ENTROPY-BASED TEST FOR TIME SERIES MODELS Siyun Park Korea University Business School, Seoul, Korea Sangyeol Lee and Jiyeon Lee Department of Statistics, Seoul National University . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . . .. .. . .. . . .. . .. .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Outline Introduce entropy-based test of fit in iid sample Test statistic and the asymptotic distribution Practical issues Extend and apply the test to time series models Autoregressive models GARCH models Simulation result and application to real data . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . .. . .. . . .. . .. . .. .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Maximum entropy principle Shannon entropy : the average unpredictability in a random variable, its information content The maximum entropy principle(Janes, 1957): Its applications successfully proven in various fields, computer vision, natural language processing. . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . . .. .. . .. . . .. . .. . .. .. . . .. . .. .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Entropy Boltzmann-Shannon entropy: ∫ ∞ H ( f ) = − f ( x ) log ( f ( x )) dx . (1) −∞ Forte and Hughes (1988) proposed the function ¯ ∑ H = − p i log ( p i / ( x i − x i − 1 )) (2) ∫ x i as the discrete analogue of (1), p i = x i − 1 f ( x ) dx . . . . . . . . . . . . . . . . . . . . . .. . . .. .. . .. . . .. . .. . .. . .. . .. . .. .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
for a variable defined in [ a , b ] , n ∑ max i | x i − x i − 1 |→ 0 − lim p i log ( p i / ( x i − x i − 1 )) = H ( f ) , (3) i = 1 ∫ x i where p i = P [ x i − 1 < X ≤ x i ] = x i − 1 f ( x ) dx , i = 1 , . . . , n − 1 and a = x 0 < . . . < x n = b . . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Simple null hypothesis case Let Y i , i = 1 , . . . , n be a random sample from F , H 0 : F = F 0 vs . F ̸ = F 0 . . . . . . . . . . . . . . . . . . . . . .. . . .. . .. .. . .. . .. . .. . .. . . .. .. . .. . .. . . .. .. . .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Generalized entropy: Lee et al.(2011) A generalization of Forte and Hughes (1988)’s: m ( F ( s i ) − F ( s i − 1 ) ) ∑ S w ( F ) = − w i ( F ( s i ) − F ( s i − 1 )) log , (4) s i − s i − 1 i = 1 where the w ′ s are appropriate weight functions with 0 ≤ w i ≤ 1 and ∑ m i = 1 w i = 1 , m is the number of disjoint intervals for partitioning the data range, and −∞ < a ≤ s 1 ≤ . . . ≤ s m ≤ b < ∞ are preassigned partition points. . . . . . . . . . . . . . . . . . . . . .. . . .. .. . .. . . .. . .. . .. . .. . .. . .. .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Test statistic the null hypothesis will be rejected if | S w ( F n ) − S w ( F 0 ) | ≥ c or, even more stringently, if | S w ( F n ) − S w ( F 0 ) | ≥ c , sup w where n ∑ F n ( x ) = n − 1 I ( Y i ≤ x ) . i = 1 . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . .. . .. . .. . .. . .. . . .. .. . .. . . .. .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Probability integral transformation U i = F 0 ( Y i ) , H 0 : F = F 0 ≡ U [ 0 , 1 ] vs . H 1 : F ̸ = F 0 ≡ U [ 0 , 1 ] . If F 0 is uniform distribution then S w ( F 0 ) = 0 . Use U i and F n ( u ) = n − 1 ∑ n i = 1 I ( U i ≤ u ) . . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . .. . .. . . .. . .. . .. .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Probability integral transformation U i = F 0 ( Y i ) , H 0 : F = F 0 ≡ U [ 0 , 1 ] vs . H 1 : F ̸ = F 0 ≡ U [ 0 , 1 ] . If F 0 is uniform distribution then S w ( F 0 ) = 0 . Use U i and F n ( u ) = n − 1 ∑ n i = 1 I ( U i ≤ u ) . . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . .. . .. . . .. . .. . .. .. . .. . . .. . .. .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Theorem (Lee et al.(2011)) Under H 0 , as n → ∞ , m √ n d ∑ | S w ( F n ) | − → | w i ( BB ( s i ) − BB ( s i − 1 )) | , sup sup { w ∈ W } { w ∈ W } i = 1 where BB ( s ) is the Brownian bridge on [0,1], W denotes the space of bounded weight functions w i : [ 0 , 1 ] → [ 0 , 1 ] with ∑ m i = 1 w i = 1 , and 0 = s 0 ≤ s 1 ≤ . . . ≤ s m = 1 . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. .. . .. . . .. . .. . .. . .. .. . . .. .. . . .. . .. . .. .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Composite null hypothesis case Y i , i = 1 , . . . , n be a random sample from F H 0 : F ∈ { F 0 ( x ; θ ); θ ∈ Θ } vs . H 1 : not H 0 , F 0 : continuous distribution, Θ : a d -dimensional parameter space. . . . . . . . . . . . . . . . . . . . . .. . . .. . .. . .. .. . . .. .. . .. . . .. . .. . .. .. . . .. . .. .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
Let ˆ θ n with n 1 / 2 (ˆ θ n − θ 0 ) = O P ( 1 ) under H 0 . Let U i = F 0 ( Y i ; θ 0 ) , ˆ U i = F 0 ( Y i ; ˆ θ n ) , n n F n ( s ) = 1 F n ( s ) = 1 ∑ I ( U i ≤ s ) and ˆ ∑ I (ˆ U i ≤ s ) , s ∈ [ 0 , 1 ] . n n i = 1 i = 1 Define the empirical process: E n ( s ) = n 1 / 2 ( F n ( s ) − s ) the estimated empirical process: ˆ E n ( s ) = n 1 / 2 (ˆ F n ( s ) − s ) . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . .. . .. . .. .. . . .. . .. . .. . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
We can express ′ √ n (ˆ ˆ E n ( s ) = E n ( s ) − h ( s ) θ n − θ 0 ) + o P ( 1 ) , ∂ F θ 0 ( F − 1 θ 0 ( s )) where h ( s ) = . ∂θ It can be seen that � m |√ nS w (ˆ � ∑ F n ) | = w i [ E n ( s i ) − E n ( s i − 1 )] � � � i = 1 � m √ n (ˆ � ′ ∑ + θ n − θ 0 ) w i [ h ( s i − 1 ) − h ( s i )] � + o P ( 1 ) . � � i = 1 Hence, the previous theorem can be applied when max 1 ≤ i ≤ m | s i − s i − 1 | → 0 and m → ∞ . . . . . . . . . . . . . . . . . . . . . . .. .. . . .. .. . . .. . .. . .. . .. . .. . .. . .. .. . .. . . .. . .. .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
We can express ′ √ n (ˆ ˆ E n ( s ) = E n ( s ) − h ( s ) θ n − θ 0 ) + o P ( 1 ) , ∂ F θ 0 ( F − 1 θ 0 ( s )) where h ( s ) = . ∂θ It can be seen that � m |√ nS w (ˆ � ∑ F n ) | = w i [ E n ( s i ) − E n ( s i − 1 )] � � � i = 1 � m √ n (ˆ � ′ ∑ + θ n − θ 0 ) w i [ h ( s i − 1 ) − h ( s i )] � + o P ( 1 ) . � � i = 1 Hence, the previous theorem can be applied when max 1 ≤ i ≤ m | s i − s i − 1 | → 0 and m → ∞ . . . . . . . . . . . . . . . . . . . . . . .. .. . . .. .. . . .. . .. . .. . .. . .. . .. . .. .. . .. . . .. . .. .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
We can express ′ √ n (ˆ ˆ E n ( s ) = E n ( s ) − h ( s ) θ n − θ 0 ) + o P ( 1 ) , ∂ F θ 0 ( F − 1 θ 0 ( s )) where h ( s ) = . ∂θ It can be seen that � m |√ nS w (ˆ � ∑ F n ) | = w i [ E n ( s i ) − E n ( s i − 1 )] � � � i = 1 � m √ n (ˆ � ′ ∑ + θ n − θ 0 ) w i [ h ( s i − 1 ) − h ( s i )] � + o P ( 1 ) . � � i = 1 Hence, the previous theorem can be applied when max 1 ≤ i ≤ m | s i − s i − 1 | → 0 and m → ∞ . . . . . . . . . . . . . . . . . . . . . . .. .. . . .. .. . . .. . .. . .. . .. . .. . .. . .. .. . .. . . .. . .. .. . .. . .. . .. . .. . S.Lee, J. Lee, and S. Park ENTROPY-BASED TEST FOR TIME SERIES MODELS
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