Rank-based testing for semiparametric VAR models: a measure transportation approach. (English) Zbl 07634391

Hallin, Marc; La Vecchia, Davide; Liu, Hang

Bernoulli 29, No. 1, 229-273 (2023).

Summary: We develop a class of tests for semiparametric vector autoregressive (VAR) models with unspecified innovation densities based on the recent measure-transportation-based concepts of multivariate center-outward ranks and signs. We show that these concepts, combined with Le Cam’s asymptotic theory of statistical experiments, yield novel testing procedures, which (a) are valid under a broad class of innovation densities (possibly non-elliptical, skewed, and/or with infinite moments), (b) are optimal (locally asymptotically maximin or most stringent) at selected ones, and (c) are robust against additive outliers. In order to show this, we establish, for a general class of center-outward rank-based serial statistics, a Hájek asymptotic representation result, of independent interest, which allows for a rank-based reconstruction of central sequences. As an illustration, we consider the problems of testing the absence of serial correlation in multiple-output and possibly non-linear regression (an extension of the classical Durbin-Watson problem) and the sequential identification of the order \(p\) of a \(\mathrm{VAR}(p)\) model. A Monte Carlo comparative study of our tests and their routinely-applied Gaussian competitors demonstrates the benefits (in terms of size, power, and robustness) of our methodology; these benefits are particularly significant in the presence of asymmetric and leptokurtic innovation densities. A real-data application concludes the paper.

MSC:

62Gxx	Nonparametric inference
62Mxx	Inference from stochastic processes
62Hxx	Multivariate analysis

Keywords:

distribution-freeness; Durbin-Watson test; Hájek representation; local asymptotic normality; multivariate ranks; VAR order selection

References:

[1]	Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (Tsahkadsor, 1971) (B.N. Petrov and F. Csaki, eds.) 267-281. · Zbl 0283.62006
[2]	Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew \(t\)-distribution. J. R. Stat. Soc. Ser. B. Stat. Methodol. 65 367-389. 10.1111/1467-9868.00391 · Zbl 1065.62094
[3]	Benghabrit, Y. and Hallin, M. (1992). Optimal rank-based tests against first-order superdiagonal bilinear dependence. J. Statist. Plann. Inference 32 45-61. 10.1016/0378-3758(92)90151-H · Zbl 0752.62036
[4]	Benghabrit, Y. and Hallin, M. (1996). Rank-based tests for autoregressive against bilinear serial dependence. J. Nonparametr. Stat. 6 253-272. 10.1080/10485259608832674 · Zbl 0885.62054
[5]	Bertsekas, D.P. (1991). Linear Network Optimization: Algorithms and Codes. Cambridge, MA: MIT Press. · Zbl 0754.90059
[6]	Boldin, M.V. (2012). Robustness of sign tests in autoregression. Moscow Math. Bull. 67 112-115. · Zbl 1417.62239
[7]	Boldin, M.V. (2013). Robustness of sign tests for testing hypotheses about order of autoregression. Theory Probab. Appl. 57 663-670. 10.1137/S0040585X97986242 · Zbl 1281.62189
[8]	Burkard, R., Dell’Amico, M. and Martello, S. (2009). Assignment Problems. Philadelphia, PA: SIAM. 10.1137/1.9780898717754 · Zbl 1196.90002
[9]	Cassart, D., Hallin, M. and Paindaveine, D. (2010). On the estimation of cross-information quantities in rank-based inference. In Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: A Festschrift in Honor of Professor Jana Jurečková (J. Antoch, M. Hušková and P.K. Sen, eds.). Inst. Math. Stat. (IMS) Collect. 7 35-45. Beachwood, OH: IMS. · Zbl 1228.62003
[10]	Chernozhukov, V., Galichon, A., Hallin, M. and Henry, M. (2017). Monge-Kantorovich depth, quantiles, ranks and signs. Ann. Statist. 45 223-256. 10.1214/16-AOS1450 · Zbl 1426.62163
[11]	Croux, C. and Joossens, K. (2008). Robust estimation of the vector autoregressive model by a least trimmed squares procedure. COMPSTAT 2008 489-501. · Zbl 1148.62067
[12]	Deb, N., Bhattacharya, B. and Sen, B. (2019). Efficiency lower bounds for distribution-free Hotelling-type two-sample tests based on optimal transport. Available at arXiv:2104.01986.
[13]	Deb, N. and Sen, B. (2022). Multivariate rank-based distribution-free nonparametric testing using measure transportation. J. Amer. Statist. Assoc. To appear. · Zbl 1514.62089
[14]	del Barrio, E., González-Sanz, A. and Hallin, M. (2020). A note on the regularity of optimal-transport-based center-outward distribution and quantile functions. J. Multivariate Anal. 180 104671. 10.1016/j.jmva.2020.104671 · Zbl 1450.62047
[15]	Dick, J. and Pillichshammer, F. (2014). Discrepancy theory and quasi-Monte Carlo integration. In A Panorama of Discrepancy Theory (W. Chen, A. Sirvastava and G. Travaglini, eds.). Lecture Notes in Math. 2107 539-619. Cham: Springer. 10.1007/978-3-319-04696-9_9 · Zbl 1358.11086
[16]	Durbin, J. and Watson, G.S. (1950). Testing for serial correlation in least squares regression. I. Biometrika 37 409-428. 10.1093/biomet/37.3-4.409 · Zbl 0039.35803
[17]	Durbin, J. and Watson, G.S. (1951). Testing for serial correlation in least squares regression. II. Biometrika 38 159-178. 10.1093/biomet/38.1-2.159 · Zbl 0042.38201
[18]	Figalli, A. (2018). On the continuity of center-outward distribution and quantile functions. Nonlinear Anal. 177 413-421. 10.1016/j.na.2018.05.008 · Zbl 1433.62132
[19]	Garel, B. and Hallin, M. (1995). Local asymptotic normality of multivariate ARMA processes with a linear trend. Ann. Inst. Statist. Math. 47 551-579. · Zbl 0841.62076
[20]	Garel, B. and Hallin, M. (1999). Rank-based autoregressive order identification. J. Amer. Statist. Assoc. 94 1357-1371. 10.2307/2669948 · Zbl 0994.62089
[21]	Ghosal, P. and Sen, B. (2022). Multivariate ranks and quantiles using optimal transportation and applications to goodness-of-fit testing. Ann. Statist. To appear.
[22]	Hájek, J. (1961). Some extensions of the Wald-Wolfowitz-Noether theorem. Ann. Math. Stat. 32 506-523. 10.1214/aoms/1177705057 · Zbl 0107.13404
[23]	Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. New York: Academic Press. · Zbl 0161.38102
[24]	Hallin, M. (1986). Nonstationary \(q\)-dependent processes and time-varying moving-average models: Invertibility properties and the forecasting problem. Adv. in Appl. Probab. 18 170-210. 10.2307/1427242 · Zbl 0597.62096
[25]	Hallin, M. (1994). On the Pitman non-admissibility of correlogram-based methods. J. Time Series Anal. 15 607-611. 10.1111/j.1467-9892.1994.tb00215.x · Zbl 0807.62068
[26]	Hallin, M. (2017). On distribution and quantile functions, ranks and signs in \[{\mathbb{R}^d} \]. ECARES. Working paper.
[27]	Hallin, M., del Barrio, E., Cuesta-Albertos, J. and Matrán, C. (2021). Distribution and quantile functions, ranks and signs in dimension \(d\): A measure transportation approach. Ann. Statist. 49 1139-1165. 10.1214/20-aos1996 · Zbl 1468.62282
[28]	Hallin, M., Hlubinka, D. and Hudecová, Š. (2022). Fully distribution-free center-outward rank tests for multiple-output regression and MANOVA. J. Amer. Statist. Assoc. To appear.
[29]	Hallin, M., Ingenbleek, J.-F. and Puri, M.L. (1985). Linear serial rank tests for randomness against ARMA alternatives. Ann. Statist. 13 1156-1181. 10.1214/aos/1176349662 · Zbl 0584.62064
[30]	Hallin, M. and La Vecchia, D. (2017). R-estimation in semiparametric dynamic location-scale models. J. Econometrics 196 233-247. 10.1016/j.jeconom.2016.08.002 · Zbl 1403.91274
[31]	Hallin, M. and La Vecchia, D. (2020). A simple R-estimation method for semiparametric duration models. J. Econometrics 218 736-749. 10.1016/j.jeconom.2020.04.036 · Zbl 1464.62505
[32]	Hallin, M., La Vecchia, D. and Liu, H. (2022). Center-outward R-estimation for semiparametric VARMA models. J. Amer. Statist. Assoc. To appear. · Zbl 1507.62286
[33]	Hallin, M. and Mélard, G. (1988). Rank-based tests for randomness against first-order serial dependence. J. Amer. Statist. Assoc. 83 1117-1128.
[34]	Hallin, M., Oja, H. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. II. Optimal \(R\)-estimation of shape. Ann. Statist. 34 2757-2789. 10.1214/009053606000000948 · Zbl 1115.62059
[35]	Hallin, M. and Paindaveine, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103-1133. 10.1214/aos/1031689019 · Zbl 1101.62348
[36]	Hallin, M. and Paindaveine, D. (2004). Rank-based optimal tests of the adequacy of an elliptic VARMA model. Ann. Statist. 32 2642-2678. 10.1214/009053604000000724 · Zbl 1076.62044
[37]	Hallin, M. and Paindaveine, D. (2004). Multivariate signed-rank tests in vector autoregressive order identification. Statist. Sci. 19 697-711. 10.1214/088342304000000602 · Zbl 1100.62577
[38]	Hallin, M. and Paindaveine, D. (2005). Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors. J. Multivariate Anal. 93 122-163. 10.1016/j.jmva.2004.01.005 · Zbl 1087.62098
[39]	Hallin, M. and Paindaveine, D. (2006). Asymptotic linearity of serial and nonserial multivariate signed rank statistics. J. Statist. Plann. Inference 136 1-32. 10.1016/j.jspi.2004.05.013 · Zbl 1082.62049
[40]	Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Statist. 34 2707-2756. 10.1214/009053606000000731 · Zbl 1114.62066
[41]	Hallin, M. and Paindaveine, D. (2008). Optimal rank-based tests for homogeneity of scatter. Ann. Statist. 36 1261-1298. 10.1214/07-AOS508 · Zbl 1360.62288
[42]	Hallin, M., Paindaveine, D. and Verdebout, T. (2010). Optimal rank-based testing for principal components. Ann. Statist. 38 3245-3299. 10.1214/10-AOS810 · Zbl 1373.62295
[43]	Hallin, M., Paindaveine, D. and Verdebout, T. (2013). Optimal rank-based tests for common principal components. Bernoulli 19 2524-2556. 10.3150/12-BEJ461 · Zbl 1457.62182
[44]	Hallin, M., Paindaveine, D. and Verdebout, T. (2014). Efficient R-estimation of principal and common principal components. J. Amer. Statist. Assoc. 109 1071-1083. 10.1080/01621459.2014.880057 · Zbl 1368.62160
[45]	Hallin, M. and Puri, M.L. (1988). Optimal rank-based procedures for time series analysis: Testing an ARMA model against other ARMA models. Ann. Statist. 16 402-432. 10.1214/aos/1176350712 · Zbl 0659.62111
[46]	Hallin, M. and Puri, M.L. (1991). Time series analysis via rank order theory: Signed-rank tests for ARMA models. J. Multivariate Anal. 39 1-29. 10.1016/0047-259X(91)90002-J · Zbl 0751.62041
[47]	Hallin, M. and Puri, M.L. (1994). Aligned rank tests for linear models with autocorrelated error terms. J. Multivariate Anal. 50 175-237. 10.1006/jmva.1994.1040 · Zbl 0805.62050
[48]	Hallin, M. and Vermandele, C. (1996). An elementary proof of asymptotic normality for serial rank statistics. In Research Developments in Probability and Statistics (E. Brunner and M. Denker, eds.). 163-191. Utrecht: VSP. · Zbl 0874.62049
[49]	Hallin, M. and Werker, B.J.M. (2003). Semi-parametric efficiency, distribution-freeness and invariance. Bernoulli 9 137-165. 10.3150/bj/1068129013 · Zbl 1020.62042
[50]	Judd, K.L. (1998). Numerical Methods in Economics. Cambridge, MA: MIT Press. · Zbl 0924.65001
[51]	Kreiss, J.-P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112-133. 10.1214/aos/1176350256 · Zbl 0616.62042
[52]	Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. New York: Springer. 10.1007/978-1-4612-4946-7 · Zbl 0605.62002
[53]	Le Cam, L. and Yang, G.L. (2000). Asymptotics in Statistics: Some Basic Concepts, 2nd ed. Springer Series in Statistics. New York: Springer. 10.1007/978-1-4612-1166-2 · Zbl 0952.62002
[54]	Lehmann, E.L. and Romano, J.P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer Texts in Statistics. New York: Springer. · Zbl 1076.62018
[55]	Machado, J.A.F. (1993). Robust model selection and \(M\)-estimation. Econometric Theory 9 478-493. 10.1017/S0266466600007775
[56]	Mancini, L., Ronchetti, E. and Trojani, F. (2005). Optimal conditionally unbiased bounded-influence inference in dynamic location and scale models. J. Amer. Statist. Assoc. 100 628-641. 10.1198/016214504000001402 · Zbl 1117.62396
[57]	Mélard, G. (2022). An indirect proof for the asymptotic properties of VARMA model estimators. Econom. Stat. 21 96-111. 10.1016/j.ecosta.2020.12.004
[58]	Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics 63. Philadelphia, PA: SIAM. 10.1137/1.9781611970081 · Zbl 0761.65002
[59]	Peyré, G. and Cuturi, M. (2019). Computational optimal transport with applications to data science. Found. Trends Mach. Learn. 11 355-607.
[60]	Pötscher, B.M. (1983). Order estimation in ARMA-models by Lagrangian multiplier tests. Ann. Statist. 11 872-885. · Zbl 0521.62072
[61]	Ronchetti, E. (1997). Robustness aspects of model choice. Statist. Sinica 7 327-338. · Zbl 0900.62172
[62]	Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461-464. · Zbl 0379.62005 · doi:10.1214/aos/1176344136
[63]	Shi, H., Hallin, M., Drton, M. and Han, F. (2021). Center-outward sign- and rank-based quadrant, Spearman, and Kendall tests for multivariate independence. Available at arXiv:2111.15567.
[64]	Shi, H., Hallin, M., Drton, M. and Han, F. (2022). On universally consistent and fully distribution-free rank tests of vector independence. Ann. Statist. To appear. · Zbl 1539.62126
[65]	Shorack, G.R. (2000). Probability for Statisticians. Springer Texts in Statistics. New York: Springer. · Zbl 0951.62005
[66]	van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press. 10 · Zbl 0910.62001