Generalizations of the Box-Jenkins Airline Model with Frequency-Specific Seasonal Coefficients and a Generalizaton of Akaike’S Maic

GENERALIZATIONS OF THE BOX-JENKINS AIRLINE MODEL WITH FREQUENCY-SPECIFIC SEASONAL COEFFICIENTS AND A GENERALIZATON OF AKAIKE’S MAIC John Aston, David Findley, Kellie Wills, U.S Census Bureau, Washington, D.C. Donald Martin, U.S Census Bureau and Howard University, Washington, D.C. Key Words: Time series, seasonal adjustment, forecasting, ss−−11 Model Selection, AIC 22jjj −=−−ε (1BBZaBbBcB ) ( )tt (1 )( ) (3) ∑∑ jj==00 Abstract. The Box-Jenkins “airline” model is the most widely used ARIMA model for seasonal time series. Findley, Martin In this model the seasonal sum polynomial has a third coefficient and Wills (2002) examined a generalization of the airline model c distinct from the coefficients associated with the other factors with a more restricted seasonal moving average factor that in the model. models only seasonal effects and with a second-order In the present paper, we investigate various generalizations nonseasonal moving average factor. Here, we generalize the of (2) and (3), which we call frequency-specific models. In these seasonal factor further by associating a subset of the frequencies s−1 1, 2, 3, 4, 5 and 6 cycles per year (in the case of monthly data) models, the final moving average factor, e.g. ∑cBj j in (3), is with one coefficient and the complementary subset with a j=0 second coefficient. A generalization of Akaike’s Minimum AIC decomposed into several factors with different coefficients. criterion is presented for choosing among subsets of a given size Restricting attention to monthly data, i.e. s =12 , the model (3) or, more generally, among generalizations of the airline model 11 having the same number of coefficients. Properties of model- can be generalized by factoring ∑cBj j in terms of frequencies based seasonal adjustment filters obtained from the new models j=0 are considered as well as forecasting performance relative to the of 1, 2, 3, 4, 5 and 6 cycles per year to obtain a general airline model. frequency-specific model, 11 (1−=BBZ )2 (j ) 1. Introduction ∑ t j=0 (4) ⎡ 5 ⎤ Box and Jenkins (1976) developed a two-coefficient time series (1−−aB bB222 )⎢() 1 + c B∏ () 1 − 2 c cos(2π j ) B + c B ⎥ε model, now known as the airline model, which is by far the most 6 jj12 t ⎢ j = 1 ⎥ widely used ARIMA model for monthly and quarterly ⎣ ⎦ macroeconomic time series. (Fischer and Planas (2000) deem it adequate for 50% of 13,232 Eurostat time series.) The Box- If the six ci ’s are distinct, this model has a different seasonal Jenkins airline model for a seasonal time series Zt observed coefficient for each seasonal frequency, for a total of eight s ≥ 2 times per year has the form coefficients. Eight moving average coefficients cannot be estimated reliably from macroeconomic time series of typical lengths. We shall consider instead the most parsimonious such (1−−BBZ )(1ss ) =−−Θ (1θε B )(1 B ) (1) tt generalizations of (2) in which there are only two distinct c ’s. i When Θ>0 , the airline model can be written That is, the seasonal frequencies are divided into two groups, with all frequencies in a group having the same coefficient. This ss−−11reduces the total number of coefficients requiring estimation in −=−−ΘΘ21//jsjsjθεthe model to four in general, and to three when we constrain the (1BBZB ) (∑∑ )tt (1 )(1 B )( B ) (2) jj==00MA(2) in (4) to have a factor whose coefficient is one of the seasonal coefficients ci , in analogy with (2). Findley, Martin and Wills (2002) substituted a general We consider two types of four-coefficient models. The first, 1/ s MA(2) polynomial for (1−−Θθ BB )(1 ) in (2), yielding a designated the 5-1(4) type, is the one in which five of the generalized airline model, frequency factors in brackets in (4) have the same coefficient c1 and the sixth has a different coefficient c . There are six such This paper is released to inform about ongoing research and to 2 encourage discussion. Any views expressed are the authors’ and models, an example being not necessarily those of the U.S. Census Bureau. John Aston is now at Academia Sinica, Taipei. Kellie Wills is now at the Corporate Executive Board, Washington, D.C. 11 −−θθχˆÂF2 2 j 2 2{lnLL ( ) ln ( )} ~ F A (8) (1−=−−BBZaBbB ) (∑ ) (1 ) dimθθ− dim t j = 0 (5) asymptotically under standard assumptions; see Taniguchi and Kakizawa (2000, p. 61). Under (8), the probability that the ⎧⎫5 ⎪⎪π ×+()112cos().cB∏ () − c2 j B + cB22 ε frequency-specific model will have a smaller AIC and thus be ⎨⎬21112 t ⎪⎪j =1 incorrectly preferred by Akaike’s Minimum AIC criterion ⎩⎭(MAIC) is The second type, designated 4-2(4), is the one in which four of the frequency factors in brackets in (4) have the same PAIC{()θθˆÂF−>= AIC ()0} coefficient c1 and the remaining two have a different coefficient (9) P{2(dimdim)}.χθθ2 >−FA dimθθFA− dim c2 . There are fifteen such models. We also consider the corresponding types of three-coefficient frequency-specific generalizations of (2), denoted 5-1(3) and 4- Thus the probability of incorrectly rejecting the airline model in 2(3) models. An example of the former is favor of a frequency-specific model is 11 pP(4)≡>={χ 2 4} 0.135 for a four-coefficient model and −=−−2 j 2 (1BBZaBcB ) (∑ ) (1 )(11 ) = t (3)≡>=χ 2 j 0 (6) pP{ 1 20.157} for a three-coefficient model. ⎧⎫5 We now describe a generalization of MAIC that is directed ⎪⎪2π j 22 ×+()112cos().cB∏ () − c B + cB ε ()i ⎨⎬21112 t toward achieving the same type I error probabilities p when ⎩⎭⎪⎪j =1 There are six 5-1(3) models and fifteen 4-2(3) models. We the minimum AIC model from a set F(i) containing several i- did not consider the twenty models that arise by grouping the coefficient models more general than (2) is compared to the frequency factors in (4) into two groups of three factors because estimated airline model, for i = 3, 4 . Our criterion is to prefer it would increase the number of models to be compared from 22 this MAIC model over the estimated airline model for a time to 42. series of length N when, for a threshold ∆=∆()ii ()(())0F i ≥ These new models cannot be estimated with standard NN ARIMA modeling software. We performed the estimation in the with a certain property, the inequality θθˆÂFi−>∆() object-oriented matrix programming environment Ox (Doornik AIC()minF∈F (i) AIC () N (10) 2001), using the state space functions in the SSFPack library holds. The property desired of ∆()i is (Koopman, Shephard and Doornik 1999). Some details are given N θθˆÂFii−>∆() () in the Appendix. PAIC{()minFN∈F (i) AIC () } p (11) Before presenting our evaluation of the new models relative when the airline model is correct. This is the property of ∆=()i 0 to the airline model for seasonal adjustment and forecasting N using Census Bureau, we present our model selection criterion when F(i) contains only one model. We call the resulting for deciding when one of the new models should be considered generalization of MAIC the GMAIC criterion. for replacing the airline model. This criterion is a generalization ∆()i Such N ’s can be obtained from the empirical distribution of Akaike’s Minimum AIC criterion (MAIC). of AIC()minθθˆÂF− AIC () when the models in F(i) are fit F∈F (i) to simulated Gaussian time series of length N generated by an 2 A Modification of Akaike’s Minimum AIC Procedure for airline model, for example, the airline model estimated from the Multiple Fixed-Dimension Comparisons to a Nested Model time series of interest, or from a nearby model for a series of approximately the same length. In this paper, for illustrative A A A ∆()i For the airline model, let θˆ , dimθ , and L()θˆ denote the purposes, we use the N ’s in Table 1 below. These values were estimated parameter vector, its dimension, and the associated obtained from simulated series from a single airline model with θ Θ maximum log-likelihood value respectively. Let θˆF , dimθ F , coefficients = 0.5 and = 0.5. These are fairly typical values. The lengths N =120 and N =150 in Table 1 are close to the and L()θˆF denote the corresponding quantities for a frequency- lengths of the Census Bureau series we model. specific model. Consider the AIC difference ∆()i Table 1. Thresholds N for the Four Model Types. ∆≡AF, θθˆˆ A − F = AIC AIC() AIC () 4-coefficient 3-coefficient . (7) Thresholds −−−−2{lnLL (θθˆÂF ) ln ( )} 2(dim θθ FA dim ) 5-1 4-2 5-1 4-2 ∆=()i Because the airline model is a special case of each type of N ,120N 1.9 2.0 1.8 2.3 frequency-specific model, when the airline model is correct, ∆=()i N ,150N 1.6 2.1 2.3 1.5 Note that the use of GMAIC always requires the fitting of the series. Among the remaining 5 series, the 4-2(4) model has the airline model to the series being modeled. minimum AIC for 4 series, and a 5-1(4) model has the minimum AIC for one series. For 7, 2, 2, and 1 of these models Remark 1. While the p()i values are large relative to respectively, the largest or smallest seasonal spectral peak of the (differenced, log-transformed) modeled series occurred at a empirically chosen significance levels of tests like .05, they are more fundamental quantities than such empirical choices frequency associated with c2 .

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