ECONOMICS FORECAST
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CHAPTER
TIME SERIES AND THEIR COMPONENTS
As we have pointed out in earlier chapters, observations of a variable Y that become available over time are called time series data, or simply a time series. These observa- tions are often recorded at fixed time intervals. For example, Y might represent sales, and the associated time series could be a sequence of annual sales figures. Other exam- ples of time series include quarterly earnings, monthly inventory levels, and weekly exchange rates. In general, time series do not behave like random samples and require special methods for their analysis. Observations of a time series are typically related to one another (autocorrelated). This dependence produces patterns of variability that can be used to forecast future values and assist in the management of business opera- tions. Consider these situations.
American Airlines (AA) compares current reservations with forecasts based on projections of historical patterns. Whether current reservations are lagging behind or exceeding the projections, AA adjusts the proportion of discounted seats accordingly. Ihe adjustments are made for each flight segment in the AA system.
A Canadian importer of cut flowers buys from growers in the United States, Mexico, Central America, and South America. However, because these sources pur- chase their growing stock and chemicals from the United States, all the selling prices are quoted in U.S. dollars at the time of the sale. An invoice is not paid immediately, and since the Canadian-US. exchange rate fluctuates, the cost to the importer in Canadian dollars is not known at the time of purchase. ff the exchange rate does not change before the invoice is paid, there is no monetary risk to the importer. If the index rises, the importer loses money for each U.S. dollar of purchase. If the index drops, the importer gains. The importer uses forecasts of the weekly Canadian-dollar-to-U.S.-dollar exchange rate to manage the inventory of cut flowers.
Although time series are often generated internally and are unique to the organi- zation, many time series of interest in business can be obtained from external sources. Publications such as Statistical Abstract of the United States, Survey of Current Business, Monthly Labor Review, and Federal Reserve Bulletin contain time series of all types. These and other publications provide time series data on prices, production, sales, employment, unemployment, hours worked, fuel used, energy produced, earn- ings, and so on, reported on a monthly, quarterly, or annual basis. Today, extensive col- lections of time series are available on the World Wide Web at sites maintained by U.S. government agencies, statistical organizations, universities, and individuals.
It is important that managers understand the past and use historical data and sound judgment to make intelligent plans to meet the demands o[ the future. Properly constructed time series forecasts help eliminate some of the uncertainty associated with the future and can assist management in determining alternative strategies.
165
1 66 Cl IA PTER 5 Time Series and Their Compo11e111s
·1hc alternative. of course, is not to plan ahead . In a dynamic bu inc emironmem. however, this Jack of planning might be disastrous. A mainframe computer manufac. turer that some years ago ignored the trend to per onal computers and work Ill~ would have lost a large part of its market share rather quickly.
Although wc will focus our attention on a model-based approach to time scrie\ analysis that relics primarily on the data. a subjective review of the foreca ting cffon
15 very important. 'The past is relevant in the search for clue about the future on!\ totb.! extent that causal conditions previously in effect continue to hold in the period ahead. In economic and business activity. causal conditions seldom remain con tant. Th. multitude of causal factors at work tends to be constantly shifting, so the connections among the pasl. the present. and the future must be continually reevaluated.
The techniques of time series provide a conceptual approach to forcca ting tha: has proved to be very useful. Forecasts arc made with the aid of a et of pecific f~rmal procedures, and judgments that follow arc indicated explicitly.
DE COM POSITION
One approach to the analysis of time series data involves an attempt to identif~ the compo~ent factors that influence each of the values in a series. Thi. identification pro- cedure IS called decomposition. Each component is identified cparatel}. Projection, of each of the components can then be combined to produce forecasts of future value. o! the ~ime series. Decomposition methods are u ed for both short-run and long-run fore- cas_11ng. They ~•re also used to simply display the underlying growth or decline of a sencs or to adJust the series by eliminating one or more of the components.
. Analyzing a time series by decomposing it into its component part has a long hist_ory. Recent I~, h~wcvcr, decomposition methods of forecasting have lost some 1,f theIT luster. ProJ ectmg the individual component into the future and recombinin~ thcs~ projec!ion_s to f~rr:1 a forecast of the underlying series often does not work \ ·1:~ w~~I m practice._ The dtffrc~lty lies in getting accurat~ forecasts of the componen!s. The d~ clo~ment of morc-!lex1ble model-based forecastmg procedures (some of which \\e ~Iscuss I~ later chapters) ha made decomposition pri marily a tool for understanding a IImc scnes rather than a forecasting method in its own right.
To un?erstand d~composition, we start with t he four components of a time serie. that were mtroduced m Chapter 3. lbese are the trend component, the cyclical compo- nent. the seasonal component. and the irregular or random component.
I.
2.
Tre,~d. ·~e trend is the component that represents the underlving growth (or declme)_ m a time series. The trend may be p roduced. for exampie. by consistent population change. inflation. technological change. and productivity increases. The trend component is denoted by T.
C.)•c/ica/. The cyclical component is a series of wavelike fl uctua tions or cycles of more than one year's duration. Changing economic conditions generally produce cycles. C denotes the cyclical component.
In practice. cycles are often difficult to identify and arc frequently regarded as part ~f the trend. In this case, the underlying general growth (or decline) compo- nent IS call ed the trend-eye/<' and denoted by T. We use the notation for the trend because the cyclical component often cannot be <,eparated from thc trend.
3. Seasonal. Seasonal fluctuations arc typically fo und in quarter!). monthly, or weekly data. easonal variation refers to a more or less tahle pa11crn of change that appears annually and repeats itself year after year. ca onal palterns occur
4 .
Cl IAPTLR 5 Time eries and their Componelll.\ 167
because of the innuence of the weather or because of calendar-related events uch as school vacations and national holiday S denotes the seasonal component. Irregular. The irregular component consi ts of unpredictable or random nuctua- tions. These fluctuations are the result of a myriad of events that individually may not be particularly important but whose combined effect could be large./ denote · the irregular component.
To study the components of a time series. the analyst must consider how the components relate to the original series. This task i accompli ·hed by SJ)\!cifying a model (mathematical relationship) that expre ses the time c;cries variable Yin term. of the components T, C, S, and / . A model that treats the time scric · values as a sum of the components is called an additiw• compone111s model. A model that treats the time cries values as the product of the components i called a multiplicative co111po11e11ts model. Both models are sometimes referred to as unobserved compo11e111s models, since, in practice, although we observe the values of the time ·cries. the values of the components arc not observed. The approach to time seric analysis de cribed in this chapter involves an attempt. given the observed series.. to c timate the value of the components. 'lhe e estimates can then be used to forcca ·tor to display the series unencumbered by the sea- sonal fluctuations. The latter proce s is called seasonal adjustmelll.
It is dif(icult to deal with the cyclical component of a time series. To the extent that cycles can be determined from historical data. both their lengths (measured in years) and their magnitudes (differences between highs and low ) are far from constant. Thi lack of a consistent wavelike pattern makes distinguishing cycles from smoothly evolv- ing trends difficult. Consequently. to keep things relatively implc. we will a<,sume an) cycle in the data is part of the trend. Initially, then, we consider only three components. T. S, and /. A brief discussion of one way to handle cyclical nuctuations in the decom- po ition approach to time series analysis i · available in the Cyclical and Irregular Variations section of this chapter (sec p. 180).
The two simplest models relating the observed value ( Y, ) of a time series to the trend (Yi), seasonal (S,), and irregular (/,) components are the additive components model
Y,=T, S1 + /1
and the multiplicative components model
Y, = T, X S1 X /1
(5.1 )
(5.2)
The additive components model works best when the time scrie being analyzed has roughly the same variabi lity throughout the length of the serie ·. That is. all the val- ues of the series fall essentially within a band of constant width centered on the tre nd.
The multiplicative components model works best when the variability of the time series increases with the level. 1 That is, the value of the ·cries spread out as the tre nd increases, and the set of observations has the appearance of a megaphone or funnel. A time series with constant variability and a time erie with variability increasing with
l 1t is po . ihle to convert a multiplicative dccoi:nrositi<ln to an additive: . dccompmi t! on by "' orking wi th the logar ithms of the data. l1~ing F.quauon 5.2 ~~d the propcrue~ of k,gar~thm,. "'c have log y = log (T x S x J) = log T log,5 ... log/. DccomptNllo n of log,ied data IS e1tplorcd tn Problem 5.15.
168 CHAfYTER 5 Time Series and Their Components
1,000
C: 0
900
g 800 e a.. =!!: ~ 700
gi
600
900
800
700
600
~ 500 >- £ 400 C: 0
~ 300
200
100
0
50
10 20 30
Month
40 Month
100
50
150
60 70
FIGURE 5-1 Time Series with Constant Variability (Top) and a Time Series with Variability Increasing with Level (Bottom)
level are shown in Figure 5-1. Both of these monthly series have an increasing trend and a clearly defined seasonal pattem.2
Trend Trends are long-term movements in a time series that can sometimes be described by a straight line or a smooth curve. Examples of the basic forces producing or affecting the ~re~d of a series are population change, price change, technological change, producth- 1ty mcreases, and product life cycles.
2 va~ ~nts of the decomposition models (sec Equations 5.1 and 5.2) exist that contain both multiplicative and
add1t1vc te rms. For example. some software packages do "multiplica1he·· decomposition using the model Y =TX S + I .
CHAPTER 5 Time Serie and Their Componenrs 1 69
A population increase may cau e retail ales of a community to ri e each year for sev~ral years. Moreover, the ales in current dollar ma:,, have been pushed upward durmg the same period b ecau e of general increa e in the price of retail goods-even though the physical volume of good· sold did not change.
Technological change ma) cau e a time erie to move upward or downward. The d~velopment of high-speed computer chip . enhanced memory device .. and improved display panels, accompanied by improvements in telecommunications technology. has r esulted in dramatic increases in the u e of persona l com puters and cellular telephones. Of course, the same technological development have led to a downward tre nd in the production of mechanical calculators and rotary telephones.
Productivity increases-which, in turn, may be due to technological change-give an upward slope to many time series. Any mea ure of total output. such as manufactur- ers' sa les, is affected by changes in productivity.
For business and economic time series. it is be l to view the trend (or trend-cycle) as smoothly changing over time. Rarely can we realistically a sume that the trend can be re presented by some simple function such as a straight line over the whole period for which the time series is observed. l lowever, it is often convenient to fit a trend curve to a time series for two reasons: (I) It provides some indication of the general direction of the observed series, and (2) it can be removed fro m the original cries to get a clearer picture of the sea. onal ity.
If the trend appears to be roughly linear-that is. if it increase. or decreases like a straight line-then it is represented by the equation
(5.3)
H e re, i; is the predicted value for the trend at time 1. The symbol I represents time. the independent variable. and ordinarily a. su mes integer value 1. 2. 3 ... . corresponding to consecutive tim e peri ods. The slope coefficient , b 1• is the average increa e or decrease in T for each o ne-period increase in time.
lime trend e quations. including the straight-line trend. can be fit to the data u ·ing the method of least squares. Recall fro m Chapter 2 that the method of least quarcs selects the values of the coefficients in the trend equation (ho and b1 in the straight-l ine case) so tha t the estimated trend va lue Cf;) are close to the actual values (Y,) as mea- sured by the s um of squared errors criterion
(5.4)
Example 5 . 1 Data on annual registrations of new passenger cars in the Un ited States from 1960 to 1992 are shown in Table 5- 1 and plotted in Figure 5-2. The values from 1960 to 1992 arc u ed to develop the trend equation. R egistrations is the dependent variable. and the independent variable is time t. coded as 1960 = I. 1961 = 2. and so on .
111e fitted trend line ha the equation
f, - 7.988 + .06871
The slope of the trend equation indicate that regi :trati?ns arc es~imated to increase an average of 68.700 each year. Figure 5-3 sho\\- t_he st~atght-hne trend fitted to the actual data. Figure 5-3 also shows forecast of new car reg1st~auon for the years I 99~ and 1994 (1 = 34 and 1 = 35). which were obtained by cxtrapolatmg the trend !me. We will say more about forecasting tre nd shortly.
The estimated trend values for passenger car regi~trations from 1960 to 1992 arc s hown in Table 5-1 under f. For example. the trend equation estimate~ registrations in 1992 (t = 33) to be
fu = 7.988 .... 0687(33 ) = 10.255
1 7 0 CHAPTER S Tim e Series and Their Components
TABLE 5 - 1 Registration of New Passenger Cars in th e United States. 1 960-1 992. for Example 5 . 1
Registrations Trend Estimates
(millions) (millions)
Year y Timet t
1960 6.577 1 8.0568
1961 5.855 2 8. 1255
1962 6.939 3 8.1942
1963 7.557 4 8.2629
1964 8.065 5 8.33 16
1965 9.314 6 8.4003
1966 9.009 7 8.4690
1967 8.357 8 8.5376
1968 9.404 9 8.6063 1969 9.447 10 8.6750 1970 8.388 11 8.7437 1971 9.831 12 8.8124 1972 10.409 13 8.8811 1973 11.351 14 8.9498 1974 8.701 15 9.0185 1975 8.168 16 9.0872 1976 9.752 17 9.1559 1977 10.826 18 9 .2246 1978 10.946 19 9.2933 1979 10.357 20 9.3620 1980 8.761 21 9 .4307 1981 8.444 22 9.4994 1982 7.754 23 9.5681 1983 8.924 24 9.6368 1984 10.118 25 9.7055 1985 10.889 26 9.7742 1986 I l.140 27 9.8429 1987 L0.183 28 9.9116 1988 10.398 29 9.9803 1989 9.833 30 10.0490 1990 9.160 31 10.1177 1991 9.234 32 10.1863 1992 8.054 33 10.2550
Error (millions)
Y- T - 1.4798 -2.2705 -1 .2552 -0.7059 -0.2666
0.9138 0.5401
-0.1 807 0.7977 0.7720
-0.3557 1.0186 1.5279 2.4012
-0.3175 -0.9192
0.5961 1.6014 1.6527 0.9950
- 0.6697 - 1.0554 -1.8141 -0.7128
0.4125 1.1148 1.2971 0.2714 0.4177
- 0.2160 - 0.9577 - 0.9524 -2.2010
Source: Data from U.S. Department of Commerce. Survey of Currell/ Business (various years).
o r l0.255poo registrations. However, registrations of new passenger cars were actuaL! 8,054,000 m 1 ~- 1lm~ for 1992, the trend equation overestimates registrations by app~ mately 2.2 ~1lhon. llus_ err<;>r and the remaining estimation errors are listed in Table~- under Y - 1 · These est11:1ation errors were used to compute the measures of fit. the MA D. MSD_, and MAP£ ~hown m Figure 5-3. lbese are the same measures of forecast accuracy th.- are given by Equaho~ 3.7, 3.8, and 3.10, r espectively, in Chapter 3. (Minitab commands u..,e,l to pr?<1uce the resultc; m Example 5.1 are given in the MinitabApplicationsscctionat tbeeod of this chapter.)
12
11
10
I ... 9 ~
~ J 8 ci::
7
• 6
1960
CHAPTER 5 Time Series and Their Components 1 71
Registrations of New Passenger Cars: 1960- 1992
1965 1970 1975 1900 1985 1990 Vear
FIGURE 5 -2 Car Registrations Time Series for Example 5. 1
12
11
~ 10
.!2 ...., g (0
~ -a,
8 & 7
6
3
Linear Trend for Annual Car Registrations Linear Trend M:Jdel
Yt = 7.988 + D.0587*t
Min it.ab Instructions Stat> Time Series> Trend Analysis
6 9 12 15 18 21 24 27 30 33 Time
Vari~e
-+-Adu~!
-- Fils Fonasts
A.:.curacy Measure< MAPE 11.2739 MAD 0,98'37 MSO 13379
FIGURE 5 -3 Trend Line for the Car Registrations Time Series for Example 5. 1
Additional Trend Curves The life cycle of a new product has three stages: introduction, growth. and maturity and saturation . A cur ve representing sales (in dollars or unit ) over a new p roduct's life cycle is s h own in Figure 5-4. Time. shown on the hor_izont~l axis. can vary from d ays to years, depe nding on the nature of the mark~t. A ~tr~1ght-lI~e trend wou ld_ not work for these data . Linear models assu m e that a variable 1s mcreasmg (or decreasing) by a con- s t a nt a mount each time period. "The increase per time period in the product's life cycle
1 7 2 CHAPTER 5 Time Series and lheir Componems
Trend
2l c ::::> ,,; Q)
~
Introduction Growth
Time
FIGURE 5-4 Life Cycle of a Typical New Product
Maturity and Saturana:
curve are quite different depending on the stage of the cycle. A curve. other tha.!1 straight line, is needed to model the trend over a new product ·s life cycle.
A simple function that aUows for curvature is the quadratic trend A 2
Tr = bu + b1 t + />it 15.S• As an illustration. Figure 5-5 shows a quadratic trend curve fit to the pa enger car ~- istrations data of Example 5.1 using the SSE criterion. The quadratic trend can be pro- jected beyond the data for, say. two additional years, 1993 and 1994. We will consider the implications of this projection in the next section, Forecasting Trend.
Quadratic Trend for Annual Car Registrations Quact-atk::: Tnn:i tvbjel
vt = 6.356 + 0.3484*t- o,ro323•t-•2 12-r---------------------,
11
7
6
3 6 9 12 ~ IB ~ ~ V ~ ~ Ttne
• •
Vort.blo -- Adwl
♦ Fonum
A=IKY ,.......,., MAPE 8.6169' MAO 0.71385 MSO 0.8!395
FIGURE 5-5 Quadratic Trend Curve for the Car Registrations Time Series for Example 5 I
CHAPTER 5 Time Series and Their Components 173
Based on the MAPE, MAD,and MSD accuracy mea ures,a quadratic trend appears to be a better representation of the general direction of the car registrations series than the linear trend in Figure 5-3. Which trend model i appropriate? Before considering this issue, we will introduce a few additional trend curves that have proved useful.
When a time series starts slowly and then appears to be increa ing at an increasing rate (see Figure 5-4) such that the percentage difference from ob ervation to ob erva- tion is constant, an exponential trend can be fitted. The exponential trend is given by
(5.6)
The coefficient b 1 is related to the growth rate. If the exponential trend is fit to annual data , the annual growth rate is estimated to be JOO(b
1 - 1)% . .
Figure 5-6 indicates the number of mutual fund salespeople employed by a partic- ular company for several consecutive years. The increase in the number of ~espeople is not constant. It appears as if increasingly larger numbers of people are being added in the later years. .
An exponential trend curve fit to the salespeople data has the equation
f; = 10.016( 1.313)'
implying an annual growth rate of about 31 %. Consequently, if the model est_imates 51 salespeople for 2005, the increase for 2006 would be 16 (51 x .31 ) for an estimated total of 67. This can be compared to the actual value of 68 salespeople.
A linear trend fit to the salespeople data would indicate a constant average increase of about nine salespeople per year. This trend overestimates the actual increase in the earlier years and underestimates the increase i~ the last year. It does not model the apparent trend in the data as wel_l as the exponential curve does. . .
It is clear that extrapolating an exponential trend with a 31 o/~ growth rate~ qwckly result in some very big numbers. This is a potential problem with an exponential trend model What happens when the economy cools off and stock prices begin to retreat? The
70 Year salespeople 2CO) 13
60 2001 18 2002 22 2003 29 2004 41 2005 so 2006 68
20
10 -l..----.----=~--:a:;;)oo!;:;13---:;2004~---:;:zc::;m;----;;:2006~ 20Xl 2001 2002 Veer
FIGURE 5 _6 Graph of Mutual Fund Salespeople
1 7 4 CHAPTER 5 Time Series and Their Componenrs
demand for mutual fund salespeople will decrease, and the number of sale people COIJ! even decline.1be trend forecast by the exponential curve will be much too high.
Growth curves of the Gompertz or logistic type represent the tendenC) of many industries and product lines to grow at a declining rate as they mature. If the plotted data reflect a situation in which sales begin low, then increase as the product catcb~oo. and finally case off as saturation is reached. the Gompertz curve or Pearl-Reed logj,x model might be appropriate. Figure 5-7 shows a comparison of the general shapt'l" (a) the Gompertz curve and (b) the Pearl- R ecd logistic model. ote that the logis:k curve is very sim~lar to the Gompertz curve but has a slightly gentler slope. figure:-- shows how the Y intercepts and maximum values for these curve are related to some the coefficients in their functional forms. "The formulas for these trend curves are COlrr pl~x. and not within the scope of this text. Many statistical oft ware packages. indudi~! Mm1tab. allow one to fit several of the trend models discussed in this section. '
Although there are some objective criteria for selecting an appropriate trend.in general. the correct choice is a matter of judgment and therefore requires experi~nct ~nd common sense on the part of the analyst. As we wi II discuss in the next section. the lme or curve tha~ best fits a set of data points might not make ense when projectedi) the trend of the future.
Forecasting Trend Suppose we a r e presently at time t = n ( end of series) and we want to use a trend model to_ fore~ast th~ value of Y.p steps ahead. The time period at which we make the forecast. n m this case, 1s called the forecast origin. The value p is called the lead rime. For the lin- ear trc~d model, we ~an produce a forecast by evaluating f,
1 + p = bo + b,(n + p).
U~mg the trend hne fitte~ to the car registration data in Example 5.1 . a forecast of the trend for 1993 (, = 34) made m 1992 (t = n = 33) would be the p = I step ahead foreabl
733 +1 === 7.988 + .0687(33 + 1) = 7.988 + .0687(3-t) = 10.324 Similarly. the p = 2 step ahead (1994) forecast is given by
7~3 ➔-2 = 7.988 + .0687(33 + 2) = 7.988 + .0687(35) = 10.393 "These two forecasts are shown in Figure 5-3 as extrapolations of the fitted trend line.
,. Ti
---- -- ------ ----
0 0
(a) Gompertz Trend Curve (b) Logistic (Pearl-Reed) Trend Curve
FIGURE 5-7 S-Shaped Growth Curves
CHAPTER 5 Time Series and Their Compo11e111s 1 7 5
. Figure 5-5 shows the fitted quadratic trend curve for the car registration data. Usmg the equation shown in the figure. we can calculate forecast of the trend for 1993 a~nd 1994 by, settin~,,_r = 33 + I = 34 and 1 = 33 + 2 = 35. The reader can verify that 7
'.13_ ... 1 = 8.688 and I 33 +2 = 8.468. These numbers were plotted in Figure 5-5 as extrapo- lations of the quadratic trend curve .
R ecalling that car registrations are measured in millions, the two forecasts of trend prod:1ced from the quadratic curve are quite different from the forecasts produced by the hnear trend equation. Moreover, they are headed in the opposite direction. If we ~ere to extrapolate the linea r and quadratic trends for additional time periods, their differences would be magnified.
The car registration example illustrates why great care must be exercised in using fitt~d trend curves for the purpose of forecasting future trends. Two equations, bot h of which may reasonably represent the observed time series, can give very different results when projected over future time periods. TI1ese differences can be substantial for large lead times (long-run forecasting).
Trend c urve models are based on the fo llowing assumptions:
1. The correct trend curve has been selected. 2. The curve that fits the past is indicative of the future.
These assumptions suggest that judgment and expertise play a substantial role in the selection and use of a trend curve. To use a trend curve for forecasting, we must be able to argue that the correct trend has been selected and that. in all likelihood, the future w ill be like the past.
There arc objective criteria for selecting a trend curve. We will discuss two of these criteria, the Akaike information criterion (A IC) and the Bayesian information crite- rion (BIC), in Chapter 9. However, although these and other criteria help to determine an appropriate model, they do not replace good judgment.
Seasonality A seasonal pattern is one that repeats itself year after year. For annual data. seasonal- ity is not an issue because there is no chance to model a within-year pattern with data recorded once per year. Time series consisting of weekly. monthly, or quarterly obser- vations. however. often exhibit seasonality.
The analysis of the seasonal component of a time series has immediate short-term implications and is of greatest importance to mid- a nd lower-level management. Marketing plans. for example. have to take into consideration expected seasonal pat- terns in consumer purchases.
Several methods for measuring seasonal variation have been developed. The basic idea in all of these methods is to first estim ate and remove the trend from the original series and then smooth out the irregular component. Keeping in mind our decomposi- tion models. this leaves data containing only seasonal variation . The seasonal values are then collected and summarized to produce a number (generally an index number) for each observed interval of the year (week. month, quarter, and so on) .
Thus. the identification of the seasonal component in a time series differs from trend analysis in a t least two ways:
I. The trend is determined directly from the original data, but the seasonal compo- nent is determined indirectly after eliminating the other components from the data. so that only the seasonality remains.
2. The trend is represented by one best-fitting curve, or equation, but a separate sea- sonal value has to be computed for each observed interval (week , month, quarter) of the year and is often in the form of an inde x number.
176 Cl !APTER 5 'lime Series and Their Compo11e111s
If an additive decomposition is employed. estimates of the trend. ·eason11l,and irregular components are added together to produce the original . cri~s. _rr a muhiplia. tive decompo ·ition is used. the individual components mu ·t be mult1phed together to reconstruct the original series. and in this form ulation. the sea onal component i rep- resented by a collection of index numbers. These numbers show which periods \\ithin the year arc relatively low a nd which periods arc relatively high. The ea onal index trace out the seasonal pattern.
lntl<'x numbers arc percentages that shcl\\ changes over time.
With monthly data. for example. a seasonal index of 1.0 for a particular month m~ the expected value for that month is l/ 12 the total for the year. An index of 1.25 for a Jc'. fcrent month implies the observation for that month is expected to be 25% more than 1/12 of the annual total. A monthly index of 0.80 indicate that the CX(h!cted level of aL1ir- ity that month is 20% less than 1/ 12 of the total for the year. and so on. lhe index numbc~ indicate the expected ups and downs in levds of activity over the course of a year aftcrthc effects due to the trend ( or trend-cycle) and irregular components have been removed.
To highlight seasonality. we must first estimate and remove the trend. The trend can be estimated with one of the trend curves we discussed pre\iously. or it can beN1· mated using a moving average. as di ·cussed in Chapter 4.
Assuming a multiplicative decomposition model. the r(l[io w moving m•era,:e i, a popular method for measuring seasonal variation. In this method. the trend is Ni· mated using a centered moving average. We illustrate the ratio-t o-mo\ing-averagc method using the monthly sales of the Cavanaugh Company. shown in Figure 5.1 (Bottom) in the next example.
Example 5 .2
lo illustrate the ratio-to-moving-average method, we use two years of the monthlv sale. of the Cavanaugh Company.
3 Table 5-2 gives the monthly sale, from January· 2004 to
Decc~bcr 2005 to illustrate the beginning of the computations. The first step for month!~ data 1s to compute a 12-month moving average (for quarter!) data. a four-month mmiog aver~ge woul~ be co~puted). Because all of the months of the year arc included in the caJ. culat1on of this rnovmg average. effects due to the '>easonal component arc rcmm·cd. and the moving average itself contains only the trend and the irregular components.
lhe steps (identified in fable 5-2) for computing seasonal indexes by the ratio-to-mming- average method follow:
Step 1. Starting at the beginning of the series, compute the 12-month moving total. and place the total for January 2004 through December 2004 between June and July 2004.
Step 2. Compute a two-year moving total so that the subsequent averages arc centt.rcd on Jul} rather than between months..
Step 3. Since the two-year total contains the data for 24 monthi. (January 2004 once. February 20()4 to December 2004 twice. and January 2005 once). this total is ccn· tcred on July 2004.
Step 4. Divi~e the two-year mm'ing total by 24 in order to ohtain the 12-monrh centered movmg average.
Step 5. The sem,onal indt!x for July is calculated by dividing the actual value for July by th 12-month centered moving average.4
)J°he unih ha\e been omiued and the datt."- and name ha\e tx.-..n ch;in~cd to pn>fect the idcn1it_\ ofrhc compan). 41bis i the ratio-to-moving-average operation that ghc~ the pro,:cJurc ir, nam·
C'HA PTE R 5 Time eries and Their Compo11e11ts 1 7 7
TABLE 5 -2 Sales of th e Cavanaug h Compa ny. 2004-2005 . for Example 5 .2
12-Month 12-Month Two-Yl'ar Cl"nll'nd
Moving Moving Moving Period Sales Total Total A•·uagt
2004 January 518 February 404 March 300 April 210 May 196 June 186 4.869 } 7 July 247 4.964 9. 3313 409.714 August 343 4.952 9,916 413.2 September 464 4.925 9.877 411.5 October 680 5,037 9.962 415.1
'ovembcr 711 5.030 10,067 419.5 December 610 10.131 422.1
2005 January 613 5.101 10.279 428.3 February 392 5,178 10,417 434.0 March 273 5.239 10.691 4-t'i.5 April 322 5.452 11.082 461.8 May 189 5,630 11.444 476.8 June 257 5.814 I 1.682 486.8 July 324 5.868 August 404 September 677 October 858 November 895 D ecember 664
Stasonal Inda
0.6015 0.83 1.13 1.64 1.69 1.45
1.43 0.90 0.61 0.70 0.40 0.53
R , t t s I to 5 beginning with the second month of the series. August 2004. and so on. ~pea s ~ ends when a full 12-month mo\.ing total can no longe r be calculated.
e ~=use there arc several estimate~ (corresponding to diff~rent years) of the scas~nal · f h onth th•·y must be summari,ed to produce a single number. The median. mdex or eac m • v • • • • 1· · I
' h h . · used a the summary measure. Usmg the median c rmmatcs t 1c rather t an t e mea n. t II A · n cnce of data for a month in a particular year that arc unusuall) large o~ ma . . su~- ~a:y of the seasonal ratios. a long with the median value for each month. 1s contamcd m
Tabl~~'~onthl seasonal indexes for each year mus_t sum to I~. ~o t~c med!a~ for each h I b a~·usted to get the final ~t of seasonal rndcxc,
5 Smee th~ mult1pher should mont musthc 1
1 .f the total of the median ratio before adjustment 1s less than 12 and
be greater an 1 · · · d fi d smaller than 1 if the total is greater than 12, the muluphcr 1s e rne as
Multiplier 12
Actual total
- . . t 12 , 0 that the e-cpect.:d annual total equal the actual annual total. 5The monthly mdexc\ must sum o
mi
178 CHAPTER 5 Time Series and Their Components
TABLE 5-3 A Summary of the Monthly Seasonal Indexes for the Cavanaugh Company for Example 5.2
Adjus1ed Seasonal
Monlh 2000 2001 2002 2003 2004 Index (.lftdi111
2()05 2006 Median X J.fNJ.14) January 1.208 l.202 1.272 1.41 l 1.431 1.272 1.278 February 0.7()(} 0 .559 0.938 1.089 0.903 0.903 0.907 March 0.524 0.564 0.785 0.800 0.613 0.613 0.616 April 0.444 0.433 0.480 0.552 0.697 0.480 0.482 May 0.424 0.365 0.488 0.503 0.396 0.424 0.426 June 0.490 0.459 0.461 0.465 0.528 0.465 0.467 July 0.639 0.904 0 .598 0.681 0.603 0.662 0.651 0.65~ August 1.115 0.913 0.889 0.799 0.830 0.830 0.860 0.864 September 1.371 1.560 1.346 l.272 1.128 1.395 1.359 1.365 October 1.792 1.863 1.796 1.574 1.638 1.771 1.782 1.790 November 1.884 2.012 J.867 1.697 1.695 1.846 1.857 1.865 December 1.519 1.088 l.224 1.282 1.445 1.282 1.288
11.948 12.002
Using the information in Table 5-3.
M 1 . /" 12 11 (Ip 1er = 1 l.
948 = 1.0044
The final column in Table 5-3 contains the final seasonal index for each month. determined by makin_g the adjustment (multiplying by 1.0044) to each of the median ratios.6The final seasonal !~~exes. shown in Figure 5-8. r epr esent the seasonal component in a multiplicative decomposn1on of the sales of the Cavanaugh Company time series. Tue seasona lity in sales
X Ql -0 E
1.9
1.4
0.9
0.4
Seasonal Indices
2 3 4 5 6 7 8 9 10 11 12
Month
FIGURE 5-8 Seasonal Indexes for the Cavanaugh Company for Example 5 .2
"The ~easonal indexes are sometimes multiplied by 100 and exprc~o;cd a, perc.!nlagcs.
CHAPTER 5 Time Series and Their Componems 1 79
is evident f~om Figure 5-8. Sale for this company are periodic. with relatively low ales in the late sprmg and relatively high sales in the late fall.
Our analysis of the sales series in Example 5.2 assumed that the seasonal pattern remained constant from year to year. If the seasonal pattern appears to change over time. then estimating the seasonal component with the entire data et can produce mis- leading results. I t is better. in this case, either (1) to use only recent data (from the last few years) to estimate the seasonal component or (2) to use a time series model that allows for evolving seasonality. We will discuss models that allow for evolving seasonal- ity in a later chapter.
The seasonal analysis illustrated in Example 5.2 is appropriate for a multiplicative decomposition model. However. the general approach outlined in steps 1 to 5 works for an additive decomposition if, in step 5. the seasonality is estimated by subtracting the trend from the original series rather than dividing by the trend (moving average) to get an index. In an additive d ecompositio n , the seasonal component is expressed in the same units as the original series.
In addition. it is apparent from our sales example that using a centered moving average to determine tre nd results in some missing values at the ends of the series. This is particularly problematic if forecasting is the objective. To forecast future values using a decomposition approach, alte rnative methods for estimating the trend must be used.
The results of a seasonal analysis can be used to (1) e limina te the seasonality in data: (2) forecast future values; (3) evaluate current positions in, for example. sales, inventory, and shipments; an d (4) schedule production.
Seasonally Adjusted Data Once the seasonal component has been isolated, it can be used to calculate seasonally adjusted da ta . For an additive decomposition. the seasonally adjusted data are com- puted by subtracting the seasonal component:
(5.7a)
For a multiplicative decomposition. the seasonally adjusted data arc computed by dividing the original observations by the seasonal component:
Y, -= T, X l , S,
(5.7b)
Most economic series published by government agencies are seasonally adjusted because seasonal variation is not of primar y interest. Rathe r, it i~ the gene~al p~ttern of economic activity. indepe nden_t of th~ no~mal seasonal ~ uctuat1ons, that ts of mt:_rcs~: For example. new car registrat10ns might mcrease by 10 Yo from May to June, but ts thts increase an indication that new car sales are completing a banner quarter? The answer is '·no·· if the 10% increase is typical at th is time of year largely due to seasonal factors.
In a survey concerned with the ac~uisition of"seasonally adjusted_ data, Bell and Hillmer (1984) found that a wide van~ty of users value seasonal adjustment. 1l1ey identified three motives for seasonal ad3ustment:
I. Seasonal adjustment allows reliable comparison of values at different points in
2.
3.
time. · b · · bl It is easier to understand the relationships among ccononuc or usmcss vana es
the complicating factor of seasonality has been removed from the data. ~:~:onal adjustment may be a useful clement in the production o f short-term fore- casts of future values of a time series.
1 80 CHAJYfER 5 Time Series and '/1ieir Componems r Bell and Hillmer concluded that "seasonal adjustment is done to simplify data sotJ-.;ii they may be more easily interpreted by statistically unsophisticated users \,ithout a~.~- nificant loss of information'· (p. 301).
Cyclical and Irregular Variations Cycles arc long-run, wavelike fluctuations that occur most frequently in macro indic.i- tor · of economic activity. As we h ave discussed. to the extent that they can be meJ- surcd. cycles do not have a consis tent pattern. However. some insight into the cyclic, behavior of a time series can be obtained by eliminating the trend and seasonal compo- nent. to give, using a multiplicative decomposition.7
(5.8)
A moving average can be used to smooth out the irregularities, /1• lea, ing the ~clical component. C,. To eliminate the centering problem encountered when a mo,in!!. a\er- age with an even number of time periods is used. the irregularities are smoothed ~smta moving average with an odd number of time periods. For monthly data, a5-. a < a 9-, or even an I I-period moving average will work. For quarterly data. an estimate of C can be computed using a three-period moving average of the values. 8
Finally, the irregular component is estimated by
C, X / 1 I = - -- 1 C
I
(5.9)
The irregular component represents the variability in the time cries after the otbe: components have been removed. It is sometimes called the residual or error. \\"ith a multip licative decomposition. both the cyclical and the irregular components are expressed as indexes.
Summary Example One re~son for decomposing a time series is to isolate and examine the componenb c' the senes. Aftc_r the ~nalyst is able to look at the trend. seasonal. cyclical. and irregular components o~ a senes one at a time. insights into the patterns in the original data \al• u~s may be ga111~d . A lso. once the components have been isolated. they can be recom· b111ed or synthesized to pr oduce forecasts of future values of the time series.
Example 5.3
In Exa~plc 3.5. Perkin Kendell. the analyst for t he Coastal Marine Corporation. lL~d au10- correlat10~ :~nalys1s to determine that sales were seasonal on a quarterlv basis. 'm, he u-..,') de_c~mpos,tio_n t<:> unders_tand the qua rterl y sales variable. Perkin us~s Minitah (.cc th, M1mtab Appltcat1ons section at the end of the chapter) to produce Table 5-4 and Figure q, In order to keep the seaso11al p·ttte · ; rv"') ' _ _ ' , 1 n current. onlv the last seven years (2000 to _wu o. sales data. Y. we re analved. '
The ori~inal dat~ arc sho,, n ii:! the ch art in the upper left of Figure 5-1 0. The trend ' computed usmg the hne·ir mod •1· T - 26 1 24 - · ' f2000 T bl . ' e · 1 ·- • + .7-:,91. Smee J represented the first quarter 0 · a _ e 5-4 sho_ws the trend value equal to 262 .000 for this time period. and estimated sales (the 7 column) increased by .759 each quarter.
7Noticc that we have added the c • •I' • I Equation 5.2. }C ic,i compo nent. C, to the multiplicati\e dccompo~ition , ho11 n U:
~For annual data. the re is no seasonal compo d • - • · -• Iii ~imp], rcmovino the trend fro th. . .
1 n~nt. an the cychcal X irregular component •~ obtatnw .
• , "' m c ong111a ~cnes.
CHAJYfER 5 Time Series and Their Components 1 81
TABLE 5-4 The Multiplicative Decomposition for Coastal Marine Sales for Example 5 .3
t Year Quarter Sales 1' SCI s TCJ CJ C I I 2000 1 232.7 262.000 0.888 0.780 298.458 1.139 2 2 309.2 262.759 1.177 l.016 304.434 1.159 1.118 1.036 3 3 310.7 263.518 1.179 1.117 278.239 1.056 1.078 0.980 4 4 293.0 264.276 1.109 1.088 269.300 1.019 1.022 0.997 5 2001 1 205.1 265.035 0.774 0.780 263.059 0.993 0.960 1.034 6 2 234.4 265.794 0.882 1.016 230.787 0.868 0.940 0.924 7 3 285.4 266.553 1.071 1.117 255.582 0.959 0.906 1.059 8 4 258.7 267.312 0.%8 1.088 237.775 0.890 0.924 0.962 9 2002 1 193.2 268.071 0.721 0.780 247.796 0.924 0.927 0.998
10 2 263.7 268.830 0.981 1.016 259.635 0.966 0.954 1.012 11 3 292.5 269.589 1.085 1.117 261.940 0.972 1.003 0.969 12 4 315.2 270.348 1.166 1.088 289.705 1.072 0.962 1.114 13 2003 1 178.3 271.107 0.658 0.780 228.686 0.844 0.970 0.870 14 2 274.5 271.866 1.0IO 1.016 270.269 0.994 0.936 1.062 15 3 295.4 272.625 1.084 1.117 264.537 0.970 0.976 0.994 16 4 286.4 273.383 1.048 l.088 263.234 0.963 0.942 1.022 17 2004 190.8 274.142 0.696 0.780 244.7 18 0.893 0.933 0.957 18 2 263.5 274.901 0.959 1.016 259.438 0.944 0.957 0.986 19 3 318 .8 275.660 1.157 1.11 7 285.492 1.036 0.998 1.038
20 4 305.3 276.419 1.104 1.088 280.605 1.015 1.058 0 .960
21 2005 242.6 277.1 78 0.875 0.780 311.156 1.123 1.089 1.031
22 2 318.8 277.937 1.147 1.016 313.886 1.129 1.104 J.023
23 3 329.6 278.696 1.183 1.117 295.164 1.059 1.100 0.963
24 4 338.2 279.455 1.2IO 1.088 310.844 1.112 1.078 1.032
25 2006 1 232.1 280.214 0.828 0.780 297.689 1.062 1.059 1.004
26 2 285.6 280.973 1.016 1.016 281.198 1.001 0 .996 1.005
27 3 291.0 281.732 1.033 I.I 17 260.597 0.925 0.947 0.977
28 4 281.4 282.491 0.996 1.088 258.639 0.916
The chart in the upper right of Figure 5-10 shows the de trended data. lbese data are also shown in the SCI column of Table 5-4. The detrended value for the first quarter of2<XX) was9
SCI = y = 232.7 = 888 T 262.<XXJ .
The seasonally adjusted data are shown in the TC/ column in Table 5-4 and in Figure 5- 10. The seasonally adjusted value for the first quarter of 2000 was
·c 232.1 1 I = _
77967 = 298.458
Sales in the first quarte r of 2005 were 242.6. However, examination of the seasonally adjusted column shows that the sales for this quarter were actually high when the data were adjusted for the fact that the first quarter is typically a very weak quarter.
<>-ro simplify the notation in this example. we omit the subscript ton the original data, Y, and each of its com- ponenL,;, T. s. c. and /. w_e also omit t~c. multiplication sign. X , between components, since it is clear we are considering a multiplicatJve decompos1t1on.
182
350
300
250
200
350
300
250
200
I I I I
I
P'1tted Tcend Equation
Tt. • 261.24 + O. 759*t
Se~onsl Ind1ce~
Per::1011 Index l O. 77967 2 l.01566 3 l. 11667 4 l.08800
Accw::ac:y 11e8m1i:es
!!APE
;:I ..:) X
:
1= 7 . 047
18.457 4 n . 59S
I Foteca!lt.3
Penod For ecast 29 220. 842 30 288 . 45S 31 317. 990 32 310 . 654
FIGURE 5-9 Minitab Output for Decomposition of Coast al Marine Quarterly Sales for Example 5.3
Component Analysis for Sales Multip licative M:ldsl
Original Data Detrended Data
fWVp 1.2
1.0
0.8
5 10 15 20 0.6
25 Quarter
5 10 15 20 Quarter
25
Seasonally Adjusted Data Seasonally Adj. and Detrended Data
40
0
5 10 15 -40
20 25 5 10 15 20 25 Qu.irter Q,o...ter
FIGURE 5 - 10 Co A · · mponent na lys1s for Coastal Manne Sales for Example 5.3
1.1
1.0
0.9
0 .8
CHAPTER 5 7ime Series and Their Componems 1 83
T he seasonal inde xes in Figure 5-9 were
First quarter = .77967--+ 78.0°0 Second qua rter - 1.0 1566--+ 101.6°0
Third qua rter = 1.l 1667--+ 111.7% Fourth quarter = 1.08800 -+ 108.8%
-The c~art on the left of Figure 5-11 shows the seasonal components relative to 1.0. We see t h:tt first-q uarter sales arc 22% below average. second-quarter ales arc about as expected, l hird-quartcr sales are almost 12% above average, and fourth-quarter sales are almost 9% above normal.
The cyclical-irr egular value for first quarter of 2000 was10
C l = ~ = 232.7 = 1.1 39 TS (262.000) (. 77967)
In o rder to calculate the cyclical column. a three-period moving average was computed. The valu e for the second quarter of 2000 was
1.1 39 1. 159 1.056 3.354 3.354/3 = 1.118
Notice how smooth t h e C column is compared to the Cl column. The reason is that using the moving average has smoothed out the irregularities.
Finally. the / column was co mputed. Fo r example. for t he second quarter of 2000,
I = C I = 1.159 = I 036 C 1.118 .
Examin ation of the / column shows that there were omc large changes in the irregular com pone nt. The irregular index dro pped from 111.4% in the fourth quarter of 2002 to 87%
Seasonal Analysis for Sales M.Jl~licatJve Model
Seasonal Indices Detrended Data by Season
□ 1..2
B □ ~ Q
D = 1.0
8 0.8 0.6
2 3 4 2 3 4
FIGURE 5 - 11 Seasona l Analysis for Coastal M arine Sales for Example 5.3
10Minitab calculate the cyclical X irregular component (o r simply the irregular component if no cyclical component is conte mplated) by_ subtrac_1ing the trend X sea5?nal com_ronent rrom th e original data. In sym- bols. Minitab sets Cl = Y - TS. The M_imtab Cl component 1s_sho'An m the low~r ngh t-h and cha~t or Figu~e 5-10. Moreover. Mini tab fits the trend hne to the seasona ll y adJusted data. That 1s. the seasonal adJu tment 1s done before the tre nd is determined.
184 CHAPTER 5 Time Series and Their Components
in the first quarter of 2003 and then increa~ed to 106.2% in the second quarter of 2003.Th:s behavior resuJts from the unusually low sales in the first quarter of 2003.
Business Indicators The cyclical indexes can be used to answer the following questions:
1. Docs the series cycle? 2. lf so, how extreme is the cycle? 3. Does the series follow the general state of the economy (business cycle)?
One way to investigate cyclical patterns is through the study of busine indiau~ A business indicator is a business-related time series that is used to help assess the gen- e ral state of the economy, particularly with reference to the business cycle. Many busi- nesspeople and economists systematically follow the movements of such statistical series to obtain economic and business information in the form of an unfolding picture that is up to date, comprehensive, relatively objective, and capable of being read and understood with a minimum expenditure of time.
Business indicators are business-related time series that are used to help a the general state of the economy.
The most important list of statistical indicators originated during the sharp busi- ness setback from 1937 to 1938. Secretary of the Treasury Henry Morgcnthau requested the National Bureau of Economic Research ( BER) to devise a system that would signal when the setback was nearing an end. Under the leadership of Wesley Mitchell and Arthur F. Burns. NBER economists selected 21 series that from past per- formance promised to be fairly reliable indicators of business revival. Since then. the business indicator list has been revised several times. The current list consists of 21 indicators- IO classified as leading, 4 as coincident, and 7 as lagging.
1. L eading indicators. In practice, components of the leading series are studied to help anticipate turning points in the economy. The Survey of Current Business publishes this list each month, along with the actual values of each series for the past several months and the most recent year. Also, a composite index of leading indicators is computed for each month and year; the most recent monthly value is frequentl} reported in the popular press to indicate the general direction of the future econom). Examples of leading indicators are building permits and an index of stock prices.
2. Coincident indicators. The four coincident indicators provide a measure of how the U.S. economy is currently performing. An index of these four series is computed each month. Examples of coincident indicators are industrial production and man- ufacturing and trade sales.
3. Lagging indicators. The lagging indicators tend to Jag behind the general state of the economy, both on the upswing and on the downswing. A composite index is also computed for this list. Examples of lagging indicators arc the prime interest rate and the unemployment rate.
Cycles imply turning points. Tbat is, turning points come into existence only as a consequence of a subsequent decline or gain in the business cycle. Leading indicators change direction ahead of turns in general busine activity, coincid~nt ~nd(cators turn at about the same time as the general economy. and turns in the lagging indicators follow
HAP !ER 5 Time \ ene\ and Thl'ir Compo11e11t\ 185
those of the general econom\. I lowever, it i, difficult to it.lentif} cyclical turning points at the time they occur. since ·all area, of the economv do not e,pand at the -.ame time during periods of expansion or contract at the ,,1me time during period'> of contraction. Hence. several month-, ma} go by before a genuine cyclical upturn or downturn i finally identified with a ny assurance.
Leading indicator-. arl! the most useful predictive tool. since they attempt to signal economic changes in advance. Coincident and lagging indicator., ar\! of minimal inter- est from a forecasting perspective. but the} arc used to a. ess the effectivenes of cur- rent and past economic policy and so to help formulate future policy.
In thcir article enti t led ··Carly Warning Signab for the Economy:· Geoffrey H. Moore and Julius Shiskin ( 1976) have thc follm\ing to say on the uscfulne. s of busine s cycle indicaton.:
It secms clear from the record tha t business qcle indicators ar\! helpful in judging the tone of current business and short-term prospects.. But becausc_of their limitations. the indicator-. mu t be used together \\<ith other data and with full awareness of the background of bu iness and consumer confidence and expectations. go\'ernmcntal policies. and internatio~al events. We_ a lso must anticipate that the indicators will often be d ifficult to mtcrpret. that mte_rprcl?- tions will sometimes \ ar) among anal) SI'-, and that the signals they g!vc \\_111 not be corrcctlv interpreted. Indicator-. provide a . en iti,e and revealing pic- ture of the cbh and OO\\ of economic t idcs that a skillful analyst of the eco- nomic. political. and international sccnc can u-.e to improve his i::hances of making a va lid forecast of ,hort-run economic tr~nds. If t~e analy,t 1~ a\~are of their limitations and alert to the world around him. he will find the md1cators useful guideposts for taking stock of the economy and its needs. (p. 81}
Cyclical components of individual t ime series_ gcn~rally conform onl)_ lo?~_ely . and sometimes not at all-to the business cycle as 1dl!nllficd by the BER md1cators. i-f o \~cver. if a cvcl ical component for a given time series i. estimated. it should a lways be plotted over.time to get some indication of the magnitude and length~ of any cycles that appear to exist. In additic~n. th~ plot can be examined for an) relation to the ups and downs of general economic actl\lt). . . _ . .
' Ihc discu;sion so far shows how the factors that create vanat1o n m ~ time se~1es can be separated and studicd individuall). Anafysi., is the process for takmg the time series apart: synthesis is the process for pulling it back tog~ther. ext. we sha ll put the components of the time serie!-> back together to do forecast mg.
FORECASTING A SEASONAL TIME SERIES
t . s, isonal time seril!s the decomposition process is reversed. In stead of In forecas mg a • c, · < · · • • • t · h · h, s ' ries into individual component<, for exammat1on. we recom )met e com-
separating td c . cl ·,he forecasts for future periods. We \\ ill US\! the multiplicative model Poncnts to e\e op · . · , · I,
. f E ,, I , s 1 to develop forecast for C oa<,tal \fanne Corporation sa cs. and the rl!sults o xamp c - ··
~xample S.r4c · . t·1l •t·1rim: Corporation ,ales for the four quarters of 2007 can be de\cl-Forccasts o O,L'> • " • opct.l using Ta hie 5--t
, arterl\ trend equa tion is f -= 261.24 .7591. The forcca,;t origin is the I. fre/1(/. llH.: _4~ f 2006_ or time period , 11 28. Sales for the fir,t quart~r of 2(Xl7
fourth qu_a, t~r O ' . ,a l I = 1x + I = 29. Thi notation ,hem-, we arc forcca,tmg p I cx:currcd m 11me pcn<,u -
186 CllAPTER 5 Time Series and Their Components
period ahead from the end of the time series. Setting t = 29. the trend proJectioca then
f 29 = 261.24 + .759(29) = 283.251
2. Seasonal. The seasonal index for the first quarter . . 77967. is given in Figure 5-9. 3. Cyclical. The cyclical projection must be determined from the estimated cyclical pa-.
tern (if any) and fro~ any other inf?rmation generated by indicators of the gen~;~ economy for 2007. Pro1ectmg the cyclical pattern for future time periods is fraught
11 ·lli
uncertainty. and as we indicated earlier. it is generally assumed. for forecasting~- poses, to be included in the trend. To demonstrate the completion of this example . set the cyclical index to 1.0. · t
4. Irregular. Irregular fluctuations rep~esent ra_ndom variation that can"t be explained bl the other components. For forecasting. the irregular component is ct to the a,erai value l.0. 11 ·rne forecast for the first quarter of 2007 is
Y29 = T29 X S29 X C29 X /z9 = (283.251)(.77967)(1.0)(1.0) = 220.842
lbe forecasts for the rest of 2007 are
Second quarter = 288.455 Third quarter = 317.990
Fourth quarter= 310.654
The multiplicative_ decomposition fit for Coastal Marine Corporat ion sales. along 11,11h the forecasts for 2007. 1s shown in Figure 5-12. We can see from the figure that the fit. con- st ructed from the tr_c~d and seasonal components, represents the actual data rcasonabil well. However. the flt 1s ~o~ good for the last two quarters of 2006. time periods 27 and 28. lbc forecasts for 2007 m1m1c the pattern of the fit.
j
""-tltip licatlve Decorrl)(>Sltlon Fit for Sales 360-r--------- ---------.,-----
320
200
240
200
... I
I J I I I I ~
3 6 9 12 15 18 21 Time
,.. t'J,,. I • ,
I j ,
Acaz.cy Menu-ti. MAPE 7/W MAD IM57
I MSD 'fTl.595 ' I I
1
24 27 3'.J
FIGURE 5-1 2 Decomposition Fit and Forecasts for Coastal Marine Sales for Example 5.4
11 For forecasts generated from an additive model. 1he irregular imfo:t is -.et ro rhe average value o.
CHAPTER 5 Time Series and Their Compone111 187
_Forecasts produced by an additive or a multiplicative decompo ition model retlect th
e import~nce of the individual components. If a variable is highly ea onal. then the forecas!s will have a strong sea onal pattern. If, in addition. there is a trend. the fore- casts will follow the seasonal pattern supcrimpo ed on the extrapolated trend. If one com ponent dominates the analysis, it alone might provide a practical, accurate short- term for ecast.
THE CENSUS II DECOMPOSITION METHOD
Time series decomposition methods have a long hi tory. In the I 920s and early 1930s. t h e Federal R eserve Board and the ational Bureau of Economic Research were heavily involved in the seasonal adjustment and smoothing of economic time series. However, before the development of computers. the decompo ition calculations were laborio u s. and practical application of the methods was limited. In the early 1950s, Julius Shisk in. chief economic statistician at the U.S. Bureau of the Census, developed a large-scale computer program to decompose time series.'TTle first computer program essentially approximated the hand methods that were used up to that time and was replaced a year later by an improved program known as Method JI. Over the years, improved variants of Method II followed. The current variant of the Census Bureau time series decomposition program is known as X-12-ARIMA. This program is avail- able from t he Census Bureau at no charge and is widely used by government agencies and private companies. 12
Census II decomposition is usually multiplicative. since most economic time series have seasonal variation that increases with the level of the series. Also. the decomposi- t ion assumes three components: trend-cycle, seasonal. and irregular.
The Census 11 method iterates through a series of steps until the components are successfully isolated. Many of the steps involve the application of weighted moving averages to the data. This results in inevitable loss of data at the beginning and end of t he series b ecause of the averaging. The AR IMA part of X-12-ARIMA provides the facil ity to extend the original series in both directions with forecasts so that more of the observations are adjusted using the full weighted moving averages. These forecasts are generated from an ARIMA time series mode l ( ·ee Chapter 9).
T h e steps for each iteration of the Census II method as implemented in X-12- A RIMA are outlined next. It may seem that the method is complicated because of the many s teps involved. However. the basic idea is quite simple-to isolate the trend- cycle, seasonal, and irregular components one by one. The various iterations are designed to improve the estimate of each component. Good references for additional s t udy are Forecasting: Methods and Applications (Makridakis, Wheelwright. and H yndman 1998), Forecasting: Practice and Process for Demand Management ( L evenbacb and Cleary 2006). and" ew Capabilities and Methods of the X - 12- A RIMA Seasonal-Adjustment Program" (Findley et al. 1998).
Step 1. Ans-period moving average is applied to the original data to get a rough estimate of the trend-cycle. (For monthly data. s = 12: for quarterly data. s = 4: and so forth.)
Step 2. lhe ratios of the original data to these moving average values are calculated as in classical multiplicative decomposition. illustrated in Example 5.2.
12The PC version of the X-12-ARIMA program can be downloaded from the U.S. Ccnsu5 Bureau website. At the time this book was wrillen. the \\eb addrc,s for the do\\nload page was www.census.gov/srd/w\\W/ x I 2a/x I 2down_pc.htm I.
188 CHAPTER 5 Time Series and Their Components
Step 3. The ratios from step 2 contain both the seasonal and the irregular com~ nents. They also include extreme values resulting from unusual C\enis such as strikes or wars. The ratios are divided by a rough estimate oftht seasonal component to give an estimate of the irregular cornponem. A large value for an irregular term indicates an extreme value in the ono. inal data. These extreme values are identified and the ratios in ste/2 adjusted accordingly. This effectively e limin ates values that do not fit the pattern of the remaining data. Missing values at the beginning and end u1 the series are also replaced by estimat es at this stage.
Step 4. 'The ratios created from the modified data (with extreme values replaced and estimates for missing values) a r e smoothed using a moving average 10 eliminate irregular variation. This creates a preliminary estimate of the seasonal component.
Step 5. lhe original data arc then divided by the preliminary seasonal component from step 4 to get the pre liminary seasonally adjusted series. This season- ally adjusted series contains the trend-cycle and irregular componenb. In symbols,
Yr T, X S, X 11 = ----- = Tr X I, st s, Step 6. The tre~d~cycle is estimated by applying a weighted moving average to
~he prehmmary seasonally adjusted series. This moving average eliminates irregular variation and gives a smooth curve that indicates the preliminan trend-cycle in the data. ·
Step 7. R epe~t _step 2 with this new estimate of the trend-cycle. That is. new ratios. c~~t~mmg only the seasonal and irregula r components, are obtained by d1v1dmg _ the original ~bservations by the tre nd-cycl e from step 6. These are the fmal seasonal-IJ"regular ratios. Mathematically,
Yr T, X St X 11 T. = T. = S1 X It
t {
Step 8. R epeat step 3 using the new ratios computed in step 7. Step 9. Repeat step 4 to get a new estimate of the seasonal component.
Step JO. R ~~eat step 5 with the seasonal component from step 9. S
t ep 11. Divi?e t~e seasonally adjusted data from step IO by the trend-cycle
obtained m step 6 to get the estimated irregular component.
S t ep 12. Extre_mc val ues of the irregular component are replaced as in step 3.
A sencs of modified data is obtained by multiplying the trend-cycle, ea- sonal component, and adjusted irregular component together. These data reproduce the original data except for the extreme values.
The preceding 12 steps are repeated, beginning with the modified data from step 12
~ather than the original data . Some of the lengths ofrhe moving averages used in the vanous st~ps are changed. de pe nding on the variability in the data.
The fm~I seasona lly adjusted series is determined by dividing the final seasonal component mto the original data. The result contains only the product of the trend- cycle and irregular components_
. The _value of each of the final components i printed out and plotted_ A series of diagnos tic tests is available to determine if the decomposition was s uccessful.
CHAPTER 5 Time Series and Their Components 189
1:1e X-12-ARIMA program contains many additional features that we have not descnbed. Fo r example, adjustments can be made for different numbers of trading days a nd for holiday effects_ missing value within the crie can be estimated and rep laced, the effects of outliers can be removed before decomposition, and ot her cha nges in trend such as shifts in level or temporary ramp effects can be modeled.
APPLICATION TO MANAGEMENT
Time series analysis is a widely used statistical tool for forecasting future events that a rc intertwined with the economy in some fashion. Manufacturers arc e xtremely inter- ested in the boom-bust cycles of our economy as well as of fore ign economies so that they can better predict demand fo r their products. which. in turn, impacts their inven- tory levels. e mployment needs. cash fl ows, and a lmost all other business acti vities within the firm.
The complexity of these problems is enormous. Take, for example, the problem of predict ing demand for oil a nd its by-p roducts. In the late l 960s, the price of oil per bar- re l was very low. and there seemed to be an insatiable worldwide demand for gas and o iL Th e n came the oil price sh ocks of the early a nd mid-1970s. What would the future demand for oil be? What about prices? Firms such as Exxon a nd General Motors were obviously very interested in the answers to these questions. If o il prices continued to escalate. would the demand for large cars diminish? What would be the demand for e lectricity? By and la rge. analysts predicted that the demand for energy, and the refore o iL would be very inelastic: thus. prices would contin ue to outstrip inflation. However, these predictions did not rnke into account a major downswing in the business cycle in the early 1980s and the greater elasticity of consumer demand for ene rgy t han pre- d icted. By 1980, the world began to sec a glut of oil on the market and radically falling pr ices_ At the time, it seemed hard to believe that consumers were actua lly benefiting o nce again from gasoline price wa rs. At the time this editi on was written. s ubstantial unrest in the Middle East created a shortage of oi l once again. lhe price of a barrel of o il and the cost of a gallon of gas in the U nited States were at record highs.
Oil demand is affected not only by long-term cyclical events but also by seasonal and random events, as arc most other forecasts of demand for any type of product or service. For instance_ consider the service and retail industries. We have witnessed a continued movement of employment away from manufacturing to the reta il and service fields. However , retailing (whethe r in-store, catalog. or web-based) is an extremely seasonal and cyclical business and demand and inventory projections are critical to retailers, so time se ries analysis will be used more widely by increasingly sophisticated retailers.
Manufacturers will h ave a cont inued need for statistical projections of future events. Witness the explosive growth in the technology and telecommunications fields during the 1990s and the s ubstantial contraction of these indus~rie~ in the early 2000s. This growth a nd contraction resulted, ~o a large extent, from projections of dema_nd that never completely materialized. Questions tha t a ll manufacturers must address include these: What will the fu ture infl ation rate be? H ow will it affect the cost-of-living adjust- ments that may be built into a company's labor contract? How will these adjustments affect prices a nd demand? What is the projected pool of managerial skills for 2025? What w ill be the effect of the government's spending and taxing strategies?
What wi ll the future population of you ng people look like? What will the ethnic mix be? These issues affect almost all segments of our economy. Demographers are closely watching the current fertility rate and us_ing alm_ost every avail~ble ti'!1e series ~oreca~t- ing techni que to try to project population variables. Very minor m1scalculat1ons will
190 CHAPTER 5 Time Series and Their Components
have major impacts on everything from the production of babie ' toys to the fin soundne s of the Social Security system. Interestingly. demographers are looking at '?Ir long-term business cycles (20 years or more per cycle) !n tl)in~ to predict \\hat thi, ~ .'. cration's population of women of childbearing age will do with regard to ha,ing _ drcn. Will they have one or two children. as families in the J960s and 197(),, did.or11 they return to hav ing two or three. as preceding generation did'? These dcci,1cin, v. determine the age compo · ition of our population for the next 50 to 75 )Cal",.
Political scientists are interested in using time scric analysi to tudy the chan • patterns of government spending on defense and social welfare programs. Ob,io~,~ these trends have great impact on the future of whole industries.
Finally, one interesting microcosm of applications of time series anal)~i ha, h~ up in the legal field. Lawyers are making increasi ng u ·c of expert witne~ e to te-.tify about the present value of a person·s or a firm·s future income. the cost incurred from the loss of a job due to discrimination. a nd the effect on a market of an illeg,11 ,tnkc These questions can often be best answered through the judiciou use of time ,;enc analys is.
Satellite technology and the World Wide Web have made the accumulation and transmission of information almost instantaneo u s. The proliferation of personal comput- ers, the availability of easy-to-use statistical oft ware programs. and increased acre-, to databases have brought information processing to the de ktop. Bu inc sunival during periods of major competitive change requires quick, data-driven deci ion making. lime series analysis and forecasting play a major role in the e decision-making proce ~'!).
APPENDIX: PRICE INDEX
Severa] of the series on production. sale , and other economic situation contain da1a available only in dollar values. These data are affected by hath the ph) ical quantit~ of goods sold and their prices. Inflation and widely varying price · over time can cau e analysis problems. For instance. an increased dollar volume may hide decrca ed ak, in units when prices a re inflated. Thus. it is frequentlv necessarv to know hO\\ much of ~he change in dollar value represents a real change i~ physical quantity and hm~ much is due to change in price because of inflation. Jt is desirable in the. e instances 1c, express dollar values in terms of constant dollars.
The concept of purchm·ing power is important. The current purcha ing power of SI is defined as follows: ~
Current purchasing power of $1 = --- _J_CJO ___ _ Consumer Price Index
(S.IO)
Thus_ if in November 2006 the consumer price index (with 2002 as 100) reache · 150. 1he current purchasing power of the November 2006 consumer dollar is
Current purchasing power of $1 = 'r> = .67 I:,()
The 2006 dollar purchased onlv two-thirds of the goods and services that could haw been purcha cd with a base period (2002) dollar. ~
To express dollar values in tcnn. of constant dol/an. Equation 5.11 i u sed.
D eflated dollar value = ( Dollar value ·' ( Pun:ha,ing power of$ I) (5.11)
Cl I APTER 5 Time S1 rie, and Their Compcmcnrs 191
Suppose that car sales rose from $300,()()() in 2005 10 350,000 in 2006, while the new-car price index (with 2002 as the ba c) ro-,e from 135 to 155. D eflated c;ale for 2005 and 2006 would be
Deflated 2005 sales ( 300.000)( :: ) = 222.222
Deflated 2006 sales = ( 350.000>( ~:) $225, 06
otc that actual dollar saJes had a sizable increase of $350.()()() - $300,000 = 50,(XXl. However. deflated sales increased by only $225,806 222.222 $3,584.
The purpose of deflating dollar values i to remove the cffecl of price changes. This adjustment is called price deflation or is referred to as expre si ng a series in con- . tant dollars.
Price deflation i · the procc.:ss of expre!>..,ing values in a series in constant dollal"'>.
The deflating process is relatively simple. To adjust price to con tant dollar~ an index number computed from the prices of commoditie \\hose value-, arc to be deflated is used. For example. shoe store sales should be deflated by an index of shoe prices. not by a general price inde x. For deflated dollar values that represent ~o~e than one type of commodity, the analy~t should devel~p a price index by combmmg the appropriate price indexes together m the proper mix.
Example 5.5 . Mr. Burnham wishes to study the long-term growth of _the 13urnhaf!1 Furn1tun.: Store. The long-term trend of his business !>hould he evaluated u !ng the physical_ volume of. ale:.._ ~f this evaluation cannot be done. price changes reflected m dollar sales Mil folio~ no :ons,1s- te nt auern and will merely obscure the real growth_ patter_n. I~ sales dollar<,_ ar~ t? bc_u :d: actu~ dollar sales need to be divided by an appropnate pnce mdcx to obtain sales that arc measured in constant dollars. . .
·The consumer price index (CPI ) is not suitable for Burnham because ll contains ele- h Cnt food and ~rsonal services not sold bv the 'itorc. but ~omc components ments sue as r . · ,·- . · · o f I , r f ·
f h .. · d . ay be appropriate. Burnham ,s aware that 70 o o a es arc rom umllure o l is m ex m, I . I h (' Pl H ·t ' po •nt bv and 30% from appliances. He can therefore mu up ) t e rctai urn, urc ~om ne _ . ' 70 It" I the appliance component by .30. and then add thc results to obwm a combmcd · . · m_u dip y ,.,ahle 5 5 illustrates this approach. in which the computation for 1999 arc pnce m ex. J, -. - -
9().1 ( .70) + 94.6(.30) - 91.45
The sales are denatcd for 1999 in terms of 2002 purcha. ing power. so that
bl 5 5 h ·s that although actual sale<, gained teadily from 1999 to 2006. physical \.OI- Ta ,e --~ain~:-rathe~ stable from 2004 to 2006. Evidently. the c;ales increa"e were due to Url_lC re k that w·erc generated in turn, h} the inflationary tendenC} of the economy. pnce mar ups , ·
192 CHAPTER 5 Time Series and Their Components
Glossary
TABLE 5-5 Burnham Furniture Sales Data, 1 999-2006, for Example 5.5
Retail Burnham Retail Furniture Applia11ce Price Price
Sales Prlcel11dex Inda Inda" Year ($1,0005) (2002=100) (2002=100) (2002=100)
1999 42.1 90.l 94.6 91.45 2000 47.2 95.4 97.2 95.94 2001 48.4 97.2 98.4 97.56 2002 50.6 100.0 100.0 100.00 2003 55.2 104.5 IOI.I 2004 57.9 108.6
103.48 103.2 106.98
2005 59.8 112.4 104.3 2006 60.7 114.0
109.97 105.6 111.48
aconstructcd for furniture (weight 70%). d r . hSa lcs divided by price index times JOO. an app iances (weight 30% ).
Deflared Salrsi t l,OOOs of 1(}()1J
46.0 49.2 49.6 50.6 53.J 54.I 54.4 54.4
B . . di · usm~ m cators. Business indkators are b . . related ti · usmess-
me series that arc used lo help assess th general state of the economy e
Index numbers. I d be.
Price de~ation. ~rice d~flation is the proce of expressing value m a senc. in constant dollars.
·h h . n e_x num rs are percentages that s ow c anges over tune.
Key Formulas
lime series additive decomposition
Yi = 7; + S, + / 1
lime series multiplicative decomposition
}; = 7; X S, X T, Unear trend
f, = ho -r- b1t Sum of squared errors criterion (with trend T)
SSE= ~ (>'; _ f,)2 Quadratic trend
ExponentiaJ trend
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
Problems
CHATTER 5 Time Serie~ and Their Cvmpo11e111s 193
SeasonulJy adjusted data (additil-e decomposition)
Seasonally adjusted data (multiplicath·e decomposition)
Y, S,
T, X I,
Cyclicul-irregulur component (multiplicathe decomposition)
Y, C, X / 1 = --=--T, X S,
Irreb'lllar component (multiplicative decomposition)
Current purchasing power of$ I
100 Consumer Price Index
Deflated dollar vaJuc
( D ollar value) X ( Purchasing power of $1 )
I. Explain the concept of decomposing a time serie .
(5.7a)
(5.7b)
(5.8)
(5.9)
(5.10)
(5.11)
2. Explain when a multiplicative decomposition may be more appropriate than an additive decomposition.
3. What are some basic forcc5 that affect the trend-cycle of most variable '?
4. What kind of trend model should be used in each of the following cases? a. The variable is increasing by a constant rate. b. The variable is increasing by a con tant rate until it reaches saturation and
levels out. c. The variable is increasing by a constant amount.
5. What are some basic forces that affect the casonal component of most variables?
6. Value Line estimates of sales and earnings growth for individual companies are derived by correlating ·ales, earnings. and dividends to appropriate components of the National Income Accounts such a · capital spending. Ja on Black, an analyst for Value L inc, is examin ing the trend of the capital ·pending variable from 1977 to 1993. The data are given in Table P-6. a. Plot the data and determine the appropriate trend model for the years 1977 to
1993. b. If the appropriate model is li near, compute the linear trend model for the years
1977 to 1993.
1 94 CHAPTER 5 Time Series and Their Components
TABLE P-6 Capital Spending (S billions) , 1977-1993
Year $Billions Year $Billions Year $Billions
1977 214 1983 357 1989 571 1978 259 1984 416 1990 578 1979 303 1985 443 1991 556 1980 323 1986 437 1992 566 1981 369 1987 443 1993 623 1982 367 1988 545 1994 ~
"Value Linc estimate.
Source: 71ie Value Line lnvesrmem Survey (New York; Value Line. 1988. 1990, 1994).
c. What has the average increase in capital spending per year been since 19T? d. Estimate the trend value for capital spending in 1994. e. Compare your trend estimate with Value Line's. f. What factor(s) influence the trend of capital spending?
7. A larg~ company is_considering cutting back on its TV advertising in favor of busi- ness VIdeos to _be given to its customers. This action is being considered after the c?mpany president read a recent article in the popular press touting business ~•deos a~ today's "hot sales weapon."' One thing the president would like to im·~- hgate_ pnor to taking this action is the history of TV advertising in this countr)". especiaJly the trend-cycle.
Table P-7 ~ntains_ the total dollars spent on U.S. TV advertising (in millions of dollars~ a. P!ot t~e time senes of U.S. TV advertising expenditures. b. Fit. a hnear trend to the advertising data and plot the fitted line on the time
senes graph. c. Forecast TV advertising dollars for 1998. d. Given the _r~sults in part b, do you think there may be a cyclical component in
TV advertismg dollars? Explain.
8. As~ume the ~ollowing specific percentage seasonal indexes for March based on the rat10-to-movmg-average method:
102.2 105.9 114.3 122.4 109.8 98.9
What is the seasonal index for March using the median?
TABLE P-7 0
Year y Year y
1980 11,424 1989 26,891 1981 12.811 1990 29,073 1982 14.566 1991 28.189 1983 16.542 1992 30.450 1984 19.670 1993 31.698 1985 20.770 1994 35,435 1986 22.585 1995 37,828 1987 23.904 1996 42.484 1988 25.686 1997 44.580
Source: Statistical AbsrraN of the united Stares. various years.
CHAPTER 5 Ttme Series and Their Components 1 95
9. !he expected trend value for October is $850. Assuming an October seasonal index of 1.12 (112%) and the multiplicative model given by Equation 5.2, what would be the forecast for October?
10. The following specific percentage seasonal indexes are given for the month of D ecember:
75.4 86.8 %.9 72.6 80.0
Assume a multiplicative decomposition model. If the expected trend for December is $900 and the median seasonal adjustment is used. what is the forecast for December?
11. A large resort near Portland, Maine, has been tracking its monthly sales for several years but has never analyzed these data. The resort computes the seasonal indexes for its monthly sales. Which of the following statements about the index are correct? a. The sum of the 12 monthly index numbers, expressed as percentages, should be
1,200. b. An index of 85 for May indicates that sales are 15% lower than the average
monthly sales. c. An index of 130 for January indicates that sales are 30% above the average
monthly sales. d. The index for any month must be between zero and 200. e. The average percent index for each of the 12 months should be 100.
12. In preparing a report for June Bancock, manager of the Kula Department Store, you include the statistics from last year's sales (in thousands of dollars) shown in Table P-12. Upon seeing them, Ms. Bancock says, "This report confirms what I've been telling you: Business is getting better and better." Is this statement accurate? Why or why not?
13. The quarterly sales levels (measured in millions of dollars) for Goodyear Tire are shown in Table P-13. Does there appear to be a significant seasonal effect in these sales levels? Analyze this time series to get the four seasonal indexes and deter- mine the extent of the seasonal component in Goodyear's sales.
TABLE P- 12
Adjusted Seasonal
Month Sales ($1,000s) Inda(%)
January 125 51 February 113 50 March 189 87 April 201 93 May 206 95 June 241 99 July 230 % August 245 89 September 271 103 October 291 120 November 320 131 D ecember 419 189
Source: Based on Kula Department Store records.
1 96 CHAPTER 5 Time Series and Their Components
TABLE P-13
Quarter
Year I 2 3 4
1985 2.292 2,450 2.363 2.477 1986 2.063 2.358 2.316 2.366 1987 2.268 2.533 2,479 2.625 1988 2.616 2.793 2.656 2.746 1989 2,643 2.811 2.679 2.736 1990 2.692 2.871 2.900 2.811 1991 2.497 2.792 2.838 2.780 1992 2.778 3.066 3.213 2.928 1993 2,874 3.000 2.913 2.916 1994 2,910 3.052 3.116 3,210 1995 3.243 3.351 3305 3.267 1996 3.246 3.330 3.34()<1 3.3003
3 Value Linc estimates.
Sour,·e: "/11e Vahw Line lm·esrmenr Surn:y (New Yori.: Value Linc. 1988. 1989. 1993. 1994. 1996).
a. Would you use _the trend component. the seasonal component. or both to forecast? b. Forecast for third and fourth quarters of l 996. c. Compare your forecasts to Value Line·s.
14 - The ~ont_hlr sales of the Cavanaugh Company. pictured in Figure 5.l (bo11om).
are given m (able P-1 4.
a. Pe~form_ ~ m~ltiplicative decomposition of the Cavanaugh Company sales time senes. assummg trend. seasonal. and irregular component..
b. Wou~d you use the trend component, the seasonal component. or both to forecast? c. Provide forecasts for the rest of 2006.
15 - ~on
st ruct a table similar to Table P-14 with the natural logarithms of monthh
sales. For example. the value for January 2000 is In( 154) = 5.037. · a Perform , dd. · d · · · an a !live ecompos1tion of ln(sales). assuming the model
TABLE P-14
Month
Ja nuary February March April May June July August September October November
December
2000
154 96 73 49 36 59 95
169 210 278 298 245
Y = T + S + I.
2001 2002 2003 2004 200.5 2006
200 223 346 518 613 628 118 104 261 40-t 392 308 90 107 224 3<XJ 273 324 79 85 141 2!0 322 24B 78 75 148 1% 189 272 91 99 145 186 257
167 135 223 247 324 169 211 272 3-13 4-04 289 335 445 -16-1 677 347 460 560 680 858 375 488 612 711 895 203 326 467 610 66-1
CHAfYfER 5 Time Series and Their Components 197
b. Would you use the trend component. the seasonal component, or both to forecast? c. Provide forecasts of ln(sales) for the remaining months of 2006. d. Take the antilogs of the forecasts calculated in part c to get forecasts of the
actual sales for the remainder of 2006. e. Compare the forecasts in part d with those in Problem 14, part c. Which set of
forecasts do you prefer? Why?
16. Table P -16 contains the quarterly sales (in millions of dollars) for the Disney Company from the first quarter of 1980 to the third quarter of 1995. a. Perform a multiplicative decomposition of the time series consisting of Disney's
quarterly sales. b. Does there appear to be a significant trend? Discuss the nature of the seasonal
component. c. Would you use both trend and seasonal components to forecast? d. Forecast sales for the fourth quarter of 1995 and the four quarters of 1996.
17. The monthly gasoline demand (in thousands of barrels/day) for Yukong Oil Company of South Korea for the period from January 1986 to September 1996 is contained in Table P-17. a. Plot the gasoline demand time series. Do you think an additive or a multiplica-
tive decomposition would be appropriate for this time series? Explain. b. Perform a decomposition analysis of gasoline demand. c. Interpret the seasonal indexes. d. Forecast gasoline demand for the last three months of 1996.
18. Table P -18 contains data values that represent the monthly sales (in billions of dol- lars) of all retail stores in the United States. Using the data through 1994, perform a decomposition analysis of this series. Comment on all three components of the series. Forecast retail sales for 1995 and compare your results with the actual val- ues provided in the table.
TABLE P- 16
Quarter
Year 1 2 3 4
1980 218.1 245.4 265.5 203.5
1981 235.1 258.0 308.4 211.8
1982 247.7 275.8 295.0 270.1
1983 315.7 358.5 363.0 302.2
1984 407.3 483.3 463.2 426.5
1985 451.5 546.9 590.4 504.2
1986 592.4 647.9 726.4 755.5
1987 766.4 819.4 630.l 734.6
1988 774.5 915.7 1,013.4 1,043.6
1989 1,037.9 1,167.6 1,345. 1 1,288.2
1990 1,303.8 J.539.5 l.712.2 1,492.4 1991 1.439.0 1,511.6 1,739.4 1,936.6
1992 1,655.1 1,853.5 2,079.1 2,391.4
1993 2,026.5 1,936.8 2,174.5 2,727.3
1994 2,275.8 2,353.6 2,698.4 3,301.7
1995 2,922.8 2.764.0 3,123.6
196 CHAPTER 5 Time Series and Their Components
TABLE P-13
Quarter
Year I 2 3 4
1985 2.292 2.450 2.363 2.477 1986 2.063 2.358 2.316 2,366 1987 2 .268 2533 2.479 2.625 1988 2.616 2 .793 2.656 2,746 1989 2.643 2,811 2 ,679 2.736 1990 2,692 2.871 2.900 2.811 1991 2.497 2.792 2,838 2.780 1992 2.778 3,066 3.213 2.928 1993 2.874 3.000 2.913 2.916 1994 2,910 3.052 3,116 3.210 1995 3.243 3.351 3 .305 3.267 1996 3.246 3.330 3.340-, 3.3001
avaluc Linc estimates.
Sourre: l11e Value Line lm·e.11111em Sun·ey (New York: Value Linc. 1988. 1989. 1993. 1994. 1996).
a. Would you use _the trend component. the seasonal component. or both to forec~? b. Forecast for third and fo urth quarters of 1996. c. Compare your forecasts to Value Line's.
14 · The ~ont_hly_sales of the Cavanaugh Company. pictured in Figure 5.1 (bollom).
are given m lablc P-14.
a. Pc~fonn_ ~ m~ltiplicative decomposition of the Cavanaugh Company sales time series, assummg trend, seasonal, and irregular components.
b. Wou~d you use the trend component. the seasonal component. or both to forecast? c. Provide forecasts for the rest of 2006.
15 · ~on
st ruct a table similar to Table P-1 4 with th e natural logarithms of monthll
~ales. For example._ t?e value for Januar y 2000 is In ( 154) = 5.037. · a. Perform an additive decomposition of ln(sales). assuming the model
TABLE P-14
Month
January February March April May June July August September October November December
2000
154 96 73 49 36 59 95
169 210 278 298 245
Y = T + S + /.
2(}0/ 2002 2003 2004 2005 2006
200 223 346 518 613 628 l 18 104 261 404 392 308 9{) 107 224 300 273 .124 79 85 141 210 322 248 78 75 148 196 189 272 91 99 I-ti 186 257
167 135 223 247 324 169 211 272 3--IJ 404 289 335 445 46--1 677 347 460 560 680 858 375 --188 612 711 895 203 326 --167 610 66--1
CHAJYfER 5 Time Series and Their Components 197
b. Would you use the trend component. the easonal component, or both to forecast? c. Provide forecasts of ln(sales) for the remai ning months of 2006. d. Take the antilogs of th e forecasts calculated in part c to get forecasts of the
actual sales for the remainder of 2006. e. Compare the forecasts in part d with those in Proble m 14, part c. Which set of
forecasts do you prefer? Why?
16. Table P -16 contains th e quarterly sales (in millions of dollars) for the Disney Company from the first quarte r o f 1980 to the third quarter of 1995. a. Perform a multiplicative decomposition of the time series consisting of Disney's
quarterly sales. b. Does there appear to be a significant trend? Discuss the nature of the seasonal
component. c. Would you use both trend and seasonal components to forecast? d. Forecast sales for the fourth quarter of 1995 and the four quarters of 1996.
17. The monthly gasoline demand (in tho usands of barrels/day) for Yukong Oil Company of South Korea for the period from January 1986 to September 1996 is contained in Table P-17. a. Plot the gasoline demand time series. Do you think an additive or a multiplica-
tive decomposition would be appropriate for this time series? Explain. b. Perform a decomposition analysis of gasoline demand. c. Interpret the seasonal indexes. d. Forecast gasoline demand for the last three months of 1996.
18. Table P -18 contains data values that represent the monthly sales (in billions of dol- lars) of all retail stores in the United States. Using the data through 1994, perform a decomposition analysis of this series. Comment on all three components of the series. Forecast retail sales for 1995 and compare your results with the actual val- ues provided in the table.
TABLE P- 16
Quarter
Year I 2 3 4
1980 218.1 245.4 265.5 203.5
1981 235.I 258.0 308.4 211.8
1982 247.7 275.8 295.0 270.1
1983 3 15.7 358.5 363.0 302.2
1984 407.3 483.3 463.2 426.5
1985 451.5 546.9 590.4 504.2
1986 592.4 647.9 726.4 755.5
1987 766.4 819.4 630.1 734.6
1988 774.5 915.7 1,013.4 1,043.6
1989 1,037.9 1,167.6 1,345.1 1,288.2 1990 1,303.8 l .539.5 J.712.2 1,492.4 1991 1,439.0 1,511.6 1,739.4 1,936.6
1992 1,655.1 1,853.5 2,079.1 2,391.4
1993 2,026.5 1,936.8 2,174.5 2,727.3
1994 2,275.8 2,353.6 2.698.4 3,301.7
1995 2,922.8 2,764.0 3,123.6
1 98 CHAPTER 5 Time Series and Their Components
TABLE P- 17
Month 1986
January 15.5 February 17.8 March 18.l April 20.5 May 21.3 June 19.8 July 20.5 August 22.3 September 22.9 October 21.1 November 22.0 December 22.8
1987 1988 1989 1990 1991 1992 1993 1994 1995 /99f
20.4 26.9 36.0 52..I 64.4 82.3 102.7 122.2 145.8 r 20.8 29.4 22.2 29.9 24.J 32.4 25.5 33.3 25.9 34.5 26.1 34.8 27.5 39.1 25.8 39.0 29.8 36.5 27.4 37.5 29.7 39.7
39.0 53 .1 68.1 83.6 102.2 121.4 42.2 56.5 68.5 85.5 104.7 125.6 44.3 58.4 72.3 91.0 l08.9 129.7 46.6 61.7 74.1 92.1 112.2 133.6 46.1 61.0 77.6 95.8 109.7 137.5 48.5 65.5 79.9 98.3 113.5 143.0 52.6 71.0 86.7 102.2 120.4 149.0 52.2 68.1 84.4 IOI .5 124.6 149.9 50.8 67.5 81.4 98.5 116.7 139.5 51.9 68.8 85.l 101.1 120.6 147.7 55.1 68.1 81.7 102.5 124.9 154.7
144.4 176J 145.2 1'42 148.6 1i6.I 153.7 lli.'iJ 157.9 1. 2: 169.7 197.0 184.2 216.1 163.2 ln.2 155.4 168.9 178.3
TABLE P-18
Month 1988 1989 1990 1991 1992 1993 1994 1995 January 113.6 February 115.0 March 13.1.6 April 130.9 May 136.0 June 137.5 July 134.1 August 138.7 September 131.9 October 133.8 November 140.2 December 171.0
122.5 132.6 130.9 142.1 148.4 154.6 167.0 118.9 127.3 128.6 143.1 145.0 155.8 16t0 141.3 148.3 149.3 154.7 164.6 184.2 192.1 139.8 145.0 148.5 159.1 170.3 181.8 1875 150.3 154.1 159.8 165.8 176.1 187.2 2014 149.0 153.5 153.9 164.6 175.7 190.1 ~02.6 144.6 148.9 154.6 166.0 177.7 185.8 194.9 153.O 157.4 159.9 166.3 177.1 193.8 204.2 144.l 145.6 146.7 160.6 171.1 ]85.9 192. 142.3 151.5 152.1 168.7 176.4 189.7 194.0 148.8 156.l 155.6 167.2 180.9 194.7 202A 176.5 179.7 181.0 204.1 218.3 233.3 238.0
Source: Based on Survey of Current Business. 1989, 1993. 1996.
TABLE P- 19
Adjusted Adjusted Month Seasonal Index Month Seasonal Index January 120 July 153 February 137 August 151 March 100 September 9.5 April 33 October 6() May 47 November 82 June 125 December 97
Sourre: Based on Mt. Spokane Resort Hotel records.
19 - The a~justed seasonal indexes presented in Table P-/ 9 renecl the changing volume
of bustness of the Mt. Spokane Resort Hotel. which carers to family tourists in the s~mmer and skiing en thusiasts during the winter months. o sharp cyclical varia- t10ns arc expected during 2007.
TABLE P-25
>tar Jan.
1993 63,344 1994 64,434 1995 65.966 1996 66.006 1997 67.640 1998 68,932 1999 69,992 2(XX) 71.862 2001 72.408 2002 71.285 2003 71.716
CHAPTER 5 Time Series and Their Components 1 99
TABLE P-24
Commodity Sales Price Inda
Volume($) (2001 = 100)
2005 January 358.235 118.0 February 297,485 118.4 March 360,32 1 118.7 April 378,904 119.2 May 394.472 119.7 June 312.589 119.6 July 401.345 119.3
a. If 600 tourists were at the resort in January 2007, what is a reasonable estimate for February?
b. The monthly trend equation is f = 140 + Sr where t = 0 represents January 15, 2001. What is the forecast for each month of2007?
c. What is the average number of new tourists per month?
20. Discuss the performance of the composite index of leading indicators as a barom- eter of business activity in recent years.
21. What is the present position of the business cycle? Is it expanding or contracting? When will the next turning point occur?
22. What is the purpose of deflating a time series that is measured in dollars?
23. [n the base period of June, the price of a selected quantity of goods was $1.289.73. In the most recent month. the price index for these goods was 284.7. How much would the selected goods cost if purchased in the most recent month?
24. Deflate the dollar sales volumes in Table P-24 using the commodity price index. These indexes are for all commoditi es, with 2001 = 100.
25. Table P-25 contains the number (in thousands) of men 16 years of age and older who were employed in the United States for the months from January
Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
63,621 64,023 64.482 65.350 66,412 67.001 66,861 65.808 65,961 65.779 65.545
64,564 64.936 65.492 66.340 67.230 67,649 67.717 66.997 67,424 67,313 67292
66.333 66.758 67.018 67.227 68,384 68.750 68,326 67.646 67.850 67.219 67,049
66,481 66.961 67.415 68-258 69298 69.819 69,533 68.614 69.099 68.565 68,434
67,981 68.573 69. 105 69.968 70.619 71,157 70,890 69.890 70.215 70,328 69,849
69,197 69.506 70.348 70,856 71,618 72.049 71,537 70,866 71.219 71.256 70,930
70,084 70.544 70,877 71.470 72.312 72,803 72,348 71.603 71,82.5 71.797 71,699
72.177 72.501 73,006 73.236 74.267 74,420 74.352 73.391 73.616 73.497 73,338
72,505 72.725 73,155 73.313 74.CXJ7 74.579 73.714 73.483 73,228 72,690 72.547
71,792 71.956 72,483 73.230 73.747 74.210 73,870 73.5% 73.513 72.718 72,437
72,237 72.304 72.905 73.131 73.894 74.269 74,032 73,715 73.979
Source: Based 011 Labor force statistics from the Current Population Survey.