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)% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa % aaleaacaWGPbaabeaakiabg2da9iabeg7aHjabgUcaRiabek7aIjaa % dIfadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaai % aadMgaaeqaaOGaaGjbVlaadMgacqGH9aqpcaaIXaGaaiilaiaaikda % caGGSaGaeSOjGSKaaiilaiaad6gaaaa!4B3A!$${Y_i} = \alpha + \beta {X_i} + {u_i}\;i = 1,2, \ldots ,n$$ 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. For example, if Y is consumption and X is disposable income then we would expect a positive relationship between these variables and the data may look like Figure 3.1 when plotted for a random sample of households. If α and β were known, one could draw the straight line (α + βX ) as shown in Figure 3.1. It is clear that not all the observations (X _i , Y _i ) lie on the straight line (α + βX). In fact, equation (3.1) states that the difference between each Y _i and the corresponding (α + βX_i) is due to a random error u _i .