Question

The demand function for good X is ${\mathbf{Q}}_{\mathbf{x}}^{\mathbf{d}}\mathbf{=}\mathbf{a}\mathbf{+}{\mathbf{bP}}_{\mathbf{x}}\mathbf{+}\mathbf{cM}\mathbf{+}{\mathbf{e}}^{\mathbf{\text{'}}}$, where role="math" localid="1657355528456" ${\mathbf{P}}_{\mathbf{x}}$is the price of good $\mathbf{X}$and $\mathbf{M}$is income. Least squares regression reveals that $\widehat{\mathbf{a}}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{27}\mathbf{,}\widehat{\mathbf{b}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{14}\mathbf{;}\mathbf{}\mathbf{and}\mathbf{}\widehat{\mathbf{c}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{36}$,${\mathbf{\sigma }}_{\widehat{\mathbf{a}}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{32}\mathbf{;}{\mathbf{\sigma }}_{\widehat{\mathbf{b}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{41}\mathbf{;}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{\sigma }}_{\widehat{\mathbf{c}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{22}$. The $\mathbf{R}\mathbf{-}\mathbf{}\mathbf{squared}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{35}\mathbf{.}\mathbf{}$.

a. Compute the t-statistic for each of the estimated coefficients.

b. Determine which (if any) of the estimated coefficients are statistically different from zero.

c. Explain, in plain words, what the R-square in this regression indicates.

Short answer

1. The value for the t-statistic parameter$\mathrm{a}$ is$\widehat{\mathrm{a}}=1.55$ , $\mathrm{b}$is$\widehat{\mathrm{b}}=-5.22$ , and$\mathrm{c}$ is$\widehat{\mathrm{c}}=1.636$ .
2. The absolute value${\mathrm{t}}_{\widehat{\mathrm{b}}}=-5.22$ is different from zero.
3. The value for $\mathrm{R}-\mathrm{squared}$implies that the regression will probably not fit in population parameter.

Step-by-step solution

Given information

The demand function forgood$\mathrm{X}$ is${\mathrm{Q}}_{\mathrm{x}}^{\mathrm{d}}=\mathrm{a}+{\mathrm{bP}}_{\mathrm{x}}+\mathrm{cM}+{\mathrm{e}}^{\text{'}}$ .

The estimated parameter values are$\widehat{\mathrm{a}}=8.27,\widehat{\mathrm{b}}=-2.14;\mathrm{and}\widehat{\mathrm{c}}=0.36$ .

The values for the standard error of the estimated parameter values are ${\mathrm{\sigma }}_{\widehat{\mathrm{a}}}=5.32;{\mathrm{\sigma }}_{\widehat{\mathrm{b}}}=0.41;\mathrm{and}{\mathrm{\sigma }}_{\widehat{\mathrm{c}}}=0.22$.

The t-statistics

a.

To determine the value of t-statistics for the coefficient value for parameter $\widehat{\mathrm{a}}$,

role="math" localid="1657356034652"

$\widehat{a}=1.55$

For parameter $\widehat{\mathrm{b}}$,

$\widehat{\mathrm{b}}=-5.22$

For parameter$\widehat{\mathrm{c}}$ ,

$\begin{array}{c}{\mathrm{t}}_{\widehat{\mathrm{c}}}=\frac{\widehat{\mathrm{c}}}{{\mathrm{\sigma }}_{\widehat{\mathrm{c}}}}\\ =\frac{0.36}{0.22}\\ =1.636\end{array}$

$\widehat{\mathrm{c}}=1.636$

Thumb basis rule to estimate the value and R-squared value

b.

Using the rule of thumb basis, we get the absolute value for the t-basis is greater than or equal to $2$. The corresponding parameter estimate will be different from zero.

Based on the above calculation, the absolute value ${\mathrm{t}}_{\widehat{\mathrm{b}}}=-5.22$is greater than $2$.

Thus,$\widehat{\mathrm{b}}=-5.22$ is different from zero.

Interpreting R-square

c.

$\mathrm{R}-\mathrm{squared}=0.35.$has a low value.

The value for $\mathrm{R}-\mathrm{squared}$implies that the regression will probably not fit in population parameter.