Nontradables and relative price levels across areas within Japan
Hidehiro Ikeno
Surugadai University
1. Introduction
This paper examines empirically the importance of tradables and nontradables indetermining Japanese local price levels. Results show that changes in relative price levels
across areas are not dominated by nontradables.

Both tradables and nontradables areimportant in determining changes in relative price levels across areas.
It examines empirically the relative price levels across ten major Japanese regions, andalso across 47 Japanese cities. The price indexes of goods and services are used as priceindexes of tradables and nontradables, respectively. The analysis is in two steps. First,
series of price indexes of goods and services for regions and cities, which are not published,
are compiled from published disaggregated CPIs. Then, the extent to which changes inrelative price levels across the regions and the cities are affected by tradable and nontradable
components is examined.
2. Methodology
Consider the general price index which is a geometric weighted average of the priceindexes of tradables and nontradables. The general price index of the 𝑖’th area in period 𝑡𝑇𝑁and 𝑝𝑖,𝑡
are the price indexes ofin log, 𝑝𝑖,𝑡 , is given by the following Equation (1), where 𝑝𝑖,𝑡tradables and nontradables in period 𝑡 in log , and 𝛼𝑖 is the share of nontradables in thegeneral price index of the 𝑖’th area:
𝑇𝑁+ 𝛼𝑖 𝑝𝑖,𝑡𝑝𝑖,𝑡 = (1 − 𝛼𝑖 )𝑝𝑖,𝑡(1)𝑞𝑖𝑗,𝑡 = 𝑥𝑖𝑗,𝑡 + 𝑦𝑖𝑗,𝑡(2)The relative general price level across areas 𝑖 and 𝑗 in period 𝑡 is denoted by 𝑞𝑖𝑗,𝑡 and isdefined as: 𝑞𝑖𝑗,𝑡 = 𝑝𝑖,𝑡 − 𝑝𝑗,𝑡 . The relative general price level can, then, be written as𝑇𝑇𝑁𝑇𝑁
𝑇− 𝑝𝑗,𝑡and 𝑦𝑖𝑗,𝑡 = 𝛼𝑖 �𝑝𝑖,𝑡− 𝑝𝑖,𝑡
� − 𝛼𝑗 �𝑝𝑗,𝑡− 𝑝𝑗,𝑡�.
where 𝑥𝑖𝑗,𝑡 = 𝑝𝑖,𝑡Here, 𝑥𝑖𝑗,𝑡 is the relative

price level of tradables across the areas in period 𝑡, and 𝑦𝑖𝑗,𝑡 is the ratio of the internal

nontradable-tradable relative prices in period 𝑡 , which is weighted by the shares ofnontradables in the general price index. Equation (2) implies that∆𝑘 𝑞𝑖𝑗,𝑡 = ∆𝑘 𝑥𝑖𝑗,𝑡 + ∆𝑘 𝑦𝑖𝑗,𝑡where ∆𝑘 𝑧𝑡 ≡ 𝑧𝑡+𝑘 − 𝑧𝑡 .

,(3)

Equation (3) indicates that the importance of the ratio of thenontradable-tradable relative price increases as the divergence of the purchasing power
parity for tradables diminishes.
Three summary statistics are calculated for both the levels and the changes in order toinvestigate the relation between the relative general price level and the nontradable -relatedcomponent.
1

𝑐𝑜𝑣�𝑞 ,𝑦 �[1] The sample correlation𝑐𝑜𝑟𝑟�𝑞𝑖𝑗 , 𝑦𝑖𝑗 � = (𝑣𝑎𝑟�𝑞 �∙ 𝑖𝑗 𝑖𝑗 �)1/2
𝑖𝑗

𝑐𝑜𝑣�∆𝑘 𝑞𝑖𝑗 ,∆𝑘 𝑦𝑖𝑗 �𝑐𝑜𝑟𝑟�∆𝑘 𝑞𝑖𝑗 , ∆𝑘 𝑦𝑖𝑗 � = (𝑣𝑎𝑟�∆𝑘 𝑞 �∙𝑖𝑗[2] The ratio of the variances𝑣𝑎𝑟�𝑦𝑖𝑗

for the change over 𝑘 periods

𝑣𝑎𝑟�∆𝑘 𝑦𝑖𝑗 �)1/2𝑣𝑎𝑟(𝑦 )𝑖𝑗𝑣𝑎𝑟𝑟𝑎𝑡𝑖𝑜�𝑥𝑖𝑗 , 𝑦𝑖𝑗 � = 𝑣𝑎𝑟�𝑦 �+𝑣𝑎𝑟(𝑥𝑖𝑗𝑣𝑎𝑟(∆𝑘 𝑦𝑖𝑗 )for the level𝑖𝑗 )for the level𝑣𝑎𝑟𝑟𝑎𝑡𝑖𝑜�∆𝑘 𝑥𝑖𝑗 , ∆𝑘 𝑦𝑖𝑗 � = 𝑣𝑎𝑟�∆𝑘 𝑦 �+𝑣𝑎𝑟(∆𝑘 𝑥 ) for the change𝑖𝑗

𝑖𝑗

𝑀𝑆𝐸(∆𝑘 𝑦𝑖𝑗 )[3] The MSE decomposition 𝑚𝑠𝑒𝑑𝑒𝑐�∆𝑘 𝑥𝑖𝑗 , ∆𝑘 𝑦𝑖𝑗 � = 𝑀𝑆𝐸�∆𝑘 𝑥 �+𝑀𝑆𝐸(∆𝑘𝑦 ) for the change
where 𝑀𝑆𝐸(∆𝑘 𝑧) ≡1𝑛−𝑘∑𝑛𝑡=𝑘+1(𝑧𝑡 − 𝑧𝑡−𝑘 )2.

𝑖𝑗𝑖𝑗

The MSE is also expressed as 𝑣𝑎𝑟(𝑧𝑡 − 𝑧𝑡−𝑘 ) + {𝑚𝑒𝑎𝑛(𝑧𝑡 − 𝑧𝑡−𝑘 )}2 , where 𝑣𝑎𝑟(𝑧𝑡 − 𝑧𝑡−𝑘 ) is
the variance of 𝑧𝑡 − 𝑧𝑡−𝑘 , and 𝑚𝑒𝑎𝑛(𝑧𝑡 − 𝑧𝑡−𝑘 ) is the mean of 𝑧𝑡 − 𝑧𝑡−𝑘 .

3. Data

There are two difficulties with the CPIs published by the Japanese Statistics Bureau forthe purpose of this paper. First, the Japanese CPIs are compiled as arithmetic means, not as
geometric means, as in most other countries. Second, the Japanese Statistical Bureau does
not provide price indexes of goods and services separately for most areas. It is, therefore,
necessary to compile price indexes in a suitable form.

The Japanese Statistics Bureau

classifies all items covered by the general CPI into 47 groups, and publishes their monthly
local price indexes. Based on the definition of goods and services used by the Japanese
Statistics Bureau, these 47 groups are classified in this paper into goods and services.
As the first step, the following regression is executed by least square estimation with the𝑛restriction ∑𝑚
𝑘=1 𝛿𝑖,𝑘 + ∑𝑘=1 𝛼𝑖,𝑘 = 1 for each area, using the general and disaggregated CPIs

published by the Japanese Statistical Bureau:𝑘𝑛𝑘∆𝑐𝑝𝑖𝑖,𝑡 = ∑𝑚
𝑘=1 𝛿𝑖,𝑘 × ∆𝑔𝑜𝑜𝑑𝑠𝑖,𝑡 + ∑𝑘=1 𝛼𝑖,𝑘𝑘 × ∆𝑠𝑒𝑟𝑣𝑖𝑐𝑒𝑖,𝑡𝑘iswhere 𝑐𝑝𝑖𝑖,𝑡 is the general CPI in the 𝑖’th area in period 𝑡 transformed into log, 𝑔𝑜𝑜𝑑𝑠𝑖,𝑡𝑘
is the CPI of
the CPI of the 𝑘’th group in the goods category transformed into log, 𝑠𝑒𝑟𝑣𝑖𝑐𝑒𝑖,𝑡the 𝑘’th group in the service category transformed into log, 𝑚 is the number of groups

classified as goods, 𝑛 (= 47 − 𝑚) is the number of groups classified as services, and∆𝑧𝑖,𝑡 ≡ 𝑧𝑖,𝑡+1 − 𝑧𝑖,𝑡 .

𝛿𝑖,𝑘 s and 𝛼𝑖,𝑘 s are weights. Next, fitted values are calculated for thegeneral price index, and also for the price indexes of goods and services, basing on the resultsof the regression as follows:
𝑘𝑛𝑘
̂
𝑐𝑝𝚤
� 𝑖,𝑡 = ∑𝑚
�𝑖,𝑘 × 𝑠𝑒𝑟𝑣𝑖𝑐𝑒𝑖,𝑡,
𝑘=1 𝛿𝑖,𝑘 × 𝑔𝑜𝑜𝑑𝑠𝑖,𝑡 + ∑𝑘=1 𝛼and𝑘𝑚
̂̂
� 𝑖,𝑡 = (∑𝑚𝑔𝑜𝑜𝑑𝑠
𝑘=1 𝛿𝑖,𝑘 × 𝑔𝑜𝑜𝑑𝑠𝑖,𝑡 )�(∑𝑘=1 𝛿𝑖,𝑘 ) ,𝑘𝑠𝑒𝑟𝑣𝚤𝑐𝑒
� 𝑖,𝑡 = (∑𝑚
�𝑖,𝑘 × 𝑠𝑒𝑟𝑣𝑖𝑐𝑒𝑖,𝑡)�(∑𝑚
�𝑖,𝑘 ) .
𝑘=1 𝛼
𝑘=1 𝛼𝑇� 𝑖,𝑡 for 𝑝𝑖,𝑡, and 𝑠𝑒𝑟𝑣𝚤𝑐𝑒
� 𝑖,𝑡 for
Below, the fitted value 𝑐𝑝𝚤
� 𝑖,𝑡 is used for the variable 𝑝𝑖,𝑡 , 𝑔𝑜𝑜𝑑𝑠

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𝑁𝑝𝑖,𝑡.

Their start is in February 1985, and end in December 2010. Seasonally unadjusted series

are used.
4. Unit-root tests
To evaluate the stationarity of the relative price levels across areas, unit-root tests areexecuted on the relative general price levels, the relative prices of goods, and the relative
� 𝑖,𝑡 /𝑔𝑜𝑜𝑑𝑠
� 𝑗,𝑡 s, and 𝑠𝑒𝑟𝑣𝚤𝑐𝑒
� 𝑗,𝑡 s, 𝑔𝑜𝑜𝑑𝑠
� 𝑖,𝑡 /𝑠𝑒𝑟𝑣𝚤𝑐𝑒
� 𝑗,𝑡 s for 𝑖 ≠ 𝑗. Theprices of services; 𝑐𝑝𝚤
� 𝑖,𝑡 /𝑐𝑝𝚤
augmented Dickey-Fuller (ADF) tests and generalized least-square detrending (denotedDF-GLS test) developed in Elliot et al. (1996) are executed. The ADF and DF-GLS tests are

performed on the following equation;𝑙𝑖𝑗𝑎𝑖𝑗,𝑘 ∆𝑧𝑖𝑗,𝑡−𝑘 + 𝑒𝑖𝑗,𝑡∆𝑧𝑖𝑗,𝑡 = 𝜇𝑖 + 𝛿𝑖𝑗 𝑧𝑖𝑗,𝑡−1 + ∑𝑘=1

where 𝑧𝑖𝑗,𝑡 is the relative price across areas 𝑖 and 𝑗, and the number of lag lengths, 𝑙𝑖𝑗 , isdetermined by Campbell and Perron’s (1991) top-down-t approach, starting with the numberof lag lengths 𝑙𝑖𝑗 = 12 .

The overall results provide only limited support for thelevel-stationarity of the three relative prices. They generally support stationarity of the firstdifferences. (See Tables 1 and 2. The results of the first differences are not indicated here
to save space.)

5. Empirical results from regional analysis
The summary statistics of levels, and also changes over one, three, and five years arecalculated for all 45 possible combinations of the ten regions. Table 5 reports the summary
statistics from these pairs. The interpretation of these results is that the importance of thenontradable-related component increases when a longer period is considered. However, thenontradable-related component never dominates the tradable component. Both components
are relevant in determining the relative general price across the regions.
6. Empirical results from city analysis
The summary statistics of levels, and also changes over one, three, and five years arecalculated for all 1081 possible pairs of the 47 prefecture capitals. Table 6 presents details of
the three summary statistics from these pairs. Essentially, the same inferences as for theregional analysis apply to the city analysis.
7. Empirical results from within-region analysis
This paper analyses statistics of relative prices between prefectural capitals which arelocated within a single region.

Following six reorganized regions are considered:Hokkaido-Tohoku, Kanto, Hokuriku-Tokai, Kinki, Chugoku-Sikoku, and Kyushu-Okinawa.
Table 7 shows statistics of relative prices between cities which are located within the same3

region. The table indicates that the relation between nontradables and relative price levels
is not generally strong even within the same region.
8. Analysis from subsample periods
The entire period from January 1985 to December 2010 is divided into two sub-periods;
from January 1985 to December 1997, and from January 1998 to December 2010.
Table 9 resents results from city analysis for the second period. (Results for the first
period are shown here to save space.) These results indicate that the relation between therelative general price level and the nontradable-related component is weaker more recently
than previously.
9. Conclusion
In changes over one year, the effects of the nontradable-related component are often
negligible across both regions and cities. The relation between the relative price level andthe nontradable-related component is rather weak, even if the timescale is extended to fiveyears. It is not the case that the nontradable-related component overwhelms the tradable
component even in the case of the 5-year change. Moreover, the relative importance of thenontradable-related component has diminished recently.
The results in this paper have two possible interpretations.

One is that arbitrage

between local markets is not perfect within Japan, because of economic factors such astransportation costs and local monopolies. The other is that goods are not perfectly tradable.
Table 1 Unit-root tests on relative price levels between regions: 1986:02 – 2010:12
Significancelevel
1% level
5% level
10% level

ADF

DF-GLSGeneral

Goods

Services

General

Goods

Services

10

10

2

1

3
4

3
56

160
2

Note: The total number of tested relative price levels is 45.

5161

Table 2 Unit-root tests on relative price levels between cities: 1986:02 – 2010:12
Significancelevel
1% level
5% level

ADF

DF-GLSGeneral

GoodsServices

General

Goods

Services

26

854

7

1611(2.4%)
79(7.3%)(0.7%)
37(3.4%)(5.0%)
128(11.8%)4(0.6%)
26(2.4%)(1.5%)
63(5.8%)(1.0%)
43(4.0%)10% level136

80(12.6%)(7.4%)

199(18.4%)

63(5.8%)Notes: The total number of tested relative price levels is 1081.
number are shown in parentheses.

119(11.0%)

76(7.0%)The ratios to the total

Table 5 Summary statistics of relative price levels between Japanese regions: 1985:01 –2010:12
Level

1-year change

3-year change

5-year change

correlations: 𝑐𝑜𝑟𝑟
median

0.53

-0.05

0.23

0.39

mean

0.40

-0.02

0.210.36

variance ratios: 𝑣𝑎𝑟𝑟𝑎𝑡𝑖𝑜
median

0.47

0.29

0.32

0.36

mean

0.44

0.30

0.34

0.38

median

0.30

0.36

0.41

mean

0.31

0.36

0.42

MSE decomposition: 𝑚𝑠𝑒𝑑𝑒𝑐

Note: The number of the samples is equal to 45.
Table 6 Relative price levels between Japanese cities: 1985:01 – 2010:12
Level

1-year change

3-year change

5-year change

correlations: 𝑐𝑜𝑟𝑟
median

0.28

-0.08

0.09

0.18

mean

0.19

-0.08

0.08

0.15

variance ratios: 𝑣𝑎𝑟𝑟𝑎𝑡𝑖𝑜
median

0.36

0.29

0.58

0.59

mean

0.38

0.30

0.58

0.60

median

0.30

0.34

0.37

mean

0.30

0.35

0.38

MSE decomposition: 𝑚𝑠𝑒𝑑𝑒𝑐

Note: The number of the samples is 1081 .
Table 7

Medians of relative price levels within each region: 1985:01 – 2010:12

correlations: 𝑐𝑜𝑟𝑟

Hokkaido-Tohoku
Kanto

Level

1-year change

3-year change

5-year change

0.57

0.14

0.34

0.41

0.18

-0.07

0.02

0.02

5

Hokuriku-Tokai
Kinki

Chugoku-Shikoku
Kyushu-Okinawa

variance ratios: 𝑣𝑎𝑟𝑟𝑎𝑡𝑖𝑜
Hokkaido-Tohoku
Kanto

Hokuriku-Tokai
Kinki

Chugoku-Shikoku
Kyushu-Okinawa

0.07

-0.08

0.02

-0.03

0.39

-0.01

0.19

0.18

-0.01

-0.14

-0.03

0.02

0.44

-0.12

0.06

0.21

0.33

0.33

0.60

0.65

0.44

0.30

0.57

0.58

0.41

0.29

0.56

0.59

0.35

0.33

0.61

0.59

0.29

0.27

0.55

0.58

0.44

0.29

0.58

0.62

0.33

0.34

0.41

0.32

0.37

0.37

0.29

0.32

0.37

0.34

0.39

0.40

0.28

0.28

0.30

0.31

0.36

0.40

MSE decomposition: 𝑚𝑠𝑒𝑑𝑒𝑐
Hokkaido-Tohoku
Kanto

Hokuriku-Tokai
Kinki

Chugoku-Shikoku
Kyushu-Okinawa

Table 9 Relative price levels among Japanese cities: 1998:1 – 2010:12Level

1-year change3-year change

5-year change

correlations: 𝑐𝑜𝑟𝑟
median

-0.24

-0.24

-0.14

-0.13

mean

-0.19

-0.20

-0.11

-0.09

variance ratios: 𝑣𝑎𝑟𝑟𝑎𝑡𝑖𝑜
median

0.29

0.27

0.30

0.31

mean

0.31

0.29

0.32

0.34

median

0.27

0.29

0.31

mean

0.29

0.32

0.33

MSE decomposition: 𝑚𝑠𝑒𝑑𝑒𝑐

Note: The number of the samples is 1081.

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