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.^[1]^[2]^[3] 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.^[4]

Structural break tests

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 $K$ for $K \in [1, T]$ .^[5]^[6] This test assesses whether the coefficients in a regression model are the same for periods $[1,2, ..., K]$ and $[K + 1, ..., T]$ .^[6]

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.^[7]^[5]^[6] Bayesian methods exist to address these difficult cases via Markov chain Monte Carlo inference.^[8]^[9]

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.^[6]^[10] 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.^[11]^[12] These tests were shown to be superior to the CUSUM test in terms of statistical power,^[11] 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.^[4] 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 $n$ as $n \to \infty$ ),^[11] and involve the assumption of homoskedasticity across break points for finite samples;^[4] 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.^[11] A method developed by Bai and Perron (2003) also allows for the detection of multiple structural breaks from data.^[13]

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.^[4]^[14] 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.^[4]