Identification and Estimation Issues in Exponential Smooth Transition Autoregressive Models
| Author | Daniel Buncic |
| Date | 01 June 2019 |
| DOI | http://doi.org/10.1111/obes.12264 |
| Published date | 01 June 2019 |
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©2018 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 3 (2019) 0305–9049
doi: 10.1111/obes.12264
Identification and Estimation Issues in Exponential
Smooth TransitionAutoregressive Models*
Daniel Buncic
Research and Modelling Division, Financial Stability Department, Sveriges Riksbank,
SE-103 37, Stockholm, Sweden (email: daniel.buncic@riksbank.se, http://www.daniel
buncic.com)
Abstract
Exponential smooth transition autoregressive (ESTAR) models are widely used in the
international finance literature, particularly for the modelling of real exchange rates. We
show that the exponential function is ill-suited as a regime weighting function because of
two undesirable properties. Firstly, it can be well approximated by a quadratic function
in the threshold variable whenever the transition function parameter , which governs the
shape of the function, is ‘small’. This leads to an identification problem with respect to
the transition function parameter and the slope vector, as both enter as a product into the
conditional mean of the model. Secondly, the exponential regime weighting function can
behave like an indicator function (or dummy variable) for very large values of . This has
the effect of ‘spuriously overfitting’ a small number of observations around the location
parameter . We show that both of these effects lead to estimation problems in ESTAR
models. We illustrate this by means of an empirical replication of a widely cited study, as
well as a simulation exercise.
I. Introduction
The Exponential Smooth Transition Autoregressive (ESTAR) model has become one of
the workhorse econometric models in the international finance literature, particulary for
the modelling of real exchange rates. ESTAR models were introduced by Granger and
Ter ¨asvirta (1993) and Ter¨asvirta (1994) into the economics literature as a generalization
JEL Classification numbers: C13, C15, C50, F30, F44.
*I am grateful to the editor, twoanonymous referees as well as Lorenzo Camponovo,Adrian Pagan, Paolo Giordani,
Timo Ter¨asvirta, Xin Zhang, Dick van Dijk, Paul Karehnke (Discussant), Pedro Barroso, Simon van Norden, Robert
Taylor,Jonas Striaukas, Leonardo Iania, Annastiina Silvennoinen, Irina Panovska, Luc Bauwens, Fabio Canova and
seminar and conference participants at the Universit´e catholique de Louvain (CORE), the University of St. Gallen,
Sveriges Riksbank, theAustralasian Finance and Banking Conference in Sydney (2017), the International Conference
onTime Series Econometrics in Granada (2017), the 23rd EBES Conference in Madrid (2017), the 26th Annual SNDE
Symposium in Tokyo (2018), and the IAAE 2018 Annual Conference in Montreal for their comments that helped
to improve the paper. Parts of this paper were written while I was visiting the Research Division of the Monetary
PolicyDepar tment at the Riksbank. I thank Jesper Lind´e and his colleagues for their hospitality and many stimulating
discussions.The opinions expressed in this article are the sole responsibility of the author and should not be interpreted
as reflecting the views of Sveriges Riksbank.
668 Bulletin
of the (nonlinear) exponential autoregressive model of Haggan and Ozaki (1981) and
threshold time series models of Tong (1983).
Despite being widely used, the exponential function employed in ESTAR models is
not suitable as a regime weighting function. The reason for this being two undesirable
features of the exponential function. The first is that for small values of the transition
function parameter , the exponential function can be well approximated by a quadratic in
the threshold variable zt.This leads to the slope vector attached to the nonlinear regime and
the parameter entering as a product into the nonlinear conditional mean of the model,
which leads to identification issues. What is particularly problematic with this scenario is
that it is not a small sample issue but rather a property of the model and is evident even in
very large samples of 5,000 observations.
The second undesirable feature is that for extremely large values of , the exponential
weighting function will be equal to unity for nearly all values of the transition variable zt,
except when ztis equal to the location parameter . The effect of this on the model is that
only a very small number of observations around receive a weight that is not equal to
unity, which leads to an ‘outlier fitting effect’of the exponential function. Although this is
evident in ‘small’ samples only, our simulation results indicate that it can be pervasive for
sample sizes as large as 500 observations, resulting in ‘large’ estimates for over 70% of
the simulations.
There exists ample evidence of these problems in empirical studies. For instance,
Michael, Nobay and Peel (1997) fit ESTAR models to real exchange rate data for a number
of countries. In panels (a) and (b) of Figure 1 on page 875 in their paper, one can see that for
the UK–US series, the weighting function remains wellbelow 0.3 for the entire range of the
data, while for the UK-France series, only 4 data points receive a weight in excess of 0.3,
with both functions being quadratic looking in shape. Taylor and Peel (2000) use monetary
fundamentals to study the evolution of exchange rates and utilize the ESTAR model to
capture nonlinearities in the data. From the regime weighting functions plotted in Figure 2
on page 45 of their paper, it can be seen that the transition function weights remain below
0.4 over the entire range of the data and are again quadratic looking. The study by Baum,
Barkoulas and Caglayan (2001) provides even stronger symptoms of a weakly identified
model. The estimates of the transition function parameter that are reported in Tables 4
and 5 on page 391 of Baum et al. (2001) are — with the exception of the WPI-based real
exchange rate for Norway — between 0.0042 and 0.0833! The corresponding transition
function plots, on pages 392 and 393, show a quadratic shape.
Similar issues are evident in Sarantis (1999), Taylor, Peeland Sar no (2001), Kilian and
Taylor (2003), Kapetanios, Shin and Snell (2003), Sarno, Leon and Valente (2006), Payaand
Peel (2006), Sollis (2008), Taylor and Kim (2009), Cerrato, Kim and Macdonald (2010),
Pavlidis, Paya and Peel (2011), Beckmann, Berger and Czudaj (2015) and many others.
The study by Beckmann et al. (2015) is particularly noteworthy to single out here, as the
estimation results reported in Table 3 of their paper provide first hand empirical evidenceof
both estimation problems that we outline above. Beckmann et al. (2015) estimate ESTAR
models on gold returns, using stock returns from 23 different equity markets as regressor
and threshold variables. The model is complicated by the addition of a GARCH type
volatility process on the error term in the ESTAR models that are fitted. From Table 3 on
page 22 in their paper, we can see that the estimates of the transition function parameter
©2018 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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