Are Parameter Estimates Unbiased If the Dependent Variable is a Per Capita Rate Based on Approximate Population Data?

Question

Suppose we want to estimate a model:

$Y_i = \beta_0 + \beta_1X_i + \epsilon_i\tag{1}$

where the $Y_i$ are averages or rates per capita for geographical regions or zones, so that if $Z_i$ is the regional aggregate variable of interest and $P_i$ is population:

$Y_i = Z_i / P_i\tag{2}$

If all the variables are accurately measured, and if there exist $β_0^*$ and $β_1^*$ such that for all i:

$E[Y_i - \beta_0^* - \beta_1^*X_i] = E[\epsilon_i] = 0\tag{3}$

then OLS estimation of model (1) should yield an unbiased estimate of $\beta_1$.

Suppose however that estimates of $P_i$ are only approximate. This is in practice likely since the values of $P_i$ will almost certainly be obtained or derived from census data, leading to several possible sources of error:

1. error in the original census data;
2. extrapolation from census data using assumptions on population trends where data are required in respect of a later date;
3. the regions used in the model may not correspond to census areas, eg concentric rings around a central point. An example is a travel cost valuation study reported in Herath (1999) (1) where populations in the range 5,000 to 35,000 for concentric ring zones are all stated in exact multiples of 5,000, suggesting that the figures are only very approximate.

The errors in $P_i$ will obviously ‘infect’ the values of $Y_i$, but not in a straightforward way, since the absolute effect on $Y_i$ of a given error in $P_i$ will depend on the size of $Z_i$ (and if $Z_i = 0$ there will be no error in $Y_i$).

Question: Given errors in $P_i$, will OLS estimation of model (1) yield an unbiased estimate of $\beta_1$, and if not, under what additional conditions would the estimate be unbiased?

Reference

1. Herath, G (1999) Estimation of Community Values of Lakes: A Study of Lake Mokoan in Victoria Australia Economic Analysis & Policy 29(1) Table 1 p 37

Answer 1

What about assuming a multiplicative error process and then using logs?

Say the data generating process were a little different: $Y_i = \beta_0 \cdot X_i^{\beta_1}\cdot E_i$

If $Y_i = Z_i / P_i$ in truth but all we could really observe was:

$\hat{P}_i = P_i \cdot \Gamma_i$, where $P_i$ was the true value and $\Gamma_i$ was a strictly positive measurement error.

We' be estimating the following equation in practice:

$\hat{Y}_i = Z_i / \hat{P}_i = \beta_0 \cdot X_i^{\beta_1}\cdot E_i$

take the logs of both sides (lower case letters are logs of individual variables):

$y_i = \beta_0 + \beta_1 \cdot x_i + \epsilon_i - \gamma_i$

If we define a variable $\xi_i = \epsilon_i - \gamma_i$ then we can have an equation $y_i = \beta_0 + \beta_1 \cdot x_i + \xi_i$ that looks very much like the one you wrote above.

If $\xi_i$ satisfies the same relationships with the log variables that $\epsilon_i$ does with the level variables it seems like everything should still work.

Answer 2

Just to complement @Bkay's answer, "measurement error" in the dependent variable is relatively "harmless" -what hurts is error in measurement in the regressors.

If we have measurement error in the dependent variable, what we need to assume in addition, in order to preserve unbiasedness, is that this error is independent of the regressors. If we can reasonably assume that (and usually we can), then @BKay 's answer shows that the effect is just a transformation of the error term of the regression. It may affect the variance, but not the parameter estimates.

On the contrary, if we have measurement error in the regressors, then they stop being strictly exogenous to the error term, and unbiasedness is lost.