of F to be 3.49 and 5.95, respectively. Since F is larger than 5.95, we can reject
the hypothesis of zero coefficients with 99 percent confidence. The equation
thus has significant explanatory power.^5STANDARD ERRORS OF THE COEFFICIENTS In addition to the values of the
coefficients themselves, we would like to have a measure of their accuracy.
Because the data themselves are subject to random errors, so too are the esti-
mated coefficients. The OLS coefficient estimates are unbiased; on average,
they neither overestimate or underestimate the true coefficients. Nonetheless,
there is a considerable dispersion of the estimate around the true value.
The standard error of a coefficientis the standard deviation of the esti-
mated coefficient. The lower the standard error, the more accurate is the esti-
mate. Roughly speaking, there is a 95 percent chance that the true coefficient
lies within two standard errors of the estimated coefficient. For example, the
estimate for the price coefficient is 2.12, and its standard error is .34. Two
times the standard error is .68. Thus, there is roughly a 95 percent chance that
the true coefficient lies in the range of 2.12 plus or minus .68—that is,
between 2.80 and 1.44. True, we would prefer greater accuracy, but remem-
ber that our estimate is computed based on only 16 observations. Increasing
the number of observations would significantly improve the accuracy of the
estimates.THE t-STATISTIC The t-statistic is the value of the coefficient estimate
divided by its standard error. The t-statistic tells us how many standard errors
the coefficient estimate is above or below zero. For example, if the statistic is 3,
then the coefficient estimate is three standard errors greater than zero. If
the t-statistic is 1.5, then the coefficient estimate is one and one-half stan-
dard errors below zero. We use the t-statistic to determine whether an indi-
vidual right-hand variable has any explanatory power. Consider the so-called
null hypothesisthat a particular variable—say, the competitor’s price—has no
explanatory power; that is, the true value of this coefficient is zero (c 0).
Of course, the regression results show this estimated coefficient to be 1.03, a
value seemingly different from zero. But is it really? The value of the coeffi-
cient’s standard error is .47. If the “true” value of c really were zero, there
would be a roughly 95 percent chance that the coefficient estimate would fall
within two standard errors of zero, that is, between .94 and .94. The
actual coefficient, 1.03, is outside this range and, therefore, appears to be
significantly different from zero.
The t-statistic tells the precise story. Its value is t 1.034/.47 2.20.
Again, this ratio says that the estimated coefficient is more than two standard144 Chapter 4 Estimating and Forecasting Demand(^5) If our computed F had been 5.0, we could reject the hypothesis of zero coefficients with 95 per-
cent confidence, but not with 99 percent confidence. F must surpass a higher threshold to justify
a higher confidence level.
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