The Bias of Certain Elasticity Estimators

In a recent post I discussed some aspects of estimating elasticities from regression models, and the interpretation of these values. That discussion should be kept in mind in reading what follows.

One thing that a lot of practitioners seem to be unaware of (or they choose to ignore it) is that in many of the common situations where we use regression analysis to estimate elasticities, these estimators are biased.

And that's true even if all of the conditions needed for the coefficient estimator (e.g., OLS) to be unbiased are fully satisfied.

Let's look at some common situations leading to the estimation of elasticities and marginal effects, and see if we can summarize what's going on.

We'll consider four simple regression models:

LIN-LIN:       y_i = α + β x_i + ε_i

LOG-LOG:  log(y_i) = α + β log(x_i) + ε_i

LIN-LOG:  y_i = α + β log(x_i) + ε_i

LOG-LIN:   log(y_i) = α + β x_i + ε_i

In each case, we'll assume x is non-random and that the ε_i's are i.i.d. [0 , σ²]; i = 1, 2, ..., n. "Log" denotes "natural logarithm".

(Although we're dealing with models that have a single regressor, all of the results below still apply in the multiple regression case.)

Now, let's consider the elasticity relating to the impact on y of a change in x.

The LOG-LOG model is easily dispensed with. In this case, as you'll know, the elasticity in question is just β. If we use an unbiased estimator for this parameter, then of course we have an unbiased estimator of the elasticity. OLS would ensure this, under the stated assumptions.

In the LOG-LIN model, for observations i, the elasticity is  η_i = (β x_i). If we estimate beta using an unbiased estimator, then the implied estimator of the elasticity will also be unbiased (as x is assumed to be non-random).

In the case of the LIN-LIN model, for observation i the elasticity is η_i = β(x_i / y_i). Now, notice that if estimate η_i by replacing β with its (unbiased) OLS estimator, b, then the elasticity estimator is random for two reasons - partly because of y_i itself, and partly because of the presence of b = Σ[(y_i - y*)(x_i - x*)] / Σ[(x_i - x*)²]. (Here, x* and y* are the sample averages of x and y.)

More specifically, η_i is a non-linear estimator of η - it's a non-linear function of the random, y, data. It's also a biased estimator, even though b is unbiased. Yes, this bias will disappear if you have a very large sample, but that may not be much comfort in some cases.

Chen and Giles (2007) derive an analytic expression for this bias in the multiple regression case, using a "small-disturbance" ("small-sigma") approximation. This expression doesn't rely on having a large value for n. The bias expression is pretty messy (even for the simple regression situation). I won't present it here, but you can check it out easily enough.

Notice that if were to evaluate the elasticity at the sample means of the data, as people often do in order to simplify the reporting of the results, then the same general points apply - the elasticity estimator is biased, even if we use an unbiased parameter estimator. The bias is somewhat different in this case. Again, the details are given by Chen and Giles (2007).

Finally, in the case of the LIN-LOG model the elasticity at observation i is η_i = (β / y_i). Even if we estimate this elasticity by using an unbiased estimator of β, the elasticity estimator will be biased, for the same reasons as for the LIN-LIN model. If you look at the results given in Chen and Giles (2007) you'll notice that they can be modified in a trivial way so that they apply to the LIN-LOG model.  

In each case, the models we've considered are linear in the parameters. If we generalize this aspect of the models, then things get even more complicated, and I'm not going to get into that in this particular post.

Also, keep in mind that none of the three model specifications considered above need be appropriate - that's also a different matter.

The bottom line -

• Just because your choice of parameter estimator ensures that it has "good" properties, these properties generally won't apply to the associated elasticity estimator.
• Even if your parameter estimator has a Normal sampling distribution, this won't be the case for your elasticity estimator (except in the LOG-LOG case if the errors are Normal). 
• So, unless you have a very large sample, you'll have to think twice before constructing conventional confidence intervals for the elasticity, or testing hypotheses.

Reference

Chen, Q. and D. E. Giles, 2007. The bias of elasticity estimators in linear regression: Some analytic results.  Economics Letters,  94, 185-191. (Download working paper version.)


© 2015, David E. Giles