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=∑VptRATIONAL 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 correspond￾ing 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