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Regression Analysis 147

OMITTED VARIABLES A related problem is that of omitted variables. Recall

that we began the analysis of airline demand with price as the only explana-

tory variable. The resulting OLS equation produced predictions that did a rea-

sonably good job of tracking actual values. However, a more comprehensive

equation, accounting for competitor’s price and income, did far better. In

short, leaving out key variables necessarily worsens prediction performance.

In fact, omission of these other variables also affects the coefficients of the

included variables. For instance, the price coefficient is 1.63 when it is the

sole explanatory variable. This is quite different from the estimated multiple-

regression coefficient, 2.12. Thus, the single-variable regression underesti-

mates the magnitude of the true price effect.

MULTICOLLINEARITY When two or more explanatory variables move

together, we say that the regression suffers from multicollinearity. In this case,

it is difficult to tell which of the variables is affecting the dependent variable.

Suppose demand for a firm’s product is believed to depend on only two factors:

price and advertising. The data show that whenever the firm initiated an aggres-

sive advertising campaign, it invariably lowered the good’s price. Sales

increased significantly as a result. When the firm decreased advertising spend-

ing it also increased price, and sales dropped. The question is: Should the

changes in sales be attributed to changes in advertising or to changes in price?

Unfortunately, it is impossible to tell, even with regression. If two right-hand

variables move together, regression cannot separate the effects. Regression does

not require that we hold one of the factors constant as we vary the other, but it

does require that the two factors vary in different ways.

What happens when the forecaster runs a regression based on these

data? If the right-hand variables are perfectly correlated, the computerized

regression program will send back an error message. If the right-hand vari-

ables are not perfectly correlated, but move very closely together (either

directly or inversely), the regression output will provide very imprecise coef-

ficient estimates with large standard errors. In this case, additional data

may improve the estimates. If not, the forecaster must live with the impre-

cise estimates.

Can the firm still use the equation to forecast? Yes and no. It can if it plans

to continue the pattern of lowering price whenever it increases advertising. In

that case, it need not care about the separate effects. However, if it plans to

lower price without an advertising campaign, or to advertise more without low-

ering price, the forecast will be very unreliable.

SIMULTANEITY AND IDENTIFICATION This brings us to a subtle, but inter-

esting and important, issue. In the preceding discussion, we assumed that the

firm had explicit control over its price. In many settings, however, price is deter-

mined by overall demand and supply conditions, not by the individual firm.

Here, the firm must take the price the market dictates or else sell nothing.

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