Nearly Unbiased Estimation of Autoregressive Models for Bounded Near‐Integrated Stochastic Processes*
| Author | María Dolores Gadea,Antonio Montañés,Josep Lluís Carrion‐i‐Silvestre |
| Date | 01 February 2021 |
| DOI | http://doi.org/10.1111/obes.12399 |
| Published date | 01 February 2021 |
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©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 83, 1 (2021) 0305–9049
doi: 10.1111/obes.12399
Nearly Unbiased Estimation of Autoregressive
Models for Bounded Near-Integrated Stochastic
Processes*
Josep Llu´
is Carrion-i-Silvestre,†Mar´
ia Dolores Gadea‡ and
Antonio Monta ˜
n´
es§
†AQR-IREA, Department of Econometrics, Statistics, and Applied Economics, University of
Barcelona, Av. Diagonal 690, Barcelona 08034, Spain (e-mail: carrion@ub.edu)
‡Department of Applied Economics, University of Zaragoza, Gran V´
ia 4, Zaragoza 50005,
Spain (e-mail: lgadea@unizar.es)
§Department of Economic Analysis, University of Zaragoza, Gran V´
ia 4, Zaragoza 50005,
Spain (e-mail: amontane@unizar.es)
Abstract
The paper investigates the estimation bias of autoregressive models for bounded near-
integrated stochastic processes and the performance of the standard procedures in the
literature that aim to correct the estimation bias. In some cases, the bounded nature of the
stochastic processes worsens the estimation bias effect.The paper extends two popular au-
toregressive estimation bias correction procedures to cover bounded stochastic processes.
Monte Carlo simulations reveal that accounting for the bounded nature of the stochastic
processes leads to improvements in the estimation of autoregressive models. Finally, an
illustration is given using the unemployment rate of the G7 countries.
I. Introduction
Since the seminal paper by Nelson and Plosser (1982), time series data-based analysis
has frequently begun with the study of the time properties of the variables. This usu-
ally implies the use of some unit root tests, and the statistical inference drawn from their
application is important for subsequent analyses. For instance, a quite popular practice
is to determine the persistence degree of shocks by means of estimating autoregres-
sive models. This provides very interesting insights about the evolution of the variable
being studied, including the analysis of persistence in variables such as real exchange
rates, where some practitioners have studied the number of periods that a shock takes to
JEL Classification numbers: C22, C32, E32, Q43.
*Wethank the editor and two referees for helpful comments and suggestions. The authors gratefully acknowledge
the financial support from the Spanish Ministerio de Ciencia y Tecnolog´
ia, Agencia Espa˜nolade Investigaci ´on (AEI)
and European Regional DevelopmentFund (ERDF, EU) under g rants ECO2015-65967-R (A. Monta˜n´es),ECO2017-
83255-C3-1-P (AEI/ERDF, EU) (J. L. Carrion-i-Silvestre and M. D.Gadea) and ECO2016-81901-REDT.
274 Bulletin
vanish – see Balli, Murray and Papell (2014), among others. Similarly, Watson (2014)
studies the effect of the Great Recession on inflation persistence. This type of analysis,
however, is not straightforward given that we should take into account that the ordinary
least squares (OLS) estimator is consistent but biased in finite samples, and this bias must
be removed in order to appropriately measure the degree of persistence.There are various
proposals in the literature which try to correct this finite sample bias. We can cite here
the contributions of Andrews (1993), Andrews and Chen (1994), Kilian (1998), Hansen
(1999), Rossi (2005) and Perron and Yabu (2009a), among others, which develop different
valid techniques to remove the estimation bias.
However, some commonly employed variables may be affected by the presence of
bounds. Macroeconomic variables such as nominal interest rates, unemployment rates,
exchange rates and the great ratios, among others, are bounded by definition, preventing
them from exhibiting a large variance. This feature generates tension in the statistical
inference associated with standard unit root tests and, hence, the estimation of the degree
of persistence of shocks. The standard order of integration analysis of time series considers
that an I(1) non-stationary stochastic process can vary freely within the limit, that is,
the constraints that impose the existence of bounds are ignored. The behaviour of these
types of variables might seem to be stationary when, in fact, they are non-stationary. In
this regard, Cavaliere (2005) and Cavaliere and Xu (2014) show that standard unit root
tests might reach misleading conclusions if the bounded nature of the time series is not
accounted for. Therefore, it is recommendable to analyse the influence of these bounds on
the determination of time series properties.
The goal of this paper is to assess whether the use of bias-corrected autoregressive
parameter estimates allows us to obtain more accurate empirical economic analyses that
build upon the computation of statistics such as shock persistence measures or long-run
variance (LRV) estimates. To address this issue, the paper investigates the performance of
some of the popular estimation bias correction methods mentioned above when applied to
bounded near-integrated stochastic processes. First, wefocus on some standard procedures,
showing that, in general, the amount of estimation bias that is corrected is small when the
bounded nature of the time series is ignored. This suggests the need to extendthese standard
procedures to incorporate the effect of the bounds on the estimation of autoregressive
models for persistent time series.
The paper proceeds as follows. Section II describes the model for bounded (near-
integrated and integrated) stochastic processes. Section III motivates the analysis show-
ing that standard bias correction methods give poor estimates when applied to bounded
stochastic processes. This leads us to propose in section IV an extension of bias cor-
rection methods that considers this feature. Section V analyses the finite sample per-
formance of the suggested approaches. Section VI provides an empirical illustration,
focusing on the unemployment rate persistence of the G7 countries. Finally, section VII
sets out the conclusions. The proofs and supplementary material are collected in the
appendix.
II. The model
Let xtbe a stochastic process with a data generating process (DGP) given by:
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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