Recent Developments in the Unit Root Problem for Moving Averages

Transcription

1 Recen Developmens in he Uni Roo Problem for Moving Averages Richard A. Davis Colorado Sae Universiy Mei-Ching Chen Chaoyang Insiue of echnology homas Miosch Universiy of Groningen

2 Non-inverible MA() Model w/o rend finie variance noise non-normal sable noise Non-inverible MA() Model Wih rend consan rend funcion general rend funcions

3 Basic Problem MA() Model: Y µ ε θ ε -, {ε } IID Ν(0,σ ) θ ( θ < inverible) Observaions: y,..., y ^, MLE of θ and µ θ ^µ For θ <, ^ θ is AN(θ, (-θ ) / Τ) and ^ µ is AN(µ, σ (-θ) / Τ). Quesion: Are hese good approximaions if θ is near or equal o? 3

4 4 Asympoics (µ0, finie variance case) Idea: build parameer normalizaion ino he lielihood funcion. Model: Y ε ( β/) ε -,,,. β ( θ), θ β/, θ 0 γ/ Lielihood: L (β) l ( β/) l (), where l () reduced log( Gaussian lielihood). heorem: Under θ 0 γ / Τ, L (β) Ζ γ (β) on C[0, ), where Ζ γ (β) 8 ) ( ln ) ( ) ( β β β γ d

5 Resuls: ( θˆ mle ) βˆ mle argmax Ζ γ (β) θˆ ) βˆ arglocalmax Ζ γ (β) ( lm lm P( θˆ ) P( βˆ 0).658 if γ 0. lm Hypohesis esing H 0 : θ 0 vs. H : θ 0 < ess: LM: Rejec if MLE: Rejec if LR: Rejec if lm LBIU (anaa): Rejec if θ ˆ < ( α) (e.g. α.05 and 50, θˆ < lm c lm θˆ mle < c mle ( α) L ˆ ( β ) > b ( α) lme glr S > s(α) lm.87) 5

6 Asympoics (µ0 case, non-normal sable case) Model: Y ε θ 0 ε -,,,, {ε } IID SαS, i.e., E[exp(i ε )] exp{ α }, 0< α <. Inverible case: ( θ 0 < ) Resuls: / α S ( /ln ) (ˆ θ ) where is he G θ0 θˆ G S M(Gaussian)LE (Miosch e al. `95) / α (ˆ θ ) η, where is he LAD (Davis `96) θlad 0 / α (ˆ θ θ ξ, where is he MLE (Calder 0) θˆ mle mle and Davis `97) 0 θˆ lad 6

7 7 Non-inverible case: Model: Y ε ( β/) ε -,,,, {ε } IID SαS β ( θ), θ β/, θ 0 γ/, Lielihood: L (β) l ( β/) l (), where l () reduced log(gaussian lielihood). heorem: Under θ 0 γ / Τ, L (β) Ζ γ (β) on C[0, ), where Ζ γ (β) and he are defined below. d, ) ( ln ~ ) ( ) ( β β β γ 8 0 / ~

8 Definiion of : 0 0 x ( M ( x)) / 0 cos( x) γ sin( x) / ( γ ) dm ( x),,,..., where M is an SαS Lévy moion on [0,] wih M() ε. Properies of ~ / : 0. Uncorrelaed.. Mean. 3. Exponenially decreasing ail. 4. Chi-square disribuion wih df when α. d 8

9 9 Idea of argumen Criical erms: ). sin( ~ )) cos( ( )) ( /( ~, ~ / /, 0 0 σ ε θ σ s Y q U s s Resul: 0, 0 / / ~ ~ ~ U d d σ α

10 Resuls: ( θˆ lm ) βˆ lm arglocalmax Ζ γ (β) ˆ P( θlm ) βlm P( ˆ 0) 0

11 Accuracy of he Asympoic Disribuion Sep. Simulae 0,,, N using he approximaion: 0 s K K s Z s, cos( s) γ sin( s) Z / ( γ ) s,,,..., N, where Z,, Z K are iid SαS random variables. Sep. runcae he infinie series for Y γ (β) Z γ (β)/ β a N. Sep 3. If Y γ (0) < 0, hen pu βˆ lm 0. βˆ lm Sep 4. If Y γ (0) > 0, hen is defined as he smalles nonnegaive zero of Y γ (β).

12 Figure. Comparison of sampling cdf s (θ 0.0, α.0 ) Limi c.d.f θ 0.0

13 Figure. Comparison of limi cdf s for differen α s α.0 α.5 α.0 α

14 able. Quaniles of ( θˆ lm ) and ˆβ 0 wih pile-up, P - U P(ˆ θ ) lm α.75 α.0 α.5 α.0.05 P-U.05 P-U.05 P-U.05 P-U Limi

15 able. Quaniles of ( θˆ lm ) for 50 and θ 0.7,.8,.9,.95 (α.0) θ P-U Limi Limi Limi Limi

16 Asympoics (consan rend, finie variance) Idea: build parameer normalizaion ino he lielihood funcion. Model: Y µ ε θ 0 ε -,,,, {ε } IID Ν(0,σ ) β ( θ), θ β/, θ 0 γ/ Lielihood: L ( β) l (- β/, µ ˆ( β)) - l (, µ ˆ(0)), where l () reduced log( Gaussian lielihood) and ˆµ( β) is he MLE of µ wih β fixed. 6

17 Resuls: d L ( β) Z γ (β) d ˆ ˆ ( θ ) β argmax Z γ (β) ˆ mle mle P( ).955 if (γ 0). θ mle If θ nown, hen ( µ ˆ µ ) N(0,σ 3/ If θ and is esimaed, hen 3/ d ( µ ˆ µ ) is NO asympoically normal. Limi disribuion is ~N(0,σ )(.955) N(0,7σ )(.045) ). 7

18 Hypohesis esing ess: H 0 : θ 0 vs. H : θ 0 < LR: Rejec if L ˆ ( β ) > b ( α). mle glr Because of he large pile-up (.955), have o use randomized es for α >.045. LBIU (anaa): Rejec if S > s(α) 8

19 Power Comparison: (cf. p.390 of anaa `96) α.0 α.05 α.0 γ LBIU LR LBIU LR LBIU LR

20 An Example : (overshors Y,..., Y 57 from an underground sorage an.) (gallons) ACF Lag 0

21 Model. Y µ ε θ ε - Problem. Esimae µ and consruc a C.I.? (Is µ < -5 gallons/day?) Esimaion. Esimaes Asympoic Var ^ µ MLE 4.78 (-θ) σ / (.905).89 ^ θ MLE.849 ( θ )/ (.070) Noe: If θ is no esimaed hen ^ µ MLE is AN(µ 0, σ / 3 ) AN(µ 0, (.36) )

22 Asympoics (general rend, finie variance)... Model: Y b 0 x b x b U, U ε ( β/) ε -,,,, {ε } IID Ν(0,σ ) Linear Model: Y b U Under growh condiions on he {x j }, limi behavior of bˆ and can be obained. θˆ lm Example: Y b 0 b ε ε -. 3/ 5/ bˆ ( bˆ ) d 0 N(, σ ) 0 ( 0 b0 b ) 70