Interpretation of coefficients in first-differenced vs level specifications with and without additional variables in levels

[Andy Berger]

Is there an intuitive explanation for why the interpretation of the coefficients in a first-differenced regression specification is the same as the for the coefficients in the specification in levels? In a regression of say house prices on income, a first-differenced specification essentially regresses price growth on income growth. Why does the interpretation of b1 remain unchanged and does not switch to the effect of income growth on price growth instead of income on price? How could I actually investigate the former? Does the interpretation change if I add an additional variable in levels, and if so, why intuitively?

log(price) =b0 + b1*ln(income) + u
b1 – a 1% increase in income is associated with a b1% increase in (house) price

Δlog(price) =b1*Δln(income) + u
b1 – same interpretation as above

If I add an additional variable in levels, say a share or interest rate variable, does the interpretation of b1 go back to growth rates?
Δlog(price) =b1*Δln(income) + b2*irate + u

I have been scratching my head for quite some time on this but have not been able to find an intuitive explanation in any of my econometrics books.

[Phil Bromiley]

y(t) = b x(t)
y(t) - y(t-1) = b x(t) - b x(t-1)
= b (x(t) - x(t-1))

Whether this retains the same percentage change interpretation that you had with the original log dependent variable I'm not sure.

[Joro Kolev]

If you add another variable in a linear fashion, and in a linear equation, the interpretation of the coefficients on the rest of the variables do not change. Because these are partial derivatives. But this does not mean that the estimator of b will stay the same.

In similar lines, there might be one big misunderstanding here, and this misunderstanding is related to wrong notation in what you have written.

Yt = a + bXt + Ut, can be also written for the past observation as
Yt-1 = a + bXt-1 + Ut-1, and the second can be subtracted from the first to obtain
Yt - Yt-1 = b(Xt - Xt-1) + (Ut - Ut-1).

Note that in your second equation (the differenced one), you used the same error u as in your first equation (the one in levels). This is incorrect. In the levels equation the error is Ut, and in the differenced equation the error is (Ut - Ut-1). These are two very different things. Similarly the regressors are different in the two equations, Xt vs. (Xt - Xt-1).

A key property of a regression equation is whether or not Cov(Xt, Ut) = 0 and Cov[(Xt - Xt-1) ,(Ut - Ut-1)]=0, and depending on this property the estimator of b might be estimating b, or might be estimating something else.

In short, one should not read too much into statements such as "the interpretation of b here and there is the same". The estimators that come from a differenced equation relative to the levels equation, or from a levels equation with another variable added relative to the original levels equation, are very different. and very generally not guaranteed to be estimating the same parameter.

[Andy Berger]

Thanks a lot for the response Joro. I take your points on the partial derivatives and missing the differenced error term in my second equation. This is much appreciated and clarifies my second question on adding additional variables.

Nevertheless, I am still not clear on my first, main question on interpretation of first differenced equations. I understand that first differencing, if warranted, can circumnavigate some statistical issues. However, only focusing on the intuition behind the interpretation of the coefficient in a univariate regression of house prices on income, will the interpretation of the beta not be different when regressing prices on in income from regressing changes in prices on changes in income?