Metals Market Efficiency in Relation
to Foreign Exchange and Financial Markets
Christopher L. Gilbert
Division Working Paper No. 1987-9
October 1987
International Commodity Markets Division
International Economics Department
The World Bank
Division Workine Papers report on work in progress and are
circulated to stimulate discussion and comment,
METALS MARKET EFFICIENCY IN RELATION TO FOREIGN EXCHANGE
AND FINANCIAL MARKETS
Christopher L. Gilbert
(Consultant)
October 1987
The World Bank does not accept responsibility for the views expressed herein
which are those of the author and should not be attributed to the World Bank
or to its affiliated organizations. The findings, interpretations, and
conclusions are -the results of research supported by the Bank; they do not
necessarily represent official policy of the Bank. The designations employed,
the presentation of material, and any maps used in this document are solely
for the convenience of the reader and do not imply the expression of any
opinion whatsoever on the part of the World Bank or its affiliates concerning
the legal status of any country, territory, city, area, or of its authorities,
or concerning the delimitation of its boundaries, or national affiliation.
TABLE OF CONTENTS
SUMMARY ................................................................1iii
I. INTRODUCTION ...................................................... 1
II. THE IMPACT OF EXCHANGE RATE CHANCES ON COMMODITY PRICES ...........5
III. DATA AND VARIABLE CONSTRUCTION...................................11
IV. ECONOMETRIC STRATEGY ............................................. 16
V. RESULTS ..........................................................21
VI. CHOICE OF EXCHANGE RATE INDEX .................................... 27
VII. CONCLUSIONS ......................................................30
REFERENCES.;............................................................39
Table
1 Estimates on the Null Hypothesis .................................32
'2 Tests for Departures from Market Efficiency ......................33
3 Price Innovation Relationships--OLS Estimates ....................34
4 Price Innovation Relationships--ARCH-OLS Estimates ............... 35
5 Innovation Relationships--SUR Estimates .......................... 36
6 Price Innovation Relationships--ARCH-SUR Estimates ................ 3'
7 Choice of Exchange Rate Index .................................... 38
SUMMARY
The paper reports on the results of an extension of the testing of
the efficiency of metal trades at the London Metal Exchange (LME). Earlier
work by the author pointed to the importance of financial market variables
such as interest rates, exchange rates and inflation to primary commodity
markets--because they affect the terms on which futures or forward traders
will be prepared to hold title to the commodity--and modelled these
relationships. This theoretical development allows an extension of the concept
of futures or forward market efficiency to include the potential for arbitrage
between commodity prices and interest rates and exchange rates. A second point
of interest is a revisiting of the question of the size of the response of
metal prices to exchange rate changes. Earlier empirical analysis by the
author had produced results which indicated, contrary to theory, that the
elasticity of primary commodity prices to exchange rate changes was greater
than one.
The period of study covered LME forward prices over the period 1978-
85 and included seven metals. Previous tests of market efficiency by the
author, using weak-form efficiency tests, had concluded that there was no
evidence of departures from efficiency in LME copper trading, but that
evidence for lead, tin and zinc was mixed. The tin market appeared to exhibit
a significant risk premium while in lead, and perhaps also in zinc, current
prices could be- predicted from previous prices. This paper extends - the.
analysis to cover aluminum, nickel and silver and to include semi-strong form
tests of efficiency of the LME in relation to foreign exchange and treasury
Sill markets.
In light of the tests carried out, none of the LME metals markets
appear to conform to the efficient market paradigm. There is clear evidence of
weak-form inefficiency through lagged dependence in the case of aluminum and
possibly also copper, lead and zinc; there is somewhat weaker evidence of bias
in aluminum, nickel, tin and zinc; there is evidence that the lagged forward
premium (backwardation) has predictive value in aluminum and lead; and some
evidence that exchange rate changes (and possibly also changes in inflation)
are not transmitted efficiently to metals prices in the case of aluminum and
possibly also lead, silver, tin and zinc.
Aluminum, nickel and zinc are seen as having negative bias over this
sample period, implying a preponderance of short hedging (typically associated
with producer sales), while tin provides evidence of positive bias. This
latter result may well be the consequence of very heavy forward support
activities by the International Tin Council during the period.
The exchange rate response parameters all take the predicted sign.
The long-run elasticities are distributed fairly widely around the theoretical
value of unity. Aluminum, nickel and zinc show much lower elasticities, which
is consistent with producer pricing on a dollar basis (moreover, these three
markets are dominated by a few large producers); while lead, silver and tin
appear oversensitive to changes in the value of the dollar (i.e., elasticity
- iv -
much larger than one). Copper alone has an estimated value close to the
theoretical value of one.
The interest, inflation and activity innovation effects are
relatively poorly defined, except in the case of copper. It is notable that
the activity variable, industrial production, is insignificant in the tin
equation; this again may demonstrate the effects of international support of
tin prices.
Tests using differently constructed exchange rate indices yielded
very different results. Theory suggests that in a multicommodity model the
weights in the index will be complicated functions of all own and cross demand
and supply elasticities and al-l market share parameters. Besides the GNP-
weighted index used in the major part of the study other constructions of the
exchange rate index included (i) the IMF's MERM index in which the weights
reflect the weights of each currency in US overseas trade; and (ii) commodity-
specific indices using weights proportional to, respectively, consumption of
the commodity in each country, production of the commodity and production plus
consumption. The commodity-specific indices gave the best fit with weighting
by consumption shares giving superior results to weighting by either
production shares or by an average of the two. The choice of the index had a
significant effect on the size of the elasticity of the response of commodity
prices to exchange rate changes. Use of the MERM index gave rise to high
elasticities, reflecting the fact that the use of trade weights gives a
particularly high. impact to currencies (in particular to the Canadian dollar)
which vary little in relation to the US dollar. Since the MERM index shows
relatively less variation than the other indices, the variation that exists
must take a higher weight in the estimated regression.
The size of the elasticity parameters should be regarded with some
caution as they have been estimated in relationships which omit most of the
fundamental determinants of price. This approach is reasonable as the study
was mainly a test of market efficiency. In further work it may be useful to
undertake the analysis using daily data which should strengthen the results on
the question of the efficiency of these markets.
I. INTRODUCTION
1. Many important primary commodities are traded on forward or futures
exchanges. These commodities may be regarded as either physical or financial
assets. Producers and consumers are concerned with physical. properties of the
commodity (location, purity, flavor, etc.), while futures traders are
concerned with likely changes in the commodity price and, in particular, with
movements in the basis (i.e. in the relative price of nearby and more distant
futures). One should therefore expect that commodity prices will be affected
not only by those factors which affect the supply and demand of the physical
commodity (Cfundamentals"), but also by financial market variables which
affect the terms on which futures traders will be prepared to hold title to
the commodity. But despite this elementary observation, scant attention has
been paid to financial mar'.et variables in the commodity modeling literature.
2. There are three sets of financial market variables that will, in
principle, be important in considering commodity price dynamics. These are
exchange rates, interest rates and inflation rates. The relationship between
exchange rates and commodity prices was discussed in an equilibrium model by
Ridler and Yandle (1972). That analysis was extended to include inflation in
Gilbert (1973). However, neither of those models apply directly to markets in
which stocks are carried and where interest rate changes will also be
important. This extension was made in Gilbert (1985). A rise in interest rates
* The work reported in this paper was completed while I was employed as a
consultant in the Commodity Studies and Projections Division of the World
Bank. I am grateful to Ron Duncan for encouragement and to Manuel
Arellano, David Hendry and Harold Cataquet for comments. The current
version of the paper has benefited from -seminar discussions at the World
Bank and at the Oxford Quantitative Economics and Finance workshops.
-2-
leads to a fall in asset prices as the given dividend stream is required to
generate a higher yield. The same should be true in commodity markets where
stock holders will be looking for a higher convenience yield. Gilbert (1985)
also shows that if futures traders are risk neutral and if futures markets are
unbiased commodity prices will respond only to the unanticipated components
(the "innovations") of the exchange rate and inflation movements.
3. This allows an extension of the concept of futures market efficiency.
A futures market is regarded as efficient if it is not possible to devise a
trading rule based on a specified information set that will have positive
expected profitability. In weak form efficiency tests, 1/ the information set
is confined to the past price history of the asset price. Here, evidence of
bias or of lagged dependence provides prima facie evidence of inefficiency,
although there is no certainty that ihe implied trading rule would have
covered transactions costs, or, in thin markets, that trades could have been
made in sufficient volumes at the quoted prices. The results derived in
Gilbert (1985) allow extension to semistrong form efficiency tests in which
the information set is extended to include exchange rates, interest rates and
inflation rates. Here one is asking whether it is possible to devise trading
rules that involve simultaneous positions in commodity and forex markets,
commodity and t-bill markets or commodity and physical markets which would
have positive expected profitability.
4. The London metals markets provide a convenient and important testing
ground for this theory. The London Metal Exchange (the LME) trades (or has
1/ See Fama (1970).
-3
recently traded) 1/ contracts in seven metals. These are silver (Ag), aluminum
(Al), copper (Cu), nickel (Ni), lead (Pb), tin (Sn) and zinc (Zn). Silver,
aluminum and copper are also traded on Comex in New York, and tin on the Kuala
Lumpur Futures Exchange (Malaysia). 2/ The LME prices of these metals
effectively constitute the free market prices, at least outside the United
States., and in copper (and previously in tin) most trade outside the United
States and the centrally planned economies takes place at or close to the LME
price.
5. Gilbert (1986a) used weak form tests to examine the efficiency of LME
trading in copper, lead, tin and zinc. He concluded that there was no evidence
of departures from efficiency in the copper market where LME trading has been
most active, but that the evidence for efficiency of the'other three markets
was more mixed. The tin market appeared to exhibit a significant risk premium,
while in lead, and perhaps also in zinc, the previous price history allowed
prediction of current movements. 3/ In this paper, we extend that analysis to
cover all seven LME metals and consider semistrong form tests of the
efficiency of the LME in relation to forex and t-bill markets.
6. These tests also allow us to throw more light on the issue of the
exchange rate response of commodity prices. The models derived in Ridler and
Yandle and Gilbert (1973, 1985) imply that the equilibrium exchange rate
elasticity of dollar commodity prices to a change in the value of the dollar
1/ Trading in the tin contract was suspended indefinitely in October 1985
after the International Tin Council default.
2/ Previously in Penang.
3/ This work was prompted by earlier discussions in Goss (1981, 1983) which
were vitiated by serious econometric problems.
-4-
will fall withirn the (0,1) interval. In practice, however, dollar commodity
prices appear to have been overresponsive to exchange rate changes during the
first half of the eighties. The model derived by Van Duyne (1979) allows the
possibility of overshooting, but does not affect the equilibrium elasticity.
Gilbert (1987) provides some evidence for the view that developing country
debt service obligations have forced supply shifts which would have the effect
of increasing this elasticity. The results reported in this paper a'low direct
estimation of the exchange rate elasticity for the LME metals from the market
price reactions to exchange rate changes.
7. The plan of the paper is as follows: in Section 2 we review the
theory relating exchange rate changes to commodity prices; in Section 3 we
discuss the data used for the efficiency tests reported in Section 5; Section
4 is devoted to econometric issues; in Section 6 we consider the implications
of alternative weighting procedures in the construction of exchange rate
indices; and Section 7 contains brief conclusions.
-5-
II. THE IMPACT OF EXCHANGE RATE CHANGES ON COMMODITY PRICES
8. We first derive the Ridler and Yandle (1972) result, as generalized
in Gilbert (1973). Write the price of commodity as p. Suppose country i's
dollar exchange rate is x (domestic units per dollar). There are n countries
with the United States taking index 1, so that xi = 1. Denote consumption of
the commodity in country i by Ci = Ci [xip/qi] where qi is the appropriate
price deflator in country i. Similarly, Let production of the commodity be
Qi =. Q'[xip/qi]. Write the country i demand and supply elasticities as ei
and c., respectively, with ei defined as positive. Let country i's share in
world consumption of the commodity be w. = C./IZC and its share in world
1 i k k
production be w. = Qi/kQk Define e and e as the weighted average demand
and supply elasticities res,t-ctively (i.e., £iwiei and iwi.)e Market
clearing, over a sufficiently long period for stock changes to be negligible,
requires
n . n
(1) z C'[xipqi] = z Ql[xip/qi]
i=1 i=l
The Ridler-Yandle result is obtained by total differentiation of (1) followed
by approximation of derivatives by first differences. This gives
n
(2) Alnp = Z vi|Alnxi - AlnqiJ
i=1
[Ridler and Yandle (1972), Gilbert (1973)]. Ridler and Yandle showed that the
weights [vi1 are given by
6-
w.e. + w. .
(3) v 1 1 -
Equations (2) and (3) may be simplified by defining (commedity specific)
exchange rate and relative price indices X and Q as
n
(4) lnX = E v.lnx.
i=2
and
n
(5) lnQ = £ v lnq
i=2
Then (2) becomes
(6) Alnp = AlnX - A1nQ
9. Note that the summation in the exchange rate and price indices X
and IT is over countries 2 to n, and thus excludes the United States. This
implies that the weights in these indices sum to 1-vl. Consider a 1000%
general appreciation of the dollar so that Alnxi = 0 for i=2,...,n. Suppose
that there are no associated changes in the levels of the price deflators.
Then (6) implies
(7) Alnp = [i-v110
and we obtain the bounding relationship 0 < Alnp < 1. The commodity price
falls by 100(1-vl)% of the dollar appreciation. Typical values, implied by
(7), for the elasticity of commodity prices with respect to changes in the
value of the dollar, would be of the order of 0.6-0.8.
-7-
10. Alternatively, suppose indices X and Q are defined with
weights vl = v./[1-v 1 which sum to unity {aver councries 2...n. Then (6)
becomes
(8) Alnp = - (1-v)[AlnX - AInQ]
which implies the same result. However, the commodity price has unit
elast city with respect to the Ridler-Yandle indices (4) and (5) defined with
weights summing (over countries 2,...,n) to 1-v , but elasticity 1-vl with
respect to the indices defined in (8) with weights summing to unity. This
implies a need for some caution in comparing the results of different studies.
In this paper we define the exchange rate indices as in (4) with the
implication that-the commodity price should exhibit a unit elasticity with
respect to changes in these indices.
11. We need to consider two extensions to the simple Ridler-Yandle model
developed above. The first is to a multicommodity world. The Ridler-Yandle
model considers the effect of a dollar devaluation on the price of (say)
copper with the price of all other commodities held constant. But it would
clearly be inconsistent to apply this result to copper while holding the price
of aluminum constant and, at the same time, to apply it to aluminum while
holding the price of copper constant. Gilbert (1987) shows that the Ridler-
Yandle result extends to a multicommodity world, and in particular that, given
gross substitutability across all traded goods, commodity price elasticities
with respect to exchange rate changes remain within the unit interval.
However, the weights [vi] implied by the exchange rate indices now depend
upon all own and cross supply and demand elasticities for all countries. This
-8-
suggests that there may be little point in attempting to construct commodity
specific exchange rate indices. We examine this suggestion in section 6e
12. The second extension is to a model in which stocks of the commodity
.are held by rational risk neutral agents. This model is discussed in Gilbert
(1985) where it is shown that the innovation (i.e. the unanticipated change)
in the commodity price is related to the unanticipated change in the Ridler-
Yandle exchange rate index X, the price index, and to a similarly weighted
index of interest rates. Assumption of covered interest parity allows the
interest rate index to be replaced by a single interest rate, and it is
natural in a model in which the US dollar is numeraire to choose the US rate.
The commrdity price change then depends on current and future expected changes
in these indices. Simple results are obtained by making appropriate constancy
assumptions. A result analogous to that developed by Ridler and Yandle follows
if changes in exchange rates, interest rates and price levels are
unanticipated. This gives
(9) Vlnp = - (VlnX - VlnQ) - aVr
(The coefficient will. depend on the expected stockholding period and may,
therefore, not be constant.) It is, however, more plausible to suppose that
the rates of inflation rather than price levels are stationary. If agents
regard all changes in inflation rates ni as unanticipated then (9) is
replaced by
(10) Vlnp = - VInX = a(Vr - VL)
where
n n
(1) I v.ir. = i v Alnqi
i=l i=l
More generally, one might model the forcing variables lnX, r and T as (nth
order) autoregressive processes
(12) xz (zt = + V (z = lnX, r, I)
where L is the lag operator Lzt = zt-l and =0 1 Vz. In this situation,
an innovation in any variable z will result in an expectation of further
changes in z over the future. The restrictions in (11) are then relaxed and
one obtains
(13) Vlnp = - 1VlnX - a2Vr + a3VII
However, if the period over which stocks are expected to be held continuously
is sufficiently long, the a coefficients in (13) may be simply expressed in
terms of the X coefficients in (12):
(14) Vlnp A VlnX - a(ArVr - AI v)
where
n
A = (1 - z -.) (z = lnX, r, i)
i=l
The expressions A zVz denote the innovations scaled by the appropriate long
run multiple. To avoid unpleasant notation we shall henceforth absorb this
scaling factor into the innovation operator V which allows us to revert to
(10). Note that the unit elasticity property is restored now that the exchange
rate innovation has been appropriately scaled.
- 10 -
13. Exchange rates, interest -rates and prices together explain only a
small proportion of commodity price movements. In a full commodity market
model one would need to model the shifts in the supply and demand functions.
In this paper, our concern is more limited and so attention is confined to the
financial market variables. However, inclusion of a demand shift (A, activity)
variable (again in scaled innovation form) increases the power. of the
efficiency tests. We therefore augment (10) as
(15) Vlnp = - VlnX - acI(Vr - V) + 2VinA
Equation (15) forms our null hypothesis. Departures from (15) imply the
possibility-of profitable trading rules.
III. DATA AND VARIABLE CONSTRUCTION
14. The LME trades metal for "prompt" (i.e., spot) and 90-day (3-month)
delivery. 1/ In this study we use a sample of 32 nonoverlapping end month LME
prices covering the period 1978-85. The selected months are January, April,
July and October--this avoids possible year-end problems. The metal price
innovation -is measured as
(16) Vlnpt = lnp lnp- I
where p f is the forward price at date t-1 for delivery at date t. 2/
~tIt-i
15. The aluminum and nickel contracts are relatively recent--taking into
account the need for a lagged forward price, data on aluminum is only
available from July 1979 and on nickel from April 1980. The tin contract was
suspended on October 24 1985--we take the price on that date as the end
October price. A longer sample would have been available if we had been
prepared to use pre-1978 data. However, forward exchange rate data, required
in the construction of the exchange rate innovations, is not easily available
before 1978; and, in any case, there is no strong reason to expect any
departures from market efficiency to be constant over long periods of time.
1/ Strictly, the 90-day contracts are forward and not futures contracts [see
Gilbert (1986a)]--they are made with an LME ring-dealing member, who acts
as principal, and not with an exchange clearing house they are canceled by
making an offsetting contract with the same delivery date with the same
broker; and they are not marked to market during the contract period.
2/ LME prices are quoted in sterling. The spot price p is converted into
dollars at the spot sterling exchange rate, and the forward price at the
three-month rate.
- 12 -
16. An alternative method of increasing the sample size would have been
to use overlapping observations--either end-month observations [see Gilbert
(1986a)] or daily observations. The use of overlapping observations in market
efficiency studies gives rise to inferential problems in the econometrics
which have been authoritatively analyzed by Hansen and Hodrick (1980). The
major difficulty with the Hansen and Hodrick procedure, noted in Gilbert
(1986a), is that it requires homoscedasticity of the innovation variance. With
financial market data, this assumption is neither theoretically reasonable nor
empirically realistic. 1/ A major advantage deriving from the use of
nonoverlapping data is the possibility of explicitly modeling the innovation
variance process. 2/
17. The exchange rate data consists of end month observations on OECD
exchange rates against the US dollar. 3/ One would wish to define the exchange
rate innovation for country j analogously to the price innovation (16):
,f
(17) Vlnx. = lnx. - lnxJ
it jt j,tlt-1
1/ See Mandelbrot (1963, 1966, 1967) and Press (1967) for theoretical
derivation of the distribution of asset price changes; and Engle and
Bollerslev (1986) for a survey of financial market applications of the
ARCH heteroscedasticity model.
2/ In principle it should be possible to extend the Hansen and Hodrick (1980)
procedure to allow an ARCH error variance process. This is, however,
beyond the scope of the current paper.
3/ Australia, Austria, Belgium-Luxembourg, Canada, Denmark, Finland, France,
Federal Republic of Germany, Ireland, Italy, Japan, The Netherlands,
Norway, Spain, Switzerland, United Kingdom. Source: spot rates--IMF,
International Financial Statistics; forward rates (see text)--Financial
Times. I am grateful to Sompheap Sem for assembling this data.
- 13 -
where xf is the 90-day forward exchange rate for currency j at date
t-1 for delivery at date t. However, a complete set of forward exchange rate
data is only easily available over this sample for a subset S of major
currencies (those of Canada, France, Federal Republic of Germany, Japan,
Switzerland and the United Kingdom). For those currencies, the exchange rate
innovations were defined as in (17). For the remaining currencies forward
rates were estimated as weighted averages of the forward discounts of the
currencies in the subset S where the weights were obtained by regressing the
actual exchange rate changes on the exchange rate changes in subset S. Thus
(18) ln(xf /x ) (xf /x )(joS)
(1)(j,t|t-l/ ,t-l) Yk( k,t|t-l/ k,t-l)1jS
where the weights- Yk are obtained from OLS estimation of the regression
(19) Alnxjt = £ YkAlnxkt + ujt
kcS
The exchange rate index X was then calculated using (4) as
n
(20) VlnXt = £ v.Vinxjt
where the weights [Vj] are equal to the share of country j in total dollar
GDP in 1978.
18. It is also possible in principle to calculate interest rate
innovations from market data. Let rt be the one period (three-month) interest
rate at date t, rf+i t be the implicit forward rate, and 2rt be the two-
period (six-month) rate. Then, in the absence of any risk premium,
- 14 -
f ~ 2
(21) t _l|t (1+ rt)
In practice, we found that these implicit forward rates for US t-bills were
not informative in the sense that, knowing the current spot rate, knowledge of
the implicit forward rate did not improve predictions of the next period's
spot rate. This is in line with the results reported by Shiller et al. (1983)
who conclude (p. 215) that "the simple theory that the slope of the term
structure can be used to forecast future changes in the interest rate seems
worthless."
19. We therefore calculated interest rate innovations from an estimated
autoregression [as (12)]. 1/ This is the same procedure that was adopted for
the inflation and industrial production variables which were computed using
the same weights [v.J as in the exchange rate index (20),
1/ The autoregressions were estimated on the aggregated inflation and
industrial production indices. In Gilbert (1987) we computed industrial
production innovations as recursive residuals. This procedure is
preferable to the use of OLS residuals since one is not obliged to suppose
that agents have access to equation estimates based on future sample
information. However, degrees of freedom constraints prevented us from
applying that procedure in this study.
- 15 -
but running-over the:complete set of countries including the United States. 1/
In each case, a first-order autoregression appeared sufficient. 2/
1/ The industrial production index (measured as an average over the quarter
previous to the month in question--i.e., the January 1978 observation
relates is the average for the fourth-quarter of 1977) covered the same
countries as entered the exchange rate index excluding Luxembourg and
Switzerland. The inflation index was calculated in terms of producer
prices (current dated) over the same set of countries. The price indices
were:
Canada: producer prices--electrical machinery;
United States: producer prices--all goods;
Japan: producer prices--manufactured goods (total);
Australia: wholesale prices--machinery and equipment;
Austria: wholesale prices--transport equipment;
Belgium: wholesale prices--manufactured goods;
Denmark: wholesale prices--machinery and transport equipment;
Finland: producer prices--investment goods;
France: wholesale prices-semimanufactured goods;
Federal Republic of Germany: producer prices--manufacturing;
Ireland: wholesale prices--manufactures;
Italy: wholesale prices--finished investment goods;
Netherlands: producer prices--nonelectrical machinery;
Norway: wholesale prices--investment goods;
Spain: wholesale prices--metals and metal products;
Sweden: producer prices--(home market)--manufactured goods;
Switzerland: wholesale prices--metals and metal products;
United Kingdom: wholesale prices--total (excluding food, drink and
tobacco).
Source (for both industrial production and producer price indices): OECD,
Historical Statistics 1960-1975 and Main Economic Indicators, various
issues. I am grateful to Sompheap Sem for assembling this data.
2/ The estimated equations are (t-statistics in parentheses)
2
AlnA = 0.0027 + 0.56AlnA R = 0.31 DW = 2.03
t (1.22) (3.65) t1
rt = 0.0049 + 0.79r R = 0.66 DW = 1.86
(1.96) (7.59)tl1
n =0.0026 + 0.73H R = 0.50 DW = 1.75
(1.62) (5.43)
The equations were also estimated by OLS over the sample 1978ql-1985q4.
- 16 -
IV.. EOONETRIC STRATEGY
20. We noted in the previous section that financial market data is prone
to exhibit heteroscedasticity. If this is not taken into account, invalid
inferences may be drawn. We adopt two procedures in the estimations reported
in this paper. The first is to compute, for the OLS regressions,
heteroscedasticity-consistent standard errors using the White (1980)
procedure. The OLS estimator in the regression
y = X$ + u with Euu' = a has variance (X'X) (x'ox)(x'x) . White showed
that, in the case in which Q is diagonal, the ijth element wij of the matrix
W = X'QX may be consistently estimated as
~ T
(22) wij = T x itujt ujt xjt
21. The second approach we adopt is to suppose that the error u follows
an ARCH (autoregressive conditional heteroscedastic) process. 1/ Confining
attention to first order ARCH processes, one posits
Et-lut 0
(23)
an 2 2~ 2 2
and E t i't a + + u t 1
t-
1/ The ARCH model was proposed by Engle (1982). Developments and applications
are surveyed in Engle et al. (1985) and Engle and Bollerslev (1986).
- 17
In this--process the. errors are uncorrelated but not independent, since they
are related through their second moments. The conditional and unconditional
error variances differ, the conditional variance being given as (23) and the
unconditional variance as (1-_2) la2
22. Engle (1982) discusses maximum likelihood estimation of ARCH models.
Alternatively, one may consider a two stage (Aitken) estimator where the ARCH
parameter(s) is estimated from an initial OLS regression and used to make a
subsequent GLS correction. Since the information matrix is block diagonal
[Engle (1982, p. 997)], this Aitken estimator will be asymptotically efficient
[Cox and Hinckley (1974, p. 308)]. Thus, if the OLS residuals are denoted as
[et] one estimates 4 from the regression
2 2- 2 2
(24) et = a + 2 e t-1 +
2 + ~22 1/2
and then scales the observations by the factor (a 2 et 1) in the
second stage regression.
23. An advantage of the data set exploited in this paper is that it
relates to identical contracts on all seven LME metals. It seems plausible,
and indeed is the case, that the price innovations are correlated across these
metals. A demand surprise, for example, that results in an upward revision in
the price of copper will have the same effect on the price of aluminum. This
suggests that efficiency gains may be obtained (in those regressions in which
the regressor sets differ across metals) by the use of the SUR (Seemingly
Unrelated Regression) estimator. Furthermore, SUR estimation has the
additional advantage that it allows straightforward imposition of cross
equation restrictions.
- 18 -
24. Consider an m. equation model, and write the ith equation as
Yi Xi i + u.. Writing E(u itui) = £ , and supposing E(u u.') = 0 for
s * t, the SUR estimator of the stacked coefficient vector
B = ( -l "P')' is
(25) 6 = [X'(E OIT)X] x'(z OIT) y
where X and y are the conformably stacked data matrices and IT is the identity
matrix of order T (the sample size). 1/ In view of the concern expressed above
about heteroscedasticity, the question arises as to whether and how (25) may
be generalized to allow for heteroscedasticity.
25. The answer to this question is somewhat mixed. There does not appear
to be any straightforward way of generalizing the White (1980) procedure to
systems estimation since the White procedure allows the stacked error variance
matrix ZOI to be unstructured while the SUR estimator requires a matrix
which may, in an appropriate normalization, be taken to be constant. However,
the ARCH procedure, which does impose a specii'Eic structure on the error
variance matrix, generalizes fairly easily to systems estimation.
1/ Because of "missing observations" for the aluminum and nickel price
series, the SUR procedure was slightly modified. The covariance matrix Z
was estimated from the subsample for which complete information was
available allowing 6 to be estimated from the entire sample with the
missing" entries for y and the corresponding entries for X set to zero. A
two-stage procedure was employed with the initial estimate of Z taken
from the OLS estimates (estimated over the entire sample for which data on
each metal was available).
- 19 -
.26. -..The. error. variance matrix Et -contains Jim(m-l) distinct elements,
and in principle one might allow each of these to follow an ARCH process. This
gives rise to a problem with regard to ensuring that Et remains positive
definite at each date; but, in any case, is ruled out in our sample by degrees
of freedom considerations. An alternative, which only expends 2m degrees of
freedom, is to suppose that St exhibits a constant correlation structure.
Write the ijth element of Et as aijt and suppose
a. h. = a + 2a
it 1t ii 1 ii,t-1
(26) and
aijt hit hjt pij (i*j)
which is a natural generalization of (23). Then, defining Pii = 1, R = (pi.)
is the constant correlation matrix. Now write h,. = (hlt^***wimt)' and Ht =
diag(ht). This allows us to express the SUR estimator of a as
(27) 0 = [X '(R 8I)X ]T X (R SIT)y
where
X = (H < I )X and y (H I )y implying X. = H1 X. and Y. = H yj.
m m
The estimation procedure is therefore the same as in the ARCH-OLS model (24):
one can obtain consistent a coefficients from the SUR estimator (25) which
allows one to estimate the scaling matrix H from (24) and to proceed to
estimate the ARCH-SUR estimator (27). 1/
1/ As with the SUR, estimator, R was estimated from the sample for which
complete data were available, the initial estimate being taken from the
ARCH-OLS estimates.
- 20 -
27.. -..In the. results.repor-ted..in-Sect-ion 5 -we- employ four -estimators: (i)
the OLS estimator with heteroscedasticity consistent standard errors; (ii) the
ARCH-OLS estimator defined by (24); (iii) the SUR estimator; and (iv) the
ARCH-SUR estimator defined by (24), (25) and (27).
- 21 -
V. RESULTS
28. In Table 1 we report estimates of the metals' price adjustment
equations (15) but without imposition of either the unit restriction on the
exchange rate innovation coefficient or the equality restriction on the
incerest rate and the inflation rate- coefficients. The generalization is
therefore to
(28) Vlnp=t ajiVlnXt - ai2Vrt + ac3Vilt ai4VlnAt ejt
(j=Ag,...,Zn)
(Vr, VI and VA are scaled innovations--see Section 3).
29. The coefficient estimates are broadly in line with the dis:ussion of
Section 2--the exchange rate innovation enters with the predicted negative
coefficient in all estimates and appears to be distributed around unity. The
interest rate coefficients, which are very poorly determined, are more often
negative than positive, and are roughly half the size of the inflation
innovation coefficients. The activity coefficients, which are much better
determined, are as expiected. Substantial ARCH effects are only apparent for
silver and zinc--in the case of silver this is almost entirely due to the
spectacular volatility exhibited during the 1979-80 Bunker Hunt corner.
Overall, the fit is only modest, but this is perhaps to be expected in
relationships which omit most of the fundamental determinants of prices.
30. It would be possible to impose restrictions on the Table 1 estimates
( = 1, a2 = a3). It is, however, preferable to look for departures from
22 -
efficiency -prior -t-o the-=-imposition of restrictions. which are predicated on
efficiency obtaining. If the markets are efficient the residuals from (28)
should be serially independent, and so one test is obtained by consideration
of the Durbin--Watson (DW) and Lagrange Multiplier (LM) 1/ tests in Table 1l
While none of these give definitive answers, the DW statistics for the
aluminum and lead equations are worryingly low.
31. IrA Table 2 we report the results of adding one variable at a time to
(28). This "simple to general" approach 2/ is dangerous since a variable may
appear insignificant in this test when it would be jointly significant with a
second variable; and because a variable may appear significant as the
consequence of correlation with a second variable, in the presence of which it
would be insignificant. However, degrees of freedom constraints prevent our
adopting the preferable full 'general to simple' approach. Nevertheless, it
should be borne in mind that the results of the tests reported in Table 2 are,
at best, indicative and not decisive.
32. There are two sets of tests and three sets of additional regressors.
The first set of additional regressors comprises the intercept and lagged
dependent variable. Addition of these variables correspond. to the weak form
efficiency tests reported in Gilbert (1986a), and asks (intercept) whether
there is a consistent bias in the metal forward price and (lagged dependent
variable) whether the price change is forecastable from its previous history.
The second set of additional regressors comprises the lagged price
innovations. Here one asks whether the response to the financial market
1/ Godfrey (1978).
2/ See Gilbert (1986b).
- 23 -
--innovations is ---noninstant-aneous. Finally we - consider addition of
noninnovational variables. This final set of tests is not altogether distinct
from those resulting from the addition of the lagged innovations since any
stationary variable may be expressed as a sum of lagged innovations (the so
called moving average representation). The first set of tests for each set of
additional regressors gives the t-statistics from the addition of each
variable to the OLS and ARCH-OLS regressions; the second (likelihood ratio
test) considers exclusion of the entire vector of variables from the
corresponding system (SUR and ARCH-SUR) estimates.
33. Inspection of the results indicates clear evidence of weak form
inefficiency through lagged dependence (aluminum; possibly also copper, lead
and zinc); -somewhat weaker evidence of bias (possibly aluminum, nickel, tin
and zinc); evidence that the lagged forward premium (backwardation) is
informative (aluminum and lead); and evidence that exchange rate changes
(possibly also changes in the rate of inflation) are not transmitted
efficiently to metals prices (aluminum; possibly also silver, lead, tin and
zinc).
34. Tables 3-6 report the results of a data-based attempt to isolate the
locations of these inefficiencies. We further generalized (28) to
(29) Vlnpjt =jO + Vlnlnpj - Oj2(npf t l-lnpj 1)
- Sj3VlnXt - aj4VlnXt-l ;5Vrt +j6 Vt
+ Sa7vlnAt + ejt (j=Ag,...,Zn)
24 -
which .nests' (28). The.:choice of...these-.add5it--ional regressors was motivated by
the results of the tests reported in Table 2. For each metal, we then
attempted to find simplifications of (29) based on the estimated equations
(not reported). The simplifications took the form of eliminating variables
which were estimated as (insignificantly) taking the opposite sign from that
predicted by the discussion of section 2 together with those variables which
were associated with very low t-statistics. Also, noting that
(30) Vlnpjt = lnpit np,t_l
= Alnp. (lnp- f -lnp,
the parameter aJ2 was restricted either to be zero or unity, the latter
value implying that the dependent variable of the regression becomes
Alnpjt. (The unit restriction was appropriate only in the case of lead). In
total, this amounts to 23 restrictions. Tables 3, 4, 5 and 6 report
respectively the OLS, ARCH-OLS, SUR and ARCH-SUR estimates of these restricted
equations.
35. An essential part of any simplification exercise is to test the
proposed simplifications against the unrestricted hypotheses [Gilbert
(1986b)]. In this case, this implies testing against the equations defined by
(29). Tables 3 and 4 give the standard F tests taking each equation
independently; and Tables 5 and 6 report the likelihood ratio tests for the
- 25 -
Xentire- se-t -of--:-restrictions. 1/ All, these. tests-,accepted the proposed
simplifications.
36. Nevertheless, other simplifications would also be possible which
might result in somewhat different interpretations. It is doubtful that
encompassing tests of rival simplifications would be very powerful on this
data given the relatively low degree of explanation obtained. The results
reported in these four tables are therefore more conjectural than those
reported in Tables 1 and 2 since they are conditional on the congruency of the
proposed simplifications. Despite this, these results do offer one plausible
interpretation of the departures from efficiency uncovered in Table 2.
37. The ARCH error variance process is only evident for silver, and it is
thus for that metal that the ARCH and conventional estimates differ most.
However, the relatively high residual correlations between silver and both
copper and nickel generalize the effects of the ARCH transformation to these
equations in the ARCH-SUR estimates.
38. Aluminum, nickel and zinc are seen as having a negative bias of the
order of 1½-2% over this sample, implying a preponderance of short hedging
(typically associated with producer sales), while tin provides evidence of a
1/ The "missing observations" for the aluminum and nickel prices complicate
the likelihood calculations. The log-likelihood L was computed as
L = ELt where
t
Lt nt ln(2) - ½lnIEtI - ½etZt_ le
where et is the vector of residuals for period t and nt is the number of
metals for which data are available. Where data were unavailable, the
associated rows and columns of t were eliminated.
- 26
posit-ive bias-of the same order. This.may-be a..consequence-of the very heavy
forward support activities operated by the International Tin Council (ITC)
over this period. 1/ The lagged price innovations appear as significant in
five of the estimated equations, confirming the results reported in Table 2,
the effect being particularly acute in the aluminum relationship. 2/
39. The exchange rate terms all take the predicted sign, although the
estimated coefficient for aluminum is very sensitive to the presence of the
ARCH adjustment. We were not very successful in isolating the lagged exchange
rate effects implied by Table 2. The long run elasticities are distributed
fairly widely around the theoretical value of unity: aluminum, nickel and zinc
all show much lower elasticities which is consistent with producer pricing on
a dollar basis (these three markets are dominated by powerful producers);
while silver, lead and tin sample which ended in 1978 found significant lagged
price change appear oversensitive to changes in the value of the dollar. Only
copper has an-estimated elasticity close to the theoretical value of unity.
40. The interest, inflation and activity innovation effects remain
relatively poorly defined, except in the copper relationship. It is notable
that the activity variable is absent from the tin equation; this again may
demonstrate the effects of the ITC support operation.
1/ See Anderson and Gilbert (1986).
2/ Gilbert (1986a), considering four of the LME metals over a sample which
ended in 1978 found significant lagged price change effects for lead (not
apparent in this sample) and zinc (sign reversed in these tests). The
biases were also positive on the earlier sample. These contrasts warn of
the danger of taking these estimated relationships as structural.
- 27 -
VI. -CHOICE OF--EXCHANGE' RATE.-INDEX
41. Equation (3) shows that the weight assigned to a particular currency
(say that of country j) in the construction of the Ridler-Yandle exchange rate
index Xi specific to commodity i should, in principle, depend on both the
commodity i supply and demand elasticities in j and on j's share of world
production and consumption of the commodity. However, Gilbert (1987) shows
that in a multicommodity model the weights will be bn;;ilicated functions of
all own and cross demand and supply elasticities and all market share
parameters. This suggests that the construction of commodity specific indices
may be of limited value. Wbwever, that study used aggregate commodity indices,
and it was therefore not possible to test that view on that data set.
42. The results reported in Section 5 are based on regressions which
employ a GNP-weighted exchange rate index. We also experimented by
substituting (in the same regressions) 1/ (i) the IMF MERM index in which the
weights reflect the weights of each currency in US overseas trade; 2/ (ii)
commodity-specific indices employing weights proportional to, respectively,
consumption of the commodity in each country, production of the commodity and
production plus consumption. 3/ Referring to equation (3), use of consumption
weights amounts to assuming that demand elasticities are uniform across
1/ The use of equation specifications chosen on the basis of estimates which
employ the GNP-weighted index will, in principle, imply a (probably small)
bias in favor of that index.
2/ IMF, International Financial Statistics, (various issues).
3/ I am grateful to Sompheap Sem for assembling the data required for these
reweighting exercises.
- 28 -
countries and--that jsupply -elasticities are-negligible by-c-omparison; the use
of production weights requires the converse assumption; and the use of the
average weights requires the assumption that supply and demand elasticities
are equal and uniform across countries.
43. The results of these tests are summarized in Table 7. An obvious
question is which choice of index gives the best fit. Since these models are
nonnested, no formal test is possible simply on the basis of these
regressions; but it is in any case doubtful whether any of the available
nonnested tests would be able to discriminate effectively between alternative
indices of this sort in regressions in which the fit is relatively poor.
However, an impressionistic test is provided by comparison of the model log-
likelihoods, which are listed in the final rows of the table. Here it is
notable that the commodity-specific indices give higher likelihood values than
the nonspecific indices implying that, despite the theoretical considerations
advanced in Gilbert (1987), there is merit in adopting a commodity-specific
weighting. Within each class, weighting by consumption shares appears superior
to weighting by either production shares or by an average of the two; and GNP
weights are superior to trade weights.
44. A second comparison relates to the size of the estimated long run
exchange rate elasticities. The theory of section 2 implies that these
elasticities should be unity, and the multicommodity generalization proposed
in Gilbert (1987) implies that these elasticities should be within the
interval
- 29 -
[0, (C-vy) ]. 1/ It is notable that the choice of index can have a
significant effect on the size of these elasticities, and that, in particular,
use of the MERM index appears to give rise to high estimated elasticities.
45. The high elasticities associated with the MERM index reflect the fact
that the use of trade weights gives a particularly high weight to currencies
(in particular to the Canadian dollar) 2/ which vary relatively little in
relation to the US dollar. The more closely a country is linked through trade
to the United States, the less its currency will move in relation to the
dollar. 3/ Since the MERM index shows relatively less variation than the other
indices, the variation that is shown must take a higher weight in the
estimated regressions. This may go some way towards explaining the
difficulties experienced by Gilbert (1987) in obtaining estimated elasticities
which satisfy the-theoretical restrictions.
I/ [0,1] in Gilbert (1987) where the MERM index is used. The weights in the
MERM index sum to unity over the set of countries 2...n which exclude the
United ptates. In this study we scaled the MERM index by the factor
[1-v1] where v, is the US weigiit in the GNP weighted index. This should
ensure that the elasticities obtained from the MERM index in this study
are comparable with those obtained from the other indices.
2/ The Canadian dollar has a weight of 0.129 in the MERM index which is to
be compared with 0.036 in the GNP-weighted index. The production-weighted
index also suffers from a problem relating to the Canadian dollar which is
apparent in the positive elasticities estimated for nickel where Canada is
the major producer--the Canadian dollar has a weight of 0.568 in the
nickel-specific production weighted index.
3/ The standard deviation of the MERM index is 0.103 over the sample 1978ql-
1985q4. Over the same sample, the GNP-weighted index has standard
deviation 0.116.
- 30 -
VII. CONCLUSIONS
46. LiThis paper has two purposes. The first is to extend standard market
efficiency tests to intermarket efficiency by looking at the extent to which
changes on the forex, t-bill and final product markets are arbitraged across
into metals markets. Secondly, we are concerned to quantify the response of
metals prices to exchange rate changes.
47. The results of the market efficiency tests have been predominantly
positive. The seven LME metals all respond as predicted to exchange rate and
inflation innovations and the evidence for noninnovational responses to these
variables appears inconclusive. On the other hand, except in the case of
copper, the interest rate responses are poorly defined and insignificanjj
48. An additional benefit of the use of these financial market variables
in the efficiency tests is that this substantially increases the power of the
standard weak form efficiency tests by reducing the equation standard errors.
The power of these tests is further increased by the use of systems
estimators. As a consequence we have been able to isolate much clearer
evidence of bias and lagged dependence than in previous work on the same
markets. In the light of these tests, none of the LME metals markets appears
to conform exactly to the efficient market paradigm.
49. The implications of this finding for policy are slight. Market
inefficiency, as evidenced by bias and lagged dependence, is generally a
consequence of thin trading; and this increases the danger of market
manipulations. The tin experience is relevant here, and there is some
suggestion that the British market regulators were insufficiently vigilant of
tin trading on the LME. 1/ The evidence of inefficiency provided in this paper
1/ See Anderson and Gilbert (1986).
31
reinforces the need for regulatory vigilance. The evidence of market
inefficiency does imply that structural econometric commodity price models may
to some extent generate predictions superior to those estimated by simply
taking the published forward price, and this provides some encouragement to
the econometric modeling program.
50. j9 n the question of the long-run exchange rate elasticity, the
estimates reported in this paper suggest wide variation across metals. The
average elasticity across all seven metals is in line with theoretical
predictions, but only a single metal (copper) exhibits an elasticity of the
predicted magnitude. One group of metals (aluminum, nickel and zinc) appears
very insensitive to changes in the value of the dollar, and this may reflect
the influence of producer pricing on a dollar basis. However, this tendency is
offset by a second group (lead, silver and tin) which appears overresponsive
to changes in the value of the dollar. Finally, we found that the estimated
elasticities are quite sensitive to the weights used in construction of the
exchange rate indeJx, that consumption-weighted commodity specific indices
appear to perform best; and that the MERM trade-weighted index appears less
satisfactory in this context.
- 32 -
- Table 1: ESTIMATES ON THE NULL HYPOTHESIS
Dependent variable VInp
VInX Vr Vii VlnA df R2 DW LM
Ag OLS -2.37 -1.53 3.37 3.88 28 0.206 2.00 0.54
(1.81) (0.32) (0.82) (1.36) (0.71)
ARCH-OLS -2.00 0.25 1.96 1.13 0.31 28 0.142 2.13 0.52
(1.39) (0.08) (0.52) (0.54) (1.77) (0.72)
Al OLS -1.15 -1.03 2.98 1.65 22 0.240 1.40 1.04
(2.36) (0.60) (2.49) (2.47) (0.41)
Cu OLS -0.93 -1.76 3.24 1.86 28 0.326 2.28 0.68
(2.19) (1.41) (2.06) (1.90) (0.61)
ARCH-OLS -0.84 -1.69 3.18 1.39 0.22 28 0.315 2.33 0.81
(1.48) (1.30) (2.32) (1.76) (1.22) (0.53)
Ni OLS -0.68 -0.81 2.57 1.86 19 0.162 1.82 0.54
(1.20) (0.51) (1.55) (3.18) (0.71)
Pb OLS -1.07 0.95 1.85 1.58 28 0.147 1.47 0.66
(1.30) (0.61) (1.22) (1.52) (0.63)
ARCH-OLS -1.11 0.98 1.77 1.56 0.02 28 0.147 1.46 0.68
(1.38) (0.56) (1.03) (1.50) (0.13) (0.62)
Sn OLS -1.39 -0.45 0.89 0.20 28 0.233 2.08 0.37
(2.00) (0.43) (1.06) (0.33) (0.83)
Zn OLS -0.21 -0.51 1.81 1.63 28 0.101 2.04 1.05
(0.32) (0.48) (1.36) (1.68) (0.40)
ARCH-OLS -0.33 -1.27 2.47 1.96 0.33 28 0.090 2.02 0.79
(0.43) (0.74) (1.54) (1.98) (1.74) (0.54)
Notes: t statistics in parentheses under coefficients--heteroscedasticity-consistent in case of OLS
estimates. The LM statistic tests for serial correlation of order up to 4 and is distributed
F(4,df-4); the marginal significance (area under the right hand tail) is given in parentheses. The
ARCH-OLS regressions are only computed if ' is estimated as positive. The statistics for the ARCH-
OLS regressions are estimated from the unscaled residuals.
- 33 -
Table 2: TESTS FOR DEPARTURES FROM MARKET EFFICIENCY
Ag Al Cu Ni Pb Sn Zn X (7)
Intercept
OLS -0.03 -1.19 -0.45 -0.94 -099 1.23 -0.68 10.34
ARCH-OLS -0.32 -0.76 -1.32 -0.45 11o35
Lagged dependent variable
OLS -0.90 4.67 -1.99 -0.18 1,42 -0.76 -?.49 8.65
ARCH-OLS -0.64 -1.90 -1.19 8.54
Lagged Innovations
VInX(-l) OLS 1.42 -0.69 -0.20 -0.43 -0.47 1.04 1.49 12.21
ARCH-OLS 0.79 -0.58 0.78 9.42
Vr(-1) OLS -0.23 -0.80 0.52 0.64 -0.11 -0.56 0.75 5.08
ARCH-OLS -1.64 0.02 - 0.10 0.42 8.41
Vi(-1) OLS -0.66 1.05 -0.82 0.42 1.27 1.21 -0.09 8.92
ARCH-OLS -0.33 0.85 -0.78 0.93 -0.64 9.17
VInA(-1) OLS -1.74 -0.69 -1.62 -0.88 -1.32 -0.30 -0.73 5.76
ARCH-OLS -1.15 -1.19 -1.06 -0.41 4.84
Noninnovational variables
fprem(-l) OLS 0.15 -1.86 0.43 0.64 -2.08 0.86 0.26 !2,76
ARCH-OLS -0.34 -0.03 -1.95 0.38 12.03
AInX OLS -0.35 2.29 -0.06 -0.43 -0.36 1.25 -0.19 11.24
ARCH-OLS 0.25 1.79 0.36 -0.30 11.64
Ar OLS -0,39 0.67 -0.13 0.64 0.38 -0.77 -0.70 4.73
ARCH-OLS 0.48 0.32 -0.10 -0.30 3.12
AII OLS -0.95 -1.32 -0.09 0.42 -0.42 -1.06 -0.18 6.43
ARCH-OLS -0.99 -1.14 0.36 -0.41 0.36 7.35
AInA OLS -1.28 -1.27 -0.94 -0.88 -0.21 -0.79 -0.44 5.75
ARCH-OLS -0.91 -1.02 -0.24 -0.23 5.28
-f
Notes 'fprem' is the forward premium Inp -Inp 1 Quoted statistics are t-statistics
t-1 t-1'
(heteroscedasticity-consistent in the OLS case) from addition of specified variable to the null
regressions reported in Tabie 1. Degrees of freedom: as Table 1, except one degree lost in LDV
regressions for Al and Ni.
The X (7) statistic tests exclusion of the entire vector of variables in respectively the SUR and
ARCH-SUR estimates. The 95% critical value for X (7) is 14.07. ARCH corrections (OLS and SUR) were
only made where 4 was estimated as positive.
- 34 -
Table 3: PRICE INNOVATION RELATIONSHIPS--OLS ESTIMATES
Ag Al Cu Ni Pb Sn Zn
Dependent variable Vlnp Vlnp Vlnp Vlnp AInp Vlnp Vlnp
Intercept -1.36 -2.68 2.04 -2.14
(xlOO) (0.77) (0.98) (1.39) (0.82)
Vlnp1 -0.23 0.43 -0.37 -0.14 -0.34
(0099) (4.84) (1.99) (1.18) (1.72)
VInX -3.26 -0.34 -1.52 -0.20 -0.77 -1.70 -0.23
(2.02) (0.68) (3.26) (0.27) (1.00) (2.85) (0.27)
VInX -0.51
(0.76)
Vr -0.53 -1.65 -0.27
(0.32) (1.62) (0.27)
Vii 5.25 1.20 4.82 1.18 1.53 1.02 2.47
(1.30) (1.04) (2.99) (0.72) (1.69) (1.33) (2.38)
VInA 3.39 1.48 1.41 1.59 1.90 1.59
(1.58) (2.49) (2.38) (1.89) (2.09) (2.16)
df 28 19 27 18 28 27 27
R2 0.233 0.419 0.410 0.143 0.238 0.282 0.205
DW 1.71 1.23 1.89 1.84 1.65 2.04 1.59
LM F(4,df-4) 0.45 0.39 0.18 0.73 1.05 0.43 0.79
ms (0.77) (0.81) (0.94) (0.58) (0.40) (0.78) (0.54)
Restrictions h 4 2 3 4 4 3 3
F(h,df-h) 0.21 0.59 0.21 0.64 0.22 0.31 1.38
ms (0.93) (0.56) (0.89) (0.65) (0.92) (0.81) (0.27)
Implied long-run -2.65 -0.59 -1.11 -0.20 -1.28 -1.49 -0.17
exchange rate elasticity
Notes: Heteroscedasticity-consistent t-statistics in parenitneses under coefficient estimates. 'ims'
(marginal significance is the area under the right-hand tail of the distribution.
- 35
-Table 4: PRICE INNOVATION RELATIONSHIPS--ARCH-OLS ESTIMATES
Ag Al Cu Ni Pb Sn Zn
Dependent variable Vlnp VInp AInp Vlnp
Intercept -1.76
(xl`00 (0.71)
Vlnp_1 -0.13 -C137 -0.30
(0.56) (1.90) (1.45)
VInX -2.24 -1.46 -0.63 -0.27
(1.48) (2.26) (0.79) (0.34)
VInX -0.49
(0.67)
Vr -1.72
(1.38)
VII - 3.25 4.81 1.29 2.41
(0,83) (3.25) (0,86) (1.50)
VInA 0,68 1.41 1.84 1.72
(0.33) (1.81) (1.88) (1.70)
' 0.39 0.06 0.04 0.08
(2.33) (0.31) (0.21) (0.35)
df 28 27 28 27
R2 0.077 0.389 0.236 0.199
DW 1.94 1.88 1.64 1.65
LM F(4,df-4) 0.60 0.16 1.09 0.76
ms (0.66) (0.87) (0.38) (0.56)
Restrictions h 4 3 4 3
F(h,df-h) 0.23 0.21 0.21 1,31
ms (0.92) (0.89) (0.93) (0.29)
Implied long-run -1.98 -1.07 -1.12 -0.23
exchange rate elasticity
Notes: t-statistics in parentheses under coefficient estimates. 'ims' (marginal significance is the area
under the right-hand tail of the distribution. ARCH-OLS estimates were only computed where ' was
estimated as positive.
- 36 -
Table 5: INNOVATION RELATIONSHIPS--SUR ESTIMATES
Ag Al Cu Ni Pb Sn Zn
Dependent variable Vlnp Vlnp Vlnp VInp AInp Vlnp Vlnp
Intercept -2.09 -1.92 2.63 -2.08
(xlOO) (1.81) (1.04) (2.09) (1.22)
Vlnp_1 -0.24 0.44 -0.27 -0.20 0.32
(2.60) (4.04) (2.87) (1.51) (2.52)
VinX -3.13 -0.01 -1.33 -0.20 -0.47 -1.81 -0.25
(2.82) (0.03) (2.89) (0.32) (0.73) (3.69) (0.41)
VInX -0.88
(1.69)
Vr -0.37 -1.64 a.!o
(0.59) (2.57) (0.11)
All 5.53 0.66 4.22 0.37 1.28 0.97 2.27
(2.47) (0.72) (4.58) (0.29) (1.04) (1.03) (1.85)
VInA 3.35 1.16 1.42 0.96 1.90 1.52
(2.46) (2.37) (2.75) (1.19) (2.51) (2.01)
Implied long-run -2.52 -0.02 -1.04 -0.20 -1.35 -1.51 -0.19
exchange rate elasticity
Residual Ag 0.52 0.56 -0.31 0.02 0.16 0.06
correlations Al 0.44 0.25 -0.19 0.14 0.04
Cu 0.26 0.15 0.41 -0.05
Ni 0.15 0011 0.24
Pb 0.36 0.39
Sn 0.21
Log-likelihood -321.1
Likelihood ratio test 2
against unrestricted model X (23) = 30.1 (ms = 0.15)
Notes: t-statistics in parentheses under coefficient estimates. The SUR estimates were estimated over
the complete sample 1978ql-1985q4 with absent data for Al and Ni (and the values for the
corresponding X variables) replaced by zeros. The error covariance matrix E was estimated over the
restricted sample 1980q3-1985q4, using a two stage procedure in which the initial estimate of E was
calculated from the residuals from the regressions reported in Table 3.
- 37 -
- Table 6: PRICE'-INNOVATION RELATIONSHIPS--ARCH-SUR ESTIMATES
Ag Al Cu N. Pb Sn Zn
Dependent variable Vlnp Vlnp Vlnp Vlnp AInp Vlnp Vlnp
Intercept -1.50 -1.58 2.67 -1.63
(xlOO) (1.30) (0.85) (2.13) (0.99)
Vlnp° -0.18 0.40 -0.33 -0.19 -0.22
(1.51) (3.61) (3.26) (1.45) (1.76)
VInX -2.40 -0.22 -1.42 -0.59 -0.54 -1.81 -0.32
(2.28) (0.48) (3.06) (0.91) (0.82) (3.71) (0.52)
VInX -0082
(1.57)
Vr -0.64 -1.81 -0.18
(0.98) (2.77) (0.20)
Vi- 5.02 1.44 4.92 1.34 1.38 1.16 2.34
(2.09) (1.62) (5.21) (1.05) (1.12) (1.24) (2.00)
VInA 1.62 1.27 1.29 1o22 1.89 1.67
(1.13) (2.71) (2.45) (1.51) (2.51) (2.25)
4) 0.38 0.07 0.01 0.08
(2.24) (0.39) (0.06) (0.36)
Implied long-run -2.04 -0.36 -1.06 -0.59 -1.36 -1,52 -0.26
exchange rate elasticity
Residual Ag 0.50 0.56 -0.29 0.04 0.24 0.13
correlations Al 0.38 0.25 -0.19 0.16 0.06
Cu 0.27 0,12 0.42 -0.05
Ni 0.17 0.09 0.25
Pb 0.37 0.38
Sn 0.19
Log-likelihood -320.8
Likelihood ratio test 2
against unrestricted model x (23) = 27.0 (ms = 0.26)
Notes: t-statistics in parentheses under coefficient estimates. The SUR estimates were estimated over
the complete sample 1978ql-1985q4 with absent data for Al and Ni (and the values for the
corresponding X variables) replaced by zeros. The error covariance matrix £ was estimated over the
restricted sample 1980q3-1985q4, using a two stage procedure in which the initial estimate of 2 was
calculated from the residuals from the regressions reported in Table 4. The ARCH coefficient 4)*
which was restricted to being nonnegative, was estimated from the residuals from the Table 5 SUR
estimates.
- 38 -
Table 7: CHOICE OF EXCHANGE RATE INDEX
LONG-RUN EXCHANGE RATE ELASTICITIES AND LOG-LIKELIHOOD
Nonspecific Indices Commodity-specific Indices
GDP Weights MERM Weights Consumption Production Consumption +
Shares Shares Production Shares
Ag SUR 2.52 -3.26 -2.79 -3.20 -2.75
ARCH-SUR -2.04 -2.66 -2.27 -3.06 -2.57
Al SUR -0.02 -0.24 -0.30 -0.04 -0.00
ARCI-i-SUR -0.36 -0.26 -0.68 -0.31 -0.06
Cu SUR -1.04 -1.26 -1.05 -1.24 -1.08
ARCH-SUR -1.06 -1.29 -1.05 -1.36 -1.08
Ni SUR -0.20 -0.14 -0.32 1.12 0.05
ARCH-SUR -0.59 -0.61 -0.66 0.86 0.04
Pb SUR -1.35 -1.37 -1.20 -0.72 -1.03
ARCH-SUR -1.36 -1.41 -1.19 -0.94 -1.11
Sn SUR -1.51 -1.79 -1.66 -1.75 -1.87
ARCH-SUR -1.52 -1.81 -1.67 -1.81 -1.86
Zn SUR -0.19 -0.21 -0.21 -0.10 -0.14
ARCH-SUR -0.26 -0.30 -0.27 -0.14 -0.14
Average SUR -0.98 -1.18 -1.08 -0.92 -0.97
ARCH-SUR -1.03 -1.19 -1.11 -0.97 -0.97
Log-likelihood SUR -321.1 -321.9 -320.1 -321.6 -320.9
ARCH-SUR -320.8 -321.6 -319.8 -319.6 -320.2
- 39-
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