More on Laspeyres price index

Economics Letters 0165.1765/93/$06.00

43 (1993) 157-162 0 1993 Elsevier

157 Science

Publishers

B.V. All rights

More on Laspeyres E.A. School Received Accepted

Selvanathan of International

reserved

price index

* Business

Relations,

Grifjith

University,

Nathan,

Queensland

4111, Australia

17 February 1993 24 June 1993

Abstract In a recent article Selvanathan (Economics Letters, 1991, 35, 35-38) uses the stochastic Laspeyres price index and its standard error. In this paper we extend this approach by allowing relative prices.

approach to derive the for systematic changes in

1. Introduction In a recent

article

Pir4rO =

Selvanathan

atP~l~~i(l

+

‘!r

(1991) 3

’

=

‘3

uses the regression ’

’

’

3

model

n

(where n is the number of commodities and lir is a zero mean, random error) to derive Laspeyres price index number and its standard error. Selvanathan showed that the GLS estimator of LY,is identical to the usual Laspeyres price index, I;, = cy=, pitqiO/c:=, pioqio. The regression procedure used in that paper also belongs to the family of stochastic index numbers [see, for example, Clements and Izan (1981, 1987), Selvanathan (1989), Prasada Rao and Selvanathan (1992), Selvanathan and Prasada Rao (1992)]. Dividing both sides of the above equation by piOqi,, and denoting pi = pi,lpio, we obtain i=l,.

p;[=a,+&,,

. . ,n.

(1)

From Eq. (1) we note that &, = (P:~ - (Y,) is the by writing &, = (pl: - 1) - ((Ye- 1) =log(pprl,l) and we have also used log(1 +x) =x, for small Taking the mathematical expectation of both

change in the ith relative price. This can be seen w h ere D is the log-change operator = D(pi,/cr,), X, with x = p:: - 1 and x = (Y,- 1. sides of (1) we have

ro 1

E +-a, [

=O,

which is the same for all i. This means that model (1) implies that the expected value of the relative price of i is a constant for all i. Clearly this is a weakness of model (1). In the context of measuring the rate of inflation, models used such as (1) were also previously criticised by Keynes (1930, pp. 85-88) as such models do not allow for systematic changes in relative prices. The main ’ I wish to thank

Professors

Ken Clements

and Prasada

Rao for their valuable

suggestions

and comments

E.A.

158

Seivanathan

I Economics

Letters 43 (1993) 1.57-162

aim of this paper is to extend the work presented in Selvanathan (1991) by using the approach of Clements and Izan (1987) and Selvanathan (1989). The procedure discussed here is also applicable to the derivation of the Paasche index. In the next section we present an extended version of model (1) which allows for systematic changes in relative prices and in the following sections we derive estimators for the Laspeyres price index and changes in relative prices. We also present an illustrative application of the results with the Australian private consumption data.

2. The extended

model

We write pii = plrlpio, the ith relative price, as the sum of the common trend commodity-specific component /3,, and a zero-mean random component &,: P;;=(Y~+/~~+[;~, where

T is the number

E[l,,l = 0 where

6.. is the

7

i=l,.

of periods.

w5,,,

Kronecker

.,n;t=l,..

.,T,

We make the following

{,,I

delta,

=

in all prices

(2) assumptions

on the disturbance

$4,>

terms:

(3)

IO

w,,, =ploq’l,l/Mo

(Y,, a

is the

budget

share

MO = c/r, pitrq10 is total expenditure in period 0. Rearranging Eq. (2) and cal expectation, we obtain pi = E[pl: - a,]. Thus p, can be interpreted as change in the ith relative price. Model (2) is not identified. This can be seen by noting that an increase number k and lowering of /3, for each i by the same k does not affect the One way of identifying the model is to impose the constraint

of i in period

0, and

taking the mathematithe expectation of the in LY,for each t by any right-hand side of (2).

Equation (4) has the simple interpretation that a budget-share-weighted-average of the systematic components of the relative price changes is zero. (4) using the In the next section we derive estimators of model (2) subject to constraint maximum likelihood method. In an appendix, available on request from the author, it has been shown that the ML estimators of LY,and p, are identical to their GLS estimators.

3. Maximum Model

likelihood

(2) together

estimators with constraint

(4) can be written i=l,...,n-1,

P1: =

i=n, 1P,+i,,,

as

E.A.

, T. We combine

for t = 1, I-

these

I Economics

equations

Y,r-1

0...010...0 . . . . . . . . 0 . . .0 1

o...o

Ynt

o...o

o...o

Yl,

where

Selvanathan

I

=

yi, = p’,),. Using

an obvious

Y,=X,y+J,,

. . .

1

Letters

in matrix

43 (1993)

159

157-162

form:

1

...

0

0

.

1

.

+

...

-wP1o

P70 notation,

W

we write the above

IlO

equation

as

T,

t=l,...,

(5)

where Y, and 5; are n-vectors, X, is a matrix of order 12X (T + n - l), and y is a (T + n - 1)-vector of parameters. In the next section we derive the following maximum likelihood estimators:

By substituting

for wit, =P~,,~~~,/C~=, ptoqlo and pp, = (pI,IpIo),

it can be easily

seen

that

the

estimator ~2, is identical to the Laspeyres price index c:=, pi,qro/C:=, ploqLo. The estimator fij of the systematic component of the change in the ith relative price is a weighted average of (py! - c?,) to n:, which in turn over all T periods. The weights in (6), c$~, . . . , #Jo, are inversely proportional is proportional to the error variance in period t; thus, less weight is accorded to those observations with a higher error variance. In the next section we also show that var &,=n:

;

var Bj =

T 2

1 [

+-1

WC)

1

(7)

Lo

As can be seen, the sampling variance of & increases with nf, which in turn rises with relative price variability. Therefore the sampling variance of the estimator of the overall price index will be higher the larger the relative price movements. The sampling variance of j$ is proportional to the difference between llwio and a constant term, so that this variance increases as w,~ falls.

4. Derivations Under assumption (3), we have var[ [,] = v:WCy ’ , with W, = diag[w,, . . . wno]. It also follows from (3) that the 5,‘s are independent over time. If we assume that & has a multinormal distribution, then the log-likelihood function of model (5) takes the form

where

c is a constant.

The first-order

derivatives

of (8) are

160

E.A.

aL

n

877:

2

-=--

Selvanathan

I Economics

Letters

43 (1993) 157-162

rlt-2+ +Tt4(y,- X,Y)‘WoK- X,Y)

(9)

and

The ML estimator

rl

=

[0

of y is obtained

0

P(W,’

-u’)

by equating

I

the right-hand

o%o(Y,,

where VI = diag[v: . . . qc], and A = c,‘=, (l/n:). Thus from the definition of y = [a1 . LYE p, . . . Pn_l]’ & = [& . . . liTI’ =

9

[

T)y2i:

i=l

This yields

wiOYi7

77T -;g

2

side of (10) to zero.

Wi"Yil . . .vi2 2

i=l

-

Ym)

we obtain

wiOYiT]’

and

s =[b, . . . /3-J’ 1' = These

~-‘w,’ -

expressions

simplify

..'~[~~~v;'w~o(Y~t

-Y,,)

. . .~~l'w,,,,(Y,,,,-Y~,)j

.

to

which is Eq. (6). As E[Y, - X,y] = E[&] = 0, f rom Eq. (10) we have information matrix of the ML estimation procedure is Consequently, the asymptotic covariance matrix of the inverse of the expectation of the derivative a2Llay ay’. again, we obtain

that E[~2L/~~~ dy’] = 0. Therefore, block-diagonal with respect to y and ML estimator of y is equal to minus If we differentiate (10) with respect

the 77:. the to y

E.A.

Selvanathan

I Economics

161

Letters 43 (1993) 157-162

(11)

Thus

it follows

from

(11) that the asymptotic T

cov 9 =

c qt-*x:woxl [

-I

1[

t=1

It follows from the definition expansion gives

0

matrix

var pi =

T 2

1 [ WC)

of 9 is

0 AP(W,‘- CL’)1 .

of y that var[ &] = q and var[p]

A

var&,=7j:,

?l

=

covariance

$1 ‘O

= A - ‘(W,’

- ~6’). Simple

matrix

1

By yriting constraint (4) as b,, = --C:Z: (wiolw,,,)&, we can show that /$ = CrTC1&(pE, - 6,) and var p, = [(ll~,~) - l]/[CT=, (ll~:)]. The above is Eq. (7). Equating the right-hand side of (9) to zero yields an ML estimator of 7:

where

it = [l,, . . . &J’ = Y, -X,+,

with &, =pp, - ~2~- pi

Table 1 Estimates of Laspeyres price index and its standard error: Australia, 1960-1981 (1960 = 1.0) 1961 1962 1063 lY64 1965 1Y66 lY67 1068 1969 lY70 1971 1972 lY73 lY74 lY75 1076 1077 lY78 lY79 1980 1981

1.004 1.022 1.027 1.060 1.096 1.128 1.168 1.205 1.244 1.318 1.404 1.493 1.685 1.973 2.271 2.539 2.780 3.035 3.346 3.668 4.005

0.083 0.077 0.076 0.073 0.069 0.065 0.060 0.055 0.051 0.041 0.033 0.026 0.027 0.031 0.051 0.069 0.086 0.093 0.107 0.139 0.177

1.004 1.007 1.018 1.053 1.092 1.120 1.159 1.187 1.225 1.285 1.373 1.454 1.644 1.919 2.166 2.468 2.701 2.923 3.218 3.521 3.887

162

E.A.

Table 2 Estimates

of commodity

dummies

Commodity

6

Food Beverages Clothing Housing Durables Medical care Transport Recreation Education Miscellaneous

-0.150 0.030 -0.120 0.177 -0.357 0.512 -0.085 0.246 0.560 0.225

Note:

Standard

errors

5. An illustrative

Selvanathan

and mean

relative

(P,”(0.019) (0.032) (0.030) (0.029) (0.036) (0.044) (0.028) (0.054) (0.129) (0.032)

I Economics

Letters 4.3 (1993) 157-162

price changes:

The United

Kingdom,

1965-1981

:I

-0.186 0.032 -0.108 0.256 -0.477 0.624 -0.099 0.188 0.632 0.270

are in parentheses.

application

In this section we present an application of model (2). We use the private final consumption expenditure data for 10 commodity groups for Australia used in Selvanathan (1991). The estimates hyIand p, and their standard errors are presented in Tables 1 and 2 based on (6) and (7). As can be seen from Table 1, for example, the general price level has increased by 32% in 1970 relative to 1960 with a standard error of 4%. The point estimates of (Y,are close to the CPI figures, which is expected, since the CPI is a Laspeyres index. Table 2 shows that the relative prices of food, clothing, durables and transport have declined while all others have increased. The estimates of pi’s are highly significant, except for beverages. The estimated relative price changes are in reasonable agreement with the mean relative price changes evaluated as (p,,, - 6) presented in the last column of Table 2.

References Clements, K.W. and H.Y. Izan, 1981, A note on estimating divisia index numbers, International Economic Review 22, 745-747. Corrigendum 23 (1982) 499. Clements, K.W. and H.Y. Izan, 1987, The measurement of inflation: A stochastic approach, Journal of Business and Economic Statistics 5, no. 3, 339-350. Keynes, J.M., 1930, A treatise on money, vol. 1 (London). Prasada Rao, D.S. and E.A. Selvanathan, 1992, Computation of standard errors for Geary-Khamis parities and international prices, Journal of Business and Economic Statistics 10, no. 1, 109-115. Selvanathan, E.A., 1989, A note on the stochastic approach to index numbers, Journal of Business and Economic Statistics 7, no. 4, 471-474. Selvanathan, E.A., 1991, Standard errors for Laspeyres and Paasche index numbers, Economics Letters 35, 35-38. Selvanathan, E.A. and D.S. Prasada Rao, 1992, An econometrics approach to the construction of generalized TheilTornqvist indices for multilateral comparisons, Journal of Econometrics 54, 335-346.