Seasonality in the U.S. motion picture industry.

There are (at least) two reasons why decay rates may not be constant over the year. First, as Figure 1 indicates, bigger movies decay slower. Bigger movies are more likely to be released in the strong-release seasons. I therefore include an interaction variable between the decay parameter and the size of the movie. I tried two specifications. The first uses production costs as a proxy for the size of the movie, which does not have much additional explanatory power. As suggested in Table 3, production costs are a noisy proxy. The second specification implements a two-step procedure. I first estimate the benchmark model and then reestimate interacting the estimated movie-fixed effects from the first step with the decay parameter. Because the interaction is endogenous, I use interactions between production costs and decay as instruments. This procedure reveals that better movies decay slower, and the model fit improved. It made little difference, however, to the estimated underlying seasonality. Allowing the decay to vary with the movie genre results in similar findings. There are slightly different decay patterns for different types of movies, but little effect on the estimated underlying seasonality.

A second possible misspecification is that the identifying assumption is incorrect. Distributors may know the decay of their movies before the actual release. If so, they may release slow-decay movies early in the summer and fast-decay movies late in the summer to make the most out of the high-demand summer season. I therefore allow the decay pattern to vary over the seasons in which they were released. I allow six different seasons: winter, spring, early summer, late summer, fall, and holidays. The estimated decay coefficients reflect the fact that bigger movies are released early in the summer and during the holidays. In particular, [lambda] is estimated to be lowest (in absolute value) for holiday releases (-0.194); higher for movies released in the winter, spring, and early in the summer (-0.206, -0.216, and -0.214, respectively); and highest for late summer and fall releases (-0.249 and -0.244, respectively). I cannot reject the hypothesis that the first four coefficients are equal, but the null that the latter two are the same as the others is rejected. Although this specification slightly improves the model fit, the estimated pattern of underlying demand does not change much. The summer coefficients are identical, while the holiday-winter periods are estimated to have somewhat higher underlying demand.

Industry trends, truncation, and Wednesday releases. In this section, I discuss other concerns and verify that the results are stable across different subsamples. First, there may be information-dissemination effects over the movie life cycle. I assume that most word-of-mouth effects occur within the first two weeks of the movie's run. The results are almost identical when the model is estimated for a subsample that includes movies only after their third week in theaters. Second, the results are virtually the same when the model is estimated for movies with a full run of 10 weeks (about 60% of the movies). This accounts for two potential concerns: (i) movies with limited or platform releases differ and (ii) movies that run less than 10 weeks in theaters bias the results. Third, the results do not change if I restrict the data to Friday releases (about 75% of the movies). A Wednesday release is likely to saturate the market for a movie faster. The (roughly) proportional decay implies that the revenue pattern of such movies shifts downward, but their decay pattern will be the same. For this reason, the estimated fixed effects for Wednesday-released movies are likely to be biased downward, but this bias should not (and does not) affect the estimates of underlying seasonality.

The main results are based on a period of 15 years. As Davis (2006) documents, the number of theaters increased during the 1990s. Capacity expansion has translated into bigger opening weekends, which may result in faster market saturation and steeper decay. I therefore estimate the benchmark model for each five-year span separately. (18) Figure 5 presents the estimates of underlying seasonality for each subperiod. Movies do decay faster later in the sample, but underlying seasonality is stable. Similarly, the results do not change much if I pool all periods together, but allow only the decay parameter to vary.

The data and the specification do not allow separate identification of the time-trend or year effects. A linear time trend is not identified. (19) A more flexible time trend will be identified only through the functional form. Similarly, year effects are identified only from movies that are released in December and continue through January. One cannot distinguish between a trend in the utility from the outside option and a trend in the average quality of movies. I can identify only the sum of the two trends, captured by the trend of the estimated movie fixed effects, which does not reveal any obvious pattern.

Instruments and market expansion. I use the number of movies in release as the instrument for the inside share. The results are similar when I use, instead or in addition, the total budget of competing movies or the total number of movies within the same genre. I also used the number of movies released within a three- and five-week window around a particular week, to account for potential endogeneity. (20) The results are fairly stable. (21)

Market expansion in the model is the result of either better or more movies. The nested logit model includes a free parameter for market expansion ([sigma]), but it imposes an implicit functional-form assumption, relating the market-expansion effect of an increase in the number of movies and the market-expansion effect of an increase in the quality of movies. Ackerberg and Rysman (2005) discuss this restriction and suggest including the logarithm of the number of products as an additional explanatory variable. The two effects can then be separately estimated. In my application, the number of products is already an instrument. Including it as an explanatory variable is possible only through functional-form restrictions or by using other characteristics as instruments (e.g., the total production costs of competing movies). Neither specification changes the results much and the estimated underlying seasonality is the same. FIGURE 5 UNDERLYING SEASONALITY OVER TIME The figure presents the estimated coefficients on the weekly dummy variables from the benchmark model, when estimated for each subperiod separately. To allow comparison of the coefficients, I impose [sigma] = 0.524 (from the benchmark model) for all sub-periods. Allowing it be separately estimated affects [lambda] but has little effect on the estimated seasonality. Additional results from the bottom panel regressions:

1985-1989 1990-1994 1995-1999 [lambda] -0.174 (0.0016) -0.206 (0.0014) -0.253 (0.0015) N 4,035 5,521 6,547 Number of titles 572 666 754 [R.sup.2] 0.402 0.430 0.499

* Implication for timing decisions. I consider the benchmark model, assuming away the disturbance term [[xi].sub.jt], and ignoring parameter uncertainty. Rewrite equation (5) to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The numerator is a scaling factor, which depends on the decay-adjusted quality of the movie. The rest of the expression depends on two elements. The first element, exp(-[[tau].sub.t]), is the market-size effect, increasing in the estimated week effect. The second element is the competition effect, as summarized by adding the decay-adjusted qualities of the competing movies. The competition effect is greater if there are more competing movies or if the average quality of the competing movies is higher. As [sigma] increases, the importance of competition is greater, as movies steal more business from other movies than they gain consumers who would otherwise not go to the movies.

One can use equation (6) to characterize the problem faced by distributors when deciding movies release dates. Because prices (and contracts) are stable and marginal costs are effectively zero, maximizing profits is equivalent to maximizing cumulative market share. The distributor of movie j chooses a release date, [r.sub.j], to maximize [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [s.sub.jt], taking the release dates of competing movies as given. H is the length of the horizon taken into account by distributors. (22) I assume that movie quality and all other parameters are common Knowledge. [[theta].sub.j] enters the optimization problem in two ways. The first is multiplicative and does not affect the optimal decision. The second enters through the competition effect.

[FIGURE 6 OMITTED]

Bigger movies will generally have a bigger competition effect and hence a stronger strategic incentive. The strategic game is analyzed in a companion article (Einav, 2003). For the remainder of the section, I assume that distributors are not strategic and I do not account for the competition effect. This is analogous to a price-taking assumption. Under this assumption, all distributors want to release when underlying demand is highest and competition is softest.