Lecture 6.1a

October 20, 2016

Scaling data

If we're estimating the model lm(height ~ weight,data) using weight in lbs and height in inches, will we find the same results when we convert to metric data?

Scaling data

The first model estimated $\frac{\Delta inches}{\Delta pounds}$.
Now we're estimating $\frac{\Delta cm}{\Delta kg}$.
The numbers should change, but they should tell the same story.
All our hypothesis tests will be exactly the same.
The units don't really matter.

Scaling data

Since the units don't really matter, if we have units that are hard to interpret, we might want to use easier to understand units.
Normalized coefficients (Wooldridge calls these Beta coefficients) are found by converting data into z-scores
- Interpretting these coefficients are interpretted as, "a one standard deviation in $x_i$ is associated with a $\beta_i$ change in $y$."
- These coefficients are easy to compare to one another…
  - They tell us about the relative strength of different effects.

Transforming the data

normalize <- function(vect){
  vbar <- mean(vect)
  z <- (vect-vbar)/sd(vect)
  return(z)
}
mtcars.n <- mtcars %>% mutate_each(funs(normalize))

Does it matter?

Regression results

Call: lm(formula = mpg ~ hp + wt + disp, data = mtcars.n)
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)  3.568e-17  7.740e-02   0.000  1.00000   
hp          -3.544e-01  1.301e-01  -2.724  0.01097 * 
wt          -6.171e-01  1.731e-01  -3.565  0.00133 **
disp        -1.927e-02  2.128e-01  -0.091  0.92851   

Residual standard error: 0.4379 on 28 degrees of freedom
Multiple R-squared:  0.8268,    Adjusted R-squared:  0.8083 
F-statistic: 44.57 on 3 and 28 DF,  p-value: < 1e-04

Regression results

library(stargazer)
fit1 <- lm(mpg ~ hp + wt + disp, mtcars)
fit2 <- lm(mpg ~ hp + wt + disp, mtcars.n)
stargazer(fit1,fit2,type = "html")

Regression results


	Dependent variable:

	mpg
	(1)	(2)

hp	-0.031^**	-0.354^**
	(0.011)	(0.130)

wt	-3.801^***	-0.617^***
	(1.066)	(0.173)

disp	-0.001	-0.019
	(0.010)	(0.213)

Constant	37.106^***	0.000
	(2.111)	(0.077)


Observations	32	32
R²	0.827	0.827
Adjusted R²	0.808	0.808
Residual Std. Error (df = 28)	2.639	0.438
F Statistic (df = 3; 28)	44.566^***	44.566^***

Note:	^p<0.1; ^p<0.05; ^**p<0.01

Logarithms

Logarithmic relationships show up when something changes at a constant rate.
- Constant rate of growth means that your bank deposit will grow $10 this year $11 next year, etc. The amount of interest payments grows in dollar terms
- But it will always take the same amount of time for your account to increase in value by $x%$.
We interpret a logarithmic coefficient as a percentage change. The underlying unit no longer matters (i.e. Canadian dollars and American dollars both grow at some percentage interest)
Log transformation often make the data more closely satisfy the Classical Linear Regression assumptions.