What is the difference between random error and residual?

[kingwinner]

1) "Simple linear regression model: Y_i = β₀ + β₁X_i + ε_i , i=1,...,n where n is the number of data points, ε_i 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 β₀,β₁ 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: Y_i = β₀ + β₁X_i + ε_i , i=1,...,n where n is the number of data points, ε_i is random error
Fitted value of Y_i for each X_i is: Y_i hat = b₀ + b₁X_i
Residual = vertical deviations = Y_i - Y_i hat = e_i
where Y_i is the actual observed value of Y, and Y_i hat is the value of Y predicted by the model"

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

Thanks for explaining!

 

[Enuma_Elish]

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.

 

[bombastic]

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.