9781118041581

STANDARD ERROR OF THE REGRESSION Finally, the standard error of the

regressionprovides an estimate of the unexplained variation in the depend-

ent variable. Thus far, we have focused on the sum of squared errors as a meas-

ure of unexplained variation. The standard error of the regression is

computed as

[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 pre-

dicting 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.

EQUATION SPECIFICATION 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

[4.9]

where, a, b, c, and d are coefficients to be estimated. One can show mathe-

matically 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

[4.10]

after taking logarithms of each side. This log-linear form is estimated using the

ordinary least-squares method.^7

log(Q)log(a)blog(P)clog(P)dlog(Y)

QaPb(P)cYd,

s 2 SSE/(Nk)

146 Chapter 4 Estimating and Forecasting Demand

(^7) Another common specification is the quadratic form, Q a bP cP (^2) , because this allows for
a curvilinear relationship among the variables.
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