Let us say we have this regression

$$\ln(y) = a + B_1(age) + B_2\ln(savings) + B_3\ln(income+1)$$

When carrying out the regression we obtain:

$$\ln(y) = 0.3445 + 0.5(age) + 0.4556 x_1 + 0.55566 x_2$$

How would one interpret the coefficients in each case? Of particular concern is the income coefficient. An increase of income of 1% would lead to an increase of how much in $y$?

This is a hypothetical example to illustrate the problem I have.

• So you just to the 0.56 ignoring the constant + 1 there? – KingKong Oct 18 '13 at 14:02

If income is typically much larger than 1, you could ignore the $+1$ for interpretation and use the usual statement for linear log-log-models: "A 1% increase in income is associated with a $100\% \cdot (1.01^{0.55566}−1)=0.5544\%$ increase in the geometric mean of $y$. Or, a bit less precise but better to understand: "A 1% increase in income is associated with about a 0.56% increase in the typical value of $y$.

Edit:

• If you do not want to ignore the $+1$ for interpretation, just say "A $1\%$ increase in $1 + \text{income}$ ..."
• If you prefer to describe the effect on the arithmetic mean of $y$ instead of its geometric mean, try a (Gamma-)GLM with log link.
• Sorry for deleting the former comment. But there was a mistake in it I couldn't correct anymore ;) – Michael M Oct 18 '13 at 14:45
• what if the item is not income and has small values! – KingKong Oct 19 '13 at 8:44
• Updated the response in this direction. – Michael M Oct 19 '13 at 10:53

Assuming everything else stays constant,

(change in y) / (y) = B3 * (change in income) / (1 + income)

LHS is your percentage change in y. Put the values in the RHS. So for a 1% increase in income, change in income is 0.01 and income is 100. Gives you a result straightaway as B3 * (1/101)% or 0.0055%.

• Forgot to include the beta coeff. – DWin Oct 19 '13 at 18:48
• extremely sorry for the error have updated the value. – htrahdis Oct 20 '13 at 13:20