Simple Linear Regression

In this chapter, we study extensively the estimation of a linear relationship between two variables, Y i and X i , of the form: 3.1 $$Y_i = \alpha + \beta X_i + ui \quad i = 1,2,\ldots n$$ where Y i denotes the i-th observation on the dependent variable Y which could be consumption, investment or output, and X i denotes the i-th observation on the independent variable X which could be disposable income, the interest rate or an input. These observations could be collected on firms or households at a given point in time, in which case we call the data a cross-section. Alternatively, these observations may be collected over time for a specific industry or country in which case we call the data a time-series. n is the number of observations, which could be the number of firms or households in a cross-section, or the number of years if the observations are collected annually. α and β are the intercept and slope of this simple linear relationship between Y and X. They are assumed to be unknown parameters to be estimated from the data. A plot of the data, i.e., Y versus X would be very illustrative showing what type of relationship exists empirically between these two variables.