San Fernando Valley State College
Convolution Integral Analysis of Economic Time Series
(�sing a Regional Business Activity Index and News­
paper Help Wanted Advertisements as· an Illustration)
A thesis submitted in partial satisfaction of the re ...
quirements for the degree of Master of Science in
Business Administration
by
Bernard Myron Van Emden
January, 1967
The thesis of Bernard Myron Van Emden i� approved:
San Fernando Valley State College
January, 1967
ii
DEDICATION
This thesis is dedicated to my Wife, Jacqueline, whose
patience and encouragement contributed greatly.
ACKNOWLEDGEMEN T
I wish to acknowledge the assistance of the Security
First National Bank in the acquisition of the source
data for this thesis.
Bernard Myron Van Emden
iii
TABLE O F CONTEN TS
Page
Title Page .
.
.
Approval Page
ii
Dedication Page .
iii
Abstract
1
Chapter
I
Chapter
II
Introduction .
The Data .
.
-2
.
4
A.
The Index of Business Activity.
B.
The Help Wanted Advertisen:tent Indexes
14
c.
The Numerical Data . .
16
.
4
Chapter I I I
The Convolution Integral Cross Correlation .
22
Chapter I V
The Analysis
28
Chapter
Conclusion
46
V
Bibliography
48
LIST OF TABLES AND FIGURES
Page
Table
1
Index of Business Activity in Southern
California .
. . .
.
.
17
.
.
•
.
.
.
.
Table
2
Number of Lines of Help Wanted Ads Index
18
Table
3
Number of Help Wanted Ads I':ldex .
19
Figure
4
"Raw" Ads Data .
20
Figure
5
Indexes With Trend Lines
21
Figure
6
Examples of Integral CalculatioJ?­
25
Figure
7
Typical Program Flow Chart
31
Figure
8
Positive Delay Program
32
Figure
9
Negative Delay Program .
33
Figure
10
Value of Cross Correlation With Trend . .
34
Table
11
Index of Business Activity in Southern
California (Without Trend)
. . .
35
.
.
.
.
.
•
.
Table
12
Number of Lines of Help Wanted Ads Index
(Without Trend) . . . . . . . . . . . .
36
Table
13
Number of Help Wanted Ads Index
(Without Trend)
37
Figure
14
Data Without Trend
38
Figure
15
Value of Cross Correlation Without Trend
40
Figure
16
Statistics From Correlation Computation
Using 18 Years Data . . . . . .
.
42
.
Figure
17
.
.
Statistics From Correlation Computation
for First 14 Years Only . . .
43
.
.
Figure
18
Comparison of Correlation Methods
45
Figure
19
Business Activity Index . .
45
.
. . .
ABSTRACT
Convolution Integral Analysis of Economic Time Series
(using a Regional Business Activity Index and News­
paper Help Wanted Advertisements as an Illustration)
by
Bernard Myron Van Emden
Master of Science in Business Administration
January, 1967
The convolution integral cross correlation was originally developed
to analyze electronic signals. Its application to the analysis of economic
This thesis describes the cross-
variables has not been explored.
correlation and shows the effect of a linear trend on the integral calculation.
The technique is applied to the analysis of the time relationships,
leading or lagging, between help wanted advertising and an index of business activity. Using cross correlation, it was determined that an increase in business activity occurred a delay of one month after an increase in newspaper help wanted advertisement. A cycle period time
of 43 months was exhibited by each of these variables. This result is
also contrasted to the resu].ts obtained using the well-known Pearsonian
product moment correlation.
1
CHAPTER I - INTRODUCTION
Cross correlation involving the convolution integral has been
used extensively in the processing of radar signals to determine the
time delay (leading or lagging) between the signals transmitted and
received.
1• 2
Use of this equation in the analysis of economic vari-
able has not been exploited.
The purpose of this thesis, therefore, is to demonstrate the applicability and usefulness of the convolution integral in the determination of lead-lag relationships between two economic variables.
In order to demonstrate the power of this data processing technique, the cross correlation between an index of Regional Business
activity in the Southern California area and help wanted advertising
indexes in the same area will be analyzed.
Employers advertise in newspapers for additional workers.
Eventually these people are hired and put to work producing goods and
and services for sale.
This process gives rise to an increase in the
index of general business activity.
An increase in business activity
1.
Y. W. Lee, T. P. Cheatham, J. R. and J. B. Weiner,
Application of Correlation Analysis to the Detection of Periodic
Signals in Noise, Proc. IRE, Vol. 38, pp 1 165-117 1, 1960
2.
H. R. Raemer and A. B. Reich, Correlation Devices Detect
Weak Si�nals, Electronics, Vol, 32, No. 2 1, pp. 58- 60,
May 22, 1959.
2
3
does not occur immediately but some time lag later.
As an illustration
of the usefulness of the convolution integral cross correlation, the
typical time lag will be found.
If the index of business activity does follow the help wanted ad­
vertising in time, then observation of the variations in advertising can
be used to predict changes in business activity.
CHAPTER II - THE DATA
In order to show the usefulness and applicability of the convolution
integral, it was decided to devise and perform a demonstration test.
The
test performed was to determine the time delays between the number of
lines of help wanted advertisements in the Los Angeles area newspapers,
the number of help wanted advertisements and the index of general business activity in the Southern California area.
will describe the data used in this test:
The following paragraphs
its history, sources, problems,
and weaknesses.
A.
The Index of Business Activity.
Calculation of business activity indicators occurred as early as
1910.
3
The early indicators were inaccurate.
The gathering of data
and the subsequent processing required a great deal of time and energy;
consequently, the results were not current.
The U. S. Government entered into the field of business indicator
4
calculation it;1 varying degrees between 1918 and 1927 eventually per.
fecting its system of national accounts, including gross national product
data.
3.
Melvin T. Copeland, ''Statistical Indices of Business Condit
ions"
Quarterly Journal of Economics, Vol. 29, pp. 522-562,
May, 19i5.
4.
U. S. Department of Commerce, Office of Business Economics,
National Income, 1954 Edition, pp. 59-60.
4
5
From that time to the present many statistics have been included
in business activity indexes and indicators.
The following is a partial
5 6 7' 8• 9
list of these statistics: • •
1.
Population
1 1.
Copper production
2.
Value added
12.
Stock market conditions
3.
Disposable income
13.
Price of iron billets
4.
Imports
14.
Bank reserves
5.
Exports
15.
Circus business
6.
Railroad gross earnings
16.
Construction awards
7.
New buildings
17.
Cotton prices
8.
Money rates
18.
Auto production
9.
Bank clearings
19.
Demand deposits
Business failures
20.
Crude oil production
10.
By 1952, in excess of 450 measures of business activity were listed
by Cole.
10
About that time the electronic digital computer came into
being; consequently, by 1958, indexes describing many more variables
were available.
5.
J. H. Brookmire, "Financial Forecasting", Moody's Magazine,
January, 19 14, p. 8, 19 13 , p. 444.
6.
E. C. May "Circuses 'as Business Barometers", Literary Digest,
Vol. 17, pp. 40-42, March 24, 1934.
7.
Time, Volume 3 4, December 25, 1939, p. 40
8.
Business Week, Various issues including July 4, 1936, p. 3 0,
November 6, 1943, p. 14, and June 5, 1948, p. 2 1.
9.
Business Week, August 7, 19.61
10.
Arthur H. Cole, Measures of Business Chan�e. Richard D.
Irwin, Inc., Homewood, Ill, 1952.
6
Most general indexes of business activity are prepared in the
following fashion:
1) Determine what variables are to be included.
2) Acquire the raw data .
3)
Process the data to insure comparability.
This processing
must eliminate the effects of calendar and season variations.
4)
Weight the data in terms of value adc;led.
The index used was provided by the Research Department of the
Security First National Bank.
11
This monthly index covers business
activity in the eight (8) Southern California Coun.ties:
Los Angeles,
Orange, San Diego, San Bernardino, Fresno, Kern, Santa Barbara,
and Ventura.
It is prepared using data derived from many sources
combined and processed as shown in the following August, 1962
description:
1 1.
Monthly Summary, Research Department, Security First National
Bank, Box 2 097, Terminal Annex, Los Angeles 90054.
DESCRIPTION OF COMPONENT SERIES OF
REVISED INDEX OF BUSINESS ACTIVITY IN
SOUTHERN CALIFORNIA
1957- 1959 = 100
1.
Bank Debits in Los Angeles
a.
Based upon debits to demand accounts in Los Angeles City,
as reported by the Research Department of the Federal
Reserve Bank of San Francisco.
Data for years prior to
1952 imputed from total debits by applying constant factor
(average ratio for year 1952) of 9627.
b.
Two-month moving totals computed, with the totals listed
for the more recent of the two months in all cases.
c.
Adjusted to daily average basis, using number of business
days.
2.
d.
Seasonally adjusted.
e.
Put on 1957-1959 base.
f.
Percentage weight in index:
10.
Bank Debits in Residential Cities
a.
Based upon debits to demand accounts in six cities (Long
Beach, Pasadena, Glendale, San Diego, Santa Barbara, and
Santa Monica) .
Glendale debits are reported by the local
branch of Security First National Bank, since they are not
reported by the Federal Reserve Bank of San Francisco.
Long Beach, Pasadena, San Diego, Santa Monica, and Santa
Barbara debits as reported by the Federal Reserve Bank.
Data for years prior to 1952 imputed from total debits by
applying constant factor (average relationship during 1952
for each city).
7
8
b.
Two-month moving totals computed (for all six cities
combined) , with the totals listed for the more recent of
the two months in all cases.
c.
Adjusted to daily average basis, using number of business
days.
3.
d.
Seasonally adjusted.
e.
Put on 1957- 1959 base.
f.
Percentage weight in index:
5.
Bank Debits in Agricultural Cities
a.
Based upon debits to demand accounts in five cities (Bakers­
field, Fresno, Riverside, San Bernardino, and Santa Ana) ,
as reported by the Federal Reserve Bank of San Francisco.
Data for years prior to 1952 imputed from total debits by
applying constant factor (average relationship during 1952
for each city) to debit figures as reported by the Federal
Reserve Bank, except for Santa Ana, which was not reported
before 1952.
Figures prior to 1952 for Santa Ana were im­
puted from total debits, as reported by the Business Men's
Association of Santa Ana.
b.
Two-month moving totals computed (for all five cities com­
bined) , with the totals listed for the more recent of the two
months in all cases.
c.
Adjusted to daily average basis, using number of business
days.
d.
Seasonally adjusted.
e.
Put on 1957- 1959 base.
f.
Percentage weight in index:
5.
9
4.
Department Store Sales
a.
We use the seasonally adjusted index of department store
sales in the Los Angeles area, -as reported by the Federal
Reserve Bank of San Francisco.
The basic data are adjusted
by the Federal Reserve Bank for the number of trading days
in each month, and then adjusted for seasonal variation.
The
index is reported on a 1957-1959 base.
b.
5.
Percentage weight in index:
15.
Building Permits Issued
a.
Based upon total dollar valuation of building permits issued
in the unincorporated areas and incorporated cities of each
of the fourteen southernmost counties of the State. All
statistics are obtained directly from local building departments.
b.
Six-month moving totals computed, with the totals listed for
the most recent of the six months in all cases.
c.
Adjusted to daily average basis, using number of business
days.
6.
d.
Seasonally adjusted.
e.
Put on 1957-1959 base.
f.
Percentage weight in index:
10.
Engineering Construction Contracts Awarded
a.
Based upon the total dollar valuation of engineering construc­
tion contracts awarded in the fourteen southernmost counties
of the State.
These statistics are compiled from the Daily
Construction Reports (Green Sheet), published by the lles­
Ayars Publishing Co.
10
b.
Twelve-month moving totals computed, with the totals listed
for the most recent of the twelve months in all cases.
c.
No adjustment for number of da_ys or seasonal variation,
since twelve-month moving average makes this unnecessary .
7.
d.
Put on 1957-1959 base.
e.
Percentage weight in index:
1.
Real Estate Sales Activity
a.
Based upon the number of deeds recorded each month with
the Los Angeles County Recorder, as reported by Realty
Tax and Service Co.
b.
Two-month moving totals computed, with the totals listed
for the more recent of the two months in all cases.
c.
Adjusted to daily average basis, using number of business
days.
8.
d.
Seasonally adjusted.
e.
Fut on 1957-1959 base.
f.
Percentage weight in index:
4.
Employment in Motion Picture Production
a.
Based upon the number of production and related workers in
motion picture production in the Los Angeles area, as re­
ported by the Division of Labor Statistics and Research of
the California Department of Industrial Relations.
b.
Seasonally adjusted.
c.
Put on 1957-1959 base.
d.
Percentage weight in inc!ex: 2.
11
9.
Employment in Manufacturing
a.
Based upon the number of wage and salary workers in manu­
facturing in the Los Angeles Me_tropolitan area, as reported
by the Division of Labor Statistics and Research of the
California Department of Industrial Relations.
10 .
b.
Seasonally adjusted.
c.
Put on 1957- 1959 base.
d.
Percentage weight in index:
8.
Man-hours Worked in Manufacturing
a.
Based upon the number of manufacturing production workers
_
in the Los Angeles area multiplied by the average number of
hours worked per week in manufacturing.
Data obtained from
the Division of Labor Statistics and Research of the California
Department of Industrial Relations.
11.
b.
Seasonally adjusted.
c.
Put on 1957-1959 base.
d.
Percentage weight in index:
10.
Industrial Power Used
a.
Based upon the total kilowatt hours of power sold to industrial
users in Southern California in areas served by ( 1) the
Department of WAter and Power of the City of Los Angeles,
and (2) the Southern California Edison Co., as reported
directly by the companies.
b.
Two-month moving totals are computed for the two organi­
zations combined, with the totals listed for the more recent
of the two months in all cases.
c.
Seasonally adjusted.
12
12.
d.
Put on 1957-1959 base.
e.
Percentage weight in index:
14.
Petroleum Production
a.
Based upon the daily average number of barrels of crude
petroleum produced in California, as reported by the
California Conservation Committee.
Total California
figure is used, since the fourteen southernmost counties
account for virtually all the State's production, and since
current data are not available on a county basis.
13.
b.
Seasonally adjusted.
c.
Put on 1957- 1959 base.
d.
Percentage weight in index:
2.
Railroad Freight Volume
a.
Based upon the number of railroad freight cars loaded and
unloaded in the Los Angeles Metropolitan Area, as reported
by the Pacific Car Demurrage Bureau.
Since figures for
194 7 were not available on this basis, they were im.puted
from a similar series for a more limited area ( 10 Los
Angeles stations), using average relationship during 1948-53.
b.
Two-month moving totals computed, with the totals listed for
the more recent "£the two months in all cases.
c.
Adjusted to daily average basis, using number of working
days at yards.
d.
Seasonally adjusted.
e.
Put on 1957- 1959 base.
f.
Percentage weight in index:
6.
13
14.
Telephones in Use
a.
Based upon the total number of telephones in use in Southern
California in areas served by the Pacific Telephone & Tele­
graph Company and General Telephone Company, as reported
directly by those companies.
b.
Seasonally adjusted.
c.
Put on 1957_.1959 base.
d.
Percentage weight in index:
8.
Although this index has been computed from January, 19 19 to
present, only data'from 1948 to 1965 will be used in the test.
(News­
paper help wanted advertisement data were available only for that period).
Since this is a general activity index, it should have some time relation­
ship to general help wanted advertising in the same area.
14
The fourteen ( 14) variables used in this index are all population dependant to some extent.
Since the data were not normalized on a per capita
basis, direct variation with the increasing population in the area was
exhibited along with the variations due to other causes.
This population
dependance and inflation produced an increasing trend in the data.
B.
The Help Wanted Advertisement Indexes
The data descriptive of help wanted advertising were also _provided
by the Security First National Bank.
The Bank now uses data provided
by the two major newspapers found in the area:
the Los Angeles Times
and the Los Angeles Herald Examiner, and has been preparing these
indexes since 1948. Two separate indexes were used in the test providing data for two experiments.
They are 1)
the Number of Help Wanted Advertisements Index and
2) the Number of Lines of Help Wanted Advertisements Index.
Both of these indexes exhibited a discontinuity in 1962.
This
variation is explained by the following note which accompanied the indexes:
In January, 1962, the four major Los Angeles newspapers
were consolidated, with two newspapers surviving-- the
Los Angeles Times and the Herald-Examiner. In prepar­
ing the index prior to this time, data from the Los Angeles
Times and the Los Angeles Examiner were used, a 94%
sample. Starting in January of 1962, figures from the Los
Angeles Times and the Herald-Examiner are used, a 1 OQ""%""
sample.
.
Due to this consolidation, and changes resulting in help wanted
advertising policy by companies, the index from January,
1962 on is not considered to be strictly comparable to the
previous indexes.
To correct the data used in both indexes for calendar and seasonal
v:ariations, the following rules were used:
The raw data were first cor­
rected for daily variations, then daily averaged, then seasonally adjusted,
15
then adjusted to the base period of 1957-1959.
The following are the
appropriate abstracts from the Security First National Bank1s notes
further explaining these calculations:
NUMBER OF HELP WANTED ADS INDEX
Calendar Correction
The number of help wanted ads averaged 32% more on Sunday than
the average for the other six days of the week.. Hence, in computing the
number of days per month, Sundays were given a value of l. 2624 days
and the other days of the week were given a value of 0. 9564 days each.
In this way, adequate adjustment could be made (or the fact that some
months have four Sundays and some five, and that the same month may
have four Sundays in one year and five in another.
Seasonal Adjustment
The following seasonal adjustment factors derived from previous
history were used:
January
Feb.
March
April
95.
93.
94.
99.
57
53
05
36
May
June
July
Aug.
104.
105.
99.
109.
84
10
75
46
Sept.
Oct.
Nov.
Dec.
119.
111.
95.
72.
30
19
49
35
Base Period Adjustment
This adjustment was made by dividing the seasonally adjusted average
for each month by 2212. l (the daily average for the base period 1957-1959).
NUMBER OF LINES OF HELP WANTED ADS INDEX
Calendar Correction
The number of lines of help wanted ads averaged 168% more on
Sunday than the average for the other six days of the week. Hence, in
computing the number of days in each month, Sundays were given a
value of 2. 1647 days and other days of the week were given a value of
16
0.8064 each day, providing the same compensation for variations in the
number of Sundays per month.
Seasonal Adjustment
The following seasonal adjustment factors were used:
January
Feb.
March
April
99.
9�.
97.
100.
30
26
61
61
May
June
July
Aug.
103. 97
103. 15
97.01
108. 28
Sept.
Oct.
Nov.
Dec.
114.
110.
97.
73.
01
32
09
83
Base Price Adjustment
This adjustment was made by dividing the seasonally adjusted averages
for each month by 27, 466 (the daily average for the base period 1957-1959) .
As these notes indicate, the index is a "daily average" adjusted to the period
1957-1959 (the same period as the business activity index).
C.
The Numerical Data
The following graphs, Figures 4 and 5, and tables 2 and 3 show
data used in various forms.
Figure 4 is a graph of the raw data used
in the calculation of the ads indexes.
Figure 5 is a graph of each of
the thre·e indexes used.� aiJ.d tables l,
.?,
a:ad 3 are the listings of the
numerical values of each of these three indexes.
for these indexes are also shown on the graphs.
The three trend lines
These trend lines are
as follows:
Business Activity Index Trend
Number of Ads Index Trend
Number of Lines of Ads Index Trend
=
=
=
48+52t/ 120
65+35t/ 120
45+40t/ 120
where t is time measured in months starting from January, 1948 as
zero (0).
These linear trends will be used in subsequent calculations.