model (1), estimated using the 2001 and 2007 HESs, to impute household expenditure into the LFSs for all available years (2001 to 2010), and then use the imputed expenditure data to estimate poverty for the entire 2001-2010 period.

We will be relying on a number of assumptions:

# **Assumption 1** : _The model is time-invariant, i.e._ βt = β _._

This assumption cannot be avoided, since we are looking to apply the model for imputing data into years other than the year for which the model can be estimated. This has the potential to introduce a bias if the two different time periods are in fact associated with different models. This bias can only be ruled out if _Assumption 1_ holds true. Note that this assumption can be tested both directly and indirectly. For a direct test one could use any test statistic that evaluates the difference between the estimate for β2001 and the estimate for β2007 . An indirect test can be obtained by estimating the model for one of the two HES years and then apply this model to obtain an imputed poverty rate for the other year. Since both years also allow us to compute the actual poverty rates based on observed data, we are able to verify how well the observed data compares to the imputed data. See section six for the test results.

**Remark 1** : The assumption of a time-invariant model also means we have to be careful with the choice of units in which both the dependent and the independent variables will be measured. One may want to avoid including independent variables that are expressed in time _t_ monetary value; we will be working with quantities and dummy variables for our model. This prevents potential model inconsistencies. Let us refer to the model parameters as betas. If one variable measures the monetary value of owned bicycles, while another variable simply checks whether the household has a car or not by equaling one or zero, then this could become problematic. The beta associated with the car ownership will measure the monetary value added of the car to the total household expenditure. The beta associated with the bicycles on the other hand has no dimension but instead simply passes on (by some factor) the monetary value of the bicycles to the value of total household expenditure. To see how this may yield an inconsistency, suppose that the dependent variable is measured in time _t_ prices (i.e. household expenditure over time is not measured at constant prices). Then, the beta attached to the car ownership will be expressed in time _t_ prices, which goes against the assumption of constant betas. Now suppose that the dependent variable is measured in constant prices. In that case, the beta attached to the bicycles will have to convert the monetary value added of the bicycles from time _t_ prices into the constant reference prices. This too is in conflict with the assumption of time-invariant betas.

# **Assumption 2** : _The error term u is homoscedastic and normal._

This assumption is not strictly necessary, and can thus be relaxed, but is adopted for ease of exposition. The model is found to perform surprisingly well as our empirical results for Morocco will show, despite these simplifying assumptions.

Since our variable of interest is the poverty rate (the probability that a randomly selected household has an expenditure level below the poverty line), we will transform the imputed consumptions into household probabilities of being poor conditional on the information contained in _x_ . Let _F_ denote the 9