An Approach to Definite Forecasting [Reprint 2016 ed.] 9781512816570

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AN APPROACH TO DEFINITE FORECASTING

AN APPROACH TO DEFINITE FORECASTING BY

L I N C O L N W. H A L L Assistant Professor of Economics, if hartón School o/ Finante and Commerce, University 0} Pennsylvania ASSISTED BY R. L E I G H

HALL

PHILADELPHIA

UNIVERSITY OF PENNSYLVANIA PRESS London Humphrey

Milford:

Oxford University

1929

Press

COFYKIGHT,

I939

UNIVERSITY OF PENNSYLVANIA PRESS

RMURRWD M TMB UMITIO R I T U o r AMBBICA LANCASTER N E U , INC. LANCASTER, PA.

To MY WIFE

PREFACE " L e a v e Hope behind, all ye who enter here"— expresses the desperate sensation of one who approaches the reading of a treatise on statistics. However valuable the material in the book may prove, the thought of so many mathematical symbols and formulas has a depressing effect. T o the more general reader especially, they are not only mystifying but stupefying. Therefore in this volume we have avoided wherever possible the use of mathematical formulas and involved mathematical discussion; however, those who are familiar with mathematics will have no difficulty in tracing the mathematical thought which is the basis for each step. As the title suggests, this book deals with the subject of "Definite Forecasting," or it may be called forecasting in definite numbers. Three approaches to definite forecasting may be distinguished, and they may be called: (1) Forecasting by demand curves, (2) Forecasting by the relations between series, (3) Forecasting by the isolation and analysis of individual time series. We are concerned in this book only with the lastnamed approach, namely "forecasting by isolation." It must be kept in mind that we are only laying down an approach and not a complete set of hard and fast rules by which to accomplish the miraculous. I am greatly indebted to Dr. M . A. Brumbaugh, Dr. C. A. R . Wardwell and Mr. W. H. Blaisdell for permitting me to use data which they had gathered for another purpose. vii

C O N T E N T S

CHATTE* I.

FAGS INTRODUCTION

The Principle of Definite Forecasting... The Experimental Approach II.

THE

DATA

The Constructed Series The Observed Series III.

CYCLICAL

9 12

INDICATORS

Use The Calculation of the Cyclical Indicator IV.

1 5

SERIES

38 38

ANALYSIS

Meaning 75 Original Components 75 Derivative Components and Relative Intensities 86 V. VI.

S H O R T - AND L O N G - T I M E

TRENDS

FORECASTING

Ultimate Object Forecasting by Original Components.... Forecasting by Derivative Components.. VII.

RESUMÉ

Complete Method Forecasting: Original Components Forecasting: Derivative Components . . . Illustration VIII.

94 96 98

CONCLUSIONS

113 126 129 132 138

ix

I INTRODUCTION T H E P R I N C I P L E OF D E F I N I T E

FORECASTING

Why definite forecasting? Is it not enough to present some general predictions—the short-time trend will be upward or downward, the long-time trend will probably be upward or downward, we may expect this or that at some unknown date—about the future? Such predictions are much like the mysterious prophecies of the Delphic oracle; they allow such a latitude of choice. They are vague, hazy, nebulous. Not that there is to be purveyed a system of black magic which, come what may, will be infallible. When a certain number is forecasted for a definite period in the future it will be erroneous to a degree depending upon the susceptibility of the series to mathematical analysis, but it will be a quantity far more definite than a quantity that any words of generalization would suggest. Furthermore by the size -of the error the forecaster will be able to evaluate the efficacy of this method, making additions or corrections which will eventually bring the forecasted data more in alignment with the original data when it occurs. The ultimate test is a pragmatic one. True, it is a severe test upon the methodology employed and the ability of whoever uses it, but at least before us will be presented the possibilities and defects of the knowledge which the method seeks to divine. ι

2

AN APPROACH

TO DEFINITE

FORECASTING

Definite forecasting then is the attempt to predict for some definite period in the future the actual numbers which will probably occur during this period. For example, if we have under consideration the sales of " X " for the next year, a definite forecast would give us the number of units of " X " which would probably be sold during the first quarter of the ensuing year, for the second quarter, and so on, until the unit sales for the entire four quarters or possibly twelve months had been forecasted. Just to say that sales will probably increase or decrease is not a definite forecast since a definite forecast requires that the actual number of units sold for each quarter of the year shall be forecasted. It is seen from the very principle of definite forecasting that any methodology which we develod must permit us to forecast in terms of the units in which the original data is expressed. If the original data is expressed in barrels of production, the forecasts must be in barrels of production—not as a percentage or any other abstract measurement. The object of the present study is to enable us to make more practicable use of knowledge concerning time series analysis. Statisticians have usually been content in their analysis of time series to use the results in a rather vague fashion which would tell what had happened heretofore in a particular series, and from that information to indicate whether this particular series would probably rise or fall in the future. This type of analysis usually took the form of a business barometer in which it was forecasted that business would be better in the next year or so, or that there

INTRODUCTION

3

might be a slump or a depression. Doubt as to the future and inability to forecast were cloaked in the words "outlook uncertain." What was meant by better or worse business conditions was not carefully defined or was left as a loose vague term. We had no way of telling how much better or how much worse business would be or whether this meant that the price of cotton would probably be 3 cents a pound higher or the price of copper be 2 cents a pound lower or that the cost of producing shoes would be 50 cents a pair higher. Furthermore the technique developed was not readily adaptable to the requirements of any particular business which might be called upon to make measurements of the data which immediately concerned it and were not or could not be studied with a general barometer. Therefore, the purpose of this study is to expound a methodology which is readily adaptable to any and all types of public and private business for their own private use, the data to be expressed in the terms of the units which the particular company supplies to the world. The methodology which will be developed is an outgrowth of cyclical measurement. In attempting to measure cycles it is immediately discovered that there are at least two and possibly three other components in a time series which are equally as important as cycles. This is unfortunate because it complicates the task, particularly since the three or four components are not of equal importance in a given series. In some series we find that the cyclical fluctuation is particularly important; in another series it may be the secular trend; and 2

4

AN APPROACH

TO DEFINITE

FORECASTING

in still another possibly the seasonal variation will be the prominent component. There is some evidence to indicate that in most series there are two components which are particularly important in determining what changes will occur in the data for the future and one component which is of comparatively little importance. 1 As the attempt to isolate cycles has led to the ascertainment of these other components there is a solid foundation upon which an approach to definite forecasting can be placed. The procedure will be to analyze a series into its component parts, namely, the cyclical fluctuation, the seasonal variation, and the secular trend (the secular trend may at times be subdivided into a short-time trend and a long-time trend) and from these component parts we shall attempt to synthesize the series for the immediate future. For example, if the probable position can be determined for the cyclical fluctuation of a series for the next year, and likewise the seasonal variation and the secular trend, then these three probable positions can be added together, and as a result there is a definite forecast of the original data for the next year. Hence, we see definite forecasting consists of making an analysis of a particular series for some time in the past and from this analysis attempting to discover the behavior of each of the components. From this behavior in the past along with any collateral information which may be considered significant each component is constructed for the future; the components are added together and the result is data which will probably materialize in the future. » See L. W. Hall, Banking Cyclts (Philadelphia, 1927), chap. 3.

5

INTRODUCTION

This process, of course, opens up the possibility of considerable error, and because of this we are particularly concerned with short-time predictions, usually not more than a year in advance and at the most not more than two years. In this way we can avoid to a considerable extent cumulative errors which otherwise might prove devastating. A long-time prediction for any series might be attempted where the secular trend of the series was the component of primary importance and where the secular trend was not erratic; but to attempt to forecast the cyclical fluctuation for any considerable length of time would be a highly dangerous procedure and likely be more a hindrance than a help. Therefore, long-time predictions are primarily trend predictions. If the cyclical fluctuation and seasonal variation are of great importance in a series, the type of long-time forecasting which this condition would force would not be of much use even though it were accurately done. Hence where we are attempting to forecast the movements of all components and where the seasonal variation or cyclical fluctuation or both are significant, we are essentially limited to short-time predictions. In this study we are not concerning ourselves with long-time predictions, but we are really trying to develop a method for short-time predictions. THE

EXPERIMENTAL

APPROACH

The procedure to be followed in developing methods for definite forecasting may be termed "experimental." In this connection we are using the word "experimental" in a rather loose manner in which it means

6

AN APPROACH TO DEFINITE

FORECASTING

" t o try out." In technical investigations we frequently come across the use of two terms, namely " i n d u c t i v e " and "experimental." B y "inductive" is usually meant the use of observed phenomena or data which is not subject to any control but which must be used just as it is found in nature. B y "experimental" is usually meant the use of data or phenomena which can be controlled at will, so that we can isolate the influence of the factor that is of particular interest. Usually these two terms are not always clearly distinguished and frequently the word "experimental" is used to cover both of these methods of approach because it is felt that what is wanted is something which works irrespective of how it is discovered. The word " experimental," as used in this study, will include both the ideas of induction and experimentation. An attempt is made to find something that will work, therefore, any means of testing its workability is "experimental." In the experimental work two types of series are used, namely, constructed series and observed series. The constructed series will be those series which will be built up by creating a secular trend and seasonal variation and cyclical fluctuation, and adding these known components to secure the original data upon which there is to be experimentation. Of course, the advantage of this type of series is that we know the components which we are using to make the original data and therefore we can test to some extent the ability of our methods to obtain these components. Also we can use increasingly difficult components so as to see how far we can go in presenting difficult situations

INTRODUCTION

7

before our methods will break down either partially or entirely. In thus learning, insofar as we can, where our methods are defective we may be able to discern why they are so defective and possibly at times eliminate the defects. The other type of series to be used is the observed series. These, as their name implies, will be series of data which have been observed, but the components of which are unknown. The purpose of using observed series is to enable the investigator to see how the methods are working after he has learned something of them from experimentation upon constructed series. The observed series were chosen primarily because they presented certain difficulties which were embarrassing in the attempts to analyze and synthesize them. The use of constructed series will help us greatly in developing our methods and will add considerably to our confidence in the ability of the methods to accomplish what is desired, but it must be kept in mind that the use of constructed series is by no means an infallible test and that the only final criteria of the methods are the actual results which are secured with them in specific problems of forecasting. And it is all too tempting to fashion the series so that it agrees with the theory upon which the methods are postulated. Seasonal variations might cancel out over a period of a year beautifully if that were the assumption of the method and the constructed series conformed to it. Trends might increase or decrease with regularity, and if it happened that this was the assumption of the method, the results would be perfect in the series constructed. The deception would

8

AN APPROACH TO DEFINITE

FORECASTING

out, however, when work was attempted in actual situations (observed series) unless the method, in truth, was infallible. We have tried to perpetrate no such juggling of numbers, rather have we constructed series with the purpose of making the methods fail if ever and whenever possible. I t has not been the desire to impart false values or to paint distorted and unreal pictures; constructed series and observed series alike have been selected for the difficulties which they present to the experimenter.

II T H E THE

DATA

CONSTRUCTED

SERIES

Series 1.—In this chapter the series which we are going to use in our experimental work will be given, and in the case of the constructed series the mythical components from which the original data is secured will also be given. The purpose in presenting all of our series in this one chapter is to make them convenient for reference when we cite them frequently in later chapters. T h e original data of this series—Constructed Series 1— was obtained from the following equation: y = 5000 + 100* + 100 sin ΤΓΧ/2 + 300 sin πχ/6. It will thus be seen that the series consists of a straight line trend with a constant positive rate of change of 100 upon which is placed a seasonal variation of one-year period which is disproportional to the trend, and finally upon these two components is placed a cyclical fluctuation with a period of three years which is also disproportional to the trend. The coefficients of the seasonal variation and cyclical fluctuation are so chosen t h a t the three components will have approximately equal influence in determining t h e movement of the original data. T h e original data, of course, is secured by adding together the corresponding items for the three components. This will be readily seen in Table 1 in which the data for one complete cycle is given. The data for all the 9

10

AN APPROACH

TO DEFINITE

FORECASTING

series, both constructed and observed, will be found at the end of this chapter. Series 2.—This series is exactly the same as Series 1 with the exception that the cyclical component has a two-year period. The data for this series will be found in Table 2. Series 3.—This series is like Series 1 with the exception that the cyclical component has a four-year period. The data for this series will be found in Table 3. Series 4.—This series is exactly the same as Series 1 with the exception that the cyclical component has a five-year period. The data will be found in Table 4. Series 5.—The equation for this series is y = 5000 + 100A; + ΛΓ2 + (5000 + 100A: + x2) .02 sin τχ/2 + (5000 + 100A: + A:2) .05 sin πχ/6. This series has a parabolic trend instead of a straight-line trend, and the seasonal variation and three-year cyclical fluctuation are both proportional to the trend. The data will be found in Table 5. Series 6.—The equation for this series is y = 5000 + 100A: + (5000 + 100A:) [.02 sin ττχ/2, .02 sin πχ/2 + 10°, · · ·, .02 sin vx/2 + 90°] + (5000 + 100A:) [.05 sin 7γ*/4, .10 sin vxj6, .07 sin πχ/8~]. The above equation shows that there is a straight-line trend, and the seasonal variation and cyclical fluctuation are proportional to this trend, but that the seasonal variation changes every year and the cyclical fluctuation changes every cycle for both amplitude and period. The data will be found in Table 6. Series 7.—The equation for this series is y = 5000 + [10Q*(3), 5Q*(2), 25*(2), 0(1), - 25*(2), - 50*(2),

THE

DATA

11

sin πχ/2 + 3 0 0 sin πχ/6. This series has a seasonal variation and cyclical fluctuation which are t h e same as Series 1, b u t t h e secular trend has d e p a r t e d considerably f r o m t h e straight-line form used u p t o this time. T h e significance of t h e secular-trend portion of t h e equation is t h a t starting with 5 0 0 0 , 1 0 0 * is used for 3 years, 50* for 2 years, and so on until finally — IOOAT is used for 3 years. T h e d a t a will be found in T a b l e 7. Series 8. T h e equation for this series is y = 5000 + [50*(2), 100x(l), 200x(l), 250χ(2), 300ΛΓ(1), 50X(1), - 300*(2), - 200Λ:(1), 100*(3)] + 100 sin πχ/2 + 300 sin πχ/6. T h i s series is t h e same as Series 7, with t h e exception t h a t we have an errative t r e n d rather t h a n t h e comparatively smooth trend of Series 7. T h e d a t a will be found in T a b l e 8. Series 9.—The equation for this series is y = 5000 + [200x(3), 400*(1), 100*(2), - 200λ;(1), - 500x(2), - 100ΛΓ(1), - 200ΛΓ(2), 100*(2)] + 100 sin πχ/2 + 300 sin ttx¡6. I t will be seen t h a t this series is the same as Series 8, except t h a t this series has a different t y p e of erratic t r e n d . T h e d a t a will be found in T a b l e 9. Series 10.—The equation for this series is y = 5000 + [100λ:(3), 25*(1), - 200*(2), - 25x(l), - 300*(3), 200*(2), - 300*(l), 25*(1)] + 100 sin πχ/2 + 300 sin τγχ/6. I t will be seen t h a t this series is exactly t h e same as Series 9 with t h e exception t h a t this series has a secular trend of a still different type. T h e d a t a will be f o u n d in T a b l e 10. Series 11.—The equation for this series is y = 5000 + 100* -f [200 sin πχ/2, 200 sin πχ/2 + 10°, 200 -

100X(3)] +

100

12

AN APPROACH TO DEFINITE

FORECASTING

sin τχ/2 + 20°, · · . ] + [400 sin τ*/10, 100 sin x*/4, 300 sin xx/8, 200 sin rx/6, 400 sin x*/10]. It will be seen that this series has a straight-line trend like Series 1, a changing seasonal variation similar to Series 6, and an erratic cyclical fluctuation in that the amplitude and period of adjacent cycles are considerably different. The data will be found in Table 11. Series 12.—This series has the same seasonal and cyclical components as Series 11, but instead of a straight-line trend we have the erratic trend of Series 10. The data for the complete series will be found in Table 12. Series 13.—In this series the secular and seasonal components are the same as in Series 11. However the cyclical component is more completely erratic than any of the preceding series. Here the first half of one cycle is hooked up with the second half of another cycle so that for a cyclical fluctuation this series has 400 sin 7r*/10 with 1000 sin ir*/4, 600 sin τχ\4 with 300 sin ΊΓχ/S, 600 sin ίγχ/S with 800 sin rx/6, 400 sin ττχ/6 with 200 sin π*/10, 400 sin τ*/10 with 1000 sin ΙΓΛΓ/4, and finally the first half of 200 sin irx/6. The complete data will be found in Table 13. Series 14.—This series has the same seasonal and cyclical components as the preceding Series 13, but it has the erratic trend of Series 10 instead of the straight-line trend of Series 13. The data for the complete series will be found in Table 14. T H E OBSERVED

SERIES

Series 1.—It should be understood at the outset that the observed series used in this study were chosen

THE

DATA

13

entirely because they represent some peculiarity of fluctuation which led to difficulty in trying to calculate their various components. Therefore, no significance should be attached to the data other than its significance as a series of numbers upon which to experiment. No attempt will be made to interpret in any way the meaning of the component parts of this series since we are interested only in our ability to calculate them, and not in discovering any relations between them or in making any interpretation concerning them. In order to keep our minds free from thinking of the data as anything other than numbers, we will henceforth always refer to the observed series by number, just as in the constructed series. For example, Observed Series 1 happens to be Loans and Discounts of National Banks from 1900 to 1925, but we are not concerned with this series as loans and discounts but only as Observed Series 1, which happens to be 26 years long. T h e series was chosen for experiment only because it represents certain considerable difficulties in the calculations of the secular trend. Table 15 at the end of this chapter contains the original data for this series. Of course, since this is an observed series, that is all the data that can be given. Series 2.—This series of data happens to be Pig Iron Production from 1903 to 1925. I t was chosen because it represents a certain amount of trend and cycle difficulties. T h e original data will be found in Table 16.1 ' T h e data for this series was secured through the courtesy of Dr. C. A. R . Wardwell.

14

AN APPROACH TO DEFINITE

FORECASTING

Series 3.—This series consists of the Liabilities of Failed Concerns from 1903 to 1925. It is important because it represents a high degree of both trend and cycle irregularity. The original data will be found in Table 17.1 Series 4.—This series consists of Stock Prices from 1903 to 1926 and represents some slight erratic element in the trend. The original data will be found in Table 18.1

Series 5.—This series consists of Vessels Commenced in the United Kingdom from 1913 to 1925. I t is interesting partly because of its erratic nature and partly for its aptness for a further analysis of secular trends. The original data will be found in Table 19.1 Series 6.—This series is the data for Anthracite Coal Production for the years 1912 to 1925 and is particularly significant because of the discontinuity which TABLE

I

CONSTRUCTED SERIES 1

1

2

3

Secular Trend

Seasonal Variation

5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100

0 + 100 0 -100 0 +100 0 -100 0 + 100 0

-100

Cyclical Fluctuation

Original Data

Original Data

Original Data

Original Data

0 +150 +261 +300 +261 + 150 0 -150 —261 -300 —261 -150

5000 5350 5461 5500 5661 5750 5600 5450 5539 5700 5739 5850

6200 6550 6661 6700 6861 6950 6800 6650 6739 6900 6939 7050

7400 7750 7861 7900 8061 8150 8000 7850 7939 8100 8139 8250

8600 8950 9061 9100

4

5

6

7

8

9

10

THE

15

DATA

occurred in 1922 because of strike conditions. T h e original data will be found in T a b l e 20. 2 Series 7.—This series represents the Overdrafts of National B a n k s from 1901 to 1925. It is significant for the purposes of experiment because it has an erratic trend and cycle, but particularly because it has a prominent seasonal variation which is also inclined to be erratic. T h e original data will be found in T a b l e 21. Series 8.—This series represents the C a s h in Vault of National B a n k s and extends from 1901 to 1925. I t was chosen because of its generally erratic nature. T h e original data will be found in T a b l e 22. Series 9.—This series represents the Loans and Discounts of Federal Reserve Banks from the years 1916 to 1925. It was chosen particularly because of its unusually great trend fluctuations which tend to completely submerge certain of the cyclical fluctuations. T h e original data will be found in T a b l e 23. TABLE

2

CONSTRUCTED SERIES 2 Secu- Seasonal lar VariTrend ation 5000 S100 5200 1

5300 5400 5500 5600

2

5700

Cyclical Fluctuation

Original Data

Original Data

Original Data

Original Data

Original Data

0 + 100

0 +212

5000

5800 6212

6600

7400 7812

8200

5412

0 -100

+300 + 212

5500 5412

6300 6212

0 + 100

0 -212

5400 5388

6200 6188

7000

7800

8600

6988

7788

0 -100

-300 -212

5300

6100

6900

7700

8588 8500

5388

3

4

6188

7012 7100 5

6

7012

6988

7

8

7812

7788

• T h e d a t a f o r this series w a s secured through the c o u r t e s y Brumbaugh.

8612

7900

8700 9

10

8612

8588

of D r . M . A .

16

AN APPROACH

TO DEFINITE

FORECASTING

Stries 10.—This series represents the Value of Coffee Imports per Pound from 1900 to 1925, and is used primarily because of the nature of its secular trend and cyclical fluctuation. The original data will be found in Table 24.« Hereafter in later chapters these observed series will be mentioned only by number so as to render them entirely apart from any interpretation. Their origin was mentioned here only in order to indicate the source of these observations. We could just as well think of any of these series as being fluctuations in costs per unit or volume of sales or volume of production or TABLE 3 CONSTRUCTED S E R I E S 3

1

2

3

4

Secular Trend

Seasonal Variation

Cyclical Fluctuation

Original Data

Original Data

Original Data

5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500

0 + 100 0 -100 0 +100 0 -100 0 +100 0 -100 0 +100 0 -100

0 +115 +212 +277 +300 +277 +212 +115 0 -115 -212 -277 -300 -277 -212 -115

5000 5315 5412 5477 5700 5877 5812 5715 5800 5885 5788 5723 5900 6123 6188 6285

6606 6915 7012 7077 7300 7477 7412 7315 7400 7485 7388 7323 7500 7723 7788 7885

8200 85 IS 8612 8677 8900 9077 9012 8915

5

6

7

8

9

10

•The data for this series was secured through the courtesy of Mr. W. H. Blaisdell.

THE

DATA

17

overhead expenses or prices of individual articles, or in fact any series of data which occurs in the actual world of business. It will no doubt be noticed that all of these observed series are of an erratic or irregular nature. The reason for this lies in the fact that knowledge concerning the effectiveness and capacity of our methods can be best secured from working with such data. It will become apparent later in connection with the analysis of constructed series that our methods will handle quite easily and accurately all series whose components are regular, and it is only when irregularities and eccentricities appear that our troubles begin. Therefore it would not seem to be of particular significance to spend our time working with series of data which are comparatively easy to analyze. Indeed, we might even thereby delude ourselves as to our ability to analyze and synthesize other forms of data which are highly important but not nearly so obligingly regular. We have in the more difficult constructed series and in these observed series about the most difficult kinds of data upon which to attempt any form of analysis. We can, therefore, feel confident that such results that we can get with these series probably represent our poorest results and that in connection with the use of our methods on particular problems we shall frequently get better results.

18

AN APPROACH

TO DEFINITE

TABLE

4

CONSTRUCTED S E R I E S

1

2

3

4

S

Secular Trend

Seasonal Variation

5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 6900

0 + 100 0 -100 0 + 100 0 -100 0 +100 0 -100 0 + 100 0 -100 0 +100 0 -100

FORECASTING

4

Cyclical Fluctuation

Original Data

Original Data

0

5000 5293 5376 5443 5685 5900 5885 5843 5976 6093 6000 5907 6024 6157 6115 6100 6315 6557 6624 6707

7000 7293 7376 7443 7685 7900 7885 7843 7976 8093 8000 7907 8024 8157 8115 8100 8315 8557 8624 8707

+ 93 + 176 +243 +285 +300 +285 +243 + 176 +

93 0

- 93 -176 -243 -285 -300 -285 -243 -176 - 93

6

7

8

9

10

SeaSecu- sonal lar Varia1 Trend tion

3

5000 5331 5430 5468 5652 5773 5636 5490 5609 5802 5835 5941

Cycli- OriRcal inal Fluc- Data tuation

6344 6469 6596 6725 6856 6989 7124 7261 7400 7541 7684 7829

Secular Trend

Seasonal Variation

7976 8490 8636 8681 8957 9135 8900 865*3 8823 9108 9140 9288 9896 10069 10244 10421

Secular Trend

Cyclical Fluctuation

Seasonal Variation

— O Ν O

+

O

Cs OJ

c-J O

+ O Φ

r^ \o

O — ' O

+

o _ o + —^ Ν S

Os

—

o

I

r^i

Ο O

O O O O · — I

1

ι—1 0

7976 8125 8276 8429 8584 8741 8900 9061 9224 9389 9556 9725

Cycli- Origcal inal FlucData tuation O

-H

^t"

00 Ό ΙΛ O 0 0 Γ-4 r s ^O Γ«-ί

+ + + + +

Ο ·—1

(Ν η

+ -182 -322 -377 -334 -196

6344 6760 6883 6927 7154 7304 7124 6934 7078 7315 7350 7476

SeaSecusonal lar VariaTrend tion Ο

+ 162 + 287 +336 + 298 + 175

0

Cycli- Origcal inal Fluc- Data tuation

9896 10522 10690 10734

Original Data

THE DATA

vO — ty-i ^ e·** Ν ·ψ Ι Λ + + + αο O Ν

I

o

Ν ι - (> (Ν Q Ö - I

+ + + + +

oo oo on O — ' O — ' O — ' O * -

ISÏ

—ι

oo

+ I +

OO

— o — o—· o — o — 1 + 1 + 1 Q -t"

^ LO

u->

^

^ ·—«

1

W ^

I I I I 1 I

O

1--.

\0

ΙΟ »Λ "O U-) O , ^O U-)

1 1 1 1 1

Ο c^

-Φ o-J

+

1

OO Ο

Ν

m

TO DEFINITE

^ Ο* Ν ft & Q Ν Λ Η Ν O ^ "ΙΙΟ'ί'ΝΟΜ'ψΜ

+ + +

8200 8300 8400 8500 8600 8700 8800 8900

I I I

Seaaonal Variation

I

+ +

ΙΛ ÛO (Λ Ν V O CS Ό fN Ν Ν ^ Ο ^ Ο

1 1 +

ι

Cyclical Fluctuation

Original Data

ο,

Secular Seaaonal Trend Variation

FORECASTING

N N O O N S O O N NSNOOOOwwOl

Secular Trend

Cyclical Fluctuation

Original DaU

AN APPROACH

ο

N O O N W i ^ u l f t ^ N n r t O O N O N H H H T l ' r t n ^ ^ Ô Ô Q Ù O Û O s S ^ W ν Ο Ό ^ Ο ί Ν Ν Ν Ν β β Ν Ν Ν Ν Ν Ν

O » O rN QG (— C-J ΟΟ η (Ν — ϊ Ο "Φ ΙΟ Ν ^ Ο) Ν Ο *-< σ\ ΟΊ »ΛνΟίηΐΛΟ'-'Μ^'ΙΛ^ΜΝΟΜΜίΛ

I

I I I

+ + + + + + +

VÛO00tntAN00V)VO^9\N00U>(SN Φ Ο Ο Ο Ο Ο Ο ^ Ο Ν Ν Ν Ν Ν ^ ΐ Λ ί Λ ν ) ^Η ^ ^Η P H

+ + I I + + I I + + I I + + I I 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 fr*·»

00

O Û O O N H H O O e ^ O w O N O f S W ^

Cyclical Fluctuation

Original Data

so

ΟΟ O >Λ

Ν Φ ΟΟ + + + »Ί

»Λ

Ν ^ (Λ ρη rs|

+ + + + +

1

^ Ό Ο Ν ^ Η Ν Ν ^ Ν Ν ^ Ο Ο ^

+

Ν Ο

Ή

1+ +

H

P H

1 1+ +

^

M

1 1+ +

P H

1

5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500

Secular Seasonal Trend Variation

I II

H

Ν

!Λ

1

21

Q 0 - 0 - > 0 0 0 f f i 0 < i 0 Ο

O LO Wì LO LO Lo

->f -»i*

888888888888

Q ^ O O N ' O i f l T W N H O w φ Ι Λ ΙΛ ΙΛ IO IO IO IO t o vo Ό T f

5989

6000

5850

O Q O O Η Η NO NO fS

Original DaU

-H

I O Q ^ O Q ^ O O O ' ^ O ^ Ì ' O

Secular Trend

©

6050

6350

6650

6711

6700

6786

» o O * o Q O O O Q O Q ^ V O C ^ O I O O O O O O ^ Ι Λ ί η ΐ ^ Ι Λ ^ ^ Π Μ Ν Ν \Q sû Ό NO NO Ό Ό NO

6800

6575

IO

Original Data

^

Secular Trend

to

5889

Secular Trend

Original Data

THE DATA

L O O ^ O O O O O O ^ O O O r ^ v o f N . O Ö O O O t - ^ ^ o i ^ o ^O NO * 0 NO NO NO NO NO NO

NO ^O ^

6314

6350

5700

5739

5850

6000

6100

5600 5600

5900

5750 5500

5539

5661 5400

5800

5500 5300

5450

5461 5200

NO

5700

5000

5350

5000

5100

Original Data Secular Trend

lo

6600

6575

6450 6450

6350

6650 6400

6550

6611 6350

6264

6500 6300

6525

6511 6250

6250

6150

6450

6150

oo

6200

Secular Trend

Original Data

r-»

NO NO NO NO

6500

NO NO Ό

0 \ 9 \ 00 00

Secular Trend

©

ί Λ Ι Λ ί Λ Ι Λ Ο ^ ί Ο ^ ^ Ό ^ 1 ^ ο ο - ^ Ν χ ο ο ^ Λ Ό η Ο Ν

oooooooooooo

Secular Trend

Original DaU

r>.

oo

os

O O ^ O ^ Q Q Q O O O Q lOvO^'ONNNNNNOOOO O O O O O O | O Oι / O O O O νι«Λ«ΛΐοΟ Λθ ι0'Λθ»ο Ο Ν ^ ^ Ι Λ Ο Ρ Ο ^ ^ Ο Ο Ο Π ^ lO^^^VONNNNOOOOOO

Original DaU

sooo 5300 5361 5350 5461 5500 5300 5100 5189 5350 5389 5500

Ό

Secular Trend

LO

o oooo o oÖo oooo Q U-Ì Ο Ο WÎ ΙΛ V) l/t UÌ ^

Μ

ΡΊ

Original Data

5000 5450 5661 5800 6061 6250 6200 6150 6339 6600 6739 6950

Secular Trend

5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 7200

Original Data

7600 8250 8661 9000 9161 9250 9100 8950 9039 9200 9239 9350

Secular Trend

7600 8000 8400 8800 8900 9000 9100 9200 9300 9400 9500 9600

Original Data

9400 9450 9261 9000 8561 8050 7300 6550 6039 5600 5039 4550

Sccular Trend 1

Original Data

4700 4850 4761 4600 4461 4250 3800 3350 3139 3000 2739 2550

Secular Trend

4700 4600 4500 4400 4200 4000 3800 3600 3400 3200 3000 2800

2900 3000 3100 3200 3300 3400 3500 3600

Secular Trend

2900 32SO 3361 3400 3S61 3650 3500 3350

Origina] Data

THE DATA

O

f-»

— ·<
0 t r > 0 0 0 0 0 0 0 0 0 i h h h n q o O ^ Ö V N O O O ' Ö

NO

- ^ r ^ - ^ r ^ t ^ T f L o h ^ O r ^ — < — H . — I l — 1 r—l ι—1 •—' ì —

Ν

^

vy"i

o o o ^ ·**« DO δ ® Ν « Ν r ^ m r - í O O Q o o c l r ^ c - ι n í n w M H O - · i c - l m f i ^ ' f t í ' i c ^ · — ι

+ + + + + +

I I I I I I I I

I

oo

\o

Original Data

SO Χ , —

(

+100 -173 -100 +173 + 68 -188 - 68 + 188 + 35 -197 - 35 + 197 0 -200 0 +200

Seasonal Cyclical VariFluctuation ation

Original Data

THE DATA

o

r ^ a o - ^ O o o r M ^ o a o f - Q f ^ o o — f">oo\ — ' O f Φ

Ο

Ο

Λ

Ν

Μ

η

Ο

Ο

^

^

Ο

Ν

Ο

Ν

Ν

Ν

Ν

Ν

Ν

ρ

Ν

Seasonal Variation

-300 -277 -212 -115 0 +100 + 174 + 200 + 174 + 100 0 -100 -174 -200 -174 -100 0 + 124 +235 +324

Cyclical Fluctuation

0 0 0 0 0 0 0 \ < 7 N O C ? \ 0 S O 0 N 0 \ O O 0 N O O O © O — «

^ ί Λ Ο Ν Λ Χ Ό Ο Ο ^ ΰ Ν Ο Ν Ο ' Λ Ν ^ Ν Ν ί Λ Ν ί ο

+ ι 7 +

+ ι 7 + + 7 7 + + 7 7 + Ϊ 7 7 +

Seasonal Variation

Cyclical Fluctuation

Original Data

1—
« N O N ' Ο

-H O Ν» Ο Ν«—«

+ + +

Ν Ν » η »-ι Ν

III

ο Ο

Ν Ν

(Ν

ΙΛ Ν Ν t ι N«.

+ + + + + + +

III

Ι Λ Ν Ι Λ Ν Ν Ο Ν Ο Ο Ο ^ Ο Ο Ό Ο Ν ^ ^ ^ Ο _

N

Q ΙΛ

N

W

N

Ν

H

f

t

^

N

O

oo

H

t/i i o i v i u-i

N

W

N

O

O

t

f

^O

O

M

N

r

i

O O CM ©

+

f ^

N

+ + + + + + + + +

O

i

l

O

Ö N

O

+ + Ι Ι + + Ι Ι + + Ι Ι + + Ι 1 +

Cyclical Fluctuation

Original Data

t-O

N N N N N N N N N C O O O O O O O O O O O O O O O O O O O O O

*0

Seasonal Variation

•·«*«

O

I

C?\

^

»-o O

' Ό

í

' ^

N

N

O

^

vo Ό

N

W

N O

N W N O O Q C O N W N

I I I I I I I I

1

O W N W N O O O O O O O O O ^ O ^ O ^ W ^ W Ο ^ ^ ^ ^ ^ Μ ^ ΰ Ο Ο υ Ν υ Ν Ν ί Λ Ν ΐ Λ 1—« « Ι , Μ

1+ + 1 1+ + 1 1 + + 1 1+ + 1 IH

Μ

ν »

1

es

co —

Is2286

2300

2768

2132

2100 2500

2088

1900

•»•


\θ

.—ι

es

m

Original Data

Original Data

^

Original Data

AN APPROACH TO DEFINITE

o o N N ^ o o p q o W n f S ^ N H N H

Original Data

Ν

Λ

Original DaU

W Ν

Ν O «

O Ν

^ Ν

Ν

fS ^ Ν

^ΟΟΝ^ΟΟννΟΟ^ΟΟ-Φ

NO Original Data

FORECASTING

fS Μ

r-» ^^

οο

Η Η ί Λ Ν η π Ν ^ )

O O ^ O N ^ O s C N r s ^ f O f ñ ^ v¿

S

Ξ

(s.

ΟΟ

ON

WJ

vO

«Ν

m

Original DaU

Original DaU

Original DaU

Original DaU

U1

Original Data

7575

8672

8384

7602

164

iH

8695

36

7096

Original Data

«η ••H

151

8966

7898

7176

6604

7S49

6906

DATA

7428

Original Data

THE

• - ^ Ο Ν Ο Ο Ι Λ Ι Λ Η ^

O

O

O

O

N

N

S

N

N

i 0 ^ N

O

O f l û ^ N ^ ^ \ 0 0 V ) O O N Ç i Ο Ο Ν Ο Ο - Ί Λ Ο Ο O O O N O U Ì N O O

O N Ο N

9058

8933

ΟΟ

6688

7790

7190

7212

7404

Original DaU

ί^-

Original Data

U->

Original Data

8453

Original Data

O

SO

N O O O O O O n N ^ ^ » Λ Ο Ο Ν Ν ν ι Ν Ν Ο Ο

Π

Ν

Ν

Ν

Ο

Ν

Ό

Ό

Μ

^ H i O t ^ C N t n C T v ^ S Ö

>

CS

Original DaU

42461 26800 31777 54941 43749 30367 29905 54473 47256 30034 32475 53735

Original DaU

Original Data

Original Data

22307 19006 18377 27460 21838 15485 17142 18797 7047 5905 5060 7211

Original Data

Original Data

13881 12421 15131 23116 19215 16406 17545 19277 12360 10770 12355 9949

Original DaU

Original DaU

11410 9352 14900 10554

Original Data

AN APPROACH TO DEFINITE

ΙΛ es

c-í Ν

ο —·

\ο

m

r-·.

FORECASTING

vû r-i

^

O

CO

es

S «Ν»

ΙΛΟΟ-ΝΙΛΝΟΟΟ^'ΛΝΝ ^Osrsf-i^Oc^OOOrMON r*-> Ν Ν

ο r--i — r^

Ο ΐ ό ι Μ ^ ν β Α Ο Λ ο ο Λ Ο " -

οο

^

ONVOO^W-^ONOvOOI-I^I t ^ M f S ^ ^ N f S m r s ^ i f ^ r j

r^

oo

NO

O O O ^ W N N ^ W t n S N i - i

co

Original Data

THE

DATA

35

Ν Q Ν O VO Φ NO ^ FL F> O

Original Data

ΙΛ Ν

fi

ΓΛ t o Γ Ί t*1 ( Ν t o t o tO t o t o

«φ t o

to Ν

^

4

Ό

Is«·

to to ^ E*·»

OO

» Ό ^ ^ Λ Η Ν Η Χ Ο Ο Ο - Ν

1

Original Data

Original Data

Original Data

rs» Ν

: Original 1 Data

CO

^t*

LO

c o r - í c o O — O s PV» ^ IR» Ν Ό ^ Ο Ο Ο Ο Η Ο ^ ^ ^ Ο O O O O O O O O O O O\ C> OÑ O\ ^

Original Data

O

η Ο ν ' Λ Ή Ό Ο Ο ^ Ο η Ο η ' φ ί Λ ( • ^ • - « Γ Ο Ο Γ ^ Ο Ο Ο Ο Ο Ο ^ Ο ' Φ N N N N O O O O » O O ^ O \ O Û O O

OO

c s N o o O N O O ' f n o o ^ n n ^ Û O O O N O N ^ W F F I N W I V Û

^

u»>

\o

»-«

CM

fO

Original Data

Original Data

f**

Original DaU

TO DEFINITE

FORECASTING

Ν Ο Ο Ο Ο Ν > Λ Ν » Λ Ν O O ' t N ' t O O N O

Original Data

o\

2

^ • ^ N n O O a ' t O N

Original Date

Ν.

00

» O t O N ^ ' t ^ V Û W ' Û H N n N l O ^ O O C A O S O ^ I ^ ^ Í N

Original Data

to

Original Data

AN APPROACH

H

»O

«

Ν

Ν

Ν

Ν

f S

w

Η

»-» Η

Ν

\ û

ft

Ν

^ G «

"Λ

Ό

NO

ρ—1

a s o

37

DATA

Γ-1 0 0 e» o r^J Ν

NO C S O ·*· CS CS CS CS e s CS CS CS psl i o

rs CS

t-. CS

NO

•Φ

ON

ON

ON

O

ON

Ν o o •Φ t O C S •»t· r o m C S C S o o LO SO CS CS CS CS

O CS

ON

. S 3 M

x Q O

Λ

(n

*o

IO

NO

ON

ON

ON

00

O o

PS (Ν t o O O O

O

O

M rt •cQ O

O ·*> c o

•>!>

00

00 t^

ON

©

O

oo

00

SO 00

to

'Φ

00

00

00

Ι Λ CS *

IO

t^

fn

LO «o

I

Original Data

ON

NO

o\

- - -

CN

Í»

Ν o

f·*

NO

"RT E 2

CS

11.8

oo

CS

n.i

«

12.6

to

10.4

a

14.5

Ό CO m

(N rs

•CD O

14.2

12.8

11.8

10.6

11.9

-