Case LOG Appendix
Case LOG
LN (natural log transformation of a variable)
1. Change of a natural log variable:
[pic]
where [pic]
Then for small value of [pic] - see the table below,
[pic]
2. Regression Using LN variables
Consider:
[pic]
Suppose that X increases to X’ and for that Y increases to Y’. The relationship between X’ and Y’ is:
[pic]
Therefore,
[pic].
Using the result in 1, the above expression can be written:
[pic]
For a small proportional change in X, i.e., for a small [pic] and [pic],
[pic] and [pic], so that the above expression can now
be written, as an approximation,
[pic].
This allows a nice interpretation for β1.
Exercises
A Suppose that a regression study of quantity supplied as a function of price yielded the following equation:
log Q = 5.2 + 1.3 log P
a. What is the elasticity of supply?
b. If price increased by 3%, by how much would the quantity supplied increase?
c. What price increase would be required to increase the quantity supplied by 10%?
B. In a study of how personal income was related to education, IQ, race and other variables, the following multiple regression was estimated from a sample of 1400 U.S. veterans (Griliches and Mason);
LINC = .046E + .0010 AFQT + . 17 WHITE + other variables
(SE) (.007) (.0004) (.05)
where LINC = log of the veteran's weekly income
E = number of years of additional education (during and after military
service)
AFQT = the veteran's percentile rating on the Armed Forces Qualification Test
Score, which was used as a rough measure of IQ
WHITE = dummy variable for race, being 1 for whites and 0 otherwise
Other variables = age, amount of military service, amount of schooling before military service, father's education and occupational status, and degree of urbanization of his childhood home
Other things equal, it was estimated that:
a. A veteran with 1 more year's additional education earned ————% more
income.
b. A veteran who was white earned ————% more income than one who was
not.
c. A veteran who rated 1 point higher on the AFQT (e.g., who was in the 51st
percentile instead of the 50th percentile) earned ————% more income.
d. A non-white veteran who scored in the 80th percentile in the AFQT earned ____% more income than a white veteran who scored in the 50th percentile.
C. For Case: Marketing a New Shampoo Formula, Tracy ran the following regression:
X1=price in cents,
X2 = promotional expenditure in $1000, and
Y=ln(unit sales).
Interpret the regression coefficients.
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[pic]
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