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  • Vanessa Berenguer-Rico (a1) and Bent Nielsen (a1)

We show that the cumulated sum of squares statistic has a standard Brownian bridge–type asymptotic distribution in nonlinear regression models with (possibly) nonstationary regressors. This contrasts with cumulated sum statistics which have been previously studied and whose asymptotic distribution has been shown to depend on the functional form and the stochastic properties, such as persistence and stationarity, of the regressors. A recursive version of the test is also considered. A local power analysis is provided, and through simulations, we show that the test has good size and power properties across a variety of situations.

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*Address correspondence to Bent Nielsen, Nuffield College, Oxford, OX1 1NF, UK; e-mail:
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Comments from Giuseppe Cavaliere and the referees are gratefully acknowledged.

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Berkes, I. & Horváth, L. (2006) Convergence of integral functionals of stochastic processes. Econometric Theory 22, 304322.
Billingsley, P. (1999) Convergence of Probability Measures. 2nd edition. Wiley.
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42, 5966.
Brown, R.L., Durbin, J., & Evans, J.M. (1975) Techniques for testing the constancy of regression relationships over time. Journal of the Royal Statistical Society B37, 149192.
Chan, N. & Wang, Q. (2015) Nonlinear regressions with nonstationary time series. Journal of Econometrics 185, 182195.
Chan, N.H. & Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367401.
Choi, I. & Saikkonen, P. (2010) Tests for non-linear cointegration. Econometric Theory 26, 682709.
Chow, Y.S. (1965) Local convergence of martingales and the law of large numbers. Annals of Mathematical Statistics 36, 552558.
Christopeit, N. (2009) Weak convergence of nonlinear transformations of integrated processes: The multivariate case. Econometric Theory 25, 11801207.
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.
de Jong, R.M. (2004) Addendum to asymptotics for nonlinear transformations of integrated time series. Econometric Theory 20, 627635.
de Jong, R.M. & Wang, C.H. (2005) Further results on the asymptotics for nonlinear transformations of integrated time series. Econometric Theory 21, 413430.
de Jong, R.M. & Hu, L. (2011) A note on nonlinear models with integrated regressors and convergence order results. Economic Letters 111, 2325.
Deng, A. & Perron, P. (2008a) A non-local perspective on the power properties of the CUSUM and CUSUM of squares tests for structural change. Journal of Econometrics 142, 212240.
Deng, A. & Perron, P. (2008b) The limit distribution of the CUSUM of squares test under general mixing conditions. Econometric Theory 24, 809822.
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427431.
Donati-Martin, C. & Yor, M. (1991) Fubini’s theorem for double Wiener integrals and the variance of the Brownian path. Annales de l’Institut Henri Poincaré 27, 181200.
Edgerton, D. & Wells, C. (1994) Critical values for the CUSUMSQ statistic in medium and large sized samples. Oxford Bulletin of Economics and Statistics 56, 355365.
Hammersley, J.M. (1956) The zeros of a random polynomial. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability 2, 89111.
Jakubowski, A., Ménin, J., & Pages, G. (1989) Convergence en loi des suites d’intégrales stochastiques sur l’espace D1 de Skorokhod. Probability Theory and Related Fields 81, 111137.
Jennrich, R.I. (1969) Asymptotic properties of non-linear least squares estimators. Annals of Mathematical Statistics 40, 633643.
Karlsen, H.A., Myklebust, T., & Tjøstheim, D. (2007) Nonparametric estimation in a non-linear cointegration type model. Annals of Statistics 35, 252299.
Kasparis, I. (2008) Detection of functional form misspecification in cointegrating relations. Econometric Theory 24, 13731403.
Kasparis, I. (2011) Functional form misspecification in regressions with a unit root. Econometric Theory 27, 285311.
Kasparis, I., Andreou, E., & Phillips, P.C.B. (2015) Nonparametric predictive regression. Journal of Econometrics 185, 468494.
Kristensen, D. & Rahbek, A. (2010) Likelihood-based inference for cointegration with nonlinear error-correction. Journal of Econometrics 158, 7894.
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 10351070.
Lai, T.L. & Wei, C.Z. (1982) Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Annals of Statistics 10, 154166.
Lai, T.L. & Wei, C.Z. (1985) Asymptotic properties of multivariate weighted sums with applications to stochastic regression in linear dynamic systems. In Krishnaiah, P.R. (ed.) Multivariate Analysis VI, Elsevier, pages 375393.
Lee, S., Na, O., & Na, S. (2003) On the CUSUM of squares test for variance change in non-stationary and non-parametric time series models. Annals of the Institute of Statistical Mathematics 55, 467485.
McCabe, B.P.M. & Harrison, M.J. (1980) Testing the constancy of regression relationships over time using least squares residuals. Applied Statistics 29, 142148.
Nielsen, B. (2005) Strong consistency results for least squares estimators in general vector autoregressions with deterministic terms. Econometric Theory 21, 534561.
Nielsen, B. & Sohkanen, J. (2011) Asymptotic behaviour of the CUSUM of squares test under stochastic and deterministic time trends. Econometric Theory 27, 913927.
Park, J.Y. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 15, 269298.
Park, J.Y. & Phillips, P.C.B. (2001) Non-linear regressions with integrated time series. Econometrica 69, 117161.
Phillips, P.C.B. (2007) Regression with slowly varying regressors and nonlinear trends. Econometric Theory 23, 557614.
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 9711001.
Pitarakis, J.Y. (2017) A simple approach for diagnosing instabilities in predictive regressions. Oxford Bulletin of Economics and Statistics 79, 851874.
Ploberger, W. & Krämer, W. (1986) On studentizing a test for structural change. Economics Letters 20, 341344.
Ploberger, W. & Krämer, W. (1990) The local power of the CUSUM and CUSUM of squares test. Econometric Theory 6, 335347.
Pötscher, B.M. (2004) Nonlinear functions and convergence to Brownian motion: Beyond the continuous mapping theorem. Econometric Theory 20, 122.
Smirnov, N. (1948) Tables for estimating the goodness of fit of empirical distributions. Annals of Mathematical Statistics 19, 279281.
Turner, P. (2010) Power properties of the CUSUM and CUSUMSQ tests for parameter instability. Applied Economic Letters 17, 10491053.
Wang, Q. (2015) Limit Theorems for Nonlinear Cointegration Regression. World Scientific.
Wang, Q. & Phillips, P.C.B. (2009) Structural nonparametric cointegrating regression. Econometrica 77, 19011948.
Wang, Q. & Phillips, P.C.B. (2012) A specification test for nonlinear nonstationary models. Annals of Statistics 40, 727758.
Wu, C.F. (1981) Asymptotic theory of nonlinear least squares estimation. Annals of Statistics 9, 501513.
Xiao, Z. & Phillips, P.C.B. (2002) A CUSUM test for cointegration using regression residuals. Journal of Econometrics 108, 4361.
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Econometric Theory
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