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Article Excerpt
I. INTRODUCTION
Price indexes play a significant role in modem economies. The consumer price index (CPI), for example, is used to index various government payments, as a target for monetary policy and as a benchmark in wage negotiations. Our focus in this paper, however, is on price indexes at a more disaggregated level, in markets where it is hard to match products from one period or region to the next. Computers and housing are notable examples of such markets. As well as being important inputs into the CPI, price indexes for such goods are often useful in their own right. Price indexes for computers play a critical role in productivity measurement across market sectors, while house price indexes provide an important indication of the state of an economy.
For the case of computers, the matching problem arises due to technological progress, which leads to the rapid evolution of products in the market, resulting in a short product cycle. For housing, the problem is that every house is different and that they tend to sell relatively infrequently. Hence, there is usually very little overlap in the houses sold from one period to the next and no overlap at all from one region to the next.
The fact that products can often not be matched across periods or regions poses a significant measurement problem in that it is therefore difficult to disentangle price differences from changes in the quality of products. In this paper, we focus primarily on the hedonic regression method for solving this problem. The hedonic method reduces the matching problem to one of comparing products on the basis of their characteristics. The "regression" aspect of hedonic regression refers to how the implicit prices for these characteristics are measured.
In the next section, we explain what is meant by the price index problem. Section III outlines more rigorously the measurement problem created by unmatched products. The hedonic imputation method is introduced in Section IV. Section V shows how the use of the hedonic imputation method complicates the price index problem. In addition to choosing between different formulas such as Fisher and Tornqvist, it is necessary to choose between different varieties of each formula. This is because index compilers have a certain amount of discretion over which prices are imputed. Possible solutions are considered in Section VI. We show that the choice of formula variety can affect the sensitivity of the results to omitted variables bias. The choice of price index formula (as opposed to variety) is also considered in this section. We show that this is intimately connected with the choice of functional form for the hedonic model. Section VII provides an empirical application of the issues raised. The case considered is the construction of house price indexes for three regions in Sydney over a 3-yr period. Section VIII concludes the paper.
II. THE PRICE INDEX PROBLEM
Let [P.sub.js,kt] denote a bilateral price index comparison between region j in time period s and region k in time period t. The price and quantity data of commodity heading n for country k in period t are denoted, respectively, by [p.sup.n.sub.kt] and [q.sup.n.sub.kt]. Six important bilateral formulas are: Paasche, Laspeyres, Fisher, geometric Paasche, geometric Laspeyres, and Tornqvist. These indexes are defined as follows:
(1) Paasche : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(2) Laspeyres : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(3) Fisher: [P.sup.F.sub.js,kt] = [square root of ([P.sup.P.sub.js,kt] x [P.sup.L.sub.js,kt])]
(4) Geometric Paasche :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(5) Geometric Laspeyres :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(6) Tornqvist : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Here, [w.sup.n.sub.kt] = [p.sup.n.sub.kt][q.sup.n.sub.kt]/[[summation].sup.N.sub.m=1] [p.sup.m.sub.kt][q.sup.m.sub.kt] denotes the expenditure share of product n in region-period kt.
These price index formulas all give the same answer if the price data satisfy the conditions for Hicks' aggregation theorem (Hicks 1946), that is, [p.sup.n.sub.kt] = [lambda][p.sup.n.sub.js] [for all]. Under this scenario, all the price relatives [p.sup.n.sub.js]/[p.sup.n.sub.kt] take the same value [lambda]; hence, there is no substitution effect. In such cases, [P.sub.js,kt] = [lambda], irrespective of the choice of formula. However, when there is some variation in the price relatives across products, the formulas diverge from each other. This is what is meant by the price index problem. It is a problem that has attracted some of the greatest minds in the economic profession over the best part of two centuries, such as Marshall, Edgeworth, Fisher, and Samuelson. Fisher (1922), for example, considers in excess of 100 different formulas.
The price index problem has been attacked from two main directions, usually referred to as the economic and the axiomatic approaches. The economic approach views quantities as utility maximizing responses to prices. This approach has culminated in the work of Diewert (1976), who proposed the concept of a superlative price index (a class of indexes that attain a second-order approximation to the underlying cost-of-living index [COLI]). Each index outlined above can be derived from a particular functional form for the cost or the utility function. Diewert's contribution was to show that some of the indexes are based upon more flexible representations of the cost function than others. The Fisher and Tornqvist index are superlative indexes as they allow for flexible substitution behavior. An alternative approach to justifying the form of index numbers is the axiomatic approach, which proposes a series of axioms that a price index should satisfy and then discriminates between them on the basis of their performance relative to these axioms (Balk 1995; Eichhorn and Voeller 1976). Fortunately, the axiomatic approach also tends to favor the Fisher and Tornqvist indexes as these usually emerge as best.
This literature, however, assumes that there is no matching problem. That is, it is assumed that all region-periods supply price and quantity data on the same list of commodity headings. Once this assumption is relaxed, the price index problem becomes more complex.
III. THEORETICAL FOUNDATIONS OF THE HEDONIC APPROACH
The problem posed by incompletely overlapping sets of products can be seen by outlining the conventional economic measurement framework. In terms of measuring price change between region j in period s and region k in period t, we want to estimate the cost for some representative consumer of obtaining a given level of utility under the two price and choice set regimes. Let the time periods be indexed by t = 1, ..., T; the set of regions by k = 1, ..., K; and the set of commodity headings by n = 1, ..., [N.sub.kt]. The price and quantity data of commodity heading n for region k in period t are denoted, respectively, by [p.sup.n.sub.kt] and q.sup.n.sub.kt] The COLI is defined as follows:
[P.sup.*.sub.js,kt] = C([p.sub.kt],[??])/C([p.sub.js],[??])
where the cost function is defined below.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The problem that arises frequently in practice is that the price vectors [p.sub.kt] and [p.sub.js] may not be comparable. For example, there might be some variety of computer that is available in region-period kt but not in region-period js. This makes the estimation of the COLI more complex.
A number of methods have been developed to tackle this problem. Hausman (1997, 1999) following Hicks (1940) suggested estimating the reservation price of the non-matched items. While this approach is conceptually appealing, it involves the estimation of demand systems and is econometrically and theoretically complex. Detailed data on both the prices paid and the quantities purchased by consumers are also required. An alternative approach suggested by Feenstra (1994) is to assume that the cost function takes the constant elasticity of substitution functional form in which case it is possible to derive the COLI exactly (see also Balk 1999; Nahm 1998). However, perhaps the most promising approach to dealing with hard-to-match products is hedonic regression. The hedonic approach dates back to Waugh (1928) and Court (1939). However, it was only with Griliches (1961) that interest in hedonics really took off (Schultze and Mackie 2002; Triplett 2004).
The conceptual basis of the hedonic approach, dating back to Lancaster (1966) and Rosen (1974), is that consumers' utility is derived from the characteristics of the goods and hence decisions also relate to these characteristics. At its most general, the hedonic approach reorients the measurement problem to one related to characteristics rather than to goods, which are bundles of characteristics.
At a conceptual level, there appears to be two main options for the application of hedonic techniques. First, we could completely reconstruct the consumers' optimization problem in terms of characteristics. That is, we could think of consumers minimizing the cost of obtaining a certain level of characteristics utility, given characteristics prices. Such an approach is at least implicit in the writing of Triplett (2004). For illustrative purposes, let us...
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