Question

For a large university, you are asked to estimate the demand for tickets to women's basketball games. You can collect time series data over 10 seasons, for a total of about 150 observations. One possible model is \(\begin{aligned} l A T T E N D_{t}=& \beta_{0}+\beta_{1} l P R I C E_{t}+\beta_{2} W I N P E R C_{t}+\beta_{3} R I V A L_{t} \\ &+\beta_{4} W E E K E N D_{t}+\beta_{5} t+u_{t} \end{aligned}\) where The l denotes natural logarithm, so that the demand function has a constant price elasticity. i. Why is it a good idea to have a time trend in the equation? ii. The supply of tickets is fixed by the stadium capacity; assume this has not changed over the 10 years. This means that quantity supplied does not vary with price. Does this mean that price is necessarily exogenous in the demand equation? (Hint: The answer is no.) iii. Suppose that the nominal price of admission changes slowly \(-\) say, at the beginning of each season. The athletic office chooses price based partly on last season's average attendance, as well as last season's team success. Under what assumptions is last season's winning percentage ( \(S E A S P E R C_{t-1}\) ) a valid instrumental variable for \(l P R I C E_{t} ?\) iv. Does it seem reasonable to include the (log of the) real price of men's basketball games in the equation? Explain. What sign does economic theory predict for its coefficient? Can you think of another variable related to men's basketball that might belong in the women's attendance equation? v. If you are worried that some of the series, particularly \(l A T T E N D\) and \(l P R I C E\), have unit roots, how might you change the estimated equation? vi. If some games are sold out, what problems does this cause for estimating the demand function? (Hint: If a game is sold out, do you necessarily observe the true demand?)

Short answer

Including a time trend accounts for temporal patterns. Price is not exogenous if affected by demand factors. Use last season's win rate as an instrument for price if uncorrelated with current errors.

Step-by-step solution

Understanding the Time Trend in the Equation

Including a time trend in the equation is crucial for capturing the natural progression and external influences on attendance that change over time sequentially. It helps account for underlying patterns in the data that are not directly associated with the other explanatory variables, such as economic growth, changes in population, or long-term changes in team popularity.

Evaluating Price Exogeneity

Even though the supply is fixed due to the stadium's capacity, price is not necessarily exogenous because it may be affected by unobserved factors that also influence attendance, such as team performance or broader economic conditions. Since price might respond to expected demand depicted by these factors, it cannot be considered purely exogenous.

Instrumental Variables and Last Season's Winning Percentage

To use last season's winning percentage as an instrument for the current price, it must be correlated with the price but not with the error term in the demand equation. This situation assumes that last season's performance influences pricing decisions but does not directly affect current attendance, except through price.

Considering Real Price of Men's Basketball Games

Including the real price of men's basketball games can be justified if men's and women's games are substitutes or complements. Economic theory suggests if they are substitutes, an increase in men's game prices could increase demand for women's games, indicating a positive sign for the coefficient. On the other hand, factors like men's team success might also correlate if they draw away potential attendees.

Addressing Unit Roots

If there are concerns that variables like \( lATTEND \) and \( lPRICE \) have unit roots, differencing these variables, such as using first differences, can make the data stationary and eliminate trends that could lead to spurious regression results. This transformation ensures that the relations we observe are not due to nonstationarity.

Issues with Sold Out Games

When games sell out, the observed attendance is censored, as it may not reflect the true demand. Demand can exceed stadium capacity, but we only observe the maximum capacity, leading to a biased estimation of the demand function. In such cases, methods to account for censoring, like Tobit models, might be necessary to accurately capture demand.

Key concepts

Demand Estimation

In econometrics, demand estimation is a crucial task which involves understanding how various factors influence the demand for a product or service. Specifically for tickets to sports games, estimating demand helps in setting appropriate prices and optimizing revenue. Collecting data over a period can reveal how variables such as ticket price, team performance, and external events affect attendance.

For efficient demand estimation, it is important to define a model that accurately represents the real-world scenario. In the case discussed, a log-linear functional form is used, where the demand function includes variables like price, winning percentage, and additional factors like rivalry and weekend status. The log transformation allows for easy interpretation of coefficients and implies constant elasticity, which is often a reasonable assumption in economics.
To gain actionable insights:

• Ensure that all relevant variables are included in the model to avoid omitted variable bias.
• Test for and address potential stationarity issues in the time series data to avoid misleading results.
• Consider the impact of external, non-modeled influences on demand trends over time.

Overall, demand estimation requires careful data selection and model specification to provide reliable estimates.

Instrumental Variables

Instrumental variables (IV) are used in econometric models to address the problem of endogenous variables, which are those influenced by other variables in the model. When estimating demand, it's essential to identify the sources of endogeneity that might affect variables like price.

A variable can be considered instrumental if it is correlated with the endogenous variable but is uncorrelated with the error term in the equation. For the women's basketball games model, last season's winning percentage is proposed as an IV for current ticket prices. It is assumed that:

• Winning percentage impacts how prices are set but doesn't directly affect current demand beyond price.
• This IV helps account for endogenous movements in ticket pricing, isolating the pure effect of price on demand.

Using such instruments is crucial because it removes bias from the estimates. Without IVs, any conclusions about price elasticity or demand responsiveness could be unreliable.
Therefore, when choosing an IV, ensure that:

• The instrument is valid and justifiably exogenous relative to the dependent variable.
• Multiple IVs might be needed to isolate effects; always test for their exclusion restrictions.

Employing IVs properly helps refine demand estimates and provides a clearer understanding of the underlying economic relationships.

Time Series Analysis

Time series analysis involves examining data points collected or recorded at specific time intervals. This analysis is vital in economics and econometrics for understanding trends, patterns, and seasonal effects over time.

For women's basketball ticket demand, time series data can reveal:

• Long-term trends in attendance which might be due to changing team popularity or economic conditions.
• Seasonal variations, for instance, an increase in attendance during the holiday season or weekends.
• The impact of exogenous shocks like management changes or economic crises.

When using time series data, it is essential to address potential challenges such as:

• Nonstationarity: Data where statistical properties like mean and variance change over time. Differencing or detrending data can help achieve stationarity, which is necessary for valid inference.
• Autocorrelation: When past and present data points are correlated, it could lead to underestimated standard errors. Applying techniques like ARIMA models can help mitigate this issue.

Applying thorough time series analysis allows for more accurate modeling and forecasting of future attendance based on past behaviors and trends.

Price Elasticity

Price elasticity measures the responsiveness of demand when there is a change in the price of a good or service. It's a significant indicator for sports teams when determining ticket pricing strategies.

In the context of women's basketball games, the model's use of the log-linear form helps to estimate constant price elasticity. The coefficient of the logged price variable in the estimation represents this elasticity, giving insights into whether demand is elastic (sensitive to price changes) or inelastic (less sensitive).
Understanding price elasticity is essential:

• An elastic demand suggests that price changes significantly affect attendance. A small increase in price may lead to a large drop in attendance.
• An inelastic demand implies attendance does not change much with price adjustments, allowing for more aggressive pricing strategies.

Considering external factors like competition from men's games or substitute events can also affect elasticity estimates. Therefore, it is crucial to:

• Regularly reevaluate elasticity as market conditions and consumer preferences evolve.
• Consider complementary or substitute goods when analyzing the elasticity of demand, such as bundling men's and women's game tickets.

Correctly estimating and interpreting price elasticity enables better pricing decisions that maximize both attendance and revenue.