Demand Estimation and Forecasting

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Learning Objectives
After reading Chapter 7 and working the problems for Chapter 7 in the textbook and in this Workbook, you should be able to:
 Specify an empirical demand function—both linear and nonlinear functional forms.  For price-setting firms with market power, you will learn to how to use least-squares regression methodology to estimate a firm’s demand function.  Forecast sales and prices using time-series regression analysis.  Employ dummy variables to account for cyclical or seasonal variation in sales.

 Discuss and explain several important problems that arise when using statistical methods to forecast demand.

Essential Concepts
1.

Empirical demand functions are demand equations derived from actual market data. Empirical demand functions are extremely useful in making pricing and production decisions.

2.

In linear form, an empirical demand function can be specified as

Q = a + bP + cM + dPR
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. In the linear form, b = ΔQ / ΔP , c = ΔQ / ΔM , and d = ΔQ / ΔPR . The expected signs of the coefficients are: (1) b is expected to be negative, (2) if good X is normal (inferior), c is expected to be positive (negative), (3) if related good R is a substitute (complement), d is expected to be positive (negative). The estimated elasticities of demand are computed as

ˆ ˆP
E=b
Q
3.

M
ˆ
ˆ
EM = c
Q

ˆP
ˆ
E XR = d R
Q

When demand is specified in log-linear form, the demand function can be written as
Q = aP b M c PRd
Chapter 7: Demand Estimation and Forecasting

144

To estimate a log-linear demand function, the above equation must be converted to logarithms:
ln Q = ln a + b ln P + c ln M + d ln PR
ˆˆˆ
ˆ
In this log-linear form, the elasticities of demand are constant: E = b , E = c , and M

ˆ
E XR

4.

ˆ
=d

When a firm possesses some degree of market power, which makes it a price-setting firm, the demand curve for the firm can be estimated using the method of least-squares estimation set forth in Chapter 4. The following steps can be followed to estimate the demand function for a price-setting firm:

Step 1: Specify the price-setting firm’s demand function.
Step 2: Collect data for the variables in demand function.
Step 3: Estimate the firm’s demand using least-squares regression.

5.

A time-series model shows how a time-ordered sequence of observations on a variable, such as price or output, is generated. The simplest form of time-series forecasting is linear trend forecasting. In a linear trend model, sales in each time period (Qt) are assumed to be linearly related to time (t):

Qt = a + bt
and regression analysis is used to estimate the values of a and b. If b > 0, sales are increasing over time, and if b < 0, sales are decreasing. If b = 0, then sales are ˆ
constant over time. The statistical significance of a trend is determined testing b ˆ
for statistical significance or by examining the p-value for b . 6.

Seasonal or cyclical variation can bias the estimation of a and b in linear trend models. In order to account for seasonal variation in trend analysis, dummy variables are added to the trend equation. Dummy variables serve to shift the trend line up or down according to the particular seasonal pattern encountered. The significance of seasonal behavior is determined by using a t-test or p-value for the estimated coefficient on the dummy variable.

7.

When using dummy variables to account for N seasonal time periods, N −1 dummy variables are added to the linear trend. 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.

8.

The following problems and limitations are inherent in forecasting: i.

The further into the future the...
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