Improvement of the Liu‐type Shiller estimator for distributed lag models

• Author: Nimet Özbay,Selahattin Kaçıranlar
• Published date: 01 November 2017
• DOI: http://doi.org/10.1002/for.2469
• Date: 01 November 2017

RESEARCH ARTICLE

Improvement of the Liu‐type Shiller estimator for distributed lag

models

Nimet Özbay | Selahattin Kaçıranlar

Department of Statistics, Faculty of Science

and Letters, Çukurova University, Adana,

Turkey

Correspondence

Nimet Özbay, Department of Statistics,

Faculty of Science and Letters, Çukurova

University, Adana 01330, Turkey.

Email: nturker@cu.edu.tr

Abstract

The problem of multicollinearity produces undesirable effects on ordinary least squares

(OLS), Almon and Shiller estimators for distributed lag models. Therefore,we introduce

aLiu‐type Shiller estimator to deal with multicollinearity for distributed lag models.

Moreover, we theoretically compare the predictive performance of the Liu‐type Shiller

estimator with OLS and the Shiller estimators by the prediction mean square error crite-

rion under the target function. Furthermore, an extensive Monte Carlo simulation study

is carried out to evaluate the predictive performance of the Liu‐type Shiller estimator.

KEYWORDS

distributed lag model, Liu‐type Shillerestimator, prediction mean square error,target function

1|INTRODUCTION

Consider the finite distributed lag model

yt¼β0xtþβ1xt−1þ…þβpxt−pþut

¼∑

p

i¼0

βixt−iþut;t¼pþ1;…;T;(1)

where β

i

are the unknown distributed lag coefficients or

weights, x

t−i

are lagged values of the explanatory variable,

u

t

are independent normal random disturbances with expecta-

tion E(u

t

) = 0 and variance Vu

t

ðÞ¼σ2

u, and pis the lag

length. This model implies that the current value of y

t

depends not just on x

t

but also on some past values (say p

in number) of x

t

. The model in Equation 1 can be written in

matrix notation as

y¼Xβþu;(2)

where

X¼

xpþ1xp…x1

xpþ2xpþ1…x2

⋮⋮⋱⋮

xTxT−1…xT−p

2

6

6

6

6

6

4

3

7

7

7

7

7

5

;

y′¼ypþ1;ypþ2;…;yT



;

β′¼β0;β1;…;βp



;and u′¼upþ1;upþ2;…;uT



The ordinary least squares (OLS) estimator of βin model

2is ^

β¼X′XðÞ

−1X′y. In economic applications, since the

invariably lagged values of an explanatory variable are highly

collinear, the OLS estimates have high variances. The most

popular method employed in this context is the incorporation

of extraneous information by specifying a lag distribution. A

lag distribution function gives the magnitude of the

coefficient of a lagged explanatory variable, expressed as a

function of the lag (see also Fisher, 1937; Kennedy, 2003;

Vinod & Ullah, 1981).

A popular method of reducing the effect of

multicollinearity is proposed by Almon (1965). It is assumed

that the lag weights can be represented by a polynomial of

degree r:

βi¼α0þα1iþα2i2þ…þαrir;p≥r≥0 (3)

or β¼Aα(4)

where α′=(α

0

,α

1

,…,α

r

) and A¼

10 0 …0

11 1 …1

⋮⋮ ⋮ ⋱ ⋮

1pp

2⋯pr

2

6

6

6

4

3

7

7

7

5

are α:(r+ 1) × 1 vector and A:(p+1)×(r+1) matr ix,

respectively. The ranks of matrices Xand Aare assumed to

be (p+1)T−p) and (r+1)p+1). If r

then the rank

of Ais r<1.

Received: 6 December 2016 Accepted: 25 February 2017

DOI: 10.1002/for.2469

776 Copyright © 2017 John Wiley & Sons, Ltd. Journal of Forecasting. 2017;36:776–783.wileyonlinelibrary.com/journal/for