Linear Regression Models (2)

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Main Question or Discussion Point

1) "Simple linear regression model: Yi = β0 + β1Xi + εi , i=1,...,n where n is the number of data points, εi is random error
We want to estimate β0 and β1 based on our observed data. The estimates of β0 and β1 are denoted by b0 and b1, respectively."


I don't understand the difference between β01 and b0,b1.
For example, when we see a scattered plot with a least-square line of best fit, say, y = 8 + 5x, then βo=8, β1=5, right? What are the b0 and b1 all about? Why do we need to introduce b0,b1?


2) "Simple linear regression model: Yi = β0 + β1Xi + εi , i=1,...,n where n is the number of data points, εi is random error
Fitted value of Yi for each Xi is: Yi hat = b0 + b1Xi
Residual = vertical deviations = Yi - Yi hat = ei
where Yi is the actual observed value of Y, and Yi hat is the value of Y predicted by the model"


Now I don't understand the difference between random error (εi) and residual (ei). What is the meaning of εi? How are εi and ei different?

Thanks for explaining!
 
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Answers and Replies

  • #2
The greek letters are for the true value of each parameter. The latin letters are for the estimated values. The values or the former do not depend on your sample. The values of the latter are sample-specific.
 
  • #3
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To answer the question on error and observables...

ε in this regression model refers to the unobserved error of the data. In the true model, it represents all the factors that the experimentor cannot see or account for. In many models, it is assumed ε is uncorrelated with X and that E[ε]=0.


A residual (e) is something totally different. It is simply as you defined it: Ybar - Yhat. It is the difference between the expected value of the dependent variable and the observed value. e is fundamentally about what you are trying to solve in the regression. Minimizing the sum of squared residuals.
 

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