732 Microeconometrics: Methods and Developments
process. Also, not all parameters may be identified. For example, regression co-
efficients may be identified only up to scale or an intercept may not be identified
while slope parameters are; Pagan and Ullah (1999) provide many examples.
A nonparametric nonlinear simultaneous equations model isr(yi,xi) = ui,
whereyanduareG×1 vectors andxisK×1. The model is nonparametric
identified if it is possible to recover the unknown functionr(·)and the distribution
ofufrom the joint distribution of(y,x). Matzkin (2008) provides identification
conditions whenuis independent ofx.
14.2.2 Partial identification
Manski (1995, 2008) and related papers emphasize partial identification or set
identification that merely provides bounds, rather than stronger point identifica-
tion or complete identification. Partial identification can be possible under weaker
assumptions about the data-generating process (DGP) than those needed for point
identification.
For example, suppose data onyare missing for 20% of the sample, potentially due
to self-selection. Then, without any further assumptions, the median is necessarily
bounded by the 0.375 and 0.625 quantiles of the nonmissing data. For example,
with 80 observed values ofysuppose that the 20 missing values are all less than
the smallest observed value. Then the median of all 100 observations is the 30th
or 31st of the 80 observed values, or the 30/ 80 =0.375 quantile. Bounding E[y],
rather than the median, is more challenging as it requires additional assumptions
on the minimum and maximum value of the mean for the missing data. Quali-
tatively similar results exist if a fraction of the data are mismeasured rather than
missing. In practice the bounds obtained can be wide, but additional information
or assumptions can tighten bounds considerably.
Manski and Pepper (2000) provide an upper bound for returns to schooling, con-
trolling for schooling level being endogenously chosen. Haile and Tamer (2003)
provide bounds on the quantiles of the distribution of bidders’ valuations using
auction outcomes. Blundellet al. (2007) provide bounds on the interquartile
range of wages, controlling for changing composition of the employed and unem-
ployed. Statistical inference, using a framework of estimation based on moment
inequalities, is presented in Chernozhukov, Hong and Tamer (2007).
Finally, it should be noted that while much of the literature focuses on identi-
fication of parameters, this may not be necessary. In particular, many studies in
microeconometrics seek to calculate the marginal effect on the conditional mean
of, say, thejth regressor,∂E[y|x]/∂xj
∣∣
∣
x=x∗, and this can be achieved by nonparamet-ric or semiparametric regression. Even where a model for E[y|x]is posited, complete
identification of E[y|x]may not be necessary. For example, consider a linear panel
fixed effects model where E[yit|xit]=x′itβandxitincludes a time-invariant vari-
able, thekth say, withxik =xk. Then even ifxkis unobserved, fixed effects
estimation provides consistent estimates of the components ofβcorresponding
to time-varying regressors, and hence the marginal effect.