Suppose we are interested in investigating how households income changes would

Suppose we are interested in investigating how

This preview shows page 11 - 13 out of 62 pages.

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:
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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.
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