Finansiell Statistik, GN, 15 hp, VT2008 Lecture 17-2: Index Numbers

Transcription

1 Finansiell Statistik, GN, 15 hp, VT2008 Lecture 17-2: Index Numbers Gebrenegus Ghilagaber, PhD, Associate Professor May 7,

2 1 Introduction Index numbers are summary measures used to compare the general level of magnitude of a group of distinct but related items in two or more situations, places, or time points. Thus, a price index number is used for comparing changes in the general level of prices of a group of commodities at di erent time periods or at the same time period for di erent situations (say di erent regions) 2

3 Example i Commodity Unit Prices Quantities P i0 P i1 Q i0 Q i1 1 Bread 1 1: Milk 2:25 2: Butter 2:50 3: Soft 0:25 0: Cloth 5:00 7: Total 11 14:05 Price relatives Pi1 P i0 1:25 1:11 1:20 1:20 1:40 1:277 Price Indexes Pi1 P i

4 1.1 Simple and Simple-Aggregate Indexes Index numbers that refer to single commodities (as in the last panel above) are called Simple Index Numbers Index numbers that are based on a group of related commodities are known as aggregate indexes. These aggregate indexes may be: Simple aggregates Weighted aggregates For the above example, 4

5 P 01 = P Pi1 P Pi0 = 14:05 11:00 = 1:277 and, thus, the corresponding Simple Aggregate Price Index is 1: = 127:7 A Simple Arithmetic Mean of the price relatives gives P 01 = 1 5! 5X P i1 P i0 1:25 + 1:11 + 1:20 + 1:20 + 1:40 = 5 = 1:232 while, a simple Geometric mean of these price relatives gives v u t P 01 = 5 5Y P i1 P i0 = 5p 1:25 1:11 1:20 1:20 1:40 = 1:228 5

6 In computing the Simple Aggregate Price Indexes no account is made for possible changes in the quantities of the commodities bought. 6

7 1.2 Weighted Aggregate Indexes P 01 = P Pi1 Q i P Pi0 Q i where Q i is the quantity of the i th commodity 7

8 1.2.1 Laspeyres Index Laspeyeres uses the quantities bought at the initial (base-) period as weights: P La 01 = P Pi1 Q i0 Expenditure at actual period relative to expenditure at base period (assuming same quantities in both periods) 8

9 Example: i Commodity Unit Prices Quantities Aggregate Costs P i0 P i1 Q i0 Q i1 P i0 Q i0 P i1 Q i0 1 Bread 1 1: :00 6:25 2 Milk 2:25 2: :25 2:50 3 Butter 2:50 3: :00 30:00 4 Soft 0:25 0: :75 2:10 5 Cloth 5:00 7: :00 28:00 Total 11 14:05 54:00 68:85 P La 01 = P Pi1 Q i0 = 68:85 54:00 = 1:275 9

10 Remark: Laspeyres Index gives more weight to items with higher changes in prices between base-period and actual period. It fails to account for possible "substitution e ect" (that as items get more expensive then they may be substituted by cheaper items and, thus, their quantity decreases). 10

11 1.2.2 Paasche s Index Paasche uses the quantities at the nal (actual) period as weights: P P a 01 = P Pi1 Q i1 P Pi0 Q i1 Expenditure at actual period (with actual prices) relative to expenditure at actual period (assuming same prices as in the base period) 11

12 Example i Commodity Unit Prices Quantities Aggregate Costs P i0 P i1 Q i0 Q i1 P i0 Q i1 P i1 Q i1 1 Bread 1 1: :00 6:25 2 Milk 2:25 2: :25 2:50 3 Butter 2:50 3: :00 36:00 4 Soft 0:25 0: :75 2:10 5 Cloth 5:00 7: :00 14:00 Total 11 14:05 49:00 60:85 P P a 01 = P Pi1 Q i1 P Pi0 Q i1 = 60:85 49:00 = 1:242 12

13 1.2.3 Fisher s Ideal Index P F 01 = qp La 01 P P a 01 = = v u t q (1:275) (1:242) = 1:258 P! P! Pi1 Q i0 Pi1 Q i1 P Pi0 Q i1 13

14 2 Relatioship between Laspeyeres & Paasches Index Consider the two indexes, and let P La 01 = P Pi1 Q i0 ; and P P a 01 = P Pi1 Q i1 P Pi0 Q i1 V i0 = P i0 Q i0 and V i1 = P i1 Q i1 be the expense on item i in the base and actual period, respectively. Thus, V 01 = P Pi1 Q i1 14

15 gives the Value Index - the total expense in actual period relative to the total expense in the base period. We can also de ne relative weights such as w i = P i0q i0 which is the relative weight of the i th item in the base period s total expense. De ne relative price and quantity of i th item between the two periods as x i = P i1 P i0, y i = Q i1 Q i0 We shall now compute a weighted covariance between x i and y i using w i as weights. 15

16 The standard formula for Cov(x; y) - with weights = 1 n is; Cov(x; y) = 1 n = 1 n (x i x) (y i y) x i y i x i y i 5 Using w i = P i0q i0 P Pi0 Q i0 in place of 1 n we get, 16

17 Cov P i1, Q! i1 P i0 Q i0 = P i1 P i0 Q i1 Q i0 P i0 Q i0 n X P i1 P i0 P i0 Q i0 n X Q i1 Q i0 P i0 Q i0 = P i1 Q i1 P i1 Q i0 P i0 Q i1 = V 01 = V 01 P La 01 P i1 Q i0 P i0 Q i1 = V 01 P La P i0 Q i1 0 P Pi1 Q i1 P i1 Q i1 1 C A 01 P i0 Q i1 = V 01 P La 01 P i0 Q i1 P Pi1 Q i1 0 P Pi0 Q i0 P i1 Q i1 1 C A 17

18 = V 01 P La 1 01 P01 P a (V 01 ) = V 01 "1 P01 La # P01 P a Thus, P La 01 P P a 01 = 1 s xy V 01 From this relatioship, we can make the following remarks: P01 La and P 01 P a will be equal (indentical) only if S xy = 0 which, in turn, means when there is no covariance between the price relatives x i = P i1 P i0 and the quantity relatives y i = Q i1 Q : i0 18

19 In reality, however, we know that changes in relative prices would be accomopanied by changes in relative quantities. Further, we know that an increase in price leads to decrease in demand and, thus, the covariance is negative (S xy < 0) We can, therefore, write the relationship between P La 01 and P P a 01 as P La 01 P P a 01 > 1 so that P01 La > P 01 P a The stronger the correlation between P i1 P and Q i1 i0 Q ; the larger the value of i0 S xy and, thus, the larger the di erence between P01 La and P 01 P a. 19

20 3 Quantity Indexes We can de ne quantity indexes in the same way as for the price indexes: Q La 01 = Q P a 01 = P Pi0 Q i1 P ; Pi0 Q P i0 Pi1 Q i1 P ; Pi1 Q i0 Q F 01 qq = La 01 QP 01 a 20

21 4 Value Index As de ned before, the value index is de ned as P Pi1 Q i1 V 01 = and re ects simultaneous changes in both prices and quantities. 21

22 5 Stock Market Indexes 5.1 Market-Value-Weighted Index 5.2 Price-Weighted Index 5.3 Equally-Weighted Index 22