2011년 12월 27일 화요일

Some sources on chain weighting



One problem with traditional “real GDP” calculations is that, since it values all goods at base year prices, it looks like prices never change. As time goes on, goods whose prices go down (and quantities usually go up) are still weighted by the old prices, and consequently get too much weight in later years’ GDP calculations. The goods don’t require a large expenditure share, but if they are valued at base year prices, it would look like a speciously high share of GDP. Let’s do the example:



Year
Expenditure
Prices
Quantity
Computers
Trucks
Computers
Trucks
Computers
Trucks
1
100
106
$1.00
$1.00
100
106
2
105
98
$0.80
$1.05
131.3
93.3
3
103
104
$0.60
$1.10
171.7
94.5
4
99
100
$0.40
$1.15
247.7
87


Notice that since Expenditure = Price*Quantity, real quantities are just Expenditure/Price. Often 
you start with quantities and prices and then derive expenditures; but you can’t do that with computers like you can with oranges, because although an orange in 1981 is the same as an orange in 2005, a computer is not – it’s harder to count up “quantity” in two different years. So, to get an idea of “quantity”, the best thing is to look at total expenditure and divide it by the price level.

It’s simple to calculate real investment (a component of real GDP) in year 1, the base year:
Yr. 1 Real Investment = (100 comp's)*($1) + (106 trucks)*($1) = $206
Now, usual calculation of real investment in year 2 would take year 2 quantities and value them at base year prices:
Yr. 2 Real Investment = (131.3 comp's)*($1) + (93.3 trucks)*($1) = $224.60
Similarly, for years 3 and 4:
Yr. 3 Real Investment = (171.7 comp's)*($1) + (94.5 trucks)*($1) = $266.20Yr. 4 Real Investment = (247.5 comp's)*($1) + (87.0 trucks)*($1) = $334.50
Look at what happened – computers represent 74% of real investment in year 4. However, this is unreasonably high since they represent less than half of actual investment expenditures that year! Why did this happen? Computers have gone down in price, but valuing them at high prices from long ago makes them look much more important than they actually are.

The “investment deflator” that’s given is the same as the GDP deflator. To calculate it, first compute nominal investment (current year investment at current year prices):
Yr. 2 Nominal Investment = (131.3 comp's)*($.80) + (93.3 trucks)*($1.05) = $203
Yr. 3 Nominal Investment = (171.7 comp's)*($.60) + (94.5 trucks)*($1.10) = $207
Yr. 4 Nominal Investment = (247.5 comp's)*($.40) + (87.0 trucks)*($1.15) = $199
The “investment deflator” is (Nominal Investment) / (Real Investment)
Yr. 2 Investment Deflator = 203/224.60 = 0.904
Yr. 3 Investment Deflator = 207/266.20 = 0.778
Yr. 4 Investment Deflator = 199/334.50 = 0.595
How can we solve this problem with real GDP? The solution that the government recently began implementing in its reporting of GDP numbers is chain-weighting. The idea is[:] 
  1. to first calculate year-on-year real rate of growth of each component separately
  2. Then, use expenditure shares for the current year to weight each component and calculate an average rate of growth
  3. Then, apply this average growth rate to the previous year’s real GDP and calculate real (cumulated) GDP in the new year.
Let’s see how this works in year 2:
We do a year-on-year comparison of real quantities in years 1 and 2 to calculate the growth rate of each component separately.
Growth_comp's = (com_new-com_old)/com_old=(131.3-100)/100= 0.313
Growth_trucks =(93.3-106)/106= -0.120
Now, in year 2, total expenditures were $203. Computers represent proportion $105/$203 = 0.517 of total expenditures, and trucks represent proportion $98/$203 = 0.483 of total expenditures.

We use this to calculate an average growth rate, weighted by expenditure shares:
avg.growth = (weight_comp's)*(growth_comp's) + (weight_trucks)*(growth_trucks) = (.517)*(.313) + (.483)*(-.120) =0.104
So, total cumulated investment is 10.4% higher than the previous year. Since investment in the first year is $206, then applying this growth rate, cumulated real investment in year 2 is:
Yr. 2 Real Investment = $206*(1+.104) = $227.42Yr. 2 Deflator = (Yr. 2 nominal) / (Yr. 2 real) = 203/227.42 = 0.893
To do this for year 3, follow the same steps and calculate the growth rate of real quantities for each component between year 2 and year 3: computers grew (171.7-131.3) / (131.3) = 0.307, or 30.7%, and real quantities of trucks grew 1.3%. In the same year, computers were 49.8% of expenditures and trucks were 50.2% of expenditures. So, the average growth rate is:
avg. growth = (0.498)*(0.307) + (0.502)*(0.013) = 0.159
We now add this growth rate to year 2’s real investment to get cumulated investment for year 3:
Yr. 3 Real Investment = $227.42*(1+.159) = $263.58Yr. 3 Deflator = (Yr. 3 nominal) / (Yr. 3 real) = 207/263.58 = 0.785
(…) 





In this exercise, we will calculate chain-weight real GDP figures from raw data.  Formally, chain-weight real GDP (Yt) for year t is calculated from the prices (p) and quantities (q) of goods at certain time periods (t):
Where the summation operator works over the implied index of goods.  Note that the figure for year t depends on the figure for year (t – 1), and year (t – 1) GDP depends on year (t – 2), etc. in a “chain” back to the base year.  Because of this chain, real GDP is most easily calculated using a multi-step process.  Note that the first term under the square root is a Laspeyres relative quantity index (1+growth rate), and the second term is a Paasche relative quantity index.  We first calculate these individual indexes (in growth rate form) for each year, and later calculate GDP levels.

(1) For reference and later use, calculate nominal GDP for each year:
(2) For each year (except the first year) calculate the growth rate of real GDP using the current year’s prices (this is a Paasche index):

(3) For each year (except the first year) calculate the growth rate of real GDP using the previous year’s prices (this is a Laspeyres index):
(4) For each year (except the first year) calculate the chain-weight growth rate of real GDP by taking the geometric average of the two fixed-weight growth rates calculated in the previous two steps.  The geometric average is the mathematically correct method of taking the average of growth rates.
(5) Declare any year a base year (b).  This means you are declaring that real GDP equals nominal GDP for that year:
(6) Calculate the level of chain-weight real GDP for the year after the base year by using the base year and one plus the chain-weight growth rate computed in step 4:
(7) Calculate the level of chain-weight real GDP for all successive years the same way:  for year (b + 2), use Yb+1 and  Y(upper do)b+2, etc.

(8) Calculate the level of chain-weight real GDP for the year before the base year rearranging the formula used in step 6, taken back one period:
(9) Calculate the level of chain-weight real GDP for each previous year the same way:  for year (b - 2), use Yb-1  and Y (upper dot) b-1 , etc.

* * *
숙제:


5. 자료: 연쇄지수(chain index)의 이해와 적용 (한국은행, 2007)

(...) 연쇄가중법에서는 고정가중법과 달리 비교년의 실질GDP를 직접 추계하는 데 사용되는 공통된 기준(기준년 또는 가중치)이 없으므로, 물량지수를 먼저 구한 후 이를 연장하여 실질GDP를 구한다. 단계별로 나누어 설명하면 다음과 같다.

  • 우선, 전년도 가격 또는 금액 가중치를 이용하여 매년도의 전년대비 물량증가율, 즉 연환지수(link factor)를 산출한다. 
  • 다음으로, 지수기준년6) (예: 2000년= 100)으로부터 각 연도의 연환지수를 누적적으로 곱하여 비교년의 지수기준년 대비 물량증가율, 즉 연쇄지수(chain index)를 도출한다. 
  • 마지막으로, 지수기준년의 GDP 금액에 당해년의 연쇄지수를 곱하여 연쇄 실질GDP 시계열을 작성한다. (...)

6. 자료: http://answers.yahoo.com/question/index?qid=20071019170226AAI8XW9
You are given the following information on an economy that produces 3 commodities
year 1: commodity x(P=$390 Q=20), commodity y(P=$40 Q=120), commodity z(P=$5 Q=1000)
year 2: commodity x (P=$410 Q=22), commodity y(P=$60 Q=138), commodity z(P=$7 Q=1050)
I've calculate the current dollar GDP to be 17600 for year one and 24650 for year two, but am confused as to how to calculate the constant dollar GDP using a chain-weighted index. What in the world is a chain-weighted index??
* * *
I believe you're in Somers class, I've been looking for answers on this as well.
I used the price from year one and multiplied it by the quantity from year two (and three), however my total differs from what he gave us on the solutions sheet, since I get:
Constant Dollar GDP
Year 2 = 19,350 (he says 19,360)
Year 3 = 20,305 (his 20,328)
--------------
I finally figured it out!
You have to take the average price of Year 1 and 2:
[390+410]/2= 400, [40+60]/2= 50, [5+7]/2= 6
Then multiply them by the quantities in Year 2
400x22, 50x138, 6x1050 = 22,000
afterward you do the same for in Year 1:
400x20, 50x120, 6x1000 =20,000
then... ([22,000-20,000) / 20,000] + 1 = 1.1
Now you take the current base year GDP=17,600 and multiply it by 1.1=19.360

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