Econometrica: Jul, 1973, Volume 41, Issue 4

Linear Regression with Error in the Deflating Variable

https://doi.org/0012-9682(197307)41:4<751:LRWEIT>2.0.CO;2-L
p. 751-759

M. C. Casson

When there is error in the deflating variable of a multiple linear regression, the ordinary least squares slope estimators are inconsistent unless one of the following conditions obtains: (i) the intercept of the ratio regression is zero, (ii) the mean of each independent ratio variable is zero, or (iii) the error in the deflator is systematic (has zero variance). The intercept estimator may be inconsistent even when the slope estimators are consistent; for the intercept estimator to be consistent it is sufficient that the slope estimators are consistent and, in addition, the expected value of the proportional error in the reciprocal of the deflator is zero. When there is inconsistency, the estimated regression tends to indicate a proportional relationship between the dependent ratio variable and each of the independent ratio variables (an important application of this result concerns the estimation of distributed lag accelerator relationships). Given assumptions about the distribution of error in the deflator, consistent estimators can be evaluated; an empirical application suggests that, as a rule, inconsistency is small even when the errors in the deflator are quite large.