Comparaison Between Different Methods to Estimate Orders in ARMA Models

Transcription

1 Comparaison Between Different Methods to Estimate Orders in MA Models Michel GRUN-REHOMME () Dominique LADIRAY (2) () IUT Département Statistique (2) INSEE Département de la Recherche Abstract It is usually difficult to determine the values of p and q for the stationnary autoregressive moving average MA(pq) models. Considerable progress over the last few years has been made in order estimation problems in time series analysis several approaches and criteria have been proposed. In the eighties two methods have been development for the identification of a model MA(pq) : the corner method ([]) and the triangle method ([8]). These two methods use the autocorrelation function. The corner method define a determinant with the autocorrelations and present an algorithm for determining simultaneously the order p and q. With the triangle method for the MA models an iterative regression procedure is applied to produce consistent estimates of the autotregressive parameters. An extended sample autocorrelation function based on these consistent estimates is then defined and used for order determination. The order determination quantity (ODQ) is an other way to solve order estimation problems in time series analysis ([567]). β γ β γ χ η 2 2 ODQ p q n σ p q n σ p q n n n n where σ n 2βγ.. denotes an estimate of the variance of the white noise; p qηis an upper bound which can be arbitrarily large but fixed and is supposed to be known a priori; n is the sample size. Finally simulations allow to compare the efficiency of these different methods by using SAS macro. χ Keywords : Time series MA models order determination criteria SAS macro.

2 Consider the usual MA(pq) model for a univariate time series {Y t } βγt βγ ϕ BY θ Bε p q where ϕβγ B φ B... φ p B and θβγ B θ B... θ q B B is the backshift operator such that BYt Yt and βγ ε t is a Gaussian white noise process. All the zeros of ϕ Bβγ and θ Bβγ are outside the unit circle and ϕ Bβγ and θ Bβγ have no commun factors. The true value of p and q are unknown. An additive constant may be elimined by the transformation : Zt Yt E Yt βγ. This article propose to compare three methods to solve this problem : the corner method by Beguin Gourieroux and monfort (98) the "triangle" method by Tiao and Tsay (984) and the order determination quantity by Koreisha and Pukkila (994). Each method has been implemented with a SAS macro.. The corner method βγ δ The autocorrelation function is defined by ρ X t r X t X t where r denotes the Pearson's coefficient between βx t γ and δx t+ ι. We consider the autocorrelation matrix Dβγ i of order i+ defined by : ( ) Di t + ( + ) ( + 2)... ( + i+ ) ρ() ρ( + )... ρ( + i) ρ ρ ρ ρ ( + i) ρ( + ) for i 2... max(p) (an a priori upper bound for the degree of ϕ Bβγ) and for 2... max(q) (an a priori upper bound for the degree of θ Bβγ). Let βγ i the determinant of Dβγ. i The authors showed that Y t is an MA(pq) process if and only if βγ i i p q iq i p βp γ q Of course the true values of βγ i are unknown and have to be estimated and it is necessary to test the nullity of them.. Student s test In a first time an estimate of the asymptotic variance of βγ i is defined as follows : ' Κϑ Φ Η Γ ( i ) ( i ) Vas ( ( i )) Σ N ρ ρ Φ Η Γ Ι ι Ι Κ ϑ

3 where N is the number of observations in the time series ( Y t ) ( i ) ρ the vector column of the partial derivates of βγ i for h... maxp + maxq + and Σ is the matrix of general term + σkl ρ( ) ρ( + k + l) + ρ( + k l) + 2ρ( ) ρ( k) ρ( l) 2ρ( l) ρ( + k) 2ρ( k) ρ( + l) ( i ) The previous sum is trimmed for max p+ maxq+. It remains to calculate the ρ for i and fixed. Resuming ( i ) ρ is the sum of the cofactors of the elements that correspond à ρ( h ). The explicit calculation of these partial derivates isn t produced in the article it s based on the following proposition Proposition Let A a matrix of order n. If ai al c x( c> i ) and if all the other elements are distinct and different from x then (det A) a lc + ai where a lc (resp. a i ) denotes the x cofactor of a lc (resp. a i ). This result is easly generalized if several elements of the matrix A are equal to x. In a second step we calculate the Student table of general term: t i ( i ) σ ( ( i )) as βγ. Also we test the nullity of i We consider the matrix T ( δ i ) defined by δ i if t i <.96 and δ i otherwise. In that case the automatic determination of p and q can be realized as follows. The matrix B ( b i ) is defined by: Ρbi δi + δi + δi i 2 2 b + + Σ δ δ 2 bi δi + δi + i 2 Τ b δ + 2 Thus the b i equal to 2 represent the corners namely b i 2 if and only if pi- and q-..2 The Pearson chi-square statistic The previous Student s test estimate separately the nullity of each βγ. i It seems preferable to test the simultaneous nullity of a corner βγ i βi+ γ and βi +γ. For i and fixed ' let ( i ) 2 ( i+ ) 3 ( i + ) and ( 2 3 ). Vectors are presumed to be column vectors unless they are transposed ( ). Denote A( a ih ) the matrix of order (3K) defined by : i aih ρ( h)

4 ' ' The quadratic form Q N ( AΣA ) converge to a chi-square distribution which three degrees of freedom. The previous step is too used to determinate an MA model. The SAS macro present and compare these two tests..3 The SAS-macro The following statements are available in %coin. DATA OUT V MAXP MAXQ NIVEAU PRINT name of the input data set containing observations. name of an output data set containing the models. names of the time series. the upper bound of the order of the autoregressive models. the upper bound of the order of the moving average models. level of the Student's test. By default niveau5 (%). prints the output. %coin(database.arma vararma_2 maxq5 maxp5 niveau5 printoui out_result_);.4 An example The same example is used for the three methods with simulations and we keep the data set. PROC IML; phi{.5 }; theta{. -.56}; donneesmasim(phitheta2); nom{'arma_2'}; CREATE essai FROM donnees [COLNAMEnom]; APPEND FROM donnees; CLOSE essai; QUIT; «Student» table MA Matrix T MA

5 Chi-square table MA Matrix of results of the chi-square test MA Specification of the order of Arma models OBS MA STUD CHI As we can see on that example each test proposes more than one model. 2. The «triangle» method This method is based on interactive regression procedure that produce consistent estimates of the autoregressive parameters. «An extended sample autocorrelation function based on these estimates is then defined and used for order determination» ([8]). As in the corner method we estimate p (resp. q) in a set of values {...maxp} (resp. {... maxq}). 2. Initialisation R denotes the matrix of the sample autocorrelation function. The general term is defined as follows : ρ ( Y t ) r( X t X t ) for...maxq +. R is a row matrix. We note r( ) ρ ( Yt) and B the backshift matrix of order (NM) where the kth column is B k Yfor k...mmaxp + maxq +. More precisely

6 B Φ Γ ΗΓ.... X... X2 X.. X X X N N 2 N M Let Φ the triangulare matrix of regression parameters of Y t on the first kth column of B. Φ ( ϕ lk ) where < l k < M+ and ϕ lk for l > k. Φ Φ ΗΓ ϕ ϕ. ϕ 2 M ϕ22. ϕ2 M.. ϕ If ϕ k denotes only the first kth columns of Φ and B k the extractive matrix of B with only the first kth column we have ' ' ϕ k ( BB k k ) BY k k where Y k is the transpose of the matrix B. 2.2 Iterative step M M Ι ϑ Κϑ Ι Κϑ The previous considerations have led us to consider a general set of iterated regressions. Let Φ ( ϕ l k ) the triangulare matrix of order M- defined by : ϕl k ϕk+ k+ ϕlk ϕlk + for l k ϕ ϕ lk ϕ k for l k kk > and k Consider the time series W kt that Wkt Xt ϕl k X k l t l for k to maxp. We now define the extended sample autocorrelation matrix A r( k) ρ ( Wk t) for k to maxp and to maxq Identification The asymptotic property of A can be exploited to help tentavely identify MA(pq) models in practice. If Y k follows an MA(q) model then r( ) > q where denotes asymptotic equivalence in probability (Student s test). If Y k follows an MA(pq) model (p>) r( k) for k p and q > k p r( k) for q and q k p This results may be presented in a two-way table E in which the rows are numbered... maxp to signify the order and the columns in a similar way for the MA order.

7 If we denotes x a non zero value and a null value an MA(pq) model is presented as follows : \MA. q. q x x x x x x x x x x x x... x x x x p.. x... x x p x x x x x We remplace x by. Let E ( δ k ) where δ k or and δk ρ( k ) for k max p+ max q+. maxq+ ) For k γ δ c and Yt MA( q) if δ q+ c 2) For k > γ k δl cwhere Uk ( l c ) k l + l k c () lc U k µ ρ The values of triangles are added in order to include the case of eventual isolated zeros. Thus we define the matrix B ( b k ) that b maxq+ b k if γ k and k 2 k max q+ b k if γ k and k 2 k max q+ Finally we denote ck bk + ( bk + bk+ ) and 2 Yt MA ( p q) cp q 2.4 The SAS-macro The following statements are available in %TsayTiao. DATA name of the input data set containing observations. OUT name of an output data set with models. V names of the time series. MAXP the upper bound of the order of the autoregressive models. MAXQ the upper bound of the order of the moving average models. NIVEAU level of the Student's test. By default niveau5 (%). PRINT prints the output. %TsayTiao(database.arma var maxp5 maxq5 niveau5 printoui out_result_); 2.5 An example The previous statements produce this output where the different matrix used are printed : + +

8 Matrix ESACF (Extended Sample Autocorrelation Function) MA Matrix E MA Matrix B MA OBS MA Order determination quantity criterion To estimate the unknown true orders of an MA model the authors introduce the ODQ that is defined by : 2 2 ODQ p q n σ p q n σ p q a where β γ β γ χ η n n n n () p p q q (2) σ n 2βγ.. denotes an estimate of the common variance of the error sequence (3) a n is any constant provited that a n an and ( β ). For example a β n n ε n (ln( n)) where < ε <. We choose an n and ( pq ) such that : µ ρ ( p q) Min ( p q) ODQ( p q) > ODQ( p q ) > ODQ( p q) <

9 3. Tentative model identification We calculate each ODQ( p p) and we stop at the first p that ODQ( p p)<. It is not necessary to calculate ODQ( p q) to each couple ( pq. ) More precisely let p and pi+ pi + we calculate successively ODQ( pi pi). If ODQ( pi pi)> we continue and if ODQ( pi pi)< we stop. Thus we obtain p such that ODQ( p p ) >. In a second step ODQ( p p ) and ODQ( p p ) are calculed. Four cases can appear according to the sign of these two terms : ) If these two terms are positive an MA ( p p ) is proposed. 2) If ODQ( p p) < and ODQ( p p ) > one seeks the greatest value of p such that : p< p and ODQ( p p) > thus one obtains an MA ( p + p ) model. 3) If ODQ( p p) > and ODQ( p p ) < one seeks the greatest value of q such that: q < p and ODQ( p q) > thus one obtains an MA ( p q + ) model. 4) If these two terms are negative one applies points ) and 2) thus two models are obtained. To determinate retained models one determine the matrix T of order ( p + q + ) such that: T ( t i ) where t i if ODQ( i ) > and t i if ODQ( i ) <. Then one replaces T by the matrix A whose general term a i is defined by : ai ti + ti + ti 2 In conclusion Yt MA ( p q) a p q The SAS-macro The following statements are available in %ODQ. + + DATA OUT V MAXPQ POWER METHOD PRINT name of the input data set containing observations. name of an output data set with models. names of the time series. the upper bound of the order of the MA models..the power parameter of n in ODQ. specifies the method for estimating parameters prints the output. %ODQ(database.arma var maxpq3 power.5 methodcls out_result_ printoui); 3.3 An example The previous statements produce this output : _NAME_ VALUE ODQ ODQ.6854 ODQ ODQ ODQ2.2546

10 ODQ2.735 _NAME_ P QMA MODELE 2 4. Comparaison between these different methods The different simulations using the corner method put in obviousness the following points: - The performance of the method is sensitive to the number n of observations of the series. The asymptotic character of the method explains these results. More precisely for n < the simulated model is recognized less once on two and models of inferior orders to the former are equally recognized. When n increases the performance of the method improves especially for n > 25 the simulated model is the most often recognized and minimal order among models proposed. Finally improvements are no longer significant when n exceeds 5. - Student s test gives better results. The following graph presents the rate of recognition of the model according to the length of the series. The nullity of the ( i ) is tested which a Student s test (curve superior) and a chi-square statistic. Rate of recognition Student's test Chi-square Number of observations In the presentation these results will be compared to these obtain by the other methods.

11 References [] Beguin J.M. Gourieroux C. and Montfort A. (98) The Corner Method : An Investigation of an Discrimination Procedure for General MA Processes J. Opl. Res. Soc. Vol [2] Box G.E.P. and Jenkins G.M. (976) Time Series Analysis- Forecasting and Control San Francisco: Holden-Day. [3] Hannan E.J. and Quin B.G. (979) The Determination of the Order of an Autoregression J.R.Statist.Soc. B [4] Hannan E.J. (98) The Estimation of the Order of an MA Process The Annals of Statistics [5] Koreisha S. Kallinen A. and Pukkila T. (99) The Identification of MA Models Biometrika [6] Koreisha S. and Pukkila T. (993) New Approaches for determining the Degree of Differencing neccesary to induce Stationarity in MA Models J. of Statistical Planning and Inference North-Holland. [7] Koreisha S. and Pukkila T. (995) A Comparaison Between Different Order-Determination Criteria for Identification of IMA Models J. of Business & Economic Statistics vol. 3 n [8] Tiao G.C. and Tsay R.S. (984) Consistent Estimates of Autoregressive Parameters and Extended Sample Autocorrelation Function for Stationary and Nonstationary MA Models JASA Vol. 79 n [9] Wang P. and Zhang Hu-Ming (994) A new Way to Estimate Orders in Time Series J; of Time Series AnalysisVol