Suppose we are interested in investigating how households’ income changes would affect average weekly expenditure on food. The answers to this question provide valuable information in long-run business planning. For example, forecasting food sections for a supermarket chain. It also helps at the macroeconomics level to predict what might happen as levels of income change for policy purpose. If we label household expenditure on food as y and the corresponding household income as x, we may express a relation in general form as y=f(x). We need be more precise about the functional form. For simplicity, we assume that it is reasonable to model the relationship as a linear function, which is also known as population regression function:
12 12yx (2.1) In this equation, βs are unknown parameters. β1is the intercept parameter, or the level of household expenditure on food when income is zero. β2is the slope parameter (day/dx) and shows by how much expenditure on food increases as household income increases. |12(|)y xE y xx(2.2)Sometimes, we are also interested in the percentage change in y brought about by a 1% change in x. In this example, it is the percentage change in food expenditure that results from a 1% increase in income, i.e., the elasticity of expenditure on food on income. It is written symbolically as: y2dy xx.dx yy (2.3) Let’s take a further look at elasticity which is widely used in economics. It is the percentage change in one variable associated with a 1% change in another variable for movements along a specific curve. 2%100(/)/.%100(/)/yxyyyyydy xxxxxxxdx yy(2.4)If we were to sample household expenditures at other levels of income, we would expect the sample values to be scattered around their mean value E(y|x). In figure 1, we depict the pdfs of food expenditure f(y|x), along with the regression line for each level of income. Figure 1 The probability density function for y at two levels of income There are likely to be many factors other than income that influences a household’s expenditure, to allow for a not-exact statistical relation, we add an unknown and unobservable random variable ε into the economic model. To collect household observations on y and x, the subscript iis introduced to the ithobservation on each of the variable. The statistical model is written as: i12iiyx (2.5) The random error term serves three main purposes: 1) to capture the combined effect of all other influences in the model; 2) to capture any approximation error due to the linear functional form, and 3) to capture ay elements of random behaviour present in each individual.