The math behind percentage change
Key facts: The log of a product is the sum of the logs.
for small
.
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
- Take a log (natural) regression.
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
- Increase x by
percent, how does ln(y) shift?
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
- How much did ln(y) shift?
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
.
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
.
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
.
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
.
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
-
.
![*](http://www-stat.wharton.upenn.edu/~waterman/icons/redpin.gif)
- Finally: percentage change in y is
.