Chapter 3, Problem 7CACQ

A Quant jock from your firm used a linear demand specification to estimate the demand for its product and sent you a hard copy of the results. Unfortunately, some entries are missing because the toner was low in her printer. Use the information presented below to find the missing values labeled ‘1’ through ‘7’ (round your answers to the nearest hundredth). Then, answer the accompanying questions.

1. Based on these estimates, write an equation that summarizes the demand for the firm’s product.
2. Which regression coefficients are statistically significant at the 5 percent level?
3. Comment on how well the regression line fits the data.

To determine

(a)

In the given table, the missing values that need to be found are:

1. R-square
2. Adjusted R-square
3. Sum of Squares of regression
4. Degree of freedom
5. Standard error of Intercept
6. t-statistic for price of X
7. coefficient estimate of income.

To find the values of these missing terms, we need to go through the following steps:

1) R-square is computed as the ratio of the explained sum of squares of the regression to the total sum of squared errors. That is,

$\text{R-square}=\frac{S{S}_{regression}}{S{S}_{total}}$

From the given information Sum of Squares of regression can be calculated,

$\begin{array}{c}S{S}_{total}=S{S}_{regression}+S{S}_{residual}\\ 73,807.49=S{S}_{regression}+63,408.62\\ S{S}_{regression}=73,807.49-63,408.62\\ =10,398.87\end{array}$

The value of Sum of Squares is 10,398.87

By putting this value in formula of R-square, we get

$\begin{array}{c}\text{R-square}=\frac{S{S}_{regression}}{S{S}_{total}}\\ =\frac{10,398.87}{73,807.49}\\ =0.141\end{array}$

The value of R-square is 0.141

2) The formula of adjusted R-square is:

$\overline{R^2}=1-\left(1-R^2\right)\frac{n-1}{n-k}$

Where r-square = 0.141

K= 3 and

n= 150

by putting these values in the formula, we get

$\begin{array}{c}\overline{R^2}=1-\left(1-R^2\right)\frac{n-1}{n-k}\\ \overline{R^2}=1-\left(1-0.141\right)\frac{150-1}{150-3}\\ =1-\left(0.859\right)\frac{149}{147}\\ =1-0.870\\ =0.129\end{array}$

Therefore, the value of the adjusted r-square is 0.129.

3) the value of Sum of Squares of regression is equal to 10,398.87.

4)the total degree of freedom is the sum of degree of freedom for regression and degree of freedom for residual.

Total degree of freedom = 147+2

= 149

Therefore, the value of the total degree of freedom is 149.

5) The standard error of intercept ${\sigma }_{\widehat{a}}$ can be calculated with the help of t-statistic derivation that is,

$t_{\widehat{a}}=\frac{\widehat{a}}{{\sigma }_{\widehat{a}}}$

In the table value of t-statistic and parameter, a is given. By putting these values in t-statistic formula we get,

$\begin{array}{c}{t}_{\widehat{a}}=\frac{\widehat{a}}{{\sigma }_{\widehat{a}}}\\ 3.84=\frac{58.87}{{\sigma }_{\widehat{a}}}\\ {\sigma }_{\widehat{a}}=\frac{58.87}{3.84}\\ =15.33\end{array}$

Therefore, the value of the standard error of the intercept is equal to 15.33.

6) By using the value of coefficient and standard error we get t-statistic.

$\begin{array}{c}{t}_{\widehat{x}}=\frac{\widehat{x}}{{\sigma }_{\widehat{x}}}\\ =\frac{-1.64}{0.85}\\ =-1.93\end{array}$

Therefore, the value of t-statistic for the price of X is equal to -1.93.

7)by using the same formula of t-statistic, the coefficient estimates of income can be found.

When we put the given information in the formula, we get

$\begin{array}{c}{t}_{\widehat{m}}=\frac{\widehat{m}}{{\sigma }_{\widehat{m}}}\\ 4.64=\frac{\widehat{m}}{0.24}\\ \widehat{m}=1.1136\end{array}$

Therefore, the value of the coefficient estimate of income is equal to 1.1136.

Based on these estimates, the linear demand function of regression model.

Explanation of Solution

The demand function of is:

$\begin{array}{l}{Q}_X^d=\widehat{a}+\widehat{b}{P}_X+\widehat{c}M\\ Q_X^d=58.87-1.64P_X+1.113M\end{array}$

Here, Px is the price of good X and M is the income.

To determine

(b)

The estimates that are statistically different from zero.

Explanation of Solution

If the absolute value of the t-statistics is greater than or equal to 2, then the corresponding parameter estimate is statistically different from zero.

From the computed values and table, we have these details:

	t-statistic
Intercept	3.84
Price of X	-1.93
Income	4.64

Since the absolute value of t-statistic for intercept and income is greater than 2, the estimated coefficient of the intercept and income at 5% significant level are statistically different from zero.

To determine

(c)

The question requires to find how well this model fit for the data.

Explanation of Solution

he values of R-square and adjusted r-square is the measurement of goodness of fit to a model. Here the value of adjusted r-square is not very different from the value of r-square. They are:

R-square = 0.141

Adjusted R-square = 0.129

The values of both the estimates are very small that means the regression equation is not able to explain the dependent variable values. Therefore, the regression line is not a good fit for the data.