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Assume we have given apples and oranges; If we only consider annual chain-linking, it is pretty simple, e.g. let us have the following values:

Year 1 Year 2 Year 3 Year 4
Apples Amount: 10, Price:3, Total 10*3=30 Amount: 8, Price:4, Total 8*4=32 Amount: 7, Price:5, Total 7*5=35 Amount: 5, Price:10, Total 5*10=50
Oranges Amount: 5, Price:4, Total 5*4=20 Amount: 7, Price:4, Total 7*4=28 Amount: 7, Price:4, Total 7*4=28 Amount: 10, Price:3, Total 10*3=30
Total 30+20=50 32+28=60 35+28=63 50+30=80

I now want to get the chain linked volumes with the reference year 1; To do so I first calculate for each year "Current volume * previous year price"

Year 1 Year 2 Year 3 Year 4
Apples Amount: 10 Amount: 8, Previous Price:3, Total 8*3=24 Amount: 7, Previous Price:4, Total 7*4=28 Amount: 5, Previous Price:5, Total 5*5=25
Oranges Amount: 5 Amount: 7, Previous Price:4, Total 7*4=28 Amount: 7, Previous Price:4, Total 7*4=28 Amount: 10, Previous Price:4, Total 10*4=40
Total 24+28=52 28+28=56 25+40=65

Now to get the chain-linked volumes with Year 1 as a reference Year I simply use that for the reference year, chain-linked volumes are equal to current prices and for Year 2,3 and 4 I can use that "Current volume * previous year price" / "Previous year's price * previous year's volume" = Chain linked volume of current year/Chain linked volume of previous year (all with same reference year);

So e.g. for Year 2, I just take 52/50 * Chain Linked volume year 1 = 52/50*50 = 52; This leads to the following table for the total of oranges+apples

Year 1 Year 2 Year 3 Year 4
Total 50 52/50*50=52 56/60*52=48.53333 65/63*48.53333= 50.07

Of course, one could do something similar not just for the total but also for oranges and apples itself but this would just give volume*price of the base year (Year 1), so chain-linking would be identical to calculating with prices of a base year;

As easy as this is, I am now trying to understand how quarterly chain-linking works; Assume we have again 4 years but with quarters 1,2,3,4 and for each quarter, we have an amount of apples, an amount of oranges, a price per apple and a price per orange;

How would I then calculate the quarterly chain-linked volume?

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It works the same way, you can use the same table but replace Year 1 with Quarter 1, Year 2 with Quarter 2, etc. And you probably want the data to be seasonally adjusted.

Side note: usually the chain-weighted approach uses both Laspeyres and Paasche indexes. To do that, you would repeat the above with following year prices, and then take the geometric average of the two results.

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  • $\begingroup$ Could you elaborate further? I mean if I consider quarters, let us say Year 2, Q1, then I need to take previous year's prices, right? But I do not take Year 1 Q1 prices but the overall (volume based) price of Year 1, so total cost of the product over the whole year 1 divided by total quantity over year 1; Therefore, I take the the same prices for Year 2, Q1 and Year 2, Q4 when calculating previous year's prices, right? But how do I proceed from there? $\endgroup$
    – Johannes
    Commented Jul 4, 2023 at 6:13

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