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Geng, Pei (1-ILS)
Estimation of functional-coefficient autoregressive models with measurement
error. (English summary)J. Multivariate Anal. 192 (2022), Paper No. 105077, 16 pp.

Consider the time series {Yt , t = 1, . . . , n} generated by the p-th order functionalcoefficient autoregressive (FAR) modelYt =pX

αj (Ut )Yt−j + εt .

j=1

It is often the case that time series data can be observed with measurement error. In lightof this, this article is concerned with estimation of the FAR model when the true timeseries {Yt , t = 1, . . . , n} is unobservable but the surrogate time series {Zt , t = 1, . . . , n}
is observable. It is assumed that the relation between Yt and Zt can be expressed as theclassical measurement error model
Zt = Yt + et ,t = 1, . . . , n,where the measurement error et is independent of Yt , while the covariate Ut that
determines the coefficients α1 (·), . . . , αp (·) is assumed to be free of measurement error.
The estimation theory for the αj (·) can be viewed as a FAR analog to those demonstrated by J. Staudenmayer and J. P. Buonaccorsi [J. Amer. Statist. Assoc. 100 (2005),
no. 471, 841–852; MR2201013] for AR(p) models in the presence of measurement error. Local linear regression smoothing is adopted as the basic estimation strategy forthe vector of functional coefficients α0 = (α1 (u0 ), . . . , αp (u0 ))> at a design point u0 .
However, naive local linear estimation using the mismeasured Zt in place of Yt is inconsistent, due to the non-vanishing bias term that depends on the measurement error
variance σe2 := E(e2t ), as in [op. cit.].
As long as σe2 is known, the bias term is consistently estimable and thus a consistent, bias-corrected estimator for α0 can be obtained. The bias-corrected estimator
bab = αb (u0 ) takes the form
−1>11bZ >
Zba =HWDu − ΩWZ,DuDZunn
where H := [Ip , 0p×p ], W := diag{Kh (U1 − u0 ), . . . , Kh (Un − u0 )}, Kh (·) := K(·/h)/he t := (Zt−1 , . . . , Zt−p )> , Z :=for a kernel function K(·) and a bandwidth h, X>
(Z1 , . . . , Zn ) ,

e>e>

X
X
11 (U1 − u0 )/h1 µ1
 .
2.
b.
.
DZ:=,
Ω:=σI
⊗
f(u),
 .
U0ue p.
µ1 µ2>>
en Xe n (Un − u0 )/hXPnfbU (u0 ) := (1/n) t=1 Kw (Ut − u0 ) is a kernel density estimator
for the stationaryR
distribution of Ut using yet another bandwidth w, and µk := uk K(u)du. Notice thatZ>
the term (1/n)(DZu ) WDu − Ω may be nonpositive definite in finite samples. Becausebb
a is not well-defined in this case, a finite sample correction of the term is also considered.
Asymptotic normality and weak uniform consistency with a rate of the estimatorb
ab are established under some regularity conditions, including strict stationarity andstrong mixing of (Ut , Xt , Yt ) for Xt = (Yt−1 , . . . , Yt−p )> . A consistent estimator of theregression error variance σε2 is also proposed.
Masayuki Hirukawa
References

1. S.B. Aruoba, F.X. Diebold, J. Nalewaik, F. Schorfheide, D. Song, Improving GDP
measurement: A measurement-error perspective, J. Econometrics 191 (2) (2016)
384–397. MR3463324
2. R. Ashley, D. Vaughan, Measuring measurement error in economic time series, J.
Bus. Econom. Statist. 4 (1) (1986) 95–103.
3. J.P. Buonaccorsi, Measurement Error: Models, Methods, and Applications, Chapman and Hall/CRC, Boca Raton, FL, 2010. MR2682774
4. J.P. Buonaccorsi, J. Staudenmayer, Statistical methods to correct for observation
error in a density-independent population model, Ecol. Monograph 79 (2) (2009)299–324.
5. Z. Cai, J. Fan, R. Li, Efficient estimation and inferences for varying-coefficientmodels, J. Amer. Statist. Assoc. 95 (451) (2000) 888–902. MR1804446
6. Z. Cai, J. Fan, Q. Yao, Functional-coefficient regression models for nonlinear timeseries, J. Amer. Statist. Assoc. 95 (451) (2000) 941–956. MR1804449
7. R. Chen, R.S. Tsay, Functional-coefficient autoregressive models, J. Amer. Statist.
Assoc. 88 (421) (1993) 298–308. MR1212492
8. J. Dedecker, A. Samson, M.-L. Taupin, Estimation in autoregressive model withmeasurement error, ESAIM Probab. Stat. 18 (2014) 277–307. MR3333991
9. H. Dong, T. Otsu, L. Taylor, Estimation of varying coefficient models with measurement error, J. Econometrics (2021). MR4466730
10. J. Fan, W. Zhang, Statistical estimation in varying coefficient models, Ann. Statist.
27 (5) (1999) 1491–1518. MR1742497
11. R.S. Gibson, U.R. Charrondiere, W. Bell, Measurement errors in dietary assessment
using self-reported 24-hour recalls in low-income countries and strategies for theirprevention, Adv. Nutr. 8 (6) (2017) 980–991.
12. G.T. Goldman, J.A. Mulholland, A.G. Russell, M.J. Strickland, M. Klein, L.A.
Waller, P.E. Tolbert, Impact of exposure measurement error in air pollution epidemiology: effect of error type in time-series studies, Environ. Health 10 (1) (2011)
1–11.
13. T. Hastie, R. Tibshirani, Varying-coefficient models, J. R. Stat. Soc. Ser. B Stat.
Methodol. 55 (4) (1993) 757–779. MR1229881
14. J.Z. Huang, H. Shen, Functional coefficient regression models for non-linear timeseries: a polynomial spline approach, Scand. J. Stat. 31 (4) (2004) 515–534.
MR2101537
15. Y.K. Lee, E. Mammen, B.U. Park, Flexible generalized varying coefficient regression
models, Ann. Statist. 40 (3) (2012) 1906–1933. MR3015048
16. L. Li, T. Greene, Varying coefficients model with measurement error, Biometrics 64(2) (2008) 519–526. MR2432422
17. O. Morgenstern, On the Accuracy of Economic Observations, Princeton UniversityPress, Princeton, NJ, 1963.
18. J. Staudenmayer, J.P. Buonaccorsi, Measurement error in linear autoregressivemodels, J. Amer. Statist. Assoc. 100 (471) (2005) 841–852. MR2201013
19. H. Wang, Y. Xia, Shrinkage estimation of the varying coefficient model, J. Amer.
Statist. Assoc. 104 (486) (2009) 747–757. MR2541592
20. D. Wied, R. Weißbach, Consistency of the kernel density estimator: a survey, Statist.

Papers 53 (1) (2012) 1–21. MR2878587
21. Y. Xia, W.K. Li, On the estimation and testing of functional-coefficient linear
models, Statist. Sinica (1999) 735–757. MR1711643
22. L. Xue, L. Yang, Additive coefficient modeling via polynomial spline, Statist. Sinica(2006) 1423–1446. MR2327498
23. J. You, Y. Zhou, G. Chen, Corrected local polynomial estimation in varyingcoefficient models with measurement errors, Canad. J. Statist. 34 (3) (2006) 391–410.
MR2328551
Note: This list reflects references listed in the original paper as
accurately as possible with no attempt to correct errors.