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Chapter 13: Problem 7

Why is the random error term included in a regression model?

Short Answer

Expert verified
The random error term is included in a regression model to account for the variation in the outcome variable that is not explained by the predictor variables included in the model. It represents the inevitable errors of prediction in the model, thus making the model realistic and bias-free to some extent.

Step by step solution

01

Introduce Regression Model

The regression model in its simplest form is given by: \( Y = \beta_0 + \beta_1X + \epsilon \) where \(\epsilon\) is the random error term, representing the unexplained variability in the outcome \(Y\) not explained by the predictor variables.
02

Explain Role of Random Error Term

The random error term is pivotal in a regression model. It accounts for the random variation in the outcome variable \(Y\) that is not captured by the predictor variables in the model. This unexplained variability could be due to various factors not included in the model.
03

Describe Consequences of Omitting Error term

Omission of the error term could lead to a biased model if the omitted factors are correlated with the predictor variables. Therefore, the error term helps in eliminating the bias to some extent and making the model more realistic.

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Most popular questions from this chapter

A sample data set produced the following information. $$ \begin{aligned} &n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680 \\ &\Sigma x^{2}=1140, \text { and } \Sigma y^{2}=25,272 \end{aligned} $$ a. Calculate the linear correlation coefficient \(r\). b. Using the \(2 \%\) significance level, can you conclude that \(\rho\) is different from zero?

Explain the difference between linear and nonlinear relationships between two variables.

Plot the following straight lines. Give the values of the \(y\) -intercept and slope for each of these lines and interpret them. Indicate whether each of the lines gives a positive or a negative relationship between \(x\) and \(y\). a. \(y=-60+8 x \quad\) b. \(y=300-6 x\)

The following table gives information on the incomes (in thousands of dollars) and charitable contributions (in hundreds of dollars) for the last year for a random sample of 10 households. $$ \begin{array}{rc} \hline \text { Income } & \text { Charitable Contributions } \\ \hline 76 & 15 \\ 57 & 4 \\ 140 & 42 \\ 97 & 33 \\ 75 & 5 \\ 107 & 32 \\ 65 & 10 \\ 77 & 18 \\ 102 & 28 \\ 53 & 4 \\ \hline \end{array} $$ a. With income as an independent variable and charitable contributions as a dependent variable, compute \(\mathrm{SS}_{x \mathrm{x}}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x v}\) b. Find the regression of charitable contributions on income. c. Briefly explain the meaning of the values of \(a\) and \(b\). d. Calculate \(r\) and \(r^{2}\) and briefly explain what they mean. e. Compute the standard deviation of errors. f. Construct a \(99 \%\) confidence interval for \(B\). g. Test at the \(1 \%\) significance level whether \(B\) is positive. h. Using the \(1 \%\) significance level, can you conclude that the linear correlation coefficient is different from zero?

A population data set produced the following information. $$ \begin{aligned} &N=460, \quad \Sigma x=3920, \quad \Sigma y=2650, \quad \Sigma x y=26,570 \\ &\Sigma x^{2}=48,530, \text { and } \Sigma y^{2}=39,347 \end{aligned} $$ Find the linear correlation coefficient \(\rho\).
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