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Abstract

Abstract An econometric model for the U. S. lodging market was developed from time series data. Estimates from this model were then combined with preliminary sales estimates. The resulting combination greatly improved the estimates of the "final sales" figures and also reduced the error in two forecasting tests. These improvements were achieved at a low cost.
Improving Current Sales Estimates with Econometric Models
Thomas H. Tessier
Arthur Andersen & Co.
Philadelphia, PA
J. Scott Armstrong
Wharton School
University of Pennsylvania
Philadelphia, PA
July 1977
Published only on the Internet. Cite as Tessier, T.H. & J. S. Armstrong (1977), “Improving
Current Sales Estimates with Econometric Models,”
http://www.fourps.wharton.upenn.edu/forecast/paperpdf/improvingsalesestimates.pdf
1
Abstract
An econometric model for the U. S. lodging market was developed from time series data.
Estimates from this model were then combined with preliminary sales estimates. The resulting
combination greatly improved the estimates of the "final sales" figures and also reduced the error
in two forecasting tests. These improvements were achieved at a low cost.
Introduction
People who have worked with statistics on industry sales levels are aware of the problems
and costs involved with obtaining reliable and valid data on a timely basis. One approach to
improving upon the statistics collected from sellers or buyers is to use econometric methods.
Econometric methods are expected to be especially useful in cases where the sales figures are
subject to much uncertainty.
The hypothesis that econometric sales estimates are useful in estimating current sales
levels was previously tested (Armstrong 1970). In that study, sales of photographic equipment in
each of 17 countries were estimated using a cross-sectional regression model. These estimates
were then combined with trade data to provide the current sales estimates for making long-range
forecasts. The resulting forecasts were superior to those obtained when only trade data were used
for estimating current status. Beyond the study cited, however, little direct evidence exists on the
value of such an approach. The current study attempts to test the same hypothesis, but differs
from (Armstrong 1970) in that:
1. a different market was studied,
2. different time periods were involved,
3. time series rather than cross-sectional data were used,
4. different procedures were used for validation.
In other words, the question is whether the hypothesis holds in a different situation.
2
The Market
The study examined the U.S. lodging market, using data from 1958 through 1970.
Interest focused on estimating lodging sales (including room, food, and beverage sales) in
current dollars.
Lodging sales are estimated annually by the U.S. Department of Commerce and
published in the U.S. Industrial Outlook. Various sources are used by the Commerce Department
in making its estimates: during the years 1958, 1963, and 1967, business census data were the
primary source: during non-census years, sales were estimated from sample surveys, tax returns,
and information from private sources. Estimates of lodging sales are made at the end of each
year. These estimates are revised in later years as additional data become available. The revisions
are often substantial. For example, at the end of 1968, the U. S. Department of Commerce
estimated that 1968 lodging sales were approximately $7.3 billion. The following year this
estimate was revised to 7.6 billion. In 1970, the amount was adjusted to $7.1 billion. The current
and presumably "final" estimate of 1968 lodging sales is $6.5 billion. In this example, the
preliminary estimate of $7.3 billion made in 1968 was 11% higher than the final estimate in
1971. The mean absolute percentage error of the Commerce Department's preliminary estimates
during the 1965-1970 time period was 10.5% (see their Exhibit 7, column 1). The substantial
revisions imply that there is much uncertainty in these estimates. This, then, is a situation where
econometric methods should improve upon the preliminary sales estimates.
3
The Econometric Model
The development of the econometric model followed standard procedures. A heavy
emphasis was placed on the a priori analysis because limited data were available for the lodging
market.
The first step in the à priori analysis was to specify the variables related to lodging sales.
In conceptual terms, lodging sales were expected to be a function of market size, ability to buy,
and consumer needs. Using this model, and considering the data that were available, we selected
five operational variables U. S. population (as a measure of market size); corporate profits and
lodging rates (as measures of ability to buy); and aircraft speed and intercity passenger miles (as
measures of needs). In addition, the problem was recast in terms of constant dollars to control for
inflation.
The direction of each relationship in the model was specified using standard economic
arguments. A positive relationship was specified for the coefficients relating sales to corporate
profits and intercity passenger miles. A negative relationship was specified for lodging rates and
aircraft speed.
Standard econometric practice was also used to select the functional form. This was the
multiplicative (log-log) model, which assumes constant elasticities.
The effect of market size was fixed à priori at 1.0; in other words, the dependent variable
was transformed to per capita figures. A range was then estimated subjectively for each of the
four remaining elasticities. Previous studies on similar products and services provided guidelines
here e.g. (Houthakker and Taylor 1970). This phase of the à priori analysis was subject to
much uncertainty, however, previous research has suggested, surprisingly, that the accuracy of
econometric models is not very sensitive to magnitudes of the relationships (Claudy 1972 and
4
Dawes and Corrigan 1974). In short, one needs only to find a reasonable estimate; beyond that,
improvements are expected to be minor.
The à priori analysis yielded a completely operational model, except for the constant
(scaling) term. A summary of this model is presented in Exhibit 1. The midpoints of the à priori
range for the coefficients are presented in the equation, and the ranges are listed in the right hand
column.
Exhibit 1
À Priori Model to Estimate Sales Level in U. S. Lodging Market
0.7
t
S
0.6
t
A
0.8
t
T
1.5
t
aB
t
Y
=
where:
À priori range
Y
?
B
T
A
S
t
=
=
=
=
=
=
=
lodging sales in constant dollars per capita
scaling constant
corporate profits per capita in constant dollars
intercity passenger miles per capita
lodging rates in constant dollars
aircraft speed in miles per hour
the year
1.0 to 2.0
0.6 to 1.0
-0.4 to -0.8
-0.4 to -1.1
Data from 1958 to 1964 were used to update the model. These data are summarized in
Exhibit 2. While limited, these data allow for an estimate of the scaling constant. Furthermore,
they provide a check on the a priori signs. Finally, they provide information on whether the
magnitudes of the coefficients are reasonable.
5
Exhibit 2
Data on the U. S. Lodging Market (Final Estimates)
Year
Lodging
Sales
a
Corporate
Profits
b
Intercity
Passenger
Miles
c
Lodging
Rates
d
Aircraft
Speed
e
Consumer
Price
Index
U.S.
Population
Y B T A S
1958
1959
1960
1961
1962
1963
1964
3644
3996
4248
4327
4616
4667
5013
22.3
28.5
26.7
27.2
31.2
33.1
38.4
702
763
782
788
815
849
892
6.95
7.40
7.76
7.92
8.27
8.59
9.58
219
223
235
253
274
287
297
0.866
0.873
0.883
0.896
0.906
0.917
0.929
175
178
181
184
187
189
192
a
U.S. Department of Commerce, U.S. Industrial Outlook. Data in millions of current dollars.
b
Council of Economic Advisors, Economic Report of the President. Data on profits are after tax profits in billions
of current dollars. Data on population given in millions. Consumer price index based on 1967 = 1.00.
c
Automobile Manufacturer's Association, Automobile Facts and Figures, Detroit, Michigan. Includes auto, air,
bus, and train miles in billions.
d
Computed from Harris, Kerr, Forster & Co, Trends in the Hotel Business and U. S. Census Bureau data:
represents daily gross income per occupied room in current dollars.
e
U.S. Department of Transportation, FAA Statistical Handbook of Aviation. Data in miles per hour.
“Conditional regression analysis” was used to update the coefficients. This approach is similar to
that described in Wold and Jureen (1953). The procedure was to first obtain an à priori
(subjective) estimate for each of the coefficients in the model. Regression analysis was used to
obtain another estimate for each coefficient. A combined estimate of one of the coefficients was
then made by taking an average of the a priori and the regression estimates. Calculations were
made to remove the effect of this first variable from the equation. (Both sides of the equation
were divided by this variable). The regression was run using the remaining variables with the
revised dependent variable. This provided a new set of regression estimates. A second variable
was then selected and the procedure was repeated until updated estimates were obtained for each
of the coefficients.
6
The conditional regression analysis required subjective inputs by the analyst; decisions had to be
made about the order the variables were introduced and the weight which was placed on the a
priori estimates. Because of this subjectivity, a number of alternative models were examined.
The model that provided the most accurate fit to the 1958-1964 data was.
0.3
t
S
0.6
t
A
0
t
T
0.9
t
B660
t
Y
(See Exhibit 1 for a description-of the variables)
A sensitivity analysis was performed to determine the effect of the subjective estimates.
Thirteen models were examined and the results are presented in Exhibit 3. The first seven models
in Exhibit 3 represent an exploratory analysis of extreme points of the à priori ranges. The final
six models represent fine tuning, examining small variations around the "best fit" of the
exploratory models (which was model number 3).
1
Note that the coefficients all have the same
signs as in the à priori model. Furthermore, the updating did not lead to substantial changes in
the magnitudes; the final estimates were within or close to the à priori ranges. Finally, the errors
for each of the 13 models were lower than the average error between the Commerce
Department's preliminary and final estimates (10.5%).
1
The criterion for best fit was the adjusted mean absolute percentage error (MAPE), which was
calculated as follows:
P)1/2(A
PA
)MAPE(
+
=
where A = lodging sales in current dollars
P = the predicted value from the econometric model
This criterion was selected because it was felt that errors in scale (percentage errors) were just as
serious on the high side as on the low side.
7
Exhibit 3
Econometric Model: Sensitivity Analysis (1958-1964 Final Estimates
Model
Number
Corporate
Profits
Intercity
Passenger
Miles
Lodging
Rates
Aircraft
Speed
MAPE
B T A S
1
2
3
4
5
6
7
1.5
2.0
1.0
1.5
1.5
1.5
1.5
0.9
0.9
0.9
0.6
1.0
0.9
0.9
-0.6
-0.6
-0.6
-0.6
-0.6
-0.4
-0.8
-0.7
-1.1
-0.4
-0.6
-0.8
-0.9
-0.6
8.1
9.8
5.1
7.7
8.1
7.9
8.3
8
*9
10
11
12
13
1.1
0.9
1.0
1.0
1.0
1.0
0.9
0.9
0.8
1.0
0.9
0.9
-0.6
-0.6
-0.6
-0.6
-0.5
-0.7
-0.4
-0.3
-0.3
-0.4
-0.4
-0.3
6.0
4.6
5.3
5.6
5.7
5.7
Average
6.8
* denotes selected model
Testing The Validity of the Econometric Estimates
Two approaches were used to test the validity of the econometric estimates. First, preliminary
survey figures were compared with a combination estimate in an attempt to predict the final
survey estimate for each of the years 1965 to 1970. The second approach compared the accuracy
of two forecasts of “final” lodging sales, the first using preliminary survey estimates for current
sales, and the second using a combined estimate.
8
Predicting “Final” Estimates
To examine whether econometric methods could improve upon the data available in year t' a
comparison was made between the Commerce Department's "preliminary" survey and their
"final" survey estimate of lodging sales for the years 1965 to 1970.
Econometric estimates were made by inserting values of the independent variables into
the econometric model for the appropriate years. Data for the independent variables used in the
econometric model are shown in Exhibit 4. These data are “preliminary” estimates that would
have been available at the time of the forecasts.
Exhibit 4
Data for Testing the Lodging Sales Model
Year
Corporate
Profits
Intercity
Passenger
Miles
Lodging
Rates
Aircraft
Speed
Consumer
Price
Index
U.S.
Population
B T A S
1965
1966
1967
1968
1969
1970
44.5
48.1
47.2
51.0
50.8
44.2
884
937
979
1070
1066
1126
9.03
10.10
11.43
12.28
12.83
13.90
315
320
354
369
390
400
a
0.937
0.963
0.989
1.028
1.088
1.165
194
197
199
201
203
205
a
subjective estimate Sources and Units: same as in Exhibit 2
It was hypothesized that the combined estimate would be more accurate than one that
relied solely on the preliminary survey. This is what occurred; as shown in Exhibit 5, the
combined estimate was off by 5.0% over this time span, while the preliminary survey was off by
10.5%. These differences were statistically significant at the .05 level (Wilcoxon matched-pairs
signed-ranks one-tail test from Dixon and Massey 1969). These improvements in accuracy were
achieved at low-cost and would appear to be of practical importance.
9
Exhibit 5
Accuracy of Preliminary vs. Combined Estimates of Current Lodging Sales
Year
Final
Estimate
Preliminary Survey
Combined Estimate
(Equal Weights)
Econometric
Estimate
($x10
6
) ($x10
6
)
MAPE
($x10
6
)
MAPE
($x10
6
)
MAPE
1965
1966
1967
1968
1969
1970
Average
5489
6365
6533
6531
6418
6801
MAPE
5200
5900
6700
7300
7800
8043
5.4
7.6
2.6
11.1
19.4
16.7
10.5
5634
6156
6342
6990
7181
7013
2.6
3.3
3.0
6.8
11.2
3.1
5.0
6067
6413
5983
6679
6562
5983
10.0
0.8
8.8
2.2
2.2
12.8
6.1
A sensitivity analysis was carried out using the 13 models from Exhibit 4 and considering
various weighting schemes. The results, presented in Exhibit 6, show that the combined
estimates reduced the error in the preliminary estimate (10.5%) in 63 of the 65 cases examined in
the columns labeled 15% to 85%. The optimum combination was achieved with equal weights.
Exhibit 6
Predicting the “Final” Estimate 1965-1970
(Entries are
MAPE
s)
Percent Econometric Model Contribution
Model 0% 15% 33% 50% 67% 85% 100%
1
2
3
4
5
6
7
8
9
10
11
12
13
10.5
10.5
10.5
10.5
10.5
10.5
10.5
10.5
10.5*
10.5
10.5
10.5
10.5
7.6
7.0
8.1
7.4
8.0
7.5
7.4
8.1
8.6
8.2
8.4
8.2
8.2
8.1
6.1
5.8
5.2
5.0
4.9
5.0
6.0
6.0
6.0
6.1
6.0
6.0
6.0
8.6
4.6
5.9
5.6
5.7
5.6
4.9
5.0*
6.5
5.1
5.0
5.0
5.5
11.4
4.8
8.0
7.8
7.6
7.6
5.8
4.6
5.0
5.2
5.0
5.0
10.4
14.5
6.0
9.7
10.0
9.8
9.9
7.5
5.3
6.5
6.7
6.4
6.4
12.4
18.0
7.2
12.6
12.4
12.4
12.4
9.9
6.1*
7.5
8.0
7.7
7.6
Average 10.5 7.9 5.9 5.7 6.4 8.4 10.3
* Detailed results for these models are provided in Exhibit 5.
10
Forecasting Tests
The accuracy of a forecast depends on two factors: first, the accuracy in estimating sales
at time t; and, second, the accuracy in forecasting changes from time t to t+f. Thus, if the
econometric estimates improve the estimates of current sales, more accurate forecasts should
result. To assess this, two forecasting tests were devised. The tests involved a comparison of the
accuracy of two forecasting models. The first model used the preliminary estimate of lodging
sales made by the U.S. Department of Commerce for “current status.” The second model used
an equally weighted average of the preliminary and econometric estimates to provide a combined
estimate for “current status.” Both models in each test used the same forecasts of “change.”
The only way that the two tests differed was in the model used to forecast change. In one
test an econometric model was used to forecast change; in the second test an extrapolation model
was employed. The two change models provided an opportunity to examine whether the results
were sensitive to the forecasts of change.
The forecasting tests were conducted for the 1965-1971 period. To obtain the largest
possible sample size, current status for each: of the years from 1964 through 1970 was used as a
starting point. This provided a total of 28 different forecasts; 7 for a one-year horizon; 6 for a
two-year horizon; etc. as shown in Exhibit 7.
11
Exhibit 7
Forecasting Scheme for Lodging Market
Year forecast was prepared
1964 1965 1966 1967 1968 1969 1970
Forecast 1965
Year 1966
1967
1968
1969
1970
1971
F
64-1
F
64-2
F
64-3
F
64-4
F
64-5
F
64-6
F
64-7
F
65-1
F
65-2
F
65-3
F
65-4
F
65-5
F
65-6
F
66-1
F
66-2
F
66-3
F
66-4
F
66-5
F
67-1
F
67-2
F
67-3
F
67-4
F
68-1
F
68-2
F
68-3
F
69-1
F
69-2
F
70-1
Key: F
i-j
is the forecast for the j
th
year from the i
th
starting year.
The econometric model for change was:
05607080011 .
t
S
ft
S
.
t
A
ft
A
.
T
ft
A
.
t
B
ft
B
t
Y
f
).(
ft
Y
+
+
+
+
=
+
(See Exhibit 1 for description of variables: f is the number of years in the future.)
It was developed with procedures similar to those used for the econometric model that
estimated current status. The coefficients in the model were not updated when each successive
starting year was used; only the initial sales level was changed. Forecasts of the independent
variables were based on linear extrapolations from data that would have been available at the
time of the forecast.
The extrapolation model was based on an average forecast from two sub-models: a
constant unit change model developed from a five-year moving average of the yearly unit
changes; and a constant percentage change model developed from a five-year moving average of
the yearly percentage changes. Data from 1958 up to the year of the forecast were used to
develop these extrapolations. Then, as the starting year was changed, data from the years 1965 to
1970 were used to update the extrapolation model. Only data that would have been available at
the time of the forecast were used. The results are summarized in Exhibit 8 where the average
MAPE
for each forecast horizon from 1 to 7 years is presented. The combined estimates yielded
12
a substantial reduction in the forecasting error: the
MAPE
s were reduced by about l/3 (from
12.7 to 8.6 for one test, and from 17.5 to 12.7 for the other test).
These improvements were each significant (p <.01 using the Wilcoxon matched-pairs
signed-ranks one-tail test). Because the econometric estimates alone would have led to further
improvements, it follows that any estimate of current sales that puts weight on the econometric
estimates would have been superior to one that used only the preliminary survey by the
Commerce Department.
Exhibit 8
Forecasting Error: Direct vs. Combined Estimate of Current Sales
(Entries are
MAPE
s)
Change predicted by:
Econometric model Extrapolation model
Current status estimated by Current status estimated by
Forecast
horizon
No. of
forecasts
Preliminary
survey
Combination
(equal
weights)
Econometric
model
Preliminary
survey
Combination
(equal
weights)
Econometric
model
1
2
3
4
5
6
7
7
6
5
4
3
2
1
15.4
16.0
14.8
10.5
8.6
10.5
13.1
8.0
11.6
11.2
8.9
6.9
4.8
8.8
6.0
7.4
9.5
9.2
9.4
6.4
4.7
14.9
16.8
16.8
14.3
11.9
17.0
30.9
7.4
11.8
12.4
12.1
11.2
11.8
22.0
4.5
7.9
11.6
13.0
15.3
15.4
14.6
Average
MAPE
12.7 8.6 7.5 17.5 12.7 11.7
Conclusions
Further tests were made on the hypotheses studied in Armstrong (1970). This study differed
substantially from that previous study in that:
(1) a different market was studied (the U. S. lodging market rather than the
international photographic market);
13
(2) different time periods were involved for model development (1958-1964 for
the current study vs. 1960-1965 for the 1970 study); and for validation (1965
to 1971 vs. 1954);
(3) time series data were used for model development in the current study vs.
cross-sectional data for the earlier study;
(4) different procedures were used for validation. The current study predicted final
estimates; in addition, two forecasting tests were made. The previous study
used one backcasting test.
Results from the study of the U. S. lodging market supported the results from Armstrong
(1970). Econometric estimates of current status provided useful information. A simple average
of preliminary survey estimates and econometric estimates reduced the error in predicting "final"
survey estimates from 10.5% to 5.7%. It also reduced the errors by about 1/3 in two forecasting
tests.
The results were not very sensitive to the weighting scheme used to combine the
econometric and direct sales estimates. A simple average of the two estimates provided nearly
optimum results, but any estimate that incorporated information from the econometric models
was superior to one that used only the preliminary survey estimates.
A sensitivity test supported the conclusion from previous studies that the accuracy of an
econometric model is not highly sensitive to the estimates of the coefficients. Each of 13
different models provided improvements in predicting the “final” sales estimates.
14
References
Armstrong, J. Scott (1970). “An Application of Econometric Models to Models to International
Marketing,” Journal of Marketing Research, 7 (May 1970), 190-198.
Claudy, John G. (1972). “A Comparison of Five Variable Weighting Procedures,” Educational
and Psychological Measurement, 32 (1972), 211-322.
Dawes, Robyn and Corrigan, Bernard (1974). “Linear Models in Decision Making,”
Psychological Bulletin, 81 (February 1974), 95-106.
Dixon, Wilfrid J. and Massey, Frank J., Jr. (1969). Introduction to Statistical Analysis. New
York: McGraw Hill, 1969.
Houthakker, H. S. and Taylor, L.D. (1970). Consumer Demand in the United States: Analyses
and Projections. Cambridge Harvard University Press, 1970.
Wold, Herman and Jureen, Lars (1953). Demand Analysis: A Study in Econometrics. New York:
John Wiley and Sons, 1953.
... While this is uncommon in the literature modeling import demand, Browning (1984) models import demand in Eastern Europe in terms of population shifts. And population has been widely used in econometric modeling as a proxy for market size (see, e.g., Tessier and Armstrong, 1977;Campbell and Hopenhayn, 2002). In addition, several studies in the trade literature have treated population size in a country's trading partners as an indicator of the market for that country's exports (see for example Podbury et al., 1995;Sharma and Chua, 2000) Apart from the fact that we want to include population in our model to facilitate comparison with Anaman and Buffong (2001), treating population as a proxy for market size in Brunei's case makes sense because since the mid-1960s, Brunei's annual population growth has been about 2.5%. ...
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A review of the literature indicates that linear models are frequently used in situations in which decisions are made on the basis of multiple codable inputs. These models are sometimes used (a) normatively to aid the decision maker, (b) as a contrast with the decision maker in the clinical vs statistical controversy, (c) to represent the decision maker "paramorphically" and (d) to "bootstrap" the decision maker by replacing him with his representation. Examination of the contexts in which linear models have been successfully employed indicates that the contexts have the following structural characteristics in common: each input variable has a conditionally monotone relationship with the output; there is error of measurement; and deviations from optimal weighting do not make much practical difference. These characteristics ensure the success of linear models, which are so appropriate in such contexts that random linear models (i.e., models whose weights are randomly chosen except for sign) may perform quite well. 4 examples involving the prediction of such codable output variables as GPA and psychiatric diagnosis are analyzed in detail. In all 4 examples, random linear models yield predictions that are superior to those of human judges. (52 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Article
This textbook (see 25: 7185) is still written for the basic course for students from all fields; and it requires only algebra. Most chapters have been changed by rewriting and the addition of new materials. Major changes have been made in the chapters on central tendency and dispersion, statistical inference, and analysis of variance. Additional material includes a new chapter on probability, 133 references at the end, and an increase from 26 to 33 tables, with many of the old tables greatly enlarged. (PsycINFO Database Record (c) 2012 APA, all rights reserved)