## Simple linear regression from raw data

Let $(x_i, y_i), i=1,2, \cdots , n$ be $n$ pairs of observations.

The simple linear regression model of $Y$ on $X$ is

$$y_i=\beta_0 + \beta_1x_i +e_i$$ where,

• $y$ is a dependent variable,
• $x$ is an independent variable,
• $\beta_0$ is an intercept,
• $\beta_1$ is the slope,
• $e$ is the error term.

## Formula

By the method of least square, the model parameters $\beta_0$ and $\beta_1$ can be estimated as

The regression coefficients $\beta_0$ (intercept) and $\beta_1$ (slope) can be estimated as

### $\hat{\beta}_0=\overline{y}-\hat{\beta}_1\overline{x}$

where,

• $\overline{x}=\dfrac{1}{n}\sum_{i=1}^n x_i$ is the sample mean of $X$,
• $\overline{y}=\dfrac{1}{n}\sum_{i=1}^n y_i$ is the sample mean of $Y$,
• $n$ is the number of data points.

Important Results

• Explained variation $SSR = \sum(\hat{y}-\overline{y})^2$
• Unexplained variation $SSE = \sum (y-\hat{y})^2$
• Total variation $SST = \sum (y-\overline{y})^2$
• Coefficient of determination $R^2 =\dfrac{SSR}{SST}$
• Standard error of estimate $S_e = \sqrt{\dfrac{\sum(y-\hat{y})^2}{n-2}}=\sqrt{\dfrac{SSE}{n-2}}$

Suggestions and comments will be appreciated.