iii) Changing the scale of y and x by the same factor: there will be no change in the reported regression results for b2, but the estimated intercept and residuals will change. t-statistics and R2are unaffected. The interpretation of the parameters is made relative to the new units of measurement.

2 2. Choosing a functional form Let’s start with the example of food expenditure. What does economics really say about the relation between food expenditure and income, holding all else constant? We expect there to be a positive relationship between these variables because food is a normal good, but nothing says the relationship must be a straight line. A very popular phenomenon is the change of marginal effect. What it means graphically is that there is not a straight-line relationship between two variables. Figure 1 A nonlinear relationship between food expenditure and income By transforming the variables y and x we can represent many curved, nonlinear relationships and still use the linear regression model. Choosing an algebraic form for the relationship means choosing transformations of the original variables. The most common are: i) Power: If x is a variable, then xpmeans raising the variable to the power p, examples include quadratic (x2) and cubic (x3). ii) Natural logarithm: If x is a variable, then its natural logarithm is ln(x). Logarithmic transformations are often used for variables that are monetary values, for example, wages, salaries, income, prices, sales, and expenditures. In general, for variables that measure the ‘‘size’’ of something. These variables have the characteristic that they are positive and often have distributions that are positively skewed, with a long tail to the right. Figure 2 Alternative function forms

3 Table 1 Some useful functions Review of the properties of natural logarithm The most popular nonlinear function in econometrics is natural logarithm, which we refer to simply as the log function, as y=log(x). The most useful property is that the difference is logs can be used to approximate proportionate changes. (using Taylor series) Let x0and x1be positive values, 1010101001xxxlog(x)log(x )log(x )log(x / x )x / xx / x Multiple both sides by 100, then we get: 100log(x)% x(a.2) for a small change in x. Therefore, a 1% increase in the value of a variable x (i.e., a 0.01 increase in relative terms) is approximately equal to a 1 log-point increase in x (0.01 increase in logx). This approximation works well when x is small. For example, a worker’s weekly wage (w) increases from $400 to $410, a 2.5% increase ((410-400)/400=0.025). Now taking logs and difference the logs, log(410)-log(400)=016-5.991=0.0247, which when multiplied by 100 is very close to 2.5. Summary of three configurations are listed as: i) In the log-log model both the dependent and independent variables are transformed by the ‘‘natural’’ logarithm, the parameter β2is the elasticity of ywith respect to x.

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- Fall '19