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The model dependence of relative mean-square error tests for market efficiency
J ECO BUSN 1986; 38:227-236
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The Model Dependence of Relative Mean-Square Error Tests for Market Efficiency Wayne Joerding
Over the past few years use of forecasting models to test the efficient markets hypothesis, EMH, has been increasing. An example of such a test is the relative mean-square error (MSE) test in which discovery of a forecasting model with a smaller MSE than that of the futures price is evidence against the EMH. Although the validity of EMH tests are dependent on the validity of a market model, in the case of relative MSE tests the degree of this dependence does not seem to have been recognized. This article shows that within a class of plausible market models models exist for which the relative MSE test is not valid. The conclusion reached is that quite specific details of a market model may be necessary in order for relative MSE tests to be valid and thus these tests are not very robust.
Introduction Tests of market efficiency have long been recognized as simultaneously tests of market equilibrium models (Fama 1970), although the degree of this dependence is sometimes ignored. The purpose of this article is to point out the importance of this dependence for interpreting relative mean-square error tests of market efficiency. Relative mean-square error tests examine the mean-square error (MSE) of market forecasts relative to forecasts based on some statistical forecasting method. If the MSE of the statistical forecast is less than the MSE of a market forecast, then it is taken as evidence against the efficient-markets hypothesis, t The problem with these tests, and the main point of the paper, is that in an efficient market in which agents have rational expectations, market forecasts do not necessarily have a smaller MSE than every individual forecast. CorneU (1977) uses a relative MSE test for efficiency in the foreign-exchange market. He
Such relative MSE tests can be tests of either the weak or semi-strong form of the efficient-markets hypothesis, because the statistical forecast uses only past prices or ineludel publicly available information.
Address reprint requests to Wayne Joerding, Deparurtent of Economies, Washington State University, Pullman, Washington 99164. The author is indebted to Russ Fuller, Jon Sonstellie, and Jeff Krautkraemer for helpful comments. All errors are the responsibility of the author. Journal of Economics and Business © 1986 Temple University
0148-6195/86/$03.50
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W. Joerding states: In an efficient market, where the forward rate incorporates all information relevant to forecasting the future spot rate, the standard deviation of these errors should be less than or equal to the standard deviation of the forecast errors derived from any alternative model. Cornell finds that, for six out of seven currencies, a simple autoregressive model outperforms the forward rate, but he considers this evidence against the efficient-markets hypothesis as tentative and possibly the result of chance. Although Cornell points out the dependence of efficient-markets tests on model specification, he does not cite this fact as justification for his reluctance to interpret the results as evidence against the efficient-markets hypothesis. Instead, he depends on a result of chance argument. Because this article demonstrates the crucial dependence of relative MSE tests on model specification, one can interpret the results of this article as an additional reason for reluctance in accepting Cornell's findings as evidence against the efficient-markets hypothesis. Leuthold and Hamnan (1979) use a relative MSE test in their study of efficiency in the hog futures market. In this case, a simple econometric model is used to forecast spot prices. The model MSE of forecasts is found to be less than the futures market MSE of forecasts, which they take to imply inefficiency in the hog futures market. Rausser and Carter (1983) obtain similar results for futures markets in the soybean complex. Reasons other than those presented here exist for concluding that the existence of a forecasting method with lower MSE than the futures market is not proof that the relevant market is inefficient. If information cost is positive, then information will only be used up to the point that marginal benefit equals marginal cost. Thus, the existence of a model that has a lower MSE than the futures market but is unable to earn an abnormal profit is fully consistent with market efficiency. 2 This article shows that information cost is not the crucial issue. Even if information cost is 0, the discovery of a model with a lower MSE than the futures market is not sufficient to conclude that the futures market is inefficient. More generally, the discovery of a forecasting method with a lower MSE than the market is not inconsistent with participation in the market by an agent with a forecast MSE as low as the discovered forecast method's MSE, and yet the agent does not earn abnormal profits. Consequently, the relative MSE method requires assessment of the risk-adjusted profits possible with the discovered forecast method. Such assessment is only possible in the context of a specific market model, is very difficult, and is often tenuous. The paper proceeds by first presenting a model of an efficient futures market and a discussion of the defin fion of market efficiency that is used. The main result is first presented using a two-person model and then generalized to a multiperson model. A conclusion presents some implications and limitations of the results.
The Model Consider a purely speculative market with two agents in which the net supply of the commodity or asset is 0.3 Each agent comes to the market with an initial expectation of the
2 For some interesting results in this area see Leuthold and Hartmann (1980), Brooks et al. (1980), and Rendleman and Carabini (1979). 3 Speculators hold uncovered buy/sell orders and profit from the difference between the price of the contract and subsequent spot price. Thus, if the forward price is less than the spot price then those speculators holding buy contracts (long position) will make a profit and those speculators holding sell contracts (short position) will lose wealth.
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future period's spot price, which she compares to the futures price and then forms demands for futures contracts. These price expectations are dependent on an information set that includes the futures price and is different for each agent. Thus, it is necessary to allow recontracting until agents are satisfied with their own private forecasts and their trades. The issues concerning the convergence of such tatontment processes are ignored here by assuming that the rational-expectations equilibrium price exists and can be achieved by the market. 4 Because the results of the article do not depend on the cost of information, it is not explicitly considered in the analysis. Suppose the market opens at the beginning of each period t = 1, 2, • .-, and establishes the equilibrium futures price Pt as a function of equilibrium private forecasts PIt and P2t. At the end of the period the spot price St is revealed and profits (positive or negative) registered. Let Dit be the equilibrium excess demand, for contracts, by agent i in period t. Although the optimization problem is not modeled, it is proper and useful to think of Dit as the demand for one type of asset (among several) and the result of maximizing expected utility subject to a budget constraint. Thus, Dit is a function of Pt, Pit, the prices of other assets, and must satisfy the usual restrictions on excess-demand functions. The condition for market equilibrium used here is that the aggregate excess demand equal 0:D1 + D2 = 0. (Henceforth, the subscript t will only be used where necessary to avoid confusion.) If D~ > 0, then i must expect the spot price to be greater than the forward price, Pi > P. The reverse is true for Di < 0. Consider the case D, > 0, then P1 > P. But this implies that D2 < 0 (because Dl + DE = 0) and/)2 < P. Clearly, P is between P, and/)2. If D1 < 0, then a similar argument again indicates that P is between Pl and P2. Conversely, if P > Pl, P2, then both agents would want to sell contracts, causing the price to decline. If P < P~, P2, then both agents would want to buy contracts, causing the price to rise. Consequently, in equilibrium it will always be possible to express P as: 5
Pt=(ttPlt+(1-ott)P2t
0-<~t-< 1,
(1)
where the weights 6t and (1 - ~t) are the relative market weights of agents 1 and 2 in period t. Equation (1) depends on our assumption of 0 net supply of the asset. In general, with positive supply and risk-averse agents the market price is less than the weighted average of individual forecasts (LeRoy 1982). The relative market weight of an agent for any particular period depends on factors that determine the demands D~ and Dz. For example, the weights are a function of the risk aversity and relative wealth of the agents. The degree of risk aversity by an agent is potentially a function of wealth, or other variables, which may be affected by stochastic variables from other markets. In general, an agent who brings greater wealth to the market than another agent has more influence on the equilibrium forward price than does an agent with less wealth, ceterus paribus. But the wealth of an agent is affected by the past random operation of all markets in the economy and thus is a random variable. Specifically, this means the weight t~t is a random variable. Consequently, Pt is unconditionally stochastic, because &t, Pit, and P2t are all stochastic. If the past history of the economy is known, then 6t is determined (in the context of a specific model) but PIt and P2t are stochastic. Given the information set available to each agent and the history of the market, Pt is nonstochastic. 4 Someexogenousnoise in the market, such as randomsupply, mustexist to preventan agent from using the futures price to infer the informationof other agents; see Grossmanand Stiglitz (1980). Here, noise is inherent in the random spot price. Also see Jordan and Radner (1982). See Figlewski(1978) for an exampleof how Equation (1) may be derived from a specificmodel of wealth maximization.
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W. Joerding Assume agents have rational expectations, then Pit = St + eit, where St is the spot price at the end of the period, E(eit) = 0, var(eit) - V/, and cov(elt, e2t) m /9.6 This assumption causes the forward price to be an unbiased forecast of the spot price, meaning that relative M S E tests reduce to relative variance tests. The forecast errors el and e2 are different because each agent is assumed to have different information and (or) to use a different forecast method. The combination of different information sets and forecast methods can result in agents with very different forecasting abilities, thus, assume I/1 _< 1/2. This assumption can be justified by differences in the risk aversion of the agents in the presence of costly information (see Verrecchia 1982). 7 Moreover, assume agents do not often radically change thgir sources of information or forecast method, so that VI and V2 are constants. The foregoing completes the description of the model; we finish this section by showing that the variance & can be arbitrarily specified, s To see the issue, note that the transition o f t~t to t~t÷l depends on the realization o f ~lt and ~2t, because o f the redistribution o f wealth. Consequently, the distribution of t~ t depends on the distribution of elt and e2t; that is, var(~) is a function of VI, V2, and p. But many other factors also affect a t , and its distribution, and these other factors allow enough flexibility in permissible values of a t so that E (&) and var(t~) can be arbitrarily specified. To see this let {Pit, PEt} be a sequence of price forecasts by agents 1 and 2. Next select an arbitrary sequence of weights {a[} with var(ett') an arbitrary constant. 9 W e can now define P / - ~[ Pit + (1 - ott' )P2t. An economy or market model is defined by specifying the preferences and resources for each agent. If the only constraint on the market is that agents maximize utility subject to a budget constraint, then it is known that Pt' is the equilibrium price for some economy. An application of Theorem 4 in Sonnenschein (1982) shows that some economy even exists for which P / i s the unique equilibrium price. In other words, we can find a sequence of market models, one for each period, subject only to budget-constrained utility maximization and Pareto optimality, for which P / is the unique equilibrium price and var(c~t') can be chosen arbitrarily. Thus, although the variance of the actual weights depends on Fl, V2, and p, no logical restriction exists on permissible values for var(t~), l0 This fact will be useful later.
M a r k e t E f f i c i e n c y Il The term efficiency has been used in several ways in the literature. In welfare economics, an efficient economy is one that results in a Pareto optimal allocation of resources. In statistics, an efficient estimator or forecaster is one with the smallest dispersion in a certain class of 6 What sign this covariance should be is not clear; a positive covariance between forecast errors seems plausible if the agents use similar forecast methods. 7 If information cost is zero, then I/1 = I"2. All of the results presented here continue to hold under this special case, which substantiates the claim made in the introduction that information cost is not crucial. Choosing agent 1 to have the lower forecast variance is arbitrary. s Of course we require var(&) _< 1/4. 9 For example, in Grossman (1977) the weights are functions only of the risk aversion of the agents. An arbitrary sequence {a/} can easily be generated by the proper specification of how risk aversion by the agents changes over time, perhaps in response to stochastic shocks from other markets. Clearly, such a specification of preferences cannot be ruled out by assuming expected utility maximization subject to a budget constraint. ~0An alternate way to look at this problem is to remember that & comes from D~ + /:)2 = 0. Suppose that Dz is a function of P, - P = el - c~e~ - (1 - ~)e2, then, for arbitrary excess demand function Di, it is not hard to believe that Vz, V2, and p impose no constraints on the possible values of var(cT). This is because of the great generality of utility maximization subject to a budget constraint. H In the case that agents demand a risk premium, the following results apply by simply taking "zero profits" to mean "zero risk-adjusted profits."
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estimators or forecasters. In financial economics a market is efficient relative to an information set if the price correctly reflects all the information in the relevant set. This definition has been interpreted in at least two ways: 1) in the statistical sense that the futures price is an efficient forecaster of the spot price (see Figlewski 1978; 1982); and 2) in the sense that the futures price is such that no trades based on available information can have a positive expected profit (see Fama 1970). Both of these interpretations are applicable in this paper. Let ere be the weight on Pit if Pt is an efficient forecaster of St in the sense of minimizing error-forecast variance. Because at is a stochastic variable, a natural characterization of the statistical interpretation for market efficiency is E (ft) --- o~ = ae, which requires that the market price be on average an efficient forecaster. The zero expected profit condition is E (~it) = 0, where ~,t is the profit of agent i in period t, and requires that average profits over time are zero. Profits for agent i in period t are ~?it = Dit(St - Pt). For example, if Pi > P and S > P, then Di > 0and $it> 0. Note that profits are dependent on f through P. To see why, consider the extreme case of f t very close to 1. In this situation, the futures price essentially equals P , , making less likely that agent one is on the correct side of the market and experiences a profitable trade. ~2Let 7~i(ft) : E (~it; fit) be expected profits for period t given the weight ft. It is assumed that there exists a unique ao that solves ri(ao) -- 0 and that oto is an attractive equilibrium in the sense that, after a random shock, market forces tend to redistribute wealth so that f t tends toward Oto.13 As an attractive equilibrium, ao is some point of central tendency for the random variable fit. Thus, the 0 expected profit interpretation is characterized by specifying Cto as the mean of at; that is, E ( 4 ) - ct = ao.
Preliminary Ideas The validity of a relative MSE test depends on the condition that a statistical model with an MSE smaller than the market's MSE implies that the information used by the statistical model is not properly incorporated in the futures price. 14 With the interpretations of market efficiency used here this means either that the market is not, on average, an efficient forecaster or some agent who used the statistical model would have positive expected profits. The rest of this paper considers the question: Is it possible in an efficient market (either interpretation) for some agent (agent 1) to use a forecaster with an MSE less than the market's MSE? Define V(a) - E ( P t - St) 2, which expresses the MSE of the market price as a function o f a . The answer to the above question is yes if either V(Ole)> V 1 or V(olo) > Vl,
(2)
remember a -= E (5). If Equation (2) holds for some market then a relative MSE test for market efficiency in that market is not valid. The analysis below shows conditions under which Equation (2) is true.
12In this model, a trade will be profitable (unprofitable) as the agent's private forecast is on the same (opposite) side of the forward price as the spot price. As P gets closer to Pi, the probability of a profitable trade diminishes. This effect can even be strong enough to prevent a more accurate agent from bankrupting a less accurate agent, see Figlewski (1982). ~3Formally, ~ri(c~t) ~ 0 as ~ ~ cto. See Figlewski (1978; 1982) for an example of a specific model that has this characteristic. ~4The informationreferred to would be knowledgeor estimates of the past relationship between the spot price and a set of predetermined variables.
W. Joerding
232
Relative MSE Tests With Two Agents A realization of c~t depends on how the past sequence of forecast errors, P/, - Sk, i = 1, 2 and k = t - 1, t - 2, • • • affected the wealth of the agents. Because P i is a rational forecast, the forecast errors are serially uncorrelated, which implies that &t is independent of Pit - St, Harvey (1981:55). t5 Rewrite Equation (1) as P - S = 6 ( P l - S ) + (1 - c~)(P2 - S ) and take the variance of both sides to obtain V(cz) = VF + var(el - e2)var(c~)
(3)
where VF = o~2Vl + (1 - o02V2 + 2c~(1 - c0 cov(el, e2). 16 S u b s t i t u t i n g Equation (3)into Equation (2), and treating cz as a variable, see that Equation (2) is satisfied if R(c0 -
var(el-e2)o~2-2( V2-p)a+
V 2 - V! + v a r ( e l - e 2 ) v a r ( ~ ) > 0 .
(4)
The quadratic R ( a ) is a parabola that opens upward and has a m i n i m u m at c~* - (V2 - p) / Var(Cl -- ~2)" 17
Because a exists for which 0 _< a _< 1 and R (a) > 0, and a can be arbitrarily specified, then markets exist (abstract) for which V ( a ) > Vl. W e are interested, however, in possible values for a that characterize our definitions of market efficiency, o~oand ae. That is, we want to know i f R ( a e ) > 0 or R(c~o) > 0 is possible. One approach to establishing the existence of efficient markets for which the relative MSE test is invalid is to formulate a specific model, derive the relationship between a , var(t~), and V1, V2, and 0, then check Equation (4). For example, consider the model of Grossman and Stiglitz (1980) with identical risk aversion by all agents and 0 = 0, then cte = V2 / (VI + V2) and var(c~) -- 0. We can see that Equation (4) is not satisfied in this case. Here we take an alternative approach of checking whether R ( . ) is positive definite by evaluating R ( a * ) . is Observe that R ( a * ) > 0 if and only if var(&) > [ ~ - O2]/[var(el - e2)] 2.
(5)
Relative M S E tests are not valid for any market that satisfies Equation (5). Note that the right side of Equation (5) can be less than 1/4, the upper bound on vat(6) imposed by the condition 0 __. 6 _< 1. Because a market exists for any arbitrary value of var(6) we can conclude that efficient markets exists for which Equation (5) is satisfied. That is, market models exist for which the relative M S E test is not valid. Recall that the demonstration that var(6) can be arbitrary, depending on the possibility of constructing a sequence of market models With the necessary distribution of 6. Specifically, because an agent's market weight depends on his or her risk aversion, the necessary variation
~5Without this condition, Equation (3) becomes more complicated. It is possible to derive conditions under which the basic argument below is valid but little insight is gained and the market would not be composed of agents with rational forecasts. ~6Equation (3) can be derived from standard formulas for var(XY), where X and Y are independent random variables. It is also necessary to observe that cov[oTel,(1 - 6)e2] = cY(l - a) cov(el, e2) - var(6) cov(el, e2), andvar(~j - ~2) = Vi + 1:2 - 2cov(~l, e2). ~7 Derivation of a* implicitly assumes that v a r ( ~ ) is not a function of its own mean. I a m not aware of any strong arguments for either a positive or negative relationship, but suppose d var(6) / da > 0, then the minimum of R(o0 would occur at an a < ~*. J8 The weight o : is also interesting because it is the weight that should be attached to agent l's forecast if P is to be a minimumvariance linear forecaster of S (see Figlewski 1982). The weight a* can also be shown to imply 0 expected profits for a model in which agents use a mean-variance analysis for investment decisions (Figlewski 1978).
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in 5 can be due to changes in risk aversion, which could occur because of random shocks to wealth from other markets in a general economy (see footnote 8). As another example, Figlewski (1978; 1982) presents a model in which 5 depends on random shocks to wealth, arising from forecast errors. Shefrin (1983) has shown that in this model the value ao is an unstable equilibrium, since 5 will almost surely not equal ao this instability may produce enough variation in 5 to satisfy Equation (5). Whether or not this could happen depends on the parameters Vn, V2, and p, among others. The point is that such possibilities cannot be eliminated by assuming expected utility maximization subject to a budget constraint without specifying a particular market model, and this model must be a general equilibrium model that limits (perhaps implicitly) the impact of shocks from other markets. Consequently, relative MSE tests of market efficiency, as other tests, depend on model specification and are not robust against alternative specifications. The above results do not depend on positive information cost. If information is free then all agents should have the same forecast ability, Vz -- V2. But this would in no way change Equation (5) or its implications. The reason that the relative MSE test is not sufficient to imply market inefficiency has nothing to do with information cost. To see this, consider the similarities and differences between a market and the weighted mean, P* = o r * P i t ( 1 a*)P2t, as forecasting mechanisms. Both are weighted averages of the private forecasts, P~ and P2. The weighted mean, however, would apply constant weights to each private forecast, the optimal weights are a* and (1 - a*), whereas the market can only apply stochastic weights. The weights must be stochastic because they are affected by prior stochastic outcomes of the market and other shocks. If the weights are independent of the private forecasts, as in our case, the mechanism with stochastic weights must have a larger variance than the constant-weight method. All we have shown is that the increased variance is large enough to cause the market-forecast variance to be greater than the forecast variance of one of the private forecasts, V > VI. In contrast, if the weights are chosen optimally, then the variance of the weighted-mean mechanism cannot be greater than the variance of any private forecast, var(P*) _< lit. The results point out the danger in casually attributing properties of optimal statistical models to market mechanisms or in assuming that the exhaustion of potential expected utility-maximizing trades causes market prices to behave like optimal statistical forecasters.
More than Two Agents If I > 2 agents exist, then Equation (1) becomes 1
e : Z ,Pi, i=1
where 1 E (~i= 1, 0 ~-~t~i~< 1. i=1
(9)
The market's forecast-error variance is now 1
var(P-S)--Z
I
1
1
Z ai°gPiJ+~ Z COV((~i, Olj)Pij,
j = l i=l
j = l i=l
(10)
W. Joerding where 0ij = cov(ei, ej) and
var
0/ie
0/i ~
E(Gti). The first term on the fight in Equation (10) is
>_ 0.
\i=1
The second term corresponds to the second term on the fight of (3). In the case of I = 2 it was possible to show that this second term is positive. I have not been able to show that this is also the case for I > 2, but we can plausibly so assume. Note cov(&i, &j)Pij = var(&i) var(ei) > 0 for i = j . For i :# j , consider the case of P,7 > 0. This means that agents i a n d j have f o ~ t errors that are positively correlated, which would sensibly indicate that their market weights would also be positively correlated. A n analogous argument for Pij < 0 suggests that p~j and cov(t~i, t~j) have the same sign and consequently cov(&i, &j) pij >- 0 for all i, j . Equation (3) will now be satisfied if RI - var(P - S ) - II1 > 0. Unfortunately, the previous analysis, finding a vector 0/* = (0/*, . . . , 0/*), which minimized R1 and then checks whether or not Rt(0/*) is positive, leads to intractable algebraic problems. It is not clear that Equation (3) can be satisfied. For example, if the weight of each agent goes to 0 as the n u m b e r of agents increases, then RI -~ - VI as I ~ oo; so Equation (3) cannot be satisfied in the limit. 19 Consider the special case Vj = V2 for j = 2, . . . , I and pij = 0 for i * j,2° and assume this results in 0/j = 0/2 f o r j = 2, . . . , L 21 With these restrictions RI(0/) can be written as22
R,(0/) = ~.~ 0/~ Vi+ ~ i
var(Si) V / - VI
i (1 --O/I)
= 0/~ Vl + V2
2
I-----f- ~"v a r ( ~ 3 Vl + ( I - 1)var(62) 1"2 - I'I1.
F r o m the first-order condition, 0/* = V2 / [V1(I - 1) + V2] minimizes RI(0/) and is also the weight that satisfies the statistical interpretation of market efficiency. Some tedious algebra shows that
Rt(0/I*) =
I"11"2
(I-1)VI+ V2
+ var(c71) V1 + ( I - 1)var(62) II2- V1.
(12)
Observe that Rt(0/~ is most likely to be satisfied for large V2, large var(&t), and small Vl. Although R l ( a * ) ~ - Vl as I ~ 00, 23 for finite I i t will always be possible to find values o f Vi, V2, var(&0, and var(&2) for which RI(0/*) > 0 and relative M S E tests would be
19If the mean oti ~ 0 as 1 ~ 0o for all i, then cov(o~i,txj) ~ 0 i,j = 1, 2, • - . , / a s I ~ 0o, because &i > 0 for all i. 2o Major sources of complications in Equation (10) are the covariance terms p~j, i =#j. Little is known about the covariance between forecast errors of different agents. The observation of hunching in forecasts suggests that these errors are positively correlated, which would make Equation (3) easier to satisfy. Reasons exist, however, to suspect that these errors are negatively correlated (Joerding and Hallagan 1982), which would make Equation (3) more difficult to satisfy. 2~ A sufficient condition for this to be true is that agents 2, " -., I have the same utility functions, so this assumption may greatly reduce the generality of the model. 22 " ' 2 Wnte EquaUon (11) as RAet) = ~xl VI + ~i~2 oti2 V~2 + var(~xl)Vi + Xi~2 var (&i)Vi - VI. Note that ~xj = a2 for allj > 2 implies % = (1 - ~l) / (I - 1) for allj >_ 2. Clearly, var(dj) = var(d2) for allj -> 2. Now use Vj = V2 for allj _.>_2 and rearrange terms to obtain Equation (11). 23 Note that this implicitly requires var(d2) --' 0 faster than (I - 1) goes to infinity. ~
Mean-Square Error Tests for Market Efficiency
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invalid. 24 An argument similar to one presented earlier indicates a sequence of plausible markets exists that would generate the necessary values of var(d0 and var(d2). Note that the results of this section cannot be used to argue that relative MSE tests can be justified by simply assuming perfectly competitive conditions. Let V(K) be the marketforecast variance for a market with K agents. The same conditions that imply limr~** RK(oq) = - Vl also mean limr~o. V(K) = 0, and relative MSE tests become impossible. The problem for relative MSE tests is not the result of being overly tendentious about assumptions. The problem is that, even in an efficient market, if the market weights are non-zero, then coming up with some sequence of market weights that invalidates a relative MSE test may be possible, whereas zero weights would not induce agents to purchase information and trade in the market: this is a conundrum similar to that investigated in a series of papers, the most recent of which is by Grossman and Stiglitz (1980).
Conclusion What are the implications of this paper for empirical studies using the relative MSE method? We showed that the discovery of a forecasting method with a smaller MSE than the market's MSE does not mean that an agent can expect positive profits from the use of such a method. In particular, we saw that it is possible some agent to exist in the market with a forecast MSE the same as the discovered forecasting method and yet not earning abnormal profits. As was mentioned in the introduction, this means that empirical studies of market efficiency using the relative MSE approach must assess the risk-adjusted profitability of the forecasting method. This assessment can only be done in the context of a specific market model. Additionally, the model must at least implicitly be a general-equilibrium model that relates variation in the market prices to the variation in other economic variables. Such models will be difficult to specify and difficult to estimate, and we would likely need to make many assumptions that make the results tenuous to many. The results of this article are not used to argue against the general approach of using model forecasts to test for market efficiency. Throop (1981) and Leuthold and Hartmann (1979) use a test suggested by Nelson (1973), with much the same results, that also compares model forecasts with market forecasts. This paper only casts doubt on the usefulness of relative MSE tests by showing the perhaps unappreciated degree to which model specification can affect the validity of the test. This should not be construed to mean that the general methods used here have no application to other tests similar to the relative MSE approach. Rather, such tests must be considered individuaUy. Comparing the results of this article to those of Verrecchia (1980), who investigated the "consensus beliefs" definition of market efficiency suggested by Rubinstein (1975), is interesting. According to Verrecchia's definition: A market is efficient with respect to an information set A, say, if the prices it generates are identical to those generated in an otherwise identical economy in which set A describes the information available to each and every market participant .... Verrecchia shows that an efficient market, as defined above, generates price forecasts that have smaller error variances than any individual agent's error variance, results that apparently
54 Note that var(ff0 > 0 is a necessary condition, as in Equation (5) earlier, because V, > Vl I/2 / [(I - 1) V~ + v:].
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W. Joerding contradict the results in this article. Several comments are in order. First, Verrecchia's model only has buyers, no sellers. This means that the model does not include forecasts by sellers of the risky asset. Second, the Verrecchia model contains rather specific assumptions about market behavior that can lead to validity of relative MSE tests. Third, the consensus-belief definition of market efficiency is very close to the statistical interpretation offered earlier, but it may lead to different results from the zero-profit interpretation. Although the point remains that the relative MSE test may depend on model specification, and plausible alternative specifications may invalidate the test, it would nevertheless be of interest to investigate this issue using the consensus-belief definition of market efficiency.
References Brooks, W., Wiebe, H., I-Iier, J., and Francis, C. Spring 1980. Testing a model for predicting commodity future prices. American Economists XXIV:24-31. Cornell, B. 1977. Spot rates, forward rates and exchange market efficiency. Journal of Financial Economics 4:55-65. Fama, E. 1970. Efficient capital markets: A review of theory and empirical work. Journal of Finance 33:383-423. Figlewski, S. 1978. Market "efficiency" in a market with heterogeneous information. Journal of Political Economy 86:581-597. - - . 1982. Information diversity and market behavior. Journal of Finance 37:87-102. Grossman, S. 1977. The existence of futures markets, noisy rational expectations and informational externalities. Review of Economic Studies 44:431-441. Grossman, S., and Stiglitz, J. 1980. The impossibility of informationally efficient markets. American Economic Review 69:393-408. Harvey, A. C. 1981. The Econometric Analysis of Time Series. New York: Halstead Press. Jordan, J. S., and Radner, R. 1982. Rational expectations in microeconomic models: An overview. Journal of Economic Theory 26:201-223. Joerding, W., and Hallagan, W. 1982. Technical and fundamental analysis: A case of polymorphic equilibrium, unpublished manuscript. LeRoy, S. F. 1982. Expectations models of asset prlces: A survey of theory. Journal of Finance 37:185-217. Leuthold, R., and Hartmann, P. 1979. A semi-strong form evaluation of the efficiency of the hog futures market. American Journal of Agricultural Economics 61:482-488. --. 1980. A semi-strong form evaluation of the efficiency of the hog futures market: Reply. American Journal of Agricultural Economics 62:585-588. Nelson, C. R. 1973. Applied Time Series Analysis. San Francisco: Holden Day. Rausser, G. C., and Carter, C. 1983. Futures market efficiency in the soybean complex. Review o f Economics and Statistics 65:469-478. Rendleman, R., and Carabini, C. 1979. The efficiency of the treasury bill futures market. Journal of Finance 34:895-914. Rubinstein, M. 1975. Securities market efficiency in an Arrow-Debreu economy. American Economic Review 65:812-824. Shefrin, H. M. 1983. Inferior Forecasters, Cycles, and the Efficient-Markets Hypothesis. A comment, forthcoming in Journal of Political Economy 92:156-161. Sonnenschein, H. 1982. Market excess demand functions. Econometrica 50:549-563. Throop, A. Fall 1981. Interest rate forecasts and market efficiency. San Francisco Federal Reserve Bulletin 29-43. Verrecchia, R. 1980. Consensus beliefs, information acquisition, and market information efficiency. American Economic Review 69:874-884. 1982. Information acquisition in a noisy rational expectations economy. Econometrica 50:1415-1431.
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The Model Dependence of Relative Mean-Square Error Tests for Market Efficiency Wayne Joerding
Over the past few years use of forecasting models to test the efficient markets hypothesis, EMH, has been increasing. An example of such a test is the relative mean-square error (MSE) test in which discovery of a forecasting model with a smaller MSE than that of the futures price is evidence against the EMH. Although the validity of EMH tests are dependent on the validity of a market model, in the case of relative MSE tests the degree of this dependence does not seem to have been recognized. This article shows that within a class of plausible market models models exist for which the relative MSE test is not valid. The conclusion reached is that quite specific details of a market model may be necessary in order for relative MSE tests to be valid and thus these tests are not very robust.
Introduction Tests of market efficiency have long been recognized as simultaneously tests of market equilibrium models (Fama 1970), although the degree of this dependence is sometimes ignored. The purpose of this article is to point out the importance of this dependence for interpreting relative mean-square error tests of market efficiency. Relative mean-square error tests examine the mean-square error (MSE) of market forecasts relative to forecasts based on some statistical forecasting method. If the MSE of the statistical forecast is less than the MSE of a market forecast, then it is taken as evidence against the efficient-markets hypothesis, t The problem with these tests, and the main point of the paper, is that in an efficient market in which agents have rational expectations, market forecasts do not necessarily have a smaller MSE than every individual forecast. CorneU (1977) uses a relative MSE test for efficiency in the foreign-exchange market. He
Such relative MSE tests can be tests of either the weak or semi-strong form of the efficient-markets hypothesis, because the statistical forecast uses only past prices or ineludel publicly available information.
Address reprint requests to Wayne Joerding, Deparurtent of Economies, Washington State University, Pullman, Washington 99164. The author is indebted to Russ Fuller, Jon Sonstellie, and Jeff Krautkraemer for helpful comments. All errors are the responsibility of the author. Journal of Economics and Business © 1986 Temple University
0148-6195/86/$03.50
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W. Joerding states: In an efficient market, where the forward rate incorporates all information relevant to forecasting the future spot rate, the standard deviation of these errors should be less than or equal to the standard deviation of the forecast errors derived from any alternative model. Cornell finds that, for six out of seven currencies, a simple autoregressive model outperforms the forward rate, but he considers this evidence against the efficient-markets hypothesis as tentative and possibly the result of chance. Although Cornell points out the dependence of efficient-markets tests on model specification, he does not cite this fact as justification for his reluctance to interpret the results as evidence against the efficient-markets hypothesis. Instead, he depends on a result of chance argument. Because this article demonstrates the crucial dependence of relative MSE tests on model specification, one can interpret the results of this article as an additional reason for reluctance in accepting Cornell's findings as evidence against the efficient-markets hypothesis. Leuthold and Hamnan (1979) use a relative MSE test in their study of efficiency in the hog futures market. In this case, a simple econometric model is used to forecast spot prices. The model MSE of forecasts is found to be less than the futures market MSE of forecasts, which they take to imply inefficiency in the hog futures market. Rausser and Carter (1983) obtain similar results for futures markets in the soybean complex. Reasons other than those presented here exist for concluding that the existence of a forecasting method with lower MSE than the futures market is not proof that the relevant market is inefficient. If information cost is positive, then information will only be used up to the point that marginal benefit equals marginal cost. Thus, the existence of a model that has a lower MSE than the futures market but is unable to earn an abnormal profit is fully consistent with market efficiency. 2 This article shows that information cost is not the crucial issue. Even if information cost is 0, the discovery of a model with a lower MSE than the futures market is not sufficient to conclude that the futures market is inefficient. More generally, the discovery of a forecasting method with a lower MSE than the market is not inconsistent with participation in the market by an agent with a forecast MSE as low as the discovered forecast method's MSE, and yet the agent does not earn abnormal profits. Consequently, the relative MSE method requires assessment of the risk-adjusted profits possible with the discovered forecast method. Such assessment is only possible in the context of a specific market model, is very difficult, and is often tenuous. The paper proceeds by first presenting a model of an efficient futures market and a discussion of the defin fion of market efficiency that is used. The main result is first presented using a two-person model and then generalized to a multiperson model. A conclusion presents some implications and limitations of the results.
The Model Consider a purely speculative market with two agents in which the net supply of the commodity or asset is 0.3 Each agent comes to the market with an initial expectation of the
2 For some interesting results in this area see Leuthold and Hartmann (1980), Brooks et al. (1980), and Rendleman and Carabini (1979). 3 Speculators hold uncovered buy/sell orders and profit from the difference between the price of the contract and subsequent spot price. Thus, if the forward price is less than the spot price then those speculators holding buy contracts (long position) will make a profit and those speculators holding sell contracts (short position) will lose wealth.
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future period's spot price, which she compares to the futures price and then forms demands for futures contracts. These price expectations are dependent on an information set that includes the futures price and is different for each agent. Thus, it is necessary to allow recontracting until agents are satisfied with their own private forecasts and their trades. The issues concerning the convergence of such tatontment processes are ignored here by assuming that the rational-expectations equilibrium price exists and can be achieved by the market. 4 Because the results of the article do not depend on the cost of information, it is not explicitly considered in the analysis. Suppose the market opens at the beginning of each period t = 1, 2, • .-, and establishes the equilibrium futures price Pt as a function of equilibrium private forecasts PIt and P2t. At the end of the period the spot price St is revealed and profits (positive or negative) registered. Let Dit be the equilibrium excess demand, for contracts, by agent i in period t. Although the optimization problem is not modeled, it is proper and useful to think of Dit as the demand for one type of asset (among several) and the result of maximizing expected utility subject to a budget constraint. Thus, Dit is a function of Pt, Pit, the prices of other assets, and must satisfy the usual restrictions on excess-demand functions. The condition for market equilibrium used here is that the aggregate excess demand equal 0:D1 + D2 = 0. (Henceforth, the subscript t will only be used where necessary to avoid confusion.) If D~ > 0, then i must expect the spot price to be greater than the forward price, Pi > P. The reverse is true for Di < 0. Consider the case D, > 0, then P1 > P. But this implies that D2 < 0 (because Dl + DE = 0) and/)2 < P. Clearly, P is between P, and/)2. If D1 < 0, then a similar argument again indicates that P is between Pl and P2. Conversely, if P > Pl, P2, then both agents would want to sell contracts, causing the price to decline. If P < P~, P2, then both agents would want to buy contracts, causing the price to rise. Consequently, in equilibrium it will always be possible to express P as: 5
Pt=(ttPlt+(1-ott)P2t
0-<~t-< 1,
(1)
where the weights 6t and (1 - ~t) are the relative market weights of agents 1 and 2 in period t. Equation (1) depends on our assumption of 0 net supply of the asset. In general, with positive supply and risk-averse agents the market price is less than the weighted average of individual forecasts (LeRoy 1982). The relative market weight of an agent for any particular period depends on factors that determine the demands D~ and Dz. For example, the weights are a function of the risk aversity and relative wealth of the agents. The degree of risk aversity by an agent is potentially a function of wealth, or other variables, which may be affected by stochastic variables from other markets. In general, an agent who brings greater wealth to the market than another agent has more influence on the equilibrium forward price than does an agent with less wealth, ceterus paribus. But the wealth of an agent is affected by the past random operation of all markets in the economy and thus is a random variable. Specifically, this means the weight t~t is a random variable. Consequently, Pt is unconditionally stochastic, because &t, Pit, and P2t are all stochastic. If the past history of the economy is known, then 6t is determined (in the context of a specific model) but PIt and P2t are stochastic. Given the information set available to each agent and the history of the market, Pt is nonstochastic. 4 Someexogenousnoise in the market, such as randomsupply, mustexist to preventan agent from using the futures price to infer the informationof other agents; see Grossmanand Stiglitz (1980). Here, noise is inherent in the random spot price. Also see Jordan and Radner (1982). See Figlewski(1978) for an exampleof how Equation (1) may be derived from a specificmodel of wealth maximization.
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W. Joerding Assume agents have rational expectations, then Pit = St + eit, where St is the spot price at the end of the period, E(eit) = 0, var(eit) - V/, and cov(elt, e2t) m /9.6 This assumption causes the forward price to be an unbiased forecast of the spot price, meaning that relative M S E tests reduce to relative variance tests. The forecast errors el and e2 are different because each agent is assumed to have different information and (or) to use a different forecast method. The combination of different information sets and forecast methods can result in agents with very different forecasting abilities, thus, assume I/1 _< 1/2. This assumption can be justified by differences in the risk aversion of the agents in the presence of costly information (see Verrecchia 1982). 7 Moreover, assume agents do not often radically change thgir sources of information or forecast method, so that VI and V2 are constants. The foregoing completes the description of the model; we finish this section by showing that the variance & can be arbitrarily specified, s To see the issue, note that the transition o f t~t to t~t÷l depends on the realization o f ~lt and ~2t, because o f the redistribution o f wealth. Consequently, the distribution of t~ t depends on the distribution of elt and e2t; that is, var(~) is a function of VI, V2, and p. But many other factors also affect a t , and its distribution, and these other factors allow enough flexibility in permissible values of a t so that E (&) and var(t~) can be arbitrarily specified. To see this let {Pit, PEt} be a sequence of price forecasts by agents 1 and 2. Next select an arbitrary sequence of weights {a[} with var(ett') an arbitrary constant. 9 W e can now define P / - ~[ Pit + (1 - ott' )P2t. An economy or market model is defined by specifying the preferences and resources for each agent. If the only constraint on the market is that agents maximize utility subject to a budget constraint, then it is known that Pt' is the equilibrium price for some economy. An application of Theorem 4 in Sonnenschein (1982) shows that some economy even exists for which P / i s the unique equilibrium price. In other words, we can find a sequence of market models, one for each period, subject only to budget-constrained utility maximization and Pareto optimality, for which P / is the unique equilibrium price and var(c~t') can be chosen arbitrarily. Thus, although the variance of the actual weights depends on Fl, V2, and p, no logical restriction exists on permissible values for var(t~), l0 This fact will be useful later.
M a r k e t E f f i c i e n c y Il The term efficiency has been used in several ways in the literature. In welfare economics, an efficient economy is one that results in a Pareto optimal allocation of resources. In statistics, an efficient estimator or forecaster is one with the smallest dispersion in a certain class of 6 What sign this covariance should be is not clear; a positive covariance between forecast errors seems plausible if the agents use similar forecast methods. 7 If information cost is zero, then I/1 = I"2. All of the results presented here continue to hold under this special case, which substantiates the claim made in the introduction that information cost is not crucial. Choosing agent 1 to have the lower forecast variance is arbitrary. s Of course we require var(&) _< 1/4. 9 For example, in Grossman (1977) the weights are functions only of the risk aversion of the agents. An arbitrary sequence {a/} can easily be generated by the proper specification of how risk aversion by the agents changes over time, perhaps in response to stochastic shocks from other markets. Clearly, such a specification of preferences cannot be ruled out by assuming expected utility maximization subject to a budget constraint. ~0An alternate way to look at this problem is to remember that & comes from D~ + /:)2 = 0. Suppose that Dz is a function of P, - P = el - c~e~ - (1 - ~)e2, then, for arbitrary excess demand function Di, it is not hard to believe that Vz, V2, and p impose no constraints on the possible values of var(cT). This is because of the great generality of utility maximization subject to a budget constraint. H In the case that agents demand a risk premium, the following results apply by simply taking "zero profits" to mean "zero risk-adjusted profits."
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estimators or forecasters. In financial economics a market is efficient relative to an information set if the price correctly reflects all the information in the relevant set. This definition has been interpreted in at least two ways: 1) in the statistical sense that the futures price is an efficient forecaster of the spot price (see Figlewski 1978; 1982); and 2) in the sense that the futures price is such that no trades based on available information can have a positive expected profit (see Fama 1970). Both of these interpretations are applicable in this paper. Let ere be the weight on Pit if Pt is an efficient forecaster of St in the sense of minimizing error-forecast variance. Because at is a stochastic variable, a natural characterization of the statistical interpretation for market efficiency is E (ft) --- o~ = ae, which requires that the market price be on average an efficient forecaster. The zero expected profit condition is E (~it) = 0, where ~,t is the profit of agent i in period t, and requires that average profits over time are zero. Profits for agent i in period t are ~?it = Dit(St - Pt). For example, if Pi > P and S > P, then Di > 0and $it> 0. Note that profits are dependent on f through P. To see why, consider the extreme case of f t very close to 1. In this situation, the futures price essentially equals P , , making less likely that agent one is on the correct side of the market and experiences a profitable trade. ~2Let 7~i(ft) : E (~it; fit) be expected profits for period t given the weight ft. It is assumed that there exists a unique ao that solves ri(ao) -- 0 and that oto is an attractive equilibrium in the sense that, after a random shock, market forces tend to redistribute wealth so that f t tends toward Oto.13 As an attractive equilibrium, ao is some point of central tendency for the random variable fit. Thus, the 0 expected profit interpretation is characterized by specifying Cto as the mean of at; that is, E ( 4 ) - ct = ao.
Preliminary Ideas The validity of a relative MSE test depends on the condition that a statistical model with an MSE smaller than the market's MSE implies that the information used by the statistical model is not properly incorporated in the futures price. 14 With the interpretations of market efficiency used here this means either that the market is not, on average, an efficient forecaster or some agent who used the statistical model would have positive expected profits. The rest of this paper considers the question: Is it possible in an efficient market (either interpretation) for some agent (agent 1) to use a forecaster with an MSE less than the market's MSE? Define V(a) - E ( P t - St) 2, which expresses the MSE of the market price as a function o f a . The answer to the above question is yes if either V(Ole)> V 1 or V(olo) > Vl,
(2)
remember a -= E (5). If Equation (2) holds for some market then a relative MSE test for market efficiency in that market is not valid. The analysis below shows conditions under which Equation (2) is true.
12In this model, a trade will be profitable (unprofitable) as the agent's private forecast is on the same (opposite) side of the forward price as the spot price. As P gets closer to Pi, the probability of a profitable trade diminishes. This effect can even be strong enough to prevent a more accurate agent from bankrupting a less accurate agent, see Figlewski (1982). ~3Formally, ~ri(c~t) ~ 0 as ~ ~ cto. See Figlewski (1978; 1982) for an example of a specific model that has this characteristic. ~4The informationreferred to would be knowledgeor estimates of the past relationship between the spot price and a set of predetermined variables.
W. Joerding
232
Relative MSE Tests With Two Agents A realization of c~t depends on how the past sequence of forecast errors, P/, - Sk, i = 1, 2 and k = t - 1, t - 2, • • • affected the wealth of the agents. Because P i is a rational forecast, the forecast errors are serially uncorrelated, which implies that &t is independent of Pit - St, Harvey (1981:55). t5 Rewrite Equation (1) as P - S = 6 ( P l - S ) + (1 - c~)(P2 - S ) and take the variance of both sides to obtain V(cz) = VF + var(el - e2)var(c~)
(3)
where VF = o~2Vl + (1 - o02V2 + 2c~(1 - c0 cov(el, e2). 16 S u b s t i t u t i n g Equation (3)into Equation (2), and treating cz as a variable, see that Equation (2) is satisfied if R(c0 -
var(el-e2)o~2-2( V2-p)a+
V 2 - V! + v a r ( e l - e 2 ) v a r ( ~ ) > 0 .
(4)
The quadratic R ( a ) is a parabola that opens upward and has a m i n i m u m at c~* - (V2 - p) / Var(Cl -- ~2)" 17
Because a exists for which 0 _< a _< 1 and R (a) > 0, and a can be arbitrarily specified, then markets exist (abstract) for which V ( a ) > Vl. W e are interested, however, in possible values for a that characterize our definitions of market efficiency, o~oand ae. That is, we want to know i f R ( a e ) > 0 or R(c~o) > 0 is possible. One approach to establishing the existence of efficient markets for which the relative MSE test is invalid is to formulate a specific model, derive the relationship between a , var(t~), and V1, V2, and 0, then check Equation (4). For example, consider the model of Grossman and Stiglitz (1980) with identical risk aversion by all agents and 0 = 0, then cte = V2 / (VI + V2) and var(c~) -- 0. We can see that Equation (4) is not satisfied in this case. Here we take an alternative approach of checking whether R ( . ) is positive definite by evaluating R ( a * ) . is Observe that R ( a * ) > 0 if and only if var(&) > [ ~ - O2]/[var(el - e2)] 2.
(5)
Relative M S E tests are not valid for any market that satisfies Equation (5). Note that the right side of Equation (5) can be less than 1/4, the upper bound on vat(6) imposed by the condition 0 __. 6 _< 1. Because a market exists for any arbitrary value of var(6) we can conclude that efficient markets exists for which Equation (5) is satisfied. That is, market models exist for which the relative M S E test is not valid. Recall that the demonstration that var(6) can be arbitrary, depending on the possibility of constructing a sequence of market models With the necessary distribution of 6. Specifically, because an agent's market weight depends on his or her risk aversion, the necessary variation
~5Without this condition, Equation (3) becomes more complicated. It is possible to derive conditions under which the basic argument below is valid but little insight is gained and the market would not be composed of agents with rational forecasts. ~6Equation (3) can be derived from standard formulas for var(XY), where X and Y are independent random variables. It is also necessary to observe that cov[oTel,(1 - 6)e2] = cY(l - a) cov(el, e2) - var(6) cov(el, e2), andvar(~j - ~2) = Vi + 1:2 - 2cov(~l, e2). ~7 Derivation of a* implicitly assumes that v a r ( ~ ) is not a function of its own mean. I a m not aware of any strong arguments for either a positive or negative relationship, but suppose d var(6) / da > 0, then the minimum of R(o0 would occur at an a < ~*. J8 The weight o : is also interesting because it is the weight that should be attached to agent l's forecast if P is to be a minimumvariance linear forecaster of S (see Figlewski 1982). The weight a* can also be shown to imply 0 expected profits for a model in which agents use a mean-variance analysis for investment decisions (Figlewski 1978).
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233
in 5 can be due to changes in risk aversion, which could occur because of random shocks to wealth from other markets in a general economy (see footnote 8). As another example, Figlewski (1978; 1982) presents a model in which 5 depends on random shocks to wealth, arising from forecast errors. Shefrin (1983) has shown that in this model the value ao is an unstable equilibrium, since 5 will almost surely not equal ao this instability may produce enough variation in 5 to satisfy Equation (5). Whether or not this could happen depends on the parameters Vn, V2, and p, among others. The point is that such possibilities cannot be eliminated by assuming expected utility maximization subject to a budget constraint without specifying a particular market model, and this model must be a general equilibrium model that limits (perhaps implicitly) the impact of shocks from other markets. Consequently, relative MSE tests of market efficiency, as other tests, depend on model specification and are not robust against alternative specifications. The above results do not depend on positive information cost. If information is free then all agents should have the same forecast ability, Vz -- V2. But this would in no way change Equation (5) or its implications. The reason that the relative MSE test is not sufficient to imply market inefficiency has nothing to do with information cost. To see this, consider the similarities and differences between a market and the weighted mean, P* = o r * P i t ( 1 a*)P2t, as forecasting mechanisms. Both are weighted averages of the private forecasts, P~ and P2. The weighted mean, however, would apply constant weights to each private forecast, the optimal weights are a* and (1 - a*), whereas the market can only apply stochastic weights. The weights must be stochastic because they are affected by prior stochastic outcomes of the market and other shocks. If the weights are independent of the private forecasts, as in our case, the mechanism with stochastic weights must have a larger variance than the constant-weight method. All we have shown is that the increased variance is large enough to cause the market-forecast variance to be greater than the forecast variance of one of the private forecasts, V > VI. In contrast, if the weights are chosen optimally, then the variance of the weighted-mean mechanism cannot be greater than the variance of any private forecast, var(P*) _< lit. The results point out the danger in casually attributing properties of optimal statistical models to market mechanisms or in assuming that the exhaustion of potential expected utility-maximizing trades causes market prices to behave like optimal statistical forecasters.
More than Two Agents If I > 2 agents exist, then Equation (1) becomes 1
e : Z ,Pi, i=1
where 1 E (~i= 1, 0 ~-~t~i~< 1. i=1
(9)
The market's forecast-error variance is now 1
var(P-S)--Z
I
1
1
Z ai°gPiJ+~ Z COV((~i, Olj)Pij,
j = l i=l
j = l i=l
(10)
W. Joerding where 0ij = cov(ei, ej) and
var
0/ie
0/i ~
E(Gti). The first term on the fight in Equation (10) is
>_ 0.
\i=1
The second term corresponds to the second term on the fight of (3). In the case of I = 2 it was possible to show that this second term is positive. I have not been able to show that this is also the case for I > 2, but we can plausibly so assume. Note cov(&i, &j)Pij = var(&i) var(ei) > 0 for i = j . For i :# j , consider the case of P,7 > 0. This means that agents i a n d j have f o ~ t errors that are positively correlated, which would sensibly indicate that their market weights would also be positively correlated. A n analogous argument for Pij < 0 suggests that p~j and cov(t~i, t~j) have the same sign and consequently cov(&i, &j) pij >- 0 for all i, j . Equation (3) will now be satisfied if RI - var(P - S ) - II1 > 0. Unfortunately, the previous analysis, finding a vector 0/* = (0/*, . . . , 0/*), which minimized R1 and then checks whether or not Rt(0/*) is positive, leads to intractable algebraic problems. It is not clear that Equation (3) can be satisfied. For example, if the weight of each agent goes to 0 as the n u m b e r of agents increases, then RI -~ - VI as I ~ oo; so Equation (3) cannot be satisfied in the limit. 19 Consider the special case Vj = V2 for j = 2, . . . , I and pij = 0 for i * j,2° and assume this results in 0/j = 0/2 f o r j = 2, . . . , L 21 With these restrictions RI(0/) can be written as22
R,(0/) = ~.~ 0/~ Vi+ ~ i
var(Si) V / - VI
i (1 --O/I)
= 0/~ Vl + V2
2
I-----f- ~"v a r ( ~ 3 Vl + ( I - 1)var(62) 1"2 - I'I1.
F r o m the first-order condition, 0/* = V2 / [V1(I - 1) + V2] minimizes RI(0/) and is also the weight that satisfies the statistical interpretation of market efficiency. Some tedious algebra shows that
Rt(0/I*) =
I"11"2
(I-1)VI+ V2
+ var(c71) V1 + ( I - 1)var(62) II2- V1.
(12)
Observe that Rt(0/~ is most likely to be satisfied for large V2, large var(&t), and small Vl. Although R l ( a * ) ~ - Vl as I ~ 00, 23 for finite I i t will always be possible to find values o f Vi, V2, var(&0, and var(&2) for which RI(0/*) > 0 and relative M S E tests would be
19If the mean oti ~ 0 as 1 ~ 0o for all i, then cov(o~i,txj) ~ 0 i,j = 1, 2, • - . , / a s I ~ 0o, because &i > 0 for all i. 2o Major sources of complications in Equation (10) are the covariance terms p~j, i =#j. Little is known about the covariance between forecast errors of different agents. The observation of hunching in forecasts suggests that these errors are positively correlated, which would make Equation (3) easier to satisfy. Reasons exist, however, to suspect that these errors are negatively correlated (Joerding and Hallagan 1982), which would make Equation (3) more difficult to satisfy. 2~ A sufficient condition for this to be true is that agents 2, " -., I have the same utility functions, so this assumption may greatly reduce the generality of the model. 22 " ' 2 Wnte EquaUon (11) as RAet) = ~xl VI + ~i~2 oti2 V~2 + var(~xl)Vi + Xi~2 var (&i)Vi - VI. Note that ~xj = a2 for allj > 2 implies % = (1 - ~l) / (I - 1) for allj >_ 2. Clearly, var(dj) = var(d2) for allj -> 2. Now use Vj = V2 for allj _.>_2 and rearrange terms to obtain Equation (11). 23 Note that this implicitly requires var(d2) --' 0 faster than (I - 1) goes to infinity. ~
Mean-Square Error Tests for Market Efficiency
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invalid. 24 An argument similar to one presented earlier indicates a sequence of plausible markets exists that would generate the necessary values of var(d0 and var(d2). Note that the results of this section cannot be used to argue that relative MSE tests can be justified by simply assuming perfectly competitive conditions. Let V(K) be the marketforecast variance for a market with K agents. The same conditions that imply limr~** RK(oq) = - Vl also mean limr~o. V(K) = 0, and relative MSE tests become impossible. The problem for relative MSE tests is not the result of being overly tendentious about assumptions. The problem is that, even in an efficient market, if the market weights are non-zero, then coming up with some sequence of market weights that invalidates a relative MSE test may be possible, whereas zero weights would not induce agents to purchase information and trade in the market: this is a conundrum similar to that investigated in a series of papers, the most recent of which is by Grossman and Stiglitz (1980).
Conclusion What are the implications of this paper for empirical studies using the relative MSE method? We showed that the discovery of a forecasting method with a smaller MSE than the market's MSE does not mean that an agent can expect positive profits from the use of such a method. In particular, we saw that it is possible some agent to exist in the market with a forecast MSE the same as the discovered forecasting method and yet not earning abnormal profits. As was mentioned in the introduction, this means that empirical studies of market efficiency using the relative MSE approach must assess the risk-adjusted profitability of the forecasting method. This assessment can only be done in the context of a specific market model. Additionally, the model must at least implicitly be a general-equilibrium model that relates variation in the market prices to the variation in other economic variables. Such models will be difficult to specify and difficult to estimate, and we would likely need to make many assumptions that make the results tenuous to many. The results of this article are not used to argue against the general approach of using model forecasts to test for market efficiency. Throop (1981) and Leuthold and Hartmann (1979) use a test suggested by Nelson (1973), with much the same results, that also compares model forecasts with market forecasts. This paper only casts doubt on the usefulness of relative MSE tests by showing the perhaps unappreciated degree to which model specification can affect the validity of the test. This should not be construed to mean that the general methods used here have no application to other tests similar to the relative MSE approach. Rather, such tests must be considered individuaUy. Comparing the results of this article to those of Verrecchia (1980), who investigated the "consensus beliefs" definition of market efficiency suggested by Rubinstein (1975), is interesting. According to Verrecchia's definition: A market is efficient with respect to an information set A, say, if the prices it generates are identical to those generated in an otherwise identical economy in which set A describes the information available to each and every market participant .... Verrecchia shows that an efficient market, as defined above, generates price forecasts that have smaller error variances than any individual agent's error variance, results that apparently
54 Note that var(ff0 > 0 is a necessary condition, as in Equation (5) earlier, because V, > Vl I/2 / [(I - 1) V~ + v:].
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W. Joerding contradict the results in this article. Several comments are in order. First, Verrecchia's model only has buyers, no sellers. This means that the model does not include forecasts by sellers of the risky asset. Second, the Verrecchia model contains rather specific assumptions about market behavior that can lead to validity of relative MSE tests. Third, the consensus-belief definition of market efficiency is very close to the statistical interpretation offered earlier, but it may lead to different results from the zero-profit interpretation. Although the point remains that the relative MSE test may depend on model specification, and plausible alternative specifications may invalidate the test, it would nevertheless be of interest to investigate this issue using the consensus-belief definition of market efficiency.
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