Generalization of the Divisia Price and Quantity Indices in a Stochastic Model with Continuous Time

Abstract

In this article, we present and discuss original price and quantity index formulas being a next step in Francois Divisia's index approach. We assume that prices and quantities of the given commodities are stochastic processes and we consider a continuous-time model. As a consequence, we obtain very general index formulas being random variables for any fixed time interval of observations. We present basic properties of the discussed formulas and show that, in the deterministic case, the general formulas lead to known classic Divisia indices.

Keywords:

• Divisia’s approach
• Price index theory
• Price indices
• Quantity indices

Mathematics Subject Classification:

• 62M99
• 37H10

Acknowledgments

The authors would like to thank the editor and the reviewers for their helpful comments on an earlier version of the manuscript which have led to an improvement of this article. The author also would like to thank Dr.Marta Małecka for her editorial remarks.

Notes

Q is derived from P by interchanging processes of prices and quantities described in (12) and (13).

In (30) we use Ito integral.

It is a direct consequence of the following fact for the Ito integral. Nevertheless, the history takes one direction only and we cannot imagine the situation when time passes back from T to 0.

We consider here t ∈ [0, 1] where, due to the total number of observations, this interval was divided into 22 subintervals (21 observations plus 1).

We generated 100,000 realizations of the P index (for the given time interval this index is a random variable) and its expected value was calculated as the arithmetic mean of these generated values.

⁶The sample figure concerns unit prices (in PLN) of one of the major open pension funds in Poland.

⁷In the case of ING we get the percentage drift α_i = −0.00028 and the percentage volatilityβ_i = 0.00584. The presented continuous line is one of the possible realizations of the unit price process.

We calculate the distances between the Pindex and chain indices as Δ = |(P_ch − P)/P|, where P_ch is one of the considered chain indices (mean or median of generated Laspeyres, Paasche of Fisher index values).

To read more about estimation of mean value and variance and the bias of this estimation in simulations see Żądło (2006), Małecka (2011), or Papież and Śmiech (2013).

If we compare medians of the generated values of indices we rule out some extreme generated values and thus we obtain a better approximation (the smaller distance Δ).

Using means for comparisons (instead of medians) we obtain even Δ > 13 %.

During the last years we observe the increasing availability of bar-code scanning data (scanner data) in Official Statistics (see, for instance, De Haan, 2002).

For example, the average daily number of quotes in the USD/EUR spot market could easily exceed 20,000 and the average daily number of observations of an actively traded NYSE stock can be even higher.