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Regression Analysis 143

The difference between R^2 and is the adjustment for the degrees of free-

domin the latter. One can show that always is smaller than R^2. In our

example, In this way, the adjusted R-squared accounts for the fact

that adding explanatory variables inevitably produces a better fit. Adding

another variable to the equation will increase (and therefore, be deemed

worthwhile) only if the additional explanatory power outweighs the loss in

degrees of freedom.

Rˆ^2

Rˆ^2 .72.

Rˆ^2

Rˆ^2

CHECK

STATION 3

Suppose the airline’s management had only eight quarters of data. For dramatic effect,

suppose it estimated Equation 4.3 using data from only odd-numbered quarters: Q1,

Q3,. ., Q15. How would this affect the quality of the regression? Would it adversely

affect R-squared? The adjusted R-squared?

THE F-STATISTIC The F-statistic is similar to the adjusted R-squared statistic.

It is computed as

[4.7]

Here, we divide the explained variation (R^2 ) by the unexplained variation

(1 R^2 ) after correcting each for degrees of freedom. The more accurate

the predictions of the regression equation, the larger the value of F.

The F-statistic has the significant advantage that it allows us to test the over-

all statistical significance of the regression equation. Consider the hypothesis

that all the coefficients of the explanatory variables in the airline regression

are zero: b c d 0. If this were true, then the regression equation would

explain nothing. The simple mean of the number of seats sold would be just

as good a predictor. Note, however, that even under this assumption both R^2

and F will almost certainly be above zero due to small accidental correlations

among the variables. In general, very low (but nonzero) values of F indicate the

great likelihood that the equation has no explanatory power; that is, we are

unable to reject the hypothesis of zero coefficients.

Under the assumption of zero coefficients, the F-statistic has a known dis-

tribution. (See Table 4B.1 in the chapter appendix for an abbreviated table of

the F distribution.) To test whether the regression equation is statistically

significant, we look up the critical value of the F-statistic with k 1 and N k

degrees of freedom. Critical values are listed for different levels of confidence,

with the 95 and 99 percent confidence levels being the most common. If the

equation’s F-value is greater than the critical value, we reject the hypothesis

of zero coefficients (at the specified level of confidence) and say that the

equation has explanatory power. In our example, the F-statistic is F 

(.776/3)/(.224/12) 13.86 and has 3 and 12 degrees of freedom. From the

F-table in the chapter appendix, we find the 95 and 99 percent critical values

F

R^2 /(k1)

(1R^2 )/(Nk)

.

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