Rational
Under general conditions, linear decision rules of agents with rational expectations are equivalent to restricted error corrections. However, empirical rejections of rational expectation restrictions are the rule, rather than the exception, in macroeconomics. Rejections often are conditioned on the assumption that agents aim to smooth only the levels of actions or are subject to geometric random delays. Generalizations of dynamic frictions on agent activities are suggested that yield closed-form, higher-order decision rules with improved statistical fits and infrequent rejections of rational expectations restrictions. Properties of these generalized `rational' error corrections are illustrated for producer pricing in manufacturing industries.
dismal. A survey of existent journal publications by Ericsson and Irons (1995) summarizes an
extensive accumulation of empirical evidence against rational expectations, including frequent
rejections of rational expectations overidentifying restrictions. A review of policy simulation
models used by central banks and international agencies, such as documented in Bryant et al.
(1993), indicates that many key rational expectations specifications are either imposed or fit by
rough empirical calibrations.
Macroeconomists have adopted a variety of responses to the absence of strong empirical
support for rational expectations. One is to maintain the rational expectations hypothesis, while
aiming to interpret a more limited subset of empirical regularities as discussed by Kydland and
(1991). Another approach is to view rational expectations as a limiting case of complete
information in a more general treatment of the information processing abilities of agents, such as
the "bounded rationality" models of learning reviewed in Sargent (1993). Closely related is the
position that rational expectations are more likely to prevail at low frequencies, a view compatible
with tests of long-run theoretical restrictions in cointegrating relationships, as discussed by Watson
(1994). Others reject the hypothesis of model-based rational expectations, such as the use in
Ericsson and Hendry (1991) of rule-of-thumb extrapolations.
This paper examines an alternative explanation for the poor empirical properties of rational
expectations models. Because most rational expectations restrictions are inherently dynamic
due to the forecasting requirements of constraints on dynamic adjustment, a plausible source
of difficulty could be the sharp friction priors typically imposed on agent responses. The
standard dynamic specification in rational expectations models utilizes geometric lead response
schedules to anticipated future events and geometric lag responses to recent "news." Because
this two-sided geometric response schedule is not a clear implication of economic theory, a
generalized polynomial frictions specification is explored in this paper. Suggested interpretations
of generalized frictions range from costs of adjusting weighted averages of current and lagged
actions to convolutions of geometric random delay distributions of agent responses.
For difference-stationary variables, the decision rules based on generalized frictions are shown
to be isomorphic to a class of "rational" error correction models. 1 The parameters of the
1 Discussion in this paper is aimed primarily at decision rules for difference-stationary variables, a specification
that is not rejected by standard tests of the integration order of most macroeconomic aggregates in postwar samples.
decision rules are subject to tight cross-coefficient restrictions due to polynomial frictions and
cross-equation restrictions due to the assumption of rational expectations. A closed-form solution
that incorporates these restrictions is derived using two companion matrix systems, one a lead
system for the forward planning required by polynomial frictions and the other a lag system
associated with the agents' forecast model. 2
Rational error corrections under polynomial frictions inherit many desirable properties of
atheoretic reduced-form time-series models, including serially independent equation residuals and
small standard errors relative to many theory-based alternatives. However, whereas conventional
error corrections and VARs have been criticized for in-sample overfitting attributable to large
numbers of estimated parameters, rational error corrections are subject to many overidentifying
restrictions which substantially reduce the number of free parameters.
To provide concrete illustrations of the consequences of restrictive priors on adjustment
costs, linear decision rules associated with alternative specifications of frictions are estimated
for producer pricing in several manufacturing industries. The basic specification is based on the
assumptions of difference-stationary producer prices and stationary price markups over costs of
production, assumptions that are not rejected for postwar U.S. producer prices. 3
A difficulty with interpreting many reported rejections of rational expectations overidentifying
restrictions is that the rejections could be due also to misspecified models. An advantage of
the side-by-side comparisons reported in this paper is that several empirical problems, including
rejections of rational expectations restrictions, are unambiguously linked to the use of second-order
Euler equations rather than higher-order Euler equations.
The paper is organized as follows. Section I summarizes dynamic properties of the restricted
error correction that is implied by the standard decision rule with geometric response schedules.
Section II derives the error correction format of rational decision rules implied by a generalized
polynomial description of frictions. Several interpretations of polynomial frictions are suggested.
Section III presents empirical estimates of decision rules for industry pricing, comparing the
geometric responses of the standard decision rule with the responses of higher-order decision
rules. A tractable method of two-stage maximum likelihood estimation of rational error corrections
2 Although this paper develops a general framework for formulating higher-order error correction decision rules,
there are several precedents for recasting decision rules as restricted error corrections or applying higher-order linear
decision rules to macroeconomic aggregates. Nickell (1985) appears to be the first paper to explore similarities
between second-order decision rules and error correction models. Generally, fourth-order decision rules have been
confined to empirical inventory models, such as Blanchard (1983), Hall, Henry, and Wren-Lewis (1986), Callen, Hall,
and Henry (1990), and Cuthbertson and Gasparro (1993). Exceptions are the applications of fourth-order decision
rules to employment in the coal industry by Pesaran (1991) and to the price of manufactured goods in Price (1992).
3 State-independent frictions in pricing would not be expected to hold over all possible states, such as episodes of
extreme hyperinflation. Nevertheless, linear time series models appear to be a useful way to analyze sticky pricing in
periods of moderate inflation, vid. Sims (1992) and Christiano, Eichenbaum, and Evans (1994).
is derived in the Appendix, including corrections required for the sampling errors of the second
stage. Section IV concludes.
I. Rational Error Correction under Geometric Frictions
Just as one-sided polynomials in the lag operator are characteristic of atheoretic, linear time
series models, two-sided polynomials in the lead and lag operators are a defining characteristic
of linear models of rational behavior. The principal vehicle of analysis in this paper is the dynamic
first-order condition linking a decision variable, y t , to its equilibrium objective, y
denotes expectations based on information at the end of t \Gamma 1; A(L) is a backwards
scalar polynomial in the lag operator, L j x polynomial in the
lead operator, F j x
In contrast to conventional backwards-looking time series models of the relationship between
y t and y
t , a notable feature of equation (1) is that the expectation of the current decision variable,
fy t
is a two-sided moving average of past and expected future values of the desired equilibrium, y   .
The agent response schedule, w i , determines the relative importance of past and future, whereP
Many models used in macroeconomics assume these relative importance weights are
adequately represented by two-sided geometric response schedules.
In the case of linear decision rules, there are two prominent rationalizations of geometric
response schedules. One is that changes in the level of the decision variable are subject to quadratic
adjustment costs, and this strictly convex friction induces geometric adjustments of the decision
variable toward its equilibrium. The second interpretation is that each agent is subjected to a
geometric distribution of random delays in adjustment, so that the level selected for a decision
variable in a given period is a weighted average of desired target settings over the expected interval
between allowable resets. As noted by Rotemberg (1996), the assumption of a geometric random
delay distribution leads to aggregate behavior that is observationally equivalent to that generated
by the assumption of quadratic costs on adjusting the level of the aggregate decision variable.
Under either interpretation, the required Euler equation is second-order, implying that the
polynomial components of equation (1) are first-order polynomials,
1. As in Tinsley (1970), the optimal decision rule that
satisfies the Euler equation and the relevant endpoint (initial and transversality) conditions implies
partial adjustment of the decision variable to a discounted weighted average of expected forward
positions of the desired equilibrium. 4 This decision rule solution is obtained by multiplying the
Euler equation (1) by the inverse of the lead polynomial, A(BF
\Deltay
where -B is the geometric discount factor over the infinite planning horizon.
To complete the derivation of the conventional decision rule, the data generating process of the
forcing term, y
t , must be provided. In linear decision rules, the decision variable, y, is cointegrated
with the conditional equilibrium target, y   , which is defined by a linear function of q variables. 5
y
Agent forecasts of the target, y
t+i , are assumed in this paper to be generated by p-order VARs in
the q arguments, x 0
Thus, the effective information set of agents is obtained by
stacking p lags of the regressors, into a single vector, z 0
t\Gammap ]. The companion form
of the agent VAR forecast model is denoted by E t fz t Consequently, agent forecasts
of the forcing term of the Euler equation are generated by
fy
where the pq \Theta 1 selector vector, '   , contains the coefficients of the cointegrating relationship, (4),
that defines the equilibrium objective.
Substituting forecasts of the forcing term from (5) into the decision rule (3) yields
Historical references and discussion of issues in formulating linear decision rules with dynamic forcing terms are
found in Tinsley (1970, 1971).
5 As needed, the q variables include deterministic trend, intercept, and seasonal dummy series. The concept of a
dynamic, frictionless equilibrium is discussed by Frisch (1936).
6 Companion forms for a variety of linear forecasting models are illustrated in Swamy and Tinsley (1980).
As shown in the first line of (6), given the matrix of the forecast model coefficients, H , and the
discount factor, B, a single friction parameter determines a geometric pattern of rational dynamic
A(1). The second line of (6) indicates that the weighted sum of expected
forward changes in the forcing term, E t f\Deltay
t+i g, can be reduced to the inner product of a restricted
coefficient vector times the industry information set
where equation (7) provides a transparent summary of rational expectations overidentifying
restrictions on the coefficient vector of the agent information vector, z t\Gamma1 .
Thus, there are two principal differences between the dynamic format of the "rational" error
correction in equation (6) and the format of a conventional error correction with p lags of each
regressor. First, only one lag of the decision variable is specified by the second-order decision
rule in equation (6), whereas up to additional lags of the first-difference of the decision
variable may appear in a conventional error correction. Second, as indicated in equation (7), the
coefficient vector, h   , is completely determined by cross-coefficient restrictions due to the friction
parameter in the error correction coefficient, -, and cross-equation restrictions due to the forecast
model coefficients, H . By contrast, the coefficients of the information vector in a conventional
error correction are unrestricted. 7
Residual independence and rational expectations overidentifying restrictions are frequently
rejected in macroeconomic studies of rational behavior. 8 One interpretation of the often
disappointing empirical performances of conventional two-root decision rules, including rejections
of rational expectation restrictions, is simply that expectations of actual agents may not be formed
under conditions required for rational expectations, such as symmetric access to full system
information by all agents. However, the limited dynamic specifications illustrated in equation
suggest another contributing factor-the arbitrary prior that agents responses are adequately
captured by two-sided geometric response schedules. The next section explores the dynamic
formats of higher-order Euler equations and rational error corrections associated with a polynomial
generalization of agent response schedules.
7 In conventional error corrections, an unrestricted coefficient vector is applied to the first-difference of the
information vector.
8 Examples of studies that report rejections of restrictions imposed by rational expectations include Sargent
(1978), Meese (1980), Rotemberg (1982), Pindyck and Rotemberg (1983), and Shapiro (1986). Significant residual
autocorrelations are indicated for rational expectations decision rules in Epstein and Denny (1983), Abel and Blanchard
(1986), and Muscatelli (1989). See also the extensive rational expectations literature review in Ericsson and Irons
(1995).
II. Rational Error Corrections under Polynomial Frictions
The standard second-order Euler equation provides a two-sided geometric description of agent
responses to anticipated and past events. The geometric schedules are determined by the roots
of the first-order component polynomials, A(L) and A(BF). This section discusses decision rules
associated with higher-order Euler equations, where the degree of the component polynomials is
increased to m ? 1. These m-order polynomials are obtained by relaxing the modeling prior that
agents aim to smooth only the levels of decision variables or, equivalently, that stochastic delays
of decision variable adjustment are generated only by geometric distributions. The first subsection
derives the closed form of rational error correction decision rules associated with higher-order
Euler equations. The second subsection briefly reviews some categories of polynomial frictions on
agent actions that are consistent with 2m-order Euler equations.
II.1 Solving for Rational Error Correction Decision Rules with Higher-Order Euler Equations
As demonstrated later in this section, the Euler equation under polynomial frictions is the same as
that initially shown in equation (1), except the factor polynomials are now m-order polynomials,
instead of first-order polynomials.
To obtain the decision rule in the case of 2m-order Euler equations, multiply by the inverse of
the lead polynomial, A(BF ) \Gamma1 , to give
The analytical solution for the forcing term of this equation, f t , is obtained by introducing a
second companion system that describes the forward motion of the (m \Gamma 1) \Theta 1 lead vector,
over the planning horizon,
where the m \Theta 1 selector vector, ' m , has a one in the mth element and zeroes
G is the m \Theta m bottom row companion matrix of the lead polynomial, A(BF ),
Substituting the solution of the forcing term, f t , from equation (9) into (8) yields the
generalized 2m-order decision rule,
\Deltay
In the first line of (10), the lag polynomial is partitioned into a level and difference format, A(L) j
is an (m-1)-order polynomial whose coefficients
are moving sums of the coefficients of A(.), as shown in the Appendix. The second line of (10)
partitions the forward path of the target into an initial level and forward differences, and uses the
to isolate the error correction ``gap,'' y
.
The final step in deriving the decision rule under polynomial frictions is to eliminate forecasts
of the equilibrium path, E t fy
t+i g, using the companion form of the forecast model. Substituting
forecasts of forward changes E t f\Deltay
t+i g, from (5) into (10) provides the closed form solution of
the generalized rational error correction decision rule 9
There are two major differences in the dynamic formats of the polynomial frictions version of
rational error correction in equation (11) and the conventional geometric frictions variant shown
earlier in (6). First, use of the m-order component polynomial, A(L), introduces lags of the
dependent variable, A   (L)\Deltay t\Gamma1 to accompany the single lag of the decision variable in the error
correction term. 10 Second, in contrast to the single forward discount factor, -B, employed by the
decision rule under geometric frictions in (3), forecasts of anticipated changes in the equilibrium
path are now discounted by the m eigenvalues embedded in the lead companion matrix, G.
Just as the rational expectations restrictions were summarized by a coefficient vector under the
geometric frictions prior, coefficient restrictions of the generalized rational error correction also
can be compactly stated. As indicated in the last line of (11), the sum of the forward-looking terms
is again equivalent to the inner product of a weighting vector, h   , and the information vector, z
9 The direct solution format using companion forms may be compared with alternative solution methods for linear
decision rules ranging from partial fractions expansions of the characteristic roots in Hansen and Sargent (1980) to
Schur decompositions in Anderson and Moore (1985) and Anderson, Hansen, McGrattan, and Sargent (1996).
will often be much smaller that the number of lags of the dependent variable in a conventional error
Two types of restrictions are imposed on the coefficient vector, h   , under polynomial frictions:
the cross-coefficient restrictions imposed by the component polynomials of the Euler equation, as
summarized by the forward companion coefficient matrix G; and the cross-equation restrictions
imposed by the agents' forecast model, summarized by the lag companion coefficient matrix H .
To reveal these restrictions, successive column stacks are applied to simplify the solution for the
coefficient vector, h   . 11
\Gamma1\Omega [H
\Gamma1\Omega [H
This definition of the restricted coefficient vector in (12) provides a closed form solution for
the linear decision rule under polynomial frictions, and a summary of differences between the
unrestricted regression coefficients in a conventional error correction and the tightly restricted
coefficients of the information vector, z t\Gamma1 , in a generalized error correction.
Finally, equation (11) indicates the friction parameters of the generalized rational error
correction are collected in A(1) and A   (:). This separable format is convenient for maximum
likelihood estimation by an iterative sequence of linear regressions, as discussed in the Appendix.
II.2 Higher-Order Euler Equations due to Polynomial Frictions
As noted earlier, standard rationalizations of the linear, second-order Euler equation are based on
the assumption of: (1) quadratic costs of adjusting the level of the decision variable, or (2) a discrete
geometric distribution of random delays in adjustments of the decision variable. Polynomial
extensions of these two prior specifications are discussed.
Adjustment costs on weighted averages of decision variables
One class of generalized frictions is associated with agent efforts to smooth weighted averages
of current and lagged values of decision variables. This smoothing is represented by quadratic
penalties on C(L)y t , where C(L) is a m-order polynomial in the lag operator,
and
Agents choose a sequence of decision variables that minimize the criterion, - t , defined by a
11 The column stack of the product of three matrices is denoted by
0\Omega A)vecB,
the Kronecker product.
second-order expansion of profits or utility around the path of equilibrium settings,
The associated Euler equation is a 2m-order equation in the lead and lag operators,
where the s k coefficients in the first line of (14) are defined by coefficients of the friction
polynomial, C(L).
Because the extended Euler equation in the first line of (14) is symmetric in L and BF , the equation
is unaffected if these two operator expressions are interchanged. This, in turn, implies that a
solution of the characteristic equation of the Euler equation, say - accompanied by the
reciprocal solution, B-. Consequently, the characteristic equation can be factored as shown in in
the second line of equation (14). The format of this Euler equation is the same as that shown earlier
in equation (1), except the factor polynomials are now explicitly identified as m-order polynomials.
The criterion and second-order Euler equation associated with the standard specification that
quadratic costs apply only to changes in the level of the decision instrument are nested in equations
and (14), respectively, for the standard prior assumption that
Smoothing levels and differences
In a frequent interpretation of higher-order Euler equations, the decision variable, y t , is an asset
stock, and adjustment costs may be applicable not only to changes in the level of the asset but also
changes in the first-difference. An example is optimal inventory planning, where y t indicates the
inventory stock at the end of period t. The change in inventories, production
less sales. Given exogenous sales, the assumption of quadratic costs on the level of production
implies a quadratic penalty on changes in the planned level of inventory, c 1 Similarly,
quadratic costs associated with changes in the rate of production can be represented by a quadratic
smoothing penalty on changes in the planned first-difference of the inventory stock, c 1
Thus, in the example of inventory modeling, it is not uncommon to assume polynomial frictions of
the general form, P 2
A generalized criterion for smoothing levels and higher-order differences,
decision variables is
with the associated 2m-order Euler equation, E t fc 0
Smoothing weighted averages
A second generalization is the case where quadratic penalties are associated with weighted moving
averages of the decision instrument. For example, let y t denote new labor hires by a firm in
period t. Suppose various job families within the firm require different durations of training by
supervisors, and the number of employees occupied by training in a given period is represented
by a fixed distribution of recent vintages of new hires, c 0 y associated
with variations in the rate of training may be approximated by the quadratic penalty,
which is a restatement of the polynomial friction
specification in equation (13). 13
Another variation is the extension of quadratic penalties from smoothed one-period changes
in the level of the decision variable, (1 \Gamma L)y t+i , to smoothed changes in moving averages,
include seasonal or term contracts where some costs are
associated with one-period averages, others with two-period averages, and so on. The criterion in
this instance takes the form,
with the associated 2m-order Euler equation, E t fc 0
Stochastic response delays
Given the tractability of linear first-order conditions, the quadratic adjustment cost specification is
widely used to characterize optimal adjustment. Applications include decision variables such as
nominal prices, extending from the seminal paper by Rotemberg (1982) to the recent example of
Hairault and Portier (1993), although the assumption of strictly increasing costs in the size of price
12 See Hall, Henry, and Wren-Lewis (1986), Callen, Hall, and Henry (1990), and Cuthbertson and Gasparro (1993).
13 In all examples, note that linear cost components can be accommodated by redefining the equilibrium target, y
of the relevant decision variable.
adjustments is often disputed. However, as noted by Rotemberg (1996), the aggregate response
arising from the quadratic adjustment cost model is equivalent to the aggregate adjustment of
agents subject to random decision delays drawn from an exponential distribution, as proposed
by Calvo (1983). Although it appears to have received little attention outside the field of
dynamic pricing, the stochastic delay model would appear to be a useful framework for modeling
adjustments in other market contexts when agent responses are dependent on unpredictable
transmissions of decisions, such as distributed production or communication networks. 14
In a discrete-time implementation of the stochastic delay approach, each agent j controls a
decision variable, y j;t , with the associated equilibrium trajectory, y
j;t . When adjustment of a
decision variable occurs, the movement to equilibrium is complete but the timing of adjustment is
stochastic. The probability of an agent adjustment in the ith period of the planning horizon, having
not adjusted in the preceding periods, is r i . The schedule of future adjustment probabilities is
represented by the lead polynomial, r(F the r i are nonnegative
and 1. Using a discrete geometric distribution as the analogue of the exponential response
distribution in Calvo (1983), the generating function is r(F is the
first-order polynomial, A(F
Given the constraint that the decision variable must remain at the level selected, say ~y
the next allowable adjustment period, the optimal setting that minimizes the expected sum of
squared deviations from the discounted path of equilibrium settings is E t f~y j;t
g.
Using simple sum aggregation, the aggregate of decision variables adjusted in t is
g. The aggregate decision variable is a normalized average of current and past vintage
decisions that survive in t. In the example of a geometric delay distribution, the survival probability
in t of a past decision variable setting from t\Gammai is proportional to r i . 15 Thus, the generating function
for the normalized survival probabilities over an infinite horizon is r(L), and the aggregate decision
variable in period t may be represented by
fy t
The lag polynomial, r(L), remains to the left of the expectations operator on the right-hand-side
of (17) to ensure that the lagged expectations embedded in past decisions are represented in
14 Effects of costly and stochastic communications in distributed production are discussed in Board and Tinsley
(1996). See also Bertsekas and Tsitsiklis (1989) for representative configurations of communication networks.
15 The hazard function, the ratio of the adjustment probability, r i , to the survival probability at lag i, is constant for
the univariate geometric distribution, Johnson, Kotz, and Kemp (1993).
the current aggregate, y t . 16 Replacing the generating functions by the polynomial components
yields the analogue to the familiar second-order decision rule for the aggregate decision variable,
g.
Aggregated delay schedules
In principle, the choice of the appropriate stochastic delay distribution should be an empirical issue.
Remaining within the polynomial frictions framework of this paper, the approach suggested below
considers higher-order polynomial approximations of more general stochastic delay distributions.
As indicated, these generalizations can be interpreted as convolutions of component geometric
distributions.
It is unlikely that agents have perfect information about the distribution of delays or stochastic
congestion in future decisions. Suppose, for example, agents may be confronted by a "low-cost"
response distribution, r 1 expected
response lag from the first distribution is smaller than that of the second. However, draws from
either delay distribution are random. In this example, the generating function of the effective
response probabilities is the product of the generating functions of the component response
probabilities,
More generally, in the case of random aggregation over m geometric response schedules, the
aggregate reset of decision variables adjusted in t is
g. As in the
case of a univariate geometric distribution, a constant-hazard approximation permits the survival
distribution of past vintage decisions to be represented by the polynomial generating function of
the stochastic delays,
Thus, under an m-order polynomial stochastic delay distribution,
As discussed in Taylor (1993), some model specifications and instrumental methods of estimating decision rules
move the equivalent of the lag polynomial, r(L), inside the expectation operator.
17 If v and w are non-negative independent random variables, the generating function of the convolution, v + w, is
the product of the generating functions of v and w, Feller (1968).
The mean absolute error of constant-hazard approximations of quarterly survival probabilities is about :02
percentage points for the first sixteen lags in the empirical decision rules using polynomial frictions discussed in
the next section. The reason for the relatively modest approximation errors can be shown using a partial fractions
representation of the approximation error. Denote the partial fractions expansion of an m-order polynomial generating
function by: A(1)A(L)
denote the characteristic roots of A(L).
The error of the constant-hazard approximation of the survival probability at lag i is
denotes the sum of the survival probabilities. In the case of the univariate geometric
distribution, -
- is equal to the single root, and the approximation error is zero. In the case of convoluted geometric
distributions with a single dominant root, as with empirical examples in this paper, - is generally very close to the
modulus of the dominant root. In addition, error components associated with smaller roots decay rapidly with lag i
because the spread for a smaller root, scaled by powers of that root, -
.
the aggregate decision variable in period t is represented by
fy t
Y
Y
Substituting in the component polynomials of the generating functions for the stochastic delay
and survival probabilities yields a solution for the aggregate decision variable that is identical to
that derived earlier for the decision rule under polynomial frictions
where the component polynomials, A(L) and A(BF ), are m-order.
III. Empirical Examples of Rational Error Corrections for Industry Pricing
Empirical contrasts of second-order and higher-order rational error corrections are discussed in
this section. The examples used are pricing decision rules of six SIC two-digit manufacturing
industries: textiles, lumber, rubber & plastics, primary metals, motor vehicles, and scientific
instruments. 19 In addition to an expected difference in statistical fits, the rational expectation
overidentifying restrictions are rejected by all but one of the second-order decision rules and by
none of the higher-order rules.
III.1 The Equilibrium Price
The equilibrium log price of the output of industry j with s j identical producers is represented by
is the log markup by producers and mc j is the log of marginal cost. Ignoring strategic
considerations, the markup is
is the price elasticity of demand,
and the monopoly and competitive solutions are obtained as s Gross production
is Cobb-Douglas in both purchased materials and rented services of primary factors. Also, returns
to scale are constant so that the log of marginal cost is proportional to the weighted average of log
input prices,
19 Motor vehicles is a large subset of the SIC two-digit industry, transportation equipment.
j is the log price of primary commodity production inputs, p i
j is the log price of
intermediate materials purchased from other industries, p v
j is the log unit cost of value added in
the j industry, and ' c
As indicated in (21), input price regressors are specific to industry j and constructed from
input-output weightings of industry producer prices. Industry producer prices at the SIC two-digit
level of aggregation are generally available only from the mid-1980s, and industry prices in earlier
periods were assembled from specific commodity prices, often at lower levels of aggregation. The
industry log unit cost of primary factors, p v
estimated by the log of hourly earnings, w j;t ,
less the log of trend productivity, ae j;t . The latter was constructed from smoothed estimates of the
log of industry industrial production less the log of industry employment hours.
Industry producer prices in the U.S. do not reject the hypothesis of difference-stationarity
over postwar samples. A common format was used to explore cointegration constructions of the
equilibrium price of each industry
Given data limitations of the trend productivity estimates, both industry log productivity trends,
ae j;t and time trends, t, were added as additional regressors, and the intercept, c 0 contains both
the log margin, - j , as well as proportional mean errors in measurements of unit cost inputs. The
relevant industry input share weights, (' c
are displayed in the initial columns of Table 1.
These share weights are not estimated but defined by benchmark input-output estimates, obtained
by manipulating the Bureau of Economic Analysis (1991) industry use and make tables.
As shown in the columns headed by f(t) in Table 1, additional trend productivity regressors
were required for cointegration in three industries. As noted above, the log price of primary factors
already incorporates trend productivity, p v
. Because this assigns a weight of \Gamma' v to
ae j , the additional positive coefficients lower the effective contribution of the trend productivity
constructions. Finally, as shown in the last column of Table 1, the hypothesis that the cointegrating
discrepancy,
j;t is I(1) is rejected at the 90% confidence level or higher for all industries.
III.2 Empirical Estimates of Pricing Decision Rules under Geometric Frictions

Table

presents summary statistics for second-order pricing rules of the six manufacturing
industries, estimated under the prior of geometric frictions. Estimated parameters of the industry
decision rules and VAR forecast model parameters were obtained by the maximum likelihood
estimator described in the Appendix. 20
Industry prices are quarterly averages of monthly, seasonally unadjusted series from the US Bureau of Labor
database on commodity and industry producer prices for the 1954-1995 sample. As noted earlier, the equilibrium
The estimated error correction coefficients, A(1) in Table 2, indicate the average quarterly
reduction rates planned for the price "gap," p
, of each industry. The proportion of
explained variation in quarterly price changes can be substantial, with R 2 in three industries
ranging from .3 to .6. The row in Table 2 labeled \DeltaR 2 (%) indicates that for four industries,
the modal source of explained variation is the sample variability of industry forecasts of future
equilibrium prices, as captured by the rational forecast term, h 0
z t\Gamma1 . Table 2 also contains the
estimated mean lag of producer responses to unexpected shocks and the estimated mean lead of
responses to anticipated events. The mean lead of the industry planning horizon is typically smaller
than the mean lag response due to discounting of forward events.
Three characteristics of these estimated equations suggest significant dynamic specification
problems. First, the mean lag responses appear to be unusually large relative to previous estimates
of response lags for manufacturing prices. 21 Second, serial independence of the residuals is
rejected for all but one industry at 95% confidence levels. Although it is possible that producers
may have serially correlated information that has not been included in the industry forecast models,
it is plausible also that residual correlations could be due to misspecifications of the frictions in
producer responses.
A final indication of potential misspecifications is indicated in the bottom row of Table 2. This
row, labeled LR(h   jz lists the rejection probabilities of a likelihood ratio test to determine
if the data prefers an unrestricted forecast model of forward equilibrium price changes to the
rational forecasts embedded in the geometric frictions version of rational error correction. With
one exception (motor vehicles), the rational expectations overidentifying restrictions are rejected
at 99% confidence levels.
III.3 Empirical Estimates of Pricing Decision Rules under Polynomial Frictions
Estimates of the industry pricing rules under the polynomial generalization of frictions are listed
in

Table

3. Because the conventional two-root decision rule, nested in the generalized
frictions model, it is interesting to note that additional lags of the dependent variable are always
price forecast model for each industry is a VAR containing the equilibrium price and the prices of production inputs.
Although seasonality is not pronounced in most industry prices (one exception is motor vehicles, as noted later), all
industry VARs contained at least four lags in regressors, and seasonal dummies were added to all VAR and error
correction equations. To reduce space, estimates of equation intercepts and seasonal dummy coefficients are not
reported in the tables. In all equations presented here, the quarterly discount factor, B, was set to :98, approximating
the postwar annual real return to equity of about 8%; empirical results are not noticeably altered by moderate variations
in B.
In analysis of the Stigler-Kindahl data of producers' prices, Carlton (1986) reports an average adjustment
frequency of about once a year. Reduced form regressions by Blanchard (1987, Table 8) for the U.S. manufacturing
price aggregate indicate a mean lag of about two quarters. By contrast, the levels friction model in Rotemberg (1982)
suggests a mean lag of about 12 quarters for the U.S. GDP deflator.
significant in the industry pricing models, generally consistent with polynomial components of
3. 22 In the case of motor vehicles, the preferred specification is
5, requiring four lags of the dependent variable. This is due to a significant seasonal pattern in
the producer price of motor vehicles which could not be adequately captured by fixed seasonal
dummies.
Without exception, all of the problems noted for the estimated decision rules under geometric
frictions in Table 2 are eliminated under polynomial frictions. The percentage of explained
considerably higher for most industries in Table 3; mean lags are more plausible;
the assumption of serially independent residuals is retained in all industries; and the rejection
probabilities in the bottom row in Table 3 indicate that the rational expectations overidentifying
restrictions are not rejected at confidence levels of 95% or higher. The latter is noteworthy because
rejections of rational expectations overidentifying restrictions are often interpreted as evidence
of non-rational forecasting by agents or of inadequate specifications of agent forecast models of
forcing terms. Because the only difference between industry model specifications used in the
side-by-side comparisons of Table 2 and Table 3 is the degree of the Euler equation polynomials,
m, the culprit, at least in these examples and for the statistical properties considered, is rigid priors
on the specification of dynamic frictions.
More intuitive insights into the dynamic effects of the higher-order lag and lead polynomials
are obtained by rearranging the Euler equation to define the current period response weights to lags
and expected leads of the forcing term, E t fp
t+i g, implied by the industry decision rules,
where negative subscripts, i ! 0, denote responses to lagged events and positive subscripts, i ? 0,
responses to anticipated events. The lag and lead weights of the six estimated industry decision
rules are displayed in the panels of Figure 1. The dotted lines are the friction weights generated by
the two-root decision rules reported in Table 2 and the solid lines are the friction weights
associated with the 2m-root decision rules (m ? 1) shown in Table 3.
Several effects of the generalization of frictions are apparent from the plots of the industry
friction weights in Figure 1. In each panel, the vertical line is positioned in the current period
22 This is not an isolated finding. Every macroeconomic aggregate to which the generalized frictions model was fit in
the FRB/US macroeconomic model also rejected the conventional prior that
As discussed in a recent literature survey by Taylor (1997), many studies of empirical staggered contract models for
wages do not support geometric response schedules, including an estimated bimodal distribution of contract lengths
in Levin (1991).
0). The mean lag of responses to unanticipated events is captured by the weighted average
of lags using the friction weights to the left of center. The friction weights are nearly symmetric
about the current period with the mean lead, associated with weights to the right of center, slightly
smaller than the mean lag due to the discounting of future events. Thus, the net mean response lag
to perfectly anticipated events is small for most industries.
Larger mean leads require longer planning horizons and are characteristic of the flatter friction
weight distributions indicated by the dotted lines in Figure 1 for the two-root decision rules,
Thus, vertical distances between the two sets of friction weight distributions in each panel are
indicative of differences between the industry mean leads of Table 2 and the corresponding mean
leads of Table 3.
As shown in the panels of Figure 1, relatively low-order friction polynomials, A(B)A(BF
can generate a variety of flexible shapes, including the seasonal weights at distances of \Sigma4 quarters
indicated for the motor vehicles industry, SIC 371. Some estimated friction distributions are
relatively flat for several quarters, while others fall off rapidly from the modal weight in the
current quarter. The plots in Figure 1 do not support the two-sided geometric distribution prior
that is consistent with two-root decision rules, In almost all industries, the drawback
of a two-sided geometric response schedule is an inability to capture relatively stronger industry
responses to events in a one - or two-quarter neighborhood of the current quarter.
Cyclical Variations in Pricing Margins
Thus far, specification of the desired industry price settings has proceeded under the assumption
that relevant arguments are difference-stationary, with estimation of industry "target" paths, p
t , by
cointegration. However, economic theory may suggest additional stationary variables as possible
arguments of the desired target. 23 If there is prior information that agents' perceptions of the
forcing terms of Euler equations are significantly influenced by additional variables, this prior
information should be introduced into the model to avoid possible distortions in estimated frictions.
An example is useful to illustrate how the distinction between friction and forecast parameters is
maintained for trial arguments of the forcing term by imposing dynamic friction restrictions.
In the present example of a price markup model, cyclical indicators such as industry capacity
utilization rates may capture variations in planned margins, - t , due to boom or bust pricing
strategies or cyclical movements in the price elasticity of demand. The Euler equation for industry
price is restated to include the effect of current and lagged industry utilization rates, u t\Gammai , 24 on the
As discussed by Wickens (1996), economic theory is required for structural interpretations of cointegrations.
Industry utilization rates are constructed by the FRB staff from surveys of capacity utilization, see Raddock
(1985). Log industry utilization rates are stationary, and the error correction responses of capacity output are either
insignificant or an order of magnitude smaller than that of industry output. Sample means are removed so u t can be
current price target,
t continues to denote I(1) arguments of the equilibrium target, and D(L) is an m 0 -order
polynomial in L. It is convenient to assume m which is the minimal order necessary to
distinguish between pricing effects of changing utilization rates and effects of higher or lower
levels of capacity utilization,
Multiplying through by the inverse of the lead polynomial, A(BF substituting in
forecasts from the agents' information set, z defines the augmented rational error correction
where the industry price decision rule now contains an infinite-horizon forecast of forward industry
utilization rates, discounted by the m eigenvalues contained in the frictions companion matrix, G.
Using the simplifying operations discussed earlier, the closed-form of the extended rational error
correction solution is
z
z
The additional coefficient vectors, h u k
discounted sums of the expected forward
path of the industry utilization rate,
m\Omega I n ][I mn \Gamma
where the n \Theta 1 selector vectors, ' u k
, locate u t\Gammak in the information vector, z t , now extended to
include current and lagged values of the industry utilization rate. Because industry log utilization
rates are stationary, the restricted coefficient vectors in (26) differ somewhat from that derived for
forecasts of difference-stationary trend prices, h   , in (12).
Effects of adding industry utilization rates to the rational error correction are reported in

Table

4. The rejection probabilities in the row labeled LR(D(L)) indicate that expected forward
utilization rates are a significant determinant of pricing at a 90% level of confidence in three of the
six industries. Cyclical markup effects associated with the level of the industry utilization rate are
indicated in the next row of Table 4, labeled D(1). Procyclical margins are indicated for primary
metals (SIC 33) and countercyclical margins for motor vehicles (SIC371).
All significant features of the rational error corrections in Table 3 are retained in Table 4,
including serially independent residuals and nonrejection of the RE overidentifying restrictions
interpreted as industry output deviations from trend or preferred utilization.
which are now extended to include forecasts of forward utilization rates. Thus, the polynomial
frictions description of industry pricing appears to be robust to the addition of a conventional
determinant of cyclical pricing.
IV. Concluding Comments
After two decades of research in macroeconomics, the rational expectations conjecture is a fixture
in theoretical macroeconomic models but is routinely rejected in empirical macroeconomic models
that test the associated overidentifying restrictions. Rather than indicting the rational expectations
assumption, it appears that the main culprit may be the arbitrary tight prior used to characterize
dynamic frictions in macroeconomic models.
The workhorse of macroeconomic descriptions of rational dynamic behavior is the
conventional linear decision rule with two characteristic roots, where one determines the geometric
discount factor of anticipated events and the other provides a geometric description of lagged
responses to unanticipated shocks. The two-sided geometric lead and lag response schedules are
generally motivated by a geometric frictions prior where agents aim to smooth levels of activity
or are subject to geometric random delays. Although it leads to tractable models of economic
behavior, the geometric frictions prior is not based on compelling economic theory and is usually
rejected by macroeconomic data.
An alternative specification of polynomial frictions is suggested in this paper which appears
to eliminate many of the empirical drawbacks of the conventional frictions specification. The
generalized frictions specification can be interpreted as the result of agents that smooth linear
combinations of current and lagged actions or a consequence of convoluted geometric distributions
of stochastic delays in decisions.
Polynomial frictions lead to higher-order Euler equations whose decision rules are solved
generally by numerical techniques. A method of obtaining closed-form solutions is presented that
uses two simple, first-order companion systems: a lead system for the forward planning required
when agent actions are restricted by frictions; and a lag system for the agents' forecast model of the
Euler equation forcing term. A tractable method of maximum likelihood estimation by a sequence
of regressions is outlined in the Appendix.
Empirical models of producer pricing are estimated for six manufacturing industries. The
second-order decision rule implied by the geometric frictions prior is nested within the polynomial
frictions specification and rejected by the data for all industries. The decision rules based on
geometric frictions had poor empirical properties, including overstatement of mean lags, strong
residual correlations, and rejections of rational expectation restrictions. Rational error corrections
using the generalized friction specification eliminated these empirical shortcomings, including
the rejection of rational expectations restrictions. The estimated generalized friction models of
industry pricing generally required fourth-order or sixth-order decision rules.
Polynomial descriptions of frictions define a rich class of rational error correction
specifications. The empirical applications indicate that higher-order decision rules provide
empirical fits comparable to reduced-form error corrections and, unlike the latter, provide useful
distinctions between lags due to forecasts of market events and lags due to constrained agent
responses to these forecasts.


--R

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--TR
error correction;producer pricing;companion systems;rational expectations