# Interpreting standardized coefficients on a natural log response variable in OLS multiple regression

I am working on a housing problem in which I use dichotomous and ratio data to predict housing production (units constructed in a year-ratio) in a 17 year time period. At this time, I am using OLS and as I get better at stats, I shall attempt this problem using time-series analysis. That said, I have used R to standardize all of my ratio predicting data and left the dichotomous data raw. And I have also transformed the response variable to a Natural log to normalize the distribution (i.e. many, many zeros>>yes, I know Poisson or Zero-populated counts in the future).

I have read the post on "interpret coefficients from a quantile regression on standardized data" and also the "convert my unstandardized independent variables to standardized." Based on those, I think that can do the following interpretation based on the following output. The variable region_id is dichotomous, supply is standardized.

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)          2.687e+00  2.171e-01  12.379  < 2e-16 ***

region_id            1.805e+00  1.383e-01  13.049  < 2e-16 ***

supply              -2.205e+01  2.204e+00 -10.005  < 2e-16 ***


Region Interpretation:
For every on city that is located in the Houston region, you can expect that annual housing production will increase by 1.8%.

Supply Interpretation:
For every one-unit increase in the standard deviation of housing supply, you can expect that annual housing production will decrease by -22.05%.

Nota bene.
I am not a stats or math person at all, but I have been using R for the past three years and I am quite familiar with OLS, but if you throw up an equation it will look "appropriately" Greek to me. :)

• Do you mean you're using the model $\log Y_i = \beta_0 + \beta_1 x_i + \beta_2 z_i + \epsilon_i$? (where $x_i$ is either 0 or 1) If so your estimated coefficients seem extremely large. Nov 15 '12 at 18:48
• @rje42, they are extremely large due to the range in the sample. However, region data is not standardized. It is simply 0 or 1.
– drm
Nov 15 '12 at 21:04

For binary predictors with large coefficients, the effect of region going from 0 to 1 is $$100 \cdot(\exp\{\beta\}-1),$$ which means the effect if more like 536% for Houston. When coefficients are small, this transformation will give you roughly the same result as merely looking at them. The intuition is that for a model like $$E[y|D]=\exp\{\alpha + \beta D\}\cdot E[\exp{u}],$$ where the second term comes from the retransformation back to levels from logs, the effect of D is $$$$\ln \frac{E[y|D=1]-E[y|D=0]}{E[Y|D=0)} \cdot 100=\frac{\exp\{\alpha + \beta\}-\exp\{\alpha\}}{\exp\{\alpha\}} \cdot 100=(\exp\{\beta\}-1) \cdot 100$$$$

Unfortunately, you can't just plug $$\hat \beta$$ in for $$\beta$$. You want to use $$100 \cdot \exp \left( \hat \beta - \frac{1}{2} \hat \sigma_{\beta}^2 \right),$$

where $$\hat \sigma_{\beta}$$ is the standard error of that coefficient. Similarly, I think you want to multiply the supply coefficient by 100 as well.

These estimates do seem very large, but I really don't know what to expect here.

Take a look at this post by David Giles for formulas and sources. The bias correction he describes (as long as you are willing to assume the errors are normal) will take your effect down a bit, but not substantively so.

• @dimitry, The estimates are large. It is because the sample includes cities as large as Houston (pop 2 million) to Pasadena TX with a (pop 141K), therefore Houston acts as a huge leverage and outlier and the data display this wide range. This is my first test of this data. ON MY NEXT go round, I plan to sample cities that are within 1 std dev in population and therefore the estimates will be more meaningful.
– drm
Nov 15 '12 at 20:41
• I wonder if you can accomplish this by including a $\ln (pop)$ with the constraint that its coefficient equals $1$. This assumes that housing production is proportional to population: $\frac{E[y \vert x]}{pop}=\exp\{x'\beta\},$ which is algebraically equivalent to a model where $E[y \vert x]=\exp\{x'\beta+\ln {pop}\}$. The term for this is a logarithmic offset. Nov 15 '12 at 22:06