Structural break

{{Short description|Econometric term}} {{Use dmy dates|date=February 2019}} [[File:Chowtest2.svg|250px|thumbnail|right|Linear regression with a structural break]] In econometrics and statistics, a '''structural break''' is an unexpected change over time in the parameters of regression models, which can lead to huge forecasting errors and unreliability of the model in general.<ref name="Panel breaks review - large N and small T">{{cite journal |last1=Antoch |first1=Jaromír |last2=Hanousek |first2=Jan |last3=Horváth |first3=Lajos |last4=Hušková |first4=Marie |last5=Wang |first5=Shixuan |title=Structural breaks in panel data: Large number of panels and short length time series |journal=Econometric Reviews |volume=38 |issue=7 |date=25 April 2018 |pages=828–855 |doi=10.1080/07474938.2018.1454378 |s2cid=150379490 | quote=Structural changes and model stability in panel data are of general concern in empirical economics and finance research. Model parameters are assumed to be stable over time if there is no reason to believe otherwise. It is well-known that various economic and political events can cause structural breaks in financial data. ... In both the statistics and econometrics literature we can find very many of papers related to the detection of changes and structural breaks.|url=http://centaur.reading.ac.uk/79661/1/final-pdf-Antoch-et-al.pdf }}</ref><ref>{{cite web |last1=Kruiniger |first1=Hugo |title=Not So Fixed Effects: Correlated Structural Breaks in Panel Data |url=http://conference.iza.org/conference_files/pada2009/kruiniger_h5168.pdf |publisher=IZA Institute of Labor Economics |access-date=20 February 2019 |pages=1–33 |date=December 2008}}</ref><ref name="Primary source on time series breaks">{{cite journal |last1=Hansen |first1=Bruce E |title=The New Econometrics of Structural Change: Dating Breaks in U.S. Labor Productivity |journal=Journal of Economic Perspectives |date=November 2001 |volume=15 |issue=4 |pages=117–128 |doi=10.1257/jep.15.4.117|doi-access=free }}</ref> This issue was popularised by David Hendry, who argued that lack of stability of coefficients frequently caused forecast failure, and therefore we must routinely test for structural stability. Structural stability − i.e., the time-invariance of regression coefficients − is a central issue in all applications of linear regression models.<ref name="Sup-MZ test">{{cite journal | last1=Ahmed | first1=Mumtaz | last2=Haider | first2=Gulfam | last3=Zaman | first3=Asad | title=Detecting structural change with heteroskedasticity|journal=Communications in Statistics – Theory and Methods | volume=46 | issue=21 | date= October 2016 | pages=10446–10455 | doi=10.1080/03610926.2016.1235200 | s2cid=126189844 | quote = The hypothesis of structural stability that the regression coefficients do not change over time is central to all applications of linear regression models.}}</ref>

==Structural break tests{{Anchor|Sup-Wald test|Sup-MZ test}}==

===A single break in mean with a known breakpoint===

For linear regression models, the Chow test is often used to test for a single break in mean at a known time period {{math|''K''}} for {{math|''K'' ∈ [1,''T'']}}.<ref name="Dating breaks in labor productivity" /><ref name="Greene Econometrics – Chow test" /> This test assesses whether the coefficients in a regression model are the same for periods {{math|[1,2, ...,''K'']}} and {{math|[''K'' + 1, ...,''T'']}}.<ref name="Greene Econometrics – Chow test" />

===Other forms of structural breaks===

Other challenges occur where there are: :Case 1: a known number of breaks in mean with unknown break points; :Case 2: an unknown number of breaks in mean with unknown break points; :Case 3: breaks in variance.

The Chow test is not applicable in these situations, since it only applies to models with a known breakpoint and where the error variance remains constant before and after the break.<ref name=guj>{{cite book |last=Gujarati |first=Damodar |title=Basic Econometrics |year=2007 |publisher=Tata McGraw-Hill |location=New Delhi |isbn=978-0-07-066005-2 |pages=278–284}}</ref><ref name="Dating breaks in labor productivity">{{cite journal | last1=Hansen | first1=Bruce E | title=The New Econometrics of Structural Change: Dating Breaks in U.S. Labor Productivity|journal=Journal of Economic Perspectives | date=November 2001 | volume=15 | issue=4 | pages=117–128 | doi=10.1257/jep.15.4.117 | doi-access=free }}</ref><ref name="Greene Econometrics – Chow test">{{cite book | last1=Greene | first1=William | title=Econometric Analysis | date=2012 | publisher=Pearson Education | isbn=9780273753568 | pages=208–211 | edition=7th | chapter=Section 6.4: Modeling and testing for a structural break | quote = An important assumption made in using the Chow test is that the disturbance variance is the same in both (or all) regressions. ...<br />6.4.4 TESTS OF STRUCTURAL BREAK WITH UNEQUAL VARIANCES ...<br />In a small or moderately sized sample, the Wald test has the unfortunate property that the probability of a type I error is persistently larger than the critical level we use to carry it out. (That is, we shall too frequently reject the null hypothesis that the parameters are the same in the subsamples.) We should be using a larger critical value. Ohtani and Kobayashi (1986) have devised a “bounds” test that gives a partial remedy for the problem.<sup>15</sup>}}</ref> Bayesian methods exist to address these difficult cases via Markov chain Monte Carlo inference.<ref>{{cite journal |last1=Erdman |first1=Chandra |last2=Emerson |first2=John W. |title=bcp: An R Package for Performing a Bayesian Analysis of Change Point Problems |journal=Journal of Statistical Software |date=2007 |volume=23 |issue=3 |page=1-1 |doi=10.18637/jss.v023.i03|s2cid=61014871 |doi-access=free }}</ref><ref name="github.com">{{cite web |last1=Li |first1=Yang |last2=Zhao |first2=Kaiguang |last3=Hu |first3=Tongxi |last4=Zhang |first4=Xuesong |title=BEAST: A Bayesian Ensemble Algorithm for Change-Point Detection and Time Series Decomposition |website=GitHub |url=https://github.com/zhaokg/Rbeast}}</ref>

In general, the CUSUM (cumulative sum) and CUSUM-sq (CUSUM squared) tests can be used to test the constancy of the coefficients in a model. The bounds test can also be used.<ref name="Greene Econometrics – Chow test" /><ref>{{cite journal |last1=Pesaran |first1=M. H. |last2=Shin |first2=Y. |last3=Smith |first3=R. J. |year=2001 |title=Bounds testing approaches to the analysis of level relationships |journal=Journal of Applied Econometrics |volume=16 |issue=3 |pages=289–326 |doi=10.1002/jae.616 |hdl=10983/25617 |s2cid=120051935 |hdl-access=free }}</ref> For cases 1 and 2, the '''sup-Wald''' (i.e., the supremum of a set of Wald statistics), '''sup-LM''' (i.e., the supremum of a set of Lagrange multiplier statistics), and '''sup-LR''' (i.e., the supremum of a set of likelihood ratio statistics) tests developed by Andrews (1993, 2003) may be used to test for parameter instability when the number and location of structural breaks are unknown.<ref name="Andrews 1993">{{cite journal | last1=Andrews | first1=Donald | title=Tests for Parameter Instability and Structural Change with Unknown Change Point | journal=Econometrica | date=July 1993 | volume=61 | issue=4 | pages=821–856 | url=https://www.ssc.wisc.edu/~bhansen/718/Andrews1993.pdf | archive-url=https://web.archive.org/web/20171106014407/https://www.ssc.wisc.edu/~bhansen/718/Andrews1993.pdf | archive-date=6 November 2017 | url-status=live | doi=10.2307/2951764| jstor=2951764 }}</ref><ref name="Andrews Corrigendum">{{cite journal | last1=Andrews | first1=Donald | title=Tests for Parameter Instability and Structural Change with Unknown Change Point: A Corrigendum | journal=Econometrica | date=January 2003 | volume=71 | issue=1 | pages=395–397 | url=https://pdfs.semanticscholar.org/780a/6188c4ef4c388e902c0872338fc24ef12b0b.pdf | archive-url=https://web.archive.org/web/20171106014215/https://pdfs.semanticscholar.org/780a/6188c4ef4c388e902c0872338fc24ef12b0b.pdf | archive-date=6 November 2017 | url-status=dead | doi=10.1111/1468-0262.00405| s2cid=55464774 }}</ref> These tests were shown to be superior to the CUSUM test in terms of statistical power,<ref name="Andrews 1993" /> and are the most commonly used tests for the detection of structural change involving an unknown number of breaks in mean with unknown break points.<ref name="Sup-MZ test" /> The sup-Wald, sup-LM, and sup-LR tests are asymptotic in general (i.e., the asymptotic critical values for these tests are applicable for sample size {{math|''n''}} as {{nowrap|{{math|''n'' → ∞}}}}),<ref name="Andrews 1993" /> and involve the assumption of homoskedasticity across break points for finite samples;<ref name="Sup-MZ test" /> however, an exact test with the sup-Wald statistic may be obtained for a linear regression model with a fixed number of regressors and independent and identically distributed (IID) normal errors.<ref name="Andrews 1993" /> A method developed by Bai and Perron (2003) also allows for the detection of multiple structural breaks from data.<ref name="Bai-Perron multi-break models">{{cite journal | last1=Bai | first1=Jushan | last2=Perron | first2=Pierre | title=Computation and analysis of multiple structural change models | journal=Journal of Applied Econometrics | date=January 2003 | volume=18 | issue=1 | pages=1–22 | doi=10.1002/jae.659| hdl=10.1002/jae.659 | hdl-access=free }}</ref>

The '''MZ test''' developed by Maasoumi, Zaman, and Ahmed (2010) allows for the simultaneous detection of one or more breaks in both mean and variance at a ''known'' break point.<ref name="Sup-MZ test" /><ref name="MZ test">{{cite journal|last1=Maasoumi|first1=Esfandiar|last2=Zaman|first2=Asad|last3=Ahmed|first3=Mumtaz|title=Tests for structural change, aggregation, and homogeneity|journal=Economic Modelling|date=November 2010|volume=27|issue=6|pages=1382–1391|doi=10.1016/j.econmod.2010.07.009}}</ref> The '''sup-MZ test''' developed by Ahmed, Haider, and Zaman (2016) is a generalization of the MZ test which allows for the detection of breaks in mean and variance at an ''unknown'' break point.<ref name="Sup-MZ test" /> <!--

Instead of assuming IID Normal errors as in standard regression, they created tests when errors form ARMA process and even more complex stochastic structures. This literature, and its applications, are reviewed in Hansen, B. (2001).<ref name="Dating breaks in labor productivity" />

HOWEVER, all of this literature makes the same assumption that structural change occurs in the regression coefficients but NOT in the error process. Even the very complex error processes, with heteroskedasticity and autocorrelation (HAC), do not change in time. Recently, Maasoumi, Zaman, and Ahmad (2010) published a paper (“Tests for Structural Change, Homogeneity, and Aggregation” Economic Modelling Vol 27 no.6 (2010): 1382–1391) which relaxes this assumption and allows variances to change at the breakpoint, while also allowing for multiple known breakpoints. This test is called the MZ test by the authors. This test requires the breakpoints to be specified in advance. On the pattern of the rolling Chow, this test can also be used to detect unknown breakpoints by using it at all possible breakpoints, and then taking the maximum of all the statistics, leading to the sup MZ test. The performance of this test was evaluated recently by Ahmed, Haider, & Zaman (2016)<ref name="Ahmed, Haider, & Zaman (2016)">{{cite journal| author = Mumtaz Ahmed and Gulfam Haider and Asad Zaman| title = Detecting structural change with heteroskedasticity| journal = Communications in Statistics – Theory and Methods| year = 2017| volume = 46|pages = 1–10|publisher = Taylor & Francis|doi = 10.1080/03610926.2016.1235200| s2cid = 126189844}}</ref> Theoretical evaluations shows that small changes in variance across the breakpoint cause significant deterioration in performance of the sup F test. Of course the sup MZ test is designed for this situation and hence works much better. Empirical evaluation was done by taking several macroeconomic series and testing them for structural change using both sup F and sup MZ. In these GNP series, tests indicated that structural change frequently involved both regression coefficients and the variance, leading to substantially superior performance of the sup MZ.

===Tests for multiple breaks=== The MZ test is designed to handle the (unlikely) case that multiple breaks all occur at known breakpoints. There is a vast literature on more complicated cases. The simpler case arises when the number of breakpoints is known but their position is unknown. The more difficult case is where both the number of breakpoints, and their position is unknown. For these cases, the sup-Wald, sup-LM, and sup-LR tests developed by Andrews (1993, 2003) may be used to test for parameter instability when the change points (the structural break locations) are unknown.<ref name="Andrews 1993" /><ref name="Andrews Corrigendum" /> -->

===Structural breaks in cointegration models===

For a cointegration model, the Gregory–Hansen test (1996) can be used for one unknown structural break,<ref>{{cite journal |last1=Gregory |first1=Allan |last2=Hansen |first2=Bruce |year=1996 |title=Tests for Cointegration in Models with Regime and Trend Shifts |journal=Oxford Bulletin of Economics and Statistics |volume=58 |issue=3 |pages=555–560 |doi=10.1111/j.1468-0084.1996.mp58003008.x }}</ref> the Hatemi–J test (2006) can be used for two unknown breaks<ref>{{cite journal |last1=Hacker |first1=R. Scott |last2=Hatemi-J |first2=Abdulnasser |year=2006 |title=Tests for Causality between Integrated Variables Using Asymptotic and Bootstrap Distributions: Theory and Application |journal=Applied Economics |volume=38 |issue=15 |pages=1489–1500 |doi=10.1080/00036840500405763 |s2cid=121999615 }}</ref> and the Maki (2012) test allows for multiple structural breaks.

==Statistical packages==

There are many statistical packages that can be used to find structural breaks, including R,<ref>{{cite book |first1=Christian |last1=Kleiber |first2=Achim |last2=Zeileis |title=Applied Econometrics with R |location=New York |publisher=Springer |year=2008 |isbn=978-0-387-77316-2 |pages=169–176 |url=https://books.google.com/books?id=86rWI7WzFScC&pg=PA169 }}</ref> GAUSS, and Stata, among others. For example, a list of R packages for time series data is summarized at the changepoint detection section of the Time Series Analysis Task View,<ref>{{cite web |last1=Hyndman|first1=Rob |last2=Killick|first2=Rebecca|author2-link=Rebecca Killick|title=CRAN Task View: Time Series Analysis. Version 2023-09-26.|url=https://cran.r-project.org/web/views/TimeSeries.html}}</ref> including both classical and Bayesian methods.<ref>{{cite web |last1=Achim|first1=Zeileis|last2=Leisch|first2=Friedrich |last3=Hornik|first3=Kurt |last4=Kleiber|first4=Christian |title=strucchange: Testing, monitoring, and dating structural changes|url=https://cran.r-project.org/web/packages/strucchange/index.html}}</ref><ref name="github.com"/>

==References== {{Reflist}}

{{DEFAULTSORT:Structural Break}} Category:Change detection Category:Time series Category:Panel data Category:Econometric modeling Category:Regression analysis