Open House, ISM in Tachikawa, 2014.6.13統計数理研究所オープンハウス（立川）、H26.6.13ポスター発

Yoshimasa Uematsu

植松 良公

English

RISM (Repository of the Institute of Statistical Mathematics) / 統計数理研究所 学術研究リポジトリ

2014年6月13日　統計数理研究所　オープンハウスPenalized Likelihood Estimationin High-Dimensional Time Series Models植松　良公 統計的機械学習研究センター　日本学術振興会特別研究員（PD）1 IntroductionAim: Construct a general estimation method for high-dim. timeseries models by penalized QML that gives sparse estimates.Examples: K-dim. VAR(r) model is defined byyt =F1yt 1+   +Fryt r+ et; (1)which has K2r parameters. K-dim. MGARCH(1,1) is given byyt = S1=2t et; St =CC>+A>yt 1y>t 1A+B>St 1B;which has K(5K+1)=2 parameters.2 General Theory2.1 The model and its PQML estimatorModel: Let fytgTt=1 be a vector stationary process with a con-tinuous conditional density g(ytjyt 1;yt 2; : : :). Consider a para-metric family of densities f f (ytjyt 1;yt 2;    : q) : q 2Qg s.t.: p := dim(q) = O(nd) for some d > 0, so possibly p> n; the “true value” q 0, the unique minimizer of the KLIC of grelative to f , is sparse.Define some notation more precisely:M0 = f j 2 f1; : : : ; pg : q 0j 6= 0g andM c0 = f1; : : : ; pgnM0; q 0M0 is the q-dim. subvector of q 0 composed of the nonzeroelements fq 0j : j 2M0g; q 0M c0 is the (p q)-dim. subvector of q 0 composed of zeros.Estimator: The PQML estimator qˆ of q 0 is defined byQn(qˆ) =maxq2QQn(q) with Qn(q) := Ln(q) Pn(q);where Ln(q) := n 1ånt=1 log f (ytjYt 1 : q) is the quasi-log-likelihood and Pn(q) := åpj=1 pl(jq jj) is the penalty term suchas L1-penalty (lasso), SCAD, MCP, etc., with l (= ln)! 0.2.2 Theoretical resultsTheorem 1 (Weak oracle property)Under regularity condi-tions, there is a local maximizer qˆ = (qˆ>M0; qˆ>M c0)> of Qn(q) s.t.:(a) P(qˆM c0 = 0)! 1; (b) kqˆM0 q 0M0k¥ = Op(n g logn).Corollary 1 (L1-penalized QML estimator)Under regularityconditions in Theorem 1, there is a local maximizer qˆ =(qˆ>M0; qˆ>M c0)> of QL1n(q) s.t. Thm. 1 (a) and (b) hold.Theorem 2 (Oracle property)Under regularity conditions,there is a local maximizer qˆ = (qˆ>M0; qˆ>M c0)> of Qn(q) s.t.:(a) P(qˆM c0 = 0)! 1; (b) kqˆM0 q 0M0k= Op(n 1=2).If a stronger assumption is added to the penalty, we have(c) (Asy. N) n1=2 qˆM0 q 0M0!d N  0; (J0M0) 1I0M0(J0>M0) 1.3 Application to VAR3.1 Theoretical result for VARConsider (1) with et  i:i:d: (0;Se). Let q 0 = vec(F01; : : : ;F0r) 2Rp with p= K2r, which is supposed sparse. Using some appro-priate S instead of unknown Se, we have:Proposition 1Under some moment and stability conditions,Thm. 2 (a) – (c) hold for qˆ in (1), where I0M0 = P>M0(GS 1SeS 1)P>M0 and J0M0= P>M0(GS 1)PM0 with G= E[xtx>t ].3.2 Empirical studyCompare performances of sparse VAR and dynamic Nelson-Siegel (DNS) model in terms of yield curve forecasting.Data: Zero-coupon US government bond yields that are:monthly from January 1986 to December 2007;made of 8 maturities t = 3;6;12;24;36;60;84;120 months.Model 1: DNS model is defined byytt = b1t +b2t1  e htthtt+b3t1  e htthtt  e htt;bit = ai+bibi;t h+uit for each i= 1;2;3:where b1t, b2t and b3t may be interpreted as latent dynamic fac-tors and ht is a sequence of tuning parameters.Model 2: In sVAR strategy, the model is specified as 8-dim.VAR(12) below and is estimated by SCAD penalized QML.0BB@Dy3;tDy6;t...Dy120;t1CCA=F10BB@Dy3;t 1Dy6;t 1...Dy120;t 11CCA+   +F120BB@Dy3;t 12Dy6;t 12...Dy120;t 121CCA+ et:Forecasting strategy: The two models are estimated recur-sively, using the data from Jan. 1986 to the time that theh(= 1;3;6;12)-month-ahead forecast is made, beginning in Jan.2001 and extending through Dec. 2007.Result: The comparison result is summarized below:Table 1: Relative RMSEs of forecasting (sVAR/DNS)h n t 3 6 12 24 36 60 84 1201 0.356 0.301 0.288 0.279 0.266 0.254 0.258 0.2753 0.418 0.393 0.358 0.345 0.333 0.324 0.329 0.3566 0.557 0.513 0.443 0.405 0.391 0.379 0.381 0.40012 0.625 0.591 0.540 0.492 0.468 0.442 0.435 0.445

Penalized Likelihood Estimation in High-Dimensional Time Series Models

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Penalized Likelihood Estimation in High-Dimensional Time Series Models

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Penalized Likelihood Estimation in High-Dimensional Time Series Models

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