|Demand Estimation and Forecasting|
This chapter presented the basic techniques of estimating demand functions and forecasting future sales and prices. Two specifications for demand, linear and log-linear, are presented in this chapter. When demand is specified to be linear in form, the coefficients on each of the explanatory variables measure the rate of change in quantity demanded as that explanatory variable changes, holding all other explanatory variables constant. In linear form, the empirical demand specification is
where Q is the quantity demanded, P is the price of the good or service, M is consumer income, and PR is the price of some related good R. The estimated demand elasticities are computed as
As in any regression analysis, the statistical significance of the parameter estimates can be assessed by performing t tests or examining p-values.
When demand is specified as log-linear, the demand function is written as
In order to estimate the log-linear demand function, it is converted to natural logarithms:
In log-linear form, the elasticities of demand are constant, and the estimated elasticities are
To choose between these two specifications of demand, a researcher should consider whether the sample data to be used for estimating demand are best represented by a demand function with varying elasticities (linear demand) or by one with constant elasticity (log-linear demand). When price and quantity observations are spread over a wide range of values, elasticities are likely to vary, and a linear specification with its varying elasticities is usually a more appropriate specification of demand. Alternatively, if the sample data are clustered over a narrow price (and quantity) range, a constant-elasticity specification of demand, such as a log-linear model, may be a better choice than a linear model.
The appropriate statistical method for estimating a demand equation depends on whether the demand to be estimated is for a single price-setting firm or for a competitive price-taking industry. In this chapter and throughout the rest of the book, we focus exclusively on estimating demand for price-setting firms since the standard regression methodology as presented in Chapter 4 can be properly employed to estimate demand for a price-setting firm. For those students who wish to see how to estimate industry demand (and supply) and how to make price and output forecasts for competitive industries, we provide a careful presentation of this methodology in Online Topic 1: Estimating and Forecasting Industry Demand for Price-Taking Firms, which can be found on the Web site for this textbook (www.mhhe.com/economics/thomas9).
Time-series forecasts use the time-ordered sequence of historical observations on a variable to develop a model for predicting future values of that variable. The simplest time-series forecast is a linear trend forecast where the generating process is assumed to be the linear model Qt = a + bt. Using time-series data on Q, regression analysis is used to estimate the trend line that best fits the data. If b is greater (less) than 0, sales are increasing (decreasing) over time. If b equals 0, sales are constant over time.
When data exhibit cyclical variation, such as seasonal patterns, dummy variables can be added to the timeseries model to account for the seasonality. If there are N seasonal time periods to be accounted for, N 1 dummy variables are added to the demand equation. Each dummy variable accounts for one of the seasonal time periods. The dummy variable takes a value of 1 for those observations that occur during the season assigned to that dummy variable and a value of 0 otherwise. This type of dummy variable allows the intercept of the demand equation to take on different values for each seasonthe demand curve can shift up and down from season to season.
When making forecasts, analysts must be careful to recognize that the further into the future the forecast is made, the wider the confidence interval or region of uncertainty. Incorrect specification of the demand equation can seriously undermine the quality of a forecast. An even greater problem for accurate forecasting is posed by the occurrence of structural changes that cause turning points in the variable being forecast. Forecasts often fail to predict turning points. While there is no satisfactory way to account for unexpected structural changes, forecasters should note that the further into the future you forecast, the more likely it is that a structural change will occur.
This chapter concludes Part II of the text on demand analysis. In Part III of the text, we will present the theory of production and cost. We also will describe the empirical techniques used to estimate production functions and the various cost equations used by managers to make output and investment decisions.