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Sample covariances of random-coefficient AR(1) panel model. (English) Zbl 1440.62336
Summary: The present paper obtains a complete description of the limit distributions of sample covariances in \(N\times n\) panel data when \(N\) and \(n\) jointly increase, possibly at different rate. The panel is formed by \(N\) independent samples of length \(n\) from random-coefficient AR(1) process with the tail distribution function of the random coefficient regularly varying at the unit root with exponent \(\beta >0\). We show that for \(\beta\in (0,2)\) the sample covariances may display a variety of stable and non-stable limit behaviors with stability parameter depending on \(\beta\) and the mutual increase rate of \(N\) and \(n\).

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62D20 Causal inference from observational studies
60F05 Central limit and other weak theorems
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62E20 Asymptotic distribution theory in statistics
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[1] Beran, J., Schützner, M. and Ghosh, S. (2010) From short to long memory: Aggregation and estimation., Comput. Stat. Data Anal. 54, 2432-2442. · Zbl 1284.62544
[2] Bhansali, R.J., Giraitis, L. and Kokoszka, P.S. (2007) Convergence of quadratic forms with nonvanishing diagonal., Statist. Probab. Lett. 77, 726-734. · Zbl 1283.62027
[3] Celov, D., Leipus, R. and Philippe, A. (2007) Time series aggregation, disaggregation and long memory., Lithuanian Math. J. 47, 379-393. · Zbl 05521796
[4] Celov, D., Leipus, R. and Philippe, A. (2010) Asymptotic normality of the mixture density estimator in a disaggregation scheme., J. Nonparametric Statist. 22, 425-442. · Zbl 1282.62086
[5] Davis, R.A. and Mikosch, T. (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH., Ann. Statist. 26, 2049-2080. · Zbl 0929.62092
[6] Davis, R.A. and Resnick, S.I. (1986) Limit theory for the sample covariance and correlation functions of moving averages., Ann. Statist. 14, 533-588. · Zbl 0605.62092
[7] Giraitis, L., Koul, H.L. and Surgailis, D. (2012), Large Sample Inference for Long Memory Processes. London: Imperial College Press. · Zbl 1279.62016
[8] Goldie, C.M. and Smith, R.L. (1987) Slow variation with remainder: theory and applications., Q. J. Math. 38, 45-71. · Zbl 0611.26001
[9] Gonçalves, E. and Gouriéroux, C. (1988) Aggrégation de processus autoregressifs d’ordre 1., Annales d’Economie et de Statistique 12, 127-149.
[10] Granger, C.W.J. (1980) Long memory relationship and the aggregation of dynamic models., J. Econometrics 14, 227-238. · Zbl 0466.62108
[11] Hall, P. and Heyde, C.C. (1980), Martingale Limit Theory and Its Applications. Academic Press, New York. · Zbl 0462.60045
[12] Horváth, L. and Kokoszka, P. (2008) Sample autocovariances of long-memory time series., Bernoulli 14, 405-418. · Zbl 1155.62323
[13] Leipus, R., Oppenheim, G., Philippe, A. and Viano, M.-C. (2006) Orthogonal series density estimation in a disaggregation scheme., J. Statist. Plan. Inf. 136, 2547-2571. · Zbl 1090.62095
[14] Leipus, R., Philippe, A., Puplinskaitė, D. and Surgailis, D. (2014) Aggregation and long memory: recent developments., J. Indian Statist. Assoc. 52, 71-101.
[15] Leipus, R., Philippe, A., Pilipauskaitė, V. and Surgailis, D. (2017) Nonparametric estimation of the distribution of the autoregressive coefficient from panel random-coefficient AR(1) data., J. Multiv. Anal. 153, 121-135. · Zbl 1353.62045
[16] Leipus, R., Philippe, A., Pilipauskaitė, V. and Surgailis, D. (2019) Estimating long memory in panel random-coefficient AR(1) data. Preprint., 1710.09735.
[17] Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002) Is network traffic approximated by stable Lévy motion or fractional Brownian motion?, Ann. Appl. Probab. 12, 23-68. · Zbl 1021.60076
[18] Oppenheim, G. and Viano, M.-C. (2004) Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results., J. Time Ser. Anal. 25, 335-350. · Zbl 1064.60066
[19] Philippe, A., Puplinskaitė, D. and Surgailis, D. (2014) Contemporaneous aggregation of triangular array of random-coefficient AR(1) processes., J. Time Series Anal. 35, 16-39. · Zbl 1301.62092
[20] Pilipauskaitė, V. and Surgailis, D. (2014) Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes., Stochastic Process. Appl. 124, 1011-1035. · Zbl 1400.62194
[21] Pilipauskaitė, V. and Surgailis, D. (2015) Joint aggregation of random-coefficient AR(1) processes with common innovations., Statist. Probab. Letters 101, 73-82. · Zbl 1325.62171
[22] Pilipauskaitė, V. and Surgailis, D. (2016) Anisotropic scaling of random grain model with application to network traffic., J. Appl. Probab. 53, 857-879. · Zbl 1351.60064
[23] Pilipauskaitė, V. and Surgailis, D. (2017) Scaling transition for nonlinear random fields with long-range dependence., Stochastic Process. Appl. 127, 2751-2779. · Zbl 1373.60089
[24] Puplinskaitė, D. and Surgailis, D. (2009) Aggregation of random coefficient AR(1) process with infinite variance and common innovations., Lithuanian Math. J. 49, 446-463. · Zbl 1194.60026
[25] Puplinskaitė, D. and Surgailis, D. (2010) Aggregation of random coefficient AR(1) process with infinite variance and idiosyncratic innovations., Adv. Appl. Probab. 42, 509-527. · Zbl 1191.62154
[26] Puplinskaitė, D. and Surgailis, D. (2015) Scaling transition for long-range dependent Gaussian random fields., Stoch. Process. Appl. 125, 2256-2271. · Zbl 1317.60062
[27] Puplinskaitė, D. and Surgailis, D. (2016) Aggregation of autoregressive random fields and anisotropic long-range dependence., Bernoulli 22, 2401-2441. · Zbl 1356.60082
[28] Rajput, B.S. and Rosinski, J. (1989) Spectral representations of infinitely divisible processes., Probab. Theory Related Fields 82, 451-487. · Zbl 0659.60078
[29] Robinson, P.M. (1978) Statistical inference for a random coefficient autoregressive model., Scand. J. Stat. 5, 163-168.
[30] Samorodnitsky, G. and Taqqu, M.S. (1994), Stable Non-Gaussian Random Processes. Chapman and Hall, New York. · Zbl 0925.60027
[31] Surgailis, D. (2004) Stable limits of sums of bounded functions of long memory moving averages with finite variance., Bernoulli 10, 327-355. · Zbl 1076.62017
[32] Zaffaroni, P. (2004) Contemporaneous aggregation of linear dynamic models in large economies., J. Econometrics 120, 75-102. · Zbl 1282.91263
[33] Zaffaroni, P. (2007) Aggregation and memory of models of changing volatility., J. Econometrics 136, 237-249. · Zbl 1418.62360
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