The effect of target function on the predictive performance of the two‐stage ridge estimator

• Date: 01 December 2019
• Author: Selma Toker,Nimet Özbay
• Published date: 01 December 2019
• DOI: http://doi.org/10.1002/for.2597

RESEARCH ARTICLE

The effect of target function on the predictive performance

of the two‐stage ridge estimator

Selma Toker | Nimet Özbay

Department of Statistics, Faculty of

Science and Letters, Çukurova University,

Adana, Turkey

Correspondence

Selma Toker, Department of Statistics,

Faculty of Science and Letters, Çukurova

University, 01330 Adana, Turkey.

Email: stoker@cu.edu.tr

Abstract

The main thrust of this study is to consider the problem of simultaneous pre-

diction of actual and average values of the simultaneous equations model

through the target function of Shalabh (Bulletin of International Statistical

Institute, 1995, 56, 1375–1390). We focus on the predictive performance of the

two‐stage ridge estimator with the motivation for eliminating the disorder aris-

ing from multicollinearity. An optimal biasing parameter of the two‐stage ridge

estimator is derived by a minimization process of prediction mean square error.

In addition, an optimal estimator for the weight of observed value in target

function is attained theoretically. The results inferred from a numerical exam-

ple and a Monte Carlo experiment provide a dramatic improvement in the pre-

dictive ability of the two‐stage ridge estimator.

KEYWORDS

data analysis, multicollinearity, prediction mean squareerror, simultaneous equations model, target

function, two stage ridge estimator

1|INTRODUCTION

Estimation techniques are widely considered in the system

of simultaneous equations, while the analysis of predictive

performance is rarely discussed. Some papers from the

literature based on the issue about estimation and predic-

tion in the simultaneous equations model are Kloek and

Mennes (1960), Amemiya (1966), Dhrymes (1973, 1974),

Hendry (1976), Kadiyala and Nunns (1976), Maasoumi

(1978), Vinod and Ullah (1981), Ullah and Srivastava

(1988), Amirkhalkhali, Ogwang, Amirkhalkhali, and Rao

(1995), Womer, Cantrell, and Mayer (1999), Skeels and

Taylor (2014), Toker, Kaçıranlar, and Güler (2018), Özbay

and Toker (2018a), and Toker (2018). Even if the predic-

tion of endogenous variable is explored, the average value

of this endogenous variable is neglected in these papers.

Accordingly, we aim to constitute simultaneous prediction

of actual and average values of the endogenous variable in

this article.

A consistent and well‐known estimator that makes use

of all the information in the system concerning the exog-

enous variables is the two‐stage least squares (TSLS) esti-

mator. When multicollinearity is severe enough not to be

ignored, a preference for the TSLS estimator is useless

since the estimates are unstable and unreliable (Hill &

Judge, 1987). A popular estimator to remedy the draw-

backs of multicollinearity is the ridge estimator (RE) of

Hoerl and Kennard (1970). A ridge‐like modification

was discussed by Maasoumi (1980) for the simultaneous

equations model. Essentially, Vinod and Ullah (1981)

recommended using this method for estimating the

coefficients. Capps and Grubbs (1991) also examined the

RE at first, second and both stages under various levels

of multicollinearity. Toker et al. (2018), Özbay and Toker

(2018a), Toker (2018), and Toker and Özbay (2018) have

recently used the two‐stage RE and developed other

biased estimators resistant to multicollinearity for this

model.

Received: 15 August 2018 Revised: 4 January 2019 Accepted: 29 March 2019

DOI: 10.1002/for.2597

Journal of Forecasting. 2019;38:749–772. © 2019 John Wiley & Sons, Ltd.wileyonlinelibrary.com/journal/for 749

Estimation performances of the TSLS estimator and the

two‐stage RE were discussed in the aforementioned papers

from a theoretical viewpoint. However, relatively little

attention has been paid to the prediction performance in

situations of strong multicollinearity. In this context, our

aim is to clarify the issue on predictive performance of

the two‐stage RE, which has many benefits for overcoming

the collinearity problem of this model. While investigating

the behavior of the predictors, we concentrate on using the

target function of Shalabh (1995). Toutenburg and Shalabh

(1996, 2000), Shalabh, Toutenburg, and Heumann (2009),

Chaturvedi and Shalabh (2014), and Özbay and Kaçıranlar

(2017a, 2017b) improved predictions by means of the target

function in some regression models. Based on these stud-

ies, in this article we will utilize the prediction mean

square error criterion and the target function to describe

the predictive ability of the TSLS estimator and two‐stage

RE for the simultaneous equations model. In addition,

the optimal estimators for the unknown parameters in

the prediction mean square error of the two‐stage RE will

be derived theoretically.

To apply the two‐stage RE to the US economy model,

which consists of two structural equations, we will use a

data set from Griffiths, Hill, and Judge (1993, p. 611). In

addition, an extensive Monte Carlo simulation experi-

ment will be performed. There are many papers

concerning Monte Carlo methods in the literature,

including: Wagnar (1958), Kmenta and Joseph (1963),

Summers (1965), Quandt (1965), Cragg (1967), Parker

(1972), Hendry (1976), Park (1982), Johnston (1984),

Capps and Grubbs (1991), Carlin, Polson, and Stoffer

(1992), Koutsoyiannis (2001), Agunbiade (2007, 2011),

Agunbiade and Iyaniwura (2010), Toker et al. (2018),

Özbay and Toker (2018a, 2018b), and Toker (2018).

The remainder of this article is organized as follows.

Section 2 gives a summary of the model and some estima-

tors. We discuss the predictive ability of the estimators

and optimal estimators of the unknown parameters in

Section 3. Section 4 includes a numerical example and

the Monte Carlo experiment. The last section presents

concluding remarks.

2|MODEL SPECIFICATION AND

TARGET FUNCTION

This section provides an introduction to the simultaneous

equations model according to the basics in Theil (1971),

Vinod and Ullah (1981), Judge, Hill, Griffiths, Lütkephol,

and Lee (1988), and Griffiths et al. (1993). Let Yand Xbe

T×Mand T×Kmatrices of observations on Mendoge-

nous and Kpredetermined variables, which are respec-

tively as follows:

Y¼

y11 …yM1

⋮⋱ ⋮

y1T…yMT

2

6

43

7

5and X¼

x11 …xK1

⋮⋱⋮

x1T…xKT

2

6

43

7

5:

Further, let Γand Bbe M×Mand K×Mmatrices of

unknown structural coefficients and Ube a T×Mmatrix

of structural disturbances, with the following respective

forms:

Γ¼

γ11 …γ1M

⋮⋱ ⋮

γM1…γMM

2

6

43

7

5;B¼

β11 …β1M

⋮⋱⋮

βK1…βKM

2

6

43

7

5;and

U¼

u11 …uM1

⋮⋱ ⋮

u1T…uMT

2

6

43

7

5:

Then, the simultaneous equations system of Mstruc-

tural equations is expressed in matrix form as

YΓþXB ¼U;(1)

where the rows u

t

(t=1, …,T)ofUare independently

and identically distributed with zero mean; these are

assumed to be homoscedastic and the elements of Xare

nonstochastic and fixed with rank(X)=K≤T.

After multiplying a nonsingular matrix Γby Equation 1

from the right‐hand side, the system becomes

Y¼−XBΓ−1þUΓ−1

:(2)

Equation 2 can be rewritten as

Y¼XΠþV(3)

and hence Equation 3 is called a reduced‐form equation,

with the following reduced‐form coefficients:

Π¼−BΓ−1(4)

and

V¼UΓ−1

:(5)

With regard to zero restrictions criterion:

y1¼Y1γ1þX1β1þu1(6)

becomes the first equation of the system. For m

1

+1

included and m*

1¼M−m1−1 excluded jointly depen-

dent variables and K

1

included, and K*

1¼K−K1

excluded predetermined variables, Y¼y1Y1Y*

1



and

X¼X1X*

1



are supposed to be variables with

dimension of T×m

1

,T×m*

1,T×K

1

and T×K*

1

corresponding to Y

1

,Y*

1,X

1

and X*

1.γ:1¼1−γ10½

′

TOKER AND ÖZBAY

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