Norris and Pendakur (2013) apply the selection model to Canadian data, and find that making use of Heckman selection model increases the imputed rent of owners by almost 30% with respect to a standard OLS model.

## **2.1.8 Stratification (Imputation Cells)**

The stratification method involves creating a number of homogeneous cells defined in terms of various characteristics, from the size of the dwelling, year of construction, quality of the building, to location or household composition. In a more sophisticated version of the approach, strata could also be defined by means of cluster or factor analysis (see, for instance, Olczyk and Lane, 2008). All households are then assigned to each strata, and the imputed rent for the home owners are then defined on the basis of rents paid by tenants in the same strata.

Statistics Finland defined 128 strata (Frick et al 2010). In a study on rent imputation in the UK, Mullan et al. (2007) define 12 strata, using as variables tenure type, council tax band, whether the property is in London or not, and the number of rooms in the property. Stratification is used in 12 out of 29 countries for estimating imputed rent in the EU Survey on Income and Living Conditions (Törmälehto and Sauli, 2013).

Typically, imputed rents are obtained by taking the mean (Crossley and Pendakur, 2006) or the median within each cell (Milligan, 2008). In case some cells are empty of renters, neighbor cells are used.

While creating the strata, the objective is to optimize the physical homogeneity of dwellings within each cluster, while ensuring a sufficient number of observations to produce a reliable rent estimate. Clearly, the more characteristics are used to define the strata, the more refined is the approach, as dwellings in the same strata will be of a more homogeneous quality. Nevertheless, increasing the number of strata reduces the average number of observations per stratum, and a very detailed stratification might raise the standard error of any rent imputation (see, for instance, Eurostat, 2013a). Moreover, seeking precision often clashes with the underlying data.

According to Malpezzi (2000), for instance, the minimum required number of observations for analysis in a given housing market is around 500, and any stratification that leads to cells with a number of observations smaller than 500 should not be considered adequate. According to the Eurostat guidelines (Eurostat, 2013b, p.210), it is not recommended to use any stratification or econometric methods for rent-imputation when the share of free-market tenants falls below 10 percent (see also Törmälehto and Sauli, 2010, p.157). According to the International Comparison Program Methodological Handbook (ICP, 2013, ch.10, p.4), instead, the standard approach (cell-stratification) should not be attempted when less than 25 percent of all dwellings in the country are rented.

# **2.2 The Capital Market Approach**

In the _capital market_ approach the implicit rent is understood as the rate of return that would have been obtained by owners if the home equity had been invested in an interest bearing account. Alternately, the owner’s annual user cost of a dwelling could be understood as the difference between purchasing a house at the beginning of the period, using it, and selling it at the end of the year. Two methodologies may be followed under this approach: the _rent-to-value_ and the _user-cost_ method.

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