Confidence Intervals For Parameters

A confidence interval for pk would be based on (4-13). We could say that

Prob(bk - ta/2Sbk < fa < bk + ta/2Sbk) = 1 - a, where 1 - a is the desired level of confidence and ta/2 is the appropriate critical value from the t distribution with (n - K) degrees of freedom.

Example 4.4 Confidence Interval for the Income Elasticity of Demand for Gasoline

Using the gasoline market data discussed in Example 2.3, we estimated following demand equation using the 36 observations. Estimated standard errors, computed as shown above, are given in parentheses below the least squares estimates.

ln(G/pop) = -7.737 - 0.05910 ln PG + 1.3733ln income (0.6749) (0.03248) (0.075628)

-0.12680 ln Pnc - 0.11871 ln Puc + e. (0.12699) (0.081337)

To form a confidence interval for the income elasticity, we need the critical value from the t distribution with n - K = 36 - 5 degrees of freedom. The 95 percent critical value is 2.040. Therefore, a 95 percent confidence interval for pi is 1.3733 ± 2.040(0.075628), or [1.2191, 1.5276].

We are interested in whether the demand for gasoline is income inelastic. The hypothesis to be tested is that pi is less than 1. For a one-sided test, we adjust the critical region and use the ta critical point from the distribution. Values of the sample estimate that are greatly inconsistent with the hypothesis cast doubt upon it. Consider testing the hypothesis

The appropriate test statistic is

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