Interpretation of the trend coefficients in log-linear model

Question

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?

Answer 1

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.

Answer 2

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 $X$.