The error of the CPI estimates in Japan runs away

A month ago I wrote about the difference between two inflation measures as expressed by the consumer price index (CPI) and the GDP deflator in Japan. It was shown that the difference between these two indices increases over time. In this post, I add couple graphs to demonstrate that the difference diverges as time squared. Figure 1 displays the difference between the GDP deflator and CPI since 1985. One may observe that the slope of the difference increases with time approximately every five years, i.e. when new basket for CPI is introduced. In Figure 2, we approximate the difference with a quadratic function of time. The difference is running away, i.e. the error in the CPI estimate runs away since the GDP deflator completely includes the CPI.
Figure 1. The difference between the CPI and GDP deflator
Figure 2. Approximation of the difference between the CPI and GDP deflator by a quadratic function of time

Compared with the recent movement of the CPI and that of the GDP deflator, the range of drop of the GDP deflator has been larger than that of the CPI. The discrepancy between the CPI and the GDP deflator is mainly ascribable to the different things they cover. Other causes of the discrepancy include, among others, the different calculation formulae employed.
(1) The Target
While the CPI focuses only on household consumption, the GDP deflator covers business investments in equipment, etc., in addition to household consumption. Since much of the investment in equipment today is made in information technology goods, whose quality is rapidly improving, price falls in such goods considerably affect the deflator. For this reason, the change ratios of the deflator tend to be lower than those of the CPI.
Also, while the prices of petroleum products and other imported goods are rising, the CPI is usually pulled up. On the other hand, the deflator tends to drop until such price hikes are all reflected in the relevant product prices. Thus, the discrepancy between the two grows wider.
If the scopes of the two indices are narrowed down to cover the same items as far as possible, i.e., if we compare the CPI for “All items” with the GDP deflator for “Final household consumption expenditure” alone, the two indices show quite similar fluctuations.

(2) The Formula
While the CPI calculation employs the Laspeyres formula, the GDP deflator employs the Paasche formula. Generally, the Paasche formula, which calculates a weighted average using the quantitative weights at the time of comparison, tends to provide a lower index, while the Laspeyres formula, which employs quantitative weights at reference period, usually produces higher values. In addition, since quality improvement is reflected in the form of an increase in volume, Paasche formula gives a larger weight to an item whose price has fallen due to quality improvement. For this reason, the rate of decline of the GDP deflator, which employs Paasche formula, tends to be getting larger.
Also note that the GDP deflator employs a “chain method” with the reference periods it updates weights annually, to minimize the bias accompanying calculation of the index. Such a chain method is also used with the CPI as well, to provide and publish an additional, referential value to the index.

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