BLS WORKING PAPERS

U.S. DEPARTMENT OF LABOR

Bureau of Labor Statistics

 

OFFICE OF PRICES AND LIVING

CONDITIONS

Estimated U.S. Manufacturing Capital and Technology

Based on an Estimated Dynamic Economic Model

Baoline Chen, Rutgers University

Peter A. Zadrozny, U.S. Bureau of Labor Statistics

Working Paper 351

November 2001

 

The views expressed are those of the authors and do not necessarily reflect the policies of the U.S. Bureau of Labor

Statistics or the views of other staff members. Baoline Chen gratefully acknowledges financial support of the Office of

Research and Sponsored Programs, Rutgers University, and the Ludwig Boltzmann Institute for Economic Policy

Analysis, Vienna, Austria. We thank Michael Binder, Timothy Erickson, Dennis Fixler, Michael Harper, and Randal

Verbrugge for comments and William Gullickson, Kurt Kunze, Lawrence Rosenblum, and Ray Wolfe for help in obtaining

data.
ESTIMATED U.S. MANUFACTURING CAPITAL AND TECHNOLOGY

BASED ON AN ESTIMATED DYNAMIC ECONOMIC MODEL*

 

*

The paper represents the authors' views and does not represent any official

positions of the Bureau of Labor Statistics. Baoline Chen gratefully

acknowledges financial support of the Office of Research and Sponsored

Programs, Rutgers University, and the Ludwig Boltzmann Institute for Economic

Policy Analysis, Vienna, Austria. We thank Michael Binder, Timothy Erickson,

Dennis Fixler, Michael Harper, and Randal Verbrugge for comments and William

Gullickson, Kurt Kunze, Lawrence Rosenblum, and Ray Wolfe for help in

obtaining data.

Baoline Chen

Department of Economics

Rutgers University

Camden, NJ 08102

e-mail: baoline@crab.rutgers.edu

Peter A. Zadrozny

Bureau of Labor Statistics

2 Massachusetts Ave., NE, Room 3105

Washington, DC 20212

e-mail: zadrozny_p@bls.gov

October 30, 2001

JEL Classification: C50, C81, D24, L60

Additional key words: Kalman smoother estimation of unobserved production

capital and total-factor productivity data.

ABSTRACT

Time series of production capital and total factor productivity (or

"technology") are fundamental to understanding the processes of output and

productivity growth. Unfortunately, capital and technology are unobserved

except at the most disaggregated levels of production units and capital

components and must be estimated prior to being used in empirical analysis.

Standard methods for estimating capital and technology were developed decades

ago and are based on analytical and computational methods of that era. We

develop and apply a new method for estimating production capital and

technology, based on advances in economics, dynamic optimization, statistics,

and computing over the intervening years. The method involves specifying and

estimating a detailed structural dynamic economic model of a representative

production firm in an industry and using the estimated model to compute

Kalman-smoothed estimates of unobserved capital and technology for the sample

period. We apply the method to annual data from 1947-97 for U.S. total

manufacturing industries and compare its capital and technology estimates

with standard estimates reported by the Bureau of Labor Statistics.
1

1. Introduction.

Time series of production capital and total factor productivity (or

"technology," as we call the latter here) are fundamental to understanding

the processes of output and productivity growth. Unfortunately, capital and

technology are unobserved except at the most disaggregated levels of

production units and capital components and must be estimated prior to being

used in empirical analysis. Standard methods for estimating capital and

technology were developed decades ago (Jorgenson, 1963; Solow, 1957) and are

based on analytical and computational methods of that era. We develop and

apply a new method for estimating production capital and technology, based on

advances in economics, dynamic optimization, statistics, and computing over

the intervening years.

We apply the method to annual data from 1947-97 for U.S. total

manufacturing industries and compare its estimates of capital and technology

with standard estimates reported by the Bureau of Labor Statistics (1998,

1999). We offer the method and results as a fresh approach for understanding

and estimating capital and technology using modern methods. The four major

findings of the application are: (1) The model-based capital estimates are 10

times more uncertain than the model-based technology estimates. (2) The trends

of the model-based capital and technology estimates broadly conform to the

trends of standard estimates. (3) The model-based capital and technology

estimates imply that above average capital growth in the 1990s -- not above

average technology growth -- explains above average growth in manufacturing

output in the 1990s. (4) In this context, changes in parameter estimates to

suit prior views, can cause large changes in the model-based capital and

technology estimates and, therefore, should be made cautiously.

We are interested in estimating aggregate capital, i.e., at the level

of total production capital (equipment and structures) of all manufacturing

industries. The present method has two major steps, a model-parameter

estimation step followed by an unobserved-variable estimation step. In the

first step, we specify and estimate by maximum likelihood a structural

dynamic economic model of a representative production firm in an industry. We

assume the firm solves a dynamic optimization problem, which is a standard

adjustment cost problem except that adjustment costs on capital and

technology are derived from a parsimoniously parameterized production

function, rather than being stated directly as is usually done. We compute

and incorporate the resulting optimal decision rules into the two estimation
2

steps. We estimate the model's structural parameters without using any

observations on capital or technology. We use only observations on prices and

quantities of output, investment, research (short for "research and

development"), labor, and materials inputs. We overcome the lack of capital

and technology data by using a missing-data variant of the Kalman filter to

compute the likelihood function and by using the overidentifying restrictions

on reduced-form parameters in terms of structural parameters implied by the

optimal decision rules. The reduced-form equations of the estimated model

imply correlations between unobserved capital and technology and the observed

variables in the model. In the second step, we use these correlations to

compute linear least squares estimates (LLSE) of capital and technology, and

their standard errors, in terms of the observed variables in the model. The

LLSEs are implemented using a version of the Kalman smoothing algorithm

(Anderson and Moore, 1979).

We now review the standard methods for estimating aggregate production

capital and technology and, then, discuss the relative advantages of the

present estimation method. Aggregate production capital stocks are often

estimated using the perpetual inventory equation (PIE), kt = δkkt-1 + it, where

kt is the capital stock being estimated, it is observed investment flow, and δk

is one minus a constant capital depreciation rate. Variants of the PIE can

accommodate non-constant or non-geometric depreciation (Bureau of Labor

Statistics, 1998). Aggregate production capital is also estimated as an index

of the service flows of capital components (equipment, structures, and other

disaggregates). The component service flows are estimated using Jorgenson's

(1963) rental prices and are indexed using expenditure weights. Accordingly,

disaggregated data are used in estimating aggregate capital, but, in either

case, the estimates depend entirely on investment flows and capital

depreciation rates and do not depend on other possible factors such as

decision errors (misallocations), which the present method accounts for

implicitly. Technology is usually estimated in percentage growth form as the

Solow (1957) residual, dτt = dqt - ∑ =

n

i 1 s , where d itdxit τ, dq, and dxit are

percentage growth of technology, output, and production inputs, and sit are

input cost shares.

Generally, the relative advantages of the present method over standard

methods are those of an elaborate econometric model over a simple econometric

model. The advantages are greater generality (fewer restrictions) and more

details, hence, more implications. The disadvantages are the need for more
3

and better data, hence, a greater risk of specification error in practice,

and greater mathematical and computational complexity. The standard methods

for estimating capital and technology, while not in theoretical conflict with

each other, are computationally independent. The present method takes the

view that capital and technology are jointly determined as the result of

purposeful, coordinated, investment and research decisions driven by the same

value-maximizing motive. Thus, the model implicitly "disembodies" technology

from capital (Jorgenson, 1966b; Hercowitz, 1998). In the standard method,

technology is an unexplained residual. Whereas the present method allows for

adjustment costs, the standard methods do not. However, the standard methods

are nonparametric, except for having to specify capital depreciation, and are

much easier to apply.

The present method automatically produces standard errors of the

estimates of capital and technology and, therefore, quantifies uncertainty

about the estimates. The standard methods have no measures of uncertainty

and, therefore, in effect, present their estimates as being certain. We

introduce uncertainty by adding disturbances to the PIEs of capital and

technology. The disturbances may be viewed as representing subjective

uncertainty or exogenous shocks. In practice, most of the uncertainty about

capital concerns its depreciation. As the paper shows, adding disturbances to

the PIEs has large consequences for the estimates of capital and technology.

When the PIE disturbances are excluded, the estimates follow smooth trends,

very similar to the standard estimates. When the disturbances are included,

the estimates exhibit short-run variations -- random noises and economic

cycles -- around their trends and the standard estimates. The economic cycles

are transmitted from observed variables through the PIE disturbances.

Recently economists have estimated technology as filtered or smoothed

estimates of an unobserved, estimated, exogenous process (Slade, 1989;

French, 2000). The present paper goes further, by treating capital and

technology as joint endogenous processes. We are unaware of other attempts to

estimate joint, endogenous, capital and technology processes using filtering

and smoothing methods, although these methods have been used to estimate

endogenous (rational) inflationary expectations (Burmeister and Wall, 1982;

Hamilton, 1985; Zadrozny, 1997). Regression methods have been used to

estimate GNP, aggregate capital, and other macroeconomic variables (Romer,

1989; Levy and Chen, 1994; Levy, 2000) but they have more limited

applicability and are less efficient. Unlike filtering and smoothing methods,

regression methods require the estimated variables to be observed in some
4

periods and cannot exploit correlations at all leads and lags. Our approach

to modelling capital and technology as joint endogenous processes could be

seen as an extension of Lucas (1967), with the benefit of modern analytical

and computational methods. Finally, we note Jorgenson, Gollop, and Fraumeni

(1987), Adams (1990), Griliches (1995), Caballero (1999), Nadiri and Prucha

(1999), and references therein as recent examples of work on production

capital and technology.

The paper continues as follows. Section 2 specifies the model and

explains how the representative firm's dynamic optimization problem is

solved. Section 3 prepares the model for estimation of parameters, capital,

and technology by assembling its equations as a vector autoregression (VAR)

and, then, restating the VAR as a state representation. Section 3 also

discusses the parameter identification and reconstructibility conditions

underlying the estimations. Section 4 discusses the application to aggregated

U.S. manufacturing data. It discusses sources and properties of the data,

statistical and economic properties of the estimated model, and compares the

estimates of capital and technology with those published by the Bureau of

Labor Statistics. Section 5 contains concluding remarks. Some technical

details are in the appendix.

2. Specification and Solution of the Model.

Following Zadrozny (1996), we describe an industry in terms of a

representative firm (henceforth, "the firm"). Except for scale differences,

firm- and industry-level variables are identical. Every period, t, the firm

maximizes the expected present value of profits,

(2.1) vt = Et∑∞

k =0 δ πt + k

k ,

with respect to a feedback decision rule, where the maximization is subject to

equations to be specified, Et denotes expectation conditional on the firm's

information in period t, δ ∈ (0,1) denotes a constant real discount factor, and

πt = rqt – (cqt + cit + crt) denotes real profits equal to revenues minus costs,

such that cqt is the cost of production and cit and crt are direct

(nonadjustment) costs of investment in capital and research in technology.

Throughout, a real value is a nominal (current dollar) value divided by the GDP
5

deflator. The firm's optimization problem is stated precisely at the end of

this section.

To obtain a competitive rational-expectations-equilibrium solution,

following Lucas and Prescott (1971), we set revenues in πt to the area under

the inverse output-demand curve as rqt = ∫ =

qt

x o p , where p q

(x,dt)dx q(⋅) is the

inverse output-demand curve, qt is the production of saleable output, and dt is

the output-demand state. Alternately, when rqt = pq(qt,dt)qt, the solution

represents the monopoly equilibrium.

To obtain linear solution equations, which facilitate estimation and to

which the Kalman smoother can be applied, we specify rqt, cqt, cit, and crt as

quadratic forms (constant and linear terms can be ignored). Accordingly, we

assume the industry's inverse output-demand curve is

(2.2) pqt = -ηqt + dt + ζpq,t,

where η > 0 is the slope parameter, dt is the demand state generated by the

second-order autoregressive (AR(2)) process

(2.3) dt = φd1dt-1 + φd2dt-2 + ζd,t,

and ζpq,t and ζd,t are disturbances. Actually, ζpq,t is introduced for purely

technical reasons. Its variance is set small enough so that it has no practical

effect on the results but large enough so that it numerically stabilizes the

Kalman filter and smoother. The full set of distributional assumptions on

disturbances is stated in section 3.

To specify cqt, we first assume that the firm uses capital (k), labor (l),

and materials (m), to produce saleable output (q), install investment goods

(i), and conduct research activities (r) (subscript t is omitted sometimes). We

assume that the "output activities," q, i, and r, are restricted according to

the separable production function

(2.4) h(q,i,r) = τ⋅g(k,l,m),

where τ is the Hicks-neutral stock of technology. Although τ is also totalfactor productivity, because g(⋅) and h(⋅) are indexes of inputs and outputs,

we refer to τ as technology. If τ were capital augmenting or labor augmenting,
6

the production function would be written as h(q,i,r) = g(τk,l,m) or h(q,i,r) =

g(k,τl,m). More specifically, following Kydland and Prescott's (1982)

treatment of the utility function, we assume g(·) and h(·) are the constant

elasticity functions,

(2.5) g(k,l,m) = (α1kβ

 + α2l

β

 + α3mβ)1/β,

 h(q,i,r) = (γ1qρ

 + γ2iρ + γ3rρ)1/ρ,

where αi > 0, α1 + α2 + α3 = 1, β < 1, γi > 0, γ1 + γ2 + γ3 = 1, and ρ > 1. CES =

(β-1)-1 is the constant elasticity of substitution among inputs, and CET =

(ρ-1)-1 is the constant elasticity of transformation among outputs. Including i

and r in h(⋅) is a parsimonious way of specifying internal adjustment costs. The

idea is that positive rates of investment and research use capital, labor, and

materials resources, which could otherwise be used to produce more output, and

that this trade-off sacrifices ever more output per unit increases in

investment and research.

We need the adjustment costs to generate dynamic decision rules for the

firm, which determine correlations among current and lagged variables, which

are used to estimate unobserved variables in terms of observed variables.

Adjustment costs are commonly specified as convex investment costs, which are

incurred in addition to purchase costs of investment goods. Here "investment"

means investment in production capital and research in technology. In the next

step, we derive a quadratic approximation of the dual variable production cost

function (DVPCF) from production function (2.4)-(2.5). The DVPCF includes

convex, investment and research, adjustment costs. Thus, having already

introduced investment and research purchase costs, pitit + prtrt, we obtain a

conventionally structured specification of investment and research adjustment

costs. Although the DVPCF is conventionally structured, it is unconventionally

parameterized. We derive the DVPCF from (2.4)-(2.5) to ensure that structural

parameters are identifiable. If we had specified a general DVPCF, subject only

to symmetry, homogeneity, and curvature restrictions, it would have 28 free

parameters, too many for the structural parameters to be identified, hence,

estimated. The identification problem arises because 4 of 13 variables in the

model are completely unobserved. The missing-data and identification problems

are solved by specifying the DVPCF in terms of the 6 free parameters of (2.4)-
7

(2.5). For recent reviews of the investment adjustment cost literature, see,

for example, Caballero (1999) and Nadiri and Prucha (1999).

Mathematically, convex internal adjustment costs arise in (2.4)-(2.5)

when, for given technology, τ, and inputs, (k,l,m), the transformation surfaces

of the outputs, (q,i,r), are concave to the origin. The adjustment costs are

"convex" because the derived DVPCF is convex in (q,i,r). Hall's (1973) analysis

shows that the division of the production function into two separate input and

output parts, g(⋅) and h(⋅), is a necessary condition for the output

transformation surfaces to be concave to the origin. Here, ρ > 1 is a necessary

and sufficient condition for the transformation surfaces to be concave. The

transformation surfaces become more curved, hence, adjustment costs increase,

as ρ increases. Similarly, β < 1 is a necessary and sufficient condition for the

input isoquants to be convex to the origin, and the isoquants become more

curved, hence, input substitutability decreases, as β decreases.

Let cq = pl

l + pmm, where pl

 is the real hiring price of labor and pm is

the real purchase price of materials. Let ci = pii and cr = prr, where pi and pr

are the real purchase prices of investment and research goods and services.

Because l and m are variable (not subject to adjustment costs) and k and τ are

quasi-fixed (subject to adjustment costs), we refer to cq as the variable cost

and to ci + cr as the fixed cost. Let cq(w) denote the dual variable cost

function: given w = (w1, ..., w7)T

 = (q, i, r, k, τ, pl, pm)T (superscript T

denotes transposition), cq(w) = minimum of pl

l + pmm, with respect to l and m,

subject to production function (2.4)-(2.5).

In the standard approach to multifactor productivity analysis (Bureau of

Labor Statistics, 1998, 1999), all inputs are treated symmetrically, as

variable flows. Accordingly, cq would include all input costs as cq = pkk + pττ +

pl

l + pmm, where pk and pτ are rental prices of capital and technology stocks,

obtained using appropriate versions of Jorgenson's (1963) formula for

converting investment purchase prices into capital rental prices. Jorgenson's

formula is based on more restrictive assumptions, notably that all inputs are

variable. In this paper, we instead work with the purchase prices of investment

and research because this allows greater flexibility for handling adjustment

costs in the firm's dynamic optimization problem. It is the explicit solution

of this problem that generates the identifying conditions that allow us to

estimate the structural parameters of the model in the face of unobserved

capital and technology.
8

The constant term in π does not affect optimal decisions in the

approximate linear-quadratic dynamic optimization problem. Linear terms in π

contribute only an additional constant term to the optimal decision rule, which

is removed by mean adjustment of the data. Therefore, ignoring constant and

linear terms, cq(wt) ≅ (1/2)wt

T

⋅∇2

cq(w0)⋅wt, where ∇2cq(w0) denotes the Hessian

matrix of second partial derivatives of cq evaluated at w = w0. ∇2cq(w0) is

stated in the appendix, for w0 = (1, 1, 1, 1, 1, α2, α3)T

, a value which results

in the simplest expression for ∇2

cq(w0). Therefore,

(2.6) πt = -(1/2)ηqt

2

 + qt(dt + ζpq,t) – (1/2)wt

T

⋅∇2

cq(w0)⋅wt – pitit – prtrt.

The Hessian matrix, ∇2

cq(w0), is symmetric (henceforth, for simplicity, we

often write ∇2

cq(w0) as ∇2cq). Ideally, (1/2)wt

T

⋅∇2

cq(w0)⋅wt should inherit the

following properties from the exact cq(w) function, for all values of w: (i)

linear homogeneity in (q,i,r,k); (ii) convexity in (q,i,r,k); (iii) strict

convexity in (q,i,r), (q,i,k), (q,r,k), and (i,r,k); (iv) linear homogeneity in

(pl

,pm); and (v) strict concavity in pl and pm. In fact, wt

T

⋅∇2

cq(w0)⋅wt satisfies

homogeneity restrictions (i) and (iv) for w = w0 and curvature restrictions

(ii), (iii), and (v) for all w.

The difference between (1/2)wt

T

⋅∇2

cq(w0)⋅wt and the translog cost function

(Christensen, Jorgenson, and Lau, 1971, 1973) is that ∇2

cq(w0) is not stated in

logs of variables and that its elements are tightly restricted in terms of the

parameters of the model, whereas the translog cost function is stated in logs

of variables and its elements are unrestricted except for the homogeneity,

convexity, and concavity restrictions. The present model could be specified in

logs of variables, but the results would be similar because the data are

standardized prior to estimation. As noted above and discussed more below,

estimating parameters without any capital and technology data and, then,

estimating the unobserved capital and technology requires having sufficient

identifying parameter restrictions on the cost function. Although we do not

know and would have difficulty determining the full set of identifying costfunction parameterizations, we do know that the general translog cost function

is not in this set.

We assume pi, pr, pl

, and pm are exogenous to the industry and are

generated by the AR(2) processes
9

(2.7) pit = φpi,1pi,t-1 + φpi,2pi,t-2 + ζpi,t,

 prt = φpr,1pr,t-1 + φpr,2pr,t-2 + ζpr,t,

 plt = φpl,1pl,t-1 + φpl,2pl,t-2 + ζpl,t,

 pmt = φpm,1pm,t-1 + φpm,2pm,t-2 + ζpm,t,

where ζpi,t, ζpr,t, ζpl,t, and ζpm,t are disturbances. Processes (2.7) need not be

stationary. A constant-coefficient autoregressive process is stationary or

asymptotically stable if and only if its characteristic roots are less than one

in absolute value. For example, the pit process is stationary if and only if the

roots, λ1 and λ2, which solve the characteristic equation, λ2

 - φpi,1λ - φpi,2 = 0,

are less than one in absolute value. The only restriction which we need on

processes (2.7) in order to solve the firm's dynamic optimization problem is

that | λ | < 1/ δ , where | λ | is the largest absolute characteristic root of any

equation in processes (2.7).

We assume that capital accumulates according to the continuous-time law

of motion

(2.8) ∂k (s)/∂s = -fk⋅k(s) + i(s) + (s) ~

k ζ ,

where fk > 0 is a depreciation parameter and (s) ~

k ζ is a continuous-time

disturbance. Integrating equation (2.8) over the sampling period s ∈ [t-1,t),

on the assumption that i(s) is constant in [t-1,t), we obtain the discrete-time

capital law of motion,

(2.9) kt = φk1kt-1 + φi0it + ζkt,

where φk1 = exp(-fk), φ10 = [(1–exp(-fk)]/fk, and ζkt = ∫ =

1

s 0 exp[-fk(1-s)] k

~

ζ (t1+s)ds is the implied discrete-time disturbance. It is customary to specify

(2.9) directly, such that φi0 ≡ 1. However, this specification understates the

depreciation of investments undertaken early in a sampling period compared to

those undertaken later in the period. The problem could be avoided by treating

φk1 and φi0 as separate parameters, but this specification is less natural and
10

introduces an additional parameter. Thus, assuming that ζkt ~ NIID(0, 2 σk ), we

parameterize (2.9) in φk1 ∈ (0,1) and 2 σk > 0, such that φi0 = (φk1-1)/ln(φk1).

Similarly, we obtain the discrete-time technology law of motion

(2.10) τt = φτ1τt-1 + φr0rt + ζτt,

parameterized in φτ1 ∈ (0,1) and 2 στ > 0, such that φr0 = (φτ1 – 1)/ln(φτ1) and ζτt ~

NIID(0, 2 στ ).

Equations (2.9)-(2.10) imply geometrical depreciation, in which most of

capital and technology's depreciation occurs in early periods of their use. A

rational-distributed-lag (RDL) specification (Jorgenson, 1966a) could describe

more general depreciation patterns, in particular, in which most depreciation

occurs in late periods of use. A RDL could also include gestation or time-tobuild lags as additional sources of capital and technology fixity. However, the

need for parsimonious parameterization precludes RDL capital and technology

equations, at least for the present data. Most RDLs could also be derived from

underlying continuous-time specifications (Zadrozny, 1988).

The model's structural components have now been specified. It remains to

explain how to solve the firm's dynamic optimization problem and how to

assemble specified laws of motion and solved optimal decision rules into a

system of linear simultaneous equations that are the equilibrium equations of

the model.

To simplify the dynamic optimization problem, we eliminate qt by

maximizing πt with respect to qt. Because qt is not a control variable in the

laws of motion of kt or τt, conditional on it and rt being at their optimal

values, the optimal value of qt is given by maximizing πt with respect to qt.

The first-order condition, ∂πt/∂qt = 0, yields the output supply rule

(2.11) qt = -(c11 + η)-1(c12it + c13rt + c14kt + c15τt + c16 plt + c17pmt - dt) + ζqt,

where (c11, ..., c17) is the first row of ∇2

cq and ζqt is an added disturbance.

In addition to adding ζpq,t to output-demand curve (2.2) and ζqt to output

supply rule (2.11), we also add disturbances to labor and materials decision

rules (2.12)-(2.13) so that each of the 13 variables in the model has its own

disturbance. Although the disturbances are added for purely technical reasons,

to ensure that the variables in the model have a nonsingular joint probability
11

distribution, as usual, they represent our specification errors or the firm's

decision errors, or both.

Similar elimination of lt and mt from the dynamic optimization problem is

justified because lt and mt are not control variables in the laws of motion of

kt or τt. Optimal values of lt and mt, conditional on qt, it and rt being at their

optimal values, are recovered using Shepard's lemma (a special case of the

envelope theorem; Diewert 1971, p. 495),

(2.12) lt = ∂cqt/∂plt = c61qt + c62it + c63rt + c64kt + c65τt + c66plt + c67pmt + ζlt,

(2.13) mt = ∂cqt/∂pmt = c71qt + c72it + c73rt + c74kt + c75τt + c76plt + c77pmt + ζmt,

where (c61, ..., c67) and (c71, ..., c77) are the sixth and seventh rows of ∇2

cq,

and ζ lt and ζmt are added disturbances.

Optimality of labor and materials decision rules (2.12) and (2.13) also

depends on cqt = (1/2)wt

T

⋅∇2

cq(w0)⋅wt being a good approximation of production

function (2.4)-(2.5). It is easy to derive decision rules for lt and mt from the

exact cost function implied by (2.4)-(2.5). However, such rules are nonlinear

in variables, which complicates parameter estimation and smoothing. Whether

exact or approximate rules are used for decisions on l and m, the approximate

linear-quadratic dynamic optimization problem remains unchanged.

To solve the remainder of the firm's dynamic optimization problem, we

restate it as a linear optimal regulator problem. We define the 2×1 control

vector ut = (it, rt)T

 and the 14×1 state vector xt = (kt, τt, pit, prt, plt, pmt, dt,

kt-1, τt-1, pi,t-1, pr,t-1, pl,t-1, pm,t-1, dt-1)T

. We assemble the laws of motion of

output demand, input prices, capital, and technology, (2.3), (2.7), (2.9), and

(2.10), as the state equation

(2.14) xt = Fxt-1 + Gut,

 F = 















7 7×7

1 2

I 0

F F

, G = 















12×2

0

0

G

,

where F1 = diag[φk1, φτ1, φpi,1, φpr,1, φpl,1, φpm,1, φd1], F2 = diag[0, 0, φpi,2, φpr,2,

φpl,2, φpm,2, φd2], G0 = diag[φi0, φτ0], Im is the m×m identity matrix, and 0m×n is the

m×n zero matrix. We suppress disturbances in equation (2.14) because the
12

regulator problem is certainty equivalent. We use the output-supply rule (2.11)

to eliminate qt from πt and write πt as the quadratic form

(2.15) πt = ut

T

Rut + 2ut

T

Sxt-1 + xt-1TQxt-1.

The matrices R, S, and Q are stated in the appendix in terms of η and the

elements of ∇2

cq.

The regulator problem maximizes expected present value, (2.1), stated in

terms of the quadratic form (2.15), with respect to the feedback matrix K in

the linear decision rule ut = Kxt-1, subject to the state equation (2.14). Under

concavity, stabilizability, and detectability conditions (Kwakernaak and Sivan,

1972), we compute the optimal K matrix by solving an algebraic matrix Riccati

equation using a Schur decomposition method (Laub, 1979). Finally, we write the

investment-research decision rule as

(2.16) ut = Kxt-1 + (ζit, ζrt)T

,

where (ζit, ζrt)T

 is an added 2×1 disturbance vector.

3. Estimation Strategy.

3.1. State Representation of the Model.

To estimate the model's parameters by maximum likelihood, using the

Kalman filter, and, then, to estimate unobserved capital and technology, using

the Kalman smoother, we express the reduced form of the model in a state

representation. To this end, we collect the variables of the model in the 13×1

vector yt = (pqt, qt, lt, mt, it, rt, kt, τt, pit, prt, plt, pmt, dt)T

 and their

disturbances in the 13×1 vector ζt = (ζpq,t, ζqt, ζlt, ζmt, ζit, ζrt, ζkt, ζτt, ζpi,t,

ζpr,t, ζpl,t, ζpm,t, ζdt)T

. We assume that the disturbances are mutually independent,

normally distributed, stationary processes, such that the first 6 disturbances

are AR(1) processes and the last 7 disturbances are serially independent. That

is, we assume ζt = (I13 – ΘL)-1εt, where εt ∼ NIID(0,Σε), L is the lag operator,

Θ = diag(θpq, θq, θl

, θm, θi, θr, 0, 0, 0, 0, 0, 0, 0), such that the θ's ∈

(-1,1), and Σε = diag( 2 σpq , 2 σq , 2 σl , 2 σm , 2 σi , 2 σr , 2 σk , 2 στ , 2 σpi, 2 σpr, 2 σpl , 2 σpm , 2 σd ).
13

The set of equations which form the basis of the parameter and capitaltechnology estimation are (2.2), (2.3), (2.7), (2.9)-(2.13), and (2.16), or

more concisely, (2.2), (2.11)-(2.14), and (2.16). These 13 scalar-level

equations constitute the complete set of linear simultaneous equations which,

for given values of parameters, past variables, and current and past

disturbances, determine unique values of the 13 variables of the model. We

assemble the equations concisely as

(3.1) A0yt = A1yt-1 + A2yt-2 + (I13 – ΘL)-1εt,

such that the elements of A0, A1, and A2 are stated in the appendix. We

premultiply equation (3.1) by A0

-1(I13 – ΘL), such that A0 is nonsingular for

admissible values of parameters. Because the autocorrelation coefficients in Θ

are nonzero only in equations with single lags of variables, the resulting

VAR(2) reduced-form system,

(3.2) yt = B1yt-1 + B2yt-2 + ξt,

has only two lags of yt, where B1 = A0

-1(A1 + ΘA0), B2 = A0-1(A2 - ΘA1), ξt = A0-1εt

∼ NIID(0,Σξ), and Σξ ∼ A0

-1ΣεA0-T. Because the input-price equations map unchanged

into equation (3.2), they are both structural and reduced-form equations.

 A complete state representation comprises a state equation, which

expresses the dynamics of the model, and an observation equation, which

accounts for how variables in the model are observed. Corresponding to state

equation (2.14), we write the reduced-form equation (3.2) as the state equation

(3.3) zt = F zt-1 + G ξt,

 F = 















13 13×13

1 2

I 0

B B

, G = 















13×13

13

0

I

,

where zt = (yt

T

, yt-1T)T is the 26×1 state vector. Associated with the state

equation is the observation equation

(3.4) y = t H zt t,
14

where y is the vector of variables observed in period t. t H is called the t

observation matrix.

Because H is completely flexible in assuming any values in any t

dimensions, including the null matrix if no observations are available,

observation equation (3.4) can account for any pattern of missing data. For

most sampling periods in the present application, H = [J, 0], where J = I t 13

with rows of unobserved variables deleted and 0 is the equivalently dimensioned

zero matrix. Thus, when variables 4, 7, 8, and 13 are unobserved, J = I13 with

rows 4, 7, 8, and 13 deleted and 0 = 09×13. Also, H accounts for observations t

on different observed variables starting and ending in different periods. We

call the Kalman filter applied to such a state representation the missing-data

Kalman filter.

The missing-data Kalman filter computes the normal distribution (or

Gaussian) likelihood function of the observations as follows. Let yt

~ = y - t

E[ y |t Yt −1 ] denote the innovation vector, where Y = ( t

T yt , ..., T y1 )T

 denotes the

vector of observations through period t, and let Ωt = E[yt

~ ⋅yt~ T

] denote the

innovation covariance matrix. In general, the reduced-form disturbance vectors,

ξt, and the innovation vectors, yt

~ , coincide only when all variables are

observed throughout the sample. Then, except for terms independent of

parameters, -2 times the log-likelihood function of the sample Y is given by N

(3.5) L(θ, Y ) = N ∑ =

N

t 1 [ln|Ωt| + yt

~ T

Ωt

-1 yt

~ ],

where θ = ( T θ0 , T θ1 , T θ2 , T θ3 )T

, θ0 = (δ, α1, α2, γ1, γ2, 2 σpq , 2 σl , 2 σm )T, θ1 = (φpi,1,

φpr,1, φpl,1, φpm,1, φpi,2, φpr,2, φpl,2, φpm,2, 2 σpi , 2 σpr , 2 σpl , 2 σpm )T, θ2 = (θpq, θq, θl, θm,

θi, θr)T

, and θ3 = (η, β, ρ, φk1, φτ1, φd1, φd2, 2 σq , 2 σi , 2 σr , 2 σk , 2 στ , 2 σd )T.

As explained further in subsection 3.2, the unidentified 8 parameters in

θ0 are normalized and the remaining 31 parameters in θ1, θ2, and θ3 are

estimated in three steps: θ1 in an ordinary-least-squares (OLS) step, θ2 in a

preliminary maximum-likelihood (ML) step, and θ3 in a final ML step. The Kalman

filtering recursions for computing (3.5), starting values for the recursions,

and other details about implementing the computations accurately and

efficiently are discussed in Anderson and Moore (1979), Zadrozny (1988, 1990),
15

and references therein. In the ML steps, L(θ, Y ) was minimized using the N

trust-region method (Mor e′ et al., 1980). Although the likelihood could be

computed in other ways, the missing-data Kalman-filter method proved to be very

effective for handling the various missing-data problems. In particular, given

the computer program (Zadrozny, 1999), we needed only to indicate missing

values in the data matrix with a missing-data indicator and did not need to

transform the reduced-form or state equations, (3.2) or (3.3), as we would

using other methods.

3.2. Parameter Identification and Reconstructibility Conditions.

The hallmark of the present method is a large number of overidentifying

restrictions on the reduced-form parameters, B1, B2, and Σξ, in terms of the

structural parameters, θ, although the structural parameters are unidentified

unless additional normalizing restrictions are imposed. Estimation of capital

and technology requires that a reconstructibility condition hold. Thus, to

estimate the model and use its estimate to estimate capital and technology, the

model must satisfy the parameter identification and reconstructibility

conditions. We comment no further on the complicated relationship between these

conditions, except to note that in our experience parameter identification

implies reconstructibility.

The parameter identification condition is standard in econometrics: the

unnormalized parameters in θ to be estimated are identified when the Hessian

matrix of L(θ, Y ) with respect to them, evaluated at the normalized and N

estimated values of parameters, is positive definite, i.e., ∇2

L( θ

ˆ , Y ) > 0. The N

challenge is to have enough identifying restrictions on reduced-form parameters

in terms of the structural parameters to compensate for the unobservability of

some variables. In this case, with θ0 normalized, the model imposes enough

restrictions to identify θ1, θ2, and θ3. The complexity of the mapping from

structural to reduced-form parameters precludes analytically deriving the

conditions under which θ1, θ2, and θ3 are identified. Fortunately, doing this is

unnecessary, because after terminating at an estimate, the ML estimation

program numerically checks if ∇2

L( θ

ˆ , Y ) > 0. N

We estimated the 31 parameters in θ1, θ2, and θ3 in three steps because

initial attempts to estimate them simultaneously resulted in numerical
16

breakdown. Although the estimation program converged successfully, it was

unable to compute standard errors of the estimated parameters because

∇2

L( θ

ˆ , Y ) was poorly conditioned for inversion. Therefore, we followed the N

three-step strategy which is consistent but (in theory) inefficient compared to

a simultaneous (or full information) estimation strategy. In all three steps,

θ0 is normalized as described below. In step 1, we estimated the 12 input-price

process coefficients and disturbance variances in θ1 using OLS. In step 2,

conditional on 1

ˆ

θ , we estimated the 19 parameters in θ2 and θ3 using ML. In

step 2, ∇2L( θ

ˆ , Y ) was virtually non-positive definite, resulting in very large N

standard errors of the autocorrelation coefficients in 2

ˆ

θ . Therefore, in step

3, conditional on 1

ˆ

θ and 2

ˆ

θ , we reestimated θ3 using ML. Thus, the final

estimates of θ are 1

ˆ

θ from step 1, 2

ˆ

θ from step 2, and 3

ˆ

θ from step 3.

We imposed normalizing restrictions on θ0 to ensure that θ1, θ2, and θ3

are identified. We emphasize that this is normalization, not calibration in the

sense of setting parameters so that the model matches selected moments in the

data. Being unidentified, the normalized parameters cannot be calibrated in

this sense. We verified numerically that the normalized parameters are

unidentified by attempting to estimate all structural parameters

simultaneously. The estimation algorithm made no moves from given initial

parameter values, indicating a flat likelihood function.

We set the discount factor to δ = .935, which corresponds to the interest

rate δ-1 - 1 = .0695. We set the weighting parameters in the production function

to the "neutral" values α1 = α2 = α3 = γ1 = γ2 = γ3 = 1/3. We considered

alternative weighting-parameter normalizations. These resulted in different

estimates of θ3 but in the same estimates of reduced-form parameters, hence, in

the same estimates of capital and technology. We expected that one disturbance

variance would have to be restricted for each unobserved variable. Three

variables are genuinely unobserved, k, τ, and d. To maintain numerical

stability of the Kalman filter and smoother, all disturbance variances must be

positive. Therefore, we set 2 σpq = 2 σl = 2 σm ≅ 10-10. Although setting 2 σpq ≅ 0 is

natural, because 2 σpq is redundant relative to 2 σd in output-demand curve (2.2),

setting 2 σl = 2 σm ≅ 10-10 is arbitrary. We could have set these disturbance

variances to other values, indeed, could have set any three disturbance

variances. It makes no difference, because each choice results in the same
17

estimated reduced form. We checked this result by estimating the model under

alternative variance normalizations. As described in section 4.1, after some

initial estimations, we decided to treat materials quantity, m, as unobserved.

It would seem, then, that another disturbance variance would have to be moved

from θ3 to θ0 and normalized. But this turned out not to be the case.

Conditional on θ0 and θ1, under the initial definitions of the θ's, θ2 and θ3

were still identified. Therefore, we conducted the final estimations using the

original normalizations.

To explain reconstructibility, let zt|s ˆ denote the linear least-squares

estimate of zt in terms of Ys, let

(3.6) Rt = [ T H , 1

T F T H , ..., ( 2t 1 F − )T T H ]t

T

,

where F is the state-transition matrix in (3.3) and H is the observation t

matrix in (3.4). The state vector, zt = (yt

T

, yt-1T)T, is said to be

reconstructible if there is a tr such that Rt has full rank equal to the

dimension of zt, for t ≥ tr. Reconstructibility means that, for t ≥ tr,

(3.7) zt|t ˆ = t

T

t

1

t

T (RtR ) R Y − ,

where t

T R is nonsingular, so that unique filtered estimates of z tR t (i.e., for

t|s = t|t), for t = 1, ..., N, can be computed. If (3.7) is feasible, an

associated formula computes the error covariance matrix E(zt-zt|t ˆ )⋅(zt-zt|t ˆ )T

 in

terms of Rt and the disturbance covariances. The smoothed estimates of zt

(i.e., for t|s = t|N), for t = 1, ..., N, may be expressed similarly. The

Kalman smoother is an accurate and efficient recursive algorithm for

computing $zt N| and E(zt-zt|N ˆ )⋅(zt-zt|N ˆ )T

, for t = 1, ..., N (Anderson and Moore,

1979).

In the application, the dimension of zt is 26, so that if Ht is time

invariant and zt is reconstructible, tr ≤ 26. This follows from the CayleyHamilton theorem, which says that every square matrix satisfies its own

characteristic equation. In such case, for t ≥ 26, the rows of t F are

linearly dependent on the rows of 25 F , 24 F , ..., F . Therefore, if H is time t

invariant, zt is reconstructible if R26, called the reconstructibility matrix,
18

has full rank 26. It is difficult to determine an upper bound for tr if H is t

time varying. For a complete discussion of reconstructibility and related

concepts, see Kwakernaak and Sivan (1972) or Anderson and Moore (1979). The

estimation algorithm numerically checks the reconstructibility condition.

4. Estimation Results.

4.1. Sources and Properties of the Data.

In estimation, we used annual U.S. total manufacturing data on prices and

quantities of output and inputs, ranging from 1947-97. Investment and GDPdeflator data were obtained from the Bureau of Economic Analysis, research data

from the National Science Foundation (1998), and all other data from the Bureau

of Labor Statistics. All data were obtained in nonseasonal form. Thus, we

obtained observations on 10 of the 13 variables in the model: pqt and qt from

1958-96, plt and lt from 1948-97, pit and it from 1947-96, prt and rt from 1953-95,

pmt from 1958-96, and mt from 1958-89.

Except for the quantity of labor, which is measured as the number of

production workers, all other prices and quantities were obtained as a nominal

price index or a real quantity index coupled with nominal expenditures. We

computed the unavailable quantity or price indexes by dividing expenditures by

the available price or quantity index, so that in each case the price index ×

quantity index = nominal expenditures. All obtained or computed nominal price

indexes were, then, converted into real form by dividing them by the GDP

deflator.

The resulting real prices and quantities of U.S. total manufacturing

output and inputs are depicted in figures 1a-j. For graphing convenience, the

data were scaled to lie between 0 and 10. The graphs suggest the following

economic interpretation, which is consistent with simulations of the model in

figures 2a-b. Increasing demand for output driven partly by a declining real

price of output induced manufacturers to increase production capacity.

Increasing quantities of investment and research built increasing stocks of

capital and technology, hence, increased production capacity. As the price of

labor increased, manufacturers saved on labor inputs, resulting in flat or

declining labor use and increasing labor productivity.

[Put figures 1a-j approximately here]
19

Initially, we considered total hours worked (total production workers

multiplied by average hours worked per worker) as an alternate labor input

measure. The graph of total hours worked (not shown) is very similar to that of

total production workers in figure 1f. The main difference is that total hours

worked is a somewhat noisier series. We chose total production workers as the

labor input because it resulted in a slightly better fitting, but

insignificantly different, estimated model. Choosing total production workers

as the labor input caused the R2

s of output price and quantity, investment, and

research to increase by .01 to .02 and that of labor to increase by .16.

Throughout, an R2

 refers to the reduced-form equation of a variable.

Initially, we estimated the model using the data described above, but

this resulted in a nearly zero R2

 for labor. The problem appeared to be

misspecification of materials in the production function. The model's

simulations and the production function’s form indicate symmetrical roles for

labor and materials, while the data in figures 1a and 1c show the time path of

materials matching closely that of output, not that of labor. The solution

options were: (i) drop materials price and quantity from the analysis; (ii)

assume materials quantity is in fixed proportions to the output good; or (iii)

keep materials price and quantity in the model, as they are, continue to use

materials price data in the parameter estimation and smoothing, but treat

materials quantity as unobserved. Options (i) and (ii) would be implemented

implicitly by measuring the output good as value added instead of shipments and

dropping materials as a production input. We chose option (iii), which was also

the easiest to implement, because it required only that the materials quantity

column in the data matrix be filled in with the missing-value indicator.

Therefore, in the final round of estimation, materials quantity was treated as

unobserved, along with actually unobserved capital, technology, and outputdemand state.

4.2. Statistical Properties of the Estimated Model.

Table 1 reports first-step OLS estimates of the input-price process

parameters in θ1. By conventional standards, the estimated equations fit the

data well, having R2

's greater than .90. Residuals show no significant

autocorrelations, having p values of Ljung-Box Q statistics greater than .25.

The estimated pi, pr, and pl

 processes have characteristic roots near one, with

maximum absolute characteristic roots, | λ |, between .785 and 1.02. A process
20

is stationary if and only if its | λ | < 1. The complete estimated reduced

form, (3.2), has five absolute characteristic roots between .98 and 1.02.

Although a cointegration analysis might seem appropriate, we did not attempt

this for two reasons. The input-price processes serve only the subsidiary

purpose of providing forecasts for the dynamic optimization problem and their

AR(2) specifications are adequate for this task. It is not clear how a

standard cointegration analysis, designed for systems in which all variables

are observed and coefficients are unrestricted except for unit-root

restrictions, applies in this case, in which parameters are restricted by the

solution of the dynamic optimization problem and 4 of 13 variables are

unobserved. We allowed unit roots insofar as residual autocorrelation

coefficients, θ, may be very close to one. Table 2 reports second-step ML

estimates of pq

ˆθ , θl

ˆ , and m

ˆθ = .999.

[Put table 1 approximately here]

Table 2 also reports third-step ML estimates of the remaining parameters

in θ3. Their absolute t statistics are less than about .50 and are not reported

because the small sample size makes them unreliable and uninformative (Sims,

1980, p. 19, fn. 19). The implied estimated reduced-form equations show

unsurprisingly good fits by conventional standards, given that the data are

used in original levels form. Moderate (≅ .50) and high (> .90) R2

's of labor

and the nonlabor variables reflect labor's noisiness and the nonlabor

variables' unit-root-like smoothness. The high estimated residual

autocorrelation coefficients ( θ

ˆ 's ≥ .84) might suggest that the residual

autocorrelation corrections and not the economic part of the model account for

most of the observed endogenous variables' sample variations, but this is not

the case. ML estimation with all θ's set to zero produced 2

pq R = .918, 2q R =

.879, 2 Rl = .436, 2 R = .772, and i

2 R = .944, so that the economic part of the r

model accounts for these fractions of the endogenous variables' sample

variations. Most importantly, as we now discuss in detail, the model's

overidentifying restrictions are not rejected by a likelihood ratio test.

[Put table 2 approximately here]
21

For large N, abstracting from terms independent of parameters, the

maximized log-likelihood function can be expressed as L( θ

ˆ , Y ) = N N ⋅ln| N Ωˆ |,

where N Ωˆ = (1/N)∑ =

N

t 1

T

tyt

~ y~ . The likelihood-ratio statistic for testing the

model's restrictions is LR = N(ln| N,R Ωˆ | - ln| N,U Ωˆ |), where N,R Ωˆ and N,U Ωˆ are N Ωˆ

based on restricted and unrestricted innovations, i.e., from maximizing the

likelihood function with the model's restrictions, respectively, imposed and

relaxed. The missing-data Kalman filter automatically produces restricted

innovations as part of the ML estimation. We obtained unrestricted innovations

as follows. We performed the test using the subsample 1960-1990, because only

during this period were observations available for the 9 observed variables.

For this period, the observation matrix, Ht, is time invariant and given by H =

[J, 09×13], where J = I13 with rows 4, 7, 8, and 13 deleted. Then, combining the

state and observation equations, (3.3)-(3.4), we obtain the infinite

autoregressive representation for y , hence, the finite p-lag approximation of t

this representation,

(4.1) y = t Φ1 yt −1 + ... + Φp yt − p + yt

~~ ,

where the residual yt

~~ is an approximation of the innovation yt

~ . We want to

test the economic restrictions of the model and not the mutual independence of

input-price processes (2.7). Therefore, except for the zero restrictions which

make the input-price processes mutually independent, we considered the Φ's to

be free parameters. For p = 2, we estimated the individual equations of (4.1)

by applying OLS to the period 1960-1990. Thus, we reestimated the input-price

processes using the shorter sample. The resulting residuals were serially

uncorrelated and were used to compute N,U Ωˆ .

LR is distributed asymptotically as χ2

(κ), in the limit as N → ∞, where

κ denotes the number of overidentifying restrictions. The statistic rejects the

null hypothesis that the overidentifying restrictions are valid when it exceeds

the critical value, cα, for the significance level α. The period 1960-1990

implies the small values N = 31 and N/κ = .15, for κ = 118. For such

situations, Sims (1980, p. 17, fn. 18) suggested replacing N with N - ν in LR,

where, in this case, ν is the number of estimated parameters divided by the

number of observed endogenous variables. Thus, N - ν = 31 - (143/9) = 15.1 and
22

κ = 118, imply LR = 142, with a p value of .067, so that the overidentifying

restrictions are not rejected at a conventional 5% significance level.

4.3. Economic Properties of the Estimated Model.

Because the estimates of capital and technology depend critically on

the economic model, to be confident in the estimates we should be confident

in the economic properties of the model. Therefore, we present and briefly

discuss some structural variance decompositions (Sims, 1986) and impulse

responses of the estimated model.

We begin by explaining how the variance decompositions are computed. Let

M = I13 with columns 1, 3, and 4 deleted. Then, combining the state and

observation equations, (3.3)-(3.4), we obtain the structural infinite movingaverage representation of y , i.e., in terms of the structural disturbance t

vector, εt,

(4.2) y = t Ψ(L)εt = (∑∞

i =0 Ψ i

iL )εt = ∑∞

i =0 Ψi εt-i,

where Ψi = J 































× 13×13

13

i

13 13 13

1 2

0

I

I 0

B B

M

and J is defined as in (4.1). M has been introduced to delete the three

structural disturbances, εpq,t, εlt, and εmt, whose variances are normalized to

near zero. Let E[ yt + k | Y ] denote the k-step-ahead forecast of t yt + k ; let yt,k

~ =

yt + k - E[ yt + k | Y ] denote the forecast error of E[ t yt + k | Y ]; and, let V t k =

Eyt,k

~ yt,k~ T

 denote the covariance matrix of yt,k

~ . Then, Vk is given by

(4.3) Vk = T

i

k

∑i =0 ΨiΣεΨ .

We decompose the k-step-ahead forecast-error variances of the 8

endogenous variables, and their sum, in terms of the 9 unnormalized estimated

structural disturbance variances. That is, we decompose vk,ii, for i = 1, ...,

8, and ∑ =

8

i 1 v , where v k,ii k,ii is the (i,i) diagonal element of Vk, in terms of
23

2 σj , for j = 2, 5, 6, ..., 13. Let sk,i,j and sk,j denote the fractions of vk,ii and

∑ =

8

i 1 v due to k,ii

2 σj ; let 1 / 2 Σε be the square-root of Σε, obtained by replacing

the diagonal elements of Σε with their positive square roots; let ei denote the

13×1 vector with one in position i and zeroes elsewhere; and, let e denote the

13×1 vector with ones in the first 8 positions and zeroes elsewhere. Then, for

i = 1, ..., 8 and j = 2, 5, 6, ..., 13, the percentage variance decompositions

of vk,ii and ∑ =

8

i 1 v are given by k,ii

(4.4) sk,i,j = i

T

i

T 1 / 2

j j

k

i 0

1 / 2

i

T ei

( Ψ Σ e e Σε Ψ )e ∑ = ε / i

T

i

k

i 0 i

T ei

(∑ Ψ Σ Ψ )e = ε ,

(4.5) sk,j = e ( e e )eT

i

T 1 / 2

j j

k

i 0

1 / 2

i

T

∑ = Ψ Σε Σε Ψ / e ( )eT

i

k

i 0 i

T

∑ = Ψ ΣεΨ .

[Put table 3 approximately here]

Table 3 shows the structural decompositions of k = 10 year ahead

forecast-error variances. Rows 2-9 show decompositions of variances of

endogenous variables; row 10 shows the decomposition of the sum of variances

of endogenous variables. For example, elements 1, 2, 6, and 10 in row 2

indicate that, according to the estimated model, 4.5, 2.8, 5.2, and 83.5

percent of the variance of pq is, respectively, due to 2 σq , 2 σi , 2 σpi, and 2 σd .

Because the model is estimated using standardized data, the decompositions are

unit free. However, different normalizations of disturbance variances in θ0

will result in different decompositions. All disturbances, except disturbances

of research, technology, price of research, and price of labor, explain

significant (> 6%) fractions of some individual variances or the summed

variances. Interestingly, the small impacts of research and technology

disturbances run contrary to the real business cycle literature which

attributes significant macroeconomic fluctuations to technology shocks.

Overall, the decompositions suggest that the capital, output-demand, and

investment-price disturbances are the leading sources of variations of the 8

endogenous variables.

[Put figures 2a-b approximately here]
24

The simulations in figures 2a-b display the dynamic adjustment-cost

behavior in the model in response to unit impulses in output-demand and

technology disturbances. The simulations in figure 2a match the general

interpretation of figures 1a-j. The simulations depict responses to unit oneperiod shock (impulse) to the output-demand state in period 1, starting from an

initial long-run equilibrium represented by the origin. The estimate ˆη = .605

implies a moderately sloped output-demand curve. The estimates β

ˆ = -9.14 and ρ

ˆ

= 267 imply CES = -.099 and CET = .004, hence, low input substitutability and

very high adjustment costs on capital and technology. High adjustment costs

imply a steep marginal-cost-of-production curve. Therefore, after the outputdemand shock occurs, the price of output rises sharply but output increases

only slightly. Initially, the extra output is produced using additional freelyadjusted labor and materials inputs and pre-shock stocks of capital and

technology. Because the shocked demand state declines moderately slowly, firms

have an incentive to increase their production capacities. Thus, they increase

their investment and research rates and substitute capital and technology for

labor and materials. Figure 2b depicts responses to a unit one-period shock to

technology in period 1, again starting from an initial long-run equilibrium at

the origin. In figure 2b, output-demand conditions remain unchanged so there is

little change in price or quantity of output. The shock mainly causes

technology to be substituted for labor and materials until the windfall

addition to technology has depreciated fully.

4.4. Model-Based versus Standard Estimates of Capital and Technology.

Figures 3a-b to 6a-b display the model-based and standard estimates of

production capital and technology of aggregated U.S. manufacturing industries

from 1958-97. The solid graphs depict the model-based estimates and their 2-

standard-error confidence bounds. The dashed graphs of capital depict the sum

of Bureau of Labor Statistics (BLS) estimates of equipment and structures

stocks, based on nonstochastic perpetual inventory equations (PIEs). The dashed

graphs of technology depict BLS estimates of multifactor productivity computed

as Solow residuals. In addition, BLS estimates equipment and structures service

flows and Bureau of Economic Analysis (BEA) produces alternate estimates of

equipment and structures stocks. These estimates are very similar and are not

displayed. Thus, we display the BLS estimates of capital and technology as

representative of standard estimates of these variables.
25

Because ML estimation of the model is tractable only if all data are

scaled similarly, the data were standardized prior to estimation, by

subtracting sample means and dividing by sample standard deviations. Therefore,

being based on standardized data, the model-based estimates are in

approximately standardized units. The BLS estimates are in arbitrarily scaled

real units. To compare the two sets of estimates, one set must be converted to

the units of the other. Therefore, prior to graphing, we exactly standardized

each set of estimates. Also, in each figure, we translated all graphs up by the

same amount so that all values are graphed as positive numbers. Because the

units of the graphs are arbitrary, movements along a graph cannot be

interpreted as percentage changes. However, differences between graphs in the

same figure are in comparable standardized units. The graphs start in 1958

because output, a critical determinant of the estimates, is first available in

1958.

[Put figures 3a-b approximately here]

Figures 3a-b depict graphs of model-based estimates based on parameter

estimates in tables 1 and 2. The capital and technology estimates,

respectively, have average standard errors of 1.03 and .089, which implies that

capital's 2-standard-error confidence intervals are over 10 times larger than

technology's. "Short-run" variations with average periodicities of less than

about 8 years are sums of unpredictable noises and business cycles. "Long-run"

variations with greater average periodicities reflect trends. The model-based

capital estimates exhibit frequent, significant, short-run variations. The

model-based technology estimates exhibit less frequent and less significant

short-run variations. Standard smoothing formulas can decompose short-run

variations into sums of noises and cycles. However, because the formulas ignore

sampling variability of parameter estimates, model misspecification, and other

uncertainties, the decompositions are themselves uncertain. To the extent that

short-run variations reflect cycles, not noises, we can often explain them in

terms of identifiable events, such as the Vietnam War (1965-73) and oil-price

increases (1973, 1979), and in terms of cyclical fluctuations of the overall

economy. The model-based estimates exhibit cycles passed by the estimation

method from the observed variables. Because they are based on nonstochastic

PIEs, the BLS capital estimates exhibit miniscule short-run variations.

[Put figures 4a-b approximately here]
26

Figures 4a-b depict alternate model-based estimates based on capital and

technology disturbances near zero ( 2 σk = 2 στ = .0001) and other structural

parameters at their table 1 and 2 values (for technical reasons, all structural

disturbance variances must be at least slightly positive). Going from figure 3a

to 4a, average standard errors of model-based capital estimates decline 5-fold,

from 1.03 to .205. Setting capital disturbance variances to near zero does not

eliminate capital standard errors because they depend on all structural

disturbance variances. Going from figure 3a to 4a, short-run variations of

capital estimates also decline 5-fold, causing the estimates to become more

trend-like and to conform better to the BLS estimates. Going from figure 3b to

4b, average standard errors of technology estimates decline slightly, from .089

to .060. Correspondingly, the technology estimates change little.

Being estimates based on PIEs, the model-based and BLS capital estimates

should be considered available capital stocks. However, apparently large shortrun variations in the model-based estimates in figure 3a might seem to

contradict this notion. Aren't available aggregate capital stocks large

relative to investment flows and capital disturbances and don't they depreciate

slowly, so that their graphs should be very smooth, like the BLS capital

estimates in figure 3a? We could informally interpret short-run variations in

the model-based capital estimates as variations in utilized capital stocks or

as variations in effective capital stocks, i.e., adjusted for misallocations.

Standard estimation methods treat all capital investments as being equally

successful, regardless of misallocations, market realizations, and market

valuations. Thus, an optimally located factory would add the same amount to

capital as a mislocated factory built using the same resources. However, in

order to formally interpret short-run capital variations as utilized or

effective capital, we would have to extend the model to include some notion of

capacity utilization or market valuation of capital.

Being Solow residuals, BLS technology estimates in figure 3b exhibit

larger short-run variations than BLS capital estimates, especially during oilprice rises in the 1970s. The technology estimates are usually considered to be

the residuals of the production function in the analysis. Here, because both

capital and technology are unobserved, either of their estimates could be

considered as the residuals. However, because the model-based capital estimates

exhibit the larger short-run variations, they are more naturally selected as

the residuals. This is consistent with capital's role as the residual income

earning factor. Presumably, technology should reflect smoothly varying
27

knowledge. Therefore, because Solow residuals are noisy, they are often

smoothed prior to being considered as technology estimates (French, 2000).

Being constructed as smoothed estimates, the model-based technology estimates

should not be smoothed further and, in fact, as seen in figures 3b to 6b, are

smoother than the BLS Solow-residual estimates.

There has been a debate about whether capital growth or technology growth

explains above average output growth in the 1990s (Gordon, 2000; Oliner and

Sichel, 2000; Stiroh, 2001). Figures 1h and 1j indicate above average growth of

investment and roughly trend growth, then, decline of research in the 1990s.

Figures 3a-b show similar growth patterns for model-based capital and

technology estimates in the 1990s. Therefore, the present estimates favor above

average capital growth as the explanation of recent above average manufacturing

output growth.

The table 1 and 2 parameter estimates seem reasonable, except possibly

for k1

ˆφ = .589 and 1ˆφτ = .161, which imply high, annual, capital and

technology, depreciation rates of 1- k1

ˆφ = .411 and 1- 1ˆφτ = .839. For example,

Jorgenson and Stephenson (1967) reported a quarterly depreciation rate, for

equipment and structures in U.S. manufacturing industries from 1947-60, which

implies k1

ˆφ = .895. Figures 5a-b depict alternate model-based estimates, based

on k1

ˆφ = 1ˆφτ = .895 and other parameters at their table 1 and 2 values. The

lower capital and technology depreciation rates result in smoother modelbased estimates, with technology estimates conformable with figures 3b and

4b, but incredible capital estimates, indicating nearly continuous decline.

Figures 6a-b depict another set of model-based estimates, based on k1

ˆφ = 1ˆφτ =

.895, 2 σk = 2 στ = .0001, and other parameters at their table 1 and 2 values.

Going from figures 5a-b to 6a-b, as expected, the estimates become smoother.

Unexpectedly, the capital estimates return to their figure 3a and 4b

patterns. The lesson of figure 5a is that, in this context, one should be

cautious about altering parameter estimates to suit prior views. Figure 6b's

model-based technology estimates are almost pure trends. The sharp drops at

sample ends are probably caused by excess sensitivity to smoother

initialization in this case.

[Put table 4 approximately here]
28

To gain further insights into the reasonableness of k1

ˆφ = .589 and 1ˆφτ =

.161, we estimated the capital and technology equations using nonlinear leastsquares (NLLS). As in the ML estimation, the equations are parameterized in

terms of their underlying continuous-time parameters. We estimated the

equations using the initial model-based and BLS, capital and technology,

estimates as real data. The results are reported in table 4. Although NLLS

estimates of φk1 and φτ1 differ from ML estimates, they are very similar for the

model-based and BLS data. As expected, the fit of the estimated equations

depends on the noisiness of the dependent variable. Thus, the capital equation

fits better when using BLS data ( 2 R = .891) rather than model-based data ( k

2 R = k

.730), and the reverse is true for the technology equation. In essence, table 4

confirms what we see in figures 3a-b and 4a-b, that the trends of the modelbased and BLS estimates are broadly conformable. Although the ML estimates, k1

ˆφ

= .589 and 1

ˆφτ = .161, might seem low economically, they are acceptable

econometrically, because, along with other parameter estimates, they imply an

acceptably fitting model, with unrejected overidentifying restrictions, which

generates smoothed estimates of capital and technology, broadly conformable

with standard estimates.

5. Conclusion.

The paper has developed a new method for estimating unobserved economic

variables based on an estimated dynamic economic model and applies it to

estimating production capital and technology (total-factor productivity) of

aggregated U.S. manufacturing industries from 1958-97. The method illustrates

how modern estimation, control, and filtering methods can be applied to a

parsimonious dynamic economic model to produce estimates and standard errors

of unobserved variables. Standard methods for estimating capital and

technology, developed forty years ago, are appealing in their theoretical and

computational simplicity, but are unnecessarily restrictive in some respects,

for example, ignore adjustment costs. The present method admits adjustment

costs of capital and technology, but is more complex analytically,

econometrically, and computationally. We regard the method as experimental and

do not advocate replacing standard methods. We urge testing the present method

further, using different models and data sets. The paper shows that the method

is feasible. In general, the method is feasible when the economic model
29

imposes enough identifying restrictions to compensate for the unobservability

of some variables.

The four major findings of the application are: (1) The model-based

capital estimates are 10 times more uncertain than the model-based technology

estimates. (2) The trends of the model-based capital and technology estimates

broadly conform to the trends of standard estimates. (3) The model-based

capital and technology estimates imply that above average capital growth in the

1990s -- not above average technology growth -- explains above average growth

in manufacturing output in the 1990s. (4) In this context, changes in parameter

estimates to suit prior views, can cause large changes in the model-based

capital and technology estimates and, therefore, should be made cautiously.

As noted previously, sorting out the competing interpretations of the

model-based capital estimates as available, utilized, or effective capital

stocks requires formally introducing some notion of capacity utilization or

market valuation of capital. The variance decompositions in table 3 assign

principal explanatory roles to capital and investment-price disturbances,

which suggests modelling investment and research decisions in more detail.

For example, the discount rate could be time-varying, as δt = 1/(1 + nt), such

that nt would be an observed exogenous interest rate whose generating process

is also estimated. The capital and technology equations could be specified as

rational distributed lags, which include time-to-built gestation lags and

non-geometrical depreciation rates of capital and technology.

6. Appendix: Statement of Cost, Profit, and Reduced-Form Parameters.

Because ∇2

cq(w0) is symmetric, it suffices to state its upper triangular

part. Let cij denote element (i,j) of ∇2

cq(w0). Then, for w0 = (1, 1, 1, 1, 1,

α2, α4)T

, we have:

c11 = γ1(1-γ1)(ρ-1) + γ1

2

α1(1-β)/(1-α1)

c12 = -γ1γ2[ρ-1 + α1(1-β)/(1-α1)]

c13 = -γ1γ3[ρ-1 + α1(1-β)/(1-α1)]

c14 = -γ1α1(1-β)/(1-α1)

c15 = -γ1(1-α1β)/(1-α1)

c16 = γ1/(1-α1)

c17 = γ1/(1-α1)

c22 = γ2(1-γ2)(ρ-1) + γ2

2

α1(1-β)/(1-α1)

c23 = -γ2γ3[ρ-1 + α1(1-β)/(1-α1)]

c24 = -γ2α1(1-β)/(1-α1)

c25 = -γ2(1-α1β)/(1-α1)

c26 = γ2/(1-α1)
30

c27 = γ2/(1-α1)

c33 = γ3(1-γ3)(ρ-1) + γ3

2

α1(1-β)/(1-α1)

c34 = -γ3α1(1-β)/(1-α1)

c35 = -γ3(1-α1β)/(1-α1)

c36 = γ3/(1-α1)

c37 = γ3/(1-α1)

c44 = α1(1-β)[1 + α1(2-α1)/(1-α1)]

c45 = -α1 + α1(2-α1-β)/(1-α1)

c46 = -α1/(1-α1)

c47 = -α1/(1-α1)

c55 = (2-α1-α1β)/(1-α1)

c56 = -1/(1-α1)

c57 = -1/(1-α1)

c66 = -α3/[α2(1-α1)(1-β)]

c77 = -α2/[α3(1-α1)(1-β)].

Next, we state the elements of the 2×2, 2×14, and 14×14 coefficient

matrices R, S, and Q, which define quadratic form (2.15). Because R and Q are

symmetric, we state only their upper-triangular parts. Rij, Sij, and Qij denote

(i,j) elements of the matrices. To eliminate the common factor 1/2, we scale πt

up by the factor of 2, which is allowable because optimal decisions are

invariant to the scale of πt. For simplicity, we state only nonzero elements of

R, S, and Q, so that all unstated elements are zero. Thus, setting c0 =

(η+c11)-1, we have

R11 = c0c122

 – c22

R12 = c0c12c13 – c23

R22 = c0c132 – c33

S11 = c0c12c14 - c24

S12 = c0c12c15 - c25

S13 = -1

S15 = c0c12c16 - c26

S16 = c0c12c17 - c27

S17 = -c0c12

S21 = c0c13c14 - c34

S22 = c0c13c15 - c35

S24 = -1

S25 = c0c13c16 – c36

S26 = c0c13c17 – c37

S27 = -c0c13

Q11 = c0c142 - c44

Q12 = c0c14c15 – c45

Q15 = c0c16c16 – c46

Q16 = c0c14c17 – c47

Q17 = -c0c14

Q22 = c0c152 – c55

Q25 = c0c15c16 – c56

Q26 = c0c15c17 - c57

Q27 = -c0c15.

Finally, we state the structural coefficient matrices Ak, for k = 0, 1,

2. Let Ak,i,j and Ki,j, respectively, denote elements (i,j) of Ak and K, the

optimal investment-research feedback matrix. As before, only nonzero elements
31

are stated. Also, because the diagonal elements of A0 are all one, they are not

stated. Proceeding row-wise across the matrices,

A0,1,2 = η

A0,1,3 = -1

A0,2,5 = c0c12

A0,2,6 = c0c13

A0,2,7 = c0c14

A0,2,8 = c0c15

A0,2,11 = c0c16

A0,2,12 = c0c17

A0,2,13 = -c0

A0,3,2 = -c16

A0,3,5 = -c26

A0,3,6 = -c36

A0,3,7 = -c46

A0,3,8 = -c56

A0,3,11 = -c66

A0,3,12 = -c67

A0,4,2 = -c17

A0,4,5 = -c27

A0,4,6 = -c37

A0,4,7 = -c47

A0,5,7 = -c57

A0,5,11 = -c67

A0,5,12 = -c77

A0,7,5 = -φi0

A0,8,6 = -φr0

[A1,5,7, ..., A1,5,13] = [K1,1, ..., K1,7]

[A1,6,7, ..., A1,6,13] = [K2,1, ..., K2,7]

[A1,7,7, ..., A1,13,13] = [φk1, φτ1, φpi,1, φpr,1, φpl,1, φpm,1, φd1]

[A2,5,7, ..., A2,5,13] = [K1,8, ..., K1,14]

[A2,6,7, ..., A2,6,13] = [K2,8, ..., K2,14]

[A2,7,7, ..., A2,13,13] = [0, 0, φpi,2, φpr,2, φpm,2, φd2].

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35

Table 1

Step 1 OLS Estimates of Input-Price Process Parameters in θ1

Var. Parameter Estimates Fit Statistics

,1

ˆφ⋅ ,2ˆφ⋅ | λ | ˆ

σ 2 R Q

pi 1.45

(11.1)

-.441

(3.30)

1.02 .178 .971 5.64

(.933)

pr

.652

(4.01)

.282

(1.81)

.949 .126 .979 4.67

(.968)

pl 1.88

(24.8)

-.883

(11.3)

1.01 .019 .999 14.8

(.254)

pm

1.49

(9.79)

-.617

(4.06)

.785 .334 .903 9.13

(.692)

Comments: Columns 2-7, respectively, show estimates of φ⋅,1 and φ⋅,2, with

their absolute t statistics in parentheses, implied maximum absolute

characteristic roots (solutions of λ2

 - ,1

ˆφ λ ⋅ - ,2ˆφ⋅ = 0), estimated standard

deviations of disturbances, unadjusted R2

s (defined as 1 - sample variance of

the innovation of a variable ÷ sample variance of the variable), and LjungBox Q statistics for testing absence of residual autocorrelations at lags

from 1 to 10, with their marginal significance levels or p values in

parentheses.
36

Table 2

Step 2 and 3 ML Estimates of Structural Parameters in θ2 and θ3

Production Function Parameters

β

ˆ = -9.14 (CES = -.099), ρ

ˆ = 267 (CET = .004)

Output-Demand Curve Parameters

ˆ

η = .605, d1

ˆφ = 1.39, d2ˆφ = -.518

Capital and Technology Equation Coefficients

k1

ˆφ = .589, i0ˆφ = .774, 1ˆφτ = .161, r0ˆφ = .459

Residual Autocorrelation Coefficients

pq

ˆθ = .999, qˆθ = .914, θl

ˆ = .999, m

ˆθ = .999, iˆθ = .840, rˆθ = .920

Structural Disturbance Standard Deviations

q σˆ = .417, i σˆ = .514, r σˆ = .362, k σˆ = .994, στ ˆ = .055, d σˆ = .465

Reduced-Form Equation Fit Statistics

2

pq R = .945, 2q R = .948, 2 Rl = .498, 2

i R = .926, 2 R = .957 r

Qpq = 10.8, Qq = 5.96, Ql

 = 5.97, Qi = 18.6, Qr = 21.4

 (.378) (.819) (.818) (.158) (.019)

Comment: The sample span is 1947-1997 (51 years). R2 and Q statistics are

defined as in table 1.
37

Table 3

Structural Variance Decomposition of the Estimated Model

2 σq

2 σi2 σr2 σk2 στ

2 σpi2 σpr2 σpl2 σpm

2 σd

s10,pq,j 4.5 2.8 .7 .2 .0 5.2 .1 .0 3.0 83.5

s10,q,j⋅ 19.4 12.2 3.1 .8 .2 27.5 .7 .0 15.9 20.2

s10,l,j .9 3.9 .0 92.7 .2 .0 .0 .0 1.6 .1

s10,m,j .9 3.9 .0 92.7 .8 .0 .0 .0 1.6 .1

s10,i,j .0 44.5 .1 14.3 .1 17.5 .4 .0 11.5 11.6

s10,r,j .0 .0 5.4 1.1 .2 39.3 1.0 .1 25.8 27.1

s10,k,j .0 4.0 .0 95.3 .0 .3 .0 .0 .2 .2

s10,τ,j .0 .0 1.9 1.1 1.6 39.9 1.1 .1 26.5 27.8

s 10,j 1.3 5.2 .7 69.6 .4 7.4 .2 .0 5.4 9.8

Comment: Rows 2-9 show the percentage decompositions of the 10-step-ahead

forecast-error variances of the 8 endogenous variables in terms of the

variances of the 10 unnormalized estimated structural disturbances. Row 10

shows the percentage decomposition of the sum of the variances of the eight

endogenous variables. Each row's numbers sum to one.
38

Table 4

Nonlinear Least-Squares Estimates of Capital and Technology Equations

Capital Equation

1

ˆφ⋅ 0ˆφ⋅

2 R ⋅

Model-Based Data k1

ˆφ = .336

 (22.2)

i0

ˆφ = .608

 (9.14)

2 R = .730 k

BLS Data k1

ˆφ = .363

 (78.1)

i0

ˆφ = .629

 (29.3)

2 R = .981 k

Technology Equation

Model-Based Data 1

ˆφτ = .376

 (118.)

r0

ˆφ = .638

 (42.2)

2 Rτ = .992

BLS Data 1

ˆφτ = .323

 (50.8)

r0

ˆφ = .599

 (21.7)

2 Rτ = .945

Comment: Columns 2-3 show estimates of the φ's, with their absolute t

statistics in parentheses. The φ's were estimated in terms of their underlying

continuous-time parameters, fk and fτ. The standard errors in the t statistics

were computed based on linear approximations of the nonlinear mappings from

the f's to the φ's.
39

Figures 1a to 1j

U.S. Total Manufacturing, Prices and Quantities of Output and Inputs, 1947-97

1a: Price of Output

47 52 57 62 67 72 77 82 87 92 97

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1c: Price of Materials

47 52 57 62 67 72 77 82 87 92 97

7.0

7.5

8.0

8.5

9.0

9.5

1e: Price of Labor

47 52 57 62 67 72 77 82 87 92 97

0.00

0.25

0.50

0.75

1.00

1.25

1g: Price of Investment

47 52 57 62 67 72 77 82 87 92 97

0.96

1.04

1.12

1.20

1.28

1.36

1.44

1.52

1i: Price of Research

47 52 57 62 67 72 77 82 87 92 97

1.08

1.10

1.12

1.14

1.16

1.18

1.20

1.22

1b: Quantity of Output

47 52 57 62 67 72 77 82 87 92 97

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

1d: Quantity of Materials

47 52 57 62 67 72 77 82 87 92 97

1.0

1.2

1.4

1.6

1.8

2.0

2.2

1f: Quantity of Labor

47 52 57 62 67 72 77 82 87 92 97

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

1h: Quantity of Investment

47 52 57 62 67 72 77 82 87 92 97

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1j: Quantity of Research

47 52 57 62 67 72 77 82 87 92 97

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0
40

Figure 2a: Responses to Impulse in Output-Demand Disturbance

Price of Output

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Investment

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Output

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Research

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Labor

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Capital

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Materials

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Technology

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Figure 2b: Responses to Impulse in Technology Disturbance

Price of Output

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Investment

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Output

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Research

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Labor

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Capital

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Materials

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Technology

5 10 15

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5
41

Figures 3a and 3b: Model-Based versus BLS Estimates of Capital and Technology

Figure 3a: Model-Based vs. BLS Estimates of Capital

phik = .589, phit = .161, sek = .994, set = .055

58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0.0

0.8

1.6

2.4

3.2

4.0

4.8

5.6

6.4

Figure 3b: Model-Based vs. BLS Estimates of Technology

phik = .589, phit = .161, sek = .994, set = .055

58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Comment: phik, phit, sek, and set denote the values of φk1, φτ1, σk, and στ used

to produce the graphs. All other parameters were either normalized or set to

values in tables 1 and 2. The same comment applies to figures 4a-b to 6a-b.
42

Figures 4a and 4b: Model-Based versus BLS Estimates of Capital and Technology

Figure 4a: Model-Based vs. BLS Estimates of Capital

phik = .589, phit = .161, sek = .0001, set = .0001

58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Figure 4b: Model-Based vs. BLS Estimates of Technology

phik = .589, phit = .161, sek = .0001, set = .0001

58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5
43

Figures 5a and 5b: Model-Based versus BLS Estimates of Capital and Technology

Figure 5a: Model-Based vs. BLS Estimates of Capital

phik = phit = .895, sek = .994, set = .055

58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 5b: Model-Based vs. BLS Estimates of Technology

phik = phit = .895, sek = .994, set = .055

58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5
44

Figures 6a and 6b: Model-Based versus BLS Estimates of Capital and Technology

Figure 6a: Model-Based vs. BLS Estimates of Capital

phik = phit = .895, sek = .0001, set = .0001

58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 6b: Model-Based vs. BLS Estimates of Technology

phik = phit = .895, sek = .0001, set = .0001

58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5