Is Aggregation Necessarily Bad? (With apologies to Grunfeld and Griliches)

Transcription

1 Is Aggregaion Necessarily Bad? (Wih apologies o Grunfeld and Griliches) Moonyoung Baek, P. Geoffrey Allen, and Bernard J. Morzuch Deparmen of Resource Economics, Universiy of Massachuses Amhers Wih high frequency daa (e.g., hourly), when decisions are based on lower frequency aggregaes (e.g., four-hourly inervals) he possibiliies are o aggregae he daa hen direcly forecas he aggregaes, or indirecly o esimae he disaggregae series hen aggregae he forecass. No clear principle has emerged concerning his choice, and pas empirical work has produced conflicing resuls wih no indicaion as o wha circumsances migh favor one approach over he oher. Aggregaing he daa amouns o hrowing away informaion. On he oher hand, if he amoun of noise in he disaggregae daa swamps any addiional signal, parameer esimaes are more difficul o make. The added uncerainy leads o less accurae forecass. For he wo series examined, hourly arrivals a a hospial emergency room, and hourly elecriciy load daa, resuls are fairly consisen. More accurae forecass are obained from esimaing and forecasing he disaggregae daa and aggregaing he forecass compared wih esimaing and forecasing using aggregae daa. The variaion over a ypical day and he day-o-day variaion hrough he week are greaer for he elecriciy daa han for he emergency room arrivals. The differences beween he series are apparenly no large enough o affec he preferred sraegy of disaggregae esimaion and forecasing.

2 Inroducion Is Aggregaion Necessarily Bad? (Wih apologies o Grunfeld and Griliches) Forecasing is abou managing uncerainy. In a pracical seing his requires including informaion ha you know for sure and leaving ou of he analysis relaionships ha are difficul o quanify. Because i is relaively parsimonious, he Hol-Winers mehod is a popular and usually effecive way of making forecass, especially when causal relaionships are eiher unclear or no esimable wih available daa. One area where uncerainy is an issue is in siuaions where daa are available a a micro-level bu forecass are needed a a higher level of aggregaion. Is i beer o forecas firs hen aggregae or aggregae firs hen forecas? The quesion arises in conemporaneous aggregaion, for example across counries, indusries, governmen expendiure caegories, and in emporal aggregaion, for example from monhs o quarers. The laer is of ineres here. Temporal aggregaion has been sudied for many years, hough he majoriy of he lieraure concerns heoreical issues, no pracical applicaions. When he daa generaing process (DGP) is known, he resul is sraighforward: beer forecass resul from using he disaggregae informaion. For an ARIMA process, he order of inegraion of he aggregaed process is he same as he order of inegraion of he original series, as is he order of he auoregressive process, bu his is no rue for he moving average componen. Generalizing o VARMA processes, Lükepohl (005) shows ha forecass from disaggregaed flow variables generally have lower MSE han forecass from he aggregae, alhough he difference disappears as he forecas horizon goes o infiniy. The wo forecass will be equal for small horizons if and only if he disaggregae series has a finie moving average srucure of order less han he lengh of season. This argues for he sraegy forecas firs, hen aggregae. In pracice, he DGPs have o be specified eiher by assumpion as wih he Hol-Winers srucure or on he basis of limied sample informaion. This adds uncerainy. Model parameers mus be esimaed again using limied sample informaion, adding furher uncerainy. In ha case quie differen resuls may be obained and, in paricular, forecass based on disaggregae processes may be inferior o hose based on he aggregae direcly. If he sample size is large enough, he par of he forecas MSE due o esimaion uncerainy will evenually be so small ha he forecas based on he disaggregae is again superior o he corresponding aggregae. If esimaed insead of known processes are used, i is possible ha forecass based on disaggregaes are worse because he MSE par due o esimaion may be larger for he former han for he laer. This siuaion is in paricular likely o occur if he DGPs are such ha effciency gains from disaggregaion do no exis or are small and he aggregaed process has a simple srucure which can be capured wih a parsimonious model (Lükepohl, 005). Man (004) performed a deailed analysis of a relaively simple periodic ARMA model. He considered when a misspecified (or adaped ) model on aggregae daa gave beer forecass in a mean squared error sense han an adaped model on disaggregaed daa. He compared his difference relaive o he MSE of he rue model forecass. The aggregae approach is preferred when he adaped disaggregae model is badly misspecified, when he aggregae daa has weak dependency (or is close o noise), when he aggregae daa has lile variabiliy, and when he

3 disaggregae moving average model has smaller noise variance a he seasonal lag han a shorer lags. A weak seasonal paern and badly misspecified disaggregae model favor an aggregae model wih simpler srucure. Table summarizes he relaively small number of sudies discovered ha compare aggregae forecass on real ime series. Mos aggregae from monhly o quarerly values and use an ARIMA model. None were found ha specifically compared Hol-Winers esimaion under emporal aggregaion, hough of course, HW wih no or addiive seasonaliy can be expressed as an equivalen ARIMA model. In addiion, here are one or wo simulaion sudies (e.g., Hoa and Neo, 993, Souza and Smih, 004). Resuls are clearly mixed wih somewha more suppor for he forecas hen aggregae sraegy and no clear indicaion of when he alernaive sraegy is preferred. Lükepohl (986) considered boh emporal and conemporaneous aggregaion simulaneously. He showed ha, under a broad range of values for number of conemporaneous series and amoun of aggregaion, emporal aggregaion reduces forecasing efficiency while conemporaneous aggregaion may improve i. In addiion, Koreisha and Fang (004) reworked Buer s (976) daa and on differenial yield raes and in addiion compared updaing he quarerly forecas wih he aid of he mos recen monhly observaion. This was highly useful for one-quarer-ahead forecass bu he value of updaing ailed off rapidly a longer horizons. Daa For he hourly emergency-room arrivals, he esimaion sample runs from midnigh April 3, 000 o 300 hours on December 3, 000 (39 weeks, 655 observaions) and he pos-sample period runs from midnigh January, 00 o 300 hours on April, 00 (3 weeks, 84 observaions). Hourly elecriciy loads for he five New England saes of he Unied Saes (Connecicu, Maine, Massachuses, New Hampshire, Rhode Island and Vermon) are repored by he Independen Service Operaor on is websie (hp:// as he second daa se. The esimaion sample and pos-sample periods are idenical o hose of he emergency room daa. Figures and show he firs four weeks of each series. Comparing he wo daa series, boh show srong daily paerns. Elecriciy loads show sronger weekly paerns han do he emergency room arrivals. Emergency room arrivals someimes have zero values while elecriciy loads do no. In each series, hourly daa are aggregaed o four-hour blocks. In conras o he siuaion analyzed by Man (004) who considered annual aggregaes of quarerly daa, we expec he aggregae daa o display seasonal paerns in much he same way as did he disaggregae daa. Models Ord, e al. (997) devised, and Koehler, e al. (00), and Hydman, e al. (00) furher developed, a complee sae-space form for he sandard Hol-Winers srucure. Wih addiive seasonal facors and addiive error in he observaion equaion he specificaion is 3

4 y = l + (a) + b + s m l = l + b + α (b) b = b + α (c) s = s m +α 3 (d) where ~ N(0, σ ) ; y is an observaion a ime of he ime series of ineres; l, b, s are he unobservable level, rend, and season (lengh m ) componens; and α, α, and α 3 are unknown parameers in he sae equaions. Under he special assumpion of perfec correlaion among he error erms, his innovaions saespace model can be esimaed by one of he sandard Hol-Winers algorihms. The wo benchmark resuls are based on his sandard single seasonaliy model, wih season lenghs m =4 and m = 68. While here are several commercial sofware packages ha accommodae season lengh of 4 we found none ha accommodaed 68 periods. Resuls are compued by a cusom program wrien in R ha was checked for m = 4 agains he oupu from he Time Series Module in SAS. Using m = 68 permis each day o have a differen paern, hough he updaing of he seasonal facors occurs only once per week. If in addiion o a daily seasonal paern here is also a weekly paern (for example where weekend days have a differen paern from weekday days) his can be modelled as muliple (in his case double) seasonaliy. Taylor (003) was he firs o propose such a developmen, by adding a second se of seasonal facors o he sandard H-W srucure. In sae-space form wih addiive seasonaliy his appears as (Gould, e al. 005): y = l b + s + s + (a) +, m, m l = l + b + α (b) = b + α (c), = s, m + α 3 (d), = s, m + α 4 (e) b s s where and are he seasonal indices wih seasonal lenghs m (= 4) for and m s, s, s, m s, (=68) for ( m ). Alhough more heavily parameerized han HW(68), requiring one addiional smoohing parameer and 4 iniial values for he shorer se of seasonal facors, Taylor found ha his double seasonal exponenial smoohing approach ouperformed he single seasonal Hol-Winers mehod, double muliplicaive seasonal ARIMA model (Taylor, 003), an arificial neural nework model, a principal-componen-analysis based regression model (Taylor, e al., 006), and a periodic ARMA model. For some daa series, i appears ha he HW(68) approach suffers by using week-old seasonal facors in he forecass, raher han day-old. 4

5 Gould, e al. (005) have developed an addiive error version of he innovaions sae-space model wih muliple seasonal paerns. The mos general version conains seven ses of seasonal facors, each of lengh 4, one facor for each day of he week. y = l r + b + xi si, m i= + (3a) l = l +α (3b) b s + b b = +α (3c) i = i m i, j j= r s, + ( x ), ( i =,, r) j There is a 7 x 7 marix of seasonal smoohing parameers, i, j each row referring o a he seasonal facors corresponding o a paricular day of he week, and each column referring o a day of he week. Only one day s smoohing parameers are used a once, so ha, for example, on Monday, he firs day of he week, smoohing parameers in column are used o updae each se of seasonal facors. Use of he appropriae column is conrolled by he dummy variable where x i = 0 if ime period occurs when sub - cycle i is in effec oherwise In conras wih he previous models, i is now essenial o keep rack of he day of he week for he curren observaion in order o apply he correc smoohing parameer o each se of seasonal facors. In he mos general case here are m k = = 7 ses of seasonal facors. Where differen m days have a common seasonal paern, hey can be grouped ino r common subcycles where r k. For example r = if all he weekdays are grouped ogeher and he weekend days are in a separae group. This would reduce he number of seasonal facors o 48 and he marix of smoohing parameers would be x. Independenly, resricions can be placed on he marix of smoohing parameers. Gould e al. * * (005) propose a common diagonal, ii = for all i, and common off-diagonal elemens ij = for all i j. If he off-diagonal parameers are non-zero, hen updaing occurs in all seasonal * * facors including hose applying o oher days of he week. A furher resricion is = and a * final one is = 0. Wih he las resricion, seasonal facors for all excep he curren day are carried forward wihou updaing, so ha updaing occurs once per week, corresponding idenically o HW(68). x i Muliplicaive seasonaliy versions of all of he above models can be consruced similarly. Muliplicaive error versions are also possible, bu do no aler he poin forecass, only predicion inervals and saisical ess. Finally, he assumpion of a single source of error can be relaxed o give an unobserved componens (UC) form of he HW srucure. Firs, he basic srucural model wih a dummy variable ype single seasonaliy appears as: (3d) 5

6 y = µ + + (4a) µ = µ + η (4b) + β β = β + ζ (4c) s j= = + ω j where ~ N(0, σ ), η ~ N(0, σ η ), ζ ~ N(0, σ ζ ), and ω ~ N (0, σ ω ), and s is he season lengh ( s = 4 for a daily paern or s = 68 for a weekly paern). The fully unresriced version of he UC model wih double seasonaliy is: y = µ + + (5a) +,, (4d) µ = µ + η (5b) + β β = β + ζ (5c) s, = + ω, j, j= s, = + ω, j, j= where s (e.g., s = 4 ) and s (e.g., s = 68)are seasonal lenghs ( s < s ), (0, ω N σ ), and ω, ~ N (0, σ ω, ~ ω ). This version of he UC model is he mos general specificaion of a srucural ime series model wih double seasonaliy because wo independen sae equaions of each seasonal paern are independenly included, and boh sae equaions capure dynamic changes in each of wo seasonal paerns. Like Taylor s double seasonaliy approach, his form of he UC model conains 9 seasonal facors. Unforunaely, alhough his srucure can be specified in PROC UCM of SAS, he program appears unable o esimae is parameers, possibly because of mulicollineariy when a shor seasonal paern is repeaed wihin a longer one. A considerably more parsimonious specificaion, and one ha can be esimaed by PROC UCM is he model wih blocked seasonaliy. I resrics each day o have a common daily paern bu allows he scale o change from day o day. (5d) (5e) y +, +, = µ + (6a) µ = µ + η (6b) + β β = β + ζ (6c) s, = + ω, j, j= (6d) 6

7 k s, = [ δi, j ] ω, i= j=0 + h where δ i 's are dummies for he blocked seasonal paerns, δ i = for he i block and δ i = 0 oherwise, k is he number of he blocks, s is he idenical block size per block ( s = 4, here). The blocked seasonal UC model has 3 (= 4 + 7) seasonal facors. Models are esimaed once wih he original hourly daa and he forecass aggregaed o give a four-hour forecas, corresponding o one sep ahead. Second, he daa are aggregaed ino fourhour blocks and he models re-esimaed. These correspond o he forecas firs hen aggregae and aggregae firs hen forecas approaches. Resuls We examined boh addiive and muliplicaive seasonaliy for he elecriciy load daa. Resuls are almos universally beer for muliplicaive seasonaliy and only hose resuls are presened. For he emergency room arrivals daa, he presence of zeroes in he original daa series precludes he use of muliplicaive seasonaliy. Resuls for boh daa series are clear cu. Regardless of he model compared, he beer sraegy is o forecas firs using he original hourly daa, hen aggregae he forecass. Wih he elecriciy load series, as shown in Table, he loss of performance from working wih aggregae daa is sriking. Forecas errors, measured by roo mean squared error pracically double when only aggregae daa are available. There are differences caused by he use of differen algorihms. In general he Muliple Seasonal model algorihm leads o larger errors compared wih equivalen models esimaed differenly. Unobserved Componens models are he bes approach wih disaggregae daa, ye wih aggegae daa heir performance is mixed. Double seasonaliy models, while never bes, perform consisenly well wih boh aggregaed and disaggregaed daa. The srong daily and weekly cycles in he hourly daa appear o be mainained in he aggregae daa. A surprisingly poor performer is he Muliple Seasonaliy model wih common weekday and separae common weekend cycles (r=), Alhough he daa display his paern visually, his parsimonious approach for some reason fails o benefi. One noiceable feaure of he elecriciy load series is ha he smoohing parameers for level end o be high, ofen exceeding 0.9. Elecriciy consumpion does display an annual paern and his possibiliy was no allowed for, and his possibly accouns for he near non-saionariy of he series. The firs sriking difference beween elecriciy loads and emergency room arrivals is ha while forecasing wih he disaggregae daa is sill desirable, he accuracy using aggregae daa is hardly much worse, and for wo of he mehods is acually slighly beer. Table 3 shows he deails. Double seasonaliy approaches perform he bes, followed by long-cycle single seasonaliy, wih shor cycle seasonaliy he wors. The difference beween mos and leas accurae is small, abou percen, compared wih he much wider range for he elecriciy daa. (6e) 7

8 Neiher he Unobserved Componens model nor he wo-cycle version of Muliple Seasonaliy perform well. Finally, he wihin-sample performance is a good guide o choice of model for he elecriciy load series, bu much less reliable for he emergency room arrivals. Conclusions The wo hourly series examined here provide addiional suppor o he principle: if you have disaggregae daa, use i, even when he forecas is needed for a more aggregaed ime period. The principle was more srongly suppored in he series ha displayed sronger daily and weekly paerns. Taylor s (003) double seasonal modificaion o he sandard Hol-Winers approach was remarkably robus. The wo series did no provide conclusive evidence concerning he value of permiing muliple sources of error when specifying sae-space models for he Hol-Winers srucure. In one case Unobserved Componens models were superior, bu no in he oher. Raher surprisingly, he principle of parameer reducion was no suppored. In paricular, he Muliple Seasonal models developed by Gould e al (005) gained nohing from imposing common cycle resricions, even when visual inspecion of he daa suggesed ha he resricion was appropriae. References Abraham, B. (98). Temporal aggregaion and ime series. Inernaional Saisical Review, 50, Buer, F. (976). The use of monhly and quarerly daa in an ARMA model. Journal of Economerics, 4, Gould, P.G., Koehler, A.B., Vahid-Araghi, V., Snyder, R. D., Ord, J. K. & Hyndman, R. J. (005). Forecasing ime-series wih muliple seasonal paerns. Monash Universiy, Dep. of Economerics and Business Saisics, Working Paper 8/05, revised Ocober 005. (Forhcoming as Gould, P.G., Koehler, A.B., Ord, J. K., Snyder, R. D., Hyndman, R. J., &Vahid- Araghi, V. (008). Forecasing ime series wih muliple seasonal paerns. European Journal of Operaional Research,9, 07.) Grunfeld, Y., & Griliches, Z. (960). Is aggregaion necessarily bad? Review of Economics and Saisics, 4, -3. Hoa L. K. & Neo, J. C. (993). The effec of aggregaion on predicion in auoregressive inegraed moving- average models. Journal of Time Series Analysis, 4, Hyndman, R. J., Koehler, A. B., Snyder, R. D., & Grose, S. (00). A sae space framework for auomaic forecasing using exponenial smoohing mehods. Inernaional Journal of Forecasing, 8,

9 Koehler, A. B., Snyder, R. D., & Ord, J. K. (00). Forecasing models and predicion inervals for he muliplicaive Hol-Winers mehod. Inernaional Journal of Forecasing, 7, Koreisha, S.G. & Fang, Y. (004). Updaing ARMA predicions for emporal aggregaes. Journal of Forecasing, 3, Kumar, V., Leone, R. P., & Gaskins, J. N. (995). Aggregae and disaggregae secor forecasing using consumer confidence measures. Inernaional Journal of Forecasing,, Lükepohl, H. (986). Comparison of predicors for emporally and conemporaneously aggregaed ime series. Inernaional Journal of Forecasing,, Lükepohl, H. (006). Forecasing wih VARMA models, chaper 6 in G. Ellio, C. Granger and A. Timmerman (eds.) Handbook of Economic Forecasing, Elsevier Norh-Holland. Ord, J. K., Koehler, A. B., & Snyder, R. D. (997). Esimaion and predicion for a class of dynamic nonlinear saisical models. Journal of he American Saisical Associaion, 9, Souza, L. R., & Smih, J. (004). Effecs of emporal aggregaion on esimaes and forecass of fracionally inegraed processes: a Mone Carlo sudy. Inernaional Journal of Forecasing, 0, Taylor, J. W. (003). Shor-erm elecriciy demand forecasing using double seasonal exponenial smoohing. Journal of he Operaional Research Sociey, 54, Taylor, J. W., de Menezes, L. M., & McSharry, P. E. (006). A comparison of univariae mehods for forecasing elecriciy demand up o a day ahead. Inernaional Journal of Forecasing,, -6. Wei, W. W. S. (978). Some consequences of emporal aggregaion in seasonal ime series models, in A. Zellner (ed.), Seasonal Analysis of Economic Time Series, U.S. Deparmen of Commerce, Bureau of he Census, pp

10 Figure : Elecriciy load daa, firs four weeks Hourly elecric load, ISO New England Load (Gw) Hour Figure : Emergency room arrivals daa, firs four weeks Hourly emergency room arrivals Number Hour 0

11 Table : Sudies ha compare forecas accuracy of forecas hen aggregae and aggregae hen forecas sraegies. Auhor/year Koreisha, Fang (004) Wei (978) Abraham (98) Variable Morgage yield - gov loan yield (From Buer 976) Inernaional airline passengers (Series G of Box & Jenkins) US employed civilian workers Canada: new dwelling unis Canada: rail freigh loaded US: civilian employed workers Canada: merchandise expors US: employmen, obacco indusry Esimaion period Mehod ARMA Forecas period Wihin sample (0,,)(0,,) s Wihin sample Aggregaion Horizons Resul Quarers D -4 A Quarers,,5,0 D 5 A (0,,)(0,,) s 974 Quarers -4 D ARIMA ARIMA ARIMA ARIMA ARIMA Lükepohl (986) US invesmen, seasonally adj AR(p) (by AIC) Kumar e al. (995) US durable goods, refrigeraors, furniure, appliances, auomobile secors UK oal consumpion, nondurables cons, workforce UK expors, impors of goods and services UK real oal invesmen, gross domesic produc BVAR Quarers D Quarers D Quarers D Quarers A Quarers A Annual A Quarers,,4 D Man (004) ARIMA ARIMA ARIMA Annual Noes: The frequency of he original disaggregaed series is indicaed by he number of digis afer he year: one for quarerly, wo for monhly. AR(I)MA means various auoregressive (inegraed) moving average models esimaed, he ones for aggregae daa do no correspond o he model derived from he disaggregae esimaed model. BVAR is Bayesian vecor auoregression. The number of seps ahead refers o he aggregae unis in he previous column. Resul D means forecas hen aggregae is more accurae, usually by mean squared error crierion, A means aggregae hen forecas is more accurae. Annual -5 D Annual -5 A D A

12 Table : Elecriciy loads: muliplicaive seasonaliy models, one-sep-ahead forecas errors and rankings, based on roo mean squared error Aggregae firs Forecas firs Mehod Wihinsample Possample Possample Rank Wihinsample Possample Possample Rank HW (shor) MS(7)[Ra] UC (shor) HW (long) MS(7)[R] UC (long) Double seasonal MS(7)[R3] MS()[R3] UC(7xs) HW = Hol-Winers mehod, MS(n)[R] = Muliple Seasonal model wih n differen seasonal cycles and R resricion on he seasonal smoohing parameers (see below), UC(m) = Unobserved Componens model. Shor = season lengh of one day (4 hours or 6 4-hour aggregaes), long = season lengh of one week (68 hours or 4 4-hour aggregaes), (7xs) = 7 seasonal blocks, conrolled by 7 dummy variables for a single shor season. For double seasonal see Taylor (003) The Muliple Seasonal model seasonal smoohing parameers form an rxr marix. For r= he marix wih * * * * resricion R0 is. Under R3 his becomes. Under R i is and under R * * * * * 0 i is. According o Gould (005) for r=7, R is equivalen o HW(long), R is equivalen o * 0 HW(shor) and R3 is equivalen o double seasonaliy.

13 Table 3: Emergency room arrivals: addiive seasonaliy models, one-sep-ahead forecas errors and rankings, based on roo mean squared error Aggregae firs Forecas firs Mehod Wihinsample Possample Possample Rank Wihinsample Possample Possample Rank HW (shor) MS(7)[Ra] UC (shor) HW (long) MS(7)[R] UC (long) Double seasonal MS(7)[R3] MS()[R3] UC(7xs) See foonoes o able 3