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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