econometrics Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, a structural break is an unexpected change over time in the

parameters A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

of regression models, which can lead to huge

forecasting Forecasting is the process of making predictions based on past and present data. Later these can be compared with what actually happens. For example, a company might Estimation, estimate their revenue in the next year, then compare it against the ...

errors and unreliability of the model in general. 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 In statistics, linear regression is a statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ...

models.

Structural break tests

A single break in mean with a known breakpoint

For

models, the

Chow test The Chow test (), proposed by econometrician Gregory Chow in 1960, is a statistical test of whether the true coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis ...

is often used to test for a single break in mean at a known time period for . This test assesses whether the coefficients in a regression model are the same for periods and .

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

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. Bayesian methods exist to address these difficult cases via

Markov chain Monte Carlo In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that ...

inference. In general, the

CUSUM In statistical process control, statistical quality control, the CUSUM (or cumulative sum control chart) is a sequential analysis technique developed by E. S. Page of the University of Cambridge. It is typically used for monitoring change detecti ...

(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. For cases 1 and 2, the sup-Wald (i.e., the

supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...

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 Andrews may refer to: Places Australia *Andrews, Queensland *Andrews, South Australia United States *Andrews, Florida (disambiguation), various places *Andrews, Indiana *Andrews, Nebraska *Andrews, North Carolina *Andrews, Oregon *Andrews, South ...

(1993, 2003) may be used to test for parameter instability when the number and location of structural breaks are unknown. These tests were shown to be superior to the CUSUM test in terms of

statistical power In frequentist statistics, power is the probability of detecting a given effect (if that effect actually exists) using a given test in a given context. In typical use, it is a function of the specific test that is used (including the choice of tes ...

, 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. The sup-Wald, sup-LM, and sup-LR tests are

asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...

in general (i.e., the asymptotic critical values for these tests are applicable for sample size as ), and involve the assumption of homoskedasticity across break points for finite samples; however, an

exact test An exact (significance) test is a statistical test such that if the null hypothesis is true, then all assumptions made during the derivation of the distribution of the test statistic are met. Using an exact test provides a significance test that ...

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. A method developed by Bai and Perron (2003) also allows for the detection of multiple structural breaks from data. 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. 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.

Structural breaks in cointegration models

For a

cointegration In econometrics, cointegration is a statistical property describing a long-term, stable relationship between two or more time series variables, even if those variables themselves are individually non-stationary (i.e., they have trends). This means ...

model, the Gregory–Hansen test (1996) can be used for one unknown structural break, the Hatemi–J test (2006) can be used for two unknown breaks 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,

GAUSS Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

, and

Stata Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose Statistics, statistical software package developed by StataCorp for data manipulation, visualization, statistics, and automated reporting. It is used by researchers ...

, 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, including both classical and Bayesian methods.

References