In general the coefficient of X2 is the derivative

dE[logY]/dX2, where E[logY] is the expectation of logY, removing the random disturbance term. With X2 being a dummy variable, this becomes: if X2 increases from zero to one, E[logY] increases with the coefficient of X2. And the other way round.

However, you note that you use 2SLS. If this is relevant for your question about X2, then apparently you assume that X2 depends on logY. But you cannot have an ordinary regression relation of X2 as a function of logY because X2 is discrete. Of course X2 cannot have, for instance, a normally distributed disturbance term.

What is X2?

Hi, Arie Ten Cate: Thank you for answering. The first portion of your answer seems similar to OLS coefficient interpretation.

The second portion is not still clear to me.

Y= wage

X2=1 if the person is male

x2=0 if the person is female.

the coefficient of X2 means most a person is male, most the chance to see his wage (Y) decreased is 86,23%

1. I do not understand why the gender dummy should be an endogenous variable.

2. The equation is easy to read. The effect is dependent on the base of the logarithm used. If it is 10, then female wage would be 7,28 (10^0,8623) times as high as that of men, which means - as Roger wrote - men's wages would be 86,23% lower. If ln (i.e. natural log) is used, women's wages would (for equal X3) be 2,37 times men's wages or these were 57,8% lower.

3. I think one can dismiss this estimate, whatever X3 should measure.

I’m assuming that -0.8623 is the 2SLS estimate, i.e., the estimate obtained after using instruments. Approximately, the difference between male and female wages is 86.23%. More precisely, if females comprise the base group, males earn 100[exp(-0.8623) – 1]% or 57.78% less. Contrarily, another way of looking at the precise effect is that if males comprise the base group, females earn 100[exp(0.8623) – 1]% or 136.86% more.

@Anton Rainer : I have made a mistake in defining x2. To make it simpler as a dummy variable, I defined x2 as gender. however, I just need to know what would be the interpretation for such a 2sls model where x2 is a dummy variable.

so the interpretation that you put is it also applies to the case if x2 is a endogenous variable other than gender?

Please explain.

2SLS is appliied when explaining variables are (partly) endogenous, i.e. when variables are somehow dependent on other explaining variables or on missing variables which cannot be included, because they are not measurable or unobserved. There could also be a feedback from the dependent variable (Y) on Xi. After the 2SLS procedure, one can hope that such disturbing effects are removed. The coefficients can, therefore interpreted as if the variables were really exogenous.

If one delogs your equation, you get

Y=b^B1/b^0,8623X2*b^(0,05X3), where b is the base of the logarithm. An increase of X3 by 1(X2 unchanged) means a division of Y by b^0,8623 and you get the results I wrote you in my first answer. If X2 is not a (0,1)-dummy, you can, of course, calculate the effect of any other change dX2 by dividing through b^0,8623dX3. You can do analogous calculations for the X2-effects (this time, of course, by multiplying). For estimations of lnY, one can, for small changes of X2, calculate the effect on Y directly from the coefficinet (0,05=5%), an increase by 1 leads to an increase of Y by about 5% (which is only slightly lower than the exact value (e^0,05-1=5,127%). For higher coefficients and higher changes of the variables this difference will grow exponentially and be to high to allow this simple calculation.

PS: 1. I think you should put more effeort in thinking over the data and their transformation and what functional relation between the variables would be the best one according to theory and/or common sense before estimating.

2. I am rather sceptical about 2SLS, mainly because it is, in general, very difficult to find good instrumental variables. Most examples I have read up to now were not really convincing.

Everything being equal, the wage of an individual decreases by 86.23% when that individual is a male

My first answer had a minor incorrectness. If one calculates in logs then a unit increase of X3 leads to a 86,27% (instead of 86,23%) lower Y. It is a strange coincidence, that the coefficient 0,8623 is so near to the value (let us call it) z=0,862871, where log(1+z)=z. Therefore one should have a look , whether the data for Y and X3 are independently ascertained.

Anton Rainer: Thanks a lot.

I asked "what is X2" because a dummy explanatory variable seldom is endogenous, requiring 2SLS. Indeed, it seems in this case odd that X2 is endogenous: gender is fixed at conception (ignoring transgender). And hence the OLS interpretation, which is clear to you, holds for X2. Or let me put the phrase differently: what are your instrumental variables for gender?

(Maybe this is one large misunderstanding?)

Arie, you are right that gender could hardly be an endogenous explaining variable, but from our discussion above you can see that X2 is not gender. Even without knowing what sort of variable it is, one can, of course, calculate the effect of a change of X2 on Y.

By the way, I want to warn Roger and Martin Paul to take the coefficient directly as a percentage change. This would only be possible for lnY and if coefficient*dX2 is small. If you take X3, then you can say that an increase of 1 leads to an increase of Y by 5% (exactly it is 5,13%=e^(1*0,05). For an increase of X3 by 10, the difference would be to large (50% against 64,9%).

I am sorry to have missed the not-gender post.

I suspect that 2SLS with a dummy endogenous explanatory variable is a applied while there is no way to tell the software that X2 is a dummy variable (like with logit or probit). In a way the software assumes that X2 coincidentally has only two values. Then the estimate is computed as if X2 can assume all values. (Admittedly this is not an answer to the original question.)

Thanks a lot.

Md. Alauddin nice question Anton Rainer well explained.