Friday 19 July 2013

The small difference between WPI and CPI

Dear Reader,

Welcome back.

Inflation is a big topic in India. With prices rising uncontrollably, everyone is concerned about the increasing prices and everyone has an opinion on how to control inflation. One look at the WPI (Wholesale Price Index), in Fig 1 below, shows this alarming trend.

Fig 1: WPI Month over Month

(Chart is dynamic and will be updated with the latest data as and when it is available)

 No surprises there. WPI is showing an increasing trend as has been widely reported in the news. Here is the link to the reported WPI data. Note that the x-axis in Fig 1 is the month in which the data is reported.
Fig 2: CPI Month over Month

(Chart is dynamic and will be updated with the latest data as and when it is available)


Again, no surprises here either... CPI is showing a steadily increasing trend similar to that of the WPI. Here is the link to the reported CPI data (Base Year 2001). Note that the x-axis in Fig 2 is the month in which the data is reported.



Fig 3: % Change in WPI and CPI Indices (Chart is dynamic and will be updated with the latest data as and when it is available)

Compare Fig 3 to the % change in CPI in countries which have a GDP in the same range (~2 USD Trillion) (in the year 2012).
Table 1: Countries listed in order of GDP (Data in table is static)

Fig 4: % Change in CPI Canada & CPI Italy
(Chart is static)

 And to the CPI in US when the US GDP was in the range of 2 USD Trillion.


Table 2: US Historical GDP
(Data in table is static)


Fig 5: % Change in CPI US (in 1976)

(Chart is static)

Visual Analysis

From Fig 4 and Fig 5, changes in CPI (month over month) in Canada and US (in the year 1976) were, on average, less than 1.00% and more or less constant, which is how one would expect variations in prices to be in a stable economy.

From Fig 3, one can also infer that an increase in WPI is causing an increase in CPI. There however must be a lag in how quickly changes in CPI react to changes in WPI, assuming there are no other external factors. One could argue that the change is 'immediate' but how 'immediate' is this affect? From the chart this lag appears to be anywhere between 1 month to 3 months.

Examining Fig 1 and Fig 2, it almost seems as if the two indices are competing neck to neck independently with an underlying trend common to both indices. We know this can't be true as consumer prices implicitly depend on wholesale prices. If this is not true, there must be something wrong.

Individual indices are reported some where in the 2nd week of the month for the previous month. So, it is safe to assume there is no reporting lag between WPI and CPI. A lag, if any, must simply be the time it takes for retailers to adjust prices for changes in wholesale prices.

For example, A reported change in wholesale prices of potatoes on 15th February for the month of January should show up in a reported change in the consumer price of potatoes on 15th March for the month of February, assuming there is a lag of 1 month for the price change to reflect. How much is this lag in reality?

 Visually inspecting Fig 3 can put this lag to anywhere between 1 month to 3 months.

Analytics Based Analysis

Basic Stats


mean - mean of CPI index (unless otherwise stated)
stddev - standard deviation of the % change in the CPI index (unless otherwise stated)
time periods for rows 1-3 are as in their corresponding figures
time periods for rows 4-5 are as in their corresponding figures going back 1 year from the latest month.

Table 3: Basic statistics
(India (WPI) and India (CPI) rows are dynamic and will be updated as when new data is available)

Table 3 shows the standard deviation of the % Change in Index, and as can be seen both the WPI and CPI in India have a high standard deviation. Which means the degree of fluctuations in WPI and CPI are high as compared to the standard deviation in row 1 or row 3.
A steadily increasing (or better still decreasing) price trend or a constant rate of change of a price is always a good sign. Such a trend in price also means that there are, a) no random factors affecting price changes, b) it is easier for end customers to adjust to price changes, and c) it is easier to be prepared for a price change in the following months.


Auto-Correlation
Just for kicks lets see what the Auto-Correlation plots for WPI and CPI look like.


Fig 6: WPI Auto-Correlation Plot (1 unit lag = 1 month)
(Chart is dynamic and is updated as and when new data is available)



Fig 7: CPI Auto-Correlation Plot (1 unit lag = 1 month)
(Chart is dynamic and is updated as and when new data is available)

Both WPI and CPI show a high degree of correlation with itself for various lags. There is also a cyclical trend here with a cycle of 12 months. A cyclical trend in WPI and CPI is interesting because one wouldn't expect prices for a basket of goods to correlate with itself 12 months ago UNLESS the basket of goods only consists of goods that are dependent on something that is dependent on an annual cycle. A cyclical 12 month trend could be attributed to goods like food grains and other agriculture related commodities that are highly dependent on the monsoon season but certainly not to fossil fuels, textiles, wood, paper, etc. Moreover food grains and agriculture related commodities have a weight of ~14% in the overall calculation of WPI so we can't say that their contribution to WPI is comparatively significant.


Cross-Correlation
 The cross correlation plot shows the lag at which the correlation between CPI and WPI is highest.


Fig 8: CPI (at lag 0) vs. WPI (at different lags) Cross-Correlation Plot (1 unit lag = 1 month)
(Chart is dynamic and is updated as and when new data is available)


Turns out (at the time of writing this blog) CPI has the highest correlation with 0-1 month lag WPI. In other words, a price change in WPI in early months reflects in a 'similar' price change in CPI in the following month.

Conclusion

A strong cross correlation between CPI and WPI in early months with very low correlation in later months is interesting because this means price changes are reflected in CPI as quickly as they happen in WPI. Also, one would certainly not expect cyclical trends in the cross correlation. However, this confirms our inference on visually inspecting Fig 3. that the lag is between 0 month and 3 months. One would have expected this lag to be higher! it probably takes more than 3 months to just move finished wholesale goods to a consumer production site let alone have the finished good ready for consumption.
The Auto-Correlation plots for CPI shows strong correlation at lower lags, in other words it has a 'short tail' (ignoring the cyclical trend). A short tail is fine if the demand is constant or demand does not change with price - otherwise, retailers would have to make significant adjustments in their inventories on a month to month basis resulting in either wasted inventory or lack of it, but we don't hear much about inventory adjustments by retailers - perhaps it is absorbed in the cost of doing business? For a country like India where the population numbers are significant, it is sort of 'safe', to assume that there is always a demand for the basic basket of goods used in calculating WPI/CPI.

 Additionally, when modeling CPI as a series of data, we can use Auto-Regressed CPI data, the 0-3 month lag WPI data as the base and additional random variables that are independent to the WPI data. For example,

 cpi[i] =                              C[i-1] * cpi[i-1]  + C[i-2] * cpi[i-2] + ....
             W[i]   * wpi[i]    + W[i-1] * wpi[i-1] + W[i-2] * wpi[i-2] +  ...
             CB[i] * consumer_behavior[i] +
             OEFIA[i] * other_external_factors_if_any[i] + 
             R[i]

 Where R[i] is a random sample of a random variable. If consumer_behavior and other_external_factors_if_any are modeled accurately, R is a Normally distributed random variable.

 Or maybe adding mobile/cell phone devices as a separate line item in the list of items when calculating WPI may change the model.

A Note on Trends in Prices

Changes in a price of a commodity can be due to numerous factors. A simplistic argument to an increasing trend in price can be attributed to a) an increase in overall demand, b) its "perceived" value increases or willingness to pay is high, c) an increase in its intrinsic value, d) it is just priced high or is selling at an inflated price. Similarly, a decreasing trend in price can be attributed to a) an overall decrease in demand, b) its "perceived" value decreases or willingness to pay is low, c) a decrease in its intrinsic value, or d) it is just priced low.

Some Assumptions

  • The data reported for calculating the indices are accurate
  • Ignore all external factors that might affect an index
  • Changes in WPI almost always affect a change in CPI

References


Feel free to leave a comment in the comments section or if you find anything amiss in the above analysis.

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