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Regression Analysis 145

errors greater than zero. We appeal to the exact distribution of the t-statistic

to pinpoint the degree of confidence with which we can reject the hypothesis

c 0. To do this, we note that the computed t-statistic has N k  16  4 

12 degrees of freedom. Using the table of the t-distribution (Table 4B.2) in

the appendix to this chapter, we note that, under the null hypothesis, there

is a 95 percent chance that the t-statistic would be between 2.18 and 2.18;

that is, there would be a 5 percent chance that the estimate would lie in one

of the tails beyond 2.18 or 2.18. Because the actual value of t is 2.20 and is

greater than 2.18, we can reject the null hypothesis c 0 with 95 percent

confidence.^6

From Table 4.6, we observe that all of the coefficients have t-values that

are much greater than 2 in absolute value. Thus, all are significantly different

than zero. Each variable has explanatory power. If additional explanatory vari-

ables were to be included, they would have to meet the same test. If we found

that a given coefficient was not significantly different than zero, we would drop

that explanatory variable from the equation (without a compelling reason to

the contrary).

Finally, we can use this same method to test other hypotheses about the

coefficients. For instance, suppose the airline’s managers have strong reasons

to predict the coefficient of the competitor’s price to be c1. The appro-

priate t-statistic for testing this hypothesis is

Since this is near zero, that is, smaller than 2, we cannot reject this hypothesis.

Applying similar tests to the other coefficients, it is clear that there is little to

choose between Equation 4.3 and the “rounded” regression equation, Q 

29 2P P3Y, used in Chapter 3.



1.04 1

.47

.085.

t

cc

standard error of c

CHECK

STATION 4

Again, suppose the demand equation is estimated using only odd-numbered quarters in

the regression. How do you think this will affect the equation’s F-statistic? The standard

errors of the coefficients?

(^6) We reject the hypothesis of a zero coefficient for positive or negative values of the t-statistic suffi-
ciently far from zero. For this reason, it usually is referred to as a two-tailed test. Thus, we select the
97.5 percent fractile of the t-distribution to construct the 95 percent confidence interval. If the
t-value lies outside this interval (either above or below), we can reject the hypothesis of a zero coef-
ficient with 95 percent confidence.
From Table 4B.2 in the chapter appendix, we note that as the number of observations increases,
the 97.5 percent fractile approaches 1.96. This justifies the benchmark of 2 as the rough bound-
ary of the 95 percent confidence interval.
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