OLS Regression
Calculating the y-intercept, b0
In linear regression, we are modeling the dependent variable using this model:
Y = β0 + β1X + ε
Here, Y is the dependent variable, X is the independent variable, β0 is the expected value of Y when X = 0 in the population, β1 is the effect of X on Y in the population, and ε is random variation unexplained by the model.
To perform statistical inference, we make the usual assumption that
ε ~ Normal(0, σ²)
The y-intercept is the estimate of Y when X = 0. It is the β0 in the model above. There are different methods for estimating this value. Ordinary least squares is one method. Its strengths are that it is easy to perform, it is exact, and that it is straight forward to create the estimators.
The Problem
Example #593: Let us examine the relationship between the thickness of the leaf on its weight. Let us specifically estimate the weight when the thickness is zero. To explore this, we collect data. The data consist of two measurements on each unit (leaf): thickness and weight. Thus, our data are
| Leaf Number | Thickness [μm] | Weight [g] |
|---|---|---|
| 1 | 3 | 7.2 |
| 2 | 3.2 | 4.9 |
| 3 | 7.8 | 3 |
| 4 | 3.6 | 5.7 |
| 5 | 3.3 | 5.8 |
| 6 | 3.9 | 6.5 |
| 7 | 6.4 | 4.4 |
| 8 | 5.6 | 6.2 |
| 9 | 6.4 | 3.3 |
With this information, estimate the value of leaf weight when leaf thickness is zero.
Note: This analysis will not make sense. Leaf thickness is never zero. More importantly, none of the values of the leaf thickness are close to zero. When it is the case that the range of the x-variable does not cover the value zero, the y-intercept should not be examined. To do so would require extrapolation.
Information given:
To summarize the above, the values of import are:
| \( \bar{x} \) | = | 4.8 |
|---|---|---|
| \( \bar{y} \) | = | 5.2222 |
| \( b_1 \) | = | -0.6349 |
Note that \( b_1 \) is important in estimating the y-intercept. If you are unsure how to calculate it, or if you would like more practice doing so, please see the OLS estimate of the slope tutorial.
Your Answer
In the box below, please enter the OLS estimate of the y-intercept, then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.
![[Correct!]](images/correct.png "You are correct. Well done!")
You got the correct answer of b0 = 8.2696. Congratulations!
![[Incorrect!]](images/incorrect.png "You are not correct.")
Unfortunately, your answer was not correct. Either try again or click on “Show Solution” below to see how to obtain the correct answer.
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