I have just started learning regression analysis and have doubts on its basic assumptions. Specifically, I do not have a clear idea over the use of the random error component $\epsilon$ in the regression model.
Assumption $1$ : The simple linear regression model is given as $$Y_i=\beta_0+\beta_1.X_i+\epsilon_i$$, where $Y$ is response variable, $X$ is regressor (deterministic), $\beta_0$ is the intercept, $\beta_1$ is slope and $\epsilon_i$ is the random error component.
Assumption $2$ : $\epsilon_i \sim \mathcal {N(0,\sigma^2)}$ and $\epsilon_i$ and $\epsilon_j$ are independent of each other.
Now let me elaborate my problem.
Suppose I am a restaurant owner and I have a huge record of numbers of meals ordered with bill amount and the tip amount for each meal ordered. I want to predict what the next tip amount should be, for any given bill amount. But there I'm stuck.
\begin{array}{| c | c | c |} \hline \\\text{Number of Meals Ordered} & \text{Bill Amount per Meal (in Dollar)} & \text{Tip Amount (in Dollar)} \\ &(X) & (Y)\\\hline 1& x_1& y_1 \\\\\hline 2& x_2& y_2\\\\\hline 3& x_3& y_3\\\\\hline 4& x_4& y_4\\\\\hline .& .& .\\. & .& .\\. & .& .\\ .& .& .\\\hline 100000 & x_{100000} & y_{100000}\\\\\hline \end{array}
If we treat this data-set as population and if we take any sample of size $100$ or of size $1000$, then we can find out $$Y_i=\widehat{\beta_0}+\widehat{\beta_1}.X_i+e_i\\\Longrightarrow Y_i=\widehat{\beta_0}+\widehat{\beta_1}.X_i+Y_i - \widehat{Y_i}\\\Longrightarrow \widehat{Y_i}=\widehat{\beta_0}+\widehat{\beta_1}.X_i$$ and later we go for model adequacy checking etc.
- Here in the diagram I just tried to understand, how does the concept of $\epsilon_i$ arise into the general model.
So, from Assumption $1$ - I have the questions
Nowhere in the analysis we are using or estimating $\epsilon_i$, then how does the concept of random error component arise in the model? If $\epsilon_i$ is unknown, then how can $e_i$ estimate $\epsilon_i$?
From Assumption $2$, we have $\epsilon_i$ and $\epsilon_j$ are independent of each other and the amount of error in $\epsilon_i$ and $\epsilon_j$ are different, therefore their variance should be different. Then $\epsilon_i$ should have heteroscadastic variance rather than homoscadastic variance.
Any help is highly appreciated.