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.