# Interpretation of the trend coefficients in log-linear model

I want to estimate price of a product, lets say a boat, over time - panel data model.

I estimated a model where I control for number of characteristics denoted as $$X$$ ( e.g. engine size, weight etc.). However, I also included a trend and trend squared variable into the model to control for "other" market developments. Let's just assume that is correct specification for now.

$$ln(price_i) = \alpha_i + \beta_i\sum X + \gamma_i~trend + \delta_i ~ trend^2$$

My estimated coefficients for $$\gamma$$ is negative (-0.5) and $$\delta$$ positive (0.002). This indicates that the initially price is decreasing with time but after around 13 years the trend reverts (convex shape).

My question is how do I interpret the coefficients (ceteris paribus), meaning I know that per one unit increase in trend (year) there is approx. $$-0.048 \approx (1- e^{-0.05})*1 + (1-e^{0.002})*1^2$$ change in price in year 1, $$-0.092$$ in year 2, $$...$$, $$-0.092$$ in year 23, $$...$$, etc. However, is it change in price relative to "year 0" or year-to-year change? In short how I would label the y-axis on the plot above, would %$$\Delta$$ Price be correct?

Personally, I don't find such coefficients to be very meaningful on their own. Here's how I would interpret these results.

The expected value for $$\ln y$$ is $$\alpha + \beta 'x + \gamma \cdot t + \delta \cdot t^2$$

Taking the derivative of that with respect to $$t$$ and applying the chain rule, you get

$$\frac{\partial y}{\partial t}\cdot \frac{1}{y}= \gamma + 2 \cdot\delta \cdot t$$

That is the very definition of semi-elasticity, and it is a function of $$t$$ in your model. It is the relative change in price $$y$$ for a one unit change in $$t$$ since you can rearrange the LHS to $$\frac{\Delta y/y}{\Delta t}$$. I would multiply things by a 100 here to convert to a percentage change in $$y$$. This curve looks like:

You can see that the percent price change is negative when $$t$$ is small at -50% percent, and shrinks somewhat over 30 years to to -38%. It is always negative over this range. In short, time always leads to boat depreciation, but the price drop grows smaller over time.

You can also ask how that semi-elasticity changes with $$t$$ by taking the derivative again:

$$\frac{\partial \varepsilon}{\partial t} = 2 \cdot\delta$$

This makes $$\delta$$ a bit more interpretable: it tells you that the elasticity falls by constant $$200 \cdot \delta$$ for each additional year.

• Great answer, thank you! – An economist Jun 21 '19 at 20:32
• +1 for sound application of discipline specific terminology and pointing out the relation between the linear model and the differential equation. You have interpreted the expected difference in price for a unit difference in time as being a linear function of time itself. Clever approach. – AdamO Jun 25 '19 at 20:14

The question boils down to: 1) how do I interpret the coefficients in a log-linear model? And 2) how do I interpret the linear and quadratic terms in a linear model?

For 1. The (exponentiated) coefficient is a relative rate. Log-linear models are usually used model count data, but they amount to using a log-transform and a mean=variance relationship (using a quasilikelihood model is considerably more general). So a $$e^\beta = 1.1$$ coefficient for a predictor $$X$$ is interpreted as a 10% difference in (price) comparing groups differing by 1-unit of $$X$$.

For 2. The intercept has the usual interpretation. The linear term is the instantaneous relative rate when $$X=0$$. The quadratic term is like an interaction term between $$X$$ and itself. You can call it a relative rate ratio, i.e. it is the ratio of relative rates comparing groups differing by 1 unit in $$X$$.