Question

Suppose the true inverse demand relation for good $\mathbf{X}\mathbf{}\mathbf{is}\mathbf{}{\mathbf{Q}}_{\mathbf{x}}^{\mathbf{d}}\mathbf{=}\mathbf{a}\mathbf{+}{\mathbf{bP}}_{\mathbf{x}}\mathbf{+}\mathbf{cM}\mathbf{+}\mathbf{e}$, and you estimated the parameters to be$\widehat{\mathbf{a}}\mathbf{=}\mathbf{22}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{\sigma }}_{\mathbf{a}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{,}\widehat{\mathbf{b}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{\sigma }}_{\mathbf{b}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{7}$Find the approximate$\mathbf{95}$percent confidence interval for the true values of a and b

Short answer

The$95\%$ confidence intervals for the true values for parameter$\mathrm{a}$ and $\mathrm{b}$are$[17,27]\mathrm{and}[-3.2,-0.4]$ .

Step-by-step solution

Given information

$\widehat{\mathrm{a}}=22\mathrm{and}{\mathrm{\sigma }}_{\mathrm{a}}=2.5,\widehat{\mathrm{b}}=1.8\mathrm{and}{\mathrm{\sigma }}_{\mathrm{b}}=0.7$

Step 2: To find the confidence level for the parameter a and  b

We use the rule of thumb principleto find the$95\%$ confident interval for the true values of the parameter$\mathrm{a}$ and b.

For parameter,$\mathrm{a}=\widehat{\mathrm{a}}\pm 2{\mathrm{\sigma }}_{\mathrm{a}}$

For parameter,$\mathrm{b}=\widehat{\mathrm{b}}\pm 2{\mathrm{\sigma }}_{\mathrm{b}}$

Put the value inthe given data.

$\begin{array}{c}\mathrm{a}=22\pm 2\times 2.5\\ \mathrm{b}=1.8\pm 2\times 0.7\end{array}$

Thus,

$\begin{array}{c}\begin{array}{l}\mathrm{a}=22\pm 2\times 2.5\\ \mathrm{a}=22\pm 5\end{array}\\ \mathrm{a}=17\mathrm{or}\mathrm{a}=27\end{array}$

$\begin{array}{l}\begin{array}{c}\mathrm{b}=1.8\pm 2\times 0.7\\ \mathrm{b}=1.8\pm 1.4\end{array}\\ \mathrm{b}=-3.2\mathrm{or}\mathrm{b}=-0.4\end{array}$

Thus, the $95\mathrm{\%}$ confidence interval for the true values for parameter$\mathrm{a}$ and$\mathrm{b}$are$[17,27]\mathrm{and}[-3.2,-0.4]$ .