Journal of Cultural Economics 26: 53–64, 2002.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

53

Research Note

The Distribution and Predictabilityof Cinema Admissions

CHRIS HANDDepartment of Economics, University of Portsmouth, Locksway Road, Milton, Southsea, Hants,PO4 8JF, U.K.

Abstract. Using a time series data set covering the period 1936–1999, this paper investigates thestatistical distribution of cinema admissions and attempts to produce a forecasting model using theARIMA methodology.

Key words: ARIMA, cinema admissions, forecasting, heavy-tailed distribution

1. Introduction

Most of the published studies of cinema demand/admissions (e.g. Cameron 1986,1988, 1990) have adopted a static model specification. Dynamic models areemployed by Fernández-Blanco and Baños-Pino (1997) who use Johansen’s proce-dure to estimate a model of Spanish cinema admissions and Cameron (2000) whoadopts a Rational Addiction framework for U.K. data from 1965 to 1983. The focusof these studies was to identify which factors have an impact on cinema admissions,through estimating price and income elasticities for example. Less effort appears tobeen focused on forecasting cinema admissions. The cinema industry’s interest insuch forecasts should be obvious, but the academic interest may not be so obvious.Recently De Vany and Walls (1999) suggested that the revenue for an individualfilm is inherently unpredictable, as the statistical distribution of revenues has aninfinite variance. This then begs the question, if individual film revenues (and henceadmissions) cannot be forecast, can total admissions be forecast with any degreeof accuracy? U.K. cinema-goers select which film they see well in advance of theirvisit to the cinema. According to a recent Cinema and Video Industry AudienceResearch survey (conducted by the Cinema Advertising Association) the majorityof people decide which film to see in advance of their visit (CAVIAR Consortium,2000). Therefore it could be argued that film choice is the driving factor behindcinema admissions and therefore admissions may be unpredictable (as the sumof unpredictable admissions to the films on offer). Alternatively, the number of“cinema prone” individuals in a given population might vary independently of film

54 CHRIS HAND

availability due to other factors such as availability/accessibility of cinemas, priorexperience of cinema-going or demographic factors such as age. The proportion ofthis population subset choosing a particular film may not be predictable, but thisneed not affect the size of the subset. This would suggest that cinema admissionsshould be forecastable, as do the models of cinema admissions identified by thestudies cited earlier.

Forecasts of cinema admissions will of course be of interest to the cinema indus-try, allowing for example the viability of expansion plans or the success of effortsto promote the cinema to be assessed. There appear to be few sources of suchforecasts in the public domain, those produced by Dodona Research (Grummetand Kouling, 2000) being perhaps the most accessible.

2. The Distribution of Admissions

The discussion of heavy-tailed distributions in economics has been largely re-stricted to the financial economics literature, with De Vany and Walls’ (1999)study of film revenues a recent exception. Their study concluded that film revenuesfollowed a stable-paretian distribution which displays unbounded higher moments(stable-paretian distributions are discussed in more detail by Mandelbrot (1963)). Itmay be reasonable to assume that if revenues follow a stable-paretian distribution,admissions may follow the same distribution. If the series were heavy-tailed, thefirst-differences of admissions would not be normally distributed with the distrib-ution showing longer tails that the normal (i.e., leptokurtic). Hence a basic test forheavy tailedness is to test for non-normality in the change in the value of the series(i.e. first differences). In the finance literature where this approach has been used,it is usual to transform the data by taking logarithms (denoted as LAdm-LAdmt inTable I). The changes in admissions in the sample period are not normally distrib-uted, but the log of the change in admissions is normally distributed according to aKolmogorov–Smirnov test (at the 5% level) as is shown in Table I.

Table I. Normality tests

LAdmt-LAdmt-1 Admt-Admt-1

Kolmogorov–Smirnov Z. 0.709 (0.696) 1.504 (0.022)

Note: Asymptotic significance level in parentheses.

The tail index of the stable paretian distribution (usually denoted as α) can beestimated from a survival-type function. A conventional survival function givesthe probability of survival beyond a given time period, denoted as Pr(X > x). Asimilar function can be calculated to show the probability of a level of admissionsbeing above a given level, where x represents admissions rather than time. Usingthe top 10 per cent of the sample (i.e., the biggest changes in admissions) the uppertail index can be obtained from the equation log (Pr(X > x)) = a + b log x.

THE DISTRIBUTION AND PREDICTABILITY OF CINEMA ADMISSIONS 55

The tail index is given by the coefficient attached to log x. If the tail index is lessthan 2 it suggests that the variance of the distribution is unbounded. Using thistechnique, assuming the top 6 observations constitute the tail, an estimate of α =1.19 was obtained from an OLS regression. A second approach, using all of theobservations has been developed by Nolan (1999). Using the quantile estimator inNolan’s STABLE program, estimates of α = 2 were obtained (for a more detaileddiscussion of these tail index estimators see Lee, 1999).

On the basis of the results above, the logarithm of the changes in admissionswould appear to be normally distributed; hence it would be appropriate to proceedusing a standard econometric technique such as ARIMA.

3. The Modelling Strategy

This study attempts to model cinema admissions using the Box–Jenkins approachotherwise known as ARIMA (Autoregressive Integrated Moving Average) method-ology (Box and Jenkins, 1976). ARIMA modelling has fallen somewhat out offavour in mainstream economic modelling, although it was widely used and isstill used as a benchmark for assessing alternative forecasting techniques. Timeseries models are increasingly being estimated using cointegration models and timeseries modellers are cautioned to test their time series for non-stationarity (Harris,1995). There are practical reasons for adopting the ARIMA framework over anyother: firstly, ARIMA is a dedicated forecasting technique; in the tourism literature,ARIMA models have been found to be better able to forecast than other econo-metric models (Dharmaratne, 1995). Secondly, cointegration requires a significantamount of data to be available. In cases where only a few reliable or completeseries are available, cointegration techniques cannot be gainfully employed, butARIMA models can be estimated and will allow forecasts to be generated. For thereasons stated above ARIMA modelling is more often employed in the tourismeconomics literature, recent examples being Dharmaratne, (1995), Kim (1999)and Dalrymple and Greenidge (1999). ARIMA models also have the intuitivelyappealing property of letting the data “speak for themselves” and does not requirethe modeller to identify each and every factor which may influence the dependentvariable. It could be argued that the aggregate series likely to be available, such asprice, income, price of substitutes and the number of cinema sites and screens, maynot adequately explain cinema admissions with individual-specific factors playinga more important role.

4. The Data and Results

It is the second reason above, the lack of complete data series, which necessitatesthe use of the ARIMA framework. Earlier studies based on U.K. data (those byCameron) used quarterly data collected for the U.K. Department of Trade andIndustry (DTI) by H.M. Customs and Excise. With the passing of the Films Act

56 CHRIS HAND

Figure 1. Cinema admissions 1936–1999 (source CAA, undated).

in 1985 the Eady Levy, which supported domestic production with funds levied onexhibition, was abolished. With the levy gone, little reason was seen in collectingstatistics. The operation of the levy required the collection of data on the number ofcinemas, seating capacity, number of admissions, gross box office takings, averageticket price, payments for film hire and payments to the British film fund. Dataon numbers of full-time and part time staff were also collected. The collectionand publication of cinema statistics was recommenced in 1987 (using a voluntarypanel, benchmarked against the Annual Business Inquiry), hence there is a 2-yeargap in the official admissions, screens, box office takings and payments for filmhire series. Given this, data collected by the cinema industry can be used; such aseries is collected by the Cinema Advertising Association (CAA, undated). Themain source for this up to 1985 was the DTI, after 1985 data were collected bymarket research companies. Currently the data are collected by AC Nielsen EDI, amajor supplier of data to the film industry and to trade publications such as Varietyand Screen International. There is a shift in the series as the DTI data defined thecinema market geographically, covering just the U.K. However, as the U.K. and theRepublic of Ireland form one market for film distributors, the industry data coversboth territories (see Allin, 1998). Hence the CAA series has recorded higher valuesthan the official series in recent years. The problem posed by the structural shift inthe data is not insurmountable, being accounted for by the introduction of a dummyvariable where necessary.

The CAA series consists of annual observations of cinema admissions from1936 to 1999. The admissions series is shown as Figure 1. Cinema admissionsfell steadily throughout the 1950s, ’60s and ’70s after the upswing in admissionsduring the Second World War and its aftermath. Various explanations have been put

THE DISTRIBUTION AND PREDICTABILITY OF CINEMA ADMISSIONS 57

Figure 2. ACF plot of admissions.

forward for this decline (for example, the introduction of television, the availabilityof credit to purchase consumer durables, improved housing) by Spraos (1962).Subsequently, home video has also been cited as a cause of the decline in admis-sions, although whether it caused a decline or just prevented an increase is a mootpoint. The decline may have been caused in part by a vicious spiral, as admissionsfell cinemas began to close, as there was less opportunity to see films (as fewercinemas were operating) admissions fell further. The upturn in cinema admissionscoincides with the expansion of the exhibition sector. The first multiplex in theU.K. opened in 1986 signalling the start of the growth in cinema screens (and thetrend towards multi-screen cinemas in the U.K.).

The data were transformed into logarithms to account for the non-normality ofthe data. The ARIMA methodology requires the data series to be stationary as it as-sumes the process generating the series remains constant over time. An AugmentedDickey–Fuller test is usually performed to assess whether a variable is stationary.However, given that the data show both a change in the underlying trend and astructural shift, the ADF test is less powerful.1 Hence, the sample autocorrelationfunction (ACF) was also plotted, and the pattern it revealed was used to discernwhether the series is stationary (Figure 2). If the series is stationary (as the ADFtest suggested), the ACF plot should show little correlation between observations.

The plot shows significant correlations after many lags, which is indicative ofnon-stationarity. After first differences are taken, only lags 1 and 2 are significantwith subsequent lags tailing off. Hence the series is identified as I(1) and modellingproceeds on the differenced series.

An initial specification for the ARIMA model can be obtained from the ACF andPartial Autocorrelation Function (PACF) with the number of significant lags in theACF and PACF suggesting the number of moving average terms and autoregressiveterms respectively, which should be included as shown in Table II.

The ACF and PACF plots (Figure 3) show significant correlations at lags 1 and2 suggesting ARIMA (2,1,2) as an initial specification.

From this initial specification, 3 models can be identified on the basis of the sig-nificance (at the 5% level) of the individual coefficients: ARIMA (1,1,1), ARIMA(2,1,0) and ARIMA (0,1,2).2 For notational convenience these models will be de-

58 CHRIS HAND

Table II. ARIMA model specification guide

Model ACF PACF

AR (p) Tails off Cuts off at lag p

MA (q) Cuts off at lag q Tails off

ARMA (p, q) Tails off Tails off

Figure 3. Autocorrelation and partial autocorrelation function plots.

noted as I, II and III respectively below. Models I and II produced non-normalerrors due to the presence of a single outlying value. This was the result of thestructural shift in the data when government-collected figures were replaced byindustry-collected figures. This was corrected for by including an impulse dummy(1985 = 1) in the estimated models (Model III did not require an impulse dummyto produce normal errors).

The models should produce residuals that are white noise. The independence ofthe residuals can be tested using either the Box–Pierce Q test (Box and Pierce,1970) or the Ljung–Box (LB) test (Ljung and Box, 1978) both producing teststatistics which are approximately χ2 distributed. However, the LB test has bettersmall sample properties (Gujarati, 1995) so it alone is employed here (see appen-dix). For lag length k = 16 the LB test produces values of 8.111 for Model I,7.549 for Model II and 12.379 for Model III, with 13 degrees of freedom. None ofthese is significant at the 5% level, suggesting all three models produce white noiseresiduals.

THE DISTRIBUTION AND PREDICTABILITY OF CINEMA ADMISSIONS 59

Table III. Goodness of fit statistics

Model I II III

rmspe 9.1% 9.3% 9.9%

If more than one specification satisfies the LB test, a number of other decisioncriteria can be used to select the best specification for the model. A variety oftests can be found in the published ARIMA studies; which tests are employedand reported appears to depend on personal preference (and on which tests areproduced by the software package used). One can test for goodness of fit to theobserved series, using the assumption that the closer the fit, the better the forecastsare likely to be. In this study, goodness of fit is tested using the Root Mean SquarePercentage Error (RMSPE) of each model (defined in the appendix). The lower theRMSPE, the better the fit to the data. The test was conducted after the fitted valueswere transformed back into levels, and the results are shown in Table III.

As is clear from Table II, all three models perform reasonably well, with ModelsI and II being the best by a small margin. There is of course no guarantee that amodel that closely follows the observed series will also generate accurate forecasts.Hence, as Pindyck and Rubinfeld (1991) suggest, where more than one specifica-tion satisfies the assumptions about the residual (i.e., produces random errors) thebest specification should be selected on the basis of its out of sample forecastingaccuracy.

The last three years (1997–1999) of the sample were excluded from the esti-mation period, allowing these observations to be used to test the accuracy of themodels’ forecasts. A three year test period was chosen to provide both a suitablylong estimation period and a sufficiently large test period to draw conclusions.The three models were used to generate forecasts for 1997–1999 and the accuracyassessed using mean absolute percentage error (MAPE), which is defined in theappendix. The lower the MAPE value, the more accurate the forecast. In general aMAPE of less than 10 per cent is regarded as highly accurate (Lewis, 1982). Theaccuracy of the forecasts was evaluated after the predicted values were transformedback from logarithmic values to actual values. The results are shown in Table IV.

The three year forecasts produced by all three models underestimate the levelof admissions for all three years. However, the last three years of the sample (keptback to test the models’ predictions) display a rather unusual pattern: admissionsbeing 139 million, 135 million and 139 million in 1997, 1998 and 1999 respec-tively. Towards the end of the sample the series changes direction more frequentlythan in earlier years. This may be the effect of particularly successful films drawingmore people to the cinema (e.g. Four Weddings and a Funeral, The Full Monty andTitanic). Model I (the ARIMA (1,1,1) model) clearly outperforms the other models

60 CHRIS HAND

Table IV. Three year forecasts

Model I II III

Forecast (millions) 1997 126.6 121.9 122.4

1998 129.9 124.6 126.3

1999 132.8 125.0 126.3

Error (millions) 1997 12.4 17.1 17.6

1998 5.1 10.4 9.7

1999 6.2 14.0 13.7

Confidence Interval 1997 104.3–153.7 100.6–147.8 100.2–149.6

(millions) 1998 94.5–178.3 91.2–170.4 90.2–176.7

1999 85.7–205.9 79.2–197.3 78.7–202.5

MAPE 5.7 10.0 9.2

Table V. One year forecasts

Model I II III

Forecast (millions) 137.4 136.7 134.9

Error (millions) 1.6 2.3 4.1

Confidence Interval (millions) 114.16–165.37 113.33–164.90 110.27–164.96

1 year MAPE 1.15 1.94 2.95

on the basis of the MAPE values. However, it should remembered that any measureof forecast accuracy is sensitive to the test period chosen.

The models were re-estimated including 1997 and 1998 in the estimation periodto generate a 1-year forecast to assess the models’ shorter term forecasting ability.The 1-year forecast values, errors, confidence intervals and MAPE values are pre-sented in Table V. On the basis of the MAPE values, all three models perform well,but again Model I outperforms the other specifications.

The forecasts are accompanied by fairly wide 95 per cent confidence intervalswhich suggest that only short term forecasts can be made with any degree ofaccuracy (longer term forecasts compound the error, so the confidence intervalswill widen over time). To some extent this illustrates that the measure of forecastaccuracy depends in part on the forecast period used.

THE DISTRIBUTION AND PREDICTABILITY OF CINEMA ADMISSIONS 61

Table VI. Cinema admissions forecasts

Year Forecast (millions) Lower Bound Upper Bound

95% C.I. 95% C.I.

2000 142 118 170

2001 144 106 196

2002 146 95 226

Figure 4. Cinema admissions, 1980–1999, and forecasts, 2000–2002 (millions).

5. Forecasted Admissions

The ARIMA (1,1,1) model is used to produce a three year forecast for the future(with the previously used test period included in the estimation period) shown inTable VI.

It has been suggested that the growth in cinema admissions is not likely to con-tinue at its average rate over the past decade of approximately 3 per cent per annum(Grummet and Couling, 2000). The growth in cinema screens has outstripped thegrowth in admissions in recent years and does not seem sustainable. The expecta-tion is that the growth in admissions will slow, as the forecasting model shows. Asthe in-sample forecasting performance of the models and the widening confidenceintervals suggest, however, confidence can only be placed in the one-step aheadforecasts. This is clearly illustrated if the forecast values and confidence values areplotted as in Figure 4.

62 CHRIS HAND

6. Conclusions

The ARIMA model used above is a fairly simple model of cinema admissions.A more accurate model might be obtained through the use of less aggregated data(quarterly or monthly). In general, as Kim (1999) argues, monthly or quarterly datamay contain fewer distortions because of their lesser degree of aggregation. Dataon quarterly admissions have been collected and published for the U.K. by theU.K. Office for National Statistics (ONS), but the series is somewhat erratic withno clear seasonal pattern emerging, so there is less reason to expect it to producea significantly more accurate model (the ONS estimated seasonal factors are fairlysmall). Disaggregated observations will also show the effects of individual filmsmore clearly; a blockbuster film released in one quarter may induce a surge ofadmissions in that quarter suggesting that the error from a quarterly model maywell be as great if not greater than that from an annual model. Hence, using lessaggregated data could introduce distortions into the model and could potentiallyencounter problems of infinite variance.

The results above would appear to suggest that, even if the performance ofindividual films cannot be forecast, the level of cinema admissions is forecastable,at least in the short term. This might appear to be something of a paradox as totaladmissions can be forecast (at least 1 year ahead), admissions to particular filmscannot, yet film choice is the driving factor behind cinema-going. Whilst it has beenshown that film revenues appear to be heavy tailed (De Vany and Walls, 1999; Lee,1999), on the basis of the results presented here, annual admissions are not. Theimplications of the stable-paretian distribution will only be important if the word ofmouth support for a film encourages those who would not otherwise have attendedto go to the cinema (i.e., it results in extra admissions). If the word of mouth effectsencourage substitution of one film for another, there would be no effect on the levelof admissions and hence on the predictability of admissions. It is likely that bothresults occur, which would appear to suggest that admissions may be forecastable,but such forecasts may be subject to large errors.

Acknowledgements

I would like to thank the two anonymous referees for their detailed and helpfulcomments on earlier drafts of this paper.

Appendix: LB, RMSPE and MAPE Statistics

The LB test is defined as:

LB = n(n + 2)

K∑k=1

(ρ2

k

n − k

), (A.1)

THE DISTRIBUTION AND PREDICTABILITY OF CINEMA ADMISSIONS 63

where n is the number of observations, K is the lag length and ρ2k is the sample

autocorrelation coefficient. The test statistic follows the χ2 distribution with k −p − q degrees of freedom (i.e. lag length, k, minus the number of autoregressiveand moving average terms, p and q minus other variables, such as the 1985 dummyif included).

The rmspe is defined below:

rmspe =√√√√ 1

T

T∑t=1

(Y

ft − Y a

t

Y at

)2

, (A.2)

where Yft is the fitted value of Y in period t , Y a

t is the actual value of Y in periodt , and T is the number of observations. The error is expressed in percentage termsto allow for easier comparison across models.

The Mean Absolute Percentage Error (MAPE) is used to asses the accuracy offorecasts and is defined as:

MAPE = 1

n

n∑t=1

|et |Nt

× 100 , (A.3)

where n is the number of forecasts, et is the forecast error and Nt is the actualobservation. Using absolute values prevents positive and negative forecast errorsfrom cancelling each other out and hence provides a better indication of the forecastaccuracy.

Notes

1. Whilst it is possible to reformulate the ADF test to accommodate structural breaks, a simplerapproach to testing for stationarity using the Autocorrelation Function (ACF) is used here. Thisapproach appears to be more usually adopted in the ARIMA literature.

2. In ARIMA notation, an ARIMA (p, i, q) model contains p autoregressive terms, is based ondata integrated of order i (i.e., the data is I (i)) and contains q moving average terms.

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