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Chapter 4
Estimating and Forecasting Demand
STANDARD ERROR OF THE REGRESSION Finally, the standard error of the regression provides an estimate of the unexplained variation in the dependent variable. Thus far, we have focused on the sum of squared errors as a measure of unexplained variation. The standard error of the regression is computed as
s 2SSE/(N k)
[4.8]
For statistical reasons, we divide the sum of squared errors (SSE) by the degrees of freedom (instead of by N) before taking the square root. The standard error is useful in constructing confidence intervals for forecasts. For instance, for regressions based on large samples, the 95 percent confidence interval for predicting the dependent variable (Q in our example) is given by the predicted value from the regression equation (Q*) plus or minus two standard errors.
Potential Problems in Regression Regression analysis can be quite powerful. Nonetheless, it is important to be aware of the limitations and potential problems of the regression approach. In our example, we assumed a linear form, and the resulting equation tracked the past data quite well. However, the real world is not always linear; relations do not always follow straight lines. Thus, we may be making an error in specification, and this can lead to poorer predictions. The constant elasticity demand equation also is widely used. This equation takes the form
EQUATION SPECIFICATION
Q aPb(P )cYd,
[4.9]
where, a, b, c, and d are coefficients to be estimated. One can show mathematically that each coefficient represents the (constant) elasticity of demand with respect to that explanatory variable. For instance, if the estimated demand equation were Q 100P 2(P ).8Y1.2, then the price elasticity of demand is 2 and the cross-price elasticity is .8. We can rewrite Equation 4.9 as log(Q) log(a) blog(P) clog(P ) dlog(Y)
[4.10]
after taking logarithms of each side. This log-linear form is estimated using the ordinary least-squares method.7 Another common specification is the quadratic form, Q a bP cP2, because this allows for a curvilinear relationship among the variables.
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