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A comparison of time series and econometric models for forecasting restaurant sales
Int. J. Hospitality Management Printed in Great Britain
Vol. 11 No. 2, pp. 129-142,
1992
0278p4319/92 $5.00 + 0.00 0 1992 Pergamon Press Ltd
A comparison of time series and econometric models for forecasting restaurant sales David A. Cranage Hotel, Restaurant and Institutional Decision Modeling, The Pennsylvania State University, University Park, PA 16802, U.S.A.
and William P. Andrew Hotel, Restaurant and Institutional Finance, The Pennsylvania State University, University Park, PA 16802, U.S.A.
Historically,
forecasting
‘judgementally’ forecasts
for
based. effective
of restaurant restaurant
casting models for accuracy actual restaurant
smoothing
econometric
model.
models)
and efficiency.
models) Since
time
series
results of this study have important
Key words:
we have
industry
has been
and long-term
investigated
various
sales fore-
The results of the study show that for the
time series models
performed
are typically more economical
the restaurant
in the hospitality of both short-term
management,
sales in this sample,
exponential
sales
Given the importance
(specifically
Box-Jenkins
as well or better in forecasting models
(especially
and
sales than an
exponential
smoothing
in terms of time and skill levels of the users, the
implications
for the use offorecasting
techniques
in
industry.
forecasting
time series
econometric
modeling
Introduction The purpose of this research is to investigate the appropriateness of various classes of forecasting models for forecasting restaurant sales. While such investigations have been performed for manufacturing and retail department store operations (see for example, Geurts and Kelly, 15X36), few studies have been published relating specifically to forecasting sales in the restaurant industry. The importance of accurate and timely forecasts of sales data is apparent at all levels of the restaurant operation. Short-term sales levels are needed for daily and weekly employee scheduling (especially where restaurants are highly dependent on part-time labor). Restaurants deal with very perishable products, where purchasing and inventory 12’)
130
David A. Cranape and William
P. Andrew
quantities need to be accurately determined. On a longer term basis, menu development decisions. employee hiring and training program decisions, and capital investment decisions including equipment, seating capacity and expansion are directly linked to predicted future business. Furthermore, effective and efficient marketing strategies and use of advertising depend upon accurate sales projections. Given the importance of sales forecasting to successful restaurant management it is interesting that much of the forecasting in the industry hasbeen ‘judgementally’based (i.e. based on the manager’s interpI-etation of current economic and environmental factors along with a certain amount of ‘gut’ feel). The reason for this may be that the majority of the restaurant industry is composed of independent restaurant owners who tend to lack sufficient resources for the development and application of more quantitative, but also more accurate, forecasting models. Since the trade-off between forecast accuracy and the cost of obtaining that accuracy (due to resource utilization, time, etc.) is important for any restaurant operation whether independent or chain, we will examine this issue in detail when we analyze the results of the forecasting models in this study. We will focus in this paper on whether statistical models, specifically: econometric, BoxJenkins, and exponential smoothing, can accurately forecast restaurant sales. If so, we will further explore which of these forecasting techniques can accomplish this task the most effectively and efficiently. Our analysis will be based on sales data from an actual restaur~~nt.
Forecasting
models and prior research
The three general classes of models used in prior studies for forecasting variables such as sales are: (1) judgemental, (2) econometric (or causal), and (3) time series. In addition, combinations of these models have been used in the past several years in attempts to improve forecasting accuracy. These three classes of models and relevant prior studies are discussed in the following brief literature review. More detailed descriptions of the techniques and prior empirical results can be found in the cited articles. 1. .Judgernental This is the technique most often used by the majority of the restaurant industry. It consists of an ‘intuitive’ forecast based on the manager’s collective experience regarding the variable in question. Numerous studies have been conducted on the accuracy of judgemental models vs statistical models. Arnistr~~ng (1983). Lorek et ui. (1976). Cleveland and Tiao (1976), Libby (19761, Cerullo and Avila (1975), Dalrymple (lY7S), and Hogarth (1975), all concluded that quantitative methods provided better forecasts than judgemental methods. Other studies, Johnston and Schmitt (1974). Critchfield etaI. (1978). and Brandon and Jarrett (1977) found that ‘analysts’ can do better than quantitative models, provided that they have accurate economic and industrial information. The key words here are analysts and accurate. As was stated above. many independent restaurant owners have limited resources. This would probably extend to being able to utilize expert analysts. The other requirement of ‘accurate’ economic and industrial information for a particular market segment may also be difficult to meet in the restaurant industry because such data
Forecasting restaurant sales
may not be available. There is also a question in this body of research as t_ vhether analysts themselves were using quantitative models to generate their forecasts.
131
the
2. Econometric Econometric models utilize a regression equation or equations to establish a casual relationship between the dependent variable (e.g. restaurant sales) and exogenous variables such as disposable income, the consumer price index and unemployment. One of the advantages of an econometric model for forecasting restaurant sales is that the decision maker can logically formulate the model based on a cause and effect relationship between the causal variables and future sales. There are, however, disadvantages in using econometric models. Geurts and Kelly (1986) note that, first, the future values of the causal variables themselves have to be predicted. Second, even when we are using a lagged causal variable to predict sales one period ahead, the reported causal variable value is often a preliminary figure that is later revised. Both of these factors can cause data in an econometric model to be inaccurate and the model to be weak in its ability to forecast. Third, the continual need to gather data can make these models expensive to use. Lastly, the relationship found between the dependent and independent variables may be a spurious one. Furthermore causal relationships can change over time, making it necessary to constantly update or totally redesign the model. 3. Time series These models differ from econometric models in that they do not attempt to incorporate a causal relationship. A time series model looks for time patterns (trends, cycles, and seasonal influences) in a single series of data and captures them in a mathematical equation(s). These mathematicaf relationships are then used to project into the future the historical time patterns in the data. In this study we will use two of the most common types of time series models: Box-Jenkins and exponential smoothing. Box-Jenkins models are of the autoregressive and/or moving average type while the exponential smoothing models are of the exponentially-weighted moving average type. Many studies have been conducted on the performance of econometric vs time series models (Schmidt, 1979; Nelson, 1972; Reid, 1971, 1975; Naylor and Seaks, 1972; Narashimham ernl.. 1974; Makridakis and Hibon, 1979; Dalrymple, 1978; Box and Tiao, 1975; and Zober, 1981) and between various time series models (Groff, 1973; Newbold and Granges, 1974; Geurts and Ibrahim, 1975; Makridakis and Hibon, 1979; Bretschneider et raf., 1979; Hollier et ai., 1981; and Poulos et af., 1987). The results of these studies are mixed and seem to depend on the particular variables, models and data used. However, in the majority of the studies the time series models performed as well as or better than the econometric models. In addition, the simplest time series models (e.g. exponential smoothing) often performed as well or nearly as well as more complex time series models (e.g. Box-Jenkins).
The data The initial data base (Fig. 1) consists of 72 monthly observations (January 1982-December 1987) of sales from an existing restaurant. An additional seven months of data (January 19X8-July 1988) was held in reserve to test the predictive ability of the models.
1.12
David A. Cranage
and William
P. Andrew
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The sales data was obtained from a full menu, full service, family style restaurant. This restaurant is independently owned and located in a medium size university town. Sales are affected by normal economic and seasonal variables as well as university seasonal factors.
Methodology Three models (two time series and one econometric) are initially fitted to the monthly sales data of the restaurant. The fit of the Box-Jenkins and econometric models is evaluated using a variety of statistical measures: (1) sum of the squared residuals, (2) adjusted R squared, and (3) Durbin Watson Statistic. The exponential smoothing model (because of the fitting technique) is evaluated using the sum of square residuals only. Also, each of the models’ ability to forecast seven periods (months) beyond the initial time period is examined. For the time series models, the forecast ability is based only on information contained in the initial time period. In contrast, the seven month forecast for the econometric model is based on information contained in the initial time period, plus monthly updates of the independent variables. Tests of the accuracy of the forecasts are based upon the sum of squared residuals. For the econometric model numerous economic variables were investigated to select the most appropriate independent variables to relate to restaurant sales. Various combinations of the variables were tried until the best model was found. After initial investigation of the significance of the variables, the list was reduced to personal consumption expenditures, unemployment rate, housing starts, and the consumers price index. All of these independent variables were seasonally adjusted and lagged one period. The model was adjusted for seasonality through the use of 11 dummy variables.
Forecasting restaurant sales
133
In the time series model category, several exponential smoothing models were tested: (1) single parameter, (2) two parameter, and (3) three parameter (Holt-Winter). The three parameter (seasonally adjusted) exponential smoothing model proved statistically superior in fit and in forecasting accuracy. The other time series mode1 tested, the Box-Jenkins model, is considerably more complex than the exponential smoothing mode1 and requires a certain amount of expertise in determining the most appropriate form of the mode1 to utilize. This is usually mentioned as being the main disadvantage of this model, although a study of Carbone et al. (1983) found that good results could be obtained by users with only a marginal understanding of the basis of the technique. As a first step in fitting the Box-Jenkins model, the data were plotted and checked for a constant mean and constant variance (stationarity). After first order differencing and differencing of period twelve (for monthly data), no acceptable model could be found on the basis of the autocorrelation and partial autocorrelation functions. Since a possible trend was detected in the plot of the data, a simple regression mode1 with eleven dummy variables was fitted to the data. These dummy variables were used to capture the seasonality of individual months. The residuals of the seasonal model were then examined and it was found that they could be modeled with an autoregressive (AR) (1. 3, 4, 5) process. In summary, the most appropriate model found using the Box-Jenkins technique for this data, is a simple seasonal regression model incorporating an AR (1,3,4,5) process for modeling the residuals.
The results 1. The econometric model The fitted econometric mode1 is shown in Fig. 2. Each of the independent economic variables is significant at a level of better than 5%. The sum of the squared residuals is 2.59 x lo”, the R* is 0.882584 and the Durbin-Watson Statistic is 2.008531. All of these measurements are within a favorable range indicating a statistically acceptable model. A graph of the fitted model, the initial data, and the residuals, is shown in Fig. 3. It should be noted that a Cochran-Orcutt adjustment was performed to reduce serial correlation in the residuals. The signs of two of the economic variables in the fitted model present an interesting anomaly, one that can represent another problem for econometric forecasting. The core of this problem is whether or not the empirical results can be taken as indicating causation or must be considered simply spurious. The two variables in question are the Consumer Price Index (CPI), lagged one period, and the Unemployment Rate (UNEMP), lagged one period. The fitted mode1 shows a negative relationship between sales and the lagged CPI. At first glance, this may seem counter-intuitive since one would expect the nominal value of sales to rise with inflation. However, the observed inverse relationship could result from the fact that wages tend to be ‘sticky’ and adjust slowly in the presence of inflation. Thus a rise in the CPI may well be associated with a decline in real income and hence a decline in demand by the customers of the particular restaurant in this study. Likewise, the positive relationship between UNEMP and sales appears to be counter-intuitive. A possible, though not the only explanation for this result, may be the fact that the unemployment
134
David A. Cranage
and William
P. Andrew
ECONOMETRIC MODEL
----_ =================================__====________=~~=============== 2-TAIL SIG. VARIABLE COEFFICIENT STD. ERROR T-STAT. ------~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-_~~~~~~~~~~------C 282170.38 107966.71 2.6134943 0.012 Dl 4712.5456 3965.2996 1.1884463 0.240 D2 7597.7157 4111.8250 1.8477721 0.070 D3 18464.575 4181.4434 4.4158375 0.000 D4 26191.036 4163.6497 6.2904034 0.000 D5 26622.087 4154.4064 6.4081567 0.000 D6 -11821.549 4142.8927 -2.8534528 0.006 D7 24644.846 4086.0441 6.0314684 0.000 D8 28602.668 4078.1794 7.0135875 0.000 D9 29733.409 4057.3120 7.3283516 0.000 D10 48013.532 4019.5201 11.945090 0.000 Dll 28671.176 3701.3117 7.7462202 0.000 CPI(-1) -3537.2568 1546.0095 -2.9348181 0.005 HOUSING (-1) 8.4955648 4.1417390 2.0512072 0.045 PCE(-1) 95.063365 24.715121 3.8463646 0.000 UNEMP(-1) 4229.2414 1716.2622 2.4642164 0.017 __-_ ____________________---_-_-_-_-________-_________-_------_______ AR(l) 0.1759298 0.1362716 1.2910236 0.202 =============================================~===================== R-squared 0.882584 Mean of dependent var 113607.9 17756.75 0.847795 S.D. of dependent var Adjusted R-squared 2.59Er09 6927.532 Sum of squared resid S.E. of regression Durbin-Watson stat 2.008531 F-statistic 25.36904 Log likelihood -718.8998 ===================================================================
c =
Constant
to
Dll = Dummy Variables m Adjust for Seasonalicy CPI(-1) = Consumer Price Index - Lagged 1 Period Housing (-1) = Housing Starts - Lagged 1 Period PCE(-1) = Personal Consumption Expenditures - Lagged I Period UNEMP(-1) = Unemployment Rate - Lagged 1 Period AR(l) = Cochron - Orcucc Adjustment to Reduce Serial Correlation DL
Fig. 2.
data used in this analysis were based on the national unemployment rate. Since the restaurant in this study is located in a college town, it may be that the local unemployment rate is not highly correlated to the national rate (indeed, some authors have suggested that demand for higher educational services actually increases during times of national recession-hence employment in a college town might be expected to follow the same pattern as the demand for such educational services). In light of these possible causative relationships, it was felt that there was not a convincing economic rationale to exclude either the CPI or the UNEMP variables. Since both were statistically significant, they were retained in the fitted econometric model. The forecasting ability of the econometric model was then tested. The seven month forecast for this model is based on information contained in the initial time period, plus monthly updates of the independent variables. A graph of the forecasted and actual sales
135
Forecasting restaurant sales
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Fig. 3.
data are shown in Fig. 4. It can be seen from this graph that this model does very well in picking up the seasonal patterns. In terms of overall accuracy, the sum of the squared residuals (SSE) is 3.56 X 10s. 2. Exponential smoothing model The results of the fitted three parameter (seasonally adjusted) exponentially smoothing model are shown in Fig. 5. With this model the sum of the squared residuals is 2.81 x lo”, which is only slightly higher than for the fitted econometric model. Again, all of the measurements are within an acceptable range signifying a relatively good model. A graph of the fitted model and the initial data, plus a plot of the residuals is shown in Fig. 6. Notice again the relatively good correspondence between the actual and fitted time series. In forecasting seven periods (months) ahead, the exponential smoothing model performed about as well as the more complicated econometric model. The sum of the squared residuals is only slightly higher at 3.89 X 10s. The graph (Fig. 7) of the forecasted vs actual data shows that the exponential smoothing model was also able to capture the seasonal fluctuations quite well. 3. The Box-Jenkins model The Box-Jenkins model (Fig. 8) appears to perform the best at producing accurate forecasts. When fitted over the initial period, the Box-Jenkins model has a sum of the Statistic of squared residuals of 1.98 x lo’, an R2 of 0.906109 and a Durbin-Watson 2.081763. All of these measures are as good or better than the econometric model and the
136
David A. Cranage and William P. Andrew
exponential smoothing model. A graph of the model and the initial data, as well as a plot of the residuals, is shown in Fig. 9. Notice how closely the fitted times series corresponds to the actual data. In forecasting for the next seven months, the results (Fig. 10) again show a high degree of correspondence between the forecasted values and the actual values. The model shows not only an ability to capture seasonal fluctuations, but also shows a stronger ability to accurately pick up the extremes of these fluctuations than the other models. As a result, the sum of the squared residuals for the forecast is the lowest at 3.03 x 10’.
Conclusions
and implications
For the monthly restaurant sales in the initial sample, the fitted time series models provided a degree of accuracy close to or better than the fitted econometric model. In forecasting beyond the time period of the initial sample, the time series models again performed close to or better than the econometric model. These results are significant in the fact that generally time series models are much less costly (from the standpoint of time, data collection, etc.) to implement than econometric models and do not suffer from many of the disadvantages previously mentioned for econometric models. These results should be taken as good news for a restaurateur who would like improved forecasting ability but does not have the resources to build complex econometric models, In comparing the two time series models tested (Box-Jenkins and exponential
Forecasting restaurant
sales
137
EXPONENTIAL SnOOTHING NODEL (THREE PAFMETER-XIJLTIPLICATIVE)
-___________________~~~~~~~~~~_____-~~~_-_____-~____-___ 0.380 0.000 0.000
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5.
smoothing) using the sales data in our sample, the Box-Jenkins model performed better in both the initial time period and in the seven month forecasting period. However, there are several considerations regarding these results which should be noted. First, the procedure used in developing a Box-Jenkins model is substantially more complex than that used for the exponential smoothing model. With appropriate software, an exponential smoothing model can be fitted literally by pushing a couple of computer keys. Thus, even a restaurant operator with virtually no background in forecast models could successfully fit this type of model. Attempts are currently under way to simplify the Box-Jenkins model fitting process by using automated software (see, for example, Poulos et al., 1987). However, at the present time the cost effectiveness of the exponential smoothing model over the BoxJenkins model may outweigh the improved accuracy of the Box-Jenkins model for many restaurateurs. Second, where there are long-term cyclical patterns in the sales data, the Box-Jenkins model may perform at an even higher level than it did for these data. This is due to the fact that the Box-Jenkins methodology is designed to explicitly model such long-term time patterns (as opposed to the exponential smoothing model which is essentially a weighted moving average). Accuracy in forecasting is often gained by including such ‘history’ in the predictive model. Again, however, the benefits of such increased accuracy must be weighed against the expense (as in employing a more complex model) of obtaining them. Third, a point should be made in regard to econometric models. When there is a turning
138
David A. Cranage
and William P. Andrew
Forecasting
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YODEL
=====================================--_============ 2-TAIL
COEFFICIENT STD. ERROR T-STAT. VARIABLE ____-_____--_----__-~------~~~~~~~_~~~~~~~___-------_---_-~~~~~11.714165 89924.608 7676.5700 C 0.6713494 99.951886 148.88207 MONTH 2.2367274 7730.3572 3456.1017 Dl 2.4048699 9437.5681 3924.3570 D2 5.9732294 21533.855 3605.0608 D3 9.3550750 30468.168 3256.8598 D4 7.0534959 27544.157 3905.0363 05 -2.4808739 -9855.3901 3972.5478 D6 6.8637762 26236.906 3822.5178 D7 9.7256613 30077.756 3092.6180 D8 8.5957694 29421.884 3422.8331 D9 13.046328 49464.338 3791.4375 D10 8.6073536 28411.025 3300.7852 Dll
0.3409622
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0.000 0.505 0.030 0.020 0.000 0.000 0.000 0.017 0.000 0.000 0.000 0.000 0.000
2.5896202
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0.3206575 0.1201090 2.6697205 AR(3) 0.2999337 0.1298408 2.3100119 ARR(4) -0.2384279 -1.8758148 0.1271063 m(5) ==================================================================== R-squared 0.906109 Mean of dependent var Adjusted R-squared 0.876064 S.D. of dependent var S.E. of regression 6293.135 Sum of squared resid F-statistic Durbin-Watson stat 2.081763 Log likelihood -671.3278 ====================================================================
c =
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point in the economy, especially due to unexpected economic events, the econometric model may respond more quickly in its predictions to the changed situation than the time series models. Of course, whether this justifies the expense of utilizing such a model must again be decided by the individual operator. A final consideration is that our data included only one restaurant. While we feel that previous research in other industries supports the generality of our findings, the relative ranking in terms of accuracy of the models which we have examined could change for different types of restaurant operations. This is a question for further research. In any case, our study shows that any of the three models examined in this paper give results that under most circumstances, would be quite acceptable. In summary, our results suggest that for a restaurant operator with limited resources, an exponential smoothing model will give a very satisfactory forecast of sales under most
140
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Forecasting restaurant sales
141
circumstances. Forecast accuracy can be improved by utilizing a Box-Jenkins model and the effects of economic turning points possibly seen more quickly with an econometric model. However, these models involve additional implementation costs which must be evaluated on a cost benefit basis by the restaurant operator.
References Armstrong, J. (1983) Relative accuracy of judgmental and extrapolation methods in forecasting annual earnings. Journal ofForecasting October/December, 437447. Box, G. E. P. and Tiao, G. C. (1975) Intervention analysis with applications to economic and environmental problems. Journal of the American Statistical Association 70, 7C79. Brandon, C. H. and Jarrett, H. E. (1977) Accuracy of externally prepared forecasts. Review of Business and Economic Research Fall, 3547. Bretschneider, S., Carbone, R. and Longini, R. L. (1979) An adaptive aproach to time-series. Decisions Sciences, April. Cerullo, M. J. and Avila, A. (1975) Sales forecasting practices-a survey. Manageriul Planning September/October, 33-39. Cleveland, W. P. and Tiao, G. C. (1976) Decomposition of seasonal time series: a mode1 for the Census X-II program. Journal of the American Statistical Association 581-587. Critchfield, T., Dyckman. T. and Lakonishok, J. (1978) An evaluation of security analysts’ forecasts. Accounting Review 651458. Dalrymple. D. J. (1975) Sales forecasting methods and accuracy. Business Horizons December, 6974. Geurts, M. D. and Ibrahim, I. B. (1975) Comparing the Box-Jenkins approach with the exponentially smoothed forecasting model application to Hawaii tourists. Journal of Marketing Research 12, 182-188. Geurts, M. and Kelly, J. (1986) Forecasting retail sales using alternative model. Internationul Journal of Forecasting 2, 261-272. Groff, G. K. (1973) Empirical comparison of models for short range forecasting. Management Science 20 (1)) 22-31. Hogarth, R. (1975) Cognitive processes and the assessment of subjective probability distributions. Journal of the American Statistical Association 70, 271-290. Hogarth, R. M. and Makridakis, S. (1981) Forecasting and planning: an evaluation. Management Science February, 115-138. Hollier, R. H., Khir, M. and Storey, R. R. (1981) A comparison of short-term adaptive forecasting methods. OMEGA International Journal of Management Science 9698. Johnson, T. E. and Schmitt, T. G. (1974) Effectiveness of earnings per share forecasts. Financial Management Summer, 64-72. Libby, R. (1976) Man versus model of man: some conflicting evidence. Organizational Behavior and Human Performance 1-12. Lorek, K. S., McDonald, C. L. and Patz, D. H. (1976) A comparative examination of management forecasts and Box-Jenkins forecasts of earnings. Accounting Review 321-330. Mahmoud, E. (1984) Accuracy in forecasting: a survey. Journal ofForecasting 139-159. Makridakis, S. and Hibon. M. (1979) Accuracy of forecasting: an empirical investigation. The Journal of the Royal Statistical Society 97-145. Narashimhan, G. V. L. (197.5) A comparison of predictive performance of alternative forecasting techniques: time series models vs. an econometric model. Proceedings of the American Statistical Association August, 459-464.
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and William
P. Andrew
Naylor, T. H. and Seaks, T. G. (1972) Box-Jenkins methods: an alternative to econometric models. international Statistical Review 113-137. Nelson, C. R. (1972) The prediction performance of the FRB-MIT PENN model of the U.S. economy. The American Economic Review December, 902-917. Newbold, P. and Granges, C. W. J. (1974) Experience with forecasting univariate time series and the combination of forecasts. Journal ofthe Royal Statistical Society, Series A 137, 131-165. Poulos, L., Kvanli, A. and Pavur. R. (1987) A comparison of the accuracy of the Box-Jenkins method with that of automated forecasting methods. International Journal of Forecasting 3. Xl267. Reid, D. J. (1971) Forecasting in action: comparison of forecasting techniques in economic time series. Procee&ngs offhe Joint Conference of ~per~~ti~ns Research Society. Reid, D. J. (1975) A review of short term projection techniques. Practical Aspects o.f F[~re~~~st~ng, Qpcrational Research Society, London. S-25. Schmidt, J. (1979) Forecasting state retail sales: econometric vs. time series models. The Annals of Regional Science 8, Yl-101. Zobcr, M. (1981) A comparison of Box-Jenkins methods of forecasting with regression. First Internatiorzal S?mposiurn on Forecasting, Quebec, Canada.
Vol. 11 No. 2, pp. 129-142,
1992
0278p4319/92 $5.00 + 0.00 0 1992 Pergamon Press Ltd
A comparison of time series and econometric models for forecasting restaurant sales David A. Cranage Hotel, Restaurant and Institutional Decision Modeling, The Pennsylvania State University, University Park, PA 16802, U.S.A.
and William P. Andrew Hotel, Restaurant and Institutional Finance, The Pennsylvania State University, University Park, PA 16802, U.S.A.
Historically,
forecasting
‘judgementally’ forecasts
for
based. effective
of restaurant restaurant
casting models for accuracy actual restaurant
smoothing
econometric
model.
models)
and efficiency.
models) Since
time
series
results of this study have important
Key words:
we have
industry
has been
and long-term
investigated
various
sales fore-
The results of the study show that for the
time series models
performed
are typically more economical
the restaurant
in the hospitality of both short-term
management,
sales in this sample,
exponential
sales
Given the importance
(specifically
Box-Jenkins
as well or better in forecasting models
(especially
and
sales than an
exponential
smoothing
in terms of time and skill levels of the users, the
implications
for the use offorecasting
techniques
in
industry.
forecasting
time series
econometric
modeling
Introduction The purpose of this research is to investigate the appropriateness of various classes of forecasting models for forecasting restaurant sales. While such investigations have been performed for manufacturing and retail department store operations (see for example, Geurts and Kelly, 15X36), few studies have been published relating specifically to forecasting sales in the restaurant industry. The importance of accurate and timely forecasts of sales data is apparent at all levels of the restaurant operation. Short-term sales levels are needed for daily and weekly employee scheduling (especially where restaurants are highly dependent on part-time labor). Restaurants deal with very perishable products, where purchasing and inventory 12’)
130
David A. Cranape and William
P. Andrew
quantities need to be accurately determined. On a longer term basis, menu development decisions. employee hiring and training program decisions, and capital investment decisions including equipment, seating capacity and expansion are directly linked to predicted future business. Furthermore, effective and efficient marketing strategies and use of advertising depend upon accurate sales projections. Given the importance of sales forecasting to successful restaurant management it is interesting that much of the forecasting in the industry hasbeen ‘judgementally’based (i.e. based on the manager’s interpI-etation of current economic and environmental factors along with a certain amount of ‘gut’ feel). The reason for this may be that the majority of the restaurant industry is composed of independent restaurant owners who tend to lack sufficient resources for the development and application of more quantitative, but also more accurate, forecasting models. Since the trade-off between forecast accuracy and the cost of obtaining that accuracy (due to resource utilization, time, etc.) is important for any restaurant operation whether independent or chain, we will examine this issue in detail when we analyze the results of the forecasting models in this study. We will focus in this paper on whether statistical models, specifically: econometric, BoxJenkins, and exponential smoothing, can accurately forecast restaurant sales. If so, we will further explore which of these forecasting techniques can accomplish this task the most effectively and efficiently. Our analysis will be based on sales data from an actual restaur~~nt.
Forecasting
models and prior research
The three general classes of models used in prior studies for forecasting variables such as sales are: (1) judgemental, (2) econometric (or causal), and (3) time series. In addition, combinations of these models have been used in the past several years in attempts to improve forecasting accuracy. These three classes of models and relevant prior studies are discussed in the following brief literature review. More detailed descriptions of the techniques and prior empirical results can be found in the cited articles. 1. .Judgernental This is the technique most often used by the majority of the restaurant industry. It consists of an ‘intuitive’ forecast based on the manager’s collective experience regarding the variable in question. Numerous studies have been conducted on the accuracy of judgemental models vs statistical models. Arnistr~~ng (1983). Lorek et ui. (1976). Cleveland and Tiao (1976), Libby (19761, Cerullo and Avila (1975), Dalrymple (lY7S), and Hogarth (1975), all concluded that quantitative methods provided better forecasts than judgemental methods. Other studies, Johnston and Schmitt (1974). Critchfield etaI. (1978). and Brandon and Jarrett (1977) found that ‘analysts’ can do better than quantitative models, provided that they have accurate economic and industrial information. The key words here are analysts and accurate. As was stated above. many independent restaurant owners have limited resources. This would probably extend to being able to utilize expert analysts. The other requirement of ‘accurate’ economic and industrial information for a particular market segment may also be difficult to meet in the restaurant industry because such data
Forecasting restaurant sales
may not be available. There is also a question in this body of research as t_ vhether analysts themselves were using quantitative models to generate their forecasts.
131
the
2. Econometric Econometric models utilize a regression equation or equations to establish a casual relationship between the dependent variable (e.g. restaurant sales) and exogenous variables such as disposable income, the consumer price index and unemployment. One of the advantages of an econometric model for forecasting restaurant sales is that the decision maker can logically formulate the model based on a cause and effect relationship between the causal variables and future sales. There are, however, disadvantages in using econometric models. Geurts and Kelly (1986) note that, first, the future values of the causal variables themselves have to be predicted. Second, even when we are using a lagged causal variable to predict sales one period ahead, the reported causal variable value is often a preliminary figure that is later revised. Both of these factors can cause data in an econometric model to be inaccurate and the model to be weak in its ability to forecast. Third, the continual need to gather data can make these models expensive to use. Lastly, the relationship found between the dependent and independent variables may be a spurious one. Furthermore causal relationships can change over time, making it necessary to constantly update or totally redesign the model. 3. Time series These models differ from econometric models in that they do not attempt to incorporate a causal relationship. A time series model looks for time patterns (trends, cycles, and seasonal influences) in a single series of data and captures them in a mathematical equation(s). These mathematicaf relationships are then used to project into the future the historical time patterns in the data. In this study we will use two of the most common types of time series models: Box-Jenkins and exponential smoothing. Box-Jenkins models are of the autoregressive and/or moving average type while the exponential smoothing models are of the exponentially-weighted moving average type. Many studies have been conducted on the performance of econometric vs time series models (Schmidt, 1979; Nelson, 1972; Reid, 1971, 1975; Naylor and Seaks, 1972; Narashimham ernl.. 1974; Makridakis and Hibon, 1979; Dalrymple, 1978; Box and Tiao, 1975; and Zober, 1981) and between various time series models (Groff, 1973; Newbold and Granges, 1974; Geurts and Ibrahim, 1975; Makridakis and Hibon, 1979; Bretschneider et raf., 1979; Hollier et ai., 1981; and Poulos et af., 1987). The results of these studies are mixed and seem to depend on the particular variables, models and data used. However, in the majority of the studies the time series models performed as well as or better than the econometric models. In addition, the simplest time series models (e.g. exponential smoothing) often performed as well or nearly as well as more complex time series models (e.g. Box-Jenkins).
The data The initial data base (Fig. 1) consists of 72 monthly observations (January 1982-December 1987) of sales from an existing restaurant. An additional seven months of data (January 19X8-July 1988) was held in reserve to test the predictive ability of the models.
1.12
David A. Cranage
and William
P. Andrew
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Fig. 1.
The sales data was obtained from a full menu, full service, family style restaurant. This restaurant is independently owned and located in a medium size university town. Sales are affected by normal economic and seasonal variables as well as university seasonal factors.
Methodology Three models (two time series and one econometric) are initially fitted to the monthly sales data of the restaurant. The fit of the Box-Jenkins and econometric models is evaluated using a variety of statistical measures: (1) sum of the squared residuals, (2) adjusted R squared, and (3) Durbin Watson Statistic. The exponential smoothing model (because of the fitting technique) is evaluated using the sum of square residuals only. Also, each of the models’ ability to forecast seven periods (months) beyond the initial time period is examined. For the time series models, the forecast ability is based only on information contained in the initial time period. In contrast, the seven month forecast for the econometric model is based on information contained in the initial time period, plus monthly updates of the independent variables. Tests of the accuracy of the forecasts are based upon the sum of squared residuals. For the econometric model numerous economic variables were investigated to select the most appropriate independent variables to relate to restaurant sales. Various combinations of the variables were tried until the best model was found. After initial investigation of the significance of the variables, the list was reduced to personal consumption expenditures, unemployment rate, housing starts, and the consumers price index. All of these independent variables were seasonally adjusted and lagged one period. The model was adjusted for seasonality through the use of 11 dummy variables.
Forecasting restaurant sales
133
In the time series model category, several exponential smoothing models were tested: (1) single parameter, (2) two parameter, and (3) three parameter (Holt-Winter). The three parameter (seasonally adjusted) exponential smoothing model proved statistically superior in fit and in forecasting accuracy. The other time series mode1 tested, the Box-Jenkins model, is considerably more complex than the exponential smoothing mode1 and requires a certain amount of expertise in determining the most appropriate form of the mode1 to utilize. This is usually mentioned as being the main disadvantage of this model, although a study of Carbone et al. (1983) found that good results could be obtained by users with only a marginal understanding of the basis of the technique. As a first step in fitting the Box-Jenkins model, the data were plotted and checked for a constant mean and constant variance (stationarity). After first order differencing and differencing of period twelve (for monthly data), no acceptable model could be found on the basis of the autocorrelation and partial autocorrelation functions. Since a possible trend was detected in the plot of the data, a simple regression mode1 with eleven dummy variables was fitted to the data. These dummy variables were used to capture the seasonality of individual months. The residuals of the seasonal model were then examined and it was found that they could be modeled with an autoregressive (AR) (1. 3, 4, 5) process. In summary, the most appropriate model found using the Box-Jenkins technique for this data, is a simple seasonal regression model incorporating an AR (1,3,4,5) process for modeling the residuals.
The results 1. The econometric model The fitted econometric mode1 is shown in Fig. 2. Each of the independent economic variables is significant at a level of better than 5%. The sum of the squared residuals is 2.59 x lo”, the R* is 0.882584 and the Durbin-Watson Statistic is 2.008531. All of these measurements are within a favorable range indicating a statistically acceptable model. A graph of the fitted model, the initial data, and the residuals, is shown in Fig. 3. It should be noted that a Cochran-Orcutt adjustment was performed to reduce serial correlation in the residuals. The signs of two of the economic variables in the fitted model present an interesting anomaly, one that can represent another problem for econometric forecasting. The core of this problem is whether or not the empirical results can be taken as indicating causation or must be considered simply spurious. The two variables in question are the Consumer Price Index (CPI), lagged one period, and the Unemployment Rate (UNEMP), lagged one period. The fitted mode1 shows a negative relationship between sales and the lagged CPI. At first glance, this may seem counter-intuitive since one would expect the nominal value of sales to rise with inflation. However, the observed inverse relationship could result from the fact that wages tend to be ‘sticky’ and adjust slowly in the presence of inflation. Thus a rise in the CPI may well be associated with a decline in real income and hence a decline in demand by the customers of the particular restaurant in this study. Likewise, the positive relationship between UNEMP and sales appears to be counter-intuitive. A possible, though not the only explanation for this result, may be the fact that the unemployment
134
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ECONOMETRIC MODEL
----_ =================================__====________=~~=============== 2-TAIL SIG. VARIABLE COEFFICIENT STD. ERROR T-STAT. ------~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-_~~~~~~~~~~------C 282170.38 107966.71 2.6134943 0.012 Dl 4712.5456 3965.2996 1.1884463 0.240 D2 7597.7157 4111.8250 1.8477721 0.070 D3 18464.575 4181.4434 4.4158375 0.000 D4 26191.036 4163.6497 6.2904034 0.000 D5 26622.087 4154.4064 6.4081567 0.000 D6 -11821.549 4142.8927 -2.8534528 0.006 D7 24644.846 4086.0441 6.0314684 0.000 D8 28602.668 4078.1794 7.0135875 0.000 D9 29733.409 4057.3120 7.3283516 0.000 D10 48013.532 4019.5201 11.945090 0.000 Dll 28671.176 3701.3117 7.7462202 0.000 CPI(-1) -3537.2568 1546.0095 -2.9348181 0.005 HOUSING (-1) 8.4955648 4.1417390 2.0512072 0.045 PCE(-1) 95.063365 24.715121 3.8463646 0.000 UNEMP(-1) 4229.2414 1716.2622 2.4642164 0.017 __-_ ____________________---_-_-_-_-________-_________-_------_______ AR(l) 0.1759298 0.1362716 1.2910236 0.202 =============================================~===================== R-squared 0.882584 Mean of dependent var 113607.9 17756.75 0.847795 S.D. of dependent var Adjusted R-squared 2.59Er09 6927.532 Sum of squared resid S.E. of regression Durbin-Watson stat 2.008531 F-statistic 25.36904 Log likelihood -718.8998 ===================================================================
c =
Constant
to
Dll = Dummy Variables m Adjust for Seasonalicy CPI(-1) = Consumer Price Index - Lagged 1 Period Housing (-1) = Housing Starts - Lagged 1 Period PCE(-1) = Personal Consumption Expenditures - Lagged I Period UNEMP(-1) = Unemployment Rate - Lagged 1 Period AR(l) = Cochron - Orcucc Adjustment to Reduce Serial Correlation DL
Fig. 2.
data used in this analysis were based on the national unemployment rate. Since the restaurant in this study is located in a college town, it may be that the local unemployment rate is not highly correlated to the national rate (indeed, some authors have suggested that demand for higher educational services actually increases during times of national recession-hence employment in a college town might be expected to follow the same pattern as the demand for such educational services). In light of these possible causative relationships, it was felt that there was not a convincing economic rationale to exclude either the CPI or the UNEMP variables. Since both were statistically significant, they were retained in the fitted econometric model. The forecasting ability of the econometric model was then tested. The seven month forecast for this model is based on information contained in the initial time period, plus monthly updates of the independent variables. A graph of the forecasted and actual sales
135
Forecasting restaurant sales
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data are shown in Fig. 4. It can be seen from this graph that this model does very well in picking up the seasonal patterns. In terms of overall accuracy, the sum of the squared residuals (SSE) is 3.56 X 10s. 2. Exponential smoothing model The results of the fitted three parameter (seasonally adjusted) exponentially smoothing model are shown in Fig. 5. With this model the sum of the squared residuals is 2.81 x lo”, which is only slightly higher than for the fitted econometric model. Again, all of the measurements are within an acceptable range signifying a relatively good model. A graph of the fitted model and the initial data, plus a plot of the residuals is shown in Fig. 6. Notice again the relatively good correspondence between the actual and fitted time series. In forecasting seven periods (months) ahead, the exponential smoothing model performed about as well as the more complicated econometric model. The sum of the squared residuals is only slightly higher at 3.89 X 10s. The graph (Fig. 7) of the forecasted vs actual data shows that the exponential smoothing model was also able to capture the seasonal fluctuations quite well. 3. The Box-Jenkins model The Box-Jenkins model (Fig. 8) appears to perform the best at producing accurate forecasts. When fitted over the initial period, the Box-Jenkins model has a sum of the Statistic of squared residuals of 1.98 x lo’, an R2 of 0.906109 and a Durbin-Watson 2.081763. All of these measures are as good or better than the econometric model and the
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David A. Cranage and William P. Andrew
exponential smoothing model. A graph of the model and the initial data, as well as a plot of the residuals, is shown in Fig. 9. Notice how closely the fitted times series corresponds to the actual data. In forecasting for the next seven months, the results (Fig. 10) again show a high degree of correspondence between the forecasted values and the actual values. The model shows not only an ability to capture seasonal fluctuations, but also shows a stronger ability to accurately pick up the extremes of these fluctuations than the other models. As a result, the sum of the squared residuals for the forecast is the lowest at 3.03 x 10’.
Conclusions
and implications
For the monthly restaurant sales in the initial sample, the fitted time series models provided a degree of accuracy close to or better than the fitted econometric model. In forecasting beyond the time period of the initial sample, the time series models again performed close to or better than the econometric model. These results are significant in the fact that generally time series models are much less costly (from the standpoint of time, data collection, etc.) to implement than econometric models and do not suffer from many of the disadvantages previously mentioned for econometric models. These results should be taken as good news for a restaurateur who would like improved forecasting ability but does not have the resources to build complex econometric models, In comparing the two time series models tested (Box-Jenkins and exponential
Forecasting restaurant
sales
137
EXPONENTIAL SnOOTHING NODEL (THREE PAFMETER-XIJLTIPLICATIVE)
-___________________~~~~~~~~~~_____-~~~_-_____-~____-___ 0.380 0.000 0.000
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2.81E+09 6250.258 117660.0 294.7486 o.aaslaa 0.901871 1.003314 1.066503 1.074733 0.733071 1.041449 1.079675 1.085802 1.248980 1.061005 0.818410
5.
smoothing) using the sales data in our sample, the Box-Jenkins model performed better in both the initial time period and in the seven month forecasting period. However, there are several considerations regarding these results which should be noted. First, the procedure used in developing a Box-Jenkins model is substantially more complex than that used for the exponential smoothing model. With appropriate software, an exponential smoothing model can be fitted literally by pushing a couple of computer keys. Thus, even a restaurant operator with virtually no background in forecast models could successfully fit this type of model. Attempts are currently under way to simplify the Box-Jenkins model fitting process by using automated software (see, for example, Poulos et al., 1987). However, at the present time the cost effectiveness of the exponential smoothing model over the BoxJenkins model may outweigh the improved accuracy of the Box-Jenkins model for many restaurateurs. Second, where there are long-term cyclical patterns in the sales data, the Box-Jenkins model may perform at an even higher level than it did for these data. This is due to the fact that the Box-Jenkins methodology is designed to explicitly model such long-term time patterns (as opposed to the exponential smoothing model which is essentially a weighted moving average). Accuracy in forecasting is often gained by including such ‘history’ in the predictive model. Again, however, the benefits of such increased accuracy must be weighed against the expense (as in employing a more complex model) of obtaining them. Third, a point should be made in regard to econometric models. When there is a turning
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David A. Cranage
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Forecasting
restaurant
BOX-JENKINS
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YODEL
=====================================--_============ 2-TAIL
COEFFICIENT STD. ERROR T-STAT. VARIABLE ____-_____--_----__-~------~~~~~~~_~~~~~~~___-------_---_-~~~~~11.714165 89924.608 7676.5700 C 0.6713494 99.951886 148.88207 MONTH 2.2367274 7730.3572 3456.1017 Dl 2.4048699 9437.5681 3924.3570 D2 5.9732294 21533.855 3605.0608 D3 9.3550750 30468.168 3256.8598 D4 7.0534959 27544.157 3905.0363 05 -2.4808739 -9855.3901 3972.5478 D6 6.8637762 26236.906 3822.5178 D7 9.7256613 30077.756 3092.6180 D8 8.5957694 29421.884 3422.8331 D9 13.046328 49464.338 3791.4375 D10 8.6073536 28411.025 3300.7852 Dll
0.3409622
AR(l)
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0.000 0.505 0.030 0.020 0.000 0.000 0.000 0.017 0.000 0.000 0.000 0.000 0.000
2.5896202
0.013 0.010 0.025 0.067
0.3206575 0.1201090 2.6697205 AR(3) 0.2999337 0.1298408 2.3100119 ARR(4) -0.2384279 -1.8758148 0.1271063 m(5) ==================================================================== R-squared 0.906109 Mean of dependent var Adjusted R-squared 0.876064 S.D. of dependent var S.E. of regression 6293.135 Sum of squared resid F-statistic Durbin-Watson stat 2.081763 Log likelihood -671.3278 ====================================================================
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point in the economy, especially due to unexpected economic events, the econometric model may respond more quickly in its predictions to the changed situation than the time series models. Of course, whether this justifies the expense of utilizing such a model must again be decided by the individual operator. A final consideration is that our data included only one restaurant. While we feel that previous research in other industries supports the generality of our findings, the relative ranking in terms of accuracy of the models which we have examined could change for different types of restaurant operations. This is a question for further research. In any case, our study shows that any of the three models examined in this paper give results that under most circumstances, would be quite acceptable. In summary, our results suggest that for a restaurant operator with limited resources, an exponential smoothing model will give a very satisfactory forecast of sales under most
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Forecasting restaurant sales
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circumstances. Forecast accuracy can be improved by utilizing a Box-Jenkins model and the effects of economic turning points possibly seen more quickly with an econometric model. However, these models involve additional implementation costs which must be evaluated on a cost benefit basis by the restaurant operator.
References Armstrong, J. (1983) Relative accuracy of judgmental and extrapolation methods in forecasting annual earnings. Journal ofForecasting October/December, 437447. Box, G. E. P. and Tiao, G. C. (1975) Intervention analysis with applications to economic and environmental problems. Journal of the American Statistical Association 70, 7C79. Brandon, C. H. and Jarrett, H. E. (1977) Accuracy of externally prepared forecasts. Review of Business and Economic Research Fall, 3547. Bretschneider, S., Carbone, R. and Longini, R. L. (1979) An adaptive aproach to time-series. Decisions Sciences, April. Cerullo, M. J. and Avila, A. (1975) Sales forecasting practices-a survey. Manageriul Planning September/October, 33-39. Cleveland, W. P. and Tiao, G. C. (1976) Decomposition of seasonal time series: a mode1 for the Census X-II program. Journal of the American Statistical Association 581-587. Critchfield, T., Dyckman. T. and Lakonishok, J. (1978) An evaluation of security analysts’ forecasts. Accounting Review 651458. Dalrymple. D. J. (1975) Sales forecasting methods and accuracy. Business Horizons December, 6974. Geurts, M. D. and Ibrahim, I. B. (1975) Comparing the Box-Jenkins approach with the exponentially smoothed forecasting model application to Hawaii tourists. Journal of Marketing Research 12, 182-188. Geurts, M. and Kelly, J. (1986) Forecasting retail sales using alternative model. Internationul Journal of Forecasting 2, 261-272. Groff, G. K. (1973) Empirical comparison of models for short range forecasting. Management Science 20 (1)) 22-31. Hogarth, R. (1975) Cognitive processes and the assessment of subjective probability distributions. Journal of the American Statistical Association 70, 271-290. Hogarth, R. M. and Makridakis, S. (1981) Forecasting and planning: an evaluation. Management Science February, 115-138. Hollier, R. H., Khir, M. and Storey, R. R. (1981) A comparison of short-term adaptive forecasting methods. OMEGA International Journal of Management Science 9698. Johnson, T. E. and Schmitt, T. G. (1974) Effectiveness of earnings per share forecasts. Financial Management Summer, 64-72. Libby, R. (1976) Man versus model of man: some conflicting evidence. Organizational Behavior and Human Performance 1-12. Lorek, K. S., McDonald, C. L. and Patz, D. H. (1976) A comparative examination of management forecasts and Box-Jenkins forecasts of earnings. Accounting Review 321-330. Mahmoud, E. (1984) Accuracy in forecasting: a survey. Journal ofForecasting 139-159. Makridakis, S. and Hibon. M. (1979) Accuracy of forecasting: an empirical investigation. The Journal of the Royal Statistical Society 97-145. Narashimhan, G. V. L. (197.5) A comparison of predictive performance of alternative forecasting techniques: time series models vs. an econometric model. Proceedings of the American Statistical Association August, 459-464.
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Naylor, T. H. and Seaks, T. G. (1972) Box-Jenkins methods: an alternative to econometric models. international Statistical Review 113-137. Nelson, C. R. (1972) The prediction performance of the FRB-MIT PENN model of the U.S. economy. The American Economic Review December, 902-917. Newbold, P. and Granges, C. W. J. (1974) Experience with forecasting univariate time series and the combination of forecasts. Journal ofthe Royal Statistical Society, Series A 137, 131-165. Poulos, L., Kvanli, A. and Pavur. R. (1987) A comparison of the accuracy of the Box-Jenkins method with that of automated forecasting methods. International Journal of Forecasting 3. Xl267. Reid, D. J. (1971) Forecasting in action: comparison of forecasting techniques in economic time series. Procee&ngs offhe Joint Conference of ~per~~ti~ns Research Society. Reid, D. J. (1975) A review of short term projection techniques. Practical Aspects o.f F[~re~~~st~ng, Qpcrational Research Society, London. S-25. Schmidt, J. (1979) Forecasting state retail sales: econometric vs. time series models. The Annals of Regional Science 8, Yl-101. Zobcr, M. (1981) A comparison of Box-Jenkins methods of forecasting with regression. First Internatiorzal S?mposiurn on Forecasting, Quebec, Canada.