Maximum Likelihood Estimation of the Multivariate Normal Mixture Model

The Hessian of the multivariate normal mixture model is derived, and estimators of the information matrix are obtained, thus enabling consistent estimation of all parameters and their precisions. The usefulness of the new theory is illustrated with two examples and some simulation experiments. The newly proposed estimators appear to be superior to the existing ones.Mixture model; Maximum likelihood; Information matrix

Estimation and inference in unstable nonlinear least squares models

In this paper, we extend Bai and Perron's (1998, Econometrica, pp. 47-78) method for detecting multiple breaks to nonlinear models. To that end, we consider a nonlinear model that can be estimated via nonlinear least squares (NLS) and features a limited number of parameter shifts occurring at unknown dates. In our framework, the break-dates are estimated simultaneously with the parameters via minimization of the residual sum of squares. Using new uniform convergence results for partial sums, we derive the asymptotic distributions of both break-point and parameter estimates and propose several instability tests. We provide simulations that indicate good finite sample properties of our procedure. Additionally, we use our methods to test for misspecification of smooth-transition models in the context of an asymmetric US federal funds rate reaction function and conclude that there is strong evidence of sudden change as well as smooth behavior.Multiple Change Points, Nonlinear Least Squares, Smooth Transition

Asymptotic Distribution Theory for Break Point Estimators in Models Estimated via 2SLS

In this paper, we present a limiting distribution theory for the break point estimator in a linear regression model estimated via Two Stage Least Squares under two different scenarios regarding the magnitude of the parameter change between regimes. First, we consider the case where the parameter change is of fixed magnitude; in this case the resulting distribution depends on distribution of the data and is not of much practical use for inference. Second, we consider the case where the magnitude of the parameter change shrinks with the sample size; in this case, the resulting distribution can be used to construct approximate large sample confidence intervals for the break point. The finite sample performance of these intervals are analyzed in a small simulation study and the intervals are illustrated via an application to the New Keynesian Phillips curve.

Inference regarding multiple structural changes in linear models estimated via two stage least squares

In this paper, we extend Bai and Perron’s (1998, Econometrica, p.47-78) framework for multiple break testing to linear models estimated via Two Stage Least Squares (2SLS). Within our framework, the break points are estimated simultaneously with the regression parameters via minimization of the residual sum of squares on the second step of the 2SLS estimation. We establish the consistency of the resulting estimated break point fractions. We show that various F-statistics for structural instability based on the 2SLS estimator have the same limiting distribution as the analogous statistics for OLS considered by Bai and Perron (1998). This allows us to extend Bai and Perron’s (1998) sequential procedure for selecting the number of break points to the 2SLS setting. Our methods also allow for structural instability in the reduced form that has been identified a priori using data-based methods. As an empirical illustration, our methods are used to assess the stability of the New Keynesian Phillips curve.unknown break points; structural change; instrumental variables; endogenous regressors; structural stability tests; new Keynesian Phillips curve

Estimation and inference in unstable nonlinear least squares models

In this paper, we extend Bai and Perron's (1998, Econometrica, pp. 47-78) method for detecting multiple breaks to nonlinear models. To that end, we consider a nonlinear model that can be estimated via nonlinear least squares (NLS) and features a limited number of parameter shifts occurring at unknown dates. In our framework, the break-dates are estimated simultaneously with the parameters via minimization of the residual sum of squares. Using new uniform convergence results for partial sums, we derive the asymptotic distributions of both break-point and parameter estimates and propose several instability tests. We provide simulations that indicate good finite sample properties of our procedure. Additionally, we use our methods to test for misspecification of smooth-transition models in the context of an asymmetric US federal funds rate reaction function and conclude that there is strong evidence of sudden change as well as smooth behavior

Maximum Likelihood Estimation of the Multivariate Normal Mixture Model

The Hessian of the multivariate normal mixture model is derived, and estimators of the information matrix are obtained, thus enabling consistent estimation of all parameters and their precisions. The usefulness of the new theory is illustrated with two examples and some simulation experiments. The newly proposed estimators appear to be superior to the existing ones

Estimation and inference in unstable nonlinear least squares models

In this paper, we extend Bai and Perron's (1998, Econometrica, pp. 47-78) method for detecting multiple breaks to nonlinear models. To that end, we consider a nonlinear model that can be estimated via nonlinear least squares (NLS) and features a limited number of parameter shifts occurring at unknown dates. In our framework, the break-dates are estimated simultaneously with the parameters via minimization of the residual sum of squares. Using new uniform convergence results for partial sums, we derive the asymptotic distributions of both break-point and parameter estimates and propose several instability tests. We provide simulations that indicate good finite sample properties of our procedure. Additionally, we use our methods to test for misspecification of smooth-transition models in the context of an asymmetric US federal funds rate reaction function and conclude that there is strong evidence of sudden change as well as smooth behavior

Inference regarding multiple structural changes in linear models estimated via two stage least squares

In this paper, we extend Bai and Perron’s (1998, Econometrica, p.47-78) framework for multiple break testing to linear models estimated via Two Stage Least Squares (2SLS). Within our framework, the break points are estimated simultaneously with the regression parameters via minimization of the residual sum of squares on the second step of the 2SLS estimation. We establish the consistency of the resulting estimated break point fractions. We show that various F-statistics for structural instability based on the 2SLS estimator have the same limiting distribution as the analogous statistics for OLS considered by Bai and Perron (1998). This allows us to extend Bai and Perron’s (1998) sequential procedure for selecting the number of break points to the 2SLS setting. Our methods also allow for structural instability in the reduced form that has been identified a priori using data-based methods. As an empirical illustration, our methods are used to assess the stability of the New Keynesian Phillips curve

Asymptotic Distribution Theory for Break Point Estimators in Models Estimated via 2SLS

In this paper, we present a limiting distribution theory for the break point estimator in a linear regression model estimated via Two Stage Least Squares under two different scenarios regarding the magnitude of the parameter change between regimes. First, we consider the case where the parameter change is of fixed magnitude; in this case the resulting distribution depends on distribution of the data and is not of much practical use for inference. Second, we consider the case where the magnitude of the parameter change shrinks with the sample size; in this case, the resulting distribution can be used to construct approximate large sample confidence intervals for the break point. The finite sample performance of these intervals are analyzed in a small simulation study and the intervals are illustrated via an application to the New Keynesian Phillips curve