An imputation method for dealing with missing data in regression

We develop two methods for imputing missing values in regression situations. We examine the standard fixed‐effects linear‐regression model Y = X β + ϵ, where the regressors X are fixed and ϵ is the error term. This research focuses on the problem of missing X values. A particular component of market‐share analysis has motivated this research where the price and other promotional instruments of each brand are allowed to have their own impact on the total sales volume in a consumer‐products category. When a brand is not distributed in a particular week, only a few of the many measures occurring in that observation are missing. ‘What values should be imputed for the missing measures?’ is the central question this paper addresses. This context creates a unique problem in the missing‐data literature, i.e. there is no true value for the missing measure. Using influence functions, from robust statistics we develop two loss functions, each of which is a function of the missing and existing X values. These loss functions turn out to be sums of ratios of low‐order polynomials. The minimization of either loss function is an unconstrained non‐linear‐optimization problem. The solution to this non‐linear optimization leads to imputed values that have minimal influence on the estimates of the parameters of the regression model. Estimates using the method for replacing missing values are compared with estimates obtained via some conventional methods.