Lecture 4Testing and Estimating Structural Breaks in Time Series.pptx

Testing and Estimating Structural
Breaks in Time Series
Hrushikesh Mallick

• Nelson and Plosser (1982) argue that almost all
macroeconomic time series one typically uses have a unit root.
• The presence or absence of unit roots helps to identify some
features of the underlying data generating process of a series.
• In the absence of unit root (stationary), the series fluctuates
around a constant long-run mean and implies that the series
has a finite variance which does not depend on time.
• On the other hand, non-stationary series have no tendency to
return to long-run deterministic path and the variance of the
series is time dependent.
• Non-stationary series suffer permanent effects from random
shocks and thus the series follow a random walk.

• The main thrust of the unit root literature concentrates
on whether time series are affected by transitory or
permanent shocks. This can be tested by the ADF model.
• which is primarily concerned with the estimate of lagged
coefficient ‘α’.
• Schwarz Bayesian Criterion (SBC) and Akaike Information
Criterion (AIC) are used to determine the optimal lag
length or k.
• Non-rejection of the null hypothesis implies that the
series is non-stationary; whereas the rejection of the
null indicates the time series is stationary.

• The debate on unit root hypothesis underwent renewed
interest following the important findings of Nelson and
Plosser (1982).
• The traditional view of the unit root hypothesis was that
the current shocks only have a temporary effect and the
long-run movement in the series is unaltered by such
shocks.
• The most important implication under the unit root
hypothesis sparked by Nelson and Plosser (1982) is that
the random shocks have permanent effects on the long-
run level of macroeconomics; that is the fluctuations are
not transitory

Identifying structural change
• Identifying structural change is a crucial step in the
analysis of time series and panel data.
• The longer the time span, the higher the likelihood
that the model parameters have changed as a result
of major disruptive events, such as the 2007–2008
financial crisis and the 2020 COVID–19 outbreak.
• Detecting the existence of breaks, and dating them is
therefore necessary not only for estimation
purposes but also for understanding drivers of
change and their effect on relationships.

Change points/structural breaks
• A key assumption here is that the coefficients do not
change over time.
• This assumption is unlikely to hold, especially for longer
periods of time, because of major disruptive events, such
as financial crises.
• Parameter instability can have a detrimental impact on
estimation and inference, and can lead to costly errors in
decision-making.
• The times in which the parameters change are called
“change points” in the statistics literature and “structural
breaks” in economics.

• The literature concerning the data generating process of
macroeconomic time series has been growing since Nelson
and Plosser (1982), the most frequently cited study on the
U. S. macroeconomic data.
• Ghatak (1996) had carried out tests of unit root hypothesis
on some macroeconomic data covering a period from 1900 -
1988 in the Indian economy by following the methodologies
developed by Nelson and Plosser (1982).
• In her exercise, she incorporated the possibility of structural
breaks by choosing break points exogenously on the basis of
some political and economic importance of the country and
by applying the methodology of Perron (1989).

• But in reality one is uncertain about when a
structural change might actually have taken place.
• There may be no guarantee, for example, that the
break point in the GDP series in India would exactly
coincide to year 1951, one of the break points
chosen by Ghatak, when economic planning was
introduced in the country.
• If break point is unknown, the use of the
conventional Chow test or the dummy variable test
for determining structural break becomes futile.

• Most of the studies on economic growth in India
have reached at some conclusions without
analyzing the time series behaviour of the data
series (Wallack 2003, Pangariya 2004, Rodrik et al.
2004, Nagaraj 2006, Balakrishnan et al. 2007).
• Testing the presence of unit root behaviour in the
macroeconomic data of a less developed country
like ours will hopefully throw a new light on the
Indian growth process.

• We have to take care of the unit root behaviour
of the time series of NDP and its major sectors,
namely, agriculture, manufacturing and services.
• This analysis can also be extended across regions
of the country by performing some tests with
the time series data on states’ domestic products
for major states.
• We would allow break points on the growth path
to be unknown

Stochastic Trend and Unit-Root Behaviour
• Any macroeconomic time series may contain
either deterministic trend or stochastic trend or
both.
• Implications of them are qualitatively different.
• A time series with deterministic trend follows
trend stationary process (TSP), while a non-
stationary time series showing stochastic trend
is a difference stationary process (DSP).

Deterministic and Stochastic trend
• A trending mean is a common violation of stationarity. There are two popular models for non-
stationary series with a trending mean.
• Trend stationary: The mean trend is deterministic. Once the trend is estimated and removed from the
data, the residual series is a stationary stochastic process.
• Difference stationary: The mean trend is stochastic. Differencing the series D times yields a stationary
stochastic process.
• The distinction between a deterministic and stochastic trend has important implications for the long-
term behavior of a process:
• Time series with a deterministic trend always revert to the trend in the long run (the effects of shocks
are eventually eliminated). Forecast intervals have constant width.
• Time series with a stochastic trend never recovers from shocks to the system (the effects of shocks are
permanent). Forecast intervals grow over time.
• Unfortunately, for any finite amount of data there is a deterministic and stochastic trend that fits the
data equally well (Hamilton, 1994). Unit root tests are a tool for assessing the presence of a stochastic
trend in an observed series.
• Time series that can be made stationary by differencing are called integrated processes. Specifically,
when D differences are required to make a series stationary, that series is said to be integrated of
order D, denoted I(D). Processes with D ≥ 1 are often said to have a unit root.
• You want to de-trend a trend stationary process ( by trend stationarity they mean that there's a slope
and it's due to time ) and difference a difference stationary process.
• Deviations from the trend are then viewed as transitory, the trend being the attractor to which the
series reverts given sufficient time for adjustment. This is the trend stationary view of the generation
of time series.

• Since testing of unit roots of a series is a precondition to the
existence of cointegration relationship, originally, the Augmented
Dickey-Fuller (1979) test was widely used to test for stationarity.
• However, Perron (1989) showed that failure to allow for an
existing break leads to a bias that reduces the ability to reject a
false unit root null hypothesis.
• To overcome this, Perron proposed allowing for a known or
exogenous structural break in the Augmented Dickey-Fuller (ADF)
tests.
• Following this development, many authors including, Zivot and
Andrews (1992) and Perron (1997) proposed determining the
break point ‘endogenously’ from the data.
• Lumsdaine and Papell (1997) extended Zivot and Andrews (1992)
model to accommodate two structural breaks.

Unit Root Tests in the presence of Structural Break
• The debate on unit root hypothesis underwent renewed
interest following the important findings of Nelson and
Plosser (1982).
• The traditional view of the unit root hypothesis was that the
current shocks only have a temporary effect and the long-
run movement in the series is unaltered by such shocks.
• The most important implication under the unit root
hypothesis sparked by Nelson and Plosser (1982) is that the
random shocks have permanent effects on the long-run level
of macroeconomics; that is the fluctuations are not
transitory.
2
-
t
Y


• These findings were challenged by Perron (1989), who argues
that in the presence of a structural break, the standard ADF
tests are biased towards the non-rejection of the null
hypothesis.
• Perron argues that most macroeconomic series are not
characterized by a unit root but rather that persistence arises
only from large and infrequent shocks, and that the economy
returns to deterministic trend after small and frequent shocks.
• According to Perron, ‘Most macroeconomic time series are
not characterized by the presence of a unit root.
• Fluctuations are indeed stationary around a deterministic
trend function. The only ‘shocks’ which have had persistent
effects are the 1929 crash and the 1973 oil price shock’

• Perron’s (1989) procedure is characterized by a single exogenous
(known) break in accordance with the underlying asymptotic
distribution theory.
• Perron uses a modified Dickey-Fuller (DF) unit root tests that
includes dummy variables to account for one known, or exogenous
structural break. The break point of the trend function is fixed
(exogenous) and chosen independently of the data.
• Perron’s (1989) unit root tests allows for a break under both the
null and alternative hypothesis.
• These tests have less power than the standard DF type test when
there is no break.
• However, Perron (2005) points out that they have a correct size
asymptotically and is consistent whether there is a break or not.
Moreover, they are invariant to the break parameters and thus
their performance does not depend on the magnitude of the break.

• Based on Perron (1989), the following three
equations are estimated to test for unit root. The
equations take into account the existence of three
kinds of structural breaks:
• a ‘crash’ model (1) which allows for a break in the
level (or intercept) of series;
• a ‘changing growth’ model (2), which allows for a
break in the slope (or the rate of growth);
• and lastly one that allows both effects to occur
simultaneously, i.e one time change in both the
level and the slope of the series (3).

)
1
.........(
)
( 1
1
1
0 t
t
p
i
t
t
t
t e
x
x
t
DTB
d
DU
x 






 
 




)
2
...(
..........
1
1
*
1
0 t
t
p
i
t
t
t e
x
x
t
DT
x 





 
 




)
3
.........(
)
( 1
1
1
1
0 t
t
p
i
t
t
t
t
t e
x
x
t
DT
DTB
d
DU
x 







 
 





Where the intercept dummy DUt represents a change in the level; DUt
=1 if (t > TB) and zero otherwise;
The slope dummy DTt (also DTt*) represents a change in the slope of
the trend function; DT* = t-TB (or DTt *= t if t > TB) and zero
otherwise;
The crash dummy (DTB) = 1 if t = TB +1, and zero otherwise; and TB is
the break date.
Perron (1989)

• Each of the three models has a unit root with a
break under the null hypothesis, as the dummy
variables are incorporated in the regression
under the null.
• The alternative hypothesis is a broken trend
stationary process.

• However, Perron’s known assumption of the break date
was criticized, most notably by Christiano (1992) as ‘data
mining’.
• Christiano argues that the data based procedures are
typically used to determine the most likely location of the
break and this approach invalidates the distribution
theory underlying conventional testing.
• Since then, several studies have developed using different
methodologies for endogenously determining the break
date.

• Some of these include Banerjee, Lumisdaine
and Stock (1992), Zivot and Andrews (1992),
Perron and Vogelsang (1992), Perron (1997)
and Lumsdaine and Papell (1998).
• These studies have shown that bias in the usual
unit root tests can be reduced by endogenously
determining the time of structural breaks.

• Zivot and Andrews (1992) endogenous structural break
test is a sequential test which utilizes the full sample
and uses a different dummy variable for each possible
break date.
• The break date is selected where the t-statistic from the
ADF test of unit root is at a minimum (most negative).
• Consequently a break date is chosen where the
evidence is least favorable for the unit root null.
• The critical values in Zivot and Andrews (1992) are
different to the critical values in Perron (1989).
• The difference is due to that the selection of the time of
the break is treated as the outcome of an estimation
procedure, rather than predetermined exogenously.

• Even though Banerjee, Lumisdaine and Stock
(1992) use endogenous structural break test,
the tests are rolling and recursive tests.
• The numbers of breaks are determined by non-
sequential tests which use sub-samples.
• This can be viewed as not having used the full
information set, which may have implications
for the power of these tests.

• This work was extended by Perron and Vogelsang (1992) and
Perron (1997) who proposed a class of test statistics that allows
for two different forms of structural break.
• These are the Additive Outlier (AO) and Innovational Outlier (IO)
models.
• The AO model allows for a sudden change in mean (crash
model) while the IO model allows for more gradual changes.
• Perron and Vogelsang (1992) argue that these tests are based
on the minimal value of t-statistics on the sum of the
autoregressive coefficients over all possible breakpoints in the
appropriate autoregression, which would reject the null of unit
root.

Issue on null hypothesis of unit root
• However, these endogenous tests were criticized for
their treatment of breaks under the null hypothesis.
• Given the breaks were absent under the null hypothesis
of unit root there may be tendency for these tests to
suggest evidence of stationarity with breaks (Lee and
Strazicich, 2003).
• Lee and Strazicich (2003) propose a two break
minimum Lagrange Multiplier (LM) unit root test in
which the alternative hypothesis unambiguously implies
the series is trend stationary.

• The issue whether a macroeconomic time series is of DSP
or TSP is extremely important because the dynamic
properties of the two processes are different (Nelson and
Plosser, 1982).
• While the former is predictable, the latter is completely
unpredictable.
• In a series following TSP, cyclical fluctuations are temporary
around a stable trend, while for DSP any random shock to
the series has a permanent effect.
• The cyclical components of a TSP originate from the
residuals of a regression of the series on the variable time,
and a DSP involves regression of a series on its own lagged
values and time.

• A TSP has a trend in the mean but no trend in the
variance, but a DSP has a trend in the variance
with or without trend in the mean.
• The most wisely used model to take over
stochastic trend is autoregressive of order p
[AR(p)]:
• Yt gives values in log form in time t and et is a
stationary series with mean zero and variance s2.
t
n
-
t
3
-
t
3
2
-
t
2
1
-
t
1
t e
Y
.....
Y
Y
Y
Y 





 n






• This model can generate the trend behaviour of
macro economic time series and the randomly
fluctuating behaviour of their growth rates.
• If, for example, Yt is generated by the model:
• which is AR(1) with â1 =1, accumulating Yt starting
with an initial value Y0 we get
• which has the same form as the conventional log-
linear trend equation, excepting for the fact that the
disturbance is not stationary.
t
1
-
t
1
t e
Y
Y 

 a

j
0
t e
Y
Y 

 t


• One important property of time series data,
not usually present in cross-sectional data, is
the existence of correlation across
observations.
• Income today is highly correlated with income
of the last year.
• Thus Yt tends to exhibit trend behaviour and is
highly correlated over time.
• The non-stationary time series containing a
unit root will give a stochastic trend.

• If β1 = 1 for an AR(1) model, then Yt has a unit
root and exhibit trend behavior, especially
when α ≠ 0 .
• Unit root series contains a so called stochastic
trend.
• The Augmented Dickey-Fuller (ADF) test is
performed for unit root hypothesis.
• The more appropriate model for testing a unit
root is the AR(p) with deterministic trend:
t
1
p
-
t
1
2
-
t
2
1
-
t
1
1
-
t
1
t e
Y
.....
Y
Y
Y
Y 










 
 t
p 






• A series belongs to the class DSP exhibiting
stochastic trend if ρ =0, δ=0, and the TSP class
if ρ < 0.
• If δ = 0, then Yt contains a unit root.
• In this case, we have to follow ADF test.
• If the t-statistics on ρ are less negative than the
Dickey-Fuller critical value, we conclude that
the series Yt has a unit root.

• To test whether the series has a unit root, we have to choose
lag length (p).
• Many sophisticated statistical criteria and testing methods are
available to determine the appropriate lag length in an AR(p)
model.
• But one can perform a simple way by choosing a maximum
lag length and then sequentially drop lag lengths if the
relevant coefficients are insignificant.
• The maximum lag length is chosen by following Schwert
(1989) rule:
• Pmax = integer part of [12(T/100)0.25
]

• AIC is also used for selecting the appropriate
lag length.
• By following such criteria, let the maximum lag
length be found as 1. Thus our model would
be
)
5
.........(
e
Y
Y
Y t
1
-
t
1
1
-
t
t 





 t





Structural Break in Output Growth
• The Dickey – Fuller test sometimes gives a wrong
signal, particularly when the t-statistic is very close to
its critical value.
• Again a time series can also appear to exhibit unit root
behaviour owing to the presence of structural change.
• The time series plot of first differences of the
logarithmic series gives a rough idea about the pattern
of growth and appearance of structural break in the
growth path.

• The most commonly used test for structural break is
attributed to Chow (1960).
• The conventional Chow test involves splitting the sample into
two or more sub-periods, estimating the parameters for each
of the sub-period and, finally, testing for the equality of these
sets of parameters using the F statistic.
• The basic weakness, however, of this test is its critical
assumption that the break point is known a priori.
• But in many cases this assumption does not match properly to
what happens in actual practice and if one picks up a break
point in an arbitrary manner to perform the test, the result
will be uninformative and even misleading.

• CUSUM and CUSUM of squares tests on the residuals may
be used for testing structural break in mean and volatility of
growth rates respectively with unknown break points .
• The power of the test, however, is rather limited compared
to the Chow test.
• Another possible way is to treat the break point as unknown
and carry out the Chow test for all the possible years and
then select the year corresponding to the largest Chow
statistic (Quandt 1960).
• Andrews and Ploberger (1994) provided the critical values of
this type of test by considering break point is unknown.

Testing of multiple structural changes
• Bai and Perron (1998) have also developed a methodology for testing
multiple structural changes.
• But in Bai and Perron, break points are determined solely on the basis
of deterministic trend.
• Thus the use of this methodology may be problematic for a series
exhibiting stochastic trend.
• One can compute a range of candidate values of F statistics, by
allowing stochastic trend in the series, corresponding to different
points ranging from .15T to .85T, T is the total number of years in the
sample, and then retain the maximum value, called the supremum
value, obtained.
• We can treat the point of time relating to the maximum value as a
break point.
• This will detect for structural break for macro variables.

• Others who have considered multiple breaks are Clemente,
Montañés and Reyes (1998) who base their approach on
Perron and Vogelsang (1992) but allow for two breaks.
• Ohara (1999) utilizes an approach based on sequential t-
tests of Zivot and Andrews to examine the case on m breaks
with unknown break dates. He provides evidence that unit
root tests with multiple trend breaks are necessary for both
asymptotic theory and empirical applications.
• Papell and Prodan (2003) propose a test based on restricted
structural change, which explicitly allows for two offsetting
structural changes.

• These endogenous break tests that allow for
the possibility of one or multiple breaks; Zivot
and Andrews, Banerjee et al., Perron (1997),
Lumsdaine and Papell (1997) and Ohara (1999)
do not allow for break(s) under the null
hypothesis of unit root and thus derive their
critical values accordingly. This may potentially
bias these tests.

• Nunes et al (1997) show that this assumption leads
to size distortions in presence of a unit root with a
break and Perron (2005) suggests that there may be
some loss of power.
• Lee and Strazicich (2003) demonstrate that when
utilizing these endogenous break unit root tests,
researchers might conclude that the time series is
trend stationary when in fact the series is non-
stationary with break(s).
• Thus ‘spurious rejections’ may occur

• Ben-David et al (2003) argue that failure to allow for
multiple breaks can cause the non-rejection of the
unit root null by these tests which incorporate only
one break.
• Lumsdaine and Papell (1997) argue that consideration
of only one endogenous break may be not sufficient
and could lead to loss of information.
• Maddala and Kim (2003) believe that allowing for the
possibility of two endogenous break points provides
further evidence against the unit root hypothesis.

• The minimum Lagrange Multiplier (LM) unit root test
proposed by Lee and Strazicich (2003) not only endogenously
determines structural breaks but also avoids the above
problems of bias and spurious rejections.
• Lee and Strazicich (2003) procedure corresponds to Perron’s
(1989) exogenous structural break (Model C) with change in
the level and the trend.
• Lee and Strazicich’s (2003) model allows for two endogenous
breaks both under the null and the alternative hypothesis.
• They show that the two-break LM unit root test statistic
which is estimated by the regression according to the LM
principle will not spuriously reject the null hypothesis of a
unit root.

Empirical Results: Presence of Unit Root
• The hypothesis of a stationary series can be evaluated by testing
whether the absolute value of ñ in equation (6) is strictly less
than zero. We have performed ADF test which takes the unit
root as the null hypothesis H0: ρ =0, against the o
• The estimated test statistics corresponding to logarithmic values,
their first and second differences of the series can be
summarized.
• The test-statistic under the null hypothesis of a unit root does
not have the conventional t-distribution.
• MacKinnon (1991) estimated the response surface using the
simulation results, permitting the calculation of Dickey-Fuller
critical values for any sample size and for any number of right-
hand variables.

Decisions
• If All the ADF t-values for the logarithmic series are statistically
insignificant even at 10% level, thus the time series of
logarithmic values of a variable (GDP) and of its sectoral
components are non-stationary, but the estimated statistics for
their first differences (implying annual growth rates), are
statistically significant.
• The pattern of data generating process of any time series can
also be revealed, although grossly, by plotting the values of first
difference of the logarithmic series (giving annual growth rates).

Structural Break
• The conventional idea of testing the existence of breakpoint is to fit
the equation separately for each sub-sample and to see whether
there are significant differences in the estimated equations.
• A significant difference indicates a structural change in the
relationship. To carry out the test for structural break, we divide the
data into two or more sub-samples. Each sub-sample must contain
more observations than the number of coefficients in the equation so
that the equation can be estimated using each sub-sample.
• The F test is based on a comparison of the sum of squared residuals
obtained by fitting a single equation to the entire sample with the
sum of squared residuals obtained when separate equations are fit to
each sub-sample of the data.

• If testing of unit root hypothesis, confirms the presence of
stochastic trend in the logarithmic values of income series, an
attempt to find out any structural break in the growth path
ignoring stochastic trend in the series may give misleading result.
• There has been discussion sorrounding the trend break in India’s
growth rate of GDP (DeLong 2001, Wallack 2003, Rodrick et al.
2004, Balakrishnan et al. 2007).
• But most of the studies did not take care of the data generating
process and ignored the presence of stochastic trend in the
estimable model.
• They considered deterministic trend in the form of conventional
log linear trend in estimating break point on the growth path.

• The estimated equation for testing structural break should
incorporate both the stochastic and the deterministic components
of trend.
• The methodology developed in Andrews and Ploberger (1994), one
can estimate F statistic by allowing stochastic trend as observed in
the data series.
• First partition the series into two sub-samples at every year
sequentially within the range between 1970 and 2025 for the series,
and retain the supremum value of the F-statistic for every series.
• The year at which this supremum value is obtained is treated as a
break point.

• Many studies reported the Indian economy
experienced a structural break in the year
1979, a long period before the initiation of
economic reforms in the country.
• In Wallack (2003), the most significant date for
the break of GDP growth was 1980, and in
Balakrishnan (2007) the break point was 1978.

Packages
• A new community contributed Stata package
called xtbreak.
• The package implements the methods
developed by Bai and Perron (1998) for the
case of pure time series, and Ditzen et al.
(2021) in case of panel data.

xtbreak
• xtbreak provides researchers with a complete toolbox for analysing multiple
structural breaks in time series and panel data.
• It can detect and date an unknown number of breaks at unknown break dates.
The toolbox is based on asymptotically valid tests for the presence of breaks, a
consistent break date estimator, and a break date confidence interval with
correct asymptotic coverage.
• xtbreak includes no less than three tests; (i) a test of no structural change
against the alternative of a specific number of changes, (ii) a test the null
hypothesis of no structural change against the alternative of an unknown
number of structural changes, and (iii) a test of the null of s changes against
the alternative of s + 1 changes.
• The package also includes an algorithm that employs the last test
consecutively in order to estimate the true number of breaks. The tested
break dates can be unknown or user-defined, as when researchers have
additional information and wish to examine whether there was a break in a
specific point in time. Once the presence of breaks has been tested and
confirmed, xtbreak estimates the locations of the breaks and provides the
associated confidence intervals.

• A large number of breaks does not translate into heavy
computational burden, as xtbreak implements an efficient
dynamic programming method described in Bai and
Perron (2003), which ensures that there are 0(T2 )
computations, even with more than two breaks.
• xtbreak can deal with models of “pure” or “partial”
structural change.
• A pure structural change model is one in which the
coefficients of all explanatory variables change, while in a
partial structural change model only a subset of the
coefficients change.

Xtbreak used under very general conditions
• xtbreak is applicable and used under very general conditions. In case of panel
data, the tools can be used even if the model errors are dependent and
heteroskedastic across both time and cross-sectional units.
• The cross-sectional dependence is assumed to have an “interactive effects”,
or “common factor”, structure, which means that the dependence is of the
strong form.
• This also allows the regressors to load on the same set of factors as the
errors, which means that they may be endogenous. The allowable
dependencies over time are also very general. The only requirement is that
they cannot be strong, as in the presence of unit roots.
• In particular, we find that an increase in the number of COVID–19 cases lead
to more deaths in the beginning of the pandemic than in later waves.
• .In particular, we find that an increase in the number of COVID–19 cases lead
to more deaths in the beginning of the pandemic than in later waves.

Model discussion
• We consider the following model with N units, T
periods and s structural breaks:
• yi,t = xi,t β + wi,t δj + ei,t,
• where t = Tj−1, ..., Tj and j = 1, ..., s + 1 with T0 = 0
and Ts+1 = T.
• Hence, there are s breaks, or s + 1 regimes with
regime j covering the observations Tj−1, ..., Tj .
• In order emphasize the break structure, we can
write (1) regime-wise;

• yi,t = x i,tβ + w i,tδ1 + ei,t for t = T0, ..., T1,
• yi,t = x i,tβ + wi,tδ2 + ei,t for t = T1, ..., T2, . . .
• yi,t = x i,tβ + w i,tδs+1 + ei,t for t = Ts, ..., Ts+1.
• For N = 1, this is a time series model, while for N > 1, it is a panel data model.
• The dependent variable yi,t and the regression error ei,t are scalars, while xi,t and wi,t
are p×1 and q×1 vectors, respectively, of regressors.
• The coefficients of the regressors in xi,t are unaffected by the breaks, while those of
wi,t are affected by the breaks.
• It is possible that all independent variables break, in which case x i,tβ is defined to be
zero. The break dates are common for all units. This is a very common assumption
that is reasonable in settings where the frequency of the data is not high.
• Let Ts = {T1, ..., Ts} be a collection of s break dates such that Tj = λjT, where λ0 = < λ1
< ... < λs < λs+1 = 1.
• By specifying the breaks in this way we ensure that they are distinct from one
another and that they are bounded away from the beginning and end of the sample.
• This is important because we need to estimate the model within each regime.

Zivot-Andrews unit root test
• The Zivot-Andrews unit root test is a statistical method
used to test for a unit root in a time series while
allowing for the possibility of a single structural break at
an unknown point in time,
• meaning it can detect if a time series appears stationary
due to a break rather than truly being stationary;
• It is considered an extension of the standard
Augmented Dickey-Fuller (ADF) test that accounts for
potential structural breaks in the data.

Key points on Zivot-Andrews test:
• Structural break consideration:
• Unlike standard unit root tests, the Zivot-Andrews test actively
searches for the most likely break point within the data, allowing for a
more accurate analysis when structural breaks are suspected.
• Endogenous break point:
• The break point is not predetermined but is estimated within the test
by finding the date that minimizes the ADF test statistic, meaning the
test identifies the break point that most strongly supports the null
hypothesis of a unit root.
• Applications:
• This test is particularly useful in economic and financial time series
analysis where significant events like policy changes or market crashes
could cause structural breaks.
• How the Zivot-Andrews test works:

• 1. Data preparation:
• The time series is initially regressed with a constant and trend
term.
• 2. Break point search:
• For each potential break date within the sample period, the data is
re-estimated with dummy variables to account for the break, and
an ADF test is performed.
• 3. Selecting the break point:
• The break date that produces the most negative ADF test statistic
(indicating the strongest evidence for a unit root) is chosen as the
"optimal" break point.
• 4. Final test:
• The ADF test is performed again using the identified break point
and associated dummy variables to determine whether the time
series has a unit root.

Important considerations:
• Power of the test:
• While the Zivot-Andrews test is powerful in detecting unit roots with a single break, its
power can be reduced if multiple structural breaks exist in the data.
• Interpretation of results:
• A significant Zivot-Andrews test result suggests that the time series exhibits a unit root even
considering a potential structural break, indicating the need for further analysis or data
transformation to address the break.
• Unit Root Tests and Structural Breaks:
• Zivot and Andrews (1992) endogenous structural break test is a sequential test which utilizes
the full sample and uses a different dummy variable for each potential break date, and
ultimately identifies the break point as the date where the ADF test statistic for a unit root is
the most negative (i.e., the minimum value) across all possible break dates; essentially
allowing the break point to be determined by the data itself.
• .
• Evidence from Zivot-Andrews and Lagrange Multiplier Unit Root Tests
• If there is a unit-root in the real exchange rate this implies that shocks to the real exchange
rate are permanent and PPP does not hold.

Structural Break models
• What are structural break models?
• Time series models estimate the relationship between
variables that are observed over a period of time. Many
models assume that the relationship between these
variables stays constant across the entire period.
• However, there are cases where changes in factors outside
of the model cause changes in the underlying relationship
between the variables in the model.
• Structural break models capture exactly these cases by
incorporating sudden, permanent changes in the
parameters of models.

• Structural break models can integrate
structural change through any of the model
parameters.
• Bai and Perron (1998) provide the standard
framework for structural breaks model in
which some, but not all, of the model
parameters are allowed to break at m possible
break points.

• Yt=xtβ+Ztδj+Ut
• t=Tj-1+1,……….,T
• There j=1,……,m+1.
• The dependent variable yt is to be modeled as
a linear combination of regressors with both
time invariant coefficients, xt, and time variant
coefficients, zt.

variance break
• Alternatively, the variance break model
assumes that breaks occur in the variance of
the error term such that
• Yt=xtβ+ut
• Var(ut) =σ1
2
t≤T1
• Var(ut) =σ2
2
t>T1

Why should I worry about structural breaks?
• Structural change is pervasive in economic
time series relationships, and it can be quite
perilous to ignore. Inferences about economic
relationships can go astray, forecasts can be
inaccurate, and policy recommendations can
be misleading or worse." -- Bruce Hansen
(2001)

• Time series models are used for a variety of
reasons -- predicting future outcomes,
understanding past outcomes, making policy
suggestions, and much more.
• Parameter instability diminishes the ability of
a model to meet any of these objectives.

Benefits of Structural break test
• First, it prevents yielding a test result which is
biased towards non-rejection, as suspected by
Perron (1989).
• Second, since this procedure can identify when the
possible presence of structural break occurred, then
it would provide valuable information for analyzing
whether a structural break on a certain variable is
associated with a particular government policy,
economic crises, war, regime shifts or other factors.

Research demonstrates:
• Many important and widely used economic
indicators have been shown to have structural
breaks.
• Failing to recognize structural breaks can lead
to invalid conclusions and inaccurate forecasts.
• Identifying structural breaks in models can
lead to a better understanding of the true
mechanisms driving changes in data.

Economic indicators with structural breaks
• In a 1996 study, Stock and Watson examined
76 monthly U.S. economic time series
relations for model instability using several
common statistical tests.
• The series analyzed encompassed a variety of
key economic measures including interest
rates, stock prices, industrial production, and
consumer expectations.

Complete group of variables studied by the authors to meet four
criteria :
• The sample included important monthly
economic aggregates and coincident indicators.
• The sample included important leading indicators.
• The series represented a number of different
types of variables, spanning different time series
properties.
• The variables had consistent historical definitions
or adjustments could easily be made if definitions
changed over time.

Stock and Watson found evidence that a "substantial
fraction of forecasting relations are unstable." Based
on this, the authors made several observations:
• Systematic stability analysis is an important part
of modeling.
• Failure to appropriate model "commonplace"
instability in models, "calls into question the
relevance of policy implications".
• There is an opportunity to improve on the
forecasts made by fixed-parameter models.

• Rossi (2013) updates the data in this study to
include data through 2000 and comes to the
same conclusions that, there is clear empirical
evidence of instabilities and that these
instabilities impact forecast performance.
• The potential empirical importance of
departures from constant parameter linear
models is undeniable" -- Koop and Potter (2011)

When should one consider structural break models?
• Structural breaks aren’t right for all data and
knowing when to use them is important for
building valid models.
• While there are statistical tests for structural
breaks, some preliminary checks are necessary
which can help determine when one may need
to consider structural breaks.

Time series plots
• Visual plots indicate a change in behavior
• It provides a quick, preliminary method for
finding structural breaks in your data.
• Visually inspecting data can provide important
insight into potential breaks in the mean or
volatility of a series.
• Don’t forget to examine both independent and
dependent variables as sudden changes in either
can change the parameters of a model

Why do instabilities matter for forecasting?
• A fixed parameter model cannot be expected to
forecast well if the true parameters of the
model change over time.
• Conversely, if your model isn't forecasting well,
it may be worth considering if model instabiliti
• Why do instabilities matter for forecasting?
Clearly, if the predictive content is not stable
over time, it will be very difficult to exploit it to
improve forecasts" es could be playing a role.

• sudden changes in policy stances including
legislative or regulatory changes, technological
changes, institutional changes, changes in
monetary or fiscal policy, or oil price shocks
can be responsible for parameter instability.

What are the statistical tests in identifying structural breaks?
• Testing for structural breaks is a rich area of research and
there is no one-size-fits-all test for structural breaks.
• which test to implement depends on several factors.
• Is the break date known or unknown?
• Is there a single break or multiple breaks?
• Knowing the statistical characteristics of both the breaks and
your data help to ensure that the correct test is
implemented.
• some of the classic tests exist in literature which have
shaped the field of structural break testing.
• The Chow Test and The Quandt Likelihood Ratio Test, The
CUSUM Test, The Hansen and Nyblom Tests

The Chow Test
• Chow (1960) test is one of the first tests which set the foundation for
structural break testing.
• It is built on the theory that if parameters are constant then out-of-
sample forecasts should be unbiased.
• It tests the null hypothesis that there is no structural break against the
alternative that there is a known structural break at time Tb.
• The test considers a linear model split into samples at a predetermined
break point such that
• The test estimates coefficients for each period and uses the out-of-
sample forecast errors to compute an F-test comparing the stability of
the estimated coefficients across the two periods.
• One key issue with the Chow test is that the break point must be
predetermined prior to implementing the test.
• Furthermore, the break point must be exogenous or the standard
distribution of the statistic is not valid.

What are some alternatives to structural break models?
• Time-varying parameter models assume that
parameters change gradually over time while
threshold models assume that model parameters
change based on the value of a specified threshold
variable.
• Markov-switching models offer an even different
solution that assumes that an underlying stochastic
Markov chain drives regime changes. Theory and
statistical tests should drive the decision of which of
these models you use.

• https://www.aptech.com/structural-breaks/
• https://www.researchgate.net/publication/30
387415_Unit_Root_Tests_and_Structural_Bre
aks_A_Survey_with_Applications

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