The Function as an Equation I: Simple Linear Regression
Tabulation has two serious limitations. In the first place, even when data have been tabulated, it is often necessary to represent the relationship, at least approximately, by an equation. A table is poorly adapted for the study of elasticities, marginal productivities, multipliers, and other quantitative aspects of economic functions. Much modern research, for example the construction of econometric models, requires that relationships be expressed by simple mathematical formulas. Secondly, tabulation requires a substantial amount of data. A small sample puts a severe limit to the number of cells that can be explored, even when the function is restricted to a single independent variable.
Linear regression is a method of approximating a statistical function by a simple linear equation. Moreover, because of its relative simplicity, it can be usefully applied even to samples too small to tabulate. These two points are illustrated by Tables 7.1 and 7.2 and the accompanying charts.
Table 7.1 contains family beef consumption tabulated as a function of income. The small standard errors indicate that the function is measured with considerable precision, and when plotted in Figure 7.1, the cell means fit tightly to a smooth curve. Although the equation of this curve is complicated, it can be usefully approximated by a straight line. What the linear approximation loses in ability to describe detail is made up, in part at least, by the simplicity of its equation.
Table 7.2 contains the total annual production and the farm price of onions for each of 11 years. Detailed tabulation of onion demand is