# Log-linear ordinary least squares specification coefficient loses its interpretation when we change an explanatory variable by a unit?

$$ln(y_i)=\beta_0+\beta_1X+u_i \Rightarrow \text{if } \Delta x = 1, \text{then } \text{%}\Delta y \approx 100\beta_1$$

I think that this is a very bad approximation, although my reasoning is probably incorrect (although I do understand the derivation of why this approx holds).

Ok, so an aside:

$$\frac{\dot{y(t)}}{y(t)} = g \Rightarrow ln(y_t) = gt + c$$

$$y_{t+1} \approx y_t (1+g \Delta t)$$

So here, if I plug in a change of t = 1, and let g = 1, y would be doubling with each unit change in t, and so we should approx y as 2^x instead of something of the form e^x. Of course, large-ish changes in x mess up the calculus.

However, isn't plugging in a unit change in t (in the econometrics textbook, x is t) what the econometrics textbooks are doing? A unit change in x -> 100% change in y (g and beta_1 are analogous, so g = 1 -> beta_1 =1) -> y approx doubles with each change in x -> y should be modelled as something of the form 2^x, not e^x, and there's a sizeable difference between the two, and so this contradicts that fact that the specification implies that y is in the form e^x (rather than 2^x).

I hope this makes sense.

The source of the approximation:

Given, $$\ln(Y_i)=\beta_0+\beta_1X_i+u_i$$, for a unit change in $$X$$, i.e., $$X_{i+1}-X_i=1$$, we have:

\begin{align} \frac{Y_{i+1}-Y_i}{Y_i} &= e^{\beta_1+\Delta u_i}-1 \end{align}

For small $$x$$, we use taylor expansion to say: $$e^x \approx1+x$$. Using this above we get:

$$\frac{\% \Delta Y}{100} \approx \beta_1+\Delta u$$

This approximation is good when $$\beta_1$$ is quite small. In your example, you have taken $$\beta_1=1$$ which makes this a bad approximation.

Consider your example with $$g=0.1$$.

\begin{align} \frac{y_{t+1}-y_t}{y_t} &= e^g-1 \\ &=1.1052 - 1 \tag{for g=0.1} \\ & \approx g \end{align}

• @1muflon1: Do you mean that it should be $u_i$ in place of $\Delta u_i$? For the expectations part, i think then the LHS should also be changed to make it in terms of $\hat{Y_i}$, isn't it? Nov 29 '20 at 2:09
• right but i think that OP just omitted hats and you can interpret only coefficients that are estimated as true betas are unobservable but I guess it’s just unnecessarily pedantic I deleted my previous comment
– 1muflon1
Nov 29 '20 at 2:15
• Right, it seems OP did omit hats in his/her example. But I guess the message is clear with this also. Nov 29 '20 at 2:17
• Thank you. I didn't stop to think that beta/g had to be approx small as well as x. And yes, I omitted hats as I thought what I was trying to convey wouldn't require them (i.e. just messing around with the true population model). Nov 29 '20 at 14:02

When

$$\ln y = \beta_0 + \beta_1 x + u \implies y = \exp\{ \beta_0 + \beta_1 x + u\}$$

$$\implies \partial y / \partial x = \beta_1 y \implies \frac{\partial y / \partial x}{y} = \beta_1.$$

So we see that $$\beta_1$$ is the marginal change in $$y$$ due to infinitesimal changes in $$x$$ as a proportion of its level. Therefore the accuracy of the approximation

$$\beta_1 = \frac{\partial y / \partial x}{y} \approx \frac{\Delta y / \Delta x }{y}$$

$$\implies \Delta x = 1: \beta_1 \approx \frac{\Delta y }{y}$$

is nothing else than the general approximation inaccuracy issue that arises when we replace a derivative (infinitesimal change) with a disrete-interval proportional change.

In the specific case we have

$$y(x+1) - y (x) = \exp\{ \beta_0 + \beta_1 x + \beta_1 +u\} - \exp\{ \beta_0 + \beta_1 x + u\}$$

$$= y(x)\cdot (e^{\beta_1 -1}) \implies \frac{\Delta y(x+1)}{y(x)} = (e^{\beta_1} -1).$$

We get and we verify the folk wisdom that the approximation is accurate enough for purpose of economic analysis if $$\beta_1 \in [-0.1,\; 0.1]$$, and maybe for a larger interval, if one percentage point is not critical for the purposes of the specific research.