ECONOMICS

VOLATILITY AND STOCK PRICE INDEXES

by

Kenneth W Clements

and

H Y Izan

and

Yihui Lan

Business School The University of Western Australia

DISCUSSION PAPER 11.16

August 2011

VOLATILITY AND STOCK PRICE INDEXES*

by

Kenneth W. Clements, H.Y. Izan, and Yihui Lan Business School

The University of Western Australia

DISCUSSION PAPER 11.16

Abstract

The stochastic approach to index numbers has been successfully applied to the estimation

of the Consumer Price Index (CPI). One distinct advantage of this approach is that it provides the

whole distribution of the index, not simply a point estimate of the rate of inflation. In this paper,

we extend the application of the stochastic approach to the estimation of a stock market

index. We demonstrate how this approach can be used to identify “redundant stocks” that do not

contribute significantly to the overall index. For index tracking purposes, these stocks can be

safely excluded.

* Kenneth Clements is a Winthrop Professor of Economics and BHP Billiton Research Fellow, H.Y. Izan is a Winthrop Professor in Accounting and Finance and Yihui Lan is an Associate Professor in Finance, Business School, UWA. This research was supported in part by BHP Billiton and the Australian Research Council. We thank Grace Gao for helpful discussion and research assistance. We have also benefited from the research assistance of Liang Li, Haiyan Liu, Jiawei Si and Stephane Verani.

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1. Introduction

Of the whole universe of economic data, the consumer price index (CPI) and the stock price

index (SPI) probably attract the most attention. While the CPI is vital for the setting of interest

rates by the central bank, for deflating nominal incomes into their real counterparts, and as the

basis of “cost escalation” clauses in many contracts, the SPI is used by individual and

institutional investors as a benchmark against which to evaluate the performance of their

portfolios, by technical analysts or chartists when they make buy and sell recommendations, and

by economists to study the relationship between financial markets and economic activity. But the

SPI also plays a broader role in society. When stock markets are down, retirees’ incomes are

squeezed, investment bankers’ bonuses evaporate, endowment earnings of universities fall,

companies cut back on expansion, and even governments feel the pressure by the generalised

anxiety caused by depressed equity prices. Given their value to such a wide group of users, it is

surprising that the theory underlying indices of stock prices has not been examined more

thoroughly.

Index numbers are summary measures of the information on individual prices and

quantities. The traditional approaches to index numbers begin and end with a single number, so

that, for example, the CPI or SPI has increased by 100 g%, end of story. We argue that this is a

limited way to summarise price behavior. Suppose price share i grows at rate ig , so that the

transition from base period 0 to current period 1 takes the form i1 i i0p 1 g p . Consider the

extreme case when all changes are identical, ig g, i 1, , n stocks. Here, as all prices change

proportionately, there can be no disputation that the index also increases by g. Panel A of Figure

1 illustrates this scenario with n=5 component prices. The graph on the far left of this panel

shows that the scatter of the current- against base-period prices lies on a straight line passing

through the origin with slope 1 g. The growth rate g is the change in the overall index, as can

be seen from the frequency distribution on the far right of Panel A. In reality however, prices do

not move proportionally and the graph on the far left of Panel B illustrates this case. Here, some

prices grow faster than average, others slower, but the average growth rate, g, is the same as in

Panel A. The index clearly lies somewhere in the middle of individual prices, as shown in the far

right of Panel B. In this case, we have i1 i0p 1 g p error , where the pricing error results from

disproportionate behavior. While the average growth is the same in Panels A and B, they

represent fundamentally different experiences. The disproptionate price movement of Panel B

implies that the overall index is less well-defined -- we can have less confidence that prices grow

on average by 100 g% in this case, as compared to Panel A when prices move in tandem. While

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this issue is at the heart of the index-number problem, surprisingly the statistical implications

have not been fully appreciated.

In the past few decades, index-number theory and practice have gone through a period of

renewed interest and new insights. One development is the stochastic approach, which has been

resurrected by Balk (1980), Clements and Izan (1981, 1987), and Selvanathan and Rao (1994).1

The overarching feature of the stochastic approach is that it provides the whole distribution of the

index, not just the point estimate. Each price change is treated as a noisy reading on the common

trend in all prices, with this trend playing the role of the index. The least-squares method

provides an elegant solution to the problem of how best to combine the individual price changes

so as to minimize the noise. Accordingly, the stochastic approach offers a richer framework that

opens up new opportunities; and it does this by using the same basic information of traditional

approaches. With randomness playing a central role, the stochastic approach is an appealing way

to analyse stock prices.

This paper investigates the use of the stochastic approach in forming a stock price index and

contributes in three aspects. First, in Section 2, we set out the basics of index-number theory and

relate it to conventional indexes of stock prices. Second, we apply the stochastic approach to

share prices in Section 3. Finally, in Section 4, we apply the framework to the issue of portfolio

tracking and investigate whether it is possible to ignore certain stocks on the basis of their

contribution to the index. Concluding remarks are provided in Section 5.

2. Index Numbers

This section sets the scene for what is to follow by providing the necessary basics of

index-number theory. The presentation is generic and stylised in order to highlight just essential

elements of the index-number approach. To focus on the analytics, we deliberately ignore a

number of practical issues, such as the chaining of index numbers, a procedure that is widely

used by statistical agencies.

1 According to Diewert (2008), the stochastic approach is now one of the four main approaches to index number theory. The other three are the fixed basket approach, the test approach and the economic approach. Applications of the stochastic approach include the measurement of inflation (Clements and Izan, 1981, 1987 and Selvanathan and Selvanathan, 2006), world interest rates (Ong et al., 1999) and international competitiveness (Clements et al., 2008). See Clements et al. (2006) for a comprehensive review.

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Three Price Indices

Suppose there are n goods in two periods, the base period and the current period, to be

denoted by 0 and 1. Let itp be the price of good i in period t, i 1, , n, t 0,1. A price index

addresses the question of how best to summarise the difference between the vector of the n prices

in the base period, 10 n0p , , p , and that in the current period, 11 n1p , , p . We could consider an

unweighted average of the n price relatives, 11 10 n1 n0p p , , p p , that is, ni 1 i1 i01 n p p . But

to recognise that some goods are more important than others, some form of a weighted average is

more appealing as that makes the index more representative of underlying expenditure patterns.

A number of popular indices are formulated in this manner and below we consider some

alternatives.

Let i0q be the quantity consumed of i in 0, ni 10 i0 i0M p q be total expenditure in 0 and

i0 i0 i0 0w p q M be the budget share of i in 0, which has a unit sum. The Laspeyres index uses as

weights these base-period budget shares:

nL i1

i0i 1 i0

pP w

p

.

This index can also be expressed as ni 1 i1 i0 0p q M , which is the ratio of two costs of the

consumption basket. The first, ni 1 i1 i0p q , is the cost of the base-period basket, 10 n0q , ,q , at

current-period prices, 10 n1p , , p . The second, 0M , is the cost of the same basket evaluated at

base-period prices. It is convenient to define tt ni 1 it itM p q as the basket in period t costed at

prices prevailing in t, where t , t 0,1 , so that observed base-period expenditure, 0M , now

becomes 00M . Accordingly, Laspeyres can equivalently be written as L 10 00P M M . This dating

notation for prices and quantities can also be applied to the budget shares, so that

tt tti it itw p q M ,

with ttni 1 iw 1, and 00

i i0w w . Thus, we have

(1) 10 n

L 00 i1i00

i 1 i0

pMP w

M p

.

The Paasche price index replaces base-period quantities with their current-period

counterparts, so that

(2) 11 n

P 01 i1i01

i 1 i0

pMP w

M p

.

4

A further way of expressing this is as follows: Write the term after the first equals sign in

equation (2) as

11

n n n01 01 11111 11 11

i0 i1 i1 i1 i0 i1 i i1 i0i 1 i 1 i 1

M 1 1 1 1

M M M p q M p q M p p w p p

.

This shows that Paasche is a weighted harmonic mean of the n price relatives:

(2 )

11n

P 11 i1i

i 1 i0

pP w

p

.

Here, the weights are the current-period budget shares, so that in this form Paasche is more

“similar” to Laspeyres.

Rather than using either base- or current-period quantities, the Fisher index strikes a

symmetric balance by taking the geometric mean of Laspeyres and Paasche:

(3)

n00 i1i10 11

i 1 i0F L P100 01 n

11 i1i

i 1 i0

pw

pM MP P P .

M M pw

p

Volume Indices

The three indices above refer to prices. The corresponding volume, or quantity, indices

are

(4)

1101 11n nL 00 P 11i1 i1

i i00 10i 1 i 1i0 i0

01 11F L P

00 10

q qM MQ w , Q w ,

M q M q

M MQ Q Q .

M M

The observed ratio of total expenditure in the two periods is 11 00M M , which is also known as

the value ratio. Indices (1) -(4) satisfy the following value-preserving identities:

(5) 11

L P P L F F00

MP Q P Q P Q

M ,

so that the product of the relevant pair of price and volume indices is the expenditure ratio.

Equation (5) shows that deflation of the expenditure ratio by the Laspeyres (Paasche) price index

gives the Paasche (Laspeyres) volume index, while deflation by Fisher gives back Fisher:

5

(6) 11 00 11 00 11 00

P L FL P F

M M M M M MQ , Q , Q

P P P ,

which again reveals the symmetric nature of Fisher.

Revealing Inequalities

It follows from equations (1) and (4) that

10 01L L

00 00

M MP Q

M M ,

so that

(7) 11 11 00

L L 100 10 01

M M MP Q Z , with Z= 1

M M M .

As 1Z 1, L L 11 00P Q M M . This shows Laspeyres is subject to an upward bias as the product

of the price and volume indices exceeds the value ratio. Similarly, there is a downward bias in

Paasche:

11 11P P

00 00

M MP Q Z

M M .

To prove the inequality in expression (7), write

(8) 11 00

0 0 0 01 1 1 110 01

1 0 0 1 0 1 1 0

M MZ=

M M

p q p qp q p q

p q p q p q p q,

t 1t nt t 1t ntwith p , , p and q , ,q p q the price and quantity vectors in period t ( t 0,1 ).

Consider the first ratio on the far right-hand side of equation (8), 1 1 0 1 . p q p q The numerator is the

observed total cost in period 1, viz. the consumption basket of period 1, 1,q evaluated at period 1

prices, 1.p The denominator, 0 1,p q is the cost of the same basket at period 0 prices, 0.p For the

consumer to minimise cost, the former is less than the latter, so the ratio is less than unity. As the

second ratio on the far right of equation (8) is also less than unity, for similar reasons, it follows

that Z<1.

A further interesting inequality follows from the above. As L L 11 00P Q M M , we have

11 00L L P

L

M MQ , or Q Q ,

P

where the second step follows from equation (6). Similarly, as P P 11 00P Q M M ,

11 00P P L

P

M MP , or P P ,

Q

6

where the second step again follows from equation (6). Thus we have the Laspeyres-Paasche

spread:

L L

P P

Q P1, 1.

Q P

A Stock Price Index

We commence this subsection with a brief discussion of current practice regarding stock-

price indexes and then recast that material in terms more familiar from index-number theory. We

show that this theory offers considerable insight and guidance to the construction of stock prices

indices.

Stock price indices are usually weighted averages of the component price relatives. S&P

Indices (2010, p. 5) state in their publication Index Mathematics: Index Methodology that

“Most of Standard and Poor’s indices, indeed most widely quoted stock indices, are capitalisation-weighted indices. Sometimes these are called value-weighted or market cap weighted instead of capitalisation weighted. Examples include the S&P 500, the S&P Global 1200 and the S&P BMI indices.”

This publication then goes on to provide more precise details. If we now interpret itp and itq as

the price of a share in company i at time t and the corresponding number of shares on issue and

make minor adjustments to be consistent with the above material, we can quote from S&P

Indices (2010, pp. 5-6) as follows:

“The formula to calculate the S&P 500 is:

(9) i i

i

p qIndex .

Divisor

=

The numerator on the right-hand side of is the price of each stock in the index multiplied by the number of shares used in the index calculation. This is summed across all stocks in the index. The denominator is the divisor…

The formula is created by a modification of the Laspeyres index, which uses base-period quantities (share counts) to calculate the price change. A Laspeyres index would be

(10) i1 i0

i

i0 i0i

p qIndex .

p q

=

In the modification to (10), the quantity measure in the numerator, 0q , is replaced

by 1q , so the numerator becomes a measure of the current market value, and the

product in the denominator is replaced by the divisor which both represents the initial market value and sets the base value for the index. The result of these modifications is equation (9) above.”

7

With it itp q now the market capitalisation of the company and ni 1t it itM p q total market

capitalisation, the above discussion makes clear that the stock-price index is the ratio of

capitalisation in period 1 relative to that in period 0:

(11) 11 n

S 00 i1 i1i00

i 1 i0 i0

p qMP w .

M p q

This is somewhat like the Laspeyres index (1), but with values, it itp q , playing the role of prices,

itp . Obviously, if the number of shares on issue is unchanged, i1 i0q q , then (11) and (1) are

identical. If the number of shares does change, we could control for this by defining the adjusted

value ratio,

i

i0i1 i1 i1 i1 i1

i0 i0 i0 i0 i1 i0dq 0

qp q p q p,

p q p q q p

so that, if 1 nq , ,q q is a vector of the number of shares, index (11) takes the form

(11 ) 11 ndS 00 i1

d i00i 1 i0

M pP w ,

M p

q=0

q=0

which is now exactly the same as Laspeyres (1). Note that it follows from equations (5) and (11)

that S L P P L F FP P Q P Q P Q , from which it is clear that the share “price” index SP can be

decomposed into more conventional price and volume indices.

An alternative way of writing index (11) is

11 11 n n

S 01i0 i1 i1 i1in00

i 1 i 1i0 i0j0 j0

j 1

p q p pMP w ,

M p pp q

where 01 nj 1i i0 i1 j0 j0w p q p q . While the terms 01

iw are all positive, unlike the earlier weights

ttiw , they do not have a unit sum, so the index is not a weighted average of the price relatives.

But if for each company, the number of shares on issue is held constant, then 01 00i iw w , and

index 11 coincides with the Laspeyres form, equation (11 ) .

8

Geometric Returns

The above indices deal with the price relatives 11 10 n1 n0p p , , p p , which are related to

percentage changes, or percentage returns of shares. This relationship is

i1 i0i1i i

i0 i0

p pp1 g with g ,

p p

the proportionate growth rate,

so that ig 100 is the percentage return. Whilst widely used, percentage returns have the defect of

being asymmetric, that is, i1 i0 i0 i0 i1 i1p p p p p p . Thus, if, for example, the price first

increases by ig 100 percent and then falls by the same ig 100 percent, then it does not arrive

back at its original value; that is, 2i0 i i i0 i i0p 1 g 1 g p 1 g p . An attractive way to avoid

this problem is to measure returns geometrically, or logarithmically, i i1 i0Dp log p p . Unlike

percentage returns, this log-change formulation is symmetric, so that

i0 i i i0exp log p Dp Dp p . The relationship between the percentage return and the log-change

is i ig 100 exp Dp 1 100, from which it follows that i ig Dp and using log z z -1 when

z 1, i ig Dp for “small” ig .2

Due to the advantages of measuring returns logarithmically, it is natural to consider an

index of these returns. As the precise specification of the weights is not important for the present

discussion, the time subscript can be omitted, so it is sufficient to denote the weight of security i

in the index by iw , with n

ii 1w 1.

Define weighted arithmetic and geometric indices of

returns as

(12) iw

nni1 i1

A i Gi 1 i 1i0 i0

p pP w , P = .

p p

2 At a more formal level, Tornqvist et al. (1985) establish that within the class of measures of relative changes, the

log difference has strong claims to priority. They show that the log difference is the only symmetric, additive and normed indicator of relative change. In the context of relative changes, these properties have the following

meanings. Define an indicator of the relative difference between the two positive numbers x and y as H y x such

that H y x 0 iff y x 1 ; H y x 0 iff y x 1 ; H y x 0 iff y x 1 ; and H is a continuous

increasing function in y x . Then this indicator is symmetric iff H y x H x y . Next, suppose in addition to

the change x y , we have the further change y z . The indicator H is then said to be additive iff it can be

expressed as the sum of the indicator of the two intermediate differences; that is, iff H z x H y x H z y .

Finally, H is normed iff its derivative at y x 1 is unity; that is, iff H 1 1 . The last property rules out the

multiplication of the indicator function by a scaling factor. For further details, see Tornqvist et al. (1985).

9

Define the proportionate deviation of i1 i0p p from its weighted arithmetic mean AP as

i1 i0 Ai

A

p p Pr ,

P

with

n

i ii 1

w r 0.

This ir is the return on security i relative to the market. We also define the weighted variance of

these relative returns as

2n n n

2 2i i i i i i

i 1 i 1 i 1

w r w r w r ,

which is a measure of the dispersion of returns. It then follows that the geometric index can be

expressed in logarithmic form as

n n

G i A i A i ii 1 i 1

logP w log P 1 r log P w log 1 r .

Using the second-order approximation that for small x, 21log 1 x x x ,

2 the last term on the

far right of the above equation can be expressed as

n n

2 2i i i i i

i 1 i 1

1 1w log 1 r w r r .

2 2

Accordingly, we have

(13) 2G A

1logP log P

2 or

21G 2

A

Pe .

P

This establishes that the geometric index is never greater than its arithmetic counterpart. This

agrees with the theorem that states that the geometric mean is never greater than the arithmetic

mean. Result (13) establishes that the divergence between the two indices will be larger, the more

is the dispersion of returns about the overall index.3 This result is consistent with Figure 1, which

highlighted the role of disproportionate price behavior as the source of uncertainty of the overall

index.

3 Interestingly, result (13) is of the same form as the expression for the mean of a log-normally distributed random

variable x 0 : If y log x and 2y ~ N , , where exp and 2exp are the geometric mean and variance,

then the logarithm of the arithmetic mean is 2log E x 2 , or 2log GM log AM 2, which is the

same as (13). Another similarity is that as p y y dy, where p y is the log normal density, the mean is

of the weighted variety, with p y as weights. Accordingly, the log normal case and the index G

logP defined in

equation (12) are both weighted means. But note the fundamental differences between the two approaches. First, result (13) is nonparametric as it does not depend on the assumption that returns are log normal. Second, result (13) is based on a second-order approximation, whereas there is no approximation for the log normal case.

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3. Application to Shares Prices

In this section, we use the data underlying the S&P/ASX20 from Datastream to develop a

stochastic stock price index.4 We model the change of the price index for the transition from t-1

and t by regressing the change in market capitalisation for stock i,

tt t 1,t 1 it it i,t 1 i,t 1M M p q p q , on capitalisation in period t-1:5

(14) it it i,t 1 i,t 1 t i,t 1 i,t 1 itp q p q p q , i 1, , n,

where the disturbance term it satisfies 2

it it jt t i ,t 1 i,t 1 ijE( ) 0,cov( , ) p q , with 2t a variance

parameter and ij the Kronecker delta. The parameter t in equation (14) is the proportionate

change in the index. Dividing both sides of (14) by S

it i,t 1 i,t 1x p q gives

(15) S Sit t it ity x , i 1, , n ,

where Sit it it i,t 1 i,t 1 i,t 1 i,t 1y p q p q p q , it it i ,t 1 i,t 1p q ,with itE( ) 0 and

2

it jt t ijcov( , ) .

The least-squares estimate of the index is

nit it i ,t 1 i,t 1 i,t 1 i,t 1t 1,t 1 t 1,t 1

t i i ni 1

i,t 1 i,t 1 i,t 1 i,t 1i 1

p q p q p qˆ w , with w .

p q p q

In words, the estimated index is a weighted average of the changes in the value of each of the n

stocks, where the weights are the market capitalisation shares. This is of the form of index (11).

The sampling variance is

22 n

it it i,t 1 i,t 12 t 1,t 1ttt tt i t

i 1 i,t 1 i,t 1

ˆ p q p qˆˆ ˆvar , with w .n 1 p q

Thus, the variance is proportional to 2t

ˆ , which is a weighted variance of the proportionate value

changes, where the weights are the same as before and the factor of proportionality is 1 n 1 .

This shows that the estimated index is less well determined when there is more disproportionate

movement in the values of the individual share. These are attractively simple interpretations of

the estimates.

4 The S&P/ASX20 is a stock market index of the 20 largest stocks by market capitalisation listed on the Australian Securities Exchange. For the purpose of this study, we use the replicated index calculated from the share price and quantity data of constituent companies. We have made adjustments that occur from company addition and deletion and changes in the investable weight factor (i.e., the percentage of total shares on issue available to ordinary investors), but not capital changes such as share buybacks, secondary issues, right issues etc. The replicated index is reasonably close to the published S&P/ASX20 index. 5 Selvanathan and Selvanathan (2006) use a similar model for the measurement of inflation.

11

We use daily data underlying the S&P/ASX20 index, so that n=20, for the period

02/01/2003 to 28/12/2008 and split this into 12 sub-periods.6 We could estimate the n equations

in (15) jointly as a system of using all n×T observations, with T being the number of days, but

this is identical to estimating each equation separately. To test the null hypothesis that the error

terms are homoscadastic, we apply the Breusch and Pagan (1979) test. The results show that for

each day in the sample, homoscedasticity is not rejected, which is not surprising given that the

error term it in model (15) is scaled by the squared root of the market capitalisation.

To visualise the results, we present in the Panel A of Figure 2 a “fan chart” of the

estimates of t . The darker line in the middle gives the point estimates. Around this darker line

are the ± k standard-error bands, where k 0.5 , 1, 1.5 and 2. Panel B gives the blow-up of the

final six months of the sample period, while Panel C is a blow-up for December 2008. As can be

seen from Panel C, the standard errors of the t -estimates for some days, such as December 8

and December 31 of 2008, are smaller, while those for some other days, e. g., December 15 2008,

are much larger, reflecting dispersion changes over time. From the change in the index, we can

compute the corresponding index level S St t 1 tˆP P (1 ) , S S

t t 1 tˆvar P P var , where St 1P is the

S&P/ASX20 index on the previous period, which is treated as a constant. Figure 3 is the

corresponding fan-chart.

4. Redundant Stocks

In this section, we show how the approach can be employed to identify redundant shares,

those that add little to the overall index. The ability to use a subset of stocks that mimic the index

is of considerable practical interest that could lead to a saving in trading costs.

The estimates of index changes were based on the model (15). For ease of notation, we now

drop the “s” superscript. To examine the influence of share j on the index, we modify model (15)

as

it t it jt it it ity x x D , i 1, , n,

where itD 1 if i j , 0 otherwise. If stock j does not exert much influence on the index for day t,

the estimate of jt is insignificantly different from zero. If the estimates of jt for all t are

jointly insignificant, then stock j is declared to be “redundant” for the index. Thus, investment

6 Sub-periods are necessary due to company additions and deletions in the S&P/ASX20 index and bonus issues. The starting dates of the 12 sub-periods are 2 January 2003, 1 July 2003, 5 July 2004, 19 September 2005, 19 December 2005, 24 July 2006, 18 September 2006, 3 July 2007, 12 November 2007, 24 December 2007, 22 September 2008 and 18 November 2008. For the transition from the end of one period to the start of the next, due to the differences in the composition of index or in the number of shares on issue, we use the S&P/ASX20 index change.

12

funds that “track the index” can employ a simpler approach by excluding this stock from their

version of “the market portfolio”.7

To test the hypothesis that j1 j2 jT , we use 2 2Tt 1 jt jt

ˆ ˆ/ (SE ) , which is distributed

as 2 (T) . The p-values of the test statistics are presented in Table 1. We can see from the column

for sub-period 1 the p-values associated with 13 shares are greater than 5 percent, so we cannot

reject the null; these 13 stocks are redundant for this sub-period. The last row of the table gives

the percentage of redundant stocks for each sub-period, which ranges from 55 to 75 percent.

Thus, efficiency can be achieved by excluding more than half the stocks and concentrating on the

remainder. The final column of the table indicates that stocks 5, 9, 12, 14, 15, 16, 19, 25, 29, and

32 are insignificant in all sub-periods and can be safely excluded from the index.

5. Summary and Conclusion

The stochastic approach to index numbers is one of four approaches to index numbers,

and has applied successfully to the measurement of inflation, world interest rates and

international competitiveness.8 In this paper, we have shown how the stochastic approach can be

used to obtain a stock price index. Uncertainty plays a central role in the approach which gives

the whole distribution of the index, not just one single number. In time of turbulence in the

market when individual stocks move disproportionately, the standard error of the index is higher,

reflecting the greater uncertainty of the average change in prices. The paper also shows how

stochastic approach can be used to identify those stocks that do not add substantially to the

overall index, which we term “redundant”. This has the interesting implication that it is possible

to safely exclude these for index-tracking purposes.

7 Giles and McCann (1994) use this approach in the context of inflation measurement. 8 See Clements and Izan (1987), Ong et al. (1999) and Clements et al. (2009).

13

References

Balk, B. M. (1980). "A Method for Constructing Price Indices for Seasonal Commodities."

Journal of the Royal Statistical Society A 143: 68-75.

Breusch, T. and A. Pagan (1979). "A Simple Test for Heteroscedasticity and Random Coefficient

Variation." Econometrica: Journal of the Econometric Society 47: 1287-94.

Clements, K. W. and H. Y. Izan (1981). "A Note on Estimating Divisia Index Numbers."

International Economic Review 22: 745-47.

Clements, K. W. and H. Y. Izan (1987). "The Measurement of Inflation: A Stochastic Approach."

Journal of Business & Economic Statistics 5: 339-50.

Clements, K. W., H. Y. Izan and Y. Lan (2009). "A Stochastic Measure of International

Competitiveness." International Review of Finance 9: 51-81.

Giles, D. and E. McCann (1994). "Price Indices: Systems Estimation and Tests." Journal of

Quantitative Economics 10: 219-25.

Ong, L. L., K. W. Clements and H. Y. Izan (1999). “The World Real Interest Rate: Stochastic Index

Number Perspectives.” Journal of International Money and Finance 18: 225-49.

S&P Indices (2010). "Index Mathematics: Index Methodology." Available at

https://www.spindexdata.com/idpfiles/citigroup/prc/active/whitepapers/Methodology_Ind

ex_Math_Web.pdf. Retrieved on 28 October 2010.

Selvanathan, E. A. and D. S. Prasada Rao (1994). Index numbers: a stochastic approach, London:

Macmillan.

Selvanathan, E. A. and S. Selvanathan (2006). "Measurement of Inflation: An Alternative

Approach." Journal of Applied Economics 9: 403-18.

Tornqvist, L., P. Vartia and Y. O. Vartia (1985). "How Should Relative Changes be Measured?"

The American Statistician 39: 43-46.

14

FIGURE 1

THE INDEX-NUMBER PROBLEM

A. Constant Relative Prices

Scatter Plot Time-series Plot  (Base-period prices normalised to unity)

Frequency Distribution

B. Variable Relative Prices

10

1

1+ g

Period

Priceitp

Period

Priceitp

1

3

4

5

10

1

ˆ1 g 2

1 g

Number

i1 i0p p

1 g i1 i0p p

Number

Base-period price i0p

1

2 3

4 5

i1 i0p (1 g)p

Current-period price i1p

Base-period price i0p

1 2 3

4

5

i1 i0ˆp (1 g)p + error

Current-period price i1p

15

FIGURE 2

SPI AND CONFIDENCE BANDS: CHANGES CHANGES

-20

-10

0

10

20

3/01/2003 15/07/2003 20/01/2004 27/07/2004 2/02/2005 11/08/2005 17/02/2006 28/08/2006 6/03/2007 12/09/2007 20/03/2008 26/09/2008

-14

-10

-6

-2

2

6

10

3/07/2008 7/08/2008 11/09/2008 16/10/2008 20/11/2008 29/12/2008

-10

-6

-2

2

6

10

1/12/2008 8/12/2008 15/12/2008 22/12/2008 31/12/2008

Change (%) Panel A

Panel B

Panel C

Date

Date

Date

16

1500

2000

2500

3000

3500

4000

4500

5000

3/01/2003 15/07/2003 20/01/2004 27/07/2004 2/02/2005 11/08/2005 17/02/2006 28/08/2006 6/03/2007 12/09/2007 20/03/2008 26/09/2008

1900

2400

2900

3400

3900

3/07/2008 7/08/2008 11/09/2008 16/10/2008 20/11/2008 29/12/2008

2700

2800

2900

3000

3100

1/12/2008 8/12/2008 15/12/2008 22/12/2008 31/12/2008 Date

Date

Date

Level

FIGURE 3 SPI AND CONFIDENCE BANDS: LEVELS

Panel A

Panel B

Panel C

17

TABLE 1 REDUNDANT STOCKS, 2003-2008

Company

p-values for sub-period Proportion of time when p>5%

(Percentage) 1 2 3 4 5 6 7 8 9 10 11 12

1. Alumina 0 7 0 0 25 2. AMCOR 0 0 0 3. AMP 100 100 44 0 72 1 0 0 0 0 0 0 33 4. ANZ Banking Group 81 100 1 96 0 38 44 33 73 97 21 98 83 5. BHP Billiton 100 100 100 100 100 100 100 100 100 100 100 100 100 6. Brambles 57 100 100 25 0 0 0 57 7. Coles Group 0 0 32 1 4 100 100 100 50 8. Commonwealth Bank of Australia 100 100 100 58 0 83 0 19 55 100 94 100 83 9. CSL 49 47 98 7 60 100

10. Foster's Group 0 100 72 0 0 100 1 0 1 0 0 0 25 11. Macquarie Group 99 98 0 1 100 100 100 9 5 67 12. Macquarie Infrastructure Group 81 26 100 13. National Australia Bank 100 100 100 100 68 2 82 70 100 100 100 38 92 14. Newcrest Mining 83 49 100 15. News Corporation 100 100 100 100 16. News Corporation Preference Shares 100 100 100 100 17. Origin Energy 0 0 18. QBE Insurance Group 2 0 56 71 100 12 19 45 3 70 70 19. Rinker Group 100 100 86 100 100 20. Rio Tinto 0 11 0 2 100 9 100 100 100 100 100 100 75 21. St. George Bank 0 0 0 0 0 0 96 0 0 26 0 18 22. Stockland 8 0 50 23. Suncorp-Metway 0 0 7 100 0 0 0 28 0 33 24. Sydney Roads Group 0 0 25. Telstra Corporation 99 100 100 100 100 98 100 89 100 100 100 100 100 26. Wesfarmers 88 0 0 3 0 11 0 0 1 10 27 1 33 27. Wesfarmers PPS 0 0 28. Westfield Holdings 0 0 0 29. Westfield Group 100 100 100 99 100 77 100 69 100 67 100 30. Westfield Trust 0 0 0 31. Westpac Banking Corporation 100 100 91 100 2 2 71 7 100 73 35 100 83 32. Woodside Petroleum 98 98 100 100 100 100 93 58 100 100 20 52 100 33. Woolworths 26 0 0 100 26 71 99 88 66 98 1 1 67 Total number of companies in sub-period 20 20 20 20 20 21 20 20 21 20 20 20 20

Percent of redundant stocks 65 65 60 55 60 71 75 75 71 75 70 60 0

Notes: 1. See footnote 6 for information rergarding sub-periods.

2. Blanks indicate that the stock in question is excluded by Standard and Poor’s in the S&P20 index.

18

ECONOMICS DISCUSSION PAPERS

2009

DP NUMBER

AUTHORS TITLE

09.01 Le, A.T. ENTRY INTO UNIVERSITY: ARE THE CHILDREN OF IMMIGRANTS DISADVANTAGED?

09.02 Wu, Y. CHINA’S CAPITAL STOCK SERIES BY REGION AND SECTOR

09.03 Chen, M.H. UNDERSTANDING WORLD COMMODITY PRICES RETURNS, VOLATILITY AND DIVERSIFACATION

09.04 Velagic, R. UWA DISCUSSION PAPERS IN ECONOMICS: THE FIRST 650

09.05 McLure, M. ROYALTIES FOR REGIONS: ACCOUNTABILITY AND SUSTAINABILITY

09.06 Chen, A. and Groenewold, N. REDUCING REGIONAL DISPARITIES IN CHINA: AN EVALUATION OF ALTERNATIVE POLICIES

09.07 Groenewold, N. and Hagger, A. THE REGIONAL ECONOMIC EFFECTS OF IMMIGRATION: SIMULATION RESULTS FROM A SMALL CGE MODEL.

09.08 Clements, K. and Chen, D. AFFLUENCE AND FOOD: SIMPLE WAY TO INFER INCOMES

09.09 Clements, K. and Maesepp, M. A SELF-REFLECTIVE INVERSE DEMAND SYSTEM

09.10 Jones, C. MEASURING WESTERN AUSTRALIAN HOUSE PRICES: METHODS AND IMPLICATIONS

09.11 Siddique, M.A.B. WESTERN AUSTRALIA-JAPAN MINING CO-OPERATION: AN HISTORICAL OVERVIEW

09.12 Weber, E.J. PRE-INDUSTRIAL BIMETALLISM: THE INDEX COIN HYPTHESIS

09.13 McLure, M. PARETO AND PIGOU ON OPHELIMITY, UTILITY AND WELFARE: IMPLICATIONS FOR PUBLIC FINANCE

09.14 Weber, E.J. WILFRED EDWARD GRAHAM SALTER: THE MERITS OF A CLASSICAL ECONOMIC EDUCATION

09.15 Tyers, R. and Huang, L. COMBATING CHINA’S EXPORT CONTRACTION: FISCAL EXPANSION OR ACCELERATED INDUSTRIAL REFORM

09.16 Zweifel, P., Plaff, D. and

Kühn, J.

IS REGULATING THE SOLVENCY OF BANKS COUNTER-PRODUCTIVE?

09.17 Clements, K. THE PHD CONFERENCE REACHES ADULTHOOD

09.18 McLure, M. THIRTY YEARS OF ECONOMICS: UWA AND THE WA BRANCH OF THE ECONOMIC SOCIETY FROM 1963 TO 1992

09.19 Harris, R.G. and Robertson, P. TRADE, WAGES AND SKILL ACCUMULATION IN THE EMERGING GIANTS

09.20 Peng, J., Cui, J., Qin, F. and

Groenewold, N.

STOCK PRICES AND THE MACRO ECONOMY IN CHINA

09.21 Chen, A. and Groenewold, N. REGIONAL EQUALITY AND NATIONAL DEVELOPMENT IN CHINA: IS THERE A TRADE-OFF?

19

ECONOMICS DISCUSSION PAPERS

2010

DP NUMBER

AUTHORS TITLE

10.01 Hendry, D.F. RESEARCH AND THE ACADEMIC: A TALE OF TWO CULTURES

10.02 McLure, M., Turkington, D. and Weber, E.J. A CONVERSATION WITH ARNOLD ZELLNER

10.03 Butler, D.J., Burbank, V.K. and

Chisholm, J.S.

THE FRAMES BEHIND THE GAMES: PLAYER’S PERCEPTIONS OF PRISONER’S DILEMMA, CHICKEN, DICTATOR, AND ULTIMATUM GAMES

10.04 Harris, R.G., Robertson, P.E. and Xu, J.Y. THE INTERNATIONAL EFFECTS OF CHINA’S GROWTH, TRADE AND EDUCATION BOOMS

10.05 Clements, K.W., Mongey, S. and Si, J. THE DYNAMICS OF NEW RESOURCE PROJECTS A PROGRESS REPORT

10.06 Costello, G., Fraser, P. and Groenewold, N. HOUSE PRICES, NON-FUNDAMENTAL COMPONENTS AND INTERSTATE SPILLOVERS: THE AUSTRALIAN EXPERIENCE

10.07 Clements, K. REPORT OF THE 2009 PHD CONFERENCE IN ECONOMICS AND BUSINESS

10.08 Robertson, P.E. INVESTMENT LED GROWTH IN INDIA: HINDU FACT OR MYTHOLOGY?

10.09 Fu, D., Wu, Y. and Tang, Y. THE EFFECTS OF OWNERSHIP STRUCTURE AND INDUSTRY CHARACTERISTICS ON EXPORT PERFORMANCE

10.10 Wu, Y. INNOVATION AND ECONOMIC GROWTH IN CHINA

10.11 Stephens, B.J. THE DETERMINANTS OF LABOUR FORCE STATUS AMONG INDIGENOUS AUSTRALIANS

10.12 Davies, M. FINANCING THE BURRA BURRA MINES, SOUTH AUSTRALIA: LIQUIDITY PROBLEMS AND RESOLUTIONS

10.13 Tyers, R. and Zhang, Y. APPRECIATING THE RENMINBI

10.14 Clements, K.W., Lan, Y. and Seah, S.P. THE BIG MAC INDEX TWO DECADES ON AN EVALUATION OF BURGERNOMICS

10.15 Robertson, P.E. and Xu, J.Y. IN CHINA’S WAKE: HAS ASIA GAINED FROM CHINA’S GROWTH?

10.16 Clements, K.W. and Izan, H.Y. THE PAY PARITY MATRIX: A TOOL FOR ANALYSING THE STRUCTURE OF PAY

10.17 Gao, G. WORLD FOOD DEMAND

10.18 Wu, Y. INDIGENOUS INNOVATION IN CHINA: IMPLICATIONS FOR SUSTAINABLE GROWTH

10.19 Robertson, P.E. DECIPHERING THE HINDU GROWTH EPIC

10.20 Stevens, G. RESERVE BANK OF AUSTRALIA-THE ROLE OF FINANCE

20

10.21 Widmer, P.K., Zweifel, P. and Farsi, M. ACCOUNTING FOR HETEROGENEITY IN THE MEASUREMENT OF HOSPITAL PERFORMANCE

10.22 McLure, M. ASSESSMENTS OF A. C. PIGOU’S FELLOWSHIP THESES

10.23 Poon, A.R. THE ECONOMICS OF NONLINEAR PRICING: EVIDENCE FROM AIRFARES AND GROCERY PRICES

10.24 Halperin, D. FORECASTING METALS RETURNS: A BAYESIAN DECISION THEORETIC APPROACH

10.25 Clements, K.W. and Si. J. THE INVESTMENT PROJECT PIPELINE: COST ESCALATION, LEAD-TIME, SUCCESS, FAILURE AND SPEED

10.26 Chen, A., Groenewold, N. and Hagger, A.J. THE REGIONAL ECONOMIC EFFECTS OF A REDUCTION IN CARBON EMISSIONS

10.27 Siddique, A., Selvanathan, E.A. and Selvanathan, S.

REMITTANCES AND ECONOMIC GROWTH: EMPIRICAL EVIDENCE FROM BANGLADESH, INDIA AND SRI LANKA

21

ECONOMICS DISCUSSION PAPERS

2011 DP NUMBER

AUTHORS TITLE

11.01 Robertson, P.E. DEEP IMPACT: CHINA AND THE WORLD ECONOMY

11.02 Kang, C. and Lee, S.H. BEING KNOWLEDGEABLE OR SOCIABLE? DIFFERENCES IN RELATIVE IMPORTANCE OF COGNITIVE AND NON-COGNITIVE SKILLS

11.03 Turkington, D. DIFFERENT CONCEPTS OF MATRIX CALCULUS

11.04 Golley, J. and Tyers, R. CONTRASTING GIANTS: DEMOGRAPHIC CHANGE AND ECONOMIC PERFORMANCE IN CHINA AND INDIA

11.05 Collins, J., Baer, B. and Weber, E.J. ECONOMIC GROWTH AND EVOLUTION: PARENTAL PREFERENCE FOR QUALITY AND QUANTITY OF OFFSPRING

11.06 Turkington, D. ON THE DIFFERENTIATION OF THE LOG LIKELIHOOD FUNCTION USING MATRIX CALCULUS

11.07 Groenewold, N. and Paterson, J.E.H. STOCK PRICES AND EXCHANGE RATES IN AUSTRALIA: ARE COMMODITY PRICES THE MISSING LINK?

11.08 Chen, A. and Groenewold, N. REDUCING REGIONAL DISPARITIES IN CHINA: IS INVESTMENT ALLOCATION POLICY EFFECTIVE?

11.09 Williams, A., Birch, E. and Hancock, P. THE IMPACT OF ON-LINE LECTURE RECORDINGS ON STUDENT PERFORMANCE

11.10 Pawley, J. and Weber, E.J. INVESTMENT AND TECHNICAL PROGRESS IN THE G7 COUNTRIES AND AUSTRALIA

11.11 Tyers, R. AN ELEMENTAL MACROECONOMIC MODEL FOR APPLIED ANALYSIS AT UNDERGRADUATE LEVEL

11.12 Clements, K.W. and Gao, G. QUALITY, QUANTITY, SPENDING AND PRICES

11.13 Tyers, R. and Zhang, Y. JAPAN’S ECONOMIC RECOVERY: INSIGHTS FROM MULTI-REGION DYNAMICS

11.14 McLure, M. A. C. PIGOU’S REJECTION OF PARETO’S LAW

11.15 Kristoffersen, I. THE SUBJECTIVE WELLBEING SCALE: HOW REASONABLE IS THE CARDINALITY ASSUMPTION?

11.16 Clements, K.W., Izan, H.Y. and Lan, Y. VOLATILITY AND STOCK PRICE INDEXES

11.17 Parkinson, M. SHANN MEMORIAL LECTURE 2011: SUSTAINABLE WELLBEING – AN ECONOMIC FUTURE FOR AUSTRALIA