Seasonal models Many business and economic ime series conain a seasonal componen ha repeas iself afer a regular period of ime. The smalles ime period for his repeiion is called he seasonal period, and is denoed by s. As examples, monhly auo sales and earnings end o decrease during Augus and Sepember because of he changeover o new models, and monhly sales of oys rise every in December. For hese cases, s 12. Quarerly ice cream sales are high each summer, and he series ends o repea iself each year, so ha he seasonal period is 4. 1
Seasonal phenomena may sem from facors such as weaher, which affecs many business and economic aciviies like ourism and home building; cusom evens such as Chrismas, which is closely relaed o sales such as jewelry, oys, cards, and samps; graduaion ceremonies in he summer monhs, which are direcly relaed o he labor force saus in hose monhs. 2
As an illusraion he figure below shows U.S. monhly employmen figures (in housands) for males aged beween 16 and 19 years of age during he period 1971 o 1981. The numbers increase dramaically in he summer monhs, wih peaks in June, and decrease in he fall when schools reopen. 3
When a series is seasonal wih a period s, o analyze he daa, i is helpful o arrange he series in a wo-dimensional able according o he period and season, and including he oals and/or averages. 4
Nex we consider radiional mehods for analyzing seasonal ime series. Convenionally, ime series have been hough o consis of a mixure of rend or rend-cycle ( T ), seasonal ( S ), and irregular ( e ) componens. If hese componens are independen and addiive, one can wrie he imes series Z as Z T S e Depending on he daa, a muliplicaive model may be used, so ha a linear model is obained afer aking logs. 5
In he regression mehod for analyzing such series, one may model T m 0 i 1 iui, where he U i are rend-cycle variables. For example, we may have T m i 0 i 1 i. Similarly, he seasonal componen may be described as a linear combinaion of seasonal dummy (indicaor) variables, or as a linear combinaion of sine-cosine funcions of various frequencies. 6
Here we may have S s1 D j1 j j, where D 1 if corresponds o he seasonal j period j, and zero oherwise. Noe ha when he seasonal period equals s, we need only s 1 seasonal dummy variables. 7
Alernaively, S can be wrien as S s /2 sin j 1 j j 2 j 2 j cos s s where s/2 is he ineger par of s /2. 8
For a given daa se Z and specified values of m and s, he sandard leas squares regression mehod may be used o obain esimaes ˆi, ˆ j, ˆ j and ˆj of he parameers i, j, j and j of he models above. The esimaed componens are hen m Tˆ ˆ ˆ U S, 0 i 1 i i s1 ˆ D or, j1 j j e Z Tˆ Sˆ. ˆ 9
In he moving average mehod o modeling seasonal imes series, a variey of moving averages are used o exrac he rend and seasonal componens ha are assumed o be fixed or change slowly. For example, he rend can be esimaed by smoohing using a moving average, and hen he seasonal esimaed by applying a moving average o smooh he de-rended series. The above esimaes may be obained ieraively by repeaedly applying various moving average operaors. 10
A successful example of uilizing his moving average procedure is he Census X- 11/12 ARIMA mehod, which has been widely used in governmen and indusry. The series wih seasonal flucuaions removed, i.e. Z ˆ S, is called he seasonally adjused series, and procedures for deermining he seasonal componen are called seasonal adjusmen. 11
The mehods presened so far are based on he assumpion ha he seasonal componen is deerminisic and independen of any nonseasonal componens. However, many series are no so well-behaved. More likely, he seasonal componen may be random and correlaed wih nonseasonal componens. We now exend he sochasic ARIMA models o seasonal ime series. 12
We have seen ha, for nonseasonal series, i may be possible o obain a parsimonious represenaion of he form d ( B)(1 B) Z ( B) e, p q wih (1 B) d rendering a polynomial rend of order d as fixed so ha (1 B) d Z is saionary. Since we expec observaions ha are s unis apar o be similar, we would expec ha he operaor s 1B Z Z Z s migh be useful in modeling. 13
As saed earlier, i is of value o hink of a seasonal series as being se down in he form of a able conaining s columns. This represenaion makes i clear ha here are wo inervals of imporance in such daa. In he case of monhly daa, hese inervals would be monhs and years. Specifically, we would expec relaionships o occur (a) beween observaions for successive monhs in a paricular year and (b) beween observaions in he same monh in successive years. 14
The siuaion is somewha like ha in a woway analysis of variance model, where similariies can be expeced beween observaions in he same column and beween observaions in he same row. Suppose ha he h observaion Z is for he monh of April. We may be able o link Z o observaions in previous Aprils by a model of he form s s D s ( B ) 1 B Z ( B ), P Q where s ( B ) and ( s B ) are polynomials P Q s in B of degrees P and Q, respecively, and saisfying causal and inveribiliy condiions. 15
Similarly, a model D s s s ( B ) 1 B Z ( B ) 1 1 P Q migh be used o link he curren behavior for March wih previous March observaions, and so on, for each of he welve monhs. Moreover, i would usually be reasonable o assume ha he parameers of and would be approximaely he same for each monh. (If no, a differen model should be used). P Q 16
The value for April of his year would likely be relaed o values in March, February, ec. of his year as well, and hus we would expec he error componens, 1, 2 ec. o be relaed. Therefore, o ake care of such relaionships, we could inroduce a second model d ( B)(1 B) ( B) e, (1) p q where now e is a whie noise process, and ( B ) and ( B) are polynomials in B of p q degrees p and q, respecively, and again saisfying causal and inveribiliy condiions. 17
We hen have P s s ( B ) 1B Z Q s ( B ) D, and muliplying hrough by ( B )(1 B ) d p and using (1) we obain a muliplicaive model d s s s ( B) ( B ) 1B 1 B Z ( B) ( B ) e. p P q Q D The resuling muliplicaive process will be said o be a seasonal ARIMA model of order pdq,, PDQ,,. s 18
The seasonal and nonseasonal polynomials are assumed o have no common roos. A similar argumen can be used o obain models o ake care of muliple seasonaliies. The mos famous of such models is he 0,1,1 0,1,1 model called he airline 12 model 12 12 1 1 1 1 B B Z B B e, so called because i was used o model naural logarihms of he number of monhly airline passengers in inernaional air ravel. 19
We now compue he auocovariance and auocorrelaion funcions of 12 1 1 W B B Z is 20
Below are 150 simulaed values from he 0,1,1 0,1,1 model 4 4 4 1 1 1 1 B B Z B B e. 21
The ACF of Z is large and decays slowly, whereas he PACF has a large spike a lag 1. These indicae ha he series is nonsaionary and ha differencing is called for. 22
To remove he nonsaionariy, he series is differenced, and he sample ACF and PACF of he differenced series B Z are shown in he nex figure. 1 23
The fac ha he ACF decays very slowly a muliples of he seasonal period 4 implies ha a seasonal differencing is also needed o achieve saionariy. I is known ha he ACF of a 0,0,1 0,0,1 model is zero excep a lags 4 1,3,4, and 5, and hus he sample ACF of he figure indicaes ha a 4 4 1B 1B Z 1 B 1B e model is appropriae for he original series. 24
One modificaion of he muliplicaive model menioned by Box and Jenkins (1976) is o allow a general moving average operaor * ( B) o replace he muliplicaive operaor s ( B) ( B ), and alernaively, or in addiion, o replace he muliplicaive auoregressive s operaor ( B) ( B ) by a more general operaor * ( B). 25
A few ips on fiing seasonal ARIMA models: In idenifying a seasonal model, he firs sep is o deermine wheher or no a seasonal difference is needed, in addiion o or perhaps insead of a non-seasonal difference. You should look a ime series plos and ACF and PACF plos for all possible combinaions of 0 and 1 non-seasonal difference and 0 or 1 seasonal difference. Don ever use more han one seasonal difference, nor more han wo oal differences (non-seasonal and seasonal combined). 26
If he seasonal paern is boh srong and sable over ime (e.g. high in he summer and low in he winer, or vice versa), hen you probably should use a seasonal difference regardless of wheher you use a non-seasonal difference, since his will preven he seasonal paern from dying ou in he longerm forecass. Noe ha we could fi a model wih fixed seasonaliy, if ha makes sense for a daa se. 27
Periodically correlaed sequences A ime series Z is called periodically correlaed wih minimal period T if he period T if is he smalles posiive ineger such ha E Z u E Z u T E Z Z E Z Z. u v ut vt A wide-sense saionary sequence is a special case of a PC sequence since i saisfies he definiion of a PC sequence wih period T 1 28
PC sequences have a lo of he properies of saionary sequences, bu are in general nonsaionary in a very simple manner ha can be useful for modeling. PC sequences have been used as models in meeorology, radio physics, and communicaions engineering. Due o he periodic naure of heir nonsaionariy, PC processes have been called cyclosaonary, periodically saionary, periodic nonsaionary, and processes wih periodic srucure. 29
Now we consider some simple models for generaing a PC sequence from a saionary Y. saionary sequence Lemma: Le C be a periodic deerminisic funcion wih period T, and Y and saionary sequence. Then he sequence C Z is PC wih periodic T. Proof: 30
Noe ha whereas here he mean and covariance funcions are periodic wih period T, T need no be he minimal period. For example, le T be even, Y have mean zero for all, and C C T /2 C has period T,. Whereas /2 /2 E Z Z EC s T Y C T Y st/2 T/2 st/2 T/2 E Z Z. s 31
Lemma: Le C and Y be as in he previous lemma. Then he sequence C Y is PC wih periodic T. Proof: 32
The period T of a PC sequence may be esimaed in several ways. One is o use a funcion of he sample auocovariances (Tian 1988). Marin (1990) poined ou problems wih his mehod, and gave an esimaion procedure ha worked in cases when he mehod above gave he wrong esimae. The mehod was based on he average magniude difference funcion 1 N j k1 s j Z Z N j for various values of j. k 33
Why use a PC model insead of an ARMA model? Several auhors have discussed he consequences of fiing a saionary ARMA model o a ime series ha has periodic auocorrelaion. These include forecasing inefficiency, inconsisen parameer esimaes, spurious seasonal facors in he auoregressive operaor, and overspecificaion of model orders. 34
Why use a PC model insead of a seasonal ARIMA model? In seasonal ARIMA models, an assumpion is made ha he correlaion srucure of he series a fixed lags is consan across he series, and ha aking seasonal differences (possibly along wih a nonseasonal differences), renders he series saionary. 35
However, for some seasonal ime series ha is no he case. For example, Vecchia and Ballerini gave examples of ime series Homesead, Florida rainfall daa and monly sreamflow for he Fraser River in Hope, Briish Columbia) ha were adjused for a esimaes of he periodic mean and variance of he series, i.e. Y Z ˆ, ˆ where Y sill displayed periodic behavior in is auocovariance funcion. They developed a es for periodic auocorrelaion based on large sample properies of a Fourier-ransformed version of he esimaed periodic auocorrelaion funcion. 36
Marin (1999) developed a es for periodic auocorrelaion based on zero-crossings, he number of imes a series crosses he x-axis. The es was applied o wo daa ses, area average precipiaion in he US over 1990 o 1992, and he number of ornadoes in he US from 1953 o 1993. Zero-crossings were used because of heir simpliciy and drasic daa reducion. 37