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Rational Expectations

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Rational Expectations

Economics 813a 2013-11-18

    ( R.H. Rasche 1997

    Rational Expectations

     The concept of rational expectations was introduced into economics by Muth [1961] in the context of microeconomic models. Muth's revolutionary idea is that

    expectations, as informed predictions of future events, are essentially the same as the

    predictions of (or forecasts from) the relevant economic theory of the process under consideration. To this he added two fundamental points:

     a) information is a scarce commodity and the economic system (and agents) does not waste it,

     b) the way expectations are formed depends specifically on the structure of the relevant system describing the economic process.

     Muth used a simple single market example:

     Demand: C = -(p tt

    e Supply: P = + u ptttt1

     Market Equilibrium: C = P tt

    where C = current consumption, P = current production, and p = the price of the ttt

    commodity sold (note the change from our conventional notation where lower case p's refer to inflation rates). There are three particular assumptions incorporated in these equations: 1) demand for the commodity is deterministic, 2) u is the supply shock term, t

    and 3) that there are no exogenous variables (including no constant terms) in the equations beyond the supply shock. The last assumption is a critical simplifying assumption which Muth rationalizes by arguing that the model as expressed can be considered as measuring deviations from deterministic equilibrium values of the endogenous variables. There is

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    really more than this going on here. Implicitly there is also the assumption that none of the “fundamentals” affecting market equilibrium (other

    than u) differs from its expected value, e.g. that there are no unexpected changes in t

    income that would move the demand curve around.

     The model consists of three equations in four endogenous variables, C, P, p and ttt

    e and one exogenous variable, u. In this particular case the model can be reduced to pttt1

    eone equation in the two endogenous variables p and and the exogenous variable u: ptttt1

    e (1) -(p = + u ptttt1

     In general such models can be reduced to a system of exogenous variables and an equal number of expectations of endogenous variables and the corresponding endogenous variables, using standard matrix techniques as long as the model is linear. The process is to reduce the dimension of the model by substituting out all endogenous variables for which there are no expectational variables.

     Serial correlation of the error terms is critical in determining the solutions to rational expectations models. The general problem is that there will be more endogenous variables than there are equations. We get the additional equations necessary from the rational expectations hypothesis that the expectation (forecast) of any variable constructed at some time, say t-1 is equal to the the mathematical expectation of that variable conditional upon all information available at the time that the expectation is formed. This

    gives the additional equation(s) necessary to complete the model. These equations say that the difference between an endogenous variable and its expectation, the expectational

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    (or forecast) error, cannot be a function of anything that is known at the time that the expectation was formed.

     Taking mathematical expectations of (1), conditional upon information known at t-1 gives:

    e E(-(p) = E() + Eu. pt-1tt-1t-1ttt1

     The typical rational expectations analysis (and Muth's examples in particular) assume that the structure of the model is known, or if it is not, that agents act as if their beliefs about the structure are absolutely correct. This latter assumption is known as certainty equivalence (see Theil [1961]). The operational effect of this assumption is that the parameters of the model (or the estimates of those parameters) can be taken outside of the conditional expectations operator:

    e -(Ep = E() + Eu = Ep + Eu pt-1tt-1t-1tt-1tt-1ttt1

    where the latter equality is possible from the rational expectations hypothesis that Ep = t-1t

    e. ptt1

     From this equation we conclude that for this model, the rational expectations hypothesis predicts that

    -1 (2) Ep = -[+(]Eu t-1tt-1t

     a) assume that u is iid (not serially correlated). Then u cannot be predicted given tt

    einformation available at t-1, hence Eu = 0 and hence = 0. From this and equation pt-1ttt1

    -1(1) we can conclude that p = -(u. tt

     b) assume that u is serially correlated so that t

     u = ~ + ~ + ... + ~ = ~ + ~(L)L t0t1t-1nt-n0tt

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    Economics 813a 2013-11-18 ( R.H. Rasche 1997

    where is an iid random variable. Then from (2) t

    -1-1-1 Ep = -[+(]Eu = -[+(]~E - [+(]~(L)L t-1tt-1t0t-1tt

    -1 = -[+(]~(L)L, since E = 0. tt-1t

    But then using (1):

    -1-1 p = (/()[+(]~(L)L - ([~ + ~(L)L] tt0tt

    -1 = -(+()~(L)L - (~/() t0t

    which is Muth equation 3.11.

     c) assume that the supply shocks are permanent: u = u + where are iid. Then tt-1ttEu = u. From (2): t-1tt-1

    -1-1 Ep = -[+(]Eu = -[+(]u t-1tt-1tt-1

    so from (1)

    -1-1 p = (-/()[-(+()]u -(u tt-1t

    -1-1 = -(+()u - ( t-1t

    ewhich is Muth (3.16). This gives and p in terms of the history of , but the ptttt1

    eadditional question that can be raised is how does relate to the history of p? pttt1Consider

    ep [(/(+()]p + [/(+()]= t-1tt;;21

    -2-1-2 -[((+()]u - [+(] - [(+()]u t-2t-1t-2

    -1-1-1e = -(+() - (+()u = - (+()u = pt-1t-2t-1tt1Therefore:

    ee = [(/(+()]p + [/(+()] ppt-1tt1tt;;21

    so

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    ( R.H. Rasche 1997

    eee = + [(/(+()][p - ] pppt-1tt1tt;;21tt;;21

    which is the expression for the adaptive expectations model. The conclusion from this

    analysis is that for one particular model with only supply shocks and for which the supply

    shocks are permanent then the rational expectations are adaptive, but the adaptation parameter, 0 < (/(+() < 1 is not arbitrary, but is the ratio of the demand elasticity to the sum of the demand and supply elasticities.

     The second model that Muth considered involved a slightly different structure. In this case, production can occur to cover current consumption demand or to accumulate inventories. Desired holdings of end-of-period inventories depend positively upon the expected price appreciation.

     Demand: C = -(p tt

    e Supply: P = + u ptttt1

    e Inventories: I = ( - p) ptttt;1

     Market Equilibrium: C + (I - I) = P ttt-1t

    where I = the end-of-period t stock of inventories so I - I is inventory accumulation ttt-1

    during period t.

     This model can be reduced to one equation in the two endogenous variables p and t

    the one period ahead expectation of prices formed at different points in time:

    ee -(p + - p = (+) - p + u ppttt-1ttt;1tt1

    Assume certainty equivalence and operate on this equation with the conditional mathematical expectation operator based on a t-1 information set, and note that if all

    einformation available at t-1 is utilized in forming expectations then E() = pt-1tt;1

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    E[Ep] = Ep: t-1tt+1t-1t+1

     -(Ep +Ep - Ep = (+)Ep - p + Eu, t-1tt-1t+1t-1tt-1tt-1t-1t

    since p is in the set of information that is available at t-1. Collecting terms and dividing t-1

    by ? 0, this equation can be written as:

     Ep - [2+((+)/]Ep + p = [1/]Eu t-1t+1t-1tt-1t-1t

     The rational expectations equation for this model is more complicated than in the model we considered previously. The problem here is that we have both leads in price

    expectations and lags prices. During the late 1970s the macroeconomics literature devoted considerable attention to techniques for solving such complex rational expectations systems. The conventional technology is to solve the problem either by explicitly extracting roots of a difference equation or using the so-called method of undetermined coefficients.

    Solution of Difference Equations in the Lag Operator

     The above relationship among the expectations on p must hold for any arbitrary t

    future period, so for all j ? 2:

     Ep - [2+((+)/]Ep + Ep = [1/]Eu t-1t+jt-1t+j-1t-1t+j-2t-1t+j-1

    which is a second order difference equation. Under the additional assumption that u is iid, t

    then Eu = 0 and we can reduce the problem to a homogeneous second order difference t-1t+j

    equation which can be written in terms of the lag operator as:

    2 [1 - {2 + ((+)}L + L]Ep = 0. t-1t+j

    2If E[p ] ? 0 then [1 - {2 + ((+)/}L + L] = 0, so we have to find the roots , of t-1t+j12

    the quadratic polynomial in L. The general solution to a second order homogenous difference equation is of the form:

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    ( R.H. Rasche 1997

    -j-j Ep = c + c t-1t+j1122

    subject in this case to the initial condition that Ep = p. t-1t-1t-1

     The roots of the quadratic polynomial in the lag operator from this model are both

    real and one is outside and one is inside the unit circle (a fairly common occurrance in this

    type of rational expectations model). From the quadratic formula:

    ?(,;(,;4?? ?;1;1;11????22(,;??

    ?(,;(,;4?? ?;11;12????22(,;??

    For this particular model:

     2222?(,;(,;4(,;(,;(,;(,;??????;;?;11;?;12;1 12????????????22(,;222??

    -1so = . 12

     Since we are interested only in non explosive solutions to the expectation problem (we want to rule out so-called “rational bubbles”) we need to eliminate from the 2

    -jsolution. To do this set c = 0. Hence Ep = c. But Ep = p = c, so c = 2t-1t+j11t-1t-1t-1111

    -1p = p. With this solution for c, Ep = p and Ep = p. 1t-12t-11t-1t2t-1tt+12t

     McCafferty [1990] (p.334) asserts that the general solution to this problem is of

    the form:

    jj Ep = c + d where < . t-1t+j2121

    But since for this problem = 1 this is equivalent to: 12

    -j-j Ep = c + d. t-1t+j12

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    Economics 813a 2013-11-18 ( R.H. Rasche 1997

    -jHe requires d = 0, so Ep = c, and his solution is the same as that developed here. His t-1t1

    approach is applicable only in the case = 1. 12

     When the expressions for Ep and Ep are substituted into the reduced form t-1ttt+1equation for p we obtain: t

     -(+()p + ,;p = (+)p - p + u t2t2t-1t-1tso

    -1-1 p = [(1-) - ,;][(1-)+(]p - [(1-)+(]u. t222t-12tTherefore for this model p is an AR(1) process. But we know that Ep = p, so it tt-1t2t-1

    -1must be that [(1-) - ,;][(1-)+(] = . 2222

    Note that

    2 ;(,,;;;;;?[()/]21022

    since is a root of this polynomial. Multiply this equation by -: 2

    2 -,; + 2,; + ((+) - = 0 222

    or:

    2 (; + ,; -,; = - ,; - ,; 22222

    so:

     [(1-) + (] = (1 - ) - ,; 2222

    and:

    -1 = [(1-) + (][(1 -) - ,;]. 2222From this the AR(1) process for p reduces to: t

     p = p - [(1-)+(]u. t2t-12t

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     Note that not all problems produce second order difference equations. One period leads in expectations plus contemporaneous variables without any lagged variables will produce a first order difference equation.

    Solution by the Method of Undetermined Coefficients (McCallum, Chapter 8)

     The problem, as above, is to find a solution to the “reduced form” equation:

    ee -(p + - p - (+)= -p + u ppttt-1ttt;1tt1

    Postulate a solution of the form:

     p = p + u t1t-12t

    Note that the things which appear as variables on the right hand side of this equation are all the lagged variables and error terms that appear in the “reduced form” equation. The key to getting the correct solution is guessing correctly on the “state” variables to include in the conjectured solution. Taking conditional expectations gives:

     Ep = p for u ? i.i.d. t-1t1t-1t

    and Ep = p = [p + u]. tt+11t11t-12t

    After the rational expectations hypothesis (and certainty equivalence) is applied to the “reduced form equation” these expressions for the conditional expectations can be substituted to give:

     -([p + u] + ,![p + u] - [p + u] - (+)p 1t-12t11t-12t1t-12t1t-1

     = -p + u t-1t

    Equate the coefficients on p and u on both sides of this equation: t-1t

    2 p: -(! + ,! - ,! = (+) - t-11111

     u: -(! + ,! - ,! = 1 t2212

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    and solve these equations for the unknown parameters of the expectation process, and 1

    : 2

    -1 = -[(1-)+(] and 21

    2 ,! + (-2-(-) + = 0 11

    and note that the quadratic for is the same quadratic that we had to solve in the lag 1

    operator approach. Let = < 1 be the smallest root of the polynomial. (Note that 12

    since the conjecture is that p = p =u, || < 1.0 is required for a stationary process). t1t-1t1

    -1Then p = p - [(1-)+(]u which is the same solution as obtained from the lag t2t-12t

    operator approach.

    Nonhomogeneous Second Order Difference Equations

     Remember that the Muth model does not have any exogenous variables (more

    importantly, the future values of the shock are not predictable from its past history) and as

    a result it produces a homogenous second order difference equation. Suppose that instead

    of a homogenous difference equation of the form:

    2 (1-L - L)Ey = 0 12t-1t+j

    we have difference equations of the form:

    2 (1-L - L)Ey = EX( 12t-1t+jt-1t+j

    where X is a vector of exogenous variables and ( is a vector of constants.

     The solution of such a difference equation is of the form:

    -1-1jj Ey = (1-L)(1-L)EX( + c + c = t-1t+j12t-1t+j1122

    -1-1jj {[/(-)][1-L] - [/(-)][1-L]}EX( + c + c 11212122t-1t+j1122

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