Structural break

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 ∈ [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 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 → ∞),[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]

For a cointegration model, the Gregory–Hansen test (1996) can be used for one unknown structural break,[15] the Hatemi–J test (2006) can be used for two unknown breaks[16] and the Maki (2012) test allows for multiple structural breaks.