Construction of confidence intervals for the Laspeyres price index

Abstract

In the paper, we present and discuss several methods of the construction of confidence intervals for the Laspeyres price index. We assume that prices of commodities are normally distributed and we consider both independent and dependent prices. Using Monte Carlo simulation, the paper compares the confidence interval computed from a simple econometric model with those obtained based on the Laspeyres density function. Our conclusions can be generalized to other price index formulas.

Keywords:

• price indices
• the Laspeyres index
• confidence interval
• Gaussian ratio distribution

Acknowledgements

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 paper. The author also would like to thank Dr Marta Małecka for her editorial remarks.

Notes

1. The Laspeyres formula is the frequently used one in official statistics. For example, the CPI is a Laspeyres-type index.[15] But let us notice that the actual CPI calculation is based on survey data. In the last years increasing availability of bar-code scanning data has provided the opportunity to switch over to the currently used Laspeyres formula compared to other price indices.[16]

2. In Section 2.2, we treat $p_i^s$, $q_i^s$ and $q_i^t$ as some fixed positive numbers and only $P_i^t$ as a random variable.

3. See also the paper of Pham-Gia et al. [17] and the general considerations about the ratio of random variables in the paper of Kermond.[18]

4. A numerical procedure in C is available at http://www.jstatsoft.org/v16/i04/.

5. We consider price processes with quite high volatilities to observe larger differences between confidence intervals. In official statistics practice, when these volatilities are smaller, the confidence intervals of the Laspeyres index would be more precise (with smaller length) than those presented in Table 2.

6. Results from the Shapiro–Wilk normality test are as follows: $\text{SW}=0.9524$, p=0.043.

7. To read more about estimation of mean value and variance in simulation studies and the bias of this estimation, see Żądło [19,20].

8. The number of class was established automatically in Mathematica 6.0, n_k denotes the number of observed values of the Laspeyres index (I_La) belonging to the kth class.

9. We consider here also negative correlations of prices. For example, in Poland (in 2010, 2011 and 2012) we observed that prices of alcoholic beverages and tobacco were strongly, negatively correlated with prices of clothing and footwear and also prices of food and non-alcoholic beverages were negatively correlated with prices of furnishings, household equipment and routine maintenance of the house.[21]

10. For example, the calculation of ${\text{CI}}_{min}$ lasted over 45 min on our computer.

Additional information

Funding

This paper is preliminary to the 6th research competition of National Bank of Poland (NBP), which is funded by NBP.