐷′𝛺𝑒

−1𝐷 +𝛺𝜀−1)−1𝐷′ 𝛺𝑒

−1⊗ 𝛤��,𝐵𝑧′𝑍 𝛤𝐵𝑧′𝑍−1 𝛤𝐵𝑧′𝑍,��

46

≡ 𝐾1

��2𝑑→(𝐴′𝛺𝑒

−1𝐴)⊗ 𝛶𝑋𝑋 − 𝐴′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧)

× [(𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) + 𝐴𝑧


−1(𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1 𝐴

⊗ 𝛶𝑋𝑍𝛶𝑍𝑍−1𝛶𝑍𝑋

≡ 𝐾2

��1𝑝→(𝐷′𝛺𝑒

−1𝐷 +𝛺𝜀−1)⊗ 𝛤𝐵𝑧′𝑍 − 𝐷

′𝛺𝑒−1𝐷⊗ 𝛤𝐵𝑧′𝑍,�� 𝛤��

−1 𝛤��,𝐵𝑧′𝑍 ≡ 𝑉1

��2𝑑→[(𝐴2𝑧 −𝐷𝐴𝑧)

′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) + 𝐴𝑧

′𝛺𝜀−1𝐴𝑧] ⊗ 𝛶𝑍𝑍

−(𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1𝐴(𝐴′𝛺𝑒

−1𝐴)−1𝐴′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) ⊗ 𝛶𝑍𝑋𝛶𝑋𝑋

−1𝛶𝑋𝑍 ≡ 𝑉2

��1𝑝→−𝛺𝑒

−1𝐷(𝐷′𝛺𝑒−1𝐷 + 𝛺𝜀

−1)−1⊗ 𝛤��,𝐵𝑧′𝑍 𝛤𝐵𝑧′𝑍−1 ≡ 𝑀1

��2𝑑→𝐴′𝛺𝑒

−1(𝐴2𝑧 − 𝐷𝐴𝑧)[(𝐴2𝑧 − 𝐷𝐴𝑧)′𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) + 𝐴𝑧


−1

⊗ 𝛶𝑋𝑍𝛶𝑍𝑍−1

≡ 𝑀2

��1𝑝→−𝐷′⊗ 𝛤𝐵𝑧′𝑍,�� 𝛤��

−1 ≡ 𝑁1

��2𝑑→(𝐴2𝑧 − 𝐷𝐴𝑧)

′𝛺𝑒−1𝐴(𝐴′𝛺𝑒

−1𝐴)−1⊗𝛶𝑍𝑋𝛶𝑋𝑋−1 ≡ 𝑁2

with

1

𝑇2∑𝑋2𝑡−1𝑋2𝑡−1

′

𝑇

𝑡=1

𝑑→ 𝛹22𝛺𝑒2

1/2∫ 𝐵𝑑(𝑢)𝐵𝑑(𝑢)

′𝑑𝑢 𝛺𝑒21/2𝛹22′

1

0

≡ 𝛶𝑍𝑍,

1

𝑇2∑𝑋2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

𝑑→ 𝛹22𝛺𝑒2

1/2∫ 𝐵𝑑(𝑢)𝐵𝑑𝑧(𝑢)

′𝑑𝑢 𝛺𝜀21/2𝛹22𝑧′

1

0

≡ 𝛶𝑋𝑍

47

1

𝑇2∑𝑋2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

𝑑→ 𝛶𝑋𝑍

′ ≡ 𝛶𝑍𝑋

1

𝑇2∑𝑍2𝑡−1𝑍2𝑡−1

′

𝑇

𝑡=1

𝑑→ 𝛹22

𝑧 𝛺𝜀21/2∫ 𝐵𝑑𝑧(𝑢)𝐵𝑑𝑧(𝑢)

′𝑑𝑢 𝛺𝜀21/2𝛹22𝑧′

1

0

≡ 𝛶𝑍𝑍

1


′

𝑡

𝑑→ 𝛹22𝛺𝑎2

1/2∫ 𝐵𝑑(𝑢)𝑑𝐵𝑚𝑦

′ (𝑢) 1

0

𝛺𝑒1/2

≡ ℬ𝑋𝑒

1


𝑒𝑦𝑡′

𝑑→ 𝛹22

𝑧 𝛺𝜀21/2∫ 𝐵𝑑𝑧(𝑢)𝑑𝐵𝑚𝑦

′ (𝑢) 1

0

𝛺𝑒1/2

≡ ℬ𝑍𝑒

1

𝑇∑𝑍2𝑡−1t

𝜀𝑥𝑡′

𝑑→ 𝛹22

𝑧 𝛺𝜀21/2∫ 𝐵𝑑𝑧(𝑢)𝑑𝐵𝑚𝑧

′ (𝑢) 1

0

𝛺𝑒1/2

≡ ℬ𝑍𝜀

where 𝑎2′ = (𝑒2

′ , 𝜀2′ ), 𝛹22 = [0, 𝐼𝑚−𝑟]𝛹[0, 𝐼𝑚−𝑟]

′ with 𝛹 = ∑ 𝛹𝑘∞𝑘=0 is sum of

infinite moving average coefficients of model (2.1),

𝛺𝑒2 = [0, 𝐼𝑚−𝑟+𝑟𝑧]𝛺𝑒[0, 𝐼𝑚−𝑟+𝑟𝑧]′, 𝛹22

𝑧 = [0, 𝐼𝑚𝑧−𝑟𝑧]𝛹𝑧

× [0, 𝐼𝑚𝑧−𝑟𝑧]′ with 𝛹𝑧 = ∑ 𝛹𝑘∞𝑘=0

𝑧 is sum of infinite moving average coefficients

of model (2.2), 𝛺𝜀2 = [0, 𝐼𝑚𝑧−𝑟𝑧]𝛺𝜖[0, 𝐼𝑚𝑧−𝑟𝑧]′ .

Finally, we obtain the results summarized in the theorem by the application of the

above lemma and the continuous mapping theorem.

48

Theorem 3.1 The asymptotic properties of the MLE that maximizes (3.19) are as

follows.

√𝑇 (𝜃𝑦 − 𝜃𝑦) 𝑑→ 𝐾1

−1[𝑅1 −𝑀1𝑅2] (3.28)

𝑇(�� − 𝛽) 𝑑→ 𝐾2

−1[𝑣𝑒𝑐{ℬ𝑋𝑒𝛺𝑒−1𝐴}

−𝑀2𝑣𝑒𝑐{ℬ𝑍𝑒𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) + ℬ𝑍𝜀𝛺𝜀

−1𝐴𝑧}] (3.29)

√𝑇 (��𝑧 − 𝛼𝑧) 𝑑→ 𝑉1

−1[𝑅2−𝑁1𝑅1] (3.30)

𝑇(��𝑧 − 𝛽𝑧) 𝑑→ 𝑉2

−1[𝑣𝑒𝑐{ℬ𝑍𝑒𝛺𝑒−1(𝐴2𝑧 − 𝐷𝐴𝑧) + ℬ𝑍𝜀𝛺𝜀

−1𝐴𝑧}

− 𝑁2𝑣𝑒𝑐{ℬ𝑋𝑒𝛺𝑒−1𝐴} ] (3.31)

as 𝑇 → ∞ , where 𝑅1 and 𝑅2 are zero mean normal random vectors with

covariance 𝛺𝑒−1⊗ 𝛤�� and (𝐷′𝛺𝑒

−1𝐷+𝛺𝜀−1) ⊗ 𝛤𝐵𝑧′𝑍 , respectively, 𝐾1, 𝑉1, 𝑀1

and 𝑁1 are functions of covariance matrices, and 𝐾2, 𝑉2, 𝑀2, 𝑁2, ℬ𝑋𝑒, ℬ𝑍𝑒, and

ℬ𝑍𝜀, are functionals of stochastic integrals of Brownian motions.

Furthermore, the asymptotic distribution of (��′, ��𝑧′)′ is a mixed normal

distribution. We sketch the proof of the mixed normality. As in Section 3.2, we

consider 𝑋𝑡∗′ = (𝑌1𝑡

′ , 𝑍1𝑡′ , 𝑌2𝑡

′ , 𝑍2𝑡′ , ) . Then we have the model ∆𝑋𝑡

∗ = 𝐴∗𝐵∗′𝑋𝑡−1∗ +

𝜀𝑡, with 𝐵∗′ = (𝐼𝑟, 𝐵0∗′). Then, it is well known that the asymptotic distribution of

the reduced rank estimator of 𝑣𝑒𝑐(𝐵0∗′) is mixed normal (for example see Theorem

4, Ahn and Reinsel, 1990). Since (𝛽′, 𝛽𝑧′)′ = 𝐻 𝑣𝑒𝑐(𝐵0

∗′), where the matrix H is a

(𝑟𝑑 − 𝑟𝑧𝑟𝑥) × 𝑟𝑑 matrix with 1’s and 0’s such that it selects the elements in 𝐵0∗′

49

corresponding to 𝐵10, 𝐵20, and 𝐵𝑥0, that is, (𝛽′, 𝛽𝑧′)′. Therefore, the asymptotic

distribution of (��′, ��𝑧′)′ is mixed normal.

3.4 NUMERICAL EXAMPLE

As a relatively simple numerical example to illustrate the methods discussed

previously, we consider the grain-meat price data plotted in Figure 3.1. Three red

lines are the prices of corn, soybeans, and wheat, and three black lines are the prices

of beef, pork, and chicken. We specify a second- order VAR model for these data

using the Akaike information criterion. With this VAR(2), we obtained the log-

likelihood test statistic of 21.67 with the p-value of 0.247 for testing whether the

0 10 20 30 40 50 60

5010

015

020

025

030

0

t

beef

pork

chicken

corn

soybean

wheat

Figure 3.1. Grain and Meat prices during the period January

1980 through December 2008

50

grain prices are exogenous. Therefore, we treat the grain prices as exogenous and

the meat prices as endogenous.

Based on the trace statistic for the cointegrating rank test (Johansen, 1988) , we

identify that the cointegrating rank of the six meat-grain price series is 3 and that the

cointegrating rank of the three grain price series is 1.

Let 𝑦1𝑡 , 𝑦2𝑡 , 𝑦3𝑡 be the prices of beef, pork, and chicken, respectively, and let

𝑧1𝑡, 𝑧2𝑡, 𝑧3𝑡 be the prices of corn, soybean, and wheat, respectively. The

estimated model based on the initial LSEs is

(

𝛥𝑦1𝑡𝛥𝑦2𝑡𝛥𝑦3𝑡

)

= (−0.218 0.151−0.020 −0.0460.007 0.026

)(1 0 −2.1290 1 −1.396

0 0.347 −0.1670 0.287 −0.135

)

(

𝑦1𝑡−1 𝑦2𝑡−1 𝑦3𝑡−1 𝑧1𝑡−1𝑧2𝑡−1𝑧3𝑡−1 )

+ (0.1410.4510.008

) (1 −0.321 −0.270)(

𝑧1𝑡−1𝑧2𝑡−1𝑧3𝑡−1

)

+(0.125 −0.029 −0.0210.204 −0.044 0.040−0.027 0.013 −0.0003

)

× [(𝛥𝑧1𝑡𝛥𝑧2𝑡𝛥𝑧3𝑡

) − (−0.2310.118−0.060

) (1 −0.321 −0.270)(


)]

51

+ (0.048 0.025 −0.518−0.152 0.038 −0.774−0.043 0.018 0.531

0.067 0.020 −0.088−0.642 0.208 −0.1300.018 0.013 −0.034

)

(

Δ𝑦1𝑡−1 Δ𝑦2𝑡−1 Δ𝑦3𝑡−1 Δ𝑧1𝑡−1Δ𝑧2𝑡−1Δ𝑧3𝑡−1 )

+𝑒𝑦𝑡 ,

(𝛥𝑧1𝑡𝛥𝑧2𝑡𝛥𝑧3𝑡

) = (−0.2310.118−0.060

) (1 −0.321 −0.270)(


)

+ (0.389 0.005 −0.136−0.216 0.299 −0.1040.481 −0.092 0.194

) (

𝛥𝑧1𝑡−1𝛥𝑧2𝑡−1𝛥𝑧3𝑡−1

) + 𝜀𝑧𝑡 .

The estimated model based on the MLEs is

(

𝛥𝑦1𝑡𝛥𝑦2𝑡𝛥𝑦3𝑡

)

= (−0.177 0.1390.136 −0.0930.008 0.026

)(1 0 −2.7870 1 −1.896

0 0.473 −0.1670 0.386 −0.029

)

(

𝑦1𝑡−1 𝑦2𝑡−1 𝑦3𝑡−1 𝑧1𝑡−1𝑧2𝑡−1𝑧3𝑡−1 )

+ (0.0840.2300.007

) (1 −0.344 −0.234)(


)

+ (0.090 −0.025 −0.0270.073 −0.029 0.018−0.027 0.013 −0.0004

)

52

× [(𝛥𝑧1𝑡𝛥𝑧2𝑡𝛥𝑧3𝑡

) − (−0.1840.239−0.085

) (1 −0.344 −0.234)(


)]

+(0.023 0.044 −0.532−0.247 0.110 −0.827−0.044 0.019 0.531

0.108 0.005 −0.091−0.488 0.150 −0.1390.018 0.013 −0.034

)

(

Δ𝑦1𝑡−1 Δ𝑦2𝑡−1 Δ𝑦3𝑡−1 Δ𝑧1𝑡−1Δ𝑧2𝑡−1Δ𝑧3𝑡−1 )

+𝑒𝑦𝑡 ,

(𝛥𝑧1𝑡𝛥𝑧2𝑡𝛥𝑧3𝑡

) = (−0.1840.239−0.085

) (1 −0.344 −0.234)(


)

+ (0.377 0.005 −0.089−0.246 0.302 −0.0080.487 −0.093 0.173

) (𝛥𝑧1𝑡−1𝛥𝑧2𝑡−1𝛥𝑧3𝑡−1

) + 𝜀𝑧𝑡

Figure 3.2 shows the cointegrating combinations among all the variables,𝑦1 −

2.787𝑦2 + 0.473𝑥2 − 0.167𝑥3 and 𝑦2 − 1.896𝑦2 + 0.386𝑥2 − 0.029𝑥3 . Figure

3.3 shows cointegrating combination among exogenous variables 𝑧1 − 0.344𝑧2 −

0.234𝑧3 . As we expected, these cointegrating combinations exhibit stationary

behavior.

Because the asymptotic distribution of the estimators of the cointegrating vector is

mixed normal as shown in the theorem in Section 3.3, the chi-squared distribution

can be used for the inferences of the cointegrating vectors. The likelihood ratio

test whose statistic is asymptotically a chi-square with 4 degrees of freedom results

in the rows of the estimate of 𝐵20′ are significantly different from zero, and thus

there is no cointegrating combination of only the meat prices.

53

Figure 3.2 Cointegrating combination of all the variable

Figure 3.3 Cointegrating combination of the exogenous variables

0 10 20 30 40 50 60

-20

020

40

t

B'Z

0 10 20 30 40 50 60

-10-5

05

t

Bx'X

54

3.5 MONTE CARLO RESULTS

In this section the finite sample properties of the estimators discussed in the

previous sections are considered through a Monte Carlo simulation. For the data

generating process we use the models in (3.11) and (3.12) with 𝑚𝑧 = 𝑚𝑦 = 2 ,

𝑟 = 2, and 𝑟𝑧 = 𝑟𝑥 = 1. The parameter values are

𝐴 = (𝑎1𝑎2) = (

−0.20.4

), 𝐵′ = [1 𝑏10 0 𝑏20 ] = [1 − 2.1 0 − 3.2],

𝐴2𝑧 = (𝑎2𝑧,1𝑎2𝑧,2

) = (0.2

−0.14)

𝐴𝑧 = (𝑎𝑧1𝑎𝑧2) = (

−0.40.12

), 𝐵𝑧 = [1 𝑏𝑧0 ] = [1 − 2.5],

𝛺 = (

25 6.1 4 −1.26.1 9 1.8 3.94 1.8 25 5.4−1.2 3.9 5.4 9

) and

𝐷 = (4 −1.21.8 3.9

) (25 5.45.4 9

)−1

= (0.2169 −0.2635−0.0248 0.4482

) (3.32)

In the simulation, various values of the length 𝑇 = 50, 100, and 200 were used,

and 5000 replications of the sample series were generated for each value of 𝑇. For

each series, estimates of the parameters were computed by LSE and MLE. The

empirical results from the simulation for the two estimation procedures are

summarized in Table 3.1, where, as the conventional measures of accuracy, the

55

means and mean squared errors (MSE’s) of the estimators are given. Brüggemann

and Lütkepohl (2005) showed through a Monte Carlo simulation study that the

initial LSE performed better than the MLE in term of smaller MSE for cointegrating

vectors. Our results are similar to theirs. However, the bias of MLE is in general

smaller than that of the initial LSE. For the stationary parameters, the MLE

performed better in terms of smaller MSE and smaller bias.

In this simulation study, when one ignores that 𝑍𝑡 is exogenous, one would

estimates the parameters using the joint model Δ𝑋𝑡 = α𝛽′𝑋𝑡−1 + 𝜀𝑡 . We let

�� = ��′ be the estimator of 𝐶 from this joint model, where �� and �� are the

reduced-rank estimators as in Ahn and Reinsel (1990). We then compare �� with

the estimator �� from the conditional and marginal models that account for 𝑍𝑡

being exogenous variables. We can estimate �� as

�� = (��′

𝑂 ��𝑧��𝑧′) + (𝑂 ��2𝑧��𝑧

′ 𝑂 𝑂

), (3.33)

where the estimators ��, ��, ��𝑧, ��2𝑧 and ��𝑧 are from the model (3.11) and (3.12).

The matrix 𝐶 in our simulation is

𝐶 = (𝐴𝐵′

𝑂 𝐴𝑧𝐵𝑧′) + (

𝑂 𝐴2𝑧𝐵𝑧′𝑂 𝑂

) = (

−0.2 0.420.4 −0.84

0.20 0.14−0.14 −0.93

𝑂−0.40 1 0.12 −0.3

) (3.34)

56

The comparison is made based on the average of the MSEs of the components of ��

and �� . The result is summarized in Table 3.2. As expected, the average MSE of the

estimator of 𝐶 is smaller when the structure of exogenous 𝑍𝑡 is incorporated for

all 𝑇. We note that the average MSE of �� decreases roughly inversely

proportional to the sample size while the average MSE of �� remains almost

constant.

Table 3.1. Means and mean squared errors (MSE) of the LSE and MLE for various

sample sizes

initial LSE MLE

𝑏10= -2.1 T =50 mean -2.0888 -2.1056

MSE 1.5789 1.6162

T =100 mean -2.0971 -2.1012

MSE 0.3478 0.3485

T =200 mean -2.0995 -2.1007

MSE 0.0915 0.0922

𝑏20= -3.2 T =50 mean -3.1789 -3.2096

MSE 5.0432 5.2189

T =100 mean -3.1943 -3.2025

MSE 1.1722 1.1569

T =200 mean -3.1988 -3.2014

MSE 0.3129 0.3054

57

𝑏𝑥0= -2.5 T =50 mean -2.4957 -2.5066

MSE 1.8579 1.9752

T =100 mean -2.4986 -2.5024

MSE 0.5255 0.4966

T =200 mean -2.4984 -2.4997

MSE 0.1599 0.1426

𝑎1= -0.2 T =50 mean -0.2503 -0.2047

MSE 1.2654 0.6940

T =100 mean -0.2254 -0.2014

MSE 0.4546 0.3122

T =200 mean -0.2145 -0.2014

MSE 0.1858 0.1447

𝑎2= 0.4 T =50 mean 0.3871 0.4042

MSE 0.2971 0.2293

T =100 mean 0.3937 0.4028

MSE 0.1191 0.1014

T =200 mean 0.3963 0.4012

MSE 0.0509 0.0455

𝑎2𝑧,1= 0.2 T =50 mean 0.2043 0.2066

MSE 2.2506 2.2730

T =100 mean 0.2019 0.2025

58

MSE 1.0111 1.0050

T =200 mean 0.2012 0.2014

MSE 0.4820 0.4779

𝑎2𝑧,2= -0.14 T =50 mean -0.1436 -0.1419

MSE 0.7527 0.7547

T =100 mean -0.1428 -0.1415

MSE 0.3368 0.3352

T =200 mean -0.1412 -0.1405

MSE 0.1587 0.1577

𝑎𝑧1=-0.4 T =50 mean -0.4170 -0.4108

MSE 1.035 1.0330

T =100 mean -0.4095 -0.4054

MSE 0.4606 0.4545

T =200 mean -0.4061 -0.4037

MSE 0.2211 0.2179

𝑎𝑧2=0.12 T =50 mean 0.1251 0.1272

MSE 0.3554 0.3439

T =100 mean 0.1205 0.1229

MSE 0.1628 0.1590

T =200 mean 0.1199 0.1214

MSE 0.0784 0.0767

59

Table 3.2. Average MSEs of �� and �� .

T ��

50 0.143708 0.005893

100 0.136918 0.002663

200 0.154145 0.001842

3.6 APPENDIX OF ASYMPTOTICS OF THE MAXIMUM

LIKELIHOOD ESTIMATOR

To get (3.24) − (3.27) we need to calculate (𝜏−1∑𝜕𝑎𝑡

′


−1 𝜕𝑎𝑡

𝜕𝜂′𝜏−1)

−1

and

(𝜏−1∑𝜕𝑎𝑡

′


−1𝑎𝑡). From (3.21),(3.22)and

𝜏 = 𝑑𝑖𝑎𝑔(√𝑇 𝐼𝑚𝑦(𝑟+𝑚𝑧), 𝑇 𝐼𝑟𝑦(𝑚−𝑟−𝑟𝑧), √𝑇 𝐼𝑚𝑧𝑟𝑧 , 𝑇 𝐼𝑟𝑧(𝑚𝑧−𝑟𝑧)), we can observe

𝜏−1∑𝜕𝑎𝑡

′

𝜕𝜂

𝑇

𝑡=1


𝜏−1 ≡ (ℳ1 ℳ2

ℳ2′ ℳ4

)

where

ℳ1 =

(

1

𝑇𝛺𝑒−1⊗∑��𝑡−1��𝑡−1

′

𝑇

𝑡=1

𝑜𝑝(1 )

𝑜𝑝(1 )1

𝑇2𝛺𝑒−1⊗∑𝑋2𝑡−1𝑋2𝑡−1

′

𝑇

𝑡=1 )

60

=: (ℳ11 𝑜𝑝(1 )

𝑜𝑝(1 ) ℳ12)

ℳ2

=

(

−1

𝑇𝛺𝑒−1𝐷⊗∑��𝑡−1𝑍𝑡−1

′ 𝐵𝑍

𝑇

𝑡=1

𝑜𝑝(1 )

𝑜𝑝(1 )1

𝑇2𝐴′𝛺𝑒

−1(𝐴2𝑧 − 𝐷𝐴𝑧) ⊗∑𝑋2𝑡−1𝑍2𝑡−1′

𝑇

𝑡=1 )

=: (ℳ21 𝑜𝑝(1 )

𝑜𝑝(1 ) ℳ22)

and

ℳ4 = (

1

𝑇(𝐷′𝛺𝑒

−1𝐷 + 𝛺𝜀

−1)⊗∑𝐵𝑧𝑍𝑡−1𝑍𝑡−1′ 𝐵𝑧

𝑇

𝑡=1

𝑜𝑝(1 )

𝑜𝑝(1 ) ℳ42

)

=: (ℳ41 𝑜𝑝(1 )

𝑜𝑝(1 ) ℳ42)

where ℳ42 =1

𝑇2[(𝐴2𝑧 − 𝐷𝐴𝑧)

′𝛺𝑒

−1(𝐴2𝑧 − 𝐷𝐴𝑧) + 𝐴𝑧

′ 𝛺𝜀−1𝐴

𝑧] ⊗ ∑ 𝑍2𝑡−1𝑍2𝑡−1

′𝑇𝑡=1

The off-block diagonal term of ℳ1,ℳ2, and ℳ4 are 𝑜𝑝(1 ) since

𝑇−3/2∑ 𝜁𝑡𝜉𝑡′

𝑡 = 𝑜𝑝(1 ) when 𝜁𝑡 is stationary process and 𝜉𝑡 is purely

nonstationary process (Lemma1 of Ahn and Reinsel, 1990). We know ℳ1−1 =

(ℳ11

−1 𝑜𝑝(1 )

𝑜𝑝(1 ) ℳ12−1 ) and similarly as ℳ2

−1 and ℳ4−1 . Using the well-known

formula of the inverse of block matrices, we have

61

(𝜏−1∑𝜕𝑎𝑡

′

𝜕𝜂

𝑇

𝑡=1


𝜏−1)

−1

= ((ℳ1 −ℳ2ℳ4

−1ℳ2′)−1

−(ℳ1 −ℳ2ℳ4−1ℳ2

′)−1ℳ2ℳ4

−1

−(ℳ4 −ℳ2′ℳ1

−1ℳ2)−1ℳ2

′ℳ1−1 (ℳ4 −ℳ2

′ℳ1−1ℳ2)

−1 )

where

(ℳ1 −ℳ2ℳ4−1ℳ2

′)−1

= ((ℳ11 −ℳ21ℳ41

−1ℳ21′)−1

𝑜𝑝(1 )

𝑜𝑝(1 ) (ℳ12 −ℳ22ℳ42−1ℳ22

′)−1),

(ℳ4 −ℳ2′ℳ1

−1ℳ2)−1

= ((ℳ41 −ℳ21

′ℳ11−1ℳ21)

−1𝑜𝑝(1 )

𝑜𝑝(1 ) (ℳ42 −ℳ22′ℳ12

−1ℳ22)−1),

ℳ2ℳ4−1 = (

ℳ21ℳ41−1 𝑜𝑝(1 )

𝑜𝑝(1 ) −ℳ22ℳ42−1) ,

and

ℳ2′ℳ1

−1 = (ℳ21

′ ℳ11−1 𝑜𝑝(1 )

𝑜𝑝(1 ) ℳ22′ ℳ12

−1),

Now, let

��1:= ℳ11 −ℳ21ℳ41−1ℳ21

′,

��2 ≔ ℳ12 −ℳ22ℳ42−1ℳ22

′,

��1:= ℳ21ℳ41−1, ��2: = ℳ22ℳ42

−1,

62

��1 ≔ℳ41 −ℳ21′ℳ11

−1ℳ21,

��2 ≔ℳ42 −ℳ22′ℳ12

−1ℳ22,

��1: = ℳ21′ ℳ11

−1, and

��2: = ℳ22′ ℳ12

−1.

It can be easily shown that these ��1, ��2, ��1, ��2, ��1, ��2, ��1, and ��2 lead to the

expressions in (3.24) − (3.27).

Also from (3.3), (3.4), and

𝜏 = 𝑑𝑖𝑎𝑔(√𝑇 𝐼𝑚𝑦(𝑟+𝑚𝑧), 𝑇 𝐼𝑟𝑦(𝑚−𝑟−𝑟𝑧), √𝑇 𝐼𝑚𝑧𝑟𝑧 , 𝑇 𝐼𝑟𝑧(𝑚𝑧−𝑟𝑧)),

we obtain

𝜏−1∑𝜕𝑎𝑡

′

𝜕𝜂

𝑇

𝑡=1

𝛺𝑎−1𝑎𝑡 =

(

−[𝛺𝑒−1⊗

1

√𝑇∑ ��𝑡−1𝑇𝑡=1 ] 𝑒𝑦𝑡

−[𝐴′𝛺𝑒−1⊗

1𝑇∑ 𝑋2𝑡−1𝑇𝑡=1 ] 𝑒𝑦𝑡

[(−𝐷′, 𝐼𝑚𝑧)𝛺𝑎−1⊗

1

√𝑇∑ 𝐵𝑧𝑍𝑡−1𝑇𝑡=1 ] 𝑎𝑡

[((𝐴2𝑧 − 𝐷𝐴𝑧)′, 𝐴𝑧

′ )𝛺𝑎−1⊗

1𝑇∑ 𝑍2𝑡−1𝑇𝑡=1 ]𝑎𝑡)

Finally, by multiplying (𝜏−1∑𝜕𝑎𝑡

′


−1 𝜕𝑎𝑡

𝜕𝜂′𝜏−1)

−1

and 𝜏−1∑𝜕𝑎𝑡

′


−1𝑎𝑡 ,

we obtain the result of (3.24) − (3.27).

References

63



Vol.85, pp. 813-823.

Burke S.P., and Hunter J. (2005), Modelling Non-Stationary Econoomic Time Series:

A Multiple Approach. Palgrave: Basingstoke.

Brüggemann, R., Lütkepohl, H., (2005), “Practical Problems with Reduced-rank

ML Estimators for Cointegration Parameters and a Simple Alternative,” Oxford

Bulletin of Economics and Statistics, Vol.67, pp. 673–690.



Gonzalo, J., and Pitarakis, J. (1999), “Dimensional Effect in Cointegration

Analysis”, in R. F. Engle and J.I. White (ed.), Cointegration, Causality and

Forecasting: Festschnift in Honour of Clive Granger, University Press, Oxford ,

OX.




64

Henry J., and Weidmann J. (1995), “Asymmetry in the EMS Revisited: Evidence

from the Causality Analysis of Daily Eurorates”, Annals of Economics and Statistics,

Vol40, pp125-160.

Hunter, J. (1990), “Cointegrating Exogeneity,” Economics Letters, Vol. 34, pp.33-35.

Hunter, J. (1992), “Tests of Cointegrating Exogeneity for PPP and Uncovered

Interest Rate Parity in the United Kingdom,” Journal of Policy Modling, Vol. 14,

pp.453-463.








65

Pettenuzzo D, and Hendry D.F. (2011) , “Exogeneity, Cointegration, and Economic

Policy Analysis”, Avaliable at SSRN 2182749.



Vol.97, pp. 293-343.



Vol.65, pp. 629-653.

Rault, C. (2000), “Non-causality in VAR-ECM Models with Purely Exogenous

Long-Run Paths”, Economics Letters, Vol.67, pp. 121-129.

Reimers, H. and Lütkepohl, H. (1992) , “Granger-causality in Cointegrated VAR

Processes: The Case of the Term Structure,” Economics Letters, Vol.40 (1992), pp.

263-268.

Shilov, G. E. (1977), Linear Algebra, Dover, Mineola, NY.

66

Chapter 4

Impact of Measurement Errors on the Vector Error

Correction Model

We study (asymptotic) properties of the reduced rank estimator of the error

correction models of vector processes observed with measurement errors.

Although it is well known that there is no asymptotic measurement error bias when

the predictor variables form an integrated process (Phillips and Durlauf, 1986) in

regression models, we investigate systematically the impact of the measurement

errors (in the dependent variables as well as in the predictor variables) on the

estimation of not only cointegrating vectors but also the speed of adjustment matrix.

We present the estimation of the model and asymptotic properties of the estimators.

We also obtain the asymptotic distribution of the likelihood ratio test for

cointegrating ranks, and investigate the impact of measurement errors on the test

through a Monte Carlo simulation study.

67

4.1 INTRODUCTION

It is well known that the measurement errors affect the results of statistical analysis.

For example, in simple regression analysis it is well known that measurement errors

cause a bias toward zero in the least squares estimator (LSE), known as attenuation

(Fuller, 1987). This bias does not vanish asymptotically resulting in the

inconsistency of the LSE. When integrated processes are used as predictor

variables in (time series) regression models, there is no measurement error bias

asymptotically (Phillips and Durlauf, 1986), although the limiting distribution is

affected by the measurement errors (Haldrup et al., 2005; Hassler and Kuzin, 2009).

When integrated processes are cointegrated and measurement errors are stationary,

the observed series are also cointegrated (Granger, 1986).

One of the main interests in cointegration analysis is the inference of the

cointegrating rank and cointegrating vectors which represent the long run

relationship among the nonstationary variables. Hassler and Kuzin (2009) studied

the effects of measurement errors in cointegration analysis. They obtained the

asymptotic distribution of the likelihood ratio (LR) test statistics for the null

hypothesis of no cointegration. They also obtained the limiting distributions of the

LSE and of the fully modified LSE by Phillips and Hansen (1990) of cointegrating

regression models where the predictor variables are non-cointegrating integrated

variables. In this chapter, we investigate asymptotic properties of estimators of

cointegrating vectors in the VECM and of LR test statistic for cointegrating rank,

68

when data are observed with measurement errors. This provides the effect of

measurement errors in cointegration analysis. Our study can be considered as an

extension of the model in Phillips and Durlauf (1986) in that our analysis

incorporates measurement errors in the dependent variables as well. It can also be

considered as an extension of Hassler and Kuzin (2009) in that our results are

applicable when the cointegrating rank is generally ( , is a dimension

of a vector process). We also investigate asymptotic properties of likelihood ratio

test statistics for the null hypothesis of cointegration rank , extension of the case of

given by Hassler and Kuzin (2009). The loading matrix, also called the speed

of adjustment matrix (Johansen and Jesulius, 1990), in the ECM provides an

interpretation of the speed of adjustment toward the equilibrium, and thus provides

useful information. However, little attention has been given to the inference of

estimators of this loading matrix, especially, when there are measurement errors.

Therefore, we also investigate asymptotic properties of estimators of the loading

matrix to assess systematically the effects of measurement errors on the estimators

of the loading matrix.

This chapter is organized as follows. The impact of the measurement errors in

VECM to LSE is considered in Section 4.2 and Section 4.3 consider the impact of

the measurement errors to MLE. We investigate the impact of the measurement

errors to the asymptotic distribution of likelihood ratio test statistic for the null

hypothesis of cointegration rank in Section 4.4. Finite sample properties of the

69

effects of the measurement errors are examined through a Monte Carlo simulation

in Section 4.5.

4.2 IMPACT OF THE MEASUREMENT ERRORS TO

LEAST SQUARE ESTIMATOR

In order to make the exposition brief, we consider the following VECM of the

vector autoregressive process of order .

∑

∑

where is an m-dimensional vector process of integrated order 1 and cointegrated

with cointegrating rank , is the first difference operator, is matrix

which has a reduced rank , and are matrices of full rank respectively,

and the error is -dimensional white noise process with mean zero and

nonsingular covariance matrix . Suppose that we cannot observe directly, but

we observe

, (4.2)

where is an independent white noise process with mean zero and nonsingular

covariance matrix representing the measurement error and uncorrelated with .

Then, the model for the observed is

70

∑

∑

where , and

is an identity matrix. We note that is a vector moving average

process of order . Our main interest of thin Chapter is the effects of the

measurement errors to the error correction matrix , or to the loading matrix and

cointegrating matrix . The effects of the measurement errors to the coefficient

matrices are the same as the effects to the multivariate regression analysis which

is introduced in Chapter 2.

We consider the case with for simplicity of exposition. Then the true model

(4.1) and fitted model (4.3) will be simplified as

In order to investigate the effects of measurement errors, we first consider the

following LSE of in (1) using the observaion

, (4.6)

where =∑ and =∑ .

71

We summarize asymptotic properties of in the following Theorem.

Theorem 4.1. As the sample size the LSE in (4.4) has the following

properties;

1) ,

2) ( ) ,

3)

(

)

,

where , is a matrix associated with the Jordan canonical form

of , is a functional of stochastic integrals of Brownian motions, and is a

matrix consisting of variance and covariance matrices, ∑(

)( ) , and . The detailed expressions of these

matrices are in the proof.

pf) We note that ∑ ∑ and

∑ ∑ . Thus, it follows

∑ ∑

. Since is of

reduced rank , we have the Jordan canonical form of as , where

, and . Let , , ,

and

As in Ahn and Reinsel

(1990), we note that for , where and are and ,

72

is nonstationary and not conintegrated, and

is stationary.

From

we get that

[ ]

∑

∑{

}

∑{

} (

)

∑{

} (

)

where ∑

∑{

}

We partition

,

and

such that

and are , and are . We first get the asymptotic results of

∑{

}

∑{

}

73

For ∑ ∑

∑

,

∑

∫

∑

where with ∑ is sum of infinite moving average

coefficients of the true model (4.4). For ∑

∑ ∑

,

∑ ∑

∑

∫

since ∑ , and

∑

Also,

∑

[(

) (

)]

Therefore,

∑{

}

74

[

[

] [

]]

Similarly, we get the asymptotic result of term by term as follows:

∑ [

∑ ∑

∑ ∑

]

[

∫

∫

]

[

]

∑

[ ∑

∑

∑ ∑

]

∑ [

∑ ∑

∑ ∑

]

[

]

Therefore,

75

[

]

and so

( )

[

]

[

( )

( )

]

Finally, our asymptotic result is given by

[ ]

[

[

] [

]]

[

( )

(

)

( )

]

[{

([

] [

] (

)

(

))}

[ (

)

( )

]]

The proof is complete with defining

76

{

([

] [

] (

)

( ))}

{

([

] [

] (

)

( ))}

where is a functional of stochastic integrals of Brownian motions defined in

Theorem 1 of Ahn and Reinsel (1990) in Section 2.1.2, and

[

] (

)

(4.24)

Next, from 2) of Theorem 1 and (4.20),

∑

{∑ ( ) (∑

) ( )

}

∑ {( ) }{

} { ( ) }

(

)

where . The proof is complete.

77

Unlike the cases with stationary time series regression models, the LSE is consistent

mainly because each component of is nonstationary as discussed in Hassler and

Kuzin (2009). But it is asymptotically biased, and its asymptotic distribution is

affected by measurement errors. The matrix P in 2) of Theorem 3.1 provides the

rearrangement of the elements in the LSE into T-consistent ones and inconsistent

ones. Our theorem with yields the same result as in Hassler and Kuzin

(2009), where a model with no cointegration is considered. This is because

when . In the absence of the measurement error, the asymptotic distribution is

the same as that in Theorem 1 of Ahn and Reinsel (1990). The LS estimator for the

variance is biased not only for but for .


MAXIMUM LIKELIHOOD ESTIMATOR

We consider the reduced-rank estimation of the model in (4.5). Although it is well

known that the closed form of the reduced rank estimators are readily available in

terms of eigenvectors of the sample moment matrices (Anderson, 1951; Johansen,

1988), we use the approach in Ahn and Reinsel (1990) for the explicit expression of

the asymptotic properties of the reduced-rank estimators. This expression displays

explicitly the impact of measurement errors on parameter estimation. To this end,

we parameterize , where is an identity matrix and is an

78

matrix of parameters. With observations for , we

obtain the MLEs of A and B in (4.5) by maximizing

| |

∑

where , , ,

and vec(.) is the vectorizing operator.

The MLE of is obtained by the iterative approximate Newton-Raphason relations

(∑

)

(∑

)

where , and is

. Note that , the last d components of , is assumed to be

purely nonstationary, that is, not cointegrated in the parameterization of

.

We summarize the asymptotic properties of the MLEs of and α, or, and in

the following theorem.

Theorem 4.2. Let denote the MLE obtained from (4.25) using the initial

consistent estimator. Then as

( )

and

79

where and are defined in Theorem 4.1, , and are matrices associated

with the Jordan canonical form of , is a matrix such that

, is a r matrix such that

, and is r matrix

such that

.

pf) With Taylor expansion of at gives us

( ) { ∑

}

{ ∑

}

∑[ ∑

∑

∑

]

[

]

Let

[

] [

∑

∑

∑

]

and we know

[

]

[

]

Therefore,

80

( )

∑[

] [

]

∑[

]

Finally, we obtain

( )

∑

{

( ∑

)

( ∑

)}( ∑

)

∑

( ∑

)

Observe that

, where and

conformably. From (4.16), we have

∑

[

[

] [

]]

that is,

∑

(

[

])

81

∑

[

]

We also have

∑

[ ∑

∑

∑

∑

]

[

]

at (4.20)

Therefore, we can obtain following asymptotic results.

∑

∑

∑

∑

∑

∑

∑

(

)

{( )

( )

}

82

∑

The equaility of is hold since

. Notice that

with

so that

and

Finally, we obtain asymptotic distributions of (4.32) and (4.33).

( )

{

[(

) (

) ]}

{

[(

) ( )

]}

83

{

( )

[(

) ( )

]}

{

{( )

(

)

}}

{ ( )

(

)}

( )

The third equality of (4.40) and second equality of (4.41) are hold from the fact that

and

[

] [

] [

]

84

The MLE of is consistent, and its asymptotic distribution is affected by the

measurement error. As can be considered as the regression coefficient matrix in

the regression of (nonstationary) on (nonstationary) , where consists of

the first r components of and of the remaining components such that

, this result is consistent with that there is no measurement error

bias asymptotically when predictors of a regression model are nonstationary

( Phillips and Durlauf, 1986). It can be considered as an extension of the result of

Phillips and Durlauf (1986) because the dependent is also subject to

measurement errors in this paper. Since the cointegrating rank, r, may be less than

, this result is considered as an extension of the cointegrating regression

result of Hassler and Kuzin (2009) which considered the case with Unlike

the results of no measurement error cases, the asymptotic distribution of is no

longer mixed-normal. This is intuitively true because and are correlated.

Formally, the two vector Brownian motions

in the asymptotic distribution of are not independent because

85

is not zero in general, where , and

.

We note that in the absence of measurement errors, this covariance is zero as

shown in Ahn and Reinsel (1990).

Given the cointegrating matrix , because the adjustment matrix can be

considered as the coefficient in the regression of on which are both

stationary, we would expect that is inconsistent according to the results of

(conventional) regression with measurement errors. This theorem confirms the

inconsistency of .


LIKELIHOOD RATIO TEST

Now, we consider asymptotic porperties of the likelihood ratio test statistic for

testing the cointegrating rank in the model (4.3). More specifically, we test the null

hypothesis versus the alternative hypothesis .

When there are no measurement errors, the asymptotic distribution of the LR test

statistic is well known (Johansen, 1988; Reinsel and Ahn, 1992). When there are

measurement errors, the asymptotic distribution of LR test statistic was obtained by

Hassler and Kuzin(2009) only the case of . Therefore, we drive the asymptotic

distribution that encompasses as well as , and thus extend the results of

Hassler and Kuzin(2009). Under the null hypothesis, the reduced rank estimator

86

maximizes the log likelihood (4.26) while LSE specified at (4.6)

maximize the log likelihood (4.26) under the alternative hypothesis. The likelihood

ratio test static is given by

| |

| |

where ∑ ( )( )

and ∑ ( )(

) We summarize asymptotic properties of in the following Theorem.

Theorem 4.3. Let denote the likelihood ratio statistic from (4.42). As the sample

size

{

}

where (

) ,

, and are defined

Theorem 4.1, and ,

and are defined at the proof of Theorem 4.1.

pf) We observe

∑( )( )

( )∑

( )

since the cross term is

87

∑( ) ( )

from ∑ ( )

.

can be represented by

| |

| |

| ( )∑

( ) |

Let denote the eigenvalues of the matrix ( )∑

( ) ,

Then under ,

∑

∑

{ ( )∑

( ) }

Therefore,

{ ( ) ( ∑

) ( ) }

{ ( ) ( )

}

First, from (4.25),

(

)

88

Next, we obtain the asymptotic distribution of ( ) and ( ) . Since

( ) ( ) ( )( )

and

( ) [ ]

{ } { }

we have

i) { }

{( ) ( ) ( )( )

( )

ii) { }

( ) ( ) ( )( )

( ) ( )

( )

(since

so that )

So,

( ) ( ) ( )

{

}

and

( ) ( ) ( )

89

( ) ( ) ( ) ( )

( )

from Theorem 4.2.

Therefore,

( )

( )

[

]

The proof is completed by combining asymptotic results of (4.20), (4.25), (4.48)

and (4.49) to (4.47).

It can be shown easily that the asymptotic distribution of in (4.42) is same as the

one in Johansen(1988) and Reinsel and Ahn (1992) when there are no measurement

error. This is mainly becuase , , , and in the

absence of measurement errors. Since under the null hypothesis of

contigration rank , it can be shown easily that the asymptotic distribution in

(4.43) is the same as in Hassler and Kuzin(2009), when there are measurement

errors. If we know measurement error variance , we can derive a consistent

90

estimates of given by Fuller(1987). Typically, the measurement error variance

is unknown, but we can still estimate consistent using estimate of

through conducting experiments (Fuller, 1987).

4.5 MONTE CARLO RESULTS

In this section, we consider the finite sample properties of the estimators and the LR

test statistic when measurement error exists through a Monte Carlo simulation. First

we consider the finite sample properties of LSE and MLE. We use the following

data generating bivariate process with cointegration rank given by

(Δ

Δ

)

(

) (

)

(

) (

) (

) (

)

The data generating process is the same as the one of Ahn and Reinsel (1990).

And we generate the observed model by

(

) (

) (

) ,

91

with . Note that ( )

. As we increase the value of effects

of measurement errors are also increasing. There are no measurement errors when

Various sample sizes are used in this simulation. For

each observed series , estimates of the parameters are computed as

and by the Newton-Raphson algorithm (4.27). is the MLE.

The initial estimator is computed using LSE where

and are the first and the second columns of

and

( )

(Ahn and Reinsel, 1990). Note that . In addition, we also

consider the classical estimate method proposed by Engle and Granger (1987),

which is obtained as

(∑

)(∑

)

and

(∑ Δ

) ∑

,

where

92

We performed 10000 replicaitons for each series. The finite sample bias and

mean squared errors (MSE) of the estimators are summarized in Table 4.1 and

Table 4.2. Ahn and Reinsel (1990) showed that the classical estimation has a

larger MSE than other estimates. Although is a consistent estimator, even there

are measurement errors (Phillips and Durlauf ,1986; Haldrup et al. ,2005; and

Hassler and Kuzin ,2009), the difference between MSE of and

s obtained

from (4.27) gets larger as the effect of measurement errors is increased. We observe

that the MSE of classical estimator is still lager than other methods even for

large sample size ( and 1000) when the effect of measurement error is large

( ). Brüggemann and Lütkepohl (2005) showed that the LSE of has

smaller MSE than the MLE for cointegrating vectors through a Monte Carlo

simulation study. We observe the similar result in Table 4.1 when , i.e., as the

iteration continues the MSE gets larger. When there are measurement errors, the

MSE of MLE of

is larger than

and

which is the same

phenomenon observed in Brüggemann and Lütkepohl (2005). Even for small

sample size, the MSE of MLE is closer to that of classical estimation method than

LSE, NR1 and NR2. The bias of tends to smaller when iteration step gets larger

with and without measurement errors.

When measurement error exists, the bias of the estimates of the loading matrix

does not decrease even sample size is large. This is a natural conclusion from

Theorem 4.2, the MLE of is inconsistent. Furthermore, the mean of the LSE of

93

( )

( )(

)

with jointly normal distribution assumption. Note that

( )

Table 4.1. Bias and MSE ( ) for LSE and MLE of cointegration vector

based on 10000 replications for Each Sample Size

T=50 T=100 T=200 T=400 T=1000

bias s=0 EG 0.1302 0.0913 0.0601 0.0365 0.0178

iLS 0.007 0.0024 0.002 0.0005 0.0001

NR1 0.007 0.0024 0.002 0.0005 0.0001

NR2 0.0003 -0.0002 0.0011 0.0002 0.0001

NRc -0.003 -0.0006 0.0011 0.0002 0.0001

s=0.5 EG 0.1796 0.1294 0.0865 0.0533 0.0261

iLS 0.0149 0.0064 0.0041 0.0017 0.0007

NR1 0.0148 0.0064 0.0041 0.0017 0.0007

NR2 0.0016 0.0006 0.002 0.001 0.0006

NRc -0.0055 -0.0006 0.0018 0.001 0.0006

s=1 EG 0.2229 0.1645 0.111 0.0691 0.0343

iLS 0.0255 0.0125 0.006 0.0022 0.0009

NR1 0.0253 0.0125 0.006 0.0022 0.0009

NR2 0.0068 0.0034 0.0026 0.001 0.0006

NRc -0.0051 0.0006 0.0022 0.0009 0.0006

94

s=1.5 EG 0.2615 0.1952 0.1343 0.0849 0.0424

iLS 0.0335 0.0155 0.0082 0.0033 0.0012

NR1 0.033 0.0154 0.0082 0.0033 0.0012

NR2 0.0087 0.0028 0.0031 0.0014 0.0007

NRc -0.0055 -0.0016 0.0023 0.0013 0.0007

MSE

s=0 EG 5.8461 2.4843 0.9374 0.3209 0.0712

iLS 1.8578 0.5559 0.1651 0.047 0.0088

NR1 1.8536 0.5561 0.1652 0.047 0.0088

NR2 1.9091 0.5638 0.1653 0.0469 0.0088

NRc 3.2891 0.5672 0.1655 0.047 0.0088

s=0.5 EG 9.5801 4.453 1.8013 0.6499 0.1498

iLS 2.5701 0.7399 0.2198 0.0623 0.0118

NR1 2.563 0.7405 0.2198 0.0623 0.0118

NR2 2.6861 0.7648 0.2217 0.0626 0.0118

NRc 3.4242 0.789 0.2227 0.0627 0.0118

s=1 EG 13.3453 6.715 2.8454 1.0673 0.2543

iLS 3.402 0.9966 0.2937 0.0774 0.0149

NR1 3.3828 0.9959 0.2936 0.0774 0.0149

NR2 3.4243 0.9585 0.2895 0.0772 0.015

NRc 4.3511 1.0191 0.2906 0.0773 0.015

s=1.5 EG 17.196 8.9713 4.0114 1.563 0.3852

iLS 4.1095 1.2015 0.3533 0.0947 0.0175

NR1 4.0757 1.1979 0.3531 0.0947 0.0175

NR2 4.0394 1.1399 0.3358 0.0934 0.0176

NRc 15.9369 1.2396 0.336 0.0937 0.0176

Table 4.2. Simulation Results : bias and MSE ( ) for LSE and MLE of

adjust matrix and based on 10000 replications for Each

Sample Size

T=50 T=100 T=200 T=400 T=1000

bias s=0 EG -0.0181 0.0043 -0.0107 0.0006 -0.0057 0.0002 -0.0032 -

0.0002 -0.0013

-

0.0001

iLS -0.0177 0.0042 -0.0106 0.0006 -0.0057 0.0002 -0.0032

-

0.0002 -0.0013

-

0.0001

95

NR1 -0.015 0.007 -0.008 0.0029 -0.0038 0.0017 -0.002 0.0008 -0.0008 0.0003

NR2 -0.0143 0.0067 -0.0077 0.0028 -0.0038 0.0017 -0.002 0.0008 -0.0008 0.0003

NRc -0.0137 0.0069 -0.0075 0.0029 -0.0037 0.0017 -0.0019 0.0008 -0.0008 0.0003

s=0.5 EG 0.0311 0.0599 0.0414 0.0582 0.049 0.0591 0.0532 0.0595 0.0562 0.0603

iLS 0.0314 0.0598 0.0415 0.0581 0.049 0.0591 0.0532 0.0595 0.0562 0.0603

NR1 0.0449 0.069 0.0517 0.0646 0.0557 0.0632 0.0573 0.0619 0.0582 0.0615

NR2 0.046 0.0684 0.0522 0.0644 0.0558 0.0631 0.0573 0.0619 0.0582 0.0615

NRc 0.0484 0.0692 0.053 0.0647 0.0561 0.0632 0.0574 0.0619 0.0582 0.0615

s=1 EG 0.0509 0.0868 0.0649 0.0869 0.0749 0.0887 0.0814 0.0903 0.0858 0.0917

iLS 0.0511 0.0865 0.0649 0.0868 0.0749 0.0887 0.0815 0.0902 0.0858 0.0917

NR1 0.0733 0.1002 0.0818 0.0966 0.0861 0.0948 0.0884 0.094 0.0892 0.0935

NR2 0.0748 0.0992 0.0826 0.096 0.0864 0.0946 0.0885 0.0939 0.0892 0.0935

NRc 0.0789 0.1005 0.0842 0.0966 0.0869 0.0948 0.0886 0.094 0.0893 0.0935

s=1.5 EG 0.0624 0.1034 0.0782 0.1042 0.0893 0.1069 0.0967 0.1089 0.1029 0.1108

iLS 0.0626 0.1032 0.0782 0.1041 0.0893 0.1068 0.0967 0.1089 0.1029 0.1108

NR1 0.0924 0.1208 0.1011 0.1168 0.1047 0.1149 0.1064 0.1138 0.1077 0.1131

NR2 0.0943 0.1193 0.1021 0.116 0.1051 0.1146 0.1065 0.1137 0.1077 0.1131

NRc 0.1005 0.1211 0.1045 0.1168 0.1059 0.1149 0.1068 0.1138 0.1078 0.1131

MSE

s=0 EG 0.9966 0.3479 0.4629 0.1592 0.2244 0.0783 0.107 0.0386 0.0429 0.0151

iLS 0.994 0.3482 0.4628 0.1592 0.2243 0.0783 0.107 0.0386 0.0429 0.0151

NR1 0.9813 0.3405 0.4561 0.1561 0.222 0.0769 0.106 0.0382 0.0426 0.015

NR2 0.9793 0.3393 0.4559 0.1558 0.222 0.0769 0.106 0.0382 0.0426 0.015

NRc 0.9848 0.3399 0.4565 0.1559 0.2221 0.0769 0.106 0.0382 0.0426 0.015

s=0.5 EG 1.415 0.7969 0.7495 0.5414 0.521 0.4498 0.4177 0.4014 0.3684 0.3827

iLS 1.4169 0.7956 0.7505 0.5408 0.5213 0.4496 0.4178 0.4012 0.3684 0.3827

NR1 1.4993 0.8835 0.8313 0.6091 0.586 0.4955 0.461 0.4298 0.3904 0.3966

NR2 1.5065 0.8729 0.8362 0.605 0.588 0.4941 0.4618 0.4293 0.3905 0.3965

NRc 1.5427 0.8809 0.8473 0.609 0.591 0.4956 0.4626 0.4298 0.3907 0.3966

s=1 EG 1.795 1.2443 1.1251 0.9846 0.8875 0.8928 0.8181 0.8662 0.7965 0.8615

iLS 1.7974 1.2407 1.126 0.9828 0.8877 0.8922 0.8182 0.866 0.7966 0.8614

NR1 1.9938 1.4459 1.3304 1.1407 1.0545 0.9983 0.931 0.9321 0.8542 0.8938

NR2 2.0047 1.4229 1.3403 1.1301 1.0594 0.9943 0.9328 0.9307 0.8547 0.8935

NRc 2.0709 1.441 1.3705 1.1402 1.0685 0.9981 0.9357 0.932 0.8552 0.8937

96

The MSE of is behave as the bias of except the MSE is decrease at large

sample size. This is from the fact that and

toward to zero but bias is not when measurement error exits. In practice, the sample

size is small and the measurement error exists. Hence, we can conclude that the

LSE is the best estimation method in practical use.

Finally, we investigate the impact of the measurement errors to the LR test for the

null hypothesis of cointegration rank . We obtained the asymptotic distribution of

the LR test statistic and summarized in Theorem 4.3. However, we cannot use this

distribution in practice because it depends on the unknown nuisance parameter .

Certain assumptions are often made about in measurement error models (Fuller,

1989) to deal with the nuisance parameter. Here, we investigate the impact of

measurement errors on the cointegrating rank test through a Monte Carlo simulation,

when they are ignored and the asymptotic distribution in Johansen (1988) or Reinsel

and Ahn (1992) is used.

For the data generating process, we consider a four dimensional vector

autoregressive process of order 1

where the eigen-values of are 1, 1, , and 0.4 with .

Therefore, the cointegratimg rank of is 2. We use and 0.9 for the

study.

We choose the matrix as

97

(

)

such that , where

and (

)

We generate the observed process

with (

)

where are considered. We note that is the ratio of the generalized

variances, ⁄ . With the above data generating process, we

generated series with and obtained the empirical rejection numbers

using 1,000 replications for testing cointegration rank is , for each

at the level of significance of 0.05. The results are summarized in Table4.3. The

columns under represent empirical sizes of finite samples for testing

. For , i.e., when there is no measurement error, the empirical sizes are not

significantly different from the nominal sizes. When there are measurement errors,

the empirical sizes are biased, in that about two to three times higher. However, the

magnitudes of the biases remain almost the same even though the ratio of the

generalized variance of the measurement error increases. The null hypothesis of no

98

cointegration, that is, , is rejected for all values of , s, and T considered.

Therefore, the existence of cointegration is detected even in the presence of

measurement errors. This is probably because the process with measurement

errors, , is also conitegrated. The null hypothsis of is rejected more often,

hence leading to the correct test of , in the presence of measurement errors.

For , the null hypothesis of is rejected twice as often in the presence

of measurement errors for , while the number of rejection is about the same

for . For , the number of rejection of in about the same

without regard to measurement errors.

Table 4.3. Empirical rejection numbers of the likelihood ratio test for the

conitegrating rank r base on 1000 replications and the significance level of 0.05.

Null hypothesis

1000 463 43 11 100 0 951 58 17

1000 779 142 21 1000 1000 104 10

1000 819 144 18 1000 1000 112 15

1000 851 153 20 1000 1000 123 12

1000 309 40 16 1000 747 73 15

1000 565 125 17 1000 960 126 19

1000 605 139 16 1000 974 132 17

1000 647 143 22 1000 989 135 19

99

If a specific cointegrating for the null hypothesis is not given, one strategy to

identify the cointegrating rank is to test , first. If this is rejected, then test

, and so on until the null hypothesis is not rejected. Then the smallest

value of for which the null hypothesis is not rejected is identified as correct rank.

Based on this strategy, we summarize in Table 4.4 the number of cases of identified

ranks at the level of significance of 0.05 for each of the different values of , s, and

T. Interestingly, one is more likely to identify the correct cointegrating rank in the

presence of measurement errors except for the case with and . In

the presence of measurement errors, it is less likely to under estimate the

cointegrating rank as one, here and more likely overestimate the cointegrating rank

as three. From this limited study, we find that the cointegrating rank is more likely

Table 4.4. Numbers of identified ranks by the likelihood ratio test for the conitegrating rank

r in based on 1000 replications and the level of significance level of 0.05.

Fitted rank

532 418 39 11 48 885 50 17

212 636 131 21 0 891 99 10

175 675 132 18 0 881 104 15

175 673 132 20 0 881 107 12

683 268 33 16 249 671 65 15

422 445 116 17 39 829 113 19

380 476 128 16 24 839 120 17

379 471 128 22 25 838 118 19

100

overestimated or correctly estimated with measurement errors than without

measurement errors, while the cointegrating rank is less likely underestimated that

with measurement errors.

References



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Roots and Reduced Rank Structure: Estimation, Likelihood Ratio Test, and

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103

Chapter 5

Concluding Remarks

In this thesis, we have investigated the impact of exogeneity and measurement

errors on the vector error correction model. Many researchers have studied these

topics with non-general assumptions. They assumed the weak exogeneity or non-

causality for the cointegration analysis with exogenous processes. The condition of

Johansen (1992), Harbo et al. (1998), and Pesaran et al. (2000), in particular, was

just a sufficient condition for the weak exogeneity and not a necessary condition.

Also many researchers who studied the impact of measurement errors on

cointegration analysis only considered the case of cointegration rank 1. The impact

of measurement errors on general VECM had not been studied.

In chapter 3, we investigate the impact of exogeneity on VECM. We show how to

change the VECM for more effective analysis, develop estimation procedure, and

derive its asymptotic properties. Since non-causality in VAR models implies the

block triangular coefficient matrices, the methods suggested in this thesis can be

104

used for the estimation of the non-causal model with cointegration under the null

hypothesis, and thus for testing non-causality among cointegrated series. This

addresses the issue of testing for Granger-causality in cointegrated systems raised in

Reimers and Lütkepohl (1992). Mosconi and Gianini (1992) studied simultaneous

testing of cointegration and non-causality. They also suggested how to choose

cointegrating ranks using the Akaike information criteria. But they did not provide

details of parameterization and suggestion of initial estimates other than the use of

zeros. However, it is important to choose good initial estimates for the numerical

iteration of parameter estimation. Moreover, Pradel and Rault (2003) studied testing

of non-causality and strong exogeneity, but did not present the details of estimation.

This thesis fills the gaps of those papers. Our methods are also applicable to the

other models considered by Johansen (1992), Harbo et al. (1998), and Pesaran et al.

(2000).

In chapter 4, we investigated the effect of measurement errors on the speed of

adjustment vector and the cointegrating vector. We found that the reduced rank

estimator of the speed of adjust matrix is inconsistent while that of the cointegrating

vector is consistent, although asymptotically biased. The latter estimator is no

longer mixed normal distributed unlike the case with no measurement error. To

adjust the inconsistency of the speed of the adjust matrix, we can use the

suggestions of Fuller (1987) and Cook and Stefanski (1994). We have found the

LSE of cointegrating vectors is the best estimation method in practical use. We also

investigated the effect of measurement errors on the cointegrating rank test

105

analytically, by obtaining the asymptotic distribution of the LR test statistic and

empirically by a Monte Carlo simulation study. For further study, it is worth

considering using an instrumental variable or Kalman filter to reduce the effect of

measurement errors of estimating and forecasting on the VECM.

106

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Anderson, T. W. (1951), “Estimating Linear Restrictions on Regression Coefficients

for Multivariate Normal Distributions,” Annals of Mathematical Statistics, Vol.22,

327-351.




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Burke S.P., and Hunter J. (2005), Modelling Non-Stationary Econoomic Time Series:

A Multiple Approach. Palgrave: Basingstoke.

107

Brüggemann, R., Lütkepohl, H., (2005), “Practical Problems with Reduced-rank

ML Estimators for Cointegration Parameters and a Simple Alternative,” Oxford

Bulletin of Economics and Statistics, Vol.67, pp. 673–690.

Cook, J. R., and Stefanski, L. A. (1994), “Simulation-extrapolation Estimation in

Parametric Measurement Error Models,” Journal of the American Statistical

Association, Vol.89, 1314-1328.

Engle R. F., and Granger C.W.J.(1987), “Co-Integration and Error Correction:

Representation, Estimation, and Testing”, Econometrica, Vol.55, pp.251-276.






Gonzalo, J., and Pitarakis, J. (1999), “Dimensional Effect in Cointegration

Analysis”, in R. F. Engle and J.I. White (ed.), Cointegration, Causality and

108

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Hunter, J. (1992), “Tests of Cointegrating Exogeneity for PPP and Uncovered

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Johansen, S. and Juselius, K. (1990), “Maximum Likelihood Estimation and

Inference on Cointegration-with Applications to the Demand for Money,” Oxford

Bulletin of Economics and Statistics, Vol.52, 169-210.




Pagan, A. (1984), “Econometric Issues in the Analysis of Regressions with

Generated Regressors,” International Econometric Review, Vol.25, 221-247.

110



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Policy Analysis”, Avaliable at SSRN 2182749.



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Phillips, B. C. B. and Hansen, B. C. (1990), “Statistical inference in instrumental

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Rault, C. (2000), “Non-causality in VAR-ECM Models with Purely Exogenous

Long-Run Paths”, Economics Letters, Vol.67, pp. 121-129.

Reimers, H. and Lütkepohl, H. (1992) , “Granger-causality in Cointegrated VAR

Processes: The Case of the Term Structure,” Economics Letters, Vol.40 (1992), pp.

263-268.

Reinsel, G. C. and Ahn, S. K. (1992), “VectorAutoregressive Models with Unit

Roots and Reduced Rank Structure: Estimation, Likelihood Ratio Test, and

Forecasting,” Journal of Time Series Analysis, Vol.13, pp.353-375.

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models with some unit roots,” Econometrica, Vol.58, pp.113-144.

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112

요약(국문초록)

본 학위논문에서는 공적분 분석에 있어서의 벡터 오차 수정 모형에 대

한 두 가지 주제를 다룬다.

3장에서는 외생성이 벡터 오차 수정 모형에 미치는 영향에 대해 알아q

보았다. 백터 시계열 과정에 외생 변수가 포함되어 있는 경우의 벡터 오

차 수정 모형의 기존 연구들을 좀 더 일반화 하여 제시하였다. 해당 모형

의 최소제곱 추정, 최대 가능도 방법 또한 제시하였다. 추정치의 점근 분

포를 유도하고, 추정치의 유한 표본에서의 성질은 몬테 카를로 모의 실험

을 통해 알아보았다.

4장에서는 측정 오차가 일반 공적분 차원을 가진 벡터 오차 수정 모형

에 주는 영향에 대해 조사하였다. 측정 오차로 인해 공적분 모형의 추정

치와 가능도비 검정 통계량의 점근적 성질에 어떤 영향을 주는지에 대해

알아보았다. 몬테 카를로 모의 실험을 통해 유한 표본에서 측정 오차의

영향 또한 알아보았다.

주 요 어: 공적분, 오차 수정 모형, 외생성, 그레인저 인과, 측정 오차,

학 번: 2008-20273

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