Simple Linear Regression Model (SLRM)
A simple linear regression model (SLRM) is based on a single independent (explanatory) variable and it fits a straight line such that the sum of squared residuals of the regression model (or vertical distances between the fitted line and points of the data set) as small as possible. This model can (usually known as statistical or probabilistic model) be written as
\begin{align*}
y_i &= \alpha + \beta x_i +\varepsilon_i\\
\text{OR} \quad y_i&=b_0 + b_1 x_i + \varepsilon_i\\
\text{OR} \quad y_i&=\beta_0 + \beta x_i + \varepsilon_i
\end{align*}
where y is dependent variable, x is independent variable. In regression context, y is called regressand and x is called the regressors. The epsilon ($\varepsilon$) is unobservable, denoting random error or the disturbance term of regression model. $\varepsilon$ (random error) has some specific importance for its inclusion in the regression model:
- Random error ($\varepsilon$) captures the effect on the dependent variable of all variables which are not included in the model under study, because the variable not included in the model may or may not be observable.
- Random error ($\varepsilon$) captures any specification error related to assumed linear-functional form.
- Random error ($\varepsilon$) captures the effect of unpredictable random component present in the dependent variable.
We can say that $\varepsilon$ is the variation in variable y not explained (unexplained) by the independent variable x included in the model.
In above equation or model $\hat{\beta_0}, \hat{\beta_1}$ are the parameters of the model and our main objective is to obtain the estimates of their numerical values i.e. $\hat{\beta_0}$ and $\hat{\beta_1}$, where $\beta_0$ is the intercept (regression constant), it passes through the ($\overline{x}, \overline{y}$) i.e. center of mass of the data points and $\beta_1$ is the slope or regression coefficient of the model and slope is the correlation between variable x and y corrected by the ratio of standard deviations of these variables. The subscript i denotes the ith value of the variable in the model.
\[y=\beta_0 + \beta_1 x_1\]
This model is called mathematical model as all the variation in y is due solely to change in x and there are no other factors affecting the dependent variable. It this is true then all the pairs (x, y) will fall on a straight line if plotted on two dimensional plane. However for observed values the plot may or may not be a straight line. Two dimensional diagram with points plotted in pair form is called scatter diagram.
