BLS WORKING PAPERS 

 

U.S. Department of Labor 

U.S. Bureau of Labor Statistics 

Office of Prices and Living Conditions 

Estimated U.S. Manufacturing Production Capital and Technology Based on an 

Estimated Dynamic Structural Economic Model 

 

Baoline Chen, Bureau of Economic Analysis

Peter Zadrozny, U.S. Bureau of Labor Statistics

Working Paper 429

September 2009

All views expressed in this paper are those of the authors and do not necessarily reflect the views or policies of the 

U.S. Bureau of Labor Statistics. 
ESTIMATED U.S. MANUFACTURING PRODUCTION CAPITAL AND TECHNOLOGY 

BASED ON AN ESTIMATED DYNAMIC STRUCTURAL ECONOMIC MODEL*

Baoline Chen 

Bureau of Economic Analysis 

1441 L Street, NW 

Washington, DC 20230 

e-mail: baoline.chen@bea.gov 

Peter A. Zadrozny**

Bureau of Labor Statistics 

2 Massachusetts Ave., NE 

Washington, DC 20212 

e-mail: zadrozny_p@bls.gov 

April 2, 2009 

JEL Classification: C50, C81, D24, L60 

Additional key words: Kalman filter estimation of latent variables 

ABSTRACT 

Production capital and total factor productivity or technology are 

fundamental to understanding output and productivity growth, but are 

unobserved except at disaggregated levels and must be estimated before being 

used in empirical analysis. In this paper, we develop estimates of production 

capital and technology for U.S. total manufacturing based on an estimated 

dynamic structural economic model. First, using annual U.S. total 

manufacturing data for 1947-1997, we estimate by maximum likelihood a dynamic 

structural economic model of a representative production firm. In the 

estimation, capital and technology are completely unobserved or latent 

variables. Then, we apply the Kalman filter to the estimated model and the 

data to compute estimates of model-based capital and technology for the 

sample. Finally, we describe and evaluate similarities and differences 

between the model-based and standard estimates of capital and technology 

reported by the Bureau of Labor Statistics.*

 

*

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

positions of the Bureau of Economic Analysis or the Bureau of Labor 

Statistics. Baoline Chen acknowledges previous financial support from Rutgers 

University and the Ludwig Boltzmann Institute for Economic Policy Analysis in 

Vienna, Austria. The paper was presented at the CESifo Conference on 

Productivity and Growth, Munich, Germany, June 22-23, 2007. We thank an 

anonymous referee, Michael Binder, Michael Harper, Peter Ireland (editor), 

James Malley, and Randal Verbrugge for comments and William Gullickson, Kurt 

Kunze, Lawrence Rosenblum, and Ray Wolfe for help in obtaining data. 

**Affiliated as Research Fellow with the Center for Financial Studies (CFS), 

Goethe University, Frankfurt, Germany, and with the Center for Economic 

Studies and Ifo Institute for Economic Research (CESifo), Munich, Germany. 
1

1. Introduction. 

Time series of production capital and total factor productivity or 

technology, as we call it here, are fundamental to understanding 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 before being used in empirical analysis. 

Standard methods for estimating capital and technology are based largely on 

work by Jorgenson (1963) and Solow (1957). We develop alternative estimates 

of production capital and technology for U.S. total manufacturing based on an 

estimated dynamic structural economic model. First, using annual U.S. total 

manufacturing data from 1947-1997, we estimate by maximum likelihood a 

dynamic structural economic model of a representative production firm. In the 

estimation, capital and technology are completely unobserved or latent 

variables. Then, we apply the Kalman filter to the estimated model and the 

data to compute model-based estimates of capital and technology for the 

sample. Finally, we describe and evaluate similarities and differences 

between the model-based estimates and standard estimates of capital and 

technology, the latter reported by the Bureau of Labor Statistics. 

We estimate aggregate capital for U.S. total manufacturing production 

capital (equipment and structures) in 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 dynamic 

structural economic model of a representative production firm in an industry. 

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

convex 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 

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 

(Jones, 1980; Ansley and Kohn, 1985; Zadrozny, 1988, 1990) to compute the 

likelihood function and by using the overidentifying restrictions on reducedform parameters in terms of structural parameters implied by optimal decision 

rules. The reduced-form equations of the estimated model imply correlations 
2

between unobserved capital and technology and the observed variables in the 

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

compute linear least squares estimates of capital and technology in terms of 

the observed variables of the model and their standard errors. 

We now broadly review standard methods for estimating aggregate 

production capital and technology and the relative advantages of the present 

model-based estimation method. 

Standard methods for estimating capital stock are usually based on 

iterating on a perpetual inventory equation (PIE), starting from a chosen 

initial value of capital. A basic PIE is kt = kt-1 + it - dt, where kt denotes 

(usually unobserved) capital stock at the end of period t (the time interval 

[t-1,t] between moments t-1 and t), it denotes (usually observed) investment 

flow over period t, and dt denotes (sometimes observed) discarded capital flow 

over period t. If dt is unobserved, then, it could be set as dt = φk1kt-1, where 

φk1 ∈ (0,1) denotes one minus a constant geometric rate of capital 

depreciation, which results in the most commonly used PIE. More generally, a 

PIE could be a rational distributed lag (Jorgenson, 1966), 

(1.1) kt = φk1kt-1 + ... + φkpkt-p + φi0it + φi1it-1 + ... + φiqit-q, 

where the φk and φi parameters determine a possibly nongeometric depreciation 

schedule. While kt = φk1kt-1 + φi0it corresponds to a first-order autoregressive 

time-series model, a rational distributed-lag PIE corresponds to a higherorder autoregressive moving-average time-series model. 

Aggregate production capital is also sometimes estimated as an index of 

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

disaggregates) which are often estimated using Jorgenson's (1963) rental 

prices and are indexed using expenditure weights. In such cases, 

disaggregated data is usually used to estimate the aggregate capital and, 

aside from chosen initial values of capital stocks, the capital estimates 

depend on observed investment minus discards or on observed investment and a 

distributed-lag depreciation schedule. 

The Solow residual (Solow, 1957) is the standard method for estimating 

technology in percentage change form: dτt = dqt - ∑ =

n

1i s , where d dxitit τt, dqt, 

and dxit denote percentage changes of technology, output, and production 

inputs, between periods t-1 and t, and sit denotes the share of input costs of 

input i in period t. Because noisy or short-run variations in output are 
3

often not considered to be variations in technology and output displays 

significant noisy variations, the Solow residual is often first smoothed by 

some method before it is considered to be an acceptable estimate of 

technological change. For example, French (2000) smooths the Solow residual 

using the Kalman filter. 

The model-based method has the relative advantages over standard 

methods that a detailed and larger econometric model has over a simpler and 

smaller econometric model: greater generality (fewer implicit or explicit 

restrictions), more details, and more implications. The disadvantages are the 

need for more data, a greater risk of model misspecification, and greater 

theoretical (mathematical) and computational complexity. Standard methods for 

estimating capital and technology, while not in theoretical conflict with 

each other, are computationally independent. The model-based method views 

capital and technology as the result of coordinated investment and research 

decisions which aim to solve a single and purposeful dynamic optimization 

problem. By constrast, in standard Solow-residual methods, technology is an 

unexplained residual. Whereas the model-based method explicitly includes 

adjustment costs, standard methods implicitly assume there are no adjustment 

costs. However, standard methods are nonparametric and, except for possibly 

having to specify and estimate a parametric capital depreciation schedule, 

are conceptually easier to understand and computationally easier to 

implement. 

The model-based method uses the Kalman filter which simultaneously 

computes estimates of unobserved capital and technology and their standard 

errors, which quantify uncertainty about the estimates. In the model-based 

method, disturbances in capital's and technology's PIEs and Kalman-filter 

estimates of the disturbances are the basis for the standard errors. By 

contrast, standard methods cannot similarly produce standard errors of 

capital and technology estimates, because they do not include disturbances in 

capital's and technology's PIEs. As figures 3a-4a below show, reducing the 

standard error of capital's disturbance in its PIE (2.9) by 10-4 changes 

capital estimates from noisy, with many and large short-run variations and 

large standard-error bounds, to smoother ones, with fewer and smaller shortrun variations and much smaller standard-error bounds. 

Recently, economists have estimated technology as filtered estimates of 

an unobserved estimated exogenous process (Slade, 1989; French, 2000). The 

present paper goes further by treating capital and technology as jointly

generated endogenous processes. We are unaware of filtering methods having 
4

been used similarly to estimate jointly generated endogenous capital and 

technology processes, although filtering methods have been used to estimate 

endogenous 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, in contrast to filtering methods, regression 

methods have more limited applicability and are less efficient. Unlike 

filtering methods, regression methods require the estimated variables to be 

observed in some periods and cannot exploit correlations at all leads and 

lags. The present model-based method for estimating capital and technology 

could be seen as an extension of Lucas (1967) based on more recently 

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

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

Prucha (2001), and references therein as more 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 annual 

U.S. total manufacturing data for 1947-1997. It discusses sources and 

properties of the data, statistical and economic properties of the estimated 

model, and compares model-based estimates of capital and technology with 

standard estimates 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∑∞

=0k πδ + kt

k , 
5

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, 

where cqt denotes the cost of production and cit and crt denote direct 

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

Throughout, a real value is a nominal (current dollar) value divided by some 

aggregate price index like the GDP 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

ox tq dx)d,x(p , where pq(⋅) is the 

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

the output-demand state. Alternatively, 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 filter 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 constant 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. Distributional assumptions on disturbances 

are 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), 
6

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, 

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 < 0 is the constant elasticity of substitution among inputs, and CET = 

(ρ-1)-1 > 0 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, 
7

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)-

(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 algebraically. 

 Let cq = pll + 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 pll + 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, 1997), all inputs are treated symmetrically, as variable 

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

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. Although 

energy is often treated as a separate input, we merge it with materials, so 

that mt denotes an aggregate of energy and materials inputs. 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 
8

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. 

 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

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

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

cq(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). This choice of w0 works acceptably in 

the application in section 4. Therefore, 

(2.6) πt = -(1/2)ηqt

2

 + qt(dt + ζpq,t) – (1/2)wt

T

⋅∇2cq(w0)⋅wt – pitit – prtrt. 

∇2cq(w0) is symmetric and ideally (1/2)wt

T

⋅∇2cq(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

⋅∇2cq(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

⋅∇2cq(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 should 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 identified costfunction parameterizations, we do know that the general translog cost function 

is not in this set. 
9

 We assume that pi, pr, pl, and pm are exogenous to the industry and are 

generated by the AR(2) processes 

(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 that 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( ~

ζk , 

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), φi0 = [(1–exp(-fk)]/fk, and ζkt = ∫ =

1

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

~

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

(2.9) directly, where φi0 ≡ 1. However, this specification understates the 
10

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 

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

parameterize (2.9) in φk1 ∈ (0,1) and 2 σk > 0, where 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, where 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, 1966) could describe 

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

occurs in late periods of use. A RDL could also include gestation 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(w0) and, for statistical reasons, 

ζqt is an added disturbance. 
11

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 statistical reasons, to 

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

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

decision errors, or some combination of 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 6th and 7th rows of ∇2

cq(w0) and, 

for statistical reasons, 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

⋅∇2cq(w0)⋅wt being a good dual representation 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 filtering. 

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 laws of motion (2.2)-(2.3) 

of output demand, (2.7) of input prices, (2.9) of capital, and (2.10) of 

technology, as the state equation 

(2.14) xt = Fxt-1 + Gut, 
12

 F = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

7 ×77

1 2

0I

FF

, G = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

×212

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 

regulator problem is certainty equivalent. We use 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(w0). 

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 a discrete-time 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, for statistical reasons, (ζit, ζrt)T

 is an added 2×1 disturbance vector. 

3. Estimation of the Model and of Capital and Technology. 

To estimate the model's parameters by maximum likelihood and, then, to 

estimate unobserved capital and technology, in both steps using the Kalman 

filter, 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 = 
13

(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), where 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 ). 

 The 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. 

Concisely, the equations are 

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

where the elements of A0, A1, and A2 are stated in the appendix in terms of η, 

φ's, elements of ∇2cq(w0), and elements of K. Rewriting (3.1), we obtain the 

reduced-form VAR(2) process 

(3.2) yt = B1yt-1 + B2yt-2 + ξt, 

where A0 is nonsingular for admissible values of parameters, B1 = 1 A0

− (A1 + ΘA0), 

B2 = 1 A0

− (A2 - ΘA1), ξt = 1 A0− εt ∼ NIID(0,Σξ), and Σξ ∼ 1 A0− ΣεA0

-T. Because the inputprice 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 reduced-form equation (3.2) as state equation 

(3.3) zt = F zt-1 + G ξt, 

 F = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

13 ×1313

1 2

0I

BB

, G = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

×1313

13

0

I

, 

where zt = ( T yt , T y −1t )T

 is a 26×1 state vector and F is a 26×26 state-transition 

matrix. Associated with the state equation is the observation equation 
14

(3.4) yt = Ht zt, 

where yt is the vector of variables observed in period t and Ht is a timevarying observation matrix. 

Because Ht is completely flexible in assuming any values in any 

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 identically 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, Ht accounts for observations 

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 (MDKF). 

The likelihood function is computed for maximum likelihood estimation 

(MLE) as follows. Let yt

~ = yt - E[ yt | Y −1t ] denote the innovation vector, where 

Yt = ( T yt , ..., T y1 )Tdenotes the vector of observations through period t, and 

let Ωt = E[yt

~ ⋅ T yt

~ ] denote the innovation covariance matrix. In general, reducedform disturbance vector, ξt, and innovation vector, yt

~ , coincide only when all 

variables are observed throughout the sample. Then, except for terms 

independent of parameters, -2 × log-likelihood function of the sample, YN , is 

given by 

(3.5) L(θ, YN ) = ∑ =

N

1t [ln|Ωt| + T yt

~ Ωt

-1 yt

~ ], 

where θ is the 39×1 vector of structural parameters, which partitions as θ = 

( T θ1 , T θ2 )T

, where θ1 = (δ, α1, α2, γ1, γ2, 2 σpq , 2 σl , 2 σm )T is 8×1 and θ2 = (φpi,1, 

φpr,1, φpl,1, φpm,1, φpi,2, φpr,2, φpl,2, φpm,2, 2 σpi , 2 σpr , 2 σpl , 2 σpm , θpq, θq, θl, θm, θi, θr, 

η, β, ρ, φk1, φτ1, φd1, φd2, 2 σq , 2 σi , 2 σr , 2 σk , 2 στ , 2 σd )T

 is 31×1. We used the MDKF 

to compute L(θ, YN ) accurately and quickly in the MLE. See Anderson and Moore 

(1979), Zadrozny (1988, 1990), and references therein for further details. For 
15

θ1 set according to identifying restrictions, L(θ, YN ) was minimized with 

respect to θ2 using a trust-region method (More′ et al., 1980). 

Because the 39 parameters in θ are not identified without further a 

priori restrictions, we imposed the following 8 identifying restrictions on θ1

to ensure that θ2 is identified and estimatable. We set δ = .935, which 

corresponds to a real interest rate of δ-1 - 1 = .0695. In production function 

(2.4)-(2.5), we set α1 = α2 = α3 = γ1 = γ2 = γ3 = 1/3. It seems we need to set one 

disturbance variance for each unobserved variable. Capital (kt), technology 

(τt), and the output-demand state (dt) are actually unobserved and, after some 

experimentation to obtain an acceptable estimated model, materials (mt) was 

also treated as unobserved. However, it turned out that setting 2 σpq = 2 σl = 2 σm

≅ 10-10 was sufficient to identify the unrestricted and estimated parameters. We 

set the 3 variances to small positive values, rather than exactly to zero, 

because doing so resulted in more accurate computations using the MDKF. We 

considered other identifying restrictions (e.g., of δ and the α's and γ's), but 

all of them resulted in approximately the same reduced-form parameter 

estimates, hence, in approximately the same estimates of capital and 

technology. 

The mapping from structural to reduced-form parameters is too complicated 

to try to derive necessary or sufficient conditions on parameters under which 

parameters are identified. Here, the parameters in θ1 were set directly, which 

rendered the parameters in θ2 identified and estimatable. Analytical 

determination of the boundaries of identification is unnecessary, because after 

terminating at an estimate, the MLE program checks for identification 

numerically by checking whether the Hessian matrix of L(θ, YN ) is numerically 

positive definite. In practice, we can at best choose a set of "reasonable" 

identifying restrictions on θ1, attempt to estimate θ2 under these 

restrictions, and, if the MLE computations converge and the Hessian matrix is 

positive definite, then, we consider the chosen restrictions to be sufficient 

for identification. 

Identifying restrictions could alternatively be considered calibrations 

based on other information, either particular economic theories or data sets or 

general notions or experience. Because the model makes predictions for all of 

its variables, the identifiable parameter space depends not just on the model's 

structure, but on the extent of unobserved variables, and expands as more 
16

unobserved variables become observed. In such case, previously unidentified 

parameters, which could only be set or calibrated, become identified and can be 

estimated. The practical challenge is imposing sufficient identifying 

restrictions to compensate for unobserved variables. 

Because input prices are assumed to be generated exogenously by 

univariate AR(2) processes (2.7), the processes can be estimated 

(asymptotically) efficiently and individually using ordinary least squares 

(OLS), which is much simpler than estimating simultaneously all parameters in 

θ2 using MLE. Thus, in the application, first, we estimated input-price 

parameters using OLS and, then, conditional on these estimates, estimated the 

remaining parameters in θ2 using MLE. The resulting set and estimated 

parameters and maximized likelihood are denoted by θ

ˆ and L( θˆ , YN ). 

Two separate general identification conditions must be satisfied in 

order to estimate the structural parameters and, then, to estimate capital and 

technology for given estimated parameters. The first parameter identification 

condition is the standard condition that the Hessian matrix of 2nd-partial 

derivatives of L(θ, YN ) with respect to θ2 is positive definite at set 1 ˆ

θ and 

estimated 2 ˆ

θ . The condition for estimating capital and technology is that 

state representation (3.3)-(3.4) is reconstructible at θ

ˆ . Briefly, let Rt = 

[ T H1 , T F T H2 , ..., ( 1t F − )T T Ht ]T

, where F and Ht denote the state-transition and 

observation matrices in state representation (3.3)-(3.4) at θ

ˆ . 

Reconstructibility holds if Rt has full column rank for a sufficiently large 

t. See Kwakernaak and Sivan (1972), Anderson and Moore (1979), and references 

therein for further details. Both the parameter identification and 

reconstructibility conditions were verified numerically in the application. 

4. Estimation Results. 

4.1. Description of the Data. 

We used annual U.S. total manufacturing data on prices and quantities of 

output and inputs for 1947-1997. Investment and GDP-deflator data were obtained 

from the Bureau of Economic Analysis (BEA), research data from the National 

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

Statistics (BLS). All data were obtained in annual form, even though all except 

research price and quantity are also available monthly or quarterly, seasonally 

adjusted or not. All data were previously released to the public and are not 
17

confidential. Thus, we obtained data on 10 of the 13 variables in the model: pqt

and qt for 1958-1996, plt and lt for 1948-1997, pit and it for 1947-1996, prt and 

rt for 1953-1995, pmt for 1958-1996, and mt for 1958-1989. 

Except for labor quantity measured by the number of production workers, 

all prices and quantities were computed as indexes based on given nominal price 

indexes, real quantity indexes, and nominal expenditures. Real quantities were 

computed as nominal expenditures divided by nominal price indexes and nominal 

prices were computed as nominal expenditures divided by real quantity indexes. 

Then, all given or computed nominal price indexes were converted to real form 

by dividing them by the GDP deflator. 

 Resulting real prices and real quantities of U.S. total manufacturing 

output and inputs are depicted in figure 1. For viewing convenience, the data 

were first standardized and were then shifted up and rescaled to lie between 0 

and 4. The graphs suggest the following brief economic interpretation: 

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 used approximately the same labor input and more materials, 

capital, and technology inputs to produce more output, which resulted in 

increased productivity. 

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). Choosing 

total production workers (a "stock" concept) instead of total hours worked (a 

"flow" concept) is theoretically more consistent with adjustment costs arising 

from production function (2.4)-(2.5). 
18

Figure 1: U.S. Total Manufacturing Prices and Quantities of Output and 

Inputs, 1947-1997 

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
19

 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 production 

function’s form and the model's simulations indicate symmetrical roles for 

labor and materials, while the data in figures 1b and 1d show that the time 

path of materials matches 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 and capital and technology 

estimation, 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 required only that the materials quantity column in the 

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

reported final estimates, materials quantity was treated as unobserved, along 

with unobserved capital, technology, and output-demand state. 

4.2. Statistical Properties of the Estimated Model. 

Table 1 reports OLS estimates of input-price-process parameters in θ2. 

The table reports estimated coefficients, their absolute t ratios in 

parentheses, implied absolute characteristic roots, R2

, Ljung-Box Q statistics 

for testing absence of residual autocorrelations at lags 1-10, and their p 

values in parentheses. Estimated equations fit expectedly for level-form 

data, having R2 ≥ .90. Residuals show no significant autocorrelations, having 

p values of Q > .25. Except for the clearly stationary materials price 

equation and the possibly nonstationary research price equation, the investment 

price and labor price equations have borderline unit roots. All estimated 

characteristic roots satisfy the growth condition | λ | < 1/ δ , which is 

necessary for solving the firm's dynamic optimization problem. Although a 

cointegration analysis might seem appropriate, we did not attempt it because 

the input-price equations are needed only to provide input-price forecasts for 

the firm's dynamic optimization problem and they appear to do this 

adequately. 
20

Table 1: Ordinary Least Squares Estimates of Input-Price Process Parameters 

in θ2

Var. Parameter Estimates Fit Statistics 

1,

ˆφ⋅ 2,ˆφ⋅ | λ | 2 R Q 

pi 1.45 

(11.1) 

-.441 

(3.30) 

1.02 .971 5.64 

(.933) 

pr 

.652 

(4.01) 

.282 

(1.81) 

.949 .979 4.67 

(.968) 

pl 1.88 

(24.8) 

-.883 

(11.3) 

1.01 .999 14.8 

(.254) 

pm 

1.49 

(9.79) 

-.617 

(4.06) 

.785 .903 9.13 

(.692) 

Columns 2-6 display estimates of φ⋅,1 and φ⋅,2, their absolute t statistics in 

parentheses, implied maximum absolute characteristic roots (solutions of λ2

 - 

1, ˆφ λ ⋅ - 2, ˆφ⋅ = 0), R2

, Ljung-Box Q statistics for testing absence of residual 

autocorrelations at lags 1 to 10, and their p values in parentheses. 

Table 2 reports MLE of the remaining structural parameters in θ2, 

conditional on OLS estimates of the input-price-process parameters, and fit 

statistics of the resulting estimated reduced-form equations of observed 

endogenous variables. The ML-estimated parameters in θ2 are individually 

insignificant and their questionable numerical standard errors, based on the 

inverse information matrix, are not reported. Possibly more accurate standard 

errors could be computed using bootstrapping (Efron and Tibshirani, 1993). 

More importantly, the ML-estimated parameters in θ2 are jointly identified in 

the sense that the Hessian matrix of L(θ, YN ) with respect to θ2, evaluated at 

set and estimated parameters, θ

ˆ , is numerically positive definite. But, most 

importantly, a likelihood-ratio test, discussed in detail below, does not 

reject overidentifying restrictions on the parameters. Reduced-form equations 

of observed endogenous variables fit expectedly for level-form data: moderate 

(≅ .50) R2

 for labor and high (> .90) R2 for other endogenous variables reflect 
21

labor's noisiness and the other variables' smoothness. Estimated 5 of 6 

residual autocorrelation parameters in θ are near one, which raises the 

question of whether residual autocorrelations or the economic part of the model 

account for most of the sample variations of observed endogenous variables. 

However, reestimation with all θs set to zero produced 2

pq R = .918, 2q R = .879, 

2 Rl = .436, 2 Ri = .772, and 2 Rr = .944 for the reduced-form equations, which 

means that the economic part of the model accounts for most of the sample 

variations of endogenous variables. 

Table 2: Maximum Likelihood Estimates of Remaining Structural Parameters in θ2

Production Function Parameters 

β

ˆ = -9.14 (CES = -.099), ρ

ˆ = 267. (CET = .004) 

Output-Demand Curve Parameters 

ˆ

η = .605, 1d ˆφ = 1.39, 2d ˆφ = -.518 

Capital and Technology Equation Coefficients 

1k ˆφ = .589, 0i ˆφ = .774, 1 ˆφτ = .161, 0r ˆφ = .459 

Residual Autocorrelation Coefficients 

pq

ˆθ = .999, qˆθ = .914, θl

ˆ = .999, mˆθ = .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 Rr = .957 

 Qpq = 10.8, Qq = 5.96, Ql = 5.97, Qi = 18.6, Qr = 21.4 

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

Ljung-Box Q statistics are as in table 1. 
22

 The following likelihood ratio (LR) test of the overall validity of the 

overidentifying restrictions on the structural parameters is the key 

statistical test in the analysis. We obtained several different estimates of 

θ2 based on several different restrictions on θ1. In each case, θ1

restrictions and θ2 estimates implied approximately the same maximized 

likelihood values and reduced-form parameter estimates, hence, approximately 

the same Kalman-filter-based capital and technology estimates. Reduced-form 

parameters are identified by the model form and the data, but structural 

parameters are identified only with additional restrictions. The fact that 

structural parameters have particular set and estimated values is less 

important here, because the goal is to obtain Kalman-filter-based capital and 

technology estimates, which depend only on the data, the model form, and the 

reduced-form parameter values. The LR test is the key statistical test in the 

analysis because the Kalman-filter-based capital and technology estimates can 

be considered empirically valid if and only if the estimated reduced form is 

empirically valid and this occurs if and only if the overidentifying 

restrictions are not rejected for set and estimated structural parameter 

values. 

 L(θ

ˆ , YN ) = N⋅ln| N Ωˆ |, where N Ωˆ = (1/N)∑ =

N

1t

T ytt

~ y~ and yt

~ denotes innovations 

of yt evaluated at θ

ˆ , so that LR = N(ln| R,N Ωˆ |-ln| U,N Ωˆ |), where R,N Ωˆ and U,N Ωˆ

denote 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 MDKF automatically produces restricted innovations as 

part of MLE. We obtained unrestricted innovations as follows. We performed the 

test using the subsample 1960-1990 because only during this period were 

observations available for all 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 state and observation 

equations (3.3)-(3.4), we obtained the infinite autoregressive representation 

of yt and its finite p-lag approximation, 

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

~~ , 

where the residual yt

~~ is an approximation of the innovation yt

~ . We want to 

test the economic restrictions of the model, excluding the zero restrictions 
23

implied by exogeneity and mutual independence of input-price processes (2.7). 

Therefore, except for these zero restrictions, we considered the Φ's to be free 

parameters. For p = 2, we estimated the endogenous-observed-variable equations 

of (4.1) by applying OLS to data for 1960-1990 and reestimated the exogenousinput-price equations using the shorter sample. The resulting residuals were 

insignificantly serially correlated and were used to compute U,N Ωˆ . 

Under the null hypothesis that the overidentifying restrictions are 

valid, LR is distributed asymptotically χ2

(κ) as N → ∞, where κ denotes the 

number of overidentifying restrictions. LR rejects the null hypothesis when it 

exceeds critical value cα for 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 

κ = 118, imply LR = 142, with a p value of 6.66% so that the overidentifying 

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

unit roots, discussed at the end of this section, could modify the test 

results, it seems unlikely that accounting for their effects would change the 

nonrejection of the test result to strong rejection. 

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 state and observation 

equations (3.3)-(3.4), we obtain the structural infinite moving-average 

representation of yt in terms of the structural disturbance vector, εt, 

(4.2) yt = Ψ(L)εt = (∑∞

=0i Ψ i

iL )εt = ∑∞

=0i Ψi εt-i, 
24

where Ψi = J ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

× ×1313

13

i

13 1313

1 2

0

I

0I

BB

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 set to near zero. Let E[ y + kt | Yt ] denote the k-step-ahead 

forecast of y + kt , let y k,t

~ = y + kt - E[ y + kt | Yt ] denote the forecast error, and 

let Vk = Ey k,t

~ T y k,t

~ denote the error covariance matrix, given by 

(4.3) Vk = T

i

1k

0i i εΨΣΨ −

∑ = . 

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

endogenous variables and their sum in terms of the 10 estimated structural 

disturbance variances. That is, we decompose vk,ii, for i = 1, ..., 8, and 

∑ =

8

1i v ii,k , where vk,ii denotes the (i,i) diagonal element of Vk, in terms of 2 σj , 

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

∑ =

8

1i v due to ii,k

2 σj ; let 2/1 Σε denote 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 zeros elsewhere. 

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

decompositions of vk,ii and ∑ =

8

1i v ii,k are given by 

(4.4) sk,i,j = i

T

i

2/1T

jj

k

0i

2/1

i

T

i

(e ε ΨΣΣΨ e)ee ∑ = ε / i

T

i

k

0i i

T

i

(e ∑ ΨΣΨ e) = ε , 

(4.5) s j,k = (e e)ee T

i

2/1T

jj

k

0i

2/1

i

T

∑ = ε ε ΨΣΣΨ / (e e)T

i

k

0i i

T

∑ = εΨΣΨ . 

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 2, 3, 4, 7, and 11 in row 2 

indicate that, according to the estimated model, 12.8, 5.2, 7.0, 5.5, and 

66.8 percent of the variance of pq is, respectively, accounted for by the 

variances of disturbances of output, investment, research, price of 
25

investment, and output demand or 2 σq , 2 σi , 2 σr , 2 σpi, and 2 σd . Because the model is 

estimated using standardized data, the decompositions are unit free. Different 

restrictions of disturbance variances in θ result in different scales of 

responses. 

Table 3: Structural Variance Decomposition of the Estimated Model 

 2 σq

2 σi2 σr2 σk2 στ2 σpi2 σpr2 σpl2 σpm2 σd

s10,pq,j 12.8 5.2 7.0 .74 .14 5.5 .29 1.1 1.1 66.8

s10,q,j 35.2 14.2 19.2 .20 .38 15.2 .81 3.0 3.0 8.8 

s10,l,j 7.9 4.0 .40 79.2 1.5 .29 .04 .40 5.7 .60 

s10,m,j 7.9 4.0 .40 79.4 1.5 .29 .04 .28 5.6 .60 

s10,i,j .00 62.8 4.8 .51 .41 15.9 .78 3.2 3.6 8.0 

s10,r,j .00 .01 58.7 .03 .50 20.0 1.1 4.2 4.5 10.9

s10,k,j .00 22.6 1.9 63.7 .09 5.8 .30 1.1 1.4 3.1 

s10,τ,j .00 .01 56.6 .03 4.3 19.1 1.1 4.0 4.4 10.5

s 10,j 5.7 17.6 8.3 46.2 .52 7.4 .39 1.5 2.7 9.6 

Rows 2-9 give percentage decompositions of 10-step-ahead forecast-error 

variances of the 8 endogenous variables in terms of the variances of the 10 

estimated structural disturbances. Row 10 gives percentage decomposition of 

the sum of the variances of the 8 endogenous variables. Each row's numbers 

sum to 100. 

All disturbances except technology, price of research, and price of labor 

disturbances account for significant fractions (say, > 5%) of individual (rows 

2-9) and overall (row 10) variations of endogenous variables. Interestingly, 

investment and capital disturbances account for slightly more of individual 

variations and signficiantly more of overall variations than do research and 

technology disturbances, which counters the real-business-cycle premise that 
26

technology disturbances are the primary source of variations of variables. 

Possibly, this could be because the research data here reflect a minor portion 

of the actual research by U.S. manufacturing firms and because the perpetualinventory technology equation misspecifies the correct pattern of technology 

depreciation. Overall, in row 10 the decompositions indicate that investment, 

capital, price of investment, price of materials, and output demand 

disturbances are the leading sources of variations of the 8 endogenous 

variables. 

The simulations in figure 2, respectively, display the dynamic 

adjustment-cost behavior in the model in response to unit impulses in outputdemand and technology disturbances. The simulations in figure 2a match the 

general interpretation of figure 1. The simulations depict responses to a unit 

one-period shock (impulse) to the output-demand state in period 1, starting 

from an initial long-run equilibrium represented by the origin. The estimate ˆ

η

= .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 output-demand shock occurs, the price of output rises sharply but 

output increases only slightly. Initially, the extra output is produced using 

additional freely-adjusted 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. Eventually, all variables return to the 

origin. 

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. Again, eventually all variables return to the origin. 
27

Figure 2: Impulse Responses of the Estimated Model 

a: To an Impulse in the 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

b: To an Impulse in the 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
28

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

We applied the missing-data Kalman filter (MDKF) to the estimated model 

and data and, thereby, computed filtered state estimates, z t|t ˆ , and their error 

covariance matrices, E(zt-z t|t ˆ )(zt-z t|t ˆ )T

, for 1958-1997. Then, we picked elements 

7 and 8 of z t|t ˆ as the model-based estimates of capital and technology, k t|t ˆ and 

t|t ˆτ , and square roots of diagonal elements 7 and 8 of the error covariance 

matrix as their estimated standard errors. Figures 3-4 depict model-based and 

standard estimates of (production) capital and technology for U.S. total 

manufacturing industries for 1958-1997. Solid lines depict model-based 

estimates and their 2-standard-error confidence bounds. For capital, dashed 

lines depict weighted sums of BLS stock estimates of equipment, structures, 

inventories, and land, based on nonstochastic perpetual inventory equations 

(PIE). In addition, BLS produces service-flow estimates and BEA produces stock 

estimates of equipment, structures, inventories and land, but weighted sums of 

these estimates are very similar to the BLS stock estimates and are, thus, not 

depicted or considered further. For technology, dashed lines depict BLS 

estimates of total factor productivity (TFP) based on Solow residuals. BLS 

capital stock and TFP estimates are graphed in figures 3-4 as examples of 

standard capital and technology estimates. 

Because MLE is tractable generally only when data are scaled similarly, 

the data were standardized prior to estimation, by subtracting sample means and 

dividing by sample standard deviations. To make standard (BLS) capital (stock) 

and technology (TFP) estimates and their standard errors comparable with modelbased ones, we standardized all estimates and their standard errors. Also, to 

make the estimates and standard-error confidence bounds look more sensible by 

being positive, before graphing them, we shifted them all up by the same 

amount. However, because the graphed values are arbitrary, vertical differences 

between them should not be interpreted as percentage differences. The graphs 

start in 1958 because output, a critical determinant of the estimates, is first 

available in 1958. 

Figure 3 shows that 1958-1997 trends of model-based and standard 

capital and technology estimates are broadly similar, which makes them 

mutually reinforcing. However, being produced by government agencies and 

commonly used, standard estimates might be considered the "truer" ones. The 

intention here is not to challenge this view but to consider alternative 

capital and technology estimates based on an estimated dynamic structural 
29

economic model which has the following key features: the variables of primary 

interest, capital and technology, are endogenous in the model; production 

firms solve an explicitly considered dynamic optimization problem; resulting 

dynamics of endogenous variables arise naturally from elementary structural 

components, in particular, adjustment costs from the CES-CET production 

function; the model is identified and estimated using real (not simulated) 

data. 

Suppose "short run" means cycles with periodicities less than about 5.6 

years long (about the average business cycle length in the U.S. after World 

War II) and "long run" means longer cycles. Some short-run variations of 

capital and technology, either model-based or standard, are correlated with 

and, hence, may be considered explained by large known events such as the 

Vietnam War (1965-73) or oil-price shocks (1973, 1979). Remaining unexplained 

short-run variations may, then, be considered random noises. Figure 3 shows 

that the model-based capital estimates have more and larger noisy short-run 

variations than the model-based technology estimates. Consequently, the 

model-based capital estimates appear to be more uncertain than the modelbased technology estimates, an interpretation supported by the standard 

errors produced by the MDKF. In figure 3, the two-standard-error confidence 

bounds of the model-based capital and technology estimates are, respectively, 

about 1.01 and .036, which says that the model-based capital estimates are 

about 28 times more uncertain than the model-based technology estimates. 

Being in unit-free standardized form, the model-based capital and technology 

estimates and their confidence bounds are comparable. 

Do investment and capital disturbances or do research and technology 

disturbances better account for variations of endogenous variables in the 

1990s, in particular, output (Gordon, 2000; Oliner and Sichel, 2000; Stiroh, 

2001)? Nonrejection of the estimated model's overidentifying restrictions by 

the likelihood-ratio test suggests that the estimated model is an 

econometrically acceptable explanation of the data for 1947-1997. Variance 

decompositions of the estimated model in table 3 indicate that investment and 

capital disturbances account for slightly more of individual variations in 

endogenous variables (rows 2-9) and significantly more overall (row 10) than do 

research and technology disturbances, which counters the real-business-cycle 

premise that research and technology disturbances are the primary source of 

variations in variables. 
30

Figure 3: Estimated Model-Based and Standard Capital (Stock) and Technology 

(TFP) Estimates for U.S. Total Manufacturing, 1958-1997 

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

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

Model-based capital and technology estimates are based on θ

ˆ , including k σˆ = 

.994 and στ ˆ = .055. Solid lines depict model-based capital and technology 

estimates and 2-standard-error confidence bounds produced by the Kalman filter. 

Dashed lines depict standard capital (stock) and technology (total factor 

productivity) estimates produced by BLS. 
31

Time-series properties of standard capital estimates depend entirely on 

time-series properties of investment and on capital depreciation in its 

perpetual inventory equation. For given time-series properties of investment, 

standard capital estimates should be smoother and more trendlike when capital 

depreciates more slowly. Time-series properties of model-based capital and 

technology estimates likewise depend on time-series properties of investment 

and research and on depreciation rates, but also on Kalman-filter estimates of 

disturbances in stochastic PIEs. For example, Kalman-filter estimates of 

capital, based on equation (2.9), are 

(4.6) k t|t ˆ = φk1k t|1t ˆ − + φi0it + t|t,k ˆζ , 

where k t|s ˆ denotes expected ks conditional on Yt . Thus, time-series properties 

of model-based capital and technology estimates also depend partly on timeseries properties of estimated disturbances, t|t,k ˆζ . 

To consider how much noise the relatively large estimated standard 

deviation of the capital disturbance, k σˆ = .99, passes to the model-based 

capital estimates through equation (4.6), we recomputed the capital and 

technology estimates for virtually no capital and technology disturbances, for 

k σˆ = στ ˆ = .0001 and the other parameters left at their set and estimated 

values. Thus, going from figure 3a to 4a, the sample average of estimated 

standard errors of model-based capital estimates declines 5-fold, from 1.03 to 

.205. Reducing k σˆ and στ ˆ in the move from figure 3a to 4a does not reduce 

capital's standard error proportionately, because it also depends on unchanged 

standard deviations of other variables' disturbances. 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, causes the sample average of the estimated standard 

error of model-based technology estimates to decline only slightly, from .089 

to .060, and, correspondingly, for the technology estimates to change little. 
32

Figure 4: Alternative Model-Based and Standard Capital (Stock) and Technology 

(TFP) Estimates for U.S. Total Manufacturing, 1958-1997 

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

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

Alternative model-based estimates are based on θ

ˆ , except for k σˆ = στ ˆ = .0001. 

Solid lines depict model-based capital and technology estimates and 2-standarderror confidence bounds produced by the Kalman filter. Dashed lines depict 

standard capital (stock) and technology (total factor productivity) estimates 

produced by BLS. 
33

Estimated annual capital and technology depreciation rates of 1- 1k ˆφ = .39 

and 1- 1 ˆφτ = .96 are very high, in particular, compared to Jorgenson and 

Stephenson's (1967) implied annual capital depreciation rate of .11. To check 

whether systemwide MLE caused the high estimated depreciation rates, we 

reestimated capital and technology equations (2.9)-(2.10) in terms of their 

underlying continuous-time parameters using nonlinear least squares (NLS) and 

the model-based and BLS capital-stock and technology estimates as data. 

Although NLS estimates of 1k ˆφ and 1 ˆφτ in table 4 differ somewhat from ML 

estimates in table 2, they are very similar for model-based and BLS data. As 

usual, estimated equation fit depends on dependent variable noisiness, so that 

the estimated capital equation fits better with BLS data ( 2 R = .891) than with k

model-based data ( 2 Rk = .730) and vice versa for the technology equation. 

Although MLE 1k ˆφ = .589 and 1 ˆφτ = .161 might seem low, they work 

econometrically, because, along with the other set and estimated parameter 

values, they imply an unrejected estimated model and model-based capital and 

technology estimates whose trends conform to the standard estimates. 

For given depreciation rates, investment, and disturbance variances, 

model-based capital and technology estimates are smoother to the extent that 

reduced-form characteristic roots are near one. A reduced-form characteristic 

root is an eigenvalue of transition matrix F of state equation (3.3). There 

are 26 such roots: 2 exogenous roots from demand-state process (2.3), 8 

exogenous roots from input-price processes (2.7), 6 exogenous roots from 

residual autocorrelations, and 10 endogenous roots. The dynamic optimization 

problem suggests that one or more endogenous roots should be near one when β is 

negatively large, so that the input-substitution elasticity CES ≅ 0, and ρ is 

positively large, so that adjustment costs are high and the outputtransformation elasticity CET ≅ 0. The estimates in tables 1-2 imply that 8 of 

16 exogenous roots are within .02 of one and table 2 indicates that β

ˆ = -9.14, 

CES = -.099, ˆρ = 275, and CET = .004, so that more than one endogenous root 

should be and is near one. To this extent, model-based capital and technology 

estimates should be smoother and more trendlike. 
34

Table 4: Nonlinear Least Squares Estimates of Capital and Technology 

Equations 

Capital Equation 

Capital Data 1k ˆφ 0i ˆφ 2 Rk

Model-Based 

Estimate 

.336 

(22.2) 

.608 

(9.14) 

.730 

BLS Estimate .363 

(78.1) 

.629 

(29.3) 

.981 

Technology Equation 

Technology Data 1 ˆφτ 0r ˆφ 2 Rτ

Model-Based 

Estimate 

.376 

(118.) 

.638 

(42.2) 

.992 

BLS Estimate .323 

(50.8) 

.599 

(21.7) 

.945 

Columns 2-3 show estimated discrete-time parameters, φ, implied by estimated 

underlying continuous-time parameters, f. Absolute t statistics in 

parenthesis are based on linear approximations of the nonlinear mappings from 

f to φ. 

5. Conclusion. 

The paper has developed and applied an economic model-based method for 

estimating unobserved or latent stocks of production capital and technology or 

total factor productivity of U.S. total manufacturing industries for 1958-1997. 

The method involves estimating a dynamic structural economic model and 

computing estimates and standard errors of capital and technology by applying 

the Kalman filter to the estimated model and the data. Although, standard 

methods for estimating capital and technology are appealing in their 

theoretical and computational simplicity, they are unnecessarily restrictive in 

important respects, for example, ignore adjustment costs. 
35

The estimated model accounts for the 1958-1997 data in the sense that 

overidentifying restrictions are not rejected and suggests that investment and 

capital disturbances better account for variations of endogenous variables than 

do research and technology disturbances because they account for significantly 

more of overall variations (table 3, row 10). Figure 3 also suggests visually 

that in the 1990s above average capital growth, not above average technology 

growth, better accounts, in particular, for above average output growth. Trends 

of model-based and standard estimates of capital and technology for 1958-1997 

are broadly similar and, therefore, reinforce each other. Model-based capital 

estimates are much noisier than standard estimates and vice versa for 

technology. Model-based capital estimates are about 10 times more uncertain 

than model-based technology estimates in terms of estimated standard errors. 

The paper has the specific objective of comparing standard capital and 

technology estimates with model-based capital and technology estimates 

computed by applying the Kalman filter to an estimated model. The key 

economic features of the particular estimated model used here are a demandsupply (partial) equilibrium in an output market in which a static outputdemand curve is specified directly and a dynamic output-supply curve is 

derived from the solution of a representative production firm's dynamic 

optimization problem arising from adjustment costs. The paper suggests at 

least the following four extensions which could be considered in the future. 

1. Capital and technology equations might be specified as rational 

distributed lags, with gestation lags and nongeometrical depreciation rates. 

For example, if in equation (1.1) φk1 = ... = φkp = 0, φi0 = ... = φir = 0, and 

φi,r+1 = ... = φiq = 1, then, a unit of investment gestates for r periods, 

becomes a unit of productive capital for the next q-r periods, and becomes 

fully depreciated thereafter. 

2. Capital embodiment of technology has been considered frequently, 

recently by Greenwood, Hercowitz, and Krusell (1997). Here, if the two 

identification conditions discussed at the end of section 3 hold, as they do in 

the application, then, the Kalman filter automatically and implicitly 

disembodies technology from capital. However, the question of the degree to 

which capital embodies technology is unlikely to be resolved theoretically 

using other specifications, only empirically using more accurate research data. 

A cross-sectionally disaggregated analysis is unlikely to resolve this question 

either, because disaggregated research data are also often inaccurate and 

incomplete. Research inputs and outputs may simply be inherently difficult to 

measure because research is largely a mental process. 
36

3. Capital and technology are usually only partly utilized and somewhat 

misallocated in any period. Both the standard and the present model-based 

capital and technology estimates treat capital and technology as fully utilized 

and optimally allocated. Including capital and technology utilization and 

misallocation in the analysis would require including capital and technology 

utilization rates and capital and technology market valuations as variables in 

the model and data on them in the empirical analysis. 

4. The analysis could be extended in several ways to a general 

equilibrium. Input prices are now treated as exogenous and determined by 

autoregressions (2.7), with a purely statistical role and no particular 

economic meanings. Input prices could also be partly determined by 

endogenously determined research and technology. Accordingly, more of the 

above average output growth in the 1990s would be attributed to research and 

technology and less to investment and capital if research and technology 

growth reduced prices of investment and, thus, increased demand for 

investment and capital. Also, the static output-demand curve could be 

extended to a dynamic, economically motivated, output-demand curve analogous 

to the dynamic output-supply curve. Such extensions should inform about how 

much the present model-based capital and technology estimates depend on the 

particular model used here. 

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

 Because ∇2cq(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) 

c27 = γ2/(1-α1) 

c33 = γ3(1-γ3)(ρ-1) + γ3

2

α1(1-β)/(1-α1) 
37

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 elements of 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 2, which 

is allowed 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 = c0c12 – c22

R12 = c0c12c13 – c23

R22 = c0 2 c13 – 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 = c0 2 c14 - c44

Q12 = c0c14c15 – c45

Q15 = c0c14c16 – c46

Q16 = c0c14c17 – c47

Q17 = -c0c14

Q22 = c0 2 c15 – c55

Q25 = c0c15c16 – c56

Q26 = c0c15c17 - c57

Q27 = -c0c15. 

Finally, we state 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 are 

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

Proceeding row-wise across the matrices, 
38

A0,1,2 = η

A0,1,13 = -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

A0,4,8 = -c57

A0,4,11 = -c67

A0,4,12 = -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, φpl,2, φpm,2, φd2]. 

References. 

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