Obtaining Linear Relationships

We have now seen how the relationship between two variables can be described by using sample data. Scatter plots provide a picture of the relationship, and correlation coefficients provide a numerical measure. In many economic and business problems a specific functional relationship is desired.

•    What mean level of sales can be expressed if the price is set at $10 per unit?
•    If 250 workers are employed, how many units should be expected?
•    If a developing country increases its fertilizer production by 1,000,000 tons, how much increase in grain production should be expected?

Economic models use specific functional relationship to indicate the effect on a dependent variable, Y, that results from various changes in an independent or input variable, X. In many cases we can adequately approximate the desired functional relationships by a linear equation:

 where Y is the dependent variable, X is the independent variable, is the Y-intercept, and is the slope of the line, or the change in Y for every unit change in X. The linear equation model computes the mean of Y for every value of X. This idea is the basis for obtaining many economic and business relationships, including demand functions, production functions, consumption functions, and sales forecasts.

We use regression to determine the best relationship between Y and X for a particular application. This requires us to the best values for the coefficients  and . Generally, we use the data available from the process to compute “estimates” or numerical values for the coefficients  and These estimates—defined as  b0 and b1 --are generally computed by the leas squares regression. Least-squares is a procedure that selects the best-fit line, given a set of data points.

The linear equation represented by the line is the best-fit linear equation. We see that individual data points are above and below the line and that the line has points with both positive and negative deviations. The distance of each point (xi, yi) from the linear equation is defined as residual, ei. We would like to choose the equation so that some function of the positive and negative residuals is as small as possible. This implies finding estimates for the coefficients and . 

Least-squares regression chooses b0 and b1 such that the sum of the squared residuals is minimized.