RATIONAL EXPECTATIONS 319 distributed random variables et with zero mean and variance o2 Ee=0, a2 if i=i Ee=10近i≠ Any desired correlogram in the us may be obtained by an appropriate hoice of the ghts wet The price will be a linear function of the same independent disturbances thus (37) 巾=∑W d-0 2E-i The expected price given only information through the (t-1)'st period has the same form as that in(3.7), with the exception that et is replaced by its expected value(which is zero). We therefore have (38) p=WEet+∑Wtet=∑We- If, in general, we let pu, L be the price expected in period t+L on the basis of information available through the tth period, the formula becomes (3.9) p ∑Wtet-t Substituting for the price and the expected price into(3. 1), which reflect the market equilibrium conditions, we obtain (310) Woet a"e=1 wet Et-t Equation(3. 10) is an identity in the es; that is, it must hold whatever values of e, happen to occur. Therefore, the coefficients of the correspond The weights Wi are therefore the followin (3.11a) W 8+y (=1,2,3,…) and price expectations functions in terms of the past history of independent shocks. The problem remains of writing the results in terms of the history of observable variables. We wish to find a relation of the form (312) p=∑Vpt
RATIONAL EXPECTATIONS 319 distributed random variables 8t with zero mean and variance a2: (3.6) co~0 r2 if ij (3.6) 6t =z Wi -Et-i, E8j = 0, E8j = (o ifi#j Any desired correlogram in the u's may be obtained by an appropriate choice of the weights wi. The price will be a linear function of the same independent disturbances; thus 00 (3.7) it- E wiet-iE i=0 The expected price given only information through the (t -1)'st period has the same form as that in (3.7), with the exception that 8t is replaced by its expected value (which is zero). We therefore have (3.800 pe O8 O0 (3.8) pt W0E6t + Wi t-i Wiet-i;E i=l1= If, in general, we let Pt,L be the price expected in period t +L on the basis of information available through the tth period, the formula becomes 00 (3.9) fit-L,L -E Wist-iE i=L Substituting for the price and the expected price into (3.1), which reflect the market equilibrium conditions, we obtain (3. 10) Wo E-t + 1 + )zwi Et-{ = - zSfet-z . A i=1 i{=0 Equation (3.10) is an identity in the e's; that is, it must hold whatever values of ej happen to occur. Therefore, the coefficients of the corresponding ej in the equation must be equal. The weights Wi are therefore the following: (3.1 la) p ze , (3.1 I1b) Wi -+w (i =1,2,3, *).. Equations (3.1 1) give the parameters of the relation between prices and price expectations functions in terms of the past history of independent shocks. The problem remains of writing the results in terms of the history of observable variables. We wish to find a relation of the form 00 (3.12) pt 1Vjfit-1
320 HN F MUTH We solve for the weights V, in terms of the weights W, in the following manner. Substituting from (3.7)and(3.8), we obtai 1+-=v,w+=(vW)4 Since the equality must hold for all shocks, the coefficients must satisfy the equations ViWi-g (=1,2,3, This is a system of equations with a triangular structure, so that it may be If the disturbances are independently distributed, as we assumed before, then we0=-1/B and all the others are zero. Equations(3. 14) therefore 少=0 (315b) pt=十 These are the results obtained before Suppose, at the other extreme, that an exogenous shock affects all future onditions of supply, instead of only the one period. This assumption would be appropriate if it represented how far technological change differed from its trend. Because ut is the sum of all the past es, ue=1(i=0, 1, 2,.) From(3.11) (316a) 1/B, (316b) From 3. 14)it can be seen that the expected price is a geometrically weighted moving average of past prices (3.17) 8 go p This prediction formula has been used by Nerlove [26] to estimate the supply elasticity of certain agricultural commodities. The only difference is that our analysis states that the coefficient of adjustment"in the ex pectations formula should depend on the demand and the supply coeffi- cients. The geometrically weighted moving average forecast is, in fact optimal under slightly more general conditions (when the disturbance is composed of both permanent and transitory components). In that case the coefficient will depend on the relative variances of the two components as well as the supply and demand coefficients. (See [24]
320 JOHN F. MUTH We solve for the weights V1 in terms of the weights Wj in the following manner. Substituting from (3.7) and (3.8), we obtain 00 00 00 00 t (3.13) WiVt- EV IWiet-i-i = V Wi 8t-ti. {=1 ?~=1 i=0 J5 =1 Since the equality must hold for all shocks, the coefficients must satisfy the equations (3.14) Wi VWiy (i = 1,2,3,...). 1=1 This is a system of equations with a triangular structure, so that it may be solved successively for the coefficients V1, V2, V3,.... If the disturbances are independently distributed, as we assumed before, then wO -1 /8 and all the others are zero. Equations (3.14) therefore imply (3.15a) t (3.15b) Pt = P+Wost - lete These are the results obtained before. Suppose, at the other extreme, that an exogenous shock affects all future conditions of supply, instead of only the one period. This assumption would be appropriate if it represented how far technological change differed from its trend. Because ut is the sum of all the past ej, wi 1 (i = 0,1,2,...). From (3.1 1), (3.16a) Wo -1/fl, (3.16b) Wi l/0 +y) From (3.14) it can be seen that the expected price is a geometrically weighted moving average of past prices: (3.17) ( ) . y pt y P t: + yJtThis prediction formula has been used by Nerlove [26] to estimate the supply elasticity of certain agricultural commodities. The only difference is that our analysis states that the "coefficient of adjustment" in the expectations formula should depend on the demand and the supply coefficients. The geometrically weighted moving average forecast is, in fact, optimal under slightly more general conditions (when the disturbance is composed of both permanent and transitory components). In that case the coefficient will depend on the relative variances of the two components as well as the supply and demand coefficients. (See [24].)
RATIONAL EXPECTATION: Deviations from Rationality. Certain imperfections and biases in the expectations may also be analyzed with the methods of this paper. Allowing for cross-sectional differences in expectations is a simple matter, because their aggregate effect is negligible as long as the deviation from the rational forecast for an individual firm is not strongly correlated with those of the others. Modifications are necessary only if the correlation of the errors is large and depends systematically on other explanatory variables. We shall examine the effect of over-discounting current information and of differences in the information possessed by various firms in the industry. Whether such biases in expectations are empirically important remains to be seen I wish only to emphasize that the methods are flexible enough to handle th Let us consider first what happens when expectations tently over- or under-discount the effect of current events. Equation(3.8), which gives the optimal price expectation, will then be replaced by (3.18) p=AW1et-1+∑Wet-4 In other words the weight attached to the most recent exogenous dis- turbance is multiplied by the factor fi, which would be greater than unity if current information is over-discounted and less than unity if it is under discounted If we use(3. 18) for the expected price instead of (3.8)to explain market orice movements, then (3. 11)is replaced by W0=--o (319b 8+fy l (3.19c) (=2,34…) The effect of the biased expectations on price movements depends on the statistical properties of the exogenous disturbances If the disturbances are independent(that is, wo= l and we=0 fori> 1) the biased expectations have no effect. The reason is that successive obser- vations provide no information about future fluctuations. On the other hand, if all the disturbances are of a permanent type(that =1), the properties of the expectations function are significantly affected. To illustrate the magnitude of the differences, the parameters of the function p=∑V
RATIONAL EXPECTATIONS 321 Deviations from Rationality. Certain imperfections and biases in the expectations may also be analyzed with the methods of this paper. Allowing for cross-sectional differences in expectations is a simple matter, because their aggregate effect is negligible as long as the deviation from the rational forecast for an individual firm is not strongly correlated with those of the others. Modifications are necessary only if the correlation of the errors is large and depends systematically on other explanatory variables. We shall examine the effect of over-discounting current information and of differences in the information possessed by various firms in the industry. Whether such biases in expectations are empirically important remains to be seen. I wish only to emphasize that the methods are flexible enough to handle them. Let us consider first what happens when expectations consistently overor under-discount the effect of current events. Equation (3.8), which gives the optimal price expectation, will then be replaced by 00 (3.18) Pt = fi Wiet-i + I Wi Et-i i=2 In other words the weight attached to the most recent exogenous disturbance is multiplied by the factor f1, which would be greater than unity if current information is over-discounted and less than unity if it is underdiscounted. If we use (3.18) for the expected price instead of (3.8) to explain market price movements, then (3.1 1) is replaced by (3.19a) Wo wo (3.19b) WW WI (3.19c) Wi Wi (i = 2,3,4,...). /3+y The effect of the biased expectations on price movements depends on the statistical properties of the exogenous disturbances. If the disturbances are independent (that is, wo =1 and wj = 0 for i > 1), the biased expectations have no effect. The reason is that successive observations provide no information about future fluctuations. On the other hand, if all the disturbances are of a permanent type (that is, w0 = w, = ... = 1), the properties of the expectations function are significantly affected. To illustrate the magnitude of the differences, the parameters of the function 00 pt - V}fit-