We establish general and versatile results regarding the limit behavior of the partial-sum process of ARMAX residuals. Illustrations include ARMA with seasonal dummies, misspecified ARMAX models with autocorrelated errors, nonlinear ARMAX models, ARMA with a structural break, a wide range of ARMAX models with infinite-variance errors, weak GARCH models and the consistency of kernel estimation of the density of ARMAX errors. Our results identify the limit distributions, and provide a general algorithm to obtain pivot statistics for CUSUM tests.
References
[2] Bai, J. (1993). On the partial sums of residuals in autoregressive and moving average models. J. Time Series Anal. 14 247–260. 0768.62073 10.1111/j.1467-9892.1993.tb00142.x[2] Bai, J. (1993). On the partial sums of residuals in autoregressive and moving average models. J. Time Series Anal. 14 247–260. 0768.62073 10.1111/j.1467-9892.1993.tb00142.x
[3] Bai, J. (1993). Estimation of structural change based on Wald-type statistics. Technical Report. Dept. Economics, Massachusetts Institute of Technology, Cambridge, MA.[3] Bai, J. (1993). Estimation of structural change based on Wald-type statistics. Technical Report. Dept. Economics, Massachusetts Institute of Technology, Cambridge, MA.
[4] Bai, J. (1994). Weak convergence of the sequential empirical processes of residuals in ARMA models. Ann. Statist. 22 2051–2061. 0826.60016 10.1214/aos/1176325771 euclid.aos/1176325771[4] Bai, J. (1994). Weak convergence of the sequential empirical processes of residuals in ARMA models. Ann. Statist. 22 2051–2061. 0826.60016 10.1214/aos/1176325771 euclid.aos/1176325771
[5] Bai, J. (1994). Least squares estimation of a shift in linear processes. J. Time Series Anal. 15 453–472. 0808.62079 10.1111/j.1467-9892.1994.tb00204.x[5] Bai, J. (1994). Least squares estimation of a shift in linear processes. J. Time Series Anal. 15 453–472. 0808.62079 10.1111/j.1467-9892.1994.tb00204.x
[6] Basrak, B. and Krizmanić, D. (2014). A limit theorem for moving averages in the $\alpha$-stable domain of attraction. Stochastic Process. Appl. 124 1070–1083. 1314.60085 10.1016/j.spa.2013.10.006[6] Basrak, B. and Krizmanić, D. (2014). A limit theorem for moving averages in the $\alpha$-stable domain of attraction. Stochastic Process. Appl. 124 1070–1083. 1314.60085 10.1016/j.spa.2013.10.006
[7] Berenguer-Rico, V. and Nielsen, B. (2015). Cumulated sum of squares statistics for non-linear and non-stationary regressions. Technical Report. Nuffield Discussion Paper 2015-W09, Univ. Oxford. 1436.62412 10.1017/S0266466618000476[7] Berenguer-Rico, V. and Nielsen, B. (2015). Cumulated sum of squares statistics for non-linear and non-stationary regressions. Technical Report. Nuffield Discussion Paper 2015-W09, Univ. Oxford. 1436.62412 10.1017/S0266466618000476
[8] Beveridge, S. and Nelson, C. (1981). A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycles. J. Monet. Econ. 7 151–174.[8] Beveridge, S. and Nelson, C. (1981). A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycles. J. Monet. Econ. 7 151–174.
[9] Bierens, H. J. (1994). Topics in Advanced Econometrics: Estimation, Testing, and Specification of Cross-Section and Time Series Models. Cambridge Univ. Press, Cambridge. 0806.62093[9] Bierens, H. J. (1994). Topics in Advanced Econometrics: Estimation, Testing, and Specification of Cross-Section and Time Series Models. Cambridge Univ. Press, Cambridge. 0806.62093
[10] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. 0172.21201[10] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. 0172.21201
[11] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. Wiley, New York. 0822.60002[11] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. Wiley, New York. 0822.60002
[12] Boldin, M. V. (1982). An estimate of the distribution of the noise in an autoregressive scheme. Teor. Veroyatn. Primen. 27 805–810. 0499.62083[12] Boldin, M. V. (1982). An estimate of the distribution of the noise in an autoregressive scheme. Teor. Veroyatn. Primen. 27 805–810. 0499.62083
[13] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. Springer, New York. 0709.62080[13] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. Springer, New York. 0709.62080
[14] Brown, R. L., Durbin, J. and Evans, J. M. (1975). Techniques for testing the constancy of regression relationships over time. J. Roy. Statist. Soc. Ser. B 37 149–192. 0321.62063 10.1111/j.2517-6161.1975.tb01532.x[14] Brown, R. L., Durbin, J. and Evans, J. M. (1975). Techniques for testing the constancy of regression relationships over time. J. Roy. Statist. Soc. Ser. B 37 149–192. 0321.62063 10.1111/j.2517-6161.1975.tb01532.x
[15] Chan, N. H. and Ling, S. (2008). Residual empirical processes for long and short memory time series. Ann. Statist. 36 2453–2470. 1205.62128 10.1214/07-AOS543 euclid.aos/1223908099[15] Chan, N. H. and Ling, S. (2008). Residual empirical processes for long and short memory time series. Ann. Statist. 36 2453–2470. 1205.62128 10.1214/07-AOS543 euclid.aos/1223908099
[16] Dahlhaus, R. and Wefelmeyer, W. (1996). Asymptotically optimal estimation in misspecified time series models. Ann. Statist. 24 952–974. 0865.62063 10.1214/aos/1032526951 euclid.aos/1032526951[16] Dahlhaus, R. and Wefelmeyer, W. (1996). Asymptotically optimal estimation in misspecified time series models. Ann. Statist. 24 952–974. 0865.62063 10.1214/aos/1032526951 euclid.aos/1032526951
[19] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1007. 0491.62099 10.2307/1912773[19] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1007. 0491.62099 10.2307/1912773
[20] Engler, E. and Nielsen, B. (2009). The empirical process of autoregressive residuals. Econom. J. 12 367–381. 1206.62147 10.1111/j.1368-423X.2009.00282.x[20] Engler, E. and Nielsen, B. (2009). The empirical process of autoregressive residuals. Econom. J. 12 367–381. 1206.62147 10.1111/j.1368-423X.2009.00282.x
[21] Epanečnikov, V. A. (1969). Nonparametric estimation of a multidimensional probability density. Teor. Veroyatn. Primen. 14 156–162. 0194.50001[21] Epanečnikov, V. A. (1969). Nonparametric estimation of a multidimensional probability density. Teor. Veroyatn. Primen. 14 156–162. 0194.50001
[22] Francq, C. and Sucarrat, G. (2017). An equation-by-equation estimator of a multivariate log-GARCH-X model of financial returns. J. Multivariate Anal. 153 16–32. 1351.62164 10.1016/j.jmva.2016.09.010[22] Francq, C. and Sucarrat, G. (2017). An equation-by-equation estimator of a multivariate log-GARCH-X model of financial returns. J. Multivariate Anal. 153 16–32. 1351.62164 10.1016/j.jmva.2016.09.010
[24] Francq, C. and Zakoïan, J.-M. (2000). Estimating weak GARCH representations. Econometric Theory 16 692–728. 0967.62065 10.1017/S0266466600165041[24] Francq, C. and Zakoïan, J.-M. (2000). Estimating weak GARCH representations. Econometric Theory 16 692–728. 0967.62065 10.1017/S0266466600165041
[26] Ghoudi, K. and Rémillard, B. (2015). Diagnostic tests for innovations of ARMA models using empirical processes of residuals. In Asymptotic Laws and Methods in Stochastics. Fields Inst. Commun. 76 239–282. 1365.62340[26] Ghoudi, K. and Rémillard, B. (2015). Diagnostic tests for innovations of ARMA models using empirical processes of residuals. In Asymptotic Laws and Methods in Stochastics. Fields Inst. Commun. 76 239–282. 1365.62340
[27] Grønneberg, S. and Holcblat, B. (2019). Supplement to “On partial-sum processes of ARMAX residuals.” DOI:10.1214/18-AOS1776SUPP.[27] Grønneberg, S. and Holcblat, B. (2019). Supplement to “On partial-sum processes of ARMAX residuals.” DOI:10.1214/18-AOS1776SUPP.
[28] Hannan, E. J. and Deistler, M. (2012). The Statistical Theory of Linear Systems. Classics in Applied Mathematics 70. SIAM, Philadelphia, PA. 1239.01122[28] Hannan, E. J. and Deistler, M. (2012). The Statistical Theory of Linear Systems. Classics in Applied Mathematics 70. SIAM, Philadelphia, PA. 1239.01122
[29] Hannan, E. J., Dunsmuir, W. T. M. and Deistler, M. (1980). Estimation of vector ARMAX models. J. Multivariate Anal. 10 275–295. 0445.62098 10.1016/0047-259X(80)90050-0[29] Hannan, E. J., Dunsmuir, W. T. M. and Deistler, M. (1980). Estimation of vector ARMAX models. J. Multivariate Anal. 10 275–295. 0445.62098 10.1016/0047-259X(80)90050-0
[30] Hannan, E. J. and Quinn, B. G. (1979). The determination of the order of an autoregression. J. Roy. Statist. Soc. Ser. B 41 190–195. 0408.62076 10.1111/j.2517-6161.1979.tb01072.x[30] Hannan, E. J. and Quinn, B. G. (1979). The determination of the order of an autoregression. J. Roy. Statist. Soc. Ser. B 41 190–195. 0408.62076 10.1111/j.2517-6161.1979.tb01072.x
[32] Klüppelberg, C. and Mikosch, T. (1996). Parameter estimation for a misspecified ARMA model with infinite variance innovations. J. Math. Sci. 78 60–65.[32] Klüppelberg, C. and Mikosch, T. (1996). Parameter estimation for a misspecified ARMA model with infinite variance innovations. J. Math. Sci. 78 60–65.
[33] Koreisha, S. G. and Fang, Y. (2001). Generalized least squares with misspecified serial correlation structures. J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 515–531. 0989.62048 10.1111/1467-9868.00296[33] Koreisha, S. G. and Fang, Y. (2001). Generalized least squares with misspecified serial correlation structures. J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 515–531. 0989.62048 10.1111/1467-9868.00296
[34] Koul, H. L. and Levental, S. (1989). Weak convergence of the residual empirical process in explosive autoregression. Ann. Statist. 17 1784–1794. 0695.60042 10.1214/aos/1176347395 euclid.aos/1176347395[34] Koul, H. L. and Levental, S. (1989). Weak convergence of the residual empirical process in explosive autoregression. Ann. Statist. 17 1784–1794. 0695.60042 10.1214/aos/1176347395 euclid.aos/1176347395
[35] Krämer, W., Ploberger, W. and Alt, R. (1988). Testing for structural change in dynamic models. Econometrica 56 1355–1369. 0655.62107 10.2307/1913102[35] Krämer, W., Ploberger, W. and Alt, R. (1988). Testing for structural change in dynamic models. Econometrica 56 1355–1369. 0655.62107 10.2307/1913102
[36] Lee, S. (1997). A note on the residual empirical process in autoregressive models. Statist. Probab. Lett. 32 405–411. 0901.62111 10.1016/S0167-7152(96)00100-9[36] Lee, S. (1997). A note on the residual empirical process in autoregressive models. Statist. Probab. Lett. 32 405–411. 0901.62111 10.1016/S0167-7152(96)00100-9
[37] Leeb, H. and Pötscher, B. M. (2008). Can one estimate the unconditional distribution of post-model-selection estimators? Econometric Theory 24 338–376. 1284.62152[37] Leeb, H. and Pötscher, B. M. (2008). Can one estimate the unconditional distribution of post-model-selection estimators? Econometric Theory 24 338–376. 1284.62152
[38] MacNeill, I. B. (1978). Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times. Ann. Statist. 6 422–433. 0375.62064 10.1214/aos/1176344133 euclid.aos/1176344133[38] MacNeill, I. B. (1978). Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times. Ann. Statist. 6 422–433. 0375.62064 10.1214/aos/1176344133 euclid.aos/1176344133
[39] Mikosch, T., Gadrich, T., Klüppelberg, C. and Adler, R. J. (1995). Parameter estimation for ARMA models with infinite variance innovations. Ann. Statist. 23 305–326. 0822.62076 10.1214/aos/1176324469 euclid.aos/1176324469[39] Mikosch, T., Gadrich, T., Klüppelberg, C. and Adler, R. J. (1995). Parameter estimation for ARMA models with infinite variance innovations. Ann. Statist. 23 305–326. 0822.62076 10.1214/aos/1176324469 euclid.aos/1176324469
[40] Nielsen, B. and Sohkanen, J. S. (2011). Asymptotic behavior of the CUSUM of squares test under stochastic and deterministic time trends. Econometric Theory 27 913–927. 1219.62139 10.1017/S0266466610000563[40] Nielsen, B. and Sohkanen, J. S. (2011). Asymptotic behavior of the CUSUM of squares test under stochastic and deterministic time trends. Econometric Theory 27 913–927. 1219.62139 10.1017/S0266466610000563
[41] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Stat. 33 1065–1076. 0116.11302 10.1214/aoms/1177704472 euclid.aoms/1177704472[41] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Stat. 33 1065–1076. 0116.11302 10.1214/aoms/1177704472 euclid.aoms/1177704472
[42] Phillips, P. C. B. and Solo, V. (1992). Asymptotics for linear processes. Ann. Statist. 20 971–1001. 0759.60021 10.1214/aos/1176348666 euclid.aos/1176348666[42] Phillips, P. C. B. and Solo, V. (1992). Asymptotics for linear processes. Ann. Statist. 20 971–1001. 0759.60021 10.1214/aos/1176348666 euclid.aos/1176348666
[43] Ploberger, W. and Krämer, W. (1987). Mean adjustment and the CUSUM test for structural change. Econom. Lett. 25 255–258. 1328.62436 10.1016/0165-1765(87)90223-0[43] Ploberger, W. and Krämer, W. (1987). Mean adjustment and the CUSUM test for structural change. Econom. Lett. 25 255–258. 1328.62436 10.1016/0165-1765(87)90223-0
[44] Ploberger, W. and Krämer, W. (1992). The CUSUM test with OLS residuals. Econometrica 60 271–285. 0744.62155 10.2307/2951597[44] Ploberger, W. and Krämer, W. (1992). The CUSUM test with OLS residuals. Econometrica 60 271–285. 0744.62155 10.2307/2951597
[45] Ploberger, W. and Krämer, W. (1996). A trend-resistant test for structural change based on OLS residuals. J. Econometrics 70 175–185. 0834.62085 10.1016/0304-4076(94)01688-7[45] Ploberger, W. and Krämer, W. (1996). A trend-resistant test for structural change based on OLS residuals. J. Econometrics 70 175–185. 0834.62085 10.1016/0304-4076(94)01688-7
[47] Robinson, P. M. (1987). Time series residuals with application to probability density estimation. J. Time Series Anal. 8 329–344. MR903762 0625.62071 10.1111/j.1467-9892.1987.tb00445.x[47] Robinson, P. M. (1987). Time series residuals with application to probability density estimation. J. Time Series Anal. 8 329–344. MR903762 0625.62071 10.1111/j.1467-9892.1987.tb00445.x
[49] Shin, D. W. and Lee, J. H. (2000). Consistency of the maximum likelihood estimators for nonstationary ARMA regressions with time trends. J. Statist. Plann. Inference 87 55–68. 0976.62090 10.1016/S0378-3758(99)00175-5[49] Shin, D. W. and Lee, J. H. (2000). Consistency of the maximum likelihood estimators for nonstationary ARMA regressions with time trends. J. Statist. Plann. Inference 87 55–68. 0976.62090 10.1016/S0378-3758(99)00175-5
[50] Shorack, G. R. and Wellner, J. A. (2009). Empirical Processes with Applications to Statistics. Classics in Applied Mathematics 59. SIAM, Philadelphia, PA. 1171.62057[50] Shorack, G. R. and Wellner, J. A. (2009). Empirical Processes with Applications to Statistics. Classics in Applied Mathematics 59. SIAM, Philadelphia, PA. 1171.62057
[51] Sucarrat, G., Grønneberg, S. and Escribano, A. (2016). Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown. Comput. Statist. Data Anal. 100 582–594. MR3505819 06918352 10.1016/j.csda.2015.12.005[51] Sucarrat, G., Grønneberg, S. and Escribano, A. (2016). Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown. Comput. Statist. Data Anal. 100 582–594. MR3505819 06918352 10.1016/j.csda.2015.12.005
[52] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, New York. 1176.62032[52] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, New York. 1176.62032
[53] Wright, J. H. (1993). The CUSUM test based on least squares residuals in regressions with integrated variables. Econom. Lett. 41 353–358. 0794.62076 10.1016/0165-1765(93)90204-P[53] Wright, J. H. (1993). The CUSUM test based on least squares residuals in regressions with integrated variables. Econom. Lett. 41 353–358. 0794.62076 10.1016/0165-1765(93)90204-P
[54] Xiao, Z. and Phillips, P. C. B. (2002). A CUSUM test for cointegration using regression residuals. J. Econometrics 108 43–61. 1020.62082 10.1016/S0304-4076(01)00103-8[54] Xiao, Z. and Phillips, P. C. B. (2002). A CUSUM test for cointegration using regression residuals. J. Econometrics 108 43–61. 1020.62082 10.1016/S0304-4076(01)00103-8
[55] Yu, H. (2007). High moment partial sum processes of residuals in ARMA models and their applications. J. Time Series Anal. 28 72–91. MR2332851 1164.62059 10.1111/j.1467-9892.2006.00499.x[55] Yu, H. (2007). High moment partial sum processes of residuals in ARMA models and their applications. J. Time Series Anal. 28 72–91. MR2332851 1164.62059 10.1111/j.1467-9892.2006.00499.x