Question

In the simple linear regression model \(y=\beta_{0}+\beta_{1} x+u,\) suppose that \(\mathrm{E}(u) \neq 0 .\) Letting \(\alpha_{0}=\mathrm{E}(u),\) show that the model can always be rewritten with the same slope, but a new intercept and error, where the new error has a zero expected value.

Short answer

Rewrite the model as \( y = \beta_0' + \beta_1 x + u' \), where \( \beta_0' = \beta_0 + \alpha_0 \) and \( \mathrm{E}(u') = 0 \).

Step-by-step solution

Original Model and Condition

The original simple linear regression model is given by \( y = \beta_0 + \beta_1 x + u \), where the expected value of the error term \( u \) is not zero, \( \mathrm{E}(u) eq 0 \). We define \( \alpha_0 = \mathrm{E}(u) \).

Express Error Term with New Mean

Since \( \alpha_0 = \mathrm{E}(u) \), we can partition the error term \( u \) into two components: its mean and a deviation from that mean. Therefore, we express \( u \) as \( u = \alpha_0 + u' \), where \( u' \) is the new error term such that \( \mathrm{E}(u') = \mathrm{E}(u - \alpha_0) = \mathrm{E}(u) - \alpha_0 = 0 \).

Substitute New Error into Model

Substitute \( u = \alpha_0 + u' \) into the original model \( y = \beta_0 + \beta_1 x + u \). This gives us:\[ y = \beta_0 + \beta_1 x + \alpha_0 + u' \].

Rewrite the Model with New Intercept

To rewrite the equation, combine \( \beta_0 \) and \( \alpha_0 \) into a new intercept \( \beta_0' \), where \( \beta_0' = \beta_0 + \alpha_0 \). Thus, the model becomes:\[ y = \beta_0' + \beta_1 x + u', \]where \( \mathrm{E}(u') = 0 \).

Conclusion

The model is now written with the same slope \( \beta_1 \), but with a new intercept \( \beta_0' \) and a new error term \( u' \), where \( \mathrm{E}(u') = 0 \), satisfying the condition for a typical linear regression error term.

Key concepts

Understanding Simple Linear Regression

In simple linear regression, we explore the relationship between two variables by fitting a linear equation to the observed data. This model takes the form:

• \( y = \beta_0 + \beta_1 x + u \)

where:

• \( y \) represents the dependent variable
• \( x \) is the independent variable
• \( \beta_0 \) is the intercept, indicating where the line crosses the y-axis
• \( \beta_1 \) is the slope, showing how much \( y \) changes on average with a one-unit change in \( x \)
• \( u \) is the error term, capturing other factors affecting \( y \)

The goal in simple linear regression is to predict \( y \) based on \( x \), optimizing \( \beta_0 \) and \( \beta_1 \) to minimize the error in predictions. This model is essential for illustrating basic relationships between two variables, making it foundational in various fields, including economics, biology, and engineering.

Decoding the Error Term

An important component of any regression model is the error term. In our simple linear regression model, the error term \( u \) represents the variation in \( y \) that is not explained by \( x \). This term captures:

• Random fluctuations in the data
• Omitted variable impacts
• Measurement errors

For effective regression analysis, we often assume that \( \mathrm{E}(u) = 0 \), meaning the errors average out to zero in the long run. This assumption ensures unbiased estimates of \( \beta_0 \) and \( \beta_1 \). However, if \( \mathrm{E}(u) eq 0 \), adjustments must be made to maintain the integrity of our model. By redefining \( u \) into \( \alpha_0 + u' \) with an adjusted error term \( u' \), which satisfies \( \mathrm{E}(u') = 0 \), our model remains valid and unbiased.

Expected Value in Regression

The expected value in regression, particularly of the error term \( u \), is a statistical concept indicating the mean or average value that \( u \) would assume over numerous samples. It is crucial for the error term in a regression model to have an expected value of zero \( \mathrm{E}(u) = 0 \). This condition ensures that the error term does not systematically bias the predictions. If the expected value of the error term deviates from zero, it implies the error is non-randomly shifting the predicted line vertically, potentially distorting the relationship between \( x \) and \( y \). Thus, when \( \mathrm{E}(u) eq 0 \), we need to conduct an intercept adjustment.

Intercept Adjustment for Optimal Predictions

Intercept adjustment is a critical step when \( \mathrm{E}(u) eq 0 \) in a regression model. In our example, we saw how the intercept was recalculated as \( \beta_0' = \beta_0 + \alpha_0 \). This modification:

• Compensates for the non-zero expectation of \( u \)
• Ensures the regression line reflects true relationships
• Preserves the slope \( \beta_1 \)
• Redefines \( u \) as \( u' \) where \( \mathrm{E}(u') = 0 \)

By making the intercept adjustment, we align the regression model with the fundamental assumption that the expected error term is zero. Such refinements allow for more accurate and unbiased insights from the regression analysis, essential for making reliable conclusions in research and applications.