[NEAS: This candidate fits an ARIMA process to box office receipts. Box office sales form a good topic for ARIMA modeling, for several reasons:
Seasonality: Box office sales are seasonal, with highs in July, August, November, and December, and lows in September and April. The seasonality is both exogenous and endogenous. The exogenous seasonality reflects vacations:
~ Summer vacations bring more young people to movies, leading to the highs in July and August. The start of school in September leads to the low in that month.
~ Holidays in November and December with many workers taking vacations lead to highs in those months.
Endogenous seasonality: The exogenous seasonal pattern from summer vacations and holidays is reinforced by the film industry. Young people are more likely to see firms in July than in September, so producers release new films in June and July, not in September. Producers heavily advertise the films released for the summer months or the year-end holidays. A moderate seasonal pattern is transformed into a strong seasonal pattern.
We use different methods to adjust for exogenous vs endogenous seasonality.
~ For daily temperature (an exogenous seasonality) we de-seasonalize the data by subtracting the long-term mean daily temperature from the observation.
~ For auto insurance premiums (an endogenous seasonality reflecting renewals), we use a 12 month autoregressive term.
For box office receipts, the preferred method is unclear. For the exogenous seasonality, we de-seasonalize the data by subtracting the long-term mean for the month.
Illustration: Suppose July’s long-term box office receipts are 50% higher than average. In 20X7, unusually successful movies caused July’s receipts to be 80% higher than average. We still expect the 20X8 July figures to be 50% higher, not 80% higher. We use the long-term mean, not the value 12 months earlier.
The endogenous seasonality of the scheduling and advertising of movies argues for a 12 month autoregressive term.
Illustration: Suppose May is an average months for box office receipts. If receipts were high in May 20X7, producers may schedule new releases for May 20X8,
You can compare the two adjustments for seasonality, seeing which produces better ARIMA fits.
ARIMA Processes: ARIMA processes are best when past values of a time series influence future values. Daily temperature is a good example, because a higher than average daily temperature one day usually means a higher than average daily temperature the next day as well. Lottery winnings are the opposite; one day has no affect on the next day. Winning the lottery one day does not make a person any more or less likely to win the lottery the next week.
Movies produce their own demand, which argues for an ARIMA process.
~ A person who sees a good movie one month is likely to come back the next month.
~ A person who sees a bad movie one month is less likely to return the next month.
This process may be modeled by either autoregressive or moving average terms. You can fit AR(1), AR(2), MA(1), and ARMA(1,1) models to either de-seasonalized data or to data with a 12 month autoregressive term to see which fits best.
Trend: Box office receipts have long-term and short-term trends. Like all sales in nominal dollars, they have an inflationary trend. To fit an ARIMA process, we divide by the CPI to get sales in real dollars.
We should also adjust for population growth, especially among young people. In many European countries, as well as Japan and Russia, young adults are a declining group. In the U.S., high immigration of young adults makes them an increasing group.
It is hard to get monthly or annual population by age group, so this is not necessary for the student project. Be aware that an apparent upward trend in receipts may be a downward trend in real dollar receipts per capita.
The trend in attendance at movies reflects easier ways to obtain the same entertainment. DVD’s, laptops, and flat screen TV’s with high resolution reduce the draw of theaters. More people are watching movies, but a smaller percentage are watching them at theaters. This is a gradual trend, and you may fit an ARIMA process to the first differences of box office receipts.
We show only an extract from this student project. This candidate examines the trends and seasonality, and then fits ARIMA processes. He uses a short time period: 6 years × 12 months. You may use a longer time period in your own student project along with inflation adjustments and seasonal corrections mentioned above.]