Question

An economist want to estimate a multiple regression equation in which the (3)
amount saved by the ith family depends on the family’s income, whether the
family is graduate or non graduate, and whether the family is headed by a male
or female. Explain how a regression equation of this sort can be estimated. What
is the dependent variable ? What are the independent variables? What assumption
must be made ?

Answer 1

Answer on Question #41402 – Math - Statistics and Probability

An economist want to estimate a multiple regression equation in which the (3) amount saved by the ith family depends on the family's income, whether the family is graduate or non graduate, and whether the family is headed by a male or female. Explain how a regression equation of this sort can be estimated. What is the dependent variable? What are the independent variables? What assumption must be made?

Solution

$$y = ax + bz + c + \varepsilon.$$

There are n observations:

$y_{i}$ is i-th family's income (a random variable that can take any real value), $i = 1, \dots, n$,

$x_{i}$ is education status (a random variable that can take either graduate or non graduate), $i = 1, \dots, n$,

$z_{i}$ is family status (a random variable that can take either male or female which is the head of the family), $i = 1, \dots, n$.

Besides,

$\varepsilon_{i}$ is an error, $c$ is intercept, $i = 1, \dots, n$.

Education status and family status are independent variables, in this problem they are categorical variables. Family's income is dependent variable. To use categorical variables in regression analysis, we convert them in dummy variables to code them as 0 or 1, for example, $x_{i} = 1$ if graduate and $x_{i} = 0$ if non-graduate. You can research model for case $x_{i} = 1$.

Leave one dummy variable out from each categorical variable. Do not use both the original categorical variable and the dummy variable. Using categorical variables, we should discuss whether interactions are present (check multicollinearity: if it is present then you can get rid of one of offenders) and heteroscedarity of the error variances. These cases will come to problems in further analysis.

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