Chapter 14, Problem 1RE

To determine

To explain: The simple regression.

Answer to Problem 1RE

Solution: The least-squares regression model can be defined as follows:

${\widehat{y}}_t={\beta }_1{x}_t+{\beta }_0+{\epsilon }_t$

Explanation of Solution

The regression equation can be defined as follows:

${\widehat{y}}_t={\beta }_1{x}_t+{\beta }_0+{\epsilon }_t$

Here, ${\widehat{y}}_t$ is the predicted value of the dependent variable and $x_t$ is the value of the independent variable. The slope of the regression line is ${\beta }_1$ and the intercept of the regression line is ${\beta }_0$. The error term is represented as ${\epsilon }_t$.

To determine

The requirements of regression.

Answer to Problem 1RE

Solution: The requirements of the least-square regression are

1. The mean of the response value is linearly dependent on the independent variable.

2. The dependent variable has a normal distribution whose mean and standard deviation are ${\mu }_{y|x}={\beta }_{1}x+{\beta }_0$ and $\sigma$, respectively.

Explanation of Solution

To draw any inference about the least-squares regression equation, the mean of the response value should be linearly dependent on the independent variable. The distribution of the dependent variable is normal and the mean is ${\mu }_{y|x}={\beta }_{1}x+{\beta }_0$ and the standard deviation is $\sigma$.

To determine

To identify: The verification process of the requirements.

Answer to Problem 1RE

Solution: The requirements can be verified from the residual plot and normal probability plot.

Explanation of Solution

To check whether the requirements are fulfilled by the data or not, the residuals are plotted and it is checked whether the distribution of the residuals is standard normal. The residual plot and the normal probability plot are used to check the requirements.