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Simple Linear Regression

Last Update: February 21, 2022

Simple linear regression is used to model linear relationship between two variables y and x . Dependent variable y is the explained one which is also known as target or response feature. Independent variable x is the explanatory one which is also known as predictor feature.

When doing simple linear regression, we can start by drawing a scatter chart with variables y and x on the vertical and horizontal axis, respectively. Then, we can draw a line which describes linear relationship between variables y and x. This line represents model fitting with formula \hat{y}_{i} = \hat{\beta}_{0} + \hat{\beta}_{1} x_{i} \; (1). Notice that we are using ˄ or hat character in formula notation because they are estimates. Regression fitted values \hat{y}_{i} are the estimated y_{i} values. Estimated constant or intercept coefficient \hat{\beta}_{0} is the \hat{y} value when x=0 or the \hat{y} value where line crosses vertical axis. Estimated slope coefficient \hat{\beta}_{1} is the estimated change in \hat{y} when x changes in one unit.

Model fitting can be done using ordinary least squares method with formula min \; \sum_{i=1}^{n} \hat{e}_{i}^{2} \; (2). This method minimizes the sum of squared estimated regression residuals \hat{e}_{i}. Estimated regression residuals \hat{e}_{i} with formula \hat{e}_{i} = y_{i} - \hat{y}_{i} \; (3) are the differences between actual y_{i} and fitted \hat{y}_{i} values.

Below, we find an example of scatter chart with simple linear regression of house price explained by its lot size [1].

Figure 1. Simple linear regression scatter chart of house price explained by its lot size.


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[1] Data Description: Sales prices of houses sold in the city of Windsor, Canada, during July, August and September, 1987.

Original Source: Anglin, P., and Gencay, R. (1996). Semiparametric Estimation of a Hedonic Price Function. Journal of Applied Econometrics, 11, 633–648.

Source: AER R package HousePrices object. Christian Kleiber and Achim Zeileis. (2008). Applied Econometrics with R. Springer-Verlag, New York.

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