Transcript of "Analyzing Emerging Market Business Cycles"
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Analyzing economic fluctuations in
emerging market economies
Carlos C. Bautista
Abstract
Economic fluctuations are analyzed by economists in order to understand how the economy
responds to shocks caused by both the external environment and local events (both political and
economic). Adequate knowledge of the causes and consequences of these shocks on the economy
allows authorities to design policies that help dampen its adverse effects. In advanced economies,
this field is a rich area of research known as empirical business cycle analysis. There is however a
dearth of studies of this kind in emerging market economies. This research adopts the
econometric techniques used in developed economy empirical business cycle research to examine
emerging market economies.
In the absence of recession dating committees like the NBER in the US, the study makes use of
Markov-switching (MS) regressions to establish and to date periods of rapid growth, moderate
growth and crisis episodes in emerging market economies. Information about the state of the
economy per period obtained from MS regressions is used to date regimes. A series representing
the state of the economy, Yt, is constructed using MS smoothed probabilities such that Yt = s if
p t (s ) = max{p t (1), p t (2 ), p t (3)}; where pt (1) + pt (2 ) + pt (3 ) = 1 . This series is used in ordered
probit regressions to model the probability of occurrence of regimes, conditional on changes in a
set of macroeconomic variables (e.g., exchange rates, interest rates, foreign exchange reserves and
money supply.) Quarterly data from 1986 to 2005 are used in the study. The countries where data
of suitable length are available to the author are used in the study. These are Korea, Malaysia,
Philippines, Chile and Mexico. The models constructed seem to perform adequately as can be
seen by their tracking ability and out-of-sample confirmation of rapid and moderate growth
phases experienced by the countries under study.
Keywords: Markov-switching, ordered probit, rapid growth, moderate growth, crisis episodes
JEL Classification: E32
January 2007
First Draft
Correspondence:
Carlos C. Bautista Tel +63 2 928 4571
College of Business Administration Fax +63 2 920 7990
University of the Philippines E–mail ccbautista@gmail.com
Diliman, Quezon City 1101, Philippines Web www.upd.edu.ph/~cba/bautista
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Analyzing economic fluctuations in
emerging market economies
Carlos C. Bautista
1 Introduction
Approximately two decades ago, one can unambiguously determine whether an economy is
highly developed or less developed. This convenient dichotomy has been rendered obsolete to a
large extent with globalization and advances in technology that led to increases in productivity.
The subsequent growth in per capita incomes in some less developed economies created new
markets and another class of economies that is neither highly developed nor less developed – the
emerging market economies. These economies that liberalized and opened their borders
experienced unprecedented growth and significant improvements in the standards of living of
their citizens.
The growth paths of these economies were not easy ones to trek however as they try to cope
with changes in the international environment given their inadequate institutional structures
(inefficient banking systems for example). For a number of them, these led to economic crises
emanating from the external sector. Researchers of aggregate economic fluctuations in emerging
market economies find this an interesting but difficult phenomenon to analyze because the
resulting patterns of aggregate fluctuations do not lend themselves well to standard statistical
analysis that assume linearity. Indeed, one will notice the dearth of studies examining emerging
market economic fluctuations. This research tries to fill in the gap and hopes to contribute to the
body of knowledge on the analysis of fluctuations in developing economies.
Recently developed macro-econometric techniques that try to deal with non-linear
relationships have been applied successfully in empirical business cycle research in highly
developed economies (See Hamilton and Raj, 2002, for a survey). This area of Macroeconomics
has been one of the most dynamic fields of research because far richer insights as to how the
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economy operates have been obtained with these techniques. This study adopts these methods to
examine economic fluctuations in emerging market economies especially those which
experienced crisis episodes.
The objectives of the study are (1) to find an adequate statistical representation of the
movements and direction of aggregate economic activity in selected emerging market economies
using techniques in non-linear time series/business cycle analysis and (2) to make these results
useful for policy analysis and in forecasting the direction of economic activity. For each country
in the study, Markov-switching (MS) regression is used to identify the state of the economy per
period over a particular time frame. The novelty in this study is the use of MS regressions in
dating the cycles in these economies. That is, the state of the economy with the highest
probability of occurrence generated by the MS model is taken to be the true state. This is a strong
assumption that needs to be made because of the absence of agencies that officially date
recessions like the NBER of the U.S. The choice of the dating mechanism may be justified by the
excellent record of the MS regression model’s in-sample predictions of recessions in highly
developed economies (See for example, Hamilton (1989) for the U.S.) Moreover, dating
mechanisms other than the NBER official dates of previous studies are not very useful when the
level of classification is more than 2 states. In this study, emerging market economies are
assumed to fall into 3 states: rapid growth, moderate growth and crisis states.
With the regime dates determined from MS regressions, an attempt is made to predict the
probability of occurrence of the state of the economy using ordered probit. These models permit
the specification of a set of predictor variables (e.g., exchange rates, interest rates, foreign
exchange reserves and money supply). This is done for the following countries where data of
suitable length are available: Chile, Korea, Malaysia, Mexico, and the Philippines. The next
section of the study reviews the methods used. Section 3 discusses the empirical strategy, the
data, and the empirical results. The last section concludes.
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2 Methods in empirical business cycle research
A huge literature on non-stationary business cycle analysis has been generated since the
seminal article of Hamilton (1989). His article is among the first to recognize the inherent
nonlinearities that are present in macroeconomic time series describing the time path of the
economy. He finds that an econometric evaluation of GDP growth with regimes endogenously
determined by Markov switches accurately describes the US business cycles and that the dates
generated by his model in fact coincides with the NBER recession dates.
Hamilton’s model was generalized by Lam (1990) to allow for a decomposition of the series
into a trend and a cycle. Lam was also able to replicate the NBER reference dates. Many other
extensions of the univariate MS regressions have been proposed and several applications to a
variety of problems in Macroeconomics and Finance can be found in the literature. Filardo (1994)
extended the Hamilton model to allow for time varying transition probabilities. In his study, the
duration of a state of the economy was made to vary with leading indicator variables. Kim (1994)
improved on Hamilton’s smoothing algorithm; Hamilton and Susmel (1994) studied ARCH
effects with Markov-switching in the US stock market and Krolzig (1997) provided an
implementation of Markov-switching VARs. A collection of more recent contributions can be
found in a volume edited by Hamilton and Raj (2002).
The original work by Burns and Mitchell emphasized 2 aspects of economic fluctuations: co-
movements of macroeconomic variables and characterization of business cycle phases into
expansions and recessions. The development of these ideas proceeded independently of each
other. The Markov-switching literature which focuses on the second aspect, developed alongside
models of co-movement: the dynamic factor models first used by Stock and Watson (1988, 1992)
in their articles on leading indicators. More recent work by Diebold and Rudebusch (1996)
allowed for regime switching in a dynamic factor model, thus allowing for joint analysis of co-
movements and business cycle phases.
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To a certain extent, the recent literature on recession forecasting using qualitative dependent
variable modeling techniques pioneered by Estrella and Mishkin (1998) is related to the
coincident and leading indicator studies by Stock and Watson by the use of exogenous variables
in forecasting regime probabilities. The difference is that the latter does not rely on official
business cycle dates but rather, on the unobserved common components of the indicators
included in the study. Recession forecasting and regime probability modeling are also closely
related to studies on nonstationary business cycle analysis reviewed above but were developed at
a much later date. Both sets of studies use official dates of recessions determined by government
agencies as prediction targets. The main difference is that the latter is a univariate time series
technique while the former allows the utilization of other variables either as leading or coincident
indicators that enter the right-hand side of the forecasting equation.
Qualitative dependent variable models are a natural choice in the recession forecasting
literature because the problem being examined can be conveniently expressed as a choice of two
regimes. For example, zero is the value assigned to a recession and 1 to expansion. Estrella and
Mishkin (1998) make use of a logit model where financial variables are used as leading indicators
to forecast the US recessions. They find that a parsimonious specification is necessary to generate
reasonable predictions of recessions. The study also finds that in-sample and out-of-sample
forecast performance can differ significantly and that out-of-sample predictive performance can
be very dependent on the forecast horizon.
A similar study was done by Bernard and Gerlach (1998) for several European countries, the
US and Japan. Instead of using several financial and aggregate macroeconomic variables as in
Estrella and Mishkin, the term structure was used as the sole predictor of a recession. Recession
dates used for the G7 countries were obtained from a study by Artis et al (1995). Using logit
analysis, Birchenhall et al (1999) and Birchenhall et al (2001) attempted to predict business cycle
regimes for the US and the UK. In the UK study, they found that real money (M4) was the best
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single leading indicator of a recession. In the case of the US, Dueker (2002) points out that there
seems to be some difficulties in predicting the 1990-91 recession even with a Markov switching
probit model.
Developed country analysis of aggregate economic activity diverges from LDC/emerging
market economies analysis because of the occurrence of crisis events which, while infrequent
relative to normal business cycle phases, renders 2-state models inadequate in providing a
realistic description of aggregate economic fluctuations. In almost all LDC studies, the focus is on
the prediction of the crisis and the formulation of early warning systems for use of policy makers
(See for example, Kaminsky and Reinhardt (2000)). A fairly recent application of nested logit in
the prediction of currency crisis was by Lau and Yan (2005). To predict speculative attacks and
determine successful defenses from attacks, they used data from 16 countries and utilized interest
differentials, and monetary and fiscal variables as explanatory variables. Liquidity and financial
fragility variables were found to be excellent predictors of a crisis. Except this last study that used
nested logit, all of the studies reviewed above assume two states of the economy – recession and
expansion - and all of them make use of reference dates and turning points determined by
government agencies of the respective countries or previous studies giving details of regime
histories.
The assumption that the state of the economy falls into two categories, while adequate for
most highly developed economies, may not be appropriate for emerging market economies
which have experienced crisis episodes. Sichel (1994) models the US economy with 3 business
cycle phases with the third phase being associated with a high-growth recovery phase. The
notion of a third phase was further developed by Kim and Murray (2002) who, extending the
Diebold and Rudebusch model, further decomposed recession into its permanent and transitory
components which are governed by Markov switching to be able to examine peak reversion
during the high-growth recovery phase.
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This study follows a different path of analysis by assuming 3 states. As shown in the next
section, 2-state models classify the economy as falling into either a crisis or a non-crisis state,
which is not really interesting. This study seeks a further categorization of the non-crisis, normal
periods into high (or rapid) growth and low (or slow) growth states. 1
As mentioned in the introduction, this study hinges heavily on the assumption that Markov
switching regressions correctly show the true state of the economy. This choice of rule to
determine the state of the economy is arguably the best given that one chooses from 3 states
instead of just 2. One can adopt rules similar to those made in studies analyzing turning points
(e.g., Birchenhall, 2001), but these procedures are not viable when the choice of regimes exceed 2.
The next section further discusses the empirical strategy and the estimation results.
3 Empirical strategy and estimation results
As outlined in the introduction above, MS regressions and ordered probit regressions are used
in sequence to examine economic fluctuations in selected emerging market economies. By
adopting these two techniques an improvement is made over the traditional way of analyzing
fluctuations especially in emerging market economies. Here, three states of the economy are
assumed instead of the two states assumed in the literature reviewed above. These states cover
periods of rapid growth, moderate growth and crisis episodes instead of the recession and
expansion phases. Hence instead of the traditional binary probit or logit regressions the study
uses ordered probit techniques. 2
1 In this study, there is no range by which a particular growth figure can be associated with rapid or
moderate growth. Here, rapid or moderate growth of a country is relative its own historical record. In this
regard cross country comparisons cannot be done using this terminology because a country’s rapid growth
figure could fall in the range of moderate growth in another.
2Detailed discussions of the econometric techniques used here can be found in graduate textbooks; very
brief expositions are relegated to the appendix.
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From the univariate MS regressions of output growth, the state of the economy per period is
obtained and is assumed to be any of the three growth states mentioned above. The latent
variable used in the ordered probit models is then mapped using the probabilities of occurrence
of the state of the economy. A simple rule is followed in the determination of the state of the
economy per period. For each period, the state with the highest probability of occurrence,
denoted by Yt, is taken to be the prevailing state of the economy for that period. That is, Yt = s if
p t (s ) = max{p t (1), p t (2 ), p t (3)}; where pt (1) + pt (2 ) + pt (3 ) = 1 .
The quarterly data used in this study come from different sources but a large portion was
obtained from the December 2005 IFS CD-ROM. The main indicator of economic activity in this
study is GDP growth. The MS regressions were estimated using GDP data (not deseasonalized)
from the earliest available data up to the last quarter of 1999. The cutoff date for the estimation
was chosen to be able to capture the effects of the Asian crisis. Post-crisis data are used for out-of-
sample prediction of the state of the economy.
Table 1 shows Hansen’s (1992, 1996) likelihood ratio tests of the null of a one-state AR(k)
model against the alternative of a 2-state Markov regime-switching model. As can be seen, for lag
lengths of 3 and 4, the tests show a rejection of the null hypothesis in most cases. As a
preliminary analysis, 2-state models were estimated and the results are shown in Table 2.
Statistically significant parameter estimates of two-state MS autoregressions for each of the five
countries under study can be seen from the table. It must be note however that 2-state models do
not seem to adequately capture normal movements in economic activity when economic crises
are deep. For example, the impact of the 1983-85 crisis in the Philippines is so huge that it dwarfs
the effects of the Asian crisis, causing a misclassification. These can be observed in Figure 1 which
shows the smoothed probabilities of each state along with actual growth rates using
deseasonalized GDP.
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Next, the study proceeds to estimate 3-state MS autoregressions with maximum lag lengths of
4 as in the 2-state models. It is important to note here that because of the highly non-linear nature
of the problem, unrestricted maximum likelihood estimates of 3-state models are more difficult to
obtain. In this regard, some values of parameters that can be reasonably assumed to take on
boundary values were determined. This is not an unusual procedure and has been employed by
Hamilton (2005) in his study of U.S. unemployment. Inspection of the time series reveals that in
most cases, the movements in GDP growth around crisis periods show no abrupt decline from a
position of rapid growth to negative growth which indicates a crisis event. Rather, growth often
slows a bit before going into a crisis episode. This is also true when the economy is coming out of
the crisis – the economy does not shift immediately to a rapid growth path but has to climb
slowly out of the bottom. Hence, letting s = 1 as the state indicating rapid growth, s = 2 indicating
moderate growth and s = 3 representing a crisis state, it seems reasonable to assume that state 1 is
never followed by state 3 and vice versa. More precisely, let pij represent the transition
probabilities. This amounts to an assumption that p13 = p31 = 0.
Table 3 reports the results of 3-state MS regression estimates. For each country, unrestricted
maximum likelihood estimates were computed and convergence was obtained for Chile,
Malaysia and the Philippines. The restrictions mentioned above were imposed for Korea and
Mexico in order to attain convergence. Table 4 shows the corresponding transition probability
matrix for each country estimate. To get an idea of how the 3-state MS model tracks economic
activity, Figure 2 plots the smoothed probabilities on the left scale and GDP growth on the right
scale. It is seen that this performs better than the 2-state model.
Table 5 presents the estimates of ordered probit models for each country. The study tries to
limit itself to major macroeconomic variables: nominal exchange rate, general price level, money
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supply, and interest rate which appear in all country estimates. 3 In some cases however, other
variables which are deemed important to the economy are included, e.g., for Mexico the price of
oil, foreign exchange reserves and country risk premium, defined as the difference between the
interest differential and the actual depreciation rate, were used. Also these variables percent
changes either on a quarterly or annually basis. The final estimates shown in the table were
chosen based on the pseudo-R2 and the significance of the coefficients.
Figure 3 shows the fitted probabilities from the ordered probit estimates. For reference, the
annual GDP growth is also plotted as broken lines. The shaded portion of the graphs covers the
out-of-sample forecasts from 2000 to 2005. It is clear from the diagrams that no crisis events took
place in the countries under study during this period. There are however fluctuations in the
growth patterns in some of them. For Chile for example, the probability of rapid growth is
highest for the first half of 2003 and the last quarter of 2004 and first quarter of 2005. Korea’s
growth pattern shifted between moderate and rapid from 2000 to the end of 2002 and then
remained in moderate growth mode. Malaysia remained at the rapid growth state until it
experienced a slowdown in exports in 2001. This is reflected in the shift from rapid to moderate
growth in the third quarter of that year. For the Philippines, deviations from rapid growth took
place in the first quarter of 2001 when a radical change in the country’s leadership occurred as
seen in the diagram.
4 Concluding remarks
This study has demonstrated that it is possible to adopt econometric tools used to analyze
fluctuations in advanced economies to emerging market economies. The results above show that
the models constructed are able to track the movements of aggregate economic activity fairly well
and have satisfactory out-of-sample performance. This exploratory model building exercise make
3 Unit root tests of these variables are shown in the Appendix.
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use data from five selected emerging market economies. This procedure outlined in this paper
can be adapted to other emerging market economies as well. The models above can be used for
policy analysis because it makes use of macroeconomic variables to track and predict growth. The
results show to some extent the usefulness of these macroeconomic variables in predicting
slowdowns. Also simulations can be done to determine under what combination of values of
these variables can lead to crisis situations – similar to early warning systems. This is not done in
this study but is an area of further research.
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countries, CEPR Discussion Paper 1137.
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Bautista C, (2003), Estimates of output gaps in four Southeast Asian countries, Economics Letters
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Analysis, Philippine Review of Economics.
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Figure 1: Two-state Markov switching smoothed probabilities and GDP growth
Chile Chile
1.0 1.0
10 10
0.5 0.5
0 0
0.0 0.0
86:1 88:1 90:1 92:1 94:1 96:1 98:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1
normal growth state GDP Growth low growth state GDP Growth
Korea Korea
1.0 1.0
10 10
0.5 0.5
0 0
0.0 0.0
75:1 80:1 85:1 90:1 95:1 75:1 80:1 85:1 90:1 95:1
normal growth state GDP Growth low growth state GDP Growth
Malaysi a Malaysi a
1.0 10 1.0 10
0.5 0 0.5 0
0.0 -10 0.0 -10
90:1 91:1 92:1 93:1 94:1 95:1 96:1 97:1 98:1 99:1 90:1 91:1 92:1 93:1 94:1 95:1 96:1 97:1 98:1 99:1
normal growth state GDP Growth low growth state GDP Growth
Mexico Mexico
1.0 1.0
0.5 0 0.5 0
0.0 0.0
82:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 82:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1
normal growth state GDP Growth low growth state GDP Growth
Philippi nes Philippi nes
1.0 10 1.0 10
0.5 0 0.5 0
0.0 -10 0.0 -10
84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1
normal growth state GDP Growth low growth state GDP Growth
19.
Figure 2: Three-state Markov switching smoothed probabilities and GDP growth
Chile Korea Malaysia
1.0 1.0 1.0 10
10
10
0.5 0.5 0.5 0
0
0
0.0 0.0 0.0 -10
86:1 88:1 90:1 92:1 94:1 96:1 98:1 75:1 80:1 85:1 90:1 95:1 90:1 91:1 92:1 93:1 94:1 95:1 96:1 97:1 98:1 99:1
rapid growth state GDP Growth rapid growth state GDP Growth rapid growth state GDP Growth
Chile Korea Malaysia
1.0 1.0 1.0 10
10
10
0.5 0.5 0.5 0
0
0
0.0 0.0 0.0 -10
86:1 87:1 88:1 89:1 90:1 91:1 92:1 93:1 94:1 95:1 96:1 97:1 98:1 99:1 72:1 74:1 76:1 78:1 80:1 82:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 90:1 91:1 92:1 93:1 94:1 95:1 96:1 97:1 98:1 99:1
moderate growth state moderate growth state moderate growth state
GDP Growth GDP Growth GDP Growth
Chile Korea Malaysi a
1.0 1.0 1.0 10
10
10
0.5 0.5 0.5 0
0
0
0.0 0.0 0.0 -10
86:1 88:1 90:1 92:1 94:1 96:1 98:1 75:1 80:1 85:1 90:1 95:1 90:1 91:1 92:1 93:1 94:1 95:1 96:1 97:1 98:1 99:1
crisis state GDP Growth crisis state GDP Growth crisis state GDP Growth
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Figure 2, continued
Mexico Philippines
1.0 1.0 10
0
0.5 0.5 0
0.0 0.0 -10
82:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1
rapid growth state GDP Growth rapid growth state GDP Growth
Mexico Philippines
1.0 1.0 10
0
0.5 0.5 0
0.0 0.0 -10
82:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1
moderate growth state moderate growth state
GDP Growth GDP Growth
Mexico Philippi nes
1.0 1.0 10
0
0.5 0.5 0
0.0 0.0 -10
82:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1
crisis state GDP Growth crisis state GDP Growth
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Figure 3: In-sample and out-of-sample fitted probit probabilities
Chile Korea Malaysia
1.0 1.0 1.0 10
10
10
0.5 0.5 0.5 0
0
0
0.0 0.0 0.0 -10
88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1 80:1 85:1 90:1 95:1 00:1 05:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1
rapid growth state GDP Growth rapid growth state GDP Growth rapid growth state GDP Growth
Chile Korea Malaysia
1.0 1.0 1.0 10
10
10
0.5 0.5 0.5 0
0
0
0.0 0.0 0.0 -10
88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1 78:1 80:1 82:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1
moderate growth state moderate growth state moderate growth state
GDP Growth GDP Growth GDP Growth
Chile Korea Malaysi a
1.0 1.0 1.0 10
10
10
0.5 0.5 0.5 0
0
0
0.0 0.0 0.0 -10
88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1 80:1 85:1 90:1 95:1 00:1 05:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1
crisis state GDP Growth crisis state GDP Growth crisis state GDP Growth
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Figure 3, continued
Mexico Philippines
1.0 1.0 10
0
0.5 0.5 0
0.0 0.0 -10
86:1 88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1
rapid growth state GDP Growth rapid growth state GDP Growth
Mexico Philippines
1.0 1.0 10
0
0.5 0.5 0
0.0 0.0 -10
86:1 88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1
moderate growth state moderate g rowth state
GDP Growth
GDP Growth
Mexico Philippi nes
1.0 1.0 10
0
0.5 0.5 0
0.0 0.0 -10
86:1 88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1 84:1 86:1 88:1 90:1 92:1 94:1 96:1 98:1 00:1 02:1 04:1
crisis state GDP Growth crisis state GDP Growth
21
23.
Appendix
Markov-Switching Regression
The univariate MS regression model used in this study is of the form:
1) ( ) ( )
y t = μ st + φ1 y t − 1 − μ st − 1 + ... + φ k y t − k − μ st − k + ε t
where yt is the variable of interest; in this study, this variable is output growth; the φks are the k
autoregression parameters and εt is a white noise process. μ st is the mean of yt when the
economy is in state st. In this study, the state of the economy is assumed to be the outcome of an
unobserved first-order 3-state Markov process (i.e., st = 1, 2, 3). Its evolution can be described by
( )
transition probabilities, Pr st = j st − 1 = i = p ij , that can be written in matrix form:
⎡ p 11 p 21 p 31 ⎤
2) P = ⎢ p 12
⎢ p 22 p 33 ⎥
⎥
⎢ p 13
⎣ p 23 p 33 ⎥
⎦
3
where ∑p
j=1
ij = 1 . Each element shows the probability that state i is followed by state j. The
process is assumed to depend on past values of yt and st only through st–1. Note that since only yt
is observed but not the state of the economy, a way must be found to form optimal inferences
about the current state based on the observed values of yt. Given the number of states, Hamilton
(1989) shows how to estimate the parameters of the model and the transition probabilities
governing the motion of the variable of interest. He provides a recursive method for drawing
probabilistic inferences about what state the economy is in (the value of st) given the history of yt.
This is the basic MS regression that is going to be utilized in the proposed study to establish
regime dates. As mentioned in the review above, several extensions of the basic model have been
done since then (see also Kim and Nelson (1999).) A three-state model has been estimated for the
Philippines by Bautista (2002).
Ordered Response Models
In ordered response models, one can specify a latent variable, y t* , that are assumed to be
influenced by a set of exogenous variables. Suppose there are K exogenous variables denoted by
xkt, where k = 1, …, K. Then one can write:
3) y t* = b 0 + b 1 x 1t + ... + b K x Kt + ε t = z t + ε t
εt is a disturbance term. The latent variable, y t* , can be mapped onto an ordered categorical
variable:
Yt = 1 if a 0 < y t* ≤ a 1
4) Yt = 2 if a 1 < y t* ≤ a 2
Yt = 3 if a 2 < y t* ≤ a 3
24.
Analyzing economic fluctuations in emerging market economies | cc bautista
where a0, a1, a2 and a3 are thresholds that serve to determine the value of Yt to be given to the
latent variable. To preserve the ordering, these thresholds that are to be estimated
econometrically along with the coefficients of equation (4) must satisfy: a0 > a1 > a2 > a3. The latent
variable’s boundary values are unknown. Hence, one can simply set the beginning and ending
thresholds to minus and plus infinity respectively (in this case, a0 = –∞ and a3 = +∞) and need not
be estimated. From the above expressions, one can write the ordered regression model as:
( )
Pr Yt = 1 x 1t , ..., x Kt = Pr(ε t ≤ a 1 − z t )
= F (a 1 − z t )
5)
( )
Pr Yt = 2 x 1t , ..., x Kt = Pr(a 1 − z t < ε t ≤ a 2 − z t )
= F (a 2 − z t ) − F (a 1 − z t )
( )
Pr Yt = 3 x 1t , ..., x Kt = Pr(ε t ≥ a 2 − z t )
= 1 − F (a 2 − z t )
where F denotes the cumulative distribution function of ε. Let there be a total of T sample
periods, (t = 1, …, T), each of which can be treated as a single draw from a multinomial
distribution. Suppose T1, T2 and T3 are the number of periods belonging to states 1, 2 and 3
respectively, with T1 +T2 + T3= T. Then the likelihood of observing the sample is given by:
6) L = F (a 1 − z t )T1 [F (a 2 − z t ) − F (a 1 − z t )]T2 [1 − F (a 2 − z t )]T3
The parameters of the model, the ah’s and the bk’s, can be estimated by maximizing the (log of the)
ˆ
likelihood function given by equation (6). z t can be computed once the b coefficients are
ˆ ˆ
estimated. With estimates of the limit coefficients, a h s , and z t , the probability of being at a
particular state can be predicted for each period t in the sample:
p 1t = F (a 1 − z t )
ˆ ˆ ˆ
7) p 2t = F (a 2 − z t ) − F (a 1 − z t )
ˆ ˆ ˆ ˆ ˆ
p 3t = 1 − F (a 2 − z t )
ˆ ˆ ˆ
ˆ ˆ ˆ
where p 1t + p 2t + p 3t = 1 . The disturbance term, εt, can be assumed to follow a normal or a
logistic distribution to produce either the ordered probit or the ordered logit model, respectively.
A good reference on qualitative and limited dependent variables regression like the above is
Davidson and MacKinnon (1993). Ordered probit and logit models are known to be well-behaved
and are easily implemented using commercially available econometric programs.
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