When the pattern of missing values _s_ is correlated with the dependent variable of (1) we are in the presence of a self-selection case.<sup>21</sup> In this case the missing values are not ignorable and we cannot get rid of incomplete observations. In this case, equation (2) must be estimated by other sample selection corrections, such as the Heckman selection model.

In what follows we discuss the first imputation mechanism proposed to deal with the problem of incomplete data; _the ICA method_ .

# **3.1 Imputation of missing values:  The ICA method**

Our method of imputing missing data, which we call the _ICA method_ , shares the expectation step of the Expectation-Maximization (EM) algorithm proposed in the seminal paper of Dempster, Laird and Rubin (1977), a method that, within the maximum likelihood approaches, has been widely applied in several scientific fields (see McLachlan and Krishnan (1997) for a review). In particular, the replacement strategy used departs from the expectation of the production function variables conditional on the industry, region and size the corresponding observation belongs to (‘ _expectation step_ ’). Or equivalently, we replace the missing value by the expectation of the distribution of the variable conditional on the information on sector, region and size according to next equation where Y, L, M and K represents output, labor, materials and capital and DR, DI and DS are vectors of region, industry and size dummies respectively. Notice that we choose (3) such that it represents the special features of the IC datasets—in IC surveys industry, region and size are the variables used to stratify the sample.

After excluding from the replacement process those observations with all the production function variables missing,<sup>22</sup> estimated values to replace incomplete data are given by Unlike the EM algorithm,<sup>23</sup> the ICA method has the advantage of separating the imputation of missing data from the estimation of the parameters of the population model. More precisely, separating the imputation mechanism of a population model is the main characteristic of the _multiple imputation_ approaches, which allows using them with virtually any kind of data and any kind of model. The _ICA method_ is, in fact, a general multiple imputation mechanism in which we assume that each imputed variable can be represented as a linear function of the

> 21 Notice that as equation (1) is equivalent to: log _Yit_  _L_ log _Lit_  _M_ log _M it_  _K_ log _Kit_   0  _IC_  _ICi_  _D_  _Dit_  _uit_ , where on the right hand side we have the productivity index. We are clearly concerned with the possible correlation of the MDM with productivity or TFP as it may induce biases in the estimators of the vector _β_ . 22 The ICA method is conservative in the sense that we do not replace missing cells for those observations with all but one PF variables unobserved. We force the industry-region-size cells to have at least 18 values to estimate consistently the sample average. Moreover, in order to avoid biases caused by outlier observations, we use the within-group median instead of the within-group mean. 23 The EM algorithm imputes missing data conditional on a given population model, and therefore chooses the candidates’ values to replace the missing cells that maximize the likelihood function conditional on a vector of parameters of that model.

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