Generalized autoregressive score (GAS) functions under Gaussian and student-t distributions

This article described and worked-out the score functions otherwise known as the in information matrixes of Generalized Autoregressive model with time-varying parameters when the error term is assumed to follow Gaussian and student-“t” conditional distributions for location and heavy-tails affected series respectively.

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Introduction

In linear time series analysis, the assumption of Gaussian and student-t distributions as imposing error terms (white noises) are indispensible for effective, efficient and consistent estimation of score functions and their standard errors of estimates with respect to their marginal distributions ( Tobias et al, 2017) [9]. Classification of linear time series models e.g., Autoregressive (AR), Moving Average (MA), Autoregressive Moving Average (ARMA) etc. in which their time-varying observations do follow either a location or heavy-tailed scale changes are dynamically driven by score functions of the estimates via their conditional marginal distributions (Harvey, 2013) [6]. Location scaled changes is conventionally modeled with Gaussian based error term while the heavy-tailed time change goes for the student-t distribution as the error term (Wintenberger, 2013) [10]. Bayesian approach has been frequently used as an alternative approach in lieu of the classical approach for parameter estimation of these score functions and their standard errors via the marginal and conditional distributions due to the former’s theoretical rigor of its parameter estimation (D’Agostino et al., 2013 [6]; Pesaran et al. 2011 [8]).

Petrella (2014) [7] affirmed the shortcomings in the Bayesian approach to restriction and imposition in order to achieve stationary of the process. Blasquesa et al. (2014) [1] claimed this imposition to do lead to large inefficiency and limit in ability to capture tails behavior if present in the time-varying observations. This behavior might be constituted by the effect of skewness, kurtosis and outliers. In order to overcome this problem, this paper duel on the theoretical parameter estimation of driven score functions of generalized autoregressive model with Gaussian and student-t distributional error terms for both location and heavy-tailed affected series respectively. 2.

Preliminaries 2.1 Generalized Autoregressive Score (GAS) model with normal distributed time-varying parameter Monache & Petrella (2014) [7] proposed that an autoregressive model of order " " p with time- varying parameters could be defined as: 0, 1, 1 , t t t t t p t p t y y y            →1 With white noise (error term) 2 (0, ) 1 t t t n    Maths 2018; 3(5): 56-61 Olanrewaju O Rasaki Department of Statistics, University of Ibadan, Ibadan, Nigeria Folorunsho A Serifat Department of Statistics, University of Ibadan, Ibadan, Nigeria Correspondence Olanrewaju O Rasaki Department of Statistics, University of Ibadan, Ibadan, Nigeria The updating rule can be defined to be variation of the vector of time-varying parameters to be ' 2 ( , ) t t t f   , ' 0, 1, , ( , , , ) t t t p t      such that equation (1) is nothing but a Markov process of the first order of 1 t t t f Kf r     , (0, ) t t r  →2 Where  is a vector of constants; K and t are the matrices containing the hyper-parameters (updated location and scale parameter respectively). tf Represents the time-varying parameters while  represents the fixed coefficient. The Generalized Autoregressive Score (GAS) model for the time-varying parameters that will be driven by the scale-function of the conditional likelihood of tf give the immediate past of 1 t  , ' 2 / 1 / 1 1 ( , ) t t t t t f       is 1/ / 1 t t t t t f Kf Gb       →3 Such that   1 1 2 1 , , , t t t Y y y y     and , , K  and G are defined same in equation (2) with the driving mechanism called the score-function. t t t b B     1 2 1 Such that   p y G p y G B I f f f   log ( / ; ) log ( / ; ) ; t t t t t t t t t t t t t 1 ' / 1 / 1 / 1                  With 1 1 tI   the information matrix,   1 , t t t G F Y   and   / 1 1/ 2 1/0 , , , t t t t t F f f f     for the vector parameters t. 2.2 Score function of the time-varying normally distributed parameters’ estimation. Equation (1) can be re-written in a matrix form as ' / 1 t t t t y J      such that 1 / 1 / (0, ) t t t t Y N f    for 1 t n  and 2 / 1 / 1 t t t t f     , where ' ' 1 / 1 0, / 1 1, / 1 , / 1 (1, , , ) & ( , , , ) t t p t t t t t t p t t J y y             Since ' / 1 t t t t y J      , ' / 1 t t t J     2 y t t   ( ) 2 t t / 1  1 ( / ; )   f t t t 2  t t p y G e f   ty   →4 2 / 1 In a matrix form and taking the conditional likelihood ( / ; ) ln ( / ; ) t t t t t t y G p y G         t t t t t t t t ' ' ' / 1 / 1 / 1 1 1 ( ) ( ) log(2 ) log 2 2 2 y J y J f f       t t / 1  So, t t t t t t t t t t        ln ( / ; ) 1 ( ) ( ) 2 2 p y G y J y J f f f      t t t t t t ' ' ' / 1 / 1 2 / 1 / 1 / 1    But, ' ' ' / 1 / 1 ( ) ( ) t t t t t t y J y J        ' ' ' ' ' ' / 1 / 1 / 1 / 1 t t t t t t t t t t t t y y y J Jy JJ            , Recall ' / 1 t t t t y J      ' ' ' ' ' ' ' ' ' ' / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 ( ) ( ) ( ) ( ) t t t t t t t t t t t t t t t t t t t t J J J J J JJ                             ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t JJ J J JJ J JJ J JJ                                       ' ' ' ' ' ' ' / 1 / 1 / 1 / 1 t t t t t t t t t t t t t t J J J J                   ' ' / 1 2 t t t t t J       So, ' ' ' '        / 1 / 1 / 1           t t t t t t t t t t t 2( ) 1 2( ) 2 2 2 J f J f f f t t t t t t    2 2 / 1 / 1 / 1 For tb to be a scale vector, ( ) 0 t           1 ( ) (2 ) ( ) ( ) 2 t t t t t t t t t J f f       ' ' / 1 / 1 2 / 1        1 2 ( ) ( ) 2 J f f      t t t t t t t t ' ' / 1 / 1 2 / 1 ' 2 2 / 1 ( ) 0, ( ) , ( ) t t t t t t t Since f            2 2 ( ) 0 0 t t t           Also, log ( / ; ) 1 (2 ) , sin ( ) 0 2 p y G J I ce f f f f       t t t t t t t t t t t t t t t t t 2 ' ' / 1 1 ' 2 3 / 1 / 1 / 1 / 1                             1 2 t 1 2 / 1 t t I f    , So 1 2 1 / 1 2 t t t t B I f    Then,                 J f b B f J f f (2 ) 2 (2 ) 2 t t t t t t t t t t t t t t t t t t t ' ' / 1 / 1 2 ' ' / 1 / 1 / 1 2 / 1         2( ) J f     ' ' / 1 / 1 The observation-driven model updates are:         , t t t t t t t 2 f t t 2 / 1  1 2 t 1 2 / 1 t t I f    So, equation (3) becomes ' ' / 1 / 1 / 1 / 1 (2 ) t t t t t t t t t t f Kf G J f                 The updated parameters equal; ' ' 1/ / 1 / 1 / 1 ((2 ) ) t t t t t t t t t t k J f              , 2 1/ 1/ / 1 (2 ) t t t t t t f f k f       2.3 Score function of GAS model with heavy-tails (student-t) time-varying parameters. From equation (1), ' / 1 t t t t y J      such that 1 / 1 / (0, , ) t t t t t Y NID f v     v 1 tv t   1 2 2 ( ) 2 ( / ; ) 1   t t t t t ty   →5 2 / 1  y p y G v v f f v  t t t t t t 2                       For location parameter t, scale parameter tf , tv degree of freedom, where ' ( , , , ) t t t t t f v    In a matrix form, v 1 tv t   1 ' ' 2 / 1       2 2 ( / ; ) 1 t t t t t t t 2 / 1  J p y G v v f f v  t t t t t t 2                         The maximum likelihood, →6     t t t t t t t t t t t t t   2 1 1 1 1 ( / ; ) log log log log 1 2 2 2 2 2 J v v v y G f v v f t t t    ' ' / 1 2 / 1 2 / 1                                            ( / ; ) t t t t t y G a h m     Such that, t t →7 1 1 1 log log log log 2 2 2 2 t v v a v                                 →8 2 / 1 1 log 2 t t t h f        ' ' / 1 →9 t t t t t t J v m v f 2 1log 1 2 t t t 2 / 1              Creal et al. (2011, 2013) [2, 3] and Harvey (2013, 2014) [5, 6] affirmed that the Degree of Freedom tv must be strictly greater than two.  t , Since   ( / ; ) t t t t t t t t t t y G f  , so   ' ' / 1 2 1log 1 2 J v m v Let 2 / 1 t t t f  t t / 1  t                   t  can be partition into two functions of & f   such that   depends on t and t m while f  depends on , , t t h m and t. m , using the function of the function rule  t t t t t f ' t t                          h m h m f f f f t t , 2 2 2 2 / 1 / 1 / 1 / 1 t t t t t t t t t       ' , t Recall,      , so       t t t t t t t t y J y J f    ' ' / 1 / 1 2 / 1 t t  J f    t t t ' 2 / 1 So,   t t t ' ' / 1  (2 ) 1 t t t t t t t t t t t = ' (2 ) 2 v J J v v      1 ' ' ' ' ' / 1 / 1 1 (2 ) (2 ) 1 2 / 1  t t t t t t t                  J m v v v J v t t                   t t t t t ' ' / 1  t t t t  = v J v v v J   1 (2 ) 2 (2 )   v J v J      1 (2 ) 2 (2 ) t t t t t t ' ' ' / 1  t t t t t ' ' ' / 1 ' ' / 1                                t t t t t t t t t t t = ' ' ' / 1 1 (2 1 2 v J J v J        1 (2 ) (2 ) 2 (2 ) v J v      ' ' ' ' ' / 1 / 1 ' ' / 1 t t t t t t t t t                          1 , 1 But, where t t , , , , ( ), ( ) t ff t t ff t I H I H     I B I I ff t ,        t t t ' , , ' v J H v 1 2      / 1 t t t t            Also, t t t t t t t t t t   1 (2 ) 2 1 1 2 2 h m v J f f f v f        ' ' ' / 1 / 1 2 2 2 / 1 / 1 / 1 / 1 t t t t t t t t t t t t                            J H v f f v f      (2 ) 1 1 1 ff t t t t t ff t t t t t t t t t t t ' ' ' , / 1 , 3 2 / 1 / 1 / 1                   ' 2 ' So, / 1 , , t t t t t t v v J f J I H v v     1 1 ( ) 2 2 t t t t                   f I H v J v J f f f f         1 1 1 ( ) 1 (2 ) 1 (2 ) t t t t f f t ff t t t t t t t t t t t t t t t t t t t 2 ' ' ' ' ' ' / 1 , , / 1 / 1 2 3 2 3 / 1 / 1 / 1 / 1                                t t  2 0 ( 1)   ' 2 , 1 / 1 So, t t t t t v I v J f B I I  f f t 1 , 0 I f f t ,                        1 t t t b I    t t t t t t  2 1 (2 ) 1 0 ( 1) 2      ' ' ' / 1 ' 2 / 1  v v J v J f v b  t t t t t t t      1 4 2 ' ' ' ' ' ' 3 / 1 / 1 / 1 / 1 / 1 2         ( 1) (2 ) 1 1 0 (2 ) ( 1) 2 f f v J J f v f v f t t t t t t t t t t t t t t t t t t t t t t t / 1 / 1                                                               The observation-driven model updates are , t t I  (2 ) 1 t t t t t       ' ' ' / 1 ' 2 / 1 t t t t t  →10 v J J f v b t       1 ( 1) (2 ) 1 (2 ) ( 1) 2 f f v J J f v f f v         t t t t t t t t t t t t t t t t t t t t t t t 1 4 2 ' ' ' ' ' ' 3 / 1 / 1 / 1 / 1 / 1 2 / 1 / 1                                                       Equation (10) is the (2x1) matrix of the Score function for GAS with heavy-tails using student-“t” distribution.

Conclusions

Having find-out that estimation of multivariate normal and student-t distributions’ score functions are usually left out in GAS related articles, working papers, manuscripts and write-ups due to its rigor. This led to the estimation of the score functions for easy computational and estimating of the function.

Acknowledgments

I appreciate Oladebo Omotayo for his useful and constructive comments; suggestions and insight in the course of the manuscript. 5. Competing interests Author declared, no potential conflict of interest exist.

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