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Standard errors for Laspeyres and Paasche index numbers
Economics Letters North-Holland
35
35 (1991) 35-38
Standard errors for Laspeyres index numbers E.A. Selvanathan
*
Griffith University, Nathan, Received Accepted
and Paasche
Qld 411 I Australia
3 May 1990 29 May 1990
In this paper we use the regression approach to index numbers to derive the standard errors for Laspeyres and Paasche index numbers. These standard errors are simple to evaluate. We show that these standard errors are linked to the variability of relative prices. When there are more changes in relative prices, the standard errors are higher.
1. Introduction Laspeyres and Paasche index numbers are well known indexes used in many countries to measure the changes in the general price level. These indexes are computed as some form of weighted average of price ratios. Allen (1975) and Banerjee (1975) used the sampling approach to construct index numbers and examined the estimation of price relatives and their standard errors. Clements and Izan (1987) and Selvanathan (1989) used the stochastic approach to estimate the rate of inflation and its standard error. Royal1 (1970) and Rao and Rao (1984) used the regression approach to derive index numbers. In this paper we use the regression approach to extend the work in this area of deriving the sampling variances of Laspeyres and Paasche index numbers.
2. The Laspeyres index Let pioqio be expenditure on commodity i(i = 1,. . . , n) in the base period 0 and let p,,qio be base-period consumption of i(q,o) valued at the current period’s prices ( p,,). Consider a regression of Pirqlo on pio4io: Pi,qio
= YtP,OqiO+
cif
3
j
=
l,. .
* I would like to thank Professor K.W. Clements University of New England for their comments. 0165-1765/91/$03.50
.2
n
(1)
3
of the University
0 1991 - Elsevier Science Publishers
of Western
B.V. (North-Holland)
Australia
and
Dr. Prasada
Rao of the
36
EA. Seluannthan / Standard errors for Lmpeyres and Paasche indexes
where yt is a constant
with respect
to commodities;
and elt is a disturbance.
where ~3,~is the Kronecker delta. Here the variance of the errors is assumed expenditures. Thus, the higher the expenditure, the larger the variance. We divide both sides of (1) by ,/‘z to give
We assume
to be related
Y,, = YtX*o + Uir,
to their
(3)
It follows from (2) that cov[uir, xl0 = /%; and G = Ed,/,/=. Therefore var[ u;,] = at, which is common for all commodities. [~/(P,,ch)l COV[~,,, $1 = ~z2k,. we can now apply LS to (3) to get the BLUE of yt, where Y,, = P,( /Pi qio/pla
u,,] = Thus
(4) i=l
i=l
where w,~ = p,oq,o/Mo is the budget share of i in period 0; and M,, = X:=, pioqio is total expenditure in period 0. The estimator (4) is the Laspeyres price index. The variance of Tt is given by
(5) where the parameter
a,’ can be estimated
l?;=l/(n-l)
t
unbiasedly
by
(yrt-?&J2.
(6)
i=l
Thus eq. (5) gives the variance (6), we obtain
var[%l= &-
I
of yi,, xiO, eqs. (4) and
~w;~[[~-l]-~l~~[~-Il]
-& i: 1 -np, n-l
price index. Using the definitions
i=l
i=l
=
of Laspeyres
w,o
log 2 [
-
5 j=l
w/o log $
1 2
(7)
where the approximation is based on log (1 + x 1 = x for small x, with x = ( p,,/piO) - 1; Dpz = log( P,,/Pio) is the log-change in the ith price from period 0 to t; DP,’ = C:=, wj,Dp,t is the Divisia price index with base-period weights; and IIF = Cl+ w,a[ Dpz - DP,‘]’ is the Divisia price variance
E.A. Seiuanathan/ Standarderrorsfor Lmpeyres andPaasche indexes
37
corresponding to II&‘. This IIF measures the degree to which prices move disproportionately. Equation (7) shows that the variance of the Laspeyres price index is approximately proportional to the degree of relative price variability. Thus, when there is more relative price variation the variance of f, will be larger.
3. The Paasche index Now consider a regression of expenditure on i in the current period, plrqlr,on current-period consumption, q,r,valued at base-year prices, pIOqr,, p,,q,, =
x*pioqir + G,
i = 1, . . . , n,
(8)
with
E[E,~;]= 0,
COV[E;, fi*,] =
a,*2pioq;,8;j.
Equations (8)-(9) are the same as (l)-(2) except that base-period consumption qLo in the former set of equations is replaced with current-period consumption qi,. As before, by dividing both sides of (8) by Jpioq,f, it can be easily shown that the BLUE of yf is
where w,: = pioqir/Mto; Mto = I:= I pioqit.The expenditure Mzo is current-period consumption valued at base-period prices, summed over all n goods; and wi is the share of commodity i in Mto with Cy= 1 wiy = 1. The right-hand side of (10) is the Paasche price index. As before, we can easily show that var Tz* = I&?/( n - l),
(11)
where IIT = X~=,w,~[Dp~ - DP,*]’ is the Divisia variance corresponding to DP,*; DP,* = Xr=, wJyDp,!!is the Divisia price index with weights w,,. ’ Equation (11) shows that the variance of the Paasche price index is also approximately proportional to the degree of relative price variability. As before, when there is more relative price variation, this variance will be larger.
4. An illustrative application We use the private consumption expenditure data of commodity groups (namely, food, beverages, clothing, housing, durables, medical care, transport, recreation, education and miscellaneous) published by the Australian Bureau of Statistics for the period 1960-1981 to compute the Laspeyres price index and its standard error with base year 1960. Table 1 presents the results. As can be seen, in most years the Laspeyres price index is significantly different from one.
38 Table 1 Estimates
E.A. Selvanaihan
of Laspeyres
/ Standard errors for Laspeyres and Paasche indexes
price index and its standard
error Australia,
1960-1980
(1960 = 1.0).
Year
LP index
Standard
1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981
1.0040 1.0217 1.0273 1.0600 1.0960 1.1283 1.1681 1.2052 1.2438 1.3178 1.4038 1.4932 1.6846 1.9731 2.2712 2.5387 2.7804 3.0349 3.3455 3.6681 4.0051
0.0089 0.0167 0.0174 0.0193 0.0241 0.0277 0.0324 0.0375 0.0407 0.0512 0.0605 0.0672 0.0791 0.1090 0.1339 0.1552 0.1720 0.1809 0.1954 0.2266 0.2652
error
5. Concluding comments In this paper we show that the regression approach can be used to derive the sampling variances of Laspeyres and Paasche price indexes. These variances are simple to evaluate. The results show that both variances are linked to the variability of relative prices and the sampling variance of both the Laspeyres and Paasche indexes will be higher the larger the relative price movements. We illustrated these results with the Australian consumption data.
References Allen, R.G.D., 1975, Index numbers in theory and practice (Macmillan Press, New York). Banerjee, K.S. 1975. Cost of living index numbers - Practice, decision and theory (Marcel Dekker, New York). Clements, K.W. and H.Y. Izan, 1987, The measurement of inflation: A stochastic approach, Journal of Business and Economic Statistics, Vol. 5, no. 3, 339-350. Rao, T.J. and P.S. Rao, 1984, Sampling bias and unbiased estimation of price index numbers, Unpublished manuscript (Indian Statistical Institute, Calcutta, India and University of New England, Armidale, Australia). Royall, R.M., 1970, On finite population sampling theory under certain linear regression models, Biometrica 57, 377-387. Selvanathan, E.A., 1989, A note on the stochastic approach to index numbers, Journal of Business and Economic Statistics 7, no. 4,471-474.
35
35 (1991) 35-38
Standard errors for Laspeyres index numbers E.A. Selvanathan
*
Griffith University, Nathan, Received Accepted
and Paasche
Qld 411 I Australia
3 May 1990 29 May 1990
In this paper we use the regression approach to index numbers to derive the standard errors for Laspeyres and Paasche index numbers. These standard errors are simple to evaluate. We show that these standard errors are linked to the variability of relative prices. When there are more changes in relative prices, the standard errors are higher.
1. Introduction Laspeyres and Paasche index numbers are well known indexes used in many countries to measure the changes in the general price level. These indexes are computed as some form of weighted average of price ratios. Allen (1975) and Banerjee (1975) used the sampling approach to construct index numbers and examined the estimation of price relatives and their standard errors. Clements and Izan (1987) and Selvanathan (1989) used the stochastic approach to estimate the rate of inflation and its standard error. Royal1 (1970) and Rao and Rao (1984) used the regression approach to derive index numbers. In this paper we use the regression approach to extend the work in this area of deriving the sampling variances of Laspeyres and Paasche index numbers.
2. The Laspeyres index Let pioqio be expenditure on commodity i(i = 1,. . . , n) in the base period 0 and let p,,qio be base-period consumption of i(q,o) valued at the current period’s prices ( p,,). Consider a regression of Pirqlo on pio4io: Pi,qio
= YtP,OqiO+
cif
3
j
=
l,. .
* I would like to thank Professor K.W. Clements University of New England for their comments. 0165-1765/91/$03.50
.2
n
(1)
3
of the University
0 1991 - Elsevier Science Publishers
of Western
B.V. (North-Holland)
Australia
and
Dr. Prasada
Rao of the
36
EA. Seluannthan / Standard errors for Lmpeyres and Paasche indexes
where yt is a constant
with respect
to commodities;
and elt is a disturbance.
where ~3,~is the Kronecker delta. Here the variance of the errors is assumed expenditures. Thus, the higher the expenditure, the larger the variance. We divide both sides of (1) by ,/‘z to give
We assume
to be related
Y,, = YtX*o + Uir,
to their
(3)
It follows from (2) that cov[uir, xl0 = /%; and G = Ed,/,/=. Therefore var[ u;,] = at, which is common for all commodities. [~/(P,,ch)l COV[~,,, $1 = ~z2k,. we can now apply LS to (3) to get the BLUE of yt, where Y,, = P,( /Pi qio/pla
u,,] = Thus
(4) i=l
i=l
where w,~ = p,oq,o/Mo is the budget share of i in period 0; and M,, = X:=, pioqio is total expenditure in period 0. The estimator (4) is the Laspeyres price index. The variance of Tt is given by
(5) where the parameter
a,’ can be estimated
l?;=l/(n-l)
t
unbiasedly
by
(yrt-?&J2.
(6)
i=l
Thus eq. (5) gives the variance (6), we obtain
var[%l= &-
I
of yi,, xiO, eqs. (4) and
~w;~[[~-l]-~l~~[~-Il]
-& i: 1 -np, n-l
price index. Using the definitions
i=l
i=l
=
of Laspeyres
w,o
log 2 [
-
5 j=l
w/o log $
1 2
(7)
where the approximation is based on log (1 + x 1 = x for small x, with x = ( p,,/piO) - 1; Dpz = log( P,,/Pio) is the log-change in the ith price from period 0 to t; DP,’ = C:=, wj,Dp,t is the Divisia price index with base-period weights; and IIF = Cl+ w,a[ Dpz - DP,‘]’ is the Divisia price variance
E.A. Seiuanathan/ Standarderrorsfor Lmpeyres andPaasche indexes
37
corresponding to II&‘. This IIF measures the degree to which prices move disproportionately. Equation (7) shows that the variance of the Laspeyres price index is approximately proportional to the degree of relative price variability. Thus, when there is more relative price variation the variance of f, will be larger.
3. The Paasche index Now consider a regression of expenditure on i in the current period, plrqlr,on current-period consumption, q,r,valued at base-year prices, pIOqr,, p,,q,, =
x*pioqir + G,
i = 1, . . . , n,
(8)
with
E[E,~;]= 0,
COV[E;, fi*,] =
a,*2pioq;,8;j.
Equations (8)-(9) are the same as (l)-(2) except that base-period consumption qLo in the former set of equations is replaced with current-period consumption qi,. As before, by dividing both sides of (8) by Jpioq,f, it can be easily shown that the BLUE of yf is
where w,: = pioqir/Mto; Mto = I:= I pioqit.The expenditure Mzo is current-period consumption valued at base-period prices, summed over all n goods; and wi is the share of commodity i in Mto with Cy= 1 wiy = 1. The right-hand side of (10) is the Paasche price index. As before, we can easily show that var Tz* = I&?/( n - l),
(11)
where IIT = X~=,w,~[Dp~ - DP,*]’ is the Divisia variance corresponding to DP,*; DP,* = Xr=, wJyDp,!!is the Divisia price index with weights w,,. ’ Equation (11) shows that the variance of the Paasche price index is also approximately proportional to the degree of relative price variability. As before, when there is more relative price variation, this variance will be larger.
4. An illustrative application We use the private consumption expenditure data of commodity groups (namely, food, beverages, clothing, housing, durables, medical care, transport, recreation, education and miscellaneous) published by the Australian Bureau of Statistics for the period 1960-1981 to compute the Laspeyres price index and its standard error with base year 1960. Table 1 presents the results. As can be seen, in most years the Laspeyres price index is significantly different from one.
38 Table 1 Estimates
E.A. Selvanaihan
of Laspeyres
/ Standard errors for Laspeyres and Paasche indexes
price index and its standard
error Australia,
1960-1980
(1960 = 1.0).
Year
LP index
Standard
1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981
1.0040 1.0217 1.0273 1.0600 1.0960 1.1283 1.1681 1.2052 1.2438 1.3178 1.4038 1.4932 1.6846 1.9731 2.2712 2.5387 2.7804 3.0349 3.3455 3.6681 4.0051
0.0089 0.0167 0.0174 0.0193 0.0241 0.0277 0.0324 0.0375 0.0407 0.0512 0.0605 0.0672 0.0791 0.1090 0.1339 0.1552 0.1720 0.1809 0.1954 0.2266 0.2652
error
5. Concluding comments In this paper we show that the regression approach can be used to derive the sampling variances of Laspeyres and Paasche price indexes. These variances are simple to evaluate. The results show that both variances are linked to the variability of relative prices and the sampling variance of both the Laspeyres and Paasche indexes will be higher the larger the relative price movements. We illustrated these results with the Australian consumption data.
References Allen, R.G.D., 1975, Index numbers in theory and practice (Macmillan Press, New York). Banerjee, K.S. 1975. Cost of living index numbers - Practice, decision and theory (Marcel Dekker, New York). Clements, K.W. and H.Y. Izan, 1987, The measurement of inflation: A stochastic approach, Journal of Business and Economic Statistics, Vol. 5, no. 3, 339-350. Rao, T.J. and P.S. Rao, 1984, Sampling bias and unbiased estimation of price index numbers, Unpublished manuscript (Indian Statistical Institute, Calcutta, India and University of New England, Armidale, Australia). Royall, R.M., 1970, On finite population sampling theory under certain linear regression models, Biometrica 57, 377-387. Selvanathan, E.A., 1989, A note on the stochastic approach to index numbers, Journal of Business and Economic Statistics 7, no. 4,471-474.