RDP 2018-08: Econometric Perspectives on Economic Measurement Appendix F: The Method of Moments Makes Sensible Multilateral Index Functions

Consider a model consisting of assumptions A1** to A4**. It is a special case of The Diewert Model.


where: f (·), ptuv, αt, εtuv and Utv are understood already; and λv is a fixed effect for varieties.

A2** Across varieties the observations are independently and identically distributed.

A3** The errors follow strict exogeneity of the form. So

A4** There is conditional heteroskedasticity of the form

where σ is some strictly positive constant.

Now consider a case that sets f (x) = x, Utv = qtvλv and Inline Equation. As the model is in levels, the heteroskedasticity assumption is natural.

The model corresponds to the conditional moment restriction

where δ and xtv is vector shorthand for the full set of coefficients and regressors (the time and product dummies) that are implicit in the model, for a given (t, v) pair.

Following Wooldridge (2010, p 542), the efficient method of moments estimators for αt and λv solve

But note that


the last of which contains the unknown parameters α and λ. The feasible method of moments estimators use Inline Equation and Inline Equation instead.

The resulting system of equations is


This is the same as the system of Geary (1958) and Khamis (1972). The result differs somewhat from the original method of moments derivation of Rao and Hajargasht (2016), who argue that an inefficient weighting system is necessary to generate the index. Although our set-ups are different, ultimately it is the introduction of units of equal interest that resolves the discrepancy.

Table F1 provides the settings needed to generate the other multilateral indices considered in Rao and Hajargasht (2016), using the method of moments. I have not explored whether there are further multilateral indices that fit the method of moments interpretation.

Table F1: Methods of Moments Interpretations of Multilateral Functions
Index name (year) Inline Equation f(x) Utv htv
Dutot-style Inline Equation x λv Inline Equation
Harmonic Inline Equation x−1 1 Inline Equation
Geometric Inline Equation ln(x) 1 1
Geary (1958)–Khamis (1972) Inline Equation x λvqtv Inline Equation
Iklé (1972) Inline Equation x−1 ptvqtv Inline Equation
Rao (1990) Inline Equation ln(x) ptvqtv 1
Hajargasht and Rao (2010, type a) Inline Equation x ptvqtv Inline Equation
Hajargasht and Rao (2010, type b) Inline Equation x 1 Inline Equation