variables used to stratify the sample (dummies of industry, region and size), and therefore the fitted values can be used to replace missing data.

Hence, the first assumption we need is that the imputed variable can be represented as a multiple linear function of other variables. The second condition that needs to be met for multiple imputation to work well is that all the variables, including those replaced and those used to replace, have normal distributions (see Allison, 2001).<sup>24</sup> According to equation (3) and (4), equation (2) represents the ‘ _maximization step_ ’, which is now given by where _y, l, m_ and _k_ with a tilde on top represent the imputed variables and _s*_ is the new pattern of missing values after the replacement process.<sup>25</sup> With identification conditions in the MAR case *  * *  *    given by _E_ ( _situit_ )  0 , _E_ [( _sit J_ )( _situit_ )]  _E_ [( _sit Juit_ )]  0 with _J_  _l_<sup></sup> _it_ , _mit_ , _kit_ , _ICi_ , _Dit_ , while in the case of _exogenous sample selection_ we need that That is, we need to control for any explanatory variable correlated with _s*_ to get consistency either in the inputs or IC variables.

When the two assumptions mentioned above (normality and linearity of imputed variables on dummies of industry, region and size) do not hold, the replacement strategy is no longer consistent. Very little can be said about the asymptotic distributions of the estimators obtained under these circumstances because they have not yet been derived. In a general fashion, in these cases we can understand our replaced variables as the _classic_ problem of _variables measured with error_ . In order to illustrate this, let our model be given by _yi_  _xi_  _ui_ , where _yi_ represents sales and _xi_ is a vector of inputs. Suppose that in the population we have that ˆ _E_ ( _ui_ | _xi_ )  0 , and that _xi_ is missing when _i_  _S_ . When we predict _xi i_  _S_ such that _xi_  _xi_  _vi_ ˆ   where _xi_ is our predicted value, then the model becomes _yi_  _xi_  _vi_  _ui_ . Where when _i_  _S_   ˆ  ˆ _xi_  _xi_ and _vi=0_ , while if _i_  _S xi_  _xi_ and _vi_  _xi_  _xi_ . Therefore, consistency of estimates of    depends on whether _E_ ( _vi_ | _xi_ )  0 . Consistency follows if the linear regression of the inputs on industry, region and size variables gives us a noisy measure of the true level of the variables. Otherwise we will have a _vi_ and the parameters obtained from regression analysis would be consequently downward biased, and the magnitude of the bias will depend on the standard deviation of the error term relative to the standard deviation of the variable and the proportion of replaced values.<sup>26</sup>

> 24 Although these are strong assumptions, the imputation method seems to works well even when the variables have distributions that are manifestly not normal, see Schafer (1997). 25 Variables included in the IC and C vectors are imputed by using the same procedure. However, by means of illustration and simplification here we only discuss the identification condition as if only PF variables were imputed. 26 We thank Ariel Pakes for useful suggestions at this point.

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