6. Measuring the Services of Durables and Owner Occupied Housing

This treatment of the purchases of durable goods dates back to Marshall (1898; 594–595) at least: “We have noticed also that though the benefits which a man derives from living in his own house are commonly reckoned as part of his real income, and estimated at the net rental value of his house; the same plan is not followed with regard to the benefits which he derives from the use of his furniture and clothes. It is best here to follow the common practice, and not count as part of the national income or dividend anything that is not commonly counted as part of the income of the individual.”.

“The third convention is that of the annual accounting period. It is this convention which is responsible for most of the difficult accounting problems. Without this convention, accounting would be a simple matter of recording completed and fully realized transactions: an act of primitive simplicity.” Gilman (1939; 26). “All the problems of income measurement are the result of our desire to attribute income to arbitrarily determined short periods of time. Everything comes right in the end; but by then it is too late to matter.” Solomons (1961; 378). Note that these authors do not mention the additional complications that are due to the fact that future revenues and costs must be discounted to yield values that are equivalent to present dollars. For more recent papers on the fundamental problem of accounting, see Cairns (2013), Diewert and Fox (2016).

This is the term used by Goodhart (2001; F350–F351).

Depreciation applies to the structure part of property value but not to the land part.

This price index may or may not include the price of the land that the new dwelling unit sits on; e.g., a new house price construction index would typically not include the land cost. The acquisitions approach concentrates on the purchases by households of goods and services that are provided by suppliers from outside the household sector. Thus if the land on which a new house sits was previously owned by the household sector, then presumably, the cost of this land would be excluded from an acquisitions type new house price index. In this case, the price index that corresponds to the acquisitions approach is basically a new house price index (excluding land) or a modification of a construction cost index where the modification takes into account builder’s margins.

See the discussion in Sect. 6.16 below on transfer costs.

The acquisitions approach is straightforward and simple for most durable goods but not for housing, if the land component of property value is regarded as out of scope. Properties are sold with a single price that includes both the land and structure components of housing and so if the land part of property value is regarded as out of scope for the index, then there is a problem in decomposing property value into land and structure components. This decomposition problem can be avoided if information on the construction costs for building a new housing unit are available. In this case, the construction cost index (including builder’s markups) can serve as the price index for newly constructed dwelling units.

Hill et al. (2017; 6) summarize the problem of variable weights as follows: “Hence the expenditure weights on OOH under the acquisitions approach can fluctuate very significantly over the housing cycle. If the weights are updated regularly this may have a destabilizing effect on the CPI. If the weights are not updated regularly, then the treatment of OOH may be highly sensitive to the choice of reference year.”.

From Hoffmann and Kurz (2002; 3–4), about 60% of German households lived in rented dwellings whereas only about 11% of Spaniards rented their dwellings in 1999.

Fenwick (2009, 2012) laid out the case for the use of the acquisitions approach as a useful measure of general inflation. He also argued for the construction of multiple consumer price indexes to suit different purposes.

This very useful publication also discusses the main methods for the treatment of OOH and it also covers the methods used to construct residential property price indexes. The latter topic is also covered in Eurostat (2013).

However, with the passage of time, it became apparent that some imputations for changes in the quality of consumer goods and services had to be made. Thus the current HICP is not completely free from imputations. See Astin (1999) for the methodological foundations of the HICP.

The use of a construction cost index also involves an imputation (but it is a reasonably straightforward one).

As will be seen in Sect. 6.16 below, the situation is not quite as simple as indicated above.

This approach is used by the Bureau of Labor Statistics (1983) in order to determine expenditure weights for owner occupied housing; i.e., homeowners are asked to estimate what their house would rent for if it were rented to a third party.

Lebow and Rudd (2003; 169) note that the US Bureau of Economic Analysis applies a benchmark rent to value ratio for rented properties to the value of the owner occupied stock of housing. It can be seen that this approach is essentially a simplified user cost method where all of the key variables in the user cost formula (to be discussed later) are held constant except the asset price of the property.

We will return to this point after we have discussed the opportunity cost approach to the valuation of OOH services.

This issue will be discussed in more detail in Sect. 6.16 below.

For example, according to the 2013 Japanese Housing and Land Survey, the average floor space (size) of owner occupied housing in Tokyo was 110.64 m\(^{2}\) for single family houses and 82.71 m\(^{2}\) for rental housing, a difference of over 30  m\(^{2}\). For condominiums, an even greater discrepancy exists: the average floor space is 65.73 m\(^{2}\) for owner-occupied housing and 37.64 m\(^{2}\) for rental housing. Moreover, in addition to the difference in floor space between rented and owned units, the quality of the owned units tends to be higher than the rented units and these quality differences need to be taken into account.

On this point, see also Lewis and Restieaux (2015).

Note that this approach to pricing the services of a durable good assumes the existence of second hand markets for units of the durable that have aged. This assumption may not be satisfied for many consumer durables including unique assets such as dwelling units and works of art, which are not bought and sold every period. We will deal with this situation later in Sect. 6.12.

This approach to the derivation of a user cost formula was used by Diewert (1974) who in turn based it on an approach due to Hicks (1946; 326).

This derivation for the user cost of a consumer durable was also made by Diewert (1974; 504).

Thus the beginning of the period user cost \(u^{0}\) discounts all monetary costs and benefits into their dollar equivalent at the beginning of period 0 whereas \(p^{0}\) discounts (or appreciates) all monetary costs and benefits into their dollar equivalent at the end of period 0. This leaves open how flow transactions that take place within the period should be treated. Following the conventions used in financial accounting suggests that flow transactions taking place within the accounting period be regarded as taking place at the end of the accounting period and hence following this convention, end of period user costs should be used by the price statistician; see Peasnell (1981).

Christensen and Jorgenson (1969) derived a user cost formula similar to (6.7) in a different way using a continuous time optimization model. If the inflation rate i equals 0, then the user cost formula (6.7) reduces to that derived by Walras (1954; 269) (first edition 1874). This zero inflation rate user cost formula was also derived by the industrial engineer A. Church (1901; 907–908), who perhaps drew on the work of Matheson: “In the case of a factory where the occupancy is assured for a term of years, and the rent is a first charge on profits, the rate of interest, to be an appropriate rate, should, so far as it applies to the buildings, be equal (including the depreciation rate) to the rental which a landlord who owned but did not occupy a factory would let it for.” Matheson (1910; 169), first published in 1884. Additional derivations of user cost formulae in discrete time have been made by Katz (1983; 408–409), Diewert (2005a). Hall and Jorgenson (1967) introduced tax considerations into user cost formulae.

The geometric model for depreciation to be explained in more detail in Sect. 6.6 below requires only a single monthly or quarterly depreciation rate. Other models of depreciation may require the estimation of a sequence of vintage depreciation rates. If the estimated annual geometric depreciation rate is \(\delta _{a}\), then the corresponding monthly geometric depreciation rate \(\delta \) can be obtained by solving the equation \((1-\delta )^{12}=1-\delta _{a}\). Similarly, if the estimated annual real interest rate is \(r_{a}^{*}\), then the corresponding monthly real interest rate \(r^{*}\) can be obtained by solving the equation \((1+r^{*})^{12}=1+r_{a}^{*}\).

Iceland uses a variant of the simplified user cost formula (6.9) to estimate the services of OOH with a real interest rate approximately equal to 4% and depreciation rate of 1.25%. The depreciation rate is relatively low because it is applied to the entire property value and not to just the structure portion of property value; see Gudnason and Jonsdottir (2011). Eurostat (2005) also uses a simplified user cost formula. Additional simplified user cost formulae have been developed by Hill et al. (2017) and many others.

Since landlords must set their rent at the beginning of the period (and in fact, they usually set their rent for an extended period of time), if the user cost approach is used to model the economic determinants of market rental rates, then the asset inflation rate \(i^{0}\) should be interpreted as an expected inflation rate rather than an after the fact actual inflation rate. This use of ex ante prices in this price measurement context should be contrasted with the preference of national accountants to use actual or ex post prices in the system of national accounts.

For housing, the situation is more complex because typically, a dwelling unit is a unique good; its location is a price determining characteristic and each housing unit has a unique location and thus is a unique good. It also changes its character over time due to renovations and depreciation of the structure. Thus the treatment of housing is much more difficult than the treatment of other durable goods. Note that the definitions (6.4) and (6.5) of the depreciation rate \(\delta \) and the asset inflation rate \(i^{0}\) implicitly assumed that prices for a new asset and a one period old asset were available in both periods 0 and 1. This assumption is not satisfied for a unique asset.

For additional material on difficulties with the user cost approach, see Diewert (1980; 475–479), Katz (1983; 415–422).

Katz (1983; 415–416) comments on the difficulties involved in determining the appropriate rate of interest to use: “There are numerous alternatives: a rate on financial borrowings, on savings, and a weighted average of the two; a rate on nonfinancial investments. e.g., residential housing, perhaps adjusted for capital gains; and the consumer’s subjective rate of time preference. Furthermore, there is some controversy about whether it should be the maximum observed rate, the average observed rate, or the rate of return earned on investments that have the same degree of risk and liquidity as the durables whose services are being valued.”.

One way for choosing the nominal interest rate for period t, \(r^{t}\), is to set it equal to \((1+r^{*})(1+\rho ^{t})-1\) where \(\rho ^{t}\) is a consumer price inflation rate for period t and \(r^{*}\) is a reference real interest rate. The Australian Bureau of Statistics has used this method for determining \(r^{t}\) with \(r^{*}\equiv 0.04\); i.e., a 4% real interest rate was chosen. Other methods for determining the appropriate interest rate that should be inserted into user cost formula are discussed by Harper et al. (1989), Schreyer (2001), Hill et al. (2017).

We will discuss geometric or declining balance depreciation and one hoss shay depreciation below. For references to the depreciation literature and for empirical methods for estimating depreciation rates, see Hulten and Wykoff (1981a, b, 1996), Beidelman (1973, 1976), Jorgenson (1996), Diewert and Lawrence (2000).

Goodhart (2001; F351) commented on the practical difficulties of using ex post user costs for housing as follows: “An even more theoretical user cost approach is to measure the cost foregone by living in an owner occupied property as compared with selling it at the beginning of the period and repurchasing it at the end ... But this gives the absurd result that as house prices rise, so the opportunity cost falls; indeed the more virulent the inflation of housing asset prices, the more negative would this measure become. Although it has some academic aficionados, this flies in the face of common sense; I am glad to say that no country has adopted this method.” As noted above, Iceland and Eurostat have in fact adopted a simplified user cost framework which seems to work well enough. Moreover, the user cost concept is used widely in production function and productivity studies and by national statisticians who construct multifactor productivity accounts for their countries.

For additional material on the difficulties involved in constructing ex ante user costs, see Diewert (1980; 475–486), Katz (1983; 419–420). For empirical comparisons of different user cost formulae, see Harper et al. (1989), Diewert and Lawrence (2000), Diewert and Fox (2018). In the latter paper, the authors calculated Jorgensonian (ex post) user costs for US land used in residential housing and found that negative user costs occurred. Diewert and Fox then replaced the ex post capital gains term in the user cost for land with the long term inflation rate for land over the previous rolling window of 25 years and this substitution led to positive user costs for land that were relatively smooth. Hill et al. (2017) also recommend the use of long run asset inflation rates to avoid chain drift in housing indexes based on user costs.

For example, property taxes are associated with the use of housing services and hence should be included in the user cost formula; see Sect. 6.16 below. As Katz (1983; 418) noted, taxation issues also impact the choice of the interest rate: “Should the rate of return be a before or after tax rate?” From the viewpoint of a household that is not borrowing to finance the purchase of the durable, an after tax rate of return seems appropriate but from the point of a leasing firm, a before tax rate of return seems appropriate. This difference helps to explain why rental equivalence prices for the durable might be higher than user cost prices; see also Sect. 6.16 below.

The opportunity cost approach to pricing the services of Owner Occupied Housing was first proposed by Diewert (2008). It was further developed by Diewert and Nakamura (2011), Diewert et al. (2009). To our knowledge, there have been only two studies that implemented the opportunity cost approach to the valuation of the services of OOH; see Shimizu et al. (2012), Aten (2018).

Another information source that could be used to identify depreciation rates for the durable good is the sequence of vintage rental or leasing prices that might exist for some consumer durables. In the closely related capital measurement literature, the general framework for an internally consistent treatment of capital services and capital stocks in a set of vintage accounts was set out by Jorgenson (1989), Hulten (1990; 127–129, 1996; 152–160).

If these second hand vintage prices depend on how intensively the durable good has been used in previous periods, then it will be necessary to further classify the durable good not only by its vintage v but also according to the intensity of its use. In this case, think of the sequence of vintage asset prices \(P_{v}^{t}\) as corresponding to the prevailing market prices of the various vintages of the good at the beginning of period t for assets that have been used at “average” intensities.

See also Jorgenson (1996) for a review of the empirical literature on the estimation of depreciation rates.

More generally, we assume that assumptions (6.15) hold for subsequent periods t as well; i.e., it is assumed that \(P_{v}^{t+1a}=(1+i^{t})P_{v}^{t}\) for \(v=0,1,2,\ldots \) and \(t=0,1,2,\ldots \) where \(P_{v}^{t+1a}\) is the anticipated price of a unit of the durable good that is v periods old at the end of period t, \(i^{t}\) is a period t expected asset inflation rate for all ages of the durable and \(P_{v}^{t}\) is the second hand market price for a unit of the durable good that is v periods old at the beginning of period t.

When \(v=0\), define \(\delta _{-1}\equiv 1\); i.e., the terms in front of the square brackets on the right hand side of (6.16) are set equal to 1.

Additional examples and discussion can be found in two recent OECD Manuals on productivity measurement and the measurement of capital; see Schreyer (2001, 2009).

In the case of one hoss shay depreciation, assumptions are made about the sequence of user costs, \(u_{v}^{t}\), as the age v varies.

A case for attributing the method to Walras (1954; 268–269) could be made but he did not lay out all of the details. Matheson (1910; 91) used the term “diminishing value” to describe the method. Hotelling (1925; 350) used the term “the reducing balance method” while Canning (1929; 276) used the term the “declining balance formula”. For a modern exposition of the geometric model of depreciation, see Jorgenson (1989).

Equation (6.20) for period t are the following ones: \(p_{v}^{t} =(1-\delta )^{v}p_{0}^{t}\) for \(v=1,2,\ldots \) and so the entire sequence of user costs by age of asset vary in a proportional manner over time under our assumptions. Thus an aggregate period t price for the entire group of assets of varying ages is \(p_{0}^{t}\) and the corresponding aggregate quantity will be \(Q^{t}\) defined by (6.23). This is an application of Hicks’ (1946; 312–313) Aggregation Theorem: “Thus we have demonstrated mathematically the very important principle, used extensively in the text, that if the prices of a group of goods change in the same proportion, that group of goods behaves just as if it were a single commodity.”.

See Jorgenson (1989) who popularized the use of the geometric model of depreciation in production function and Total Factor Productivity studies.

This model of depreciation dates back to the late 1800s; see Matheson (1910; 55), Garcke and Fells (1893; 98) or Canning (1929; 265–266).

Thus as the price of a new unit of the durable good changes over time, the value of depreciation will also change in line with the change in the price of the new unit. Thus economic depreciation as we have defined it is different from historical cost accounting depreciation which does not adjust depreciation allowances for changes in the levels of asset prices over time.

Diewert and Lawrence (2000) noted this problem with the straight line model of depreciation; i.e., that in general, an index number formula should be used to aggregate over the different ages of the asset in order to obtain an aggregate of the capital services of the different vintages of the asset.

However, if one is willing to assume that the reference interest rate for period t, \(r^{t}\), and the expected asset inflation rate over all ages of the asset, \(i^{t}\), both remain constant, then all reasonable index number formula will estimate the overall rate of user cost inflation between periods 0 and t as the new good price ratio, \(P_{0}^{t}/P_{0}^{0} \). However, the assumption that \(r^{t}\) and \(i^{t}\) remain constant over time is only a rough approximation to reality. Note that in order to calculate real and nominal consumption of the durable (over all ages of the durable), it will be necessary to use the vintage user costs defined by (6.27) for a constant r and i to weight up past purchases of the durable good. Thus define the constants \(\alpha _{v}\equiv [(r-i)(L-v)L^{-1}+(1+i)L^{-1}]\) for \(v=0,1,2,\ldots ,L-1\) and \(\alpha _{v}\equiv 0\) for \(v\ge L\). Then the period t nominal value of durable services is defined as \(v^{t}\equiv p_{0}^{t}q^{t}+p_{1}^{t}q^{t-1}+p_{2}^{t}q^{t-2}+\cdots +p_{L-1}^{t}q^{t-L+1}=\alpha _{0}P_{0}^{t}q^{t}+\alpha _{1}P_{0}^{t} q^{t-1}+\alpha _{2}P_{0}^{t}q^{t-2}+\cdots +\alpha _{L-1}P_{0}^{t}q^{t-L+1} =P_{0}^{t}Q^{t}\) where \(Q^{t}\) is the real value or volume of durable services defined as \(Q^{t}\equiv \alpha _{0}q^{t}+\alpha _{1}q^{t-1}+\alpha _{2} q^{t-2}+\cdots +\alpha _{L-1}q^{t-L+1}\). Define \(\beta _{v}\equiv (L-v)/L\) for \(v=0,1,2,\ldots ,L-1\). The period t asset value of consumer holdings of the durable good is defined as \(V^{t}\equiv P_{0}^{t}q^{t}+P_{1}^{t} q^{t-1}+P_{2}^{t}q^{t-2}+\cdots +P_{L-1}^{t}q^{t-L+1}=P_{0}^{t}[\beta _{0} q^{t}+\beta _{1}q^{t-1}+\beta _{2}q^{t-2}+\cdots +\beta _{L-1}q^{t-L+1}]=P_{0} ^{t}Q^{t*}\) where we have used assumptions (6.26) applied to period t and the real value of durable stocks held by households at the end of period t is defined as \(Q^{t*}\equiv \beta _{0}q^{t}+\beta _{1}q^{t-1} +\beta _{2}q^{t-2}+\cdots +\beta _{L-1}q^{t-L+1}\). The decomposition of \(V^{t}\) into \(P_{0}^{t}Q^{t*}\) does not require the assumption of constant \(r^{t}\) and \(i^{t}\).

This model can be traced back to Böhm-Bawerk (1891; 342). For a more comprehensive exposition, see Hulten (1990; 124) or Diewert (2005a).

The assumption of a single life L for a durable can be relaxed using a methodology due to Hulten: “We have thus far taken the date of retirement T to be the same for all assets in a given cohort (all assets put in place in a given year). However, there is no reason for this to be true, and the theory is readily extended to allow for different retirement dates. A given cohort can be broken into components, or subcohorts, according to date of retirement and a separate T assigned to each. Each subcohort can then be characterized by its own efficiency sequence, which depends among other things on the subcohort’s useful life \(T_{i} \).” Hulten (1990; 125).

If \(\gamma \ge 1\), then use the second equation in (6.32) to express \(u^{0}\) in terms of \(P_{0}^{0}\) and the various powers of \(\gamma \).

In the national income accounting literature, this measure is sometimes called the gross capital stock.

Using Eq. (6.31), it can be shown that \(P_{v}^{0}=u^{0}[1+(\gamma ^{0})+(\gamma ^{0})^{2} +\cdots +(\gamma ^{0})^{L-1-v}]\) for \(v=0,1,2,\ldots ,L-1\) where \(\gamma ^{0}\equiv (1+i^{0})/(1+r^{0})\) and \(P_{v}^{0}=0\) for \(v\ge L\). Thus the period 0 value of the stock of consumer durables is \(\sum _{v=0}^{L-1} P_{v}^{0}q^{-v}\). The corresponding asset prices for period t are equal to \(P_{v}^{t}=u^{t}[1+(\gamma ^{t})+(\gamma ^{t})^{2}+\cdots +(\gamma ^{t})^{L-1-v}]\) for \(v=0,1,2,\ldots ,L-1\) where \(u^{t}\equiv [1-(\gamma ^{t})]P_{0} ^{t}/[1-(\gamma ^{t})^{L}]\), \(\gamma ^{t}\equiv (1+i^{t})/(1+r^{t})\) and \(P_{v}^{t}=0\) for \(v\ge L\). The period t value of the stock of consumer durables is \(\sum _{v=0}^{L-1}P_{v}^{t}q^{t-v}\). An index number formula will have to be used to form aggregate price and quantity indexes for the stocks of consumer durables using the one hoss shay model of depreciation.

These three classes of methods were noted in Malpezzi et al. (1987; 373–375) in the housing context.

A length of life L is can be converted into an equivalent geometric depreciation rate \(\delta \) by setting \(\delta \) equal to a number between 1/L and 2/L.

The following material is based on Diewert (2002).

Let \(r^{0*}=0.03\), \(g=0.01\) and \(\delta =0.2\). Under these assumptions, using (6.39), we find that \(V_{U}^{0}/V_{A}^{0}=1.11\); i.e., using a geometric depreciation rate of 20%, the user cost approach leads to an estimated value of consumption that is 11% higher than the acquisitions approach under the conditions specified. Thus the acquisitions approach for consumer durables with high depreciation rates is probably satisfactory. However, for longer lived durables such as houses, automobiles and household furnishings, it would be useful for a national statistical agency to produce user costs for these goods and for the national accounts division to produce the corresponding consumption flows as “analytic series”. This would extend the present national accounts treatment of housing to other long lived consumer durables. Note also that this revised treatment of consumption in the national accounts would tend to make rich countries richer, since poorer countries hold fewer long lived consumer durables on a per capita basis.

In the international System of National Accounts, these unique goods are listed as valuables.

The nominal imputed user cost for period t is \((1+i)^{t}u^{t}\).

If \(P^{0}-P^{T}(1+r)^{-T}<0\), then as before, \(u^{0}\) becomes negative (and \(u^{1},\ldots ,u^{T-1}\) become negative as well) and again, the services of the unique durable are free of charge and \(-(1+i)^{t-1}(1-\delta )^{t-1}u^{t}=-(1+i)^{t-1}u^{0}>0\) becomes an addition to household income for period t.

The cost of the demolition should be added to the purchase price for the land to get the overall land price for the land plot.

Other papers that have suggested hedonic regression models that lead to additive decompositions of property values into land and structure components include Clapp (1980; 257–258), Bostic et al. (2007; 184), Francke and Vos (2004), Diewert (2008; 19–22, 2010), Francke (2008; 167), Koev and Santos Silva (2008), Rambaldi et al. (2010), Diewert et al. (2011, 2015), Eurostat (2013), Diewert and Shimizu (2015, 2016, 2017a), Burnett-Issacs et al. (2016), Diewert et al. (2017).

We will pursue a demand side model in Sect. 6.14 below.

See Schwann (1998), Diewert et al. (2011, 2015) on the multicollinearity problem.

This formulation follows that of Diewert (2010), Diewert et al. (2011, 2015, 2017), Eurostat (2013), Diewert and Shimizu (2015, 2016, 2017a), Burnett-Issacs et al. (2016). These authors assume that property value is the sum of land and structure components but movements in the price of structures are proportional to an exogenous structure price index. Note that the index \(p_{St}\) should be a levels price that gives the period t cost of building one square meter of structure.

Equivalently, one could make the normalization \(\alpha _{1}=1\) and not normalize the \(\omega _{j}\). The resulting estimated \(\alpha _{t}\) for \(t=2,3,\ldots ,T\) can then be interpreted as a constant quality land price index for the entire region relative to period 1 where \(\alpha _{1}\equiv 1\). In this section, we are drawing heavily on Diewert et al. (2017) and using the normalization used in that paper.

In order to obtain sensible parameter estimates in our final (quite complex) nonlinear regression model, it is absolutely necessary to follow our procedure of sequentially estimating gradually more complex models, using the final coefficients from the previous model as starting values for the next model. The models that are being described in this section were implemented in Diewert et al. (2017) where the econometric software Shazam was used to perform the nonlinear regressions; see White (2004).

This brings up an important point that has not been mentioned up to now. Panel data on the selling prices of properties and on the characteristics of the properties are subject to tremendous variations in the ratio of the say highest price property to the lowest price property, to the largest lot size to the smallest lot size, to the largest floor space area to the smallest floor space area and so on. The observations that appear in the tales of the distribution of prices and in the distributions of property characteristics are inevitably sparse and subject to measurement error. Thus in order to obtain sensible estimates in running these hedonic regressions, it is typically necessary to delete the observations that are in the tales of these distributions.

For the example in Diewert et al. (2017) where the models described in this section were estimated, the log likelihood increased by 1762 log likelihood points and the \(R^{2}\) jumped from 0.7662 for Model 2 to 0.8283 for Model 3 for the addition of 6 new \(\lambda _{k}\) parameters.

At this stage of the sequential estimation procedure, it is usually not necessary to impose a normalization on the parameters \(\mu _{1}{-}\mu _{M}\). This lack of a normalization means that the scale of the exogenous structure price levels \(p_{St}\) is allowed to change; i.e., essentially, allowance is now made to quality adjust the exogenous index to a certain extent. However, if the resulting estimated structure values turn out to be unreasonably large or small, then it will be necessary to set one of the \(\mu _{m}\) to equal 1.

For the example in Diewert et al. (2017), the log likelihood increased by 935 log likelihood points and the \(R^{2}\) jumped from 0.8283 for Model 3 to 0.8520 for Model 4 for the addition of 5 new \(\mu _{M}\) parameters.

Every country will have a national residential construction deflator because this deflator is required in order to form estimates of real investment in residential structures. However, this national deflator may not be entirely appropriate for the type of buildings in a particular neighbourhood.

It is also possible to estimate more general models of depreciation using the builder’s model; see Diewert and Shimizu (2017a), Diewert et al. (2017).

It is a net estimate since renovation and replacement investments in the building tend to extend the life of the building or augment its value. Thus the gross wear and tear depreciation rate for the structure will tend to be larger than the estimated net depreciation rate.

Crosby et al. (2012; 230) distinguish the two types of depreciation and in addition, they provide a comprehensive survey of the depreciation literature as it applies to commercial properties.

What has been labeled as wear and tear depreciation could be better described as anticipated amortization of the structure rather than wear and tear depreciation. Once a structure is built, it becomes a fixed asset which cannot be transferred to alternative uses (like a truck or machine). Thus amortization of the cost of the structure should be proportional to the cash flows or to the service flows of utility that the building generates over its expected lifetime. However, technical progress, obsolescence or unanticipated market developments can cause the building to be demolished before it is fully amortized. See Diewert and Fox (2016) for a more complete discussion of the fixity problem.

The two period time dummy variable hedonic regression (and its extension to many periods) was first considered explicitly by Court (1939; 109–111) as his hedonic suggestion number two. Court used adjacent period time dummy hedonic regressions as links in a longer chain of comparisons extending from 1920 to 1939 for US automobiles: “The net regressions on time shown above are in effect price link relatives for cars of constant specifications. By joining these together, a continuous index is secured.” If the two periods being compared are consecutive years, Griliches (1971; 7) coined the term “adjacent year regression” to describe this method for updating the index as new information becomes available. Diewert (2005b) looked at the axiomatic properties of adjacent year time dummy hedonic regressions.

For the details on how the mean splice method works, see Diewert and Fox (2017).

For additional hedonic regression models for detached houses, see Verbrugge (2008), Garner and Verbrugge (2011), Eurostat (2013, 2017), Hill (2013), Hill et al. (2018), Rambaldi and Fletcher (2014), Silver (2018).

The analysis in this section follows that of Diewert and Shimizu (2016).

Diewert and Shimizu (2016; 303) constructed estimates of Tokyo total building private floor space to total building floor space for each observation nt as \(N_{tn}S_{tn}/TS_{tn}\), where \(N_{tn}\) is the number of units in the building which contained condo sale n in period t, \(S_{tn}\) is the private floor space of the sold unit and \(TS_{tn}\) is the total floor space of the building. The sample wide average of these ratios was 0.899. Thus the first imputation method in definitions (6.67) was changed from \(L_{Stn}\equiv (S_{tn}/TS_{tn})TL_{tn}\) to \(L_{Stn}\equiv (1/0.899)(S_{tn}/TS_{tn})TL_{tn}=(1.1)(S_{tn}/TS_{tn})TL_{tn}\). Burnett-Issacs et al. (2016) estimated a similar condo model and consulted with construction experts and determined that on average, the ratio of total space to private space for Ottawa condominium apartments was approximately 1.33. Thus they changed \(L_{Stn}\equiv (S_{tn}/TS_{tn})TL_{tn}\) to \(L_{Stn}\equiv (1.33)(S_{tn}/TS_{tn})TL_{tn}\).

Diewert and Shimizu (2016) assumed \(\delta =0.03\) and Burnett-Issacs et al. (2016) assumed \(\delta =0.02\) where the age variable \(A_{tn}\) is measured in years. Later, \(\delta \) will be estimated.

As usual, we need a normalization on the parameters such as \(\alpha _{1}=1\) in order to identify all of the remaining parameters, \(\alpha _{2},\ldots ,\alpha _{T},\omega _{1},\ldots ,\omega _{J}\). Note that this regression uses the first method of land imputation defined by (6.67). Later, the second method will also be considered.

Again normalizations like \(\alpha _{1}\equiv 1;\chi _{1}\equiv 1\) are required in order to identify the remaining parameters. If all \(\chi _{s}=1\), then the model defined by (6.72) collapses down to the model defined by (6.70).

Normalizations like \(\alpha _{1} \equiv 1;\chi _{1}\equiv 1\) need to be imposed in order to identify the remaining parameters.

The studies that have implemented this model found that the estimated \(\gamma \) was in the 2–4% range. Thus the imputed land value of a unit increases by 2–4% for each story above the threshold level of 3.

For the DS Tokyo condo data, the estimated \(\lambda \) turned out to be \(\lambda ^{*}=0.3636\) \((t=9.84)\) so that the very simple land imputation method that just divided the total land plot size by the number of units in the building got a higher weight (0.6364) than the weight for the floor space allocation method (0.3636). For the Ottawa condo data, the estimated \(\lambda \) turned out to be \(\lambda ^{*}=0.2525\) \((t=12.10)\).

Recall the hedonic regression model defined by (6.59) in the previous section which introduced linear splines on the valuation of the land area of a stand alone housing unit. This introduction also greatly increased the log likelihood of the regression. In the present context, the excess land dummy variables take the place of the linear spline functions in (6.59).

Attempting to estimate the parameters in (6.83) without good starting values for the nonlinear regression will not lead to sensible parameter estimates. Thus it is necessary to obtain good starting values for (6.83) by estimating the rather long sequence of regressions explained above, starting with a very simple model and gradually introducing additional explanatory variables. Each regression in the sequence contains the previous one as a special case so that the final estimates of one regression can be used as starting values for the subsequent one.

The early analysis in this section follows that of Diewert and Fox (2017), McMillen (2003; 289–290), Shimizu et al. (2010a; 795). McMillen assumed that \(\alpha +\beta =1\). We follow Shimizu, Nishimura and Watanabe in allowing \(\alpha \) and \(\beta \) to be unrestricted.

Log price hedonic regressions for property prices date back to Bailey et al. (1963).

See for example the estimated model in Diewert et al. (2017).

To obtain this approximation result, it is also necessary that the depreciation rate that is estimated by the log price time dummy model be reasonable.

For examples of studies where it was found that this approximate equality held, see Diewert (2010; 21), Diewert and Shimizu (2015; 1692), Diewert et al. (2017; 32).

\(P_{Stn}\) is the price of a square meter of new structure of the type used by rental unit n at the beginning of period t.

On this point, see Genesove (2003), Verbrugge (2008), Shimizu et al. (2010b), Diewert and Nakamura (2011), Garner and Verbrugge (2011), Suzuki et al. (2018).

The expected land inflation rate \(i_{Lt}\) should be an average of land price inflation over the past 15–25 years to reflect the long holding periods that investors have for rental properties and the high transactions costs of buying and selling properties. Diewert and Fox (2018) used a rolling window annualized 25 year inflation rate for land for the 25 years prior to period t to generate very smooth estimates for the expected land inflation rate in their user costs for land in the US.

\(\mu _{t}\) is also known as a capitalization rate; i.e., it is the ratio of the rental price of the structure to its capital value.

If multicollinearity becomes a problem, it may be necessary to set \(\mu _{t}=\mu \) or assume that that the \(\mu _{t}\) are slowly trending over time.

“The average [annual] depreciation rate for rental property is remarkably constant, ranging from 0.58 to 0.60% over the 25 year period. Depreciation rates for owner occupied units show more variation than the estimated rates for renter occupied units. The average depreciation rate for owner occupied housing ranges from 0.9% in year 1 to 0.28% in year 20.” Stephen Malpezzi et al. (1987; 382). Note that these depreciation rates are underestimates for the “true” rates since demolition depreciation is not taken into account using this methodology. Put another way, the geometric model of depreciation may not be the “right” model of depreciation for rental housing.

For example, see Malpezzi et al. (1987), Crone et al. (2000, 2011), Verbrugge (2008), Shimizu et al. (2010a), Garner and Verbrugge (2011).

Our discussion here is similar to that of Hill et al. (2017; 7): “The services a household obtains from renting a dwelling are not the same as the services obtained by owner-occupying.” They consider some additional factors that can cause rents to differ from user costs. They also assert that since OOH services are derived from both the structure and land, it follows that there is no need to try and separate land from structure in the rental house price index. However, depreciation affects only the structure part of rents and if one attempts to adjust a market rent for this aging factor, it is necessary to apply the depreciation adjustment to only the structure part of rents.

On the stickiness of rents, see Shimizu et al. (2010b), Lewis and Restieaux (2015; 72–75), Suzuki et al. (2018), Hill et al. (2017; 9). Lewis and Restieaux label their three categories as (i) Occupied Let, (ii) Renewal and (iii) New Let. Their category (i) is a stock measure that includes all occupied rental units while their categories (ii) and (iii) match up with categories (ii) and (iii) in the text above. Rents in categories (ii) and (iii) may be subject to rent controls which means that rents in these categories do not reflect current opportunity costs.

However, when constructing a rental price index for renters, rents for all 3 categories should be used.

Crone et al. (2000) used hedonic techniques to estimate both a rent index and a selling price index for housing in the U.S. They also suggested that capitalization rates (i.e., the ratio of the market rent of a housing property to its selling price) can be applied to an index of housing selling prices in order to obtain an imputed rent index for OOH. As will be shown below, capitalization rates are functions of many variables, some of which can change considerably over time. Also it will be seen that capitalization rates for rented houses are not exactly appropriate as estimators for capitalization rates for owned houses.

Older structures will probably have higher \(m_{tn}\) ratios.

The algebra will be different for different models of depreciation but the same conclusion will follow.

See Gudnason and Jonsdottir (2011; 148). Note that as in the case of Iceland, the depreciation rate is applied to total property value and not to just the structure value. This may be an acceptable approximation if the shares of land and structure in total property value remain roughly constant over time.

See Lewis and Restieaux (2015; 156). We have changed their notation to match up with our notation.

Often high end houses that are not being used by their owners are rented out at prices that are far below their user costs just so someone will be in the house to maintain it and deter theft and vandalism.

See Heston and Nakamura (2011). Hill et al. (2017; 8) find similar results for Australia and Aten (2018) finds similar results for the US. Shimizu et al. (2012) found that user cost valuations for OOH in Tokyo were about 1.7 times as big as the equivalent rent estimates.

For a more comprehensive decomposition of the user cost formula for an owned dwelling unit with a mortgage on the unit, see Diewert et al. (2009), Diewert and Nakamura (2011).

Thus the UK still uses the payments approach to value OOH in its Retail Prices Index.

Fenwick (2009, 2012) has argued strongly that statistical agencies responsible for consumer price indexes should produce a range of indexes that suit different purposes. Thus the payments approach to OOH could be produced by statistical agencies that provide multiple consumer price indexes to suit different purposes. However, the payments approach cannot serve as a reliable guide for pricing the services of OOH.

The period t user cost valuation \(v_{t}\) for a unit of the durable good that is t periods old can be converted into an equivalent amount of a new unit of a durable good if the geometric or one hoss shay model of depreciation is applicable for the durable good under consideration. Otherwise, units of the durable good of different ages at the same point in time need to be aggregated using an index number formula.

However, a household that does not pay off its balance owing in a timely fashion will find itself in Case 3 above.

Hill et al. (2017), using Australian data, found substantial differences using the three main approaches to the valuation of OOH. This emphasizes the need for statistical agencies to produce estimates for all three approaches if possible.

See Rambaldi and Fletcher (2014) on various smoothing methods that could be used. Diewert and Shimizu (2017b) suggested a very simple method which worked well in their empirical application.

The rental equivalence approach could be used for durables that are rented or leased but typically, most consumer durables are not rented. Depreciation rates will in most cases be based on educated guesses. Durable stock estimates can be made once depreciation rates have been determined. The current value of household stocks of consumer durables should also be constructed and added to household balance sheets.

However, the equivalent rents should be based on new contract rents if possible in order to provide a current opportunity cost for using the services of an owned dwelling unit; recall the discussion on this point in Sect. 6.16.

Recall the evidence on this point in Heston and Nakamura (2011).

If the acquisitions approach is used in the headline CPI, the alternative approaches can be published as experimental or supplementary series.

However, for housing, the “comparable” rental property may not be exactly the same as the owned unit. Moreover, the observed rents may include insurance services and the services of some utilities and possibly furniture. It will be difficult to extract these costs from the observed rent.

The long run asset inflation rate over the past 20 or 25 years or the long run rate of inflation in housing rents could be used to predict future asset inflation rates. Many other prediction methods could be used; see for example Verbrugge (2008). However, the focus should be on predicting long run asset inflation rather than period to period inflation.

Long run user costs and rents will tend to be approximately equal to each other for lower end housing units since this type of housing unit will be built by property developers who provide rental housing and they need to set rents that are approximately equal to their long run user costs. However, short run dynamics can cause user costs and rents to diverge even for lower end housing units.

It is not a “true” acquisitions price that is observed in the marketplace since it involves imputations to subtract the land value from the property sale. The resulting acquisitions price obviously does not reflect the total services provided by the purchase.