1 CHANGES IN THE UNITS OF MEASUREMENT Suppose that the units of measurement of Y or X are changed. How will this affect the regression results? Intuitively, we would anticipate that nothing of substance will be changed, and this is correct.

2 We will demonstrate this for the estimates of the regression coefficients in this section, and we will trace the implications for the rest of the regression output in due course. We begin by supposing that the true and fitted models are as shown above. CHANGES IN THE UNITS OF MEASUREMENT

3 We now suppose that the units of measurement of Y are changed, with the new measure, Y*, being a linear function of the old one. CHANGES IN THE UNITS OF MEASUREMENT

4 Typically, a change of measurement involves a simple multiplicative scaling, such as when we convert pounds into grams. CHANGES IN THE UNITS OF MEASUREMENT

5 However, one occasionally encounters a full linear transformation. Conversion of temperatures from degrees Celsius to degrees Fahrenheit is an example. CHANGES IN THE UNITS OF MEASUREMENT

6 Regressing Y* on X, the new slope coefficient b 2 * is as shown. CHANGES IN THE UNITS OF MEASUREMENT

7 We substitute for Y* from the linear relationship defining it. CHANGES IN THE UNITS OF MEASUREMENT

8 The 1 terms cancel. 2 is a common factor in the numerator. CHANGES IN THE UNITS OF MEASUREMENT

9 We find that the new slope coefficient is equal to the original one, multiplied by 2. CHANGES IN THE UNITS OF MEASUREMENT

10 This is logical. A unit change in Y is the same as a change of 2 units in Y*. According to the regression equation, a unit change in X leads to a change of b 2 units in Y, so it should lead to a change of 2 b 2 units in Y*. CHANGES IN THE UNITS OF MEASUREMENT

11 The effect on the intercept will be left as an exercise. The effect of a change in the units of measurement of X will also be left as an exercise. CHANGES IN THE UNITS OF MEASUREMENT

12 However, we will consider a special case of a change in the measurement of X. Often the intercept in a regression equation has no sensible interpretation because X = 0 is distant from the data range. CHANGES IN THE UNITS OF MEASUREMENT

13 CHANGES IN THE UNITS OF MEASUREMENT The earnings function discussed in the previous slideshow is an example, with the intercept actually being negative. EARNINGS = – S ^

14 CHANGES IN THE UNITS OF MEASUREMENT Here is the output for the regression.. reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons |

15 CHANGES IN THE UNITS OF MEASUREMENT The intercept is a meaningless negative number. Literally, it is the predicted level of hourly earnings for those with zero years of schooling, but nobody in the sample had fewer than seven years of schooling and only 31 had fewer than twelve years.. reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons |

16 CHANGES IN THE UNITS OF MEASUREMENT Sometimes it is useful to deal with this problem by defining X* as the deviation of X about its sample mean.

17 Note that, by definition, the sample sum of X i * is 0, and hence the mean value of X* is zero.. CHANGES IN THE UNITS OF MEASUREMENT

18 If we regress Y on X* instead of X, the slope coefficient is not affected. CHANGES IN THE UNITS OF MEASUREMENT

19 The intercept is now the fitted value of Y at the sample mean of X. This is the sample mean of Y and it may be more informative for analytical purposes. CHANGES IN THE UNITS OF MEASUREMENT

20 We will investigate the effect of demeaning schooling int he wage equation example. First, we find the sample mean. (In Stata, we use the ‘sum’ command.) We find that the mean is years, and we define a new variable SDEV by subtracting from S sum S Variable | Obs Mean Std. Dev. Min Max S | gen SDEV = S CHANGES IN THE UNITS OF MEASUREMENT

21 Here is the output using SDEV instead of S. The intercept, 19.63, now gives the predicted earnings of those with mean schooling.. reg EARNINGS SDEV Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] SDEV | _cons | CHANGES IN THE UNITS OF MEASUREMENT

22 Comparing the new output with the original, we see that, apart from the standard error and t statistic of the intercept, nothing else has changed. CHANGES IN THE UNITS OF MEASUREMENT. reg EARNINGS SDEV Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] SDEV | _cons | reg EARNINGS S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons |

23 This is exactly as one would expect. The only effect of the demeaning of S is to move the vertical axis to the point that was formerly As a consequence, the intercept becomes Otherwise, the regression line is unaffected. CHANGES IN THE UNITS OF MEASUREMENT EARNINGS = S ^

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 1.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course EC2020 Elements of Econometrics