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

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"

• I updated the title of this post to better reflect the question you're asking. If you don't like the change, feel free to roll back the edit. Welcome to the site! Commented Jan 15, 2016 at 15:55

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 $X$.

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.

• You could also add why this interpretation is correct. Commented Jan 15, 2016 at 14:37
• This part is answered in the answer of @fcop. Commented Jan 18, 2016 at 9:03