Empirical likelihood and self-weighting approach for hypothesis testing of infinite variance processes and its applications

Transcription

1 Empirical likelihood and self-weighting approach for hypothesis testing of infinite variance processes and its applications Fumiya Akashi Research Associate Department of Applied Mathematics Waseda University August 17, 2016 Boston University/Keio University Workshop 2016 This work was supported by Grant-in-Aid for Young Scientists (B) (16K16022, Fumiya Akashi) 1 / 34

2 Outline Outline 1 Introduction 2 Fundamental settings 3 Main results 4 Numerical examples 5 Concluding remarks 2 / 34

3 Introduction Outline 1 Introduction 2 Fundamental settings 3 Main results 4 Numerical examples 5 Concluding remarks 3 / 34

4 Introduction Introduction P(y t > x) cx α (x ) Electrical engineering, Hydrology, Finance, physical systems Heavy-tailed data y 1,, y n α: tail-index 0 < α < 2 E[y 2 t ] =. 4 / 34

5 Introduction Introduction (cont.) Figure: Log-stock return process of Ford and Hill-plot for y t Log return X t Date H k,n n {y t : t Z} : Infinite variance process Example: Stable AR-model y t = β X t 1 + e t, X t 1 = (y t 1,..., y t p ) E[ X t 1 2 ] = 5 / 34

6 Introduction Introduction (cont.) Figure: Log-stock return process of Ford and Hill-plot for y t Log return X t Date H k,n n {y t : t Z} : Infinite variance process Example: Stable AR-model y t = β X t 1 + e t, X t 1 = (y t 1,..., y t p ) E[ X t 1 2 ] = 5 / 34

7 Introduction Introduction (cont.) Figure: Atmospheric pressure data in Chicago, 2015 (mbar) Jan. Feb. Mar. Apr Jan05 Jan12 Jan19 Jan26 Feb02 Feb09 Feb16 Feb23 Mar02 Mar09 Mar16 Mar23 Mar30 Apr06 Apr13 Apr20 Apr May. Jun. Jly. Aug. May04 May11 May18 May25 Jun01 Jun01 Jun08 Jun15 Jun22 Jun29 Jul06 Jul13 Jul20 Jul27 Aug03 Aug10 Aug17 Aug24 Aug31 Sep. Oct Nov Dec Sep07 Sep14 Sep21 Sep28 Oct05 Oct12 Oct19 Oct26 Nov02 Nov09 Nov16 Nov23 Nov30 Dec07 Dec14 Dec21 Dec28 6 / 34

8 EL & SW approach for hypothesis testing of IV-processes and its applications Introduction Introduction (cont.) Figure: QQ-plot for atmospheric pressure data in Chicago, 2015 (mbar) Jan. Feb. Mar. Apr Jun. May. Aug. Jly. Sep. Oct Dec Nov / 34

9 EL & SW approach for hypothesis testing of IV-processes and its applications Introduction Introduction (cont.) Figure: QQ-plot for atmospheric pressure data in Chicago, 2015 (mbar) Jan. Feb. Mar. Apr Jun. May. Aug. Jly. Sep. Oct Dec Nov Gaussian tail or Heavy tail?? (or Another tail??) 7 / 34

10 Introduction Introduction (cont.) EL approach for... Problem: Akashi, Liu & Taniguchi (2015) Stable process + FD-SN-EL r n (θ 0 ) Limit distributions contain α & σ (unknown). Self-Weighting (SW) approach (Ling (2005), Pan et al. (2007)) Least Absolution Deviation (LAD) based statistic (Chen et al. (2008)) Main aim L 1 -SW-EL based statistic for IV-process: ρ n := inf Rβ=c { 2 log r n(β)} inf β B { 2 log r n(β)} (r n: L 1 -SW-EL function) ρ n L χ 2 (pivotal limit distribution) 8 / 34

11 Fundamental settings Outline 1 Introduction 2 Fundamental settings 3 Main results 4 Numerical examples 5 Concluding remarks 9 / 34

12 Fundamental settings Fundamental settings Model: ARMA(p, q) process y t = p q b j y t j + e t + a j e t j, j=1 j=1 where β = (b 1,..., b p, a 1,..., a q ) Int(B) (B: compact parameter space), {e t : t Z}: i.i.d. r.v.s. with med(e 1 ) = 0. Remark 1 {e t : t Z} can be infinite variance r.v.s. 10 / 34

13 Fundamental settings Fundamental settings (cont.) Nested linear hypothesis: where H : Rβ = c { R : r (p + q) c : r 1 (r < p + q). Examples (i) R = (O q p, I q q ) & c = 0 q H : a j = 0 for j = 1,..., q (test for serial correlation) (ii) R = (I p1 p 1, O p1 (p+q p 1 )) & c = 0 p1 H : b j = 0 for j = 1,..., p 1 (variable selection) 11 / 34

14 Fundamental settings Fundamental settings (cont.) p q e t = y t b j y t j a j e t j & med(e t ) = 0 j=1 j=1 Self-weighted LAD estimator (Pan et al. (2007)) where ˆβ n = arg min β B Q(β), u max{p, q} + 1: some starting point, n Q(β) = w t ε t (β), t=u+1 { 0 (t 0) ε t (β) = y t p j=1 b jy t j q j=1 a jε t j (β) (1 t n) w t : self-weights of the form w t = w(y t 1, y t 2,...), 12 / 34

15 Fundamental settings Fundamental settings (cont.) Remark 2 Because of the truncation, ε t (β 0 ) e t but ε t (β 0 ) e t. Figure: Example of ε t (β) for AR(p) case (left) & general ARMA case 13 / 34

16 Fundamental settings Fundamental settings (cont.) Pan et al. (2007) showed that ˆβ n is asymptotically normal & SW-Wald test: W n := n ( Rˆβ n c ) Ĥ ( Rˆβ n c ) L χ 2 r. Problem: Ĥ contains unknown quantity f (0). Main aim Remove unknown quantities from statistic/limit distribution. 14 / 34

17 Fundamental settings Fundamental settings (cont.) Y 1,..., Y n i.i.d., F(x) Hypothesis: H : E F [Y i ] = 0 Nonparametric likelihood: Empirical distribution function L(F) = n {F(Y i ) F(Y i )} i=1 F n (y) := 1 n maximize L(F) (L(F n ) = n n ). Profile nonparametric likelihood ratio under H: where F H = F(y) = n I(Y i y) i=1 r n = sup{l(f) : F F H}, L(F n ) n v i I(Y i y) : i=1 n n v i Y i = 0, v i = 1, 0 v i 1 i=1 i=1 15 / 34

18 Fundamental settings Fundamental settings (cont.) For EL, we need moment condition E[ g t (β 0 )] = 0 p+q. Q(β): NOT differentiable w.r.t β. Q(β) = n t=u+1 ε t (β) 2 Minimizer of Q(β): Solution to n t=u+1 ε t (β) ε t(β) β = 0 p+q 16 / 34

19 Fundamental settings Fundamental settings (cont.) Q(β) = n t=u+1 w t ε t (β) Minimizer of Q(β): Solution to n t=u+1 w t sign {ε t (β)} ε t(β) β = 0 p+q Definition 1 (L 1 -based Self-weighted moment function) g t (β) := w t sign {ε t (β)} A t (β) (t = u + 1,..., n), where u max{p, q} + 1 (starting point), A t (β) = ε t(β) β. 17 / 34

20 Fundamental settings Fundamental settings (cont.) Example of self-weights (Pan et al. (2007)) t 1 w t = 1 + k γ y t k k=1 2 (γ > 2) Remark 3 Numerical results are not sensitive w.r.t. choice of γ or w t s. 18 / 34

21 Fundamental settings Fundamental settings (cont.) Definition 2 (L 1 -SW-EL statistic) n rn(β) = sup nv t : t=u+1 n n v t g t (β) = 0 p+q, v t = 1, 0 v t 1 t=u+1 t=u+1 Remark 4 r n(β): Nonparametric likelihood ratio function 19 / 34

22 Main results Outline 1 Introduction 2 Fundamental settings 3 Main results 4 Numerical examples 5 Concluding remarks 20 / 34

23 Main results Main results Let Q t = (V t 1,..., V t p, W t 1,..., W t q ), V t b 01 V t 1 b 0p V t p = e t, W t + a 01 W t a 0q W t q = e t. Assumption 1 (i) β 0 : the unique solution to E[g t (β 0 )] = 0 p+q. (ii) β 0 Int(B) & B is compact in R p+q. (ii) b(z) = 1 + β 1 z + + β p z p 0 & a(z) = 1 + a 1 z + + a q z q 0 in {z : z 1}. (iii) b(z) & a(z) have no common zeros. (iv) δ > 0 s.t. E[ e t δ ] <. (iv) E[(w t + w 2 t )( Q t (β 0 ) 2 + Q t (β 0 ) 3 )] <. (v) Ω = E[w 2 t Q t Q t ] is nonsingular. 21 / 34

24 Main results Main results (cont.) Theorem 1 Suppose that Assumption 1 holds. Then, under H : Rβ = c, ρ n L χ 2 r as n. 22 / 34

25 Main results Main results (cont.) Remark 5 Limit distribution pivotal does not contain any unknown quantity. Test for nested linear hypothesis SN: ĝ(β 0 ) = 1 n u n 1 { n s=1 y 2 s } g t(β 1/2 0 ) t=u+1 Limit distribution contains α (tail-index of e t ) Rate of convergence contains α & n SW: n n nĝ (β 0 ) = w t sign {ε t (β 0 )} A t (β 0 ) L N(0 p+q, Ω) n u t=u+1 23 / 34

26 Numerical examples Outline 1 Introduction 2 Fundamental settings 3 Main results 4 Numerical examples 5 Concluding remarks 24 / 34

27 Numerical examples Example 1: Test of serial correlation Model: ARMA(1,1) process where b < 1, a < 1 & b + a 0 (a) N(0, 1) (b) (5/3) e t 1/2 t 5 (c) (3) 1/2 t 3 (d) 2 1/2 L(0, 1) Testing problem: H : a = 0 b: a nuisance parameter y t = by t 1 + e t + ae t 1, Sample size: n = 200, 400 Number of iteration: Nominal level: δ = 0.10, 0.05 SW-Wald-test (Pan et al. (2007)) 25 / 34

28 Numerical examples Example 1: Test of serial correlation (cont.) Figure: Rejection rate of the tests (b = 0.5, n = 200) rejection rate 1.0 (a) N(0, 1) (b) (5/2) 1/2 t 5 rejection rate elr (delta =0.10) 0.8 elr (delta =0.10) 0.6 wld (delta =0.10) 0.6 wld (delta =0.10) elr (delta =0.05) elr (delta =0.05) 0.4 wld (delta =0.05) 0.4 wld (delta =0.05) { { rejection rate 1.0 (c) (3) 1/2 t 3 (d) 2 1/2 L(0, 1) rejection rate elr (delta =0.10) 0.8 elr (delta =0.10) 0.6 wld (delta =0.10) 0.6 wld (delta =0.10) elr (delta =0.05) elr (delta =0.05) 0.4 wld (delta =0.05) 0.4 wld (delta =0.05) { { 26 / 34

29 Numerical examples Example 1: Test of serial correlation (cont.) Figure: Rejection rate of the tests (b = 0.5, n = 400) rejection rate 1.0 (a) N(0, 1) (b) (5/2) 1/2 t 5 rejection rate elr (delta =0.10) 0.8 elr (delta =0.10) 0.6 wld (delta =0.10) 0.6 wld (delta =0.10) elr (delta =0.05) elr (delta =0.05) 0.4 wld (delta =0.05) 0.4 wld (delta =0.05) { { rejection rate 1.0 (c) (3) 1/2 t 3 (d) 2 1/2 L(0, 1) rejection rate elr (delta =0.10) 0.8 elr (delta =0.10) 0.6 wld (delta =0.10) 0.6 wld (delta =0.10) elr (delta =0.05) elr (delta =0.05) 0.4 wld (delta =0.05) 0.4 wld (delta =0.05) { { 27 / 34

30 Numerical examples Example 1: Test of serial correlation (cont.) Figure: Rejection rate of the tests (b = 0.9, n = 200) rejection rate 1.0 (a) N(0, 1) (b) (5/2) 1/2 t 5 rejection rate elr (delta =0.10) 0.8 elr (delta =0.10) 0.6 wld (delta =0.10) 0.6 wld (delta =0.10) elr (delta =0.05) elr (delta =0.05) 0.4 wld (delta =0.05) 0.4 wld (delta =0.05) { { rejection rate 1.0 (c) (3) 1/2 t 3 (d) 2 1/2 L(0, 1) rejection rate elr (delta =0.10) 0.8 elr (delta =0.10) 0.6 wld (delta =0.10) 0.6 wld (delta =0.10) elr (delta =0.05) elr (delta =0.05) 0.4 wld (delta =0.05) 0.4 wld (delta =0.05) { { 28 / 34

31 Numerical examples Example 1: Test of serial correlation (cont.) Figure: Rejection rate of the tests (b = 0.9, n = 400) rejection rate 1.0 (a) N(0, 1) (b) (5/2) 1/2 t 5 rejection rate elr (delta =0.10) 0.8 elr (delta =0.10) 0.6 wld (delta =0.10) 0.6 wld (delta =0.10) elr (delta =0.05) elr (delta =0.05) 0.4 wld (delta =0.05) 0.4 wld (delta =0.05) { { rejection rate 1.0 (c) (3) 1/2 t 3 (d) 2 1/2 L(0, 1) rejection rate elr (delta =0.10) 0.8 elr (delta =0.10) 0.6 wld (delta =0.10) 0.6 wld (delta =0.10) elr (delta =0.05) elr (delta =0.05) 0.4 wld (delta =0.05) 0.4 wld (delta =0.05) { { 29 / 34

32 Concluding remarks Outline 1 Introduction 2 Fundamental settings 3 Main results 4 Numerical examples 5 Concluding remarks 30 / 34

33 Concluding remarks Concluding remarks Pivotal limit distribution We do not need to estimate unknown parameters (e.g., α & σ of stable distribution) { } (i) heavy-tailed noise case L 1 -SW-EL-test improves power of test in. (ii) near unit-root Extension 1: SW-GEL test statistics (EL, ET, CU) Extension 2: FD-SW-GEL test statistics Nonparametric model & quantities 31 / 34

34 Concluding remarks Thank you for listening. 32 / 34

35 Concluding remarks Reference [1] Akashi, F. (Submitted for publication). Self-weighted empirical likelihood method for testing problem of infinite variance processes. [2] Akashi, F., Liu, Y., and Taniguchi, M. (2015). An empirical likelihood approach for symmetric α-stable processes. Bernoulli, 21(4): [3] Chen, K., Ying, Z., Zhang, H., and Zhao, L. (2008). Analysis of least absolute deviation. Biometrika, 95(1): [4] Hall, P. and Heyde, C. (1980). Martingale limit theory and applications. Academic, New York. 33 / 34

36 Concluding remarks Reference (cont.) [5] Ling, S. (2005). Self-weighted least absolute deviation estimation for infinite variance autoregressive models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(3): [6] Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75(2): [7] Pan, J., Wang, H., and Yao, Q. (2007). Weighted least absolute deviations estimation for arma models with infinite variance. Econometric Theory, 23(05): [8] Parente, P. M. and Smith, R. J. (2011). Gel methods for nonsmooth moment indicators. Econometric Theory, 27(01): / 34

37 Appendix: SW-Wald-test (Pan et al. (2007)) { } 1 1 W n = n(rˆβ n c) 4ˆf (0) R ˆΣ 1 ˆΩˆΣ 1 R (Rˆβ n c) = nâ2 n 2 ŵ, where ˆβ n = (ˆb n, â n ) = arg min β B ˆf (0) = n 1/5 n t=u+1 w t n w t ε t (β), t=u+1 n t=u+1 ( w t K ε t (ˆβ n ) 1.06n 1/5 exp( x) K(x) = {1 + exp( x)}, ŵ = 1 2 4ˆf (0) R ˆΣ 1 ˆΩˆΣ 1 R, 2 ˆΣ = 1 n w t A t (ˆβ n )A t (ˆβ n ), ˆΩ = 1 n w 2 t A t (ˆβ n )A t (ˆβ n ). n u n u t=u+1 ), t=u+1 W n L χ 2 r 35 / 34