Sub-Regional Workshop on International Merchandise Trade Statistics Compilation and Export and Import Unit Value Indices 21 – 25 November 2016. Guam Reef & Olive Spa Resort, Tamuning, Guam Price Indices and UVIs Presentation by Rens Hendriks Economic Statistician, Pacific Community Email: NilimaL@spc.int

𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 0 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0 The Laspeyres index 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 0 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0 L = Laspeyres t = current period 0 = base period 𝑃 𝐿 𝑡/0 = Laspeyres price index between current and base period p = price q = quantity i = item 𝑖=1 𝑛 = sum over item 1 to n

The Laspeyres index The Laspeyres index compares the costs of buying the same goods today as we bought yesterday: $5,00 $5,50

An example 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 0 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 0 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0 𝑃 𝐿 𝑡/0 = €6.00 €5.00 = 1.2 Base period (t = 0) Item Quantity Price Expenditure Apples (i = 1) 2 € 1.00 € 2.00 Pears (i = 2) € 1.50 € 3.00 Total   € 5.00 Comparison period (t = 1)   Price Expenditure Apples (i = 1) € 1.50 € 3.00 Pears (i = 2) Total € 6.00

Exercise 1 Calculate the Index in exercise 1

A more practical way In practice, the index as it is written is very hard to calculate Detailed quantities are usually not available. Luckily, the index can also alternatively be written as: 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝑖 0 𝑝 𝑖 𝑡 𝑝 𝑖 0 That is: The index can be calculated as the weighted average of the price changes of the individual products

Laspeyres index - example Prices Jan Feb Mar Apples $3.29 Pears $1.99 $2.19 Oranges $2.79 $2.49 Price relatives Weights Contribu- tions 1.00 20% 0.20 1.10 70% 0.70 0.77 0.89 10% 0.10 0.09 Index 1.07 1.06

Exercise 2 Calculate the Index in exercise 2

Sampling of products "Simple random sampling" by Dan Kernler - Own work. Licensed under CC BY-SA 4.0 via Commons - https://commons.wikimedia.org/wiki/File:Simple_random_sampling.PNG#/media/File:Simple_random_sampling.PNG

Elementary indices 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝑖 0 𝑝 𝑖 𝑡 𝑝 𝑖 0 Weights 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝑖 0 𝑝 𝑖 𝑡 𝑝 𝑖 0 Weights Apples Pears Price relatives Pink ladies Granny smiths Bartlet D’anjous Mismatch between price relatives and weights

Elementary indices Apples 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝐸𝐴 0 𝑝 𝑖 𝑡 𝑝 𝑖 0 𝑃 𝐸𝐴 𝑡/0 Pears 𝑃 𝐸𝐴 𝑡/0 But how to calculate? 𝑃 𝐿 𝑡/0 = 𝑖=1 𝑛 𝑤 𝐸𝐴 0 𝑝 𝑖 𝑡 𝑝 𝑖 0 𝑃 𝐸𝐴 𝑡/0 EA: Elementary aggregate Pink ladies Granny smiths Bartlet D’anjous

The Jevons index 𝑃 𝐽 𝑡/0 = 1 𝑛 𝑝 𝑖 𝑡 1 𝑛 1 𝑛 𝑝 𝑖 0 1 𝑛 Toothpaste Jan 𝑃 𝐽 𝑡/0 = 1 𝑛 𝑝 𝑖 𝑡 1 𝑛 1 𝑛 𝑝 𝑖 0 1 𝑛 Toothpaste Jan April Colgate € 1.99 € 1.49 Aquafresh € 2.49 € 2.59 Signal € 2.19 € 2.99 Geometric mean prices € 2.21 € 2.26 Jevons index =   1.021 𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑒𝑎𝑛 𝐽𝑎𝑛= 1.99∙2.49∙2.19 1 3 =2.21 𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑒𝑎𝑛 𝐴𝑝𝑟𝑖𝑙= 1.49∙2.59∙2.99 1 3 =2.26 𝑃 𝐽 𝑡/0 = 2.26 2.21 =1.021

Exercise 3 Calculate the Elementary Indices for pears and apples Using the elementary indices, calculate the index for ‘fruit’

Summary The process just treated results in ‘pure’ price indices The prices of a fixed basket of goods is followed through time. A variety of methods is available to deal with situations where the representativity of the sample changes Downsides: The process is labour intensive and expensive

Alternative Replace the elementary indices based on samples with unit value indices. For trade: unit value indices are a by product of IMTS compilation. Benefits: Cheap and easy to produce In theory full coverage: no sampling error However, they must be used with caution because: Often issues with data quality Unit value indices can have bias. Requires extensive outlier detection routines

The Unit Value Index 𝑃 𝑈𝑉𝐼 𝑡/0 = 𝑖=1 𝑛 𝑝 𝑖 𝑡 𝑞 𝑖 𝑡 𝑖=1 𝑛 𝑞 𝑖 𝑡 𝑖=1 𝑛 𝑝 𝑖 0 𝑞 𝑖 0 𝑖=1 𝑛 𝑞 𝑖 0 Formula looks complicated However, really it is simply the average price in the current period divided by the average price in the previous period: 𝑃 𝑈𝑉𝐼 𝑡/0 = 𝑝 𝑡 𝑝 0

Example - Apples Jan Feb Apples Price Quantity Value Red delicious $2.00 4 $8.00 $2.20 5 $11.00 Anjou $1.50 $6.00 $1.65 3 $4.95 Golden Delicious $1.80 $7.20 $1.89 $5.67 Total   12 $21.20 11 $21.62 Average Price $21.2/12 = $1.77 $29.81/15 = $1.97 Unit value Index $1.99/$1.77 = 1.11

Unit Value Bias 𝑃 𝐿 𝑡/0 = $8∙1.1+$6∙1.1+$7.2∙1.05 $21.20 =1.08 Jan Feb Price relatives Apples Price Quantity Value Red delicious $2.00 4 $8.00 $2.20 5 $11.00 1.1 Anjou $1.50 $6.00 $1.65 3 $4.95 Golden Delicious $1.80 $7.20 $1.89 $5.67 1.05 Total   12 $21.20 11 $21.62 Average Price $21.2/12 = $1.77 $29.81/15 = $1.97 Unit value Index $1.99/$1.77 = 1.11 𝑃 𝐿 𝑡/0 = $8∙1.1+$6∙1.1+$7.2∙1.05 $21.20 =1.08

Exercise 4 Exercise 4 has example data for imports of vehicles for quarters For each quarter calculate The total value of imports The total quantity of vehicles imported The unit value of the imports (value/quantity) Calculate the index (Q2 UVI / Q1 UVI) Tip: Use a pivot table to do the calculations quickly Do you think the index could be biased?