1 Lecture 4 Data Transformation and Functional Forms This chapter will cover several topics in multiple regression that we could not conveniently study in previous chapters, but they are important for empirical problems. We will cover: how changing the units of measurement affect OLS estimates, how to incorporate popular functional forms into regression models; and how to examine model specifications. We focus on the model specification. 1. Effects of scaling the data Data we obtain are not always in convenient form. When the scale of the data is not convenient, it can be altered without changing any of the real underlying relationships between variables. For example, the gross domestic product at market price in the UK as the 1stquarter of 2012 is £380,756,000,000. While we could use this long form of number in a regression analysis, there is no advantage of doing so. Simply choosing the units of measurement to be “billions of dollars,” we have taken this long number and made it comprehensible. What are the effects of scaling the variables in a regression model? Example: Consider the food expenditure example, we report weekly expenditures in dollars, but when we report income in $100 units, a weekly income of $2,000 is reported as x = 20. 2***_83.4210.210.385se 43.412.09FOODEXPINCOMER(1) If we had estimated the regression using income in dollars, the results would have been: 2***_83.420.1021$ 0.385se 43.410.0209FOODEXPINCOMER(2) Comparing (2) to what we have used in simple regression (1), we notice the changes as: 1) the estimated coefficient of income is now 0.1021; 2) the standard error becomes smaller, by a factor of 100; and 3) since the estimated coefficient is smaller by a factor of 100 also, this leaves the t-statistic and all other results unchanged. Such a change in the units of measurement is called scaling the data. The choice of the scale depends on how to make interpretation meaningful and convenient. Scaling does not affect the measurement of the underlying relationship, but it does affect the interpretations and some summary measures as listed below: i) Changing the scale of x: the coefficient of x must be multiplied by c, the scaling factor. When the scale of x is altered, the only other change occurs in the standard error of the regression coefficient, but it changes by the same multiplicative factor as the coefficient, so that their ratio, the t-statistic, is unaffected. All other regression statistics are unchanged. ii) Changing the scale of y: If we change the units of measurement of y, but not x, then all the coefficients must change in order for the equation to remain valid. Because the error term is scaled in this process the least squares residuals will also be scaled. This will affect the standard errors of the regression coefficients, but it will not affect t-statistics or R2.
- Fall '19