How to interpret effect of log-inventory on log-price?

Question

I have an autoregressive model that explains house prices.

• The dependent variable is the log of the house prices, in which the house prices are an index number (lprice)
• The independent variables are the log of real housing investment (linv) and the dependent variable one period lagged (l.lprice). And time to control for spurious regression etc. Model: $\log({\rm price}_t) = b_0 + b_1\log({\rm inv}_t) + b_2\log({\rm price}_{t-1}) + b_3t +\epsilon_t$

How should I interpret the effect of linv on lprice? I only need to know how to interpret it. Is it: "a one percent increase in investment will cause price to increase by $b_1$ percent" or is it "a one percent increase in investment will cause prices to increase by $b_1$ percentage point"

Answer 1

If $log(Y)=\beta_0+\beta_1 log(X)=\beta_0+log(X^{\beta_1})$ then, after exponentiating, you get $Y=e^{\beta_0} X^ {\beta_1}$.

Therefore $\frac{dY}{Y}=\frac{e^{\beta_0} \beta_1 X^ {\beta_1-1}dX}{Y}=\frac{e^{\beta_0} \beta_1 X^ {\beta_1-1}dX}{e^{\beta_0} X^ {\beta_1}}=\beta_1 \frac{dX}{X}$.

$\frac{dY}{Y}$ is the percentage change in $Y$ and $\frac{dX}{X}$ is the percentage change in $X$, so we find that the percentage change in $Y$ is equal to $\beta_1$ times the percentage change in $X$.

In your case $Y$ is the price index and $X$ is the investment.

Note that, from the above it follows that $\beta_1=\frac{\frac{dY}{Y}}{\frac{dX}{X}}$ meaning that, as @user89073 says, $\beta_1$ is the elasticity of $Y$ with respect to $X$, i.e. the percentage change in $Y$ for a 1% change in $X$.

Answer 2

First off, the inclusion of the time trend does not correct for spurious regression.

Secondly, when using a log-log model as in your case, the interpretation of $b_1$ is that it the elasticity of prices to investment in housing.

Therefore, if the investment in housing increases by 1%, the prices will increase by $b_1$ percent.