Temporal aggregation in diffusion models of first-time purchase: Does choice of frequency matter?

NORTH- ~

Temporal Aggregation in Diffusion Models of First-Time Purchase: Does Choice of Frequency Matter? W I L L I A M P. P U T S I S , JR.

ABSTRACT Consistent with work in the advertisingresponse literature, the author addresses the time-intervalbias present when estimatinginnovationmodelsof new product growth and diffusionwith discretetime-seriesdata. Specifically, the author explores the theoretical and empiricalimplicationsof using varyingdata frequencies when estimatingdiffusionmodels usingboth nonlinearleast squares (NLLS) and ordinaryleast squares (OLS). Parameter estimatesacross fiveconsumerdurablesare obtainedusingannual, quarterly,and monthlydata. The centralconclusionis that the informationgainedand bias minimizedby usingseasonallyadjusted quarterly data results in empiricalestimates that are an improvementover those obtained by using annual data. This is true for both the NLLS and OLS estimates. In contrast, the move from quarterly to monthly data produces only marginalstatistical improvement.

Introduction Since the introduction of diffusion models in the early 1960s [e.g., 7, 16], studies addressing the issue of new product growth and diffusion have been voluminous [e.g., 5, 10, 14, 15, 17, 19, 21]. Most of these studies center around the seminal work of Bass [1], which has its origins in epidemiology, and this study follows in that tradition. It does so by extending some of the work on data-interval bias seen in the advertising response literature to diffusion models. Work on advertising-sales response has produced a general consensus that shorter data intervals are generally better than longer ones. For example, m a n y of the advertising response models estimated with a n n u a l data produce estimates of advertising carryover effects (on sales) that are 20 to 50 times larger than those estimated with monthly data [9, p. 221; 4]. ~ Whereas work subsequent to Clarke [4] using microparameter recovery procedures has produced mixed results [e.g., 2, 3, 24-27], a clear implication to be drawn from this literature is that using shorter interval data is preferred to longer interval data. WILLIAM P. PUTSIS, JR. is an Associate Professor of Marketingat the Yale School of Management, Yale University,New Haven, Connecticut. The early seminalwork in this area is the work by Clarke [4]. The reader is referred to Hansseus [9, pp. 221-223] and Leone [13] for an excellentreviewof this literature. Address reprintrequests to WilliamP. Putsis, Jr., AssociateProfessor of Marketing,Yale School of Management, Box 208200, Yale UniversityNew Haven, CT 06520-8200. Technological Forecasting and Social Change 51, 265-279 (1996) © 1996Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

0040-1625/96/$15.00 SSDI 0040-1625(95)00252-9

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Despite this, the overwhelming majority of empirical studies using new product growth and diffusion models use annual data in the derivation of the parameter estimates (see [23] for a notable exception). This seems to run counter to more traditional statistics, which suggests using as many (relevant) data points as possible in estimation, to the observations of Tigert and Farivar [23] regarding the reliability of parameter estimates obtained with a small number of data points, and to findings in the advertising-sales response literature. Accordingly, this study attempts to evaluate theoretically and empirically (using forecasting performance as the evaluation criteria) the impact of varying levels of temporal aggregation (i.e., various data intervals) on diffusion parameter estimates. Examination of this bias (referred to as temporal aggregation bias, data-interval bias, and time-interval bias interchangeably throughout) 2 is important for a number of reasons. First, from a forecasting perspective, a researcher should have some understanding of the impact that choice of data frequency (i.e., annual, quarterly, or monthly) has on empirical estimates. Data on sales in markets where diffusion models are typically applied are often available much more frequently than once a year. Thus, the primary contribution of this study is the comparison between forecasts obtained for each of five products using a specific estimation technique (nonlinear least squares), but varying data frequencies. Whether annual, quarterly, or monthly data produce more reliable forecasts using nonlinear least squares is an empirical question addressed in the Empirical Results section. To provide the proper perspective on the empirical results, the direction, magnitude, and source of the bias is discussed in detail in the next section. Both the analytical and empirical results suggest that the effect of varying data frequencies in the context of simple diffusion models is not nearly as large as has been speculated in the advertising response literature [4, 9]. Second, one must understand the direction and the magnitude of the bias present when estimating diffusion models via ordinary least squares (OLS) in order to compare appropriately more recent studies using the nonlinear least squares (NLLS) procedure with previous studies using O L S ) Thus, a further point of comparison (and a secondary contribution o f this research) will be between the OLS estimates and the NLLS estimates across the product categories holding data frequency constant. The results below contradict the findings of Mahajan, Mason, and Srinivasan in [15, chapter 8] that the forecasts using NLLS parameter estimates consistently outperform those obtained using OLS parameter estimates. Finally, from a more general perspective, now that the diffusion literature has been so well developed, we have begun to form a consensus on a variety of issues related to diffusion models. Indeed, Sultan, Farley, and Lehmann [22] in their meta-analysis of diffusion models do an excellent job relating estimated model parameters in a diffusion model to a variety of factors found in 213 applications. One factor not addressed in the Sultan, Farley and Lehmann [22] meta-analysis, however, is the frequency of data used in the study. This factor was not addressed in their analysis for an obvious r e a s o n virtually all of the studies in their analysis used annual data. The following research addresses the question: Does it matter? Research in the advertising-response literature suggests that it does. 2 Clarke [4] and Hanssens [9], consistent with work in advertising response, use the term data-interval bias. Schmittlein and Mahajan [19] and Srinivasan and Mason [21] use the term time-interval bias. Here, l will use the term temporal aggregationbias. All can be viewed as interchangeable in the current context. 3 Mahajan, Mason, and Srinivasanin [15,chapter 8] provideevidencethat suggeststhat the NLLSestimator outperforms the maximum likelihood (ML) estimator across a wide array of product categories. It is for this reason that the NLLS estimator is used as the basis of comparison here.

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A systematic and complete analysis of the empirical implications of using varying data frequencies is needed within the context of diffusion models. This study, which uses the Bass [1] model as a simple starting point, represents a modest attempt at such an analysis. The study begins with a short theoretical explanation of the bias that exists and concludes with an empirical examination of the use of varying data frequencies.

The Time-Interval Bias and New Product Growth and Diffusion Models THE TIME INTERVAL BIAS AND THE BASS [1] FORMULATION The seminal work of Bass [1] set out a model of product diffusion that utilizes a generalized logistic curve. Concentrating on first-time purchase only, the basic model can be succinctly stated as: P(T) = p + [(q/m) N(T)]

(1.1a)

S(T) = 0N(T)/0T = P(T)[m - N(T)I = pm + [ ( q - p) N(T)] - [(q/m) N(T) 2]

(1.1b)

where S(T) represents sales at time T, P(T) is the probability of purchase at time T given that no purchase has yet been made, N(T) is cumulative number of "adopters" to time T, and m is the maximum number of potential adopters of this product. The constant p equals the probability of purchase at time t = 0 since N(0) = 0. In addition, q reflects the word-of-mouth or "infectious" communication. As we do not observe sales, S(T), and cumulative sales, N(T), continuously, the Bass model has traditionally been estimated over a discrete time interval. This suggests that the probability of purchase over the tth time interval (from time t - 1 to t) is a linear function of the number of previous buyers until the beginning of the tth time interval: Pt = P + [(q/m)Nt-d, S, = P, [m

(1.2a)

- Nt-i]

= pm + [(q - p) Nt_ t] - [(q/m) N,_ 2].

(1.2b)

N t - l represents cumulative sales at the outset of the tth interval. Expected sales in the tth time interval (S,) is simply the probability of purchase (Pt) multiplied by the number of potential adopters at the outset of the tth interval, (m - N t - 1). Note that subscripts, t, will be used throughout to denote an interval representation (from time t - 1 to t), whereas brackets, (T), will be used to denote a "point" representation, i.e., at time T. In estimating the Bass [1] model and related extensions [11, 18] via OLS, there is a "time-interval" bias created by attempting to estimate equation 1.2b with discrete time series data. One way of viewing this bias is to note that the right-hand side (RHS) of equation 1.2b is theoretically the partial derivative of N(T) with respect to T evaluated at time t - 1, whereas the left-hand side (LHS) is the difference [N(t) - N ( t - 1)]. As a result, St will overestimate S(T) when sales are growing quickly (before peak sales) and underestimate S(T) when sales are growing slowly (after peak sales). This bias, which results from approximating any continuous function with discrete-period observations, is commonly referred to as the time-interval bias. It is also important to point out that the more frequently we observe sales, the smaller the time-interval bias. Finally, also, note that, by definition, the derivative of N(T), sales at time T, equals the limit as 1/n goes to 0 (or n goes to oo) of [N(t - 1 + (l/n)) - N ( t - 1)]/(1/n). Hence, as the frequency of data used to approximate the continuous function increases, the LHS

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of equation 1.2b approaches the RHS of equation 1.2b (the derivative of N(T)) and the time-interval bias approaches 0 as n approaches co. THE EFFECT OF VARYING DATA FREQUENCIESAND THE MAGNITUDE OF THE TIME-INTERVALBIAS (OLS) To begin, note that the probability of purchase over some interval I t - 1, t] clearly depends upon the length of the interval: the probability of purchase is considerably greater in a year than it would be in a month, for example. Hence, the measured parameters p and q in the interval representation of the Bass [1] model (equation 1.2) must be different when the model is measured with data of different frequencies. In particular, it can be shown that parameters p and q estimated using data with unit frequency (i.e., yearly data) will always have values greater than (n p3 and (n qg, where p' and q' are the Bass model parameters estimated with data of frequency n. Equation 2.1 is derived in the

Appendix: (p - np~ = [n(N(t) - N ( t - 1 + (1/n)))]/m >0.

(2.1)

Because (p - n p ' ) is greater than zero, parameter p in the discrete approximation using annual data is larger than n times the parameter p' in the discrete approximation using data with frequency n. This expression represents the magnitude of the bias found in the parameter p. This bias will vary directly with the level of cumulative sales, N(t), and will be different for different frequency data. We can make a similar statement about Bass' coefficient of imitation, q. In particular, (q - nq3 =

[N(t) - n N ( t - 1 + (l/n))] + [ ( n - 1)N(t- 1)]

[ N ( t - 1) - ( N ( t - 1)2/m)] _ [m - N ( t - 1)][[nN(t) - n N ( t - 1 + (1/n))]/m] I N ( t - 1) - ( N ( t - 1y/m)]

(2.2)

Equation 2.2 is derived and shown to be positive in the Appendix. If equation 1.2 is estimated via OLS, we would expect that the estimates of p and q using yearly data (p, q) would approximately equal the estimates of p and q obtained using data with frequency n (p', q9 multiplied by a factor of n. As with parameter p, it is crucially important to note that this expression represents the magnitude of the bias found in the parameter q. This bias will again vary as the level of cumulative sales, N(t), changes and will be different for different frequency data. Theoretically, the difference between p and (n p') is given by equation 2.1, and the difference between q and (n q') is given by equation 2.2. In practice, the parameter pair (p', q3 does not exactly equal n(p, q) for two reasons: (1) there is information provided by the n partitioned intervals between t - 1 and t, and (2) p and q have a higher degree of bias than does p' and q', as discussed earlier. This might suggest that the Bass [1] model estimated by OLS using monthly or quarterly data should provide estimates of p and q with a smaller degree of bias and better predictive ability than the estimates obtained by estimating the Bass model by OLS using annual data. However, it is not entirely clear that the information provided by the more frequently observed data is useful. The issue of relative performance between the Bass model using different data frequencies, phrased in this manner, becomes an empirical issue. The empirical results for five consumer durables, discussed in the next section, suggest that the estimates obtained with quarterly and monthly data are statistically superior to those obtained using annual data in a majority of cases.

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THE EFFECT OF VARYING DATA FREQUENCIES AND THE MAGNITUDE OF THE TIME-INTERVAL BIAS (NLLS) Srinivasan and Mason [21] suggested an interesting solution to the problem presented by equation 1.2b. In short, the time interval bias is created by RHS of the Bass formulation (equation 1.2b) being equal to the partial derivative of the sales curve evaluated at time t - 1, whereas the LHS is equal to the difference in sales over the tth interval (from time t - 1 to t). In order to "correct" for this, they proposed a formulation such that the RHS equals the LHS of equation 1.2b in a somewhat unusual fashion. Instead of attempting to set the LHS equal to the appropriate derivative, they derived an expression for the RHS, such that the RHS equals the same difference as the LHS. They recognize the fact that there is a continuous-form expression available for the difference between N(t) and N ( t - 1) and estimated parameters p, q, and m via the difference formulation using an NLLS procedure. An additional benefit is that the NLLS procedure provides estimates of the parameter standard errors, whereas the OLS procedure does not. Srinivasan and Mason's [21] NLLS suggestion has become the standard estimation technique for estimating diffusion models. The comparison between the NLLS estimates obtained using annual data and those obtained using data with frequency n is much simpler than for the OLS case. To illustrate this, first note that if we define a unit time period as equaling 1 year, then each time period has length l / n if the data has frequency n, for all n > 1. Thus, from the perspective of the time-interval bias, we would expect that p = p' and q = q', where again, the pair (p, q) are the estimates of p and q using data of frequency 1 and (p', q~ are the estimates of p and q using data of frequency n. Simply stated, because time is measured consistently, we are still estimating the function 1.2 and the estimates of p and q should be equal. Differences in (p, q) versus (p', q~, therefore, are solely attributable to the information added to the estimations by the data of higher frequency. It should also be noted that when using data of frequency n and estimating via NLLS it is possible to define a time interval as one unit instead of 1/n units. For example, if one is using monthly data, it is natural to define one observation to be one time period (i.e., 1 month) rather than 1/12 a time period (i.e., 1/12 of a year). In this case, the parameter pair (p*, q*) obtained with a frequency n and defined in this fashion exactly equals n (p', q~. As mentioned previously, understanding any empirical differences that may exist is crucial to being able to compare earlier studies that have used OLS with annual data and more recent (future) studies that have used (will use) NLLS with quarterly or monthly data. The presence (or absence) of these differences, as well as the magnitude of the differences, and their implications on the forecast performance of the model, are important pieces of information to anyone using new product growth models to forecast future sales of a given consumer durable.

Empirical Results The Bass model and variations of the Bass model have been estimated numerous times in practice and in academic research, invariably using annual data. The purpose of previous direct applications of the Bass model has generally been to forecast eventual first-time sales given data on first-time purchase early on in a product's life-cycle. The empirical section of this study attempts to apply the Bass model to product categories early on in their life-cycle and evaluate within-sample and forecast performance of the Bass model using annual, quarterly, and monthly data, respectively.

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Accordingly, the Bass [l ] model was estimated using both OLS and the NLLS procedure suggested by Srinivasan and Mason [21] using monthly, quarterly, and annual data on color television sets over the years 1964-1970, video cassette recorder (VCRs) sales from 1978-1985, total microwave oven sales from 1975-1979, household trash compactor sales over the period 1972-1976, and projection televisions over the years 1981-1985. In addition, as a n n u a l data o n microwave sales were also available over the years 19721974, the Bass model was estimated for microwave ovens using annual data over the period 1972-1979 via both OLS and NLLS. The data for color televisions, projection televisions, a n d video cassette records were obtained from Television Digest with Consumer Electronics, an industry trade association publication. The data for microwave ovens and trash compactors were obtained from Appliance magazine and from the Association of H o m e Appliance Manufacturers (AHAM). The interval of estimation for each product was chosen using a variety of criteria. First, each sample began at the earliest possible date at which data were available. 4Second, the final year of each sample was chosen to minimize the a m o u n t of replacement purchases present in the s a m p l e - since the focus of this study is on the within-sample and short-term forecast performance of the Bass [1] model, there are better models available for forecasting long-term demand into the "replacement period" [see, for example, 12]. Third, every attempt was made to keep the years estimated consistent with earlier studies that applied the Bass model to these categories using OLS and annual data. 5 Finally, in the spirit of Tigert and Farivar [23], varying sample lengths could provide some intuition about the robustness of any forecasts made based upon the parameter estimates. Consequently, the products represented contain product categories where sales are only observed prior to point of inflection (VCR, microwave ovens, and projection televisions) and product categories where sales are observed beyond the inflection point (color televisions and trash compactors). Using actual monthly and quarterly sales figures presents problems for diffusion models because of the inherent seasonal nature of such data. However, seasonal components can be removed relatively easily from time-series data. As diffusion models are concerned with making inferences about general sales trends over time, seasonally adjusted data should provide reasonable estimates in this regard, whereas nonseasonally adjusted data can produce estimates biased by the seasonal factors largely irrelevant to general sales trends. Accordingly, each monthly and quarterly data series was seasonally adjusted using the census X-11 moving average method prior to estimating each of the parameters. 6 The parameter estimates obtained in this fashion were then used to forecast future sales. 4 Sales figures beginning2 to 3 years before the first year of published data were obtained from industry trade associations(ElectronicsindustriesAssociationvia TelevisionDigest withHomeElectronicsand the Association of Home ApplianceManufacturers) for color televisions,microwave ovens, and household trash compactors. 5 For example, growth models for color televisionswere estimated by Schmittleinand Mahajan [19] over the years 1963-1970, by Nevers [17] over the years 1962-1969, and by Easingwood, Mahajan, and Muller [6] over the years 1963-1970. 6 Becauseresultsshould(ideally)not be dependentupon the seasonaladjustmentprocedureUsed,two things were done to address this issue. First, the census X-II procedure was used as it has become a standard for seasonallyadjustinggovernmentaldata and, largelyas a consequence,is widelyavailablethrough moststatistical packages, l wouldexpect that most applicationsusingmonthlyor quarterlydata would use the X-I 1 procedure (employingSAS or RATS, for example) to seasonallyadjust the data. Second, these results were compared to those obtained using the Sims [20] spectral seasonal adjustment technique. There was little difference found between the results obtained using the X-11 technique and those obtained from using the Sims procedure, suggestingthat the biases observed are not particularlysensitiveto the choice of seasonaladjustmentprocedure used. One should not generalizethese results to nonseasonallyadjusted data, however.

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Forecasting future sales and comparing the forecasts obtained using different data frequencies presents a statistical problem analogous to comparing apples and oranges. As it was believed a priori that the estimates using the monthly data would provide better forecasts than those obtained with quarterly and yearly data, evaluation procedures and criteria were set out to favor the annual data whenever there was any question about methodology. Because the primary goal of this section is to evaluate how the parameter estimates obtained from monthly and quarterly data compare with those obtained from annual data, the predicted sales using the annual data results were used as a basis for comparison. Specifically, the parameter values obtained using annual data were used to predict sales for each year of the sample period and for 3 years beyond the sample period. In addition, the quarterly and monthly data parameter estimates were used to predict quarterly and monthly sales, respectively, over the same time period. The quarterly predictions were then used to calculate a predicted annual sales figure equal to the sum of the predicted quarterly sales figures for the four quarters of each year. This annual sales figure was calculated for each year, beginning with the first year of the sample period and ending with 3 years into a forecast period. Similar calculations were conducted using the monthly parameter estimates. These annualized predicted values were then compared across data frequencies. Specifically, the mean square error (MSE) and mean absolute deviation (MAD) of the predicted values over the sample period and over a 2-year and 3-year forecast period were calculated from the annualized predictions discussed in the previous paragraph. These calculations are presented in Tables 1 and 2. As the 2- and 3-year forecasts put the higher frequency results at a distinct disadvantage, the 2-year and 3-year forecasts were envisioned as a more stringent test of the a priori belief that the higher frequency data would produce more reliable forecasts. Table 1 presents the general results over the sample period for each of the products. Recall that p represents the coeffcient of innovation, or the strength of innovative effects, whereas q represents the coefficient of imitation, or the strength of imitative effects. Also, m represents the eventual number of adopters and t* denotes the time period in which the model predicts peak sales will occur. Because the vast majority of parameter estimates were statistically significant at all relevant confidence levels, individual standard errors and t-statistics are left out of the table to keep it as uncluttered as possible. Parameter estimates marked with a " + " denote values that are n o t statistically different from zero at the 95°70 level of confidence. Before proceeding, an important note is in order regarding the interpretation of the values of p, q, and t* above. The theoretical relationship between each of these parameters under OLS and NLLS was discussed in the last section. In short, the theory suggests that the values of p and q estimated with quarterly data under OLS should be approximately three times those estimated using monthly data. In general, the annual estimates of p and q should be approximately n times the parameter estimates using data of frequency n. NLLS estimates for p and q should be approximately equal across data frequencies. The value of t* under NLLS are expressed on constant time units (years), whereas the OLS estimates are expressed in the estimation frequency. For example, the first two lines of Table 1 indicate that the estimated t* using monthly data equals 57.3 months (4.8 years), whereas the quarterly estimate of t* is 18.8 quarters (4.8 years). As with p and q, the NLLS estimates of t* are expressed in years, and are, therefore, directly comparable across data frequencies. Finally, note that, as predicted, OLS annual parameter estimates for p and q are approximately four times the quarterly parameter estimates. The quarterly

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TABLE 1 Bass [1] Model Parameter Estimates - Sample Period Product (est.)

Years

Frequency

p

Color TV (OLS)

64-70 64-70 64-70

Monthly Quarterly Annually

0.00299 0.00979 0.02644

0.04385 0.12571 0.46485

40,080,785 41,485,052 50,772,776

57.3 9.44E + 10 2.45E + 05 18.8 1.59E + 11 3.33E + 05 5.8 5.38E + 11 6.37E + 05

Color TV (NLLS)

64-70 64-70 64-70

Monthly Quarterly Annually

0.03489 0.03558 0.03573

0.54707 0.54707 0.54539

38,670,450 39,006,020 38,951,540

4.7 9.38E + 10 2.67E + 05 4.7 9.15E + 10 2.26E + 05 4.7 9.02E + 10 2.45E + 05

VCR (OLS)

78-85 78-85 78-85

Monthly Quarterly Annually

(*) (*) (*)

(*) (*) (*)

VCR (NLLS)

78-85 78-85 78-85

Monthly Quarterly Annually

0.00116 0.00116 0.00100

0.69054 0.71540 0.66820

91,079,690 81,749,700 116,693,900

9.2 9.99E + 10 2.69E + 05 9.0 9.12E + 10 2.67E + 05 9.7 7.31E + 10 2.35E + 05

Microwave (OLS)

75-79 75-79 75-79

Monthly Quarterly Annually

0.00336 0.01046 0.00790

0.03095 0.08942 0.47070

25,124,155 25,752,997 22,389,652

64.7 1.04E + 10 7.24E + 04 21.5 1.55E + 10 9.51E + 04 8.5 2.18E + 10 1.05E + 05

Microwave (NLLS) 75-79 75-79 75-79

Monthly Quarterly Annually

0.04024 0.04013 0.00660

0.38207 0.37191 0.50200

24,633,910 25,253,480 21,282,940

5.3 1.0BE + 10 7.71E + 04 5.4 1.08E + 10 7.78E + 04 8.5 1.82E + 11 3.16E + 05

Trashcomp (OLS)

72-76 72-76 72-76

Monthly Quarterly Annually

0.00992 0.03090 0.09520

0.03698 0.09734 0.38192

1,790,844 1,891,005 2,156,276

28.0 1.71E + 09 3.63E + 04 8.9 1.92E + 09 3.75E + 04 2.9 2.14E + 09 4.41E + 04

Trashcomp (NLLS) 72-76 72-76 72-76

Monthly Quarterly Annually

0.12206 0.12343 0.06980

0.47183 0.44185 0.55462

1,710,288 1,763,263 1,850,546

2.3 1.67E + 09 3.63E + 04 2.3 3.51E + 09 1.81E + 04 3.3 4.90E + 09 5.84E + 04

Proj TV (OLS)

81-85 81-85 81-85

Monthly Quarterly Annually

0.00193 0.00588 0.01619

0.02664 0.08015 0.31502

3,490,986 3,477,124 3,794,813

91.9 2.91E + 08 1.51E + 04 30.4 2.76E + 08 1.57E + 04 9.0 5.75E + 08 5.70E + 04

Proj TV (NLLS)

81-85 81-85 81-85

Monthly Quarterly Annually

0.02381 0.33390 0.00165t 0.17636 0.00159t 0.19737f

(*) (*) (*)

q

m

t*

(*) (*)" (*)

MSE

(*) (*) (*)

MAD

(*) (*) (*)

3,344,035 7.4 2.73E + 08 1.49E + 04 6,380,497t 24.2 2.66E + 08 1.87E + 04 4,726,311t 24.5 4.14E + 08 2.liE + 04

* The OLS procedure yielded the incorrect sign for some of the parameters in equation 1.2; hence, it is not possible to derive estimates of p, q, and m, respectively. t Not statistically significant at the 5% level of confidence (NLLS estimates only).

estimates are, in turn, approximately three times the monthly parameter estimates. The NLLS estimates are generally consistent across data frequencies, as expected. Table 2 presents the estimated 2-year and 3-year forecast errors. Note that this comparison between the predictions made using annual data and those made using higher frequency data puts the quarterly and monthly data predictions at a distinct disadvantage as a 2-year annual forecast is actually a 24-step forecast using monthly data. 7 Despite A comparison was also made among the forecast errors of a two-step forecast using annual data (a 2-year forecast), the two-step forecast using quarterly data at an annualized rate (a 6-month forecast) and the two-step forecast using monthly data at an annualized rate (a two-month forecast). The MSE and the MAP of the forecast errors were considerably higher for the annual data predictions in this case. In addition, the quarterly data predictions had considerably higher MSE and MAD than the monthly results. Although valid statistically, this comparison tells us little about the relative predictive ability of the models estimated with different levels of temporal aggregation.

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TABLE 2 B u s [ll Model 2-Yelr and 3-Year F o r e c ~ Errors Product (est.)

Years

Frequency

MSE (2)

MAD (2)

MSE (3)

MAD (3)

Color TV (OLS)

64-70 64-70 64-70

Monthly Quarterly Annually

4.37E + 11 4.03E + 11 7.73E + 12

6.08E + 05 6.06E + 05 2.46E + 06

2.59E + 12 2.39E + 12 1.96E + 13

1.28E + 06 1.25E + 06 3.83E + 06

Color TV (NLLS)

64-70 64-70 64-70

Monthly Quarterly Annually

1.86E + 13 1.83E + 13 1.87E + 13

4.92E + 06 4.09E + 06 4.94E + 06

3.33E + 13 3.30E + 13 5.35E + 13

1.25E + 06 1.36E + 06 2.41E + 06

VCR (OLS)

78-85 78-85 78-85

Monthly Quarterly Annually

(*) (*) (*)

(*) (*) (*)

(*) (*) (*)

(*) (*) (*)

VCR (NLLS)

78-85 78-85 78-85

Monthly Quarterly Annually

3.53E + 12 7.04E + 11 3.87E + 13

1.85E + 06 8.29E + 05 6.06E + 06

2.43E + 12 9.91E + 11 5.91E + 12

2.87E + 06 1.01E + 06 8.81E + 06

Microwave (OLS)

75-79 75-79 75-79

Monthly Quarterly Annually

1.62E + 12 1.63E + 12 2.27E + 12

1.19E + 06 1.20E + 06 1.42E + 06

2.04E + 12 2.07E + 12 3.00E + 12

1.36E + 06 1.37E + 06 1.65E + 06

Microwave (NLLS)

75-79 75-79 75-79

Monthly Quarterly Annually

1.62E + 12 1.56E + 12 1.96E + 12

1.19E + 06 1.17E + 06 1.33E + 06

2.08E + 12 1.97E + 12 2.55E + 12

1.367 + 06 1.34E + 06 1.53E + 06

Trashcomp (OLS)

72-76 72-76 72-76

Monthly Quarterly Annually

2.70E + 10 2.43E + 10 2.60E + 10

1.59E + 05 1.51E + 05 1.20E + 05

3.61E + 10 3.28E + 10 3.40E + 10

1.84E + 05 1.75E + 05 1.47E + 05

Trashcomp (NLLS)

72-76 72-76 72-76

Monthly Quarterly Annually

3.08E + l0 3.18E + 10 3.60E + 10

1.71E + 05 2.48E + 05 2.20E + 05

4.01E + 10 6.66E + 10 6.51E + l0

1.95E + 05 2.57E + 05 2.86E + 05

Proj TV (OLS)

81-85 81-85 81-85

Monthly Quarterly Annually

2.55E + 13 2.99E + 13 4 . l I E + 13

5.43E + 06 5.41E + 06 7.65E + 06

4.12E + 13 4.34E + 13 4.94E + 13

5.91E + 06 5.82E + 06 5.99E + 06

Proj TV (NLLS)

81-85 81-85 81-85

Monthly Quarterly Annually

2.31E + 13 2.59E + 13 3.87E + 13

4.98E + 06 5,11E + 06 6.82E + 06

3.45E + 13 3.21E + 13 3.66E + 13

4.13E + 06 4.59E + 06 4.67E + 06

this disadvantage, the forecast errors are lower for the monthly and quarterly estimates across all of the product categories, with the exception of the series for trash compactors (OLS only). A statistical comparison of the squared forecast errors of the predictions made when using different data frequencies was also conducted, using a traditional test for zero correlation based on the sample correlation coefficient as suggested by Granger and Newbold [8, p. 281]. Over the sample period, the sales predictions made using the quarterly and monthly data were generally not statistically distinguishable at most relevant confidence levels. However, the predicted sales in the sample period based upon the monthly data estimates did have a statistically (a = 0.05) lower squared error than the sales based upon the quarterly estimates for trash compactors (NLLS only). In the forecast period, there was no statistical difference (a = 0.05) between the predictions made using the quarterly and monthly data for color televisions (OLS and NLLS), microwave ovens (OLS and NLLS), household trash compactors (OLS only), and projection televisions (OLS and NLLS). The predictions made into the forecast period using monthly data are

274

W.P. PUTSIS, JR.

statistically superior to (again, a = 0.05) those obtained using quarterly data for only two products: VCRs (NLLS) and household trash compactors (NLLS only). Comparing the forecasts made using annual data with those made using higher frequency data, all of the annual forecast errors were higher (with the exception of the OLS trash compactor results) than those using either monthly or quarterly data. These differences were not always statistically significant, however. Specifically, the two-step forecast errors were statistically higher for the annual data for color televisions (OLS only), VCRs, and for projection televisions (OLS and NLLS). All other differences in the squared errors were statistically insignificant (a = 0.05). With respect to the three-step errors, the annual results were dominated by the higher frequency data results for color televisions (OLS and NLLS), VCRs, and trash compactors (NLLS months only). Again, all other differences in the squared forecast errors were not statistically significant at the 95 070level of confidence. In summary, a total of nine comparisons (five products times two estimation techniques minus one, as OLS for the VCR data failed to produce reasonable parameter estimates) were made between monthly and annual data forecasts. Similarly, an additional nine comparisons were made between quarterly and annual data forecasts. For the two-step forecasts, the higher frequency (monthly and quarterly) forecast errors were significantly lower than the annual forecasts in four out o f the nine comparisons. Likewise, for the three-step forecasts, the higher frequency (again monthly and quarterly) forecast errors were smaller in four out of the nine comparisons (although not the same four as in the two-step case). It is important to note that the higher frequency data had smaller forecast errors for all products and for all estimation techniques, with the exception of the OLS results for trash compactors. 8 As these differences were not always statistically different, one is left with the conclusion that forecast performance did not deteriorate when moving from lower to higher frequency data, whereas a statistically improved forecast was observed in approximately 45°70 o f the cases. Figures 1 and 2 present a graphical representation of the information contained in Table 2. 9 This analysis of the forecast (Table 2) and sample (Table 1) period prediction errors suggest a number of conclusions. First, there is generally an improvement in the model's performance when one switches from annual data to quarterly data, but the jump to monthly data from quarterly data produces a forecast improvement that is marginal at best. However, the improvement in the within-sample and forecast period errors when one goes from annual to quarterly or monthly data is statistically significant in only a subset of the products (despite the fact that the within-sample and forecast period errors are almost universally higher for the annual data, they are not always significantly higher). Second, unlike Mahajan, Mason, and Srinivasan [15, chapter 81, and despite the conclusions in the theoretical section, we find no pattern o f improvement in the model's performance when using NLLS instead of OLS. However, there are three distinct advantages of the NLLS estimator observed in this study: (1) parameter standard errors are easily obtainable, (2) the OLS procedure sometimes produces parameter estimates that have an incorrect sign so that it is not possible to obtain estimates of p, q, and m, and (3) s The anomaly of the OLS results for trash compactors may be the result of the instability of parameter estimates that result from applying OLS to only 5 years worth of data (see [23]). This is supported by the fact that the NLLS estimations did not result in the same (apparently) anomalous result. 9 Some of the figures in Table 2 have been multiplied by either a factor of 10 or 1.000 so that the various product forecast MSEs could be represented on the samegraph. The followingrepresents the adjustment factors for Figure 1: microwaveovens (10), trash compactors (1,000). The followingrepresents the adjustment factors for Figure 2: VCR (10), microwaveovens (10), trash compactors (1,000).

TEMPORAL AGGREGATION IN DIFFUSION MODELS

275

50

--O4

4O

+ LU ¢/)

E 30 LU O9 20

6

OLS NLLS NLLS OLS NLLS OLS NLLS OLS NLLS C'IV VCR M-W Ovens Tr. Comp. Proj. TV

k

]

I

Fig. 1. Two-step MSE forecast errors by product and estimation technique.

the NLLS estimator has a much sounder theoretical foundation. Third, both the NLLS and OLS procedures do a good job estimating sales over the sample period across all product categories, and, as expected, the model's performance deteriorates appreciably as we move into the forecast period. Finally, it is important to note that each model was also estimated over shorter time intervals for each of the five durables. Over the first few years of the sample for each product, the models were not estimable using annual data, whereas estimation with quarterly and monthly data produced empirical predictions qualitatively similar to those presented earlier.

Conclusions and Limitations The empirical results are quite interesting when viewed in light of the bias derived in the time-interval bias section and in the Appendix. Theoretical examination of the bias (equations 2.1 and 2.2) suggests that the bias is relatively small in magnitude for both p and q. This theoretical finding contrasts sharply with the empirical findings in the advertising response literature [4, 9], which suggests that carryover effects estimated using annual data can be 20 to 50 times larger than those estimated using monthly data. The empirical results from this study suggest that (at least for some products) the difference between forecasts obtained when applying the Bass [ 1] model using annual versus monthly data can also be quite large (see Figures 1 and 2). So if the bias is analytically quite small, why then are at least some of the empirical results consistent with the advertising response literature?

276

W . P . PUTSIS, JR.

8O

LU

A

60

W

E v

LU CO 40

.~ 20 e-

OLS NLLS NLLS OLS NLLS OLS NLLS OLS NLLS CTV VCR M-W Ovens Tr. Comp. Proj."rv

]

I

Fig. 2. Three-step MSE forecast errors by product and estimation technique.

To answer this question, it is important to note that there are two general comparisons taking place in the empirical section. The first comparison is between OLS and NLLS results holding data frequency constant. This comparison tells us about the bias present in the empirical results as the bias is only present in the OLS parameter estimates; there is no discernable pattern of difference between the OLS and NLLS results. The second comparison is between the data frequencies holding estimation technique constant; as discussed earlier, this comparison suggests that higher frequency data generally produces a lower degree of forecast error (see Figures 1 and 2). All of this suggests that the bias observed both empirically and theoretically is in fact quite small. It appears that most of the improvement in terms of forecast performance comes from choosing a higher frequency data, n o t from using NLLS rather than OLS. Simply put, the information gained appears to swamp any reduction in bias that occurs. In more general terms, every attempt was made to be consistent with previous studies that have estimated the Bass model for some of these product categories. Further, every attempt was made to minimize the "con[amination" of the data by replacement sales through limiting the length of the data series. New product growth and diffusion models such as the Bass [1] model are generally not the best models to use to do long-term forecasting of product sales where replacement demand is becoming a significant part of total product demand. Their usefulness is their ability to forecast short-term sales in the early years of a product's life and to provide an initial estimate of the ultimate market for the product. Unfortunately, 5 years of annual data leaves only two degrees of freedom.

TEMPORAL AGGREGATION IN DIFFUSION MODELS

277

Accordingly, using annual data to estimate such models often makes little sense. Before one can appropriately utilize d a t a o f higher frequency, however, it is i m p o r t a n t to understand the theoretical and empirical relationships between the models estimated with annual versus higher frequency data. Highlighting these relationships has been the focus o f this study. Nonetheless, it should be recognized that for some of the products, replacement sales did exist, but they were generally kept to a minimum. Further, the comparisons used in this study are made on the basis o f seasonally adjusted quarterly and monthly data. When using quarterly or monthly data, seasonal adjustment becomes a necessary part o f estimation. Unfortunately, using seasonal adjustment techniques can often increase the length o f the data series required for estimation. This fact could often weigh heavily against any benefits o f using higher frequency data. Most published versions o f innovation models o f new product growth and diffusion use annual data. Evidence is presented in this study suggesting that the same models estimated with seasonally adjusted quarterly data o u t p e r f o r m those estimated with annual data. In addition, there does not appear to be a significant gain (yet no consistent loss) to using monthly data in estimating these models. As quarterly and monthly data are often available from a variety o f sources for a variety o f products and as seasonal adjustment techniques are easily accessible in both the frequency and time domain, future studies should use higher frequency data wherever possible.

The author thanks W. Keith Bryant, Subrata K. Sen, and seminar participants at Corner University f o r helpful comments on earlier versions o f the article, and Craig Schuitz o f the Association o f Home Appliance Manufacturers for providing an important portion o f the data. Comments from three anonymous reviewers improved the manuscript substantially. Support o f the Yale School o f Organization and Management Research Fund is gratefully acknowledged.

References 1. Bass, Frank M., A New Product Growth Model for Consumer Durables, ManagementScience 15(5), 215227 (1969). 2. Bass, Frank M., and Leone, Robert P., Estimating Micro Relationshipsfrom Macro Data: A Comparative Studyof Two Approximationsof the Brand LoyalModelunder Temporal Agsregation, Journalof Marketing Research 23, 291-297 (1986). 3. Bass, Frank M., and Leone, Robert P., Estimation of BimonthlyRelations from Annual Data, Management Science 29, 1-11 (1983). 4. Clarke, Darral G., Econometric Measurement of the Duration of Advertising Effect on Sales, Journalof Marketing Research 13, 345-357 (1976). 5. Dodson, Jr., Joe A., and Muller, Eitan, Models of New Product Diffusionthrough Advertisingand Word of Mouth, ManagementScience24(15), 1568-1597 (1978). 6. Easingwood, Christopher J., Mahajan, Vijay, and Muller, Eitan, A Non-Uniform Influence Innovation Diffusion Model of New Product Acceptance, Marketing Science 2(3), 273-295, (1983). 7. Fourt, L. A., and Woodlock, J. W. Early Prediction of Market Success for Grocery Products, Journalof Marketing 25, 31-38 (1960). 8. Granger• C. W. J.• and Newb••d, Pan•• F•recasting Econ•mic TimeSeries• AcademicPress• NewY•rk• 1977. 9. Hanssens, Dominique,Parsons, Leonard J., and Schultz, Randall L., MarketResponseModels:Econometric and Time SeriesModels, Kluwer Academic, Boston, MA, 1990. 10. HeUer, Roger M., and Hustad, Thomas P., Problems in Predicting New Product Growth for Consumer Durables, ManagementScience 26(10), 1007-1020. 11. Horsky, D., and Simon, Leonard S., Advertising and the Diffusionof New Products, Marketing Science 2, 1-18 (1983). 12. Kamakura, W. A., and Balasubramanian, S. K. (1987). Long-Term Forecasting with Innovation Diffusion Models: The Impact of Replacement Purchases, Journal of Forecasting 6(1), 1-20 (1987).

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W . P . PUTSIS, JR.

13. Leone, Robert P., Generalizing What Is Known about Temporal Aggregation and Advertising Carryover, Marketing Science 14(3), Part 2, 141-150 (1995). 14. Mahajan, Vijay, and Muller, Eitan, Innovation Diffusion and New Product Growth Models in Marketing, Journal o f Marketing 43, 55-68 (1979). 15. Mahajan, Vijay, andWind, Yoram, eds.,lnnovationD~ffusionModelsofNewProductAcceptance, Ballinger Publishing, Cambridge, MA, 1986. 16. Mansfield, Edwin, Technical Change and the Rate of Imitation, Econometrica 29(4), 741-766 (1961). 17. Nevers, John V., Extensions of a New Product Growth Model, Sloan Management Review Winter, 7791 (1972). 18. Robinson, B., and Lakhani, C. Dynamic Price Models for New-Product Planning, Management Science 12, 1113-1122 (1975). 19. Schmittlein, D., and Mahajan, V., Maximum Likelihood Estimation for an Innovation Diffusion Model of New Product Acceptance, Marketing Science 1, 57-78 (1982). 20. Sims, C. A., Seasonality in Regression, Journal of the American StatisticalAssociation 69, 618-626 (1974). 21. Srinivasan, V., and Mason, C. H., Nonlinear Least Squares Estimation of New Product Diffusion Models, Marketing Science 5, 169-178 (1986). 22. Sultan, Fareena, Farley, John U., and Lehmann, Donald R., A Meta-Analysis of Applications of Diffusion Models, Journal o f Marketing Research 27, 70-77 (1990). 23. Tigert, Douglas, and Farivar, Behrooz, The Bass New Product Growth Model: A Sensitivity Analysis for a High Technology Product, Journal o f Marketing 45, 81-90 (1981). 24. Vanhonacker, Wilfred R., Estimation and Testing of a Dynamic Sales Response Model with Data Aggregated Over Time: Some Results for the Autoregressive Current Effects Model, Journal o f Marketing Research 21,445-455 (1984). 25. Vanhonacker, Wilfred R., Carryover Effects and Temporal Aggregation in a Partial Adjustment Framework, Marketing Science 2, 297-317 (1983). 26. Weiss, Doyle L., Weinberg, Charles B., and Windal, Pierre M., The Effects of Serial Correlation and Data Aggregation on Advertising Measurement. Journal of Marketing Research 20, 268-279. 27. Winer, Russell S., An Analysis of the Time-Varying Effects of Advertising: The Case of Lydia Pinkham, Journal of Business 52(4), 563-576. Received 14 June 1995; accepted 5 October 1995

APPENDIX The Relationship between Parameters p and q Estimated by OLS When Using Annual Data versus Data with Frequency n Note that we can state the continuous-form model and the discrete approximation as follows: 0N(T) I limit [ N ( t - 1 + ( l / n ) ) - N ( t - 1)]/[l/n] Continuous: ~ T = t - 1 = n ~ o,

(1)

I

= p"m + ( q " - p " ) N ( t - 1 ) - ( q " / m ) N ( t - 1 ) 2. Annual: [N(t) - N ( t - 1)]/1

(2)

= p m + ( q - p ) N ( t - 1) - ( q / m ) N ( t - 1)2. Freq = n: [ N ( t - 1 + (l/n)) - N ( t - 1)]/[I/n]

(3)

= n[p'm + (q'-p')N(t- 1) - ( q ' / m ) N ( t - 1)2]. Conceptually, the difference between the parameter pair CO, q) and the pair Co', q~ results from a different definition on the LHS o f equations 2 and 3 above, which implies different discrete approximations to the continuous function I. To compare parameters CO,q) with (p', q'), subtract equation 3 from equation 2: [N(t) - N ( t - 1)] - n [ N ( t - 1 + ( l / n ) ) - N ( t -

1)]

(4)

= [pm + ( q - p ) N ( t - 1 ) - ( q / m ) N ( t - 1 ) 2] - n[p'm + ( q ' - p g N ( t - 1) - q ' / m N ( t - 1)2] = mco - n p ' ) + ( n p ' - p)N(t - 1) + (q - nq~)N(t - 1) + ( n q ' - q)[N(t - 1)2/m]

(5)

= [ m - N(t - l ) ] [ p - np']+ [q - nq'] [ N ( t - 1 ) - ( N ( t - l)2/m)].

(6)

T E M P O R A L A G G R E G A T I O N IN D I F F U S I O N M O D E L S

279

To solve for ( p - np'), note that at time t = 0, N(0) = 0. Therefore, Co- np')= [n(N(t)-N(t-

1 +(1/n)))]/m.

(7)

E q u a t i o n 7 is greater t h a n 0 before the inflection point since the second derivative of the cumulative function is greater t h a n 0 until this point. Therefore, as long as peak sales occur after the first period, which it certainly should, we could expect p, the coefficient o f innovation with annual data, to be greater than n p', where p' is the coefficient of innovation obtained with data of frequency n. Turning o u r attention to the coefficient o f innovation a n d ignoring equation 7 for the m o m e n t , we can solve expression 6 for (q - nq~: (q - nq') =

[N(t) - nN(t - 1 + ( l / n ) ) ] + [ ( n - l)N(t - 1)] [N(t- 1)- [N(t-

1)Vm]]

-

[m - N(t - l)][p - n p ' ] [ N ( t - 1 ) - [ N ( t - 1)2/m]]

(8)

Substituting equation 7 into equation 8, (q - nq') =

[N(t) - nN(t - 1 + ( l / n ) ) ] + [ ( n - 1)N(t - 1)] [ N ( t - 1) - [N(t - l)2/m]]

(9)

[m - N(t - l)] [[nN(t) - nN(t - 1 + ( 1 / n ) ) ] / m ] [ N ( t - 1 ) - [ N ( t - l)Vm]] Hence, ( q - nq') is >/0 if (10) is >/0: [N(t) - nN(t - 1 + ( l / n ) ) + (n - l)N(t - 1)] - [m - N(t - l)][[nN(t) - nN(t - 1 + (l/n))]/m]>~0

(10)

E q u a t i o n 10 simplifies to: (n - 1)[N(t - 1) + N(t - 1 + ( l / n ) ) l >I[N(t - 1)/ml [n(N(t - 1 + (1/n)) - N(t))]. Since N(t) > N ( t - l + ( l / n ) ) v n ( q - nq') > 0. q.e.d.

(11 )

> 1, R H S of ( l l ) < 0, while L H S of 0 1 ) > 0, which implies that