Estimation and inference in unstable nonlinear least squares models

Abstract

There is compelling evidence that many macroeconomic and financial variables are not generated by linear models. This evidence is based on testing linearity against either smooth nonlinearity or piece-wise linearity, but there is no framework that encompasses both. This paper provides an econometric framework that allows for both breaks and smooth nonlinearity in-between breaks.We estimate the unknown break-dates simultaneously with other parameters via nonlinear least-squares. Using new central limit results for nonlinear processes, we provide inference methods on break-dates and parameter estimates and several instability tests. We illustrate our methods via simulated and empirical smooth transition models with breaks.JEL classification: C12, C13, C22

The asymptotic behaviour of the residual sum of squares in models with multiple break points

“…From the arguments of Boldea and Hall (2013), it follows that (12) and (13) continue to apply, but now with…”

“…A considerable literature now exists concerned with least square-based estimation and testing in models with multiple discrete breaks in the parameters, see inter alia Bai andPerron (1998), Hall et al (2012), and Boldea and Hall (2013). In these contexts, if the model is assumed to have m breaks, then the break points (the points at which the parameters change) are estimated by minimizing the residual sum of squares over all possible data partitions involving m breaks.…”

“…To avoid excessive notation, rede ne the estimators and residual sum of squares analogously to Section 2.1, replacing x ′ t β i by f t (β i ) in (3). Compared with the OLS case, the consistency and large sample distribution ofλ andβ({T i } m i=1 ) have been established to date in the NLS setting only under more restrictive conditions on the dynamic structure of the data and also the rate of shrinkage between regimes; see Boldea and Hall (2013)[Assumptions 2-8]. These additional restrictions arise because of the inherent nonlinearity of the model; see Boldea and Hall (2013) for further discussion.…”

“…The seminal paper by Bai and Perron (1998) developed a framework for estimation and inference in linear regression models estimated via ordinary least squares (OLS) that has served as the template for similar frameworks in more general models, including systems of linear regression models (Perron and Qu, 2006), linear models with endogenous regressors estimated via two stage least squares (2SLS, Hall et al, 2012), and nonlinear regression models estimated by Nonlinear Least Squares (NLS, Boldea and Hall, 2013).…”

Approximate p-Values of Certain Tests Involving Hypotheses About Multiple Breaks

“…Leading statistics are the so-called Sup-, UDmax-and sequential tests that are used respectively to test stability versus a fixed number of breaks, stability against up to a fixed number of breaks, and l breaks versus l + 1 breaks. These statistics have been shown to have the same limiting distributions in linear models (with exogenous regressors) estimated by Ordinary Least Squares (Bai and Perron 1998), linear models (with endogenous regressors) estimated via Two Stage Least Squares (Hall, Han, and Boldea 2012), and nonlinear regression models estimated by nonlinear least squares (Boldea and Hall 2013). The limiting distributions in question are non-standard but depend only on the number of unknown breaks, a trimming parameter, and the number of time varying regression parameters in each regime.…”