Interpreting coefficients for transformed variables

Transcription

1 Interpreting coefficients for transformed variables! Recall that when both independent and dependent variables are untransformed, an estimated coefficient represents the change in the dependent variable expected when the independent variable changes by one unit, holding the other variables constant.! The substantive interpretation of coefficients in such situations is accordingly fairly straightforward.! Interpreting coefficients when variables have been transformed can be somewhat trickier.! The most straightforward case involves transforms with logarithms.! We will deal with situation first, and talk about how to deal with some of the others later.

2 Logged variables! There are two common bases that are used for logarithmic transformations.! A natural logarithm is in base e. e, you may know, is a mathematical constant. Its first few digits are y! The natural log of x is y such that e = x.! In STATA, log(x) and ln(x) both return the natural log of x.! Another common base for the logarithm is 10. y! The log 10 of x is y such that 10 = x.! In STATA, log (x) returns the log of x ! One property of logarithms is that multiplying x by some constant a adds log a to its log.! Thus if the natural log of a variable increases by 1, that implies that the original variable has been multiplied by e.! If log of a variable increases by 1, the original variable 10 has been multiplied by 10.

3 Either the independent or dependent variable is logged! If the dependent variable is raw, and the independent variable is logged, the estimated coefficient b is the absolute change in the dependent variable expected when the original independent variable is multiplied by e or 10, depending on the base of the transform.! In this situation, you can work out the expected change in the dependent variable associated with a x percent increase in the independent variable by multiplying the coefficient by log([100+x]/100). Make sure to keep the bases the same.! To work out the expected change associated with a 10% increase in the independent variable, therefore, multiply by log(110/100) = log(1.1).! ln(1.1) = ! log 10(1.1) = ! If the dependent variable is logged, and the independent variable is not, every unit change in the independent variable is expected to multiply the original dependent b b variable by e or 10, depending on the base of the transform. b is the estimated coefficient.

4 When both independent and dependent variables are logged! If both the independent and dependent variables are logged, multiplying the original independent variable by e or 10 b b will multiply the original dependent variable by e or 10, depending on the base.! In the latter situation, where a proportional change in the independent variable is associated with a proportional change in the dependent variable, the coefficient is referred to as an elasticity.! To get the proportional change in the dependent variable associated with a x percent increase in the independent ab variable, calculate a = log([100+x]/100) and take e or ab 10, depending on the base.! The predicted proportional change can be converted to a predicted percentage change by subtracting 1 and multiplying by 100.! Be careful in all these calculations to keep your bases consistent.

5 Some examples! Let's consider the relationship between the percentage urban and per capita GNP: 100 % urban 95 (World Bank) United Nations per capita GDP! This doesn't look too good. Let's try transforming the per capita GNP by logging it: 100 % urban 95 (World Bank) lpcgdp95

6 ! That looked pretty good. Now let's quantify the association between percentage urban and the logged per capita income:. regress urb95 lpcgdp95 Source SS df MS Number of obs = F( 1, 130) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = urb95 Coef. Std. Err. t P> t [95% Conf. Interval] lpcgdp _cons ! The implication of this coefficient is that multiplying capita income by e, roughly , 'increases' the percentage urban by percentage points.! Increasing per capita income by 10% 'increases' the percentage urban by 10.43* = percentage points.

7 What about the situation where the dependent variable is logged?! We could just as easily have considered the 'effect' on logged per capita income of increasing urbanization: lpcgdp % urban 95 (World Bank). regress lpcgdp95 urb95 Source SS df MS Number of obs = F( 1, 130) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lpcgdp95 Coef. Std. Err. t P> t [95% Conf. Interval] urb _cons ! Every one point increase in the percentage urban multiplies per capita income by e = In other words, it increases per capita income by 5.4%.

8 Logged independent and dependent variables! Let's look at infant mortality and per capita income: limr lpcgdp95. regress limr lpcgdp95 Source SS df MS Number of obs = F( 1, 192) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = limr Coef. Std. Err. t P> t [95% Conf. Interval] lpcgdp _cons ! Thus multiplying per capita income by multiplies the infant mortality rate by e = 0.607! A 10% increase in per capita income multiplies the infant *ln(1.1) mortality rate e = ! In other words, a 10% increase in per capita income reduces the infant mortality rate by 4.6%.

9 What about other transformations?! The power and root transformations don't lead to such intuitive interpretations.! The coefficient represents the effect, after all, of a change in the power or root of the original variable.! One of the best things to do in such situations is to look at predicted values of the dependent variable for a range of values of the independent variable, most likely through a graphical plot of the predicted variable against the untransformed variable.! Consider the relationship between IMR and the square root of the percentage of houses with running water: 149 IMR water2. regress IMR water2 Source SS df MS Number of obs = F( 1, 90) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = IMR Coef. Std. Err. t P> t [95% Conf. Interval] water _cons ! So increasing the square root of the percentage of

10 households with running water by 1 lowers the infant mortality rate by 20 per 1000.! Let's vary the percentage from 0 to 100, predict values of the IMR, and look at the results:. replace water95 = _n - 1 (216 real changes made). replace water2 = sqrt(water95) (216 real changes made). predict pimr. graph pimr water95 if water95 <= pimr Water (World Bank)! Another approach is to consider derivatives.! The prediction equation from the above estimation is:

11 ŷ' %&20.17 x! If we differentiate that with respect to x, we get 1 dy ˆ dx '&0.5(20.17x& 2 '&10.085x & 1 2! If we evaluate that at a few locations: x (%) dy/dx ! The effect of an increase in the percentage in houses with running water is much stronger when the percentage is small than when it is large.! Typically, a root transformation of an independent variable implies 'diminishing returns.'