Bounds for the expected value of the stochastic Divisia's price index

ABSTRACT

In this article, we estimate bounds for the expected value of the stochastic Divisia's price index, that is, we assume that prices and quantities of the given commodities are stochastic processes with continuous time. We consider some special case of the stochastic model in which prices and quantities are described by the geometric Brownian motion. It is shown that the precision of this estimation depends rather on the volatility of prices than quantities volatilities.

KEYWORDS:

• Divisia's approach
• Geometric Brownian motion
• Price indices
• Price index theory

JEL CLASSIFICATION:

• C43

Notes

1 In the next part of the paper, we consider only the price index formula.

2 Even for yearly observations, the relative changes in budget shares of goods from CPI do not exceed several percents.

3 Since each p_i(0) from formula (17(17) ) is some real initial value, the assumption about independent prices leads to independence of partial price indices P_i = p_i(T)/p_i(0).

4 HereP(0, t) denotes the stochastic price index calculated on the time interval [0, t].

5 Since v*₁(t) + v₂*(t) + v*₃(t) + v₄*(t) = 1, we have and thus v*₁ + v₂* + v*₃ + v₄* = 1.

6 We use highly aggregated data taking into account price indices of the following group of consumer goods and services in Poland: food and non alcoholic beverages (X1); alcoholic beverages, tobacco (X2); clothing and footwear (X3); housing, water, electricity, gas, and other fuels (X4); furnishings, household equipment, and routine maintenance of the house (X5); health (X6); transport (X7); communications (X8); recreation and culture (X9); education (X10); restaurants and hotels (X11); and miscellaneous goods and services (X12).

7 According to the CPI methodology in Poland, we take constant weights from the previous year, namely: 24%, 5.7%, 5.2%, 20.7%, 4.9%, 4.8%, 9.1%, 4.5%, 7.8%, 1.3%, 6.8%, 5.2%.

8 Główny Urząd Statystyczny (GUS) in Poland.

9 We have no data about quantities on the official GUS webpage and thus we assume this quantity approximation.

10 We use maximal likelihood estimation (MLE). The mean absolute error (MAE) for each repetition was smaller than 8.5%.

11 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.

12 The presented consideration can be generalized into the case with any other type of portfolio, including shares, bonds, currencies, etc. In the next part of the paper prices of the considered shares are denoted by ALIOR, ING, and MBANK, respectively.

13 We collected data from the webpage: http://www.money.pl/gielda/archiwum/spolki/.

14 This normalization does not influence on the final price index result.