BLS WORKING PAPERS 

 

U.S. Department of Labor 

U.S. Bureau of Labor Statistics 

Office of Prices and Living Conditions 

Further Model-Based Estimates of U.S. Total Manufacturing Production Capital and

Technology, 1949-2005 

 

Baoline Chen, Bureau of Economic Analysis

Peter Zadrozny, U.S. Bureau of Labor Statistics

Working Paper 430

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. 
FURTHER MODEL-BASED ESTIMATES OF U.S. TOTAL MANUFACTURING 

PRODUCTION CAPITAL AND TECHNOLOGY, 1949-2005 

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.peter@bls.gov 

April 1, 2009 

JEL Classification: C50, C81, D24, L60 

Key words: Kalman filter estimation of unobserved variables 

ABSTRACT 

Production capital and technology (i.e., total factor productivity) in U.S. 

manufacturing are fundamental for understanding output and productivity 

growth of the U.S. economy but are unobserved at this level of aggregation 

and must be estimated before being used in empirical analysis. Previously, we 

developed a method for estimating production capital and technology based on 

an estimated dynamic structural economic model and applied the method using 

annual SIC data for 1947-1997 to estimate production capital and technology 

in U.S. total manufacturing. In this paper, we update this work by 

reestimating the model and production capital and technology using annual SIC 

data for 1949-2001 and partly overlapping NAICS data for 1987-2005.*

 

*

The paper represents the authors' views and does not necessarily represent 

any official positions of the Bureau of Economic Analysis or the Bureau of 

Labor Statistics. The paper was presented at the CESifo Conference on 

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

Malley for his discussion of the paper at the conference and Sung Ahn for 

help with an econometric problem. 

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

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

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

1. Introduction. 

Production capital and total factor productivity or, more simply, 

capital and technology in U.S. manufacturing are fundamental for 

understanding output and productivity growth of the U.S. economy but are 

unobserved at this level of aggregation and must be estimated before being 

used in empirical analysis. Standard methods for estimating aggregate capital 

and technology are based on Solow (1957) and Jorgenson (1963). In level form, 

technology is usually called total factor productivity and, in percentagegrowth form, the Solow residual. Chen and Zadrozny (2005) developed a modelbased method for estimating unobserved capital and technology by, first, 

estimating a dynamic structural economic model by maximum likelihood and, 

then, using the estimated model, the data, and the Kalman filter to estimate 

capital and technology. Chen and Zadrozny (2005) applied the method to annual 

Standard Industrial Classifications (SIC) data for 1947-1997 and obtained 

estimates of capital and technology for U.S. total manufacturing. Current and 

previous applications illustrate the model-based method as a general method 

for accounting for the growth of output which is grounded in dynamic economic 

modelling and econometric practice. 

The present paper extends Chen and Zadrozny (2005) by reestimating 

their model and reestimating capital and technology for U.S. total 

manufacturing using more contemporary data, a combination of annual Standard 

Industrial Classification (SIC) data for 1949-2001 and partly overlapping 

annual North American Industrial Classification System (NAICS) data for 1987-

2005. The application shows: (1) small changes in the estimated model; (2) 

for 1949-2000, trends of model-based and standard capital and technology are 

broadly similar; (3) for 2000-2005, model-based and standard estimates 

diverge significantly: model-based capital continues its previous growth; 

standard capital levels off; model-based technology declines and levels off; 

and, standard technology continues its previous growth; and, (4) for 1949-

2005, model-based capital is noisy and uncertain and model-based technology 

is smooth and certain. The overall continuity of present results with those 

in Chen and Zadrozny (2005) supports the model-based method as a general 

method for output-growth accounting. 

The paper contributes in several ways. Updating model-based capital and 

technology using data through 2005 is significant not only because more and 

more recent data are used, but because during 2000-2005 manufacturing output 

and input quantities leveled off or declined, despite the fact that the 
2

estimated parameters of the model changed little. Figure 1 shows these 

patterns: overall, output and research leveled off, materials and investment 

declined slightly, and labor declined significantly. 

Standard capital and technology are based minimally on economic 

motivations for accumulating capital and technology. By contrast, the present 

model-based estimates, graphed in figure 3, aim to more fully incorporate 

these motivations. This is done by explicitly including the motivations in 

the estimated model which is, then, used to estimate capital and technology. 

Thus, unlike the present model-based method, standard methods are unsuitable 

for an economic analysis of the effects for policy changes on capital and 

technology estimates. For example, suitably expanded the present model could 

predict the effects of a change in tax rules on investment in capital and 

technology. Apparently, Slade (1989) is the only previous work that similarly 

applies Kalman filtering to an estimated economic model with explicit 

optimization in order to estimate technology (but not capital). However, by 

treating technology as exogenous, Slade's model disconnects technology from 

economic decisions and, thus, is unsuitable for such economic policy analysis 

of technology accumulation. Because it treats technology as an unexplained 

residual, the standard Solow-residual analysis is similarly unsuitable for 

such economic policy analysis. 

Finally, in contrast to standard methods, the model-based method 

produces standard errors of capital and technology as a normal product of the 

Kalman filter, which quantify uncertainty about capital and technology due to 

model disturbances (but not uncertainty due to model and parameter 

uncertainty). The model-based capital and technology standard errors in 

figure 3 are about 1.02 and .036, respectively, so that the capital 

confidence bounds in figure 3 are about 28 times wider than the technology 

confidence bounds. Being in unit-free standardized form, the model-based 

capital and technology estimates and their confidence bounds are comparable. 

A similar numerical measure of uncertainty does not exist for standard 

capital and technology. 

The partly overlapping SIC and NAICS data for 1987-2001 required some 

method for merging them. We solved this problem by averaging growth rates of 

SIC and NAICS data (additively and with equal weights) where they overlapped. 

Despite being based on extended, newly classified, partly revised, and partly 

averaged data, current model estimates and model-based capital and technology 

estimates for 1949-2005 are close to previous ones for 1947-1997, as is seen 

by comparing current and previous tables 1 and 2 and comparing current and 
3

previous figures 3. Thus, at least for the present application, the new SIC 

and NAICS data are compatible with the previous SIC data. 

The paper continues as follows. Section 2 describes the structure of 

the model. In the model, U.S. total manufacturing is treated as a single 

industry whose demand side is modeled by a conventional static demand curve 

but whose supply side is modeled in explicit dynamic detail. Following Lucas 

and Prescott (1971), the industry is considered to be competitive and its 

output supply is modeled by a representative firm which solves a dynamic 

optimization problem. The problem's explicit solution is used to impose 

overidentifying restrictions during estimation on the reduced-form parameters 

in terms of the structural parameters. The restrictions ensure that the 

estimated structural parameters are identified when key variables in the 

model (capital, technology, and demand state) are completely unobserved. 

Section 3 first discusses how, following Chen and Zadrozny (2005), the 

model's equations are assembled into a state representation for estimating 

the structural parameters and for computing filtered estimates of capital and 

technology based on the estimated model. Section 3 then discusses the 

application, first, sources and properties of the SIC and NAICS data and how 

they were merged and, then, properties of the estimated structural model and 

comparison of model-based and standard estimates of capital and technology. 

Section 4 contains concluding remarks.

2. Specification and Solution of the Model. 

Every period, t, the representative firm of the industry being 

considered maximizes the expected present value of profits, 

(2.1) vt = Et∑∞

=0k πδ + kt

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 or revenues minus costs, where 

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 deflator. The firm's 

optimization problem is stated precisely in this section. 
4

 To obtain a competitive rational expectations equilibrium solution, 

following Lucas and Prescott (1971), we set revenues 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. 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. Accordingly, we assume the 

industry's inverse output-demand curve is 

(2.2) pqt = -ηqt + dt + ζpqt, 

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

the 2nd-order autoregressive (AR(2)) process 

(2.3) dt = φd1dt-1 + φd2dt-2 + ζdt, 

and ζpqt and ζdt are disturbances. Preliminary experimentation with alternative 

specifications during model estimation showed that dt is specified adequately 

as generated by an AR(2) process. The full set of distributional assumptions on 

disturbances is stated in section 3. 

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

materials (m), to produce saleable output (q), installs investment goods (i), 

and conducts 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 stock of technology or total factor productivity. 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, 
5

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. Thus, we 

call (2.4)-(2.5) the CES-CET production function. 

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 of increases in investment and research. 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. 

 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). 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 (becomes more 

negative). 

 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 total factor 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 would be prices of renting capital and technology stocks, 
6

obtained using Jorgenson's (1963) results for converting purchase prices of 

investment and research flows to rental prices of capital and technology 

stocks, under the more restrictive assumption that all inputs are variable. In 

this paper, we instead work with purchase prices of investment and research 

flows 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 restrictions that allow us to identify, 

hence, to estimate the structural parameters of the model when capital and 

technology stocks are completely unobserved. 

 Ignoring constant and linear terms, which contribute only an additional 

constant term to the optimal decision rule that is removed by mean adjustment 

of the data, 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 explicitly in terms of the α's, β, γ's, and ρ in Chen and Zadrozny 

(2005) for w0 = (1, 1, 1, 1, 1, α2, α3)T

, a value which results in the simplest 

expression for ∇2

cq(w0) and works econometrically for the present and previous 

data. Therefore, 

(2.6) πt = -(1/2)ηqt

2

 + qt(dt + ζpq,t) – (1/2)wt

T

⋅∇2cq(w0)⋅wt – pitit – prtrt. 

The resulting symmetric Hessian matrix, ∇2

cq(w0), has the standard local 

properties of homogeneity, convexity and concavity with respect to w. 

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

industry being modeled and preliminary experimentation showed that the input 

prices are specified adequately as generated by independent, univariate, AR(2) 

processes, 

(2.7) pit = φpi1pi,t-1 + φpi2pi,t-2 + ζpit, 

 prt = φpr1pr,t-1 + φpr2pr,t-2 + ζprt, 

 plt = φpl1pl,t-1 + φpl2pl,t-2 + ζplt, 

 pmt = φpm1pm,t-1 + φpm2pm,t-2 + ζpmt, 
7

where ζpit, ζprt, ζplt, and ζpmt are mutually uncorrelated disturbances. Input-price 

processes (2.7) are both structural and reduced form and have no particular 

structural interpretations. They are needed to provide forecasts of input 

prices for the firm's dynamic optimization problem. 

Let λ and | λ | denote a characteristic root and the largest absolute 

characteristic root of an input-price process. For example, a characteristic 

root of the investment-price process solves λ2

 - φpi1λ - φpi2 = 0 and is 

stationary if it is less than one in absolute value. Whether or not the 

estimated model has all stationary or some nonstationary roots (in practice, 

near unity), estimates of the structural parameters will be consistent and 

efficient and estimates of capital and technology will be linear least squares 

estimates based on the estimated model. The main difference when the estimated 

model has nonstationary roots is that the estimated structural parameters will 

no longer be asymptotically normally distributed, but will have a nonstandard 

asymptotic distribution. To solve the dynamic optimization problem, each inputprice process must satisfy the growth condition | λ | < 1/ δ (Hansen and 

Sargent, 1980) and the estimated model satisfies this condition. 

 We assume that capital accumulates according to the stochastic 

perpetual inventory equation (PIE) 

(2.8) kt = φk1kt-1 + φi0it + ζkt, 

where 0 < φk1 < 1, φi0 = (φk1 - 1)/ln(φk1), ζkt ~ NIID(0, 2 σk ), and 2 σk > 0. The 

restriction φi0 = (φk1 - 1)/ln(φk1) comes from an antecedent continuous-time 

formulation and reflects the property that investments undertaken earlier in a 

period depreciate more by the end of the period than investments undertaken 

later in the period (Chen and Zadrozny, 2005). Similarly, we assume that 

technology accumulates according to the stochastic PIE 

(2.9) τt = φτ1τt-1 + φr0rt + ζτt, 

where 0 < φτ1 < 1, φr0 = (φτ1 – 1)/ln(φτ1), and ζτt ~ NIID(0, 2 στ ), and 2 στ > 0. 

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 the specified laws of motion and the solved optimal decision rules 
8

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 

(2.10) 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 a disturbance added for 

statistical reasons. 

 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 obtained using the envelope theorem, 

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

(2.12) 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, and ζ lt

and ζmt are disturbances added for statistical reasons. 

 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 output-demand, inputprice, capital, and technology processes, (2.3) and (2.7)-(2.9), as the state 

equation 

(2.13) xt = Fxt-1 + Gut, F = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

7 ×77

1 2

0I

FF

, G = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

×212

0

0

G

, 

where F1 = diag[φk1, φτ1, φpi1, φpr1, φpl1, φpm1, φd1], F2 = diag[0, 0, φpi2, φpr2, φpl2, 

φpm2, φd2], G0 = diag[φi0, φτ0], Im is the m×m identity matrix, and 0m×n is the m×n 
9

zero matrix. We may suppress disturbances in equation (2.13) because the 

regulator problem is certainty equivalent. We use output-supply rule (2.10) to 

eliminate qt from πt and write πt as the quadratic form 

(2.14) πt = ut

T

Rut + 2ut

T

Sxt-1 + xt-1TQxt-1. 

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

of ∇2

cq in Chen and Zadrozny (2005). 

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

terms of the quadratic form (2.14), with respect to the feedback matrix K in 

the linear decision rule ut = Kxt-1, subject to the state equation (2.13). Under 

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

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

matrix Riccati equation using a Schur decomposition method (Laub, 1979). 

Finally, we write the optimal investment-research decision rule as 

(2.15) ut = Kxt-1 + (ζit, ζrt)T

, 

where (ζit, ζrt)T

 is a 2×1 disturbance vector added for statistical reasons. 

 Further details about the model's specification, including comparison 

with the more familiar translog DVPCF (Christensen, Jorgenson, and Lau, 1973), 

are discussed in Chen and Zadrozny (2005). 

3. Estimation of the Model and Capital and Technology. 

3.1 Maximum Likelihood Estimation of the Model. 

To estimate the model's structural parameters by maximum likelihood 

(MLE), using the Kalman filter, and, then, to estimate unobserved capital and 

technology, also 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 = (ζpqt, ζqt, ζlt, ζmt, ζit, ζrt, 

ζkt, ζτt, ζpit, ζprt, ζplt, ζpmt, ζdt)T

. We assume that the disturbances are generated 

by mutually uncorrelated, normally distributed, stationary processes, where the 

first 6 disturbances are generated by AR(1) processes and the last 7 

disturbances are serially uncorrelated. The AR(1) and white-noise 
10

specifications of the disturbance processes are based on preliminary 

experimentation and their adequacy is verified by the insignificant Q 

statistics in table 2. Thus, we assume that 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), 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), (2.10)-(2.13), and (2.15). These 13 scalarlevel equations constitute the complete set of linear simultaneous equations 

which, 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 A0, A1, and A2, which depend on η, the φ's, and the elements of ∇2

cq and 

K, are stated in detail in Chen and Zadrozny (2005). Rewriting (3.1), we obtain 

the reduced-form VAR(2) process 

(3.2) yt = B1yt-1 + B2yt-2 + ξt, 

where B1 = 1 A0

− (A1 + ΘA0), B2 = 1 A0− (A2 - ΘA1), ξt = 1 A0− εt ∼ NIID(0,Σξ), and Σξ = 

1 A0− ΣεA0

-T. 

Finally, following Chen and Zadrozny (2005), we write process (3.2) in 

state-space form and use this form, in conjunction with the missing-data Kalman 

filter (MDKF), to compute the normal likelihood function of the observations. 

Let L(θ) denote the resulting likelihood function, where θ is the vector of the 

39 structural parameters, defined by θ = ( T θ1 , T θ2 )T

, θ1 = (δ, α1, α2, γ1, γ2, 2 σpq , 

2 σl , 2 σm )T

, and θ2 = (φpi1, φpr1, φpl1, φpm1, φpi2, φpr2, φpl2, φpm2, 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. 

Under the following identifying restrictions on θ1, we used the MDKF to compute 

L(θ) in maximum likelihood estimation (MLE). We needed to use the MDKF as 

opposed to the ordinary KF, which requires that all variables are observed in 

every sample period, because variables in the model are either not observed 

over the same periods or, like kt, τt, and dt, are not observed at all. A future 
11

extension of the present analysis could further exploit the MDKF by using a 

sample of mixed-frequency observations, analogous to Zadrozny (1990). Details 

about implementing the MDKF accurately and quickly in the MLE are discussed in 

Anderson and Moore (1979), Zadrozny (1988, 1990), and references therein. 

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

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

which ensure that θ2 is locally identified and, hence, is estimatable by MLE. 

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 

would seem that 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 materials (mt) was treated as unobserved (for reasons 

explained in Chen and Zadrozny, 2005). However, it turned out that setting 2 σpq

= 2 σl = 2 σm = 10-10 was sufficient to identify the unrestricted and estimated 

structural parameters. We set the 3 variances to small positive values, rather 

than to zero, because doing so resulted in more accurate computations using the 

MDKF. We considered different identifying restrictions and obtained different 

structural-parameter estimates but obtained similar reduced-form-parameter and 

capital and technology estimates. 

Because input prices are assumed to be generated by exogenous 

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 using MLE to estimate simultaneously all 

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

parameters using OLS and, then, conditional on these estimates, used MLE to 

estimate the remaining parameters in θ2. 

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 one that the Hessian matrix of L(θ, YN ) with respect 

to estimated parameters in θ2, evaluated at restricted and estimated parameter 

values in θ, is positive definite. The second capital-and-technologyidentification condition is that the model's state representation is 

reconstructible. See Chen and Zadrozny (2005) and references therein for 

further details. Both the parameter identification and reconstructibility 

conditions were verified numerically in the application. 
12

3.2. Description of the Data. 

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

output and inputs from 1949-2005. Investment and GDP-deflator data were 

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

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

Statistics (BLS). All data were obtained from their producers in annual form, 

even though all except research price and quantity data are also available 

monthly or quarterly and seasonally adjusted or not. All data were previously 

released to the public and are not confidential. Thus, we obtained data on 10 

of 13 variables in the model: pqt, qt, lt, mt, it, pit, plt, and pmt for 1949-2005 

and prt and rt for 1953-2005. 

Until 1997, U.S. industries were classified according to the Standard 

Industrial Classification (SIC), with the highest (cross-sectionally) 

aggregated manufacturing industries numbered 20-39. After 1997, the North 

American Industry Classification System (NAICS) replaced SIC, with the highest 

aggregated manufacturing industries numbered 31-33. As expected, data for more 

aggregated industry groups were affected less by the switch than data for less 

aggregated industry groups and, for each degree of aggregation, levels data 

were affected more than growth-rate data. After the switch, BLS produced SIC 

manufacturing data until 2001 and NAICS manufacturing data back to 1987. Thus, 

overlapping SIC and NAICS manufacturing data are available for 1987-2001. 

In MLE with the MDKF, variables can have any pattern of missing 

observations subject only to having enough observations so that parameters can 

be identified and estimated. The MDKF can automatically process multiple 

observations per variable when a state representation is set up to allow this 

possibility. However, it comes at the cost of specifying an observation-error 

process, imposing additional restrictions on parameters to ensure 

identification, and estimating additional observation-error parameters. 

Instead, we more simply transformed the two observations per variable to a 

single one in the overlapping period 1987-2001 as follows. When both SIC and 

NAICS observations were available, we arithmetically averaged their growth 

rates, with equal weights, to a single observation; when one observation was 

available, we chose it as the single observation. 

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, as follows. Real 
13

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 scaled 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, which resulted in increased 

labor productivity. Current and previous figures 1 show that current merged SIC 

and NAICS data and previous SIC data are similar in the overlapping 1949-1997 

period. The notable exception is research prices which in the previous SIC data 

are slightly noisier and follow a mostly concave path but in the current SIC 

and NAICS data initially follow a convex path which later switches to a concave 

path. 

 [Put figure 1 approximately here] 

3.3. Properties of the Estimated Model. 

 Following Chen and Zadrozny (2005) and for reasons explained there, when 

estimating the model, we treated materials inputs as unobserved, because 

including their observations in the estimation resulted in a very poor fit of 

the labor equation. Table 1 reports OLS estimates of input-price processes 

(2.7): estimated coefficients, their absolute t ratios in parentheses, 

implied absolute characteristic roots, R2

 statistics, and Q statistics for 

testing absence of residual autocorrelations at lags 1-10, with marginal 

significance levels or p values in parentheses. Estimated equations fit 

typically well for level-form data, having R2 ≥ .81. Residuals show no 

significant autocorrelations, having p values of Q statistics greater than 

1%. Except for the clearly stationary materials price equation, the other price 

equations have borderline unit roots. All characteristic roots satisfy the 

growth condition | λ | < 1/ δ , which is required for solving the firm's dynamic 

optimization problem. The only notable difference between current and previous 
14

table 1 estimates of the input-price processes is that the maximum absolute 

characteristic root of the estimated research-price equation increased from a 

borderline stationary value of .949 to a firmly unit-root value of .999. 

[Put table 1 approximately here] 

Table 2 reports MLE of non-input-price parameters in θ2 (conditional on 

θ1 set by identifying restrictions and on OLS-estimated input-price parameters 

in θ2) and fit statistics of implied reduced-form equations of observed 

endogenous variables. Standard errors of ML-parameter estimates were very large 

and are not reported because the nonuniqueness of the structural parameter 

estimates vitiates the usual meaning of their standard errors. In particular, 

experimentation showed that different identifying restrictions resulted in 

different structural-parameter estimates, but similar reduced-form-parameter, 

capital, and technology estimates. However, if the principal goal is obtaining 

model-based capital and technology estimates, then, as happened, obtaining 

essentially unique reduced-form-parameter, capital, and technology estimates is 

primary and obtaining nonunique structural-parameter estimates is secondary. 

Thus, because nonunique structural-parameter estimates depend on somewhat 

arbitrary identifying restrictions, their dubious standard errors are not 

reported. 

[Put table 2 approximately here] 

R2

 statistics in table 2 show that reduced-form equations of observed 

endogenous variables have typically good fits for level-form data: moderate (> 

.65) R2

 of the labor equation and high (> .93) R2s of the other equations 

reflect labor's noisiness and the other variables' smoothness. Estimated 

residual autocorrelation parameters, θ, are high, which raises the question of 

whether residual autocorrelations or the economic part of the model account for 

most of the sample variations of the observed endogenous variables. However, 

reestimation with all θ's set to zero produced 2

pq R = .921, 2q R = .929, 2 Rl = 

.604, 2 Ri = .905, and 2 Rr = .961 for the reduced-form equations, which 

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

variations of the endogenous variables. 
15

Comparing current and previous tables 2, only 1 ˆφτ differs significantly, 

declining from .161 to .043, which indicates increased per annum depreciation 

of technology from 83.9% to 95.7%. Although differences in other parameter 

estimates might seem insignificant, they imply quantitatively different 

behavior: although current and previous impulse responses in figures 2 are 

qualitatively similar, variance decompositions in current and previous tables 3 

are quantitatively different. 

Following Chen and Zadrozny (2005), we used a likelihood ratio to test 

the validity of the model's overidentifying restrictions. The test is important 

because if the estimated model's overidentifying restrictions are not rejected, 

then, the estimated model and the capital and technology estimates derived from 

it can be considered consistent with the data. To test the economic or 

behavioral implications of the model, we excluded exogeneity zero restrictions 

on the input-price processes. Under standard assumptions on the data generating 

process and the assumption that the null hypothesis that the overidentifying 

restrictions are valid, the likelihood-ratio statistic, LR, is distributed 

asymptotically as χ2

(κ), where κ = 118 denotes the degrees of freedom and the 

number of overidentifying restrictions being tested. For details on how LR was 

constructed, see Chen and Zadrozny (2005) and Sims (1980, p. 17, fn. 18). 

For the 1960-1990 SIC-NAICS data (the sample period for the LR test in 

Chen and Zadrozny, 2005), under conventional χ2

(118) evaluation, LR = 151 with 

p = .0205 indicates either weak rejection (p < 5%) or weak nonrejection (p > 

1%) of the null hypothesis. For the 1960-2005 SIC-NAICS data, under 

conventional χ2

(118) evaluation, LR = 181 with p = .000169 indicates strong 

rejection of the null hypothesis. Although unit roots might affect these p 

values (see the discussion at the end of section 3.4), it is unclear in which 

direction. More significantly, because 118 degrees of freedom is large relative 

to 57 sample periods and 13 variables, the distribution of LR might not be 

close to χ2

(118). Despite these doubts, LR is a useful quantitative indicator 

of the consistency of the model with the data. In sum, given good fit (high 

R2

's and high p values of Q's) and economically rationalizable impulse 

responses in terms of adjustment costs, we accept the estimated model and the 

capital and technology estimates derived from it as an acceptable economicstatistical accounting, surely for the 1960-1990 SIC-NAICS data, more 

tentatively for the 1991-2005 SIC-NAICS data. 

[Put figure 2 approximately here] 
16

Figure 2 illustrates some adjustment-cost features of the estimated model 

in terms of responses to unit impulses in output-demand and technology 

disturbances. Upper graphs 2a depict responses to a unit one-period shock to 

output demand (ζdt) in period 1, starting from an initial long-run equilibrium 

at the origin. High adjustment costs arising from ρ

ˆ = 267 imply a steep 

marginal-cost-of-production curve, so that 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 sufficiently slowly, firms have an incentive 

to increase their production capacities, by increasing their investment and 

research rates and substituting capital and technology for labor and materials. 

Eventually, all variables return to the origin. Lower graphs 2b depict 

responses to a unit one-period shock to technology (ζτt) in period 1, again 

starting from an initial long-run equilibrium at the origin. 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. 

[Put table 3 approximately here] 

Table 3 reports variance decompositions (Sims, 1986) of the estimated 

model. Each number in the table, in percentage form, indicates the fraction of 

the variance of one of the 8 endogenous variables (rows 2-9 indicated by 

s10,pq,j, ..., s10,τ,j) or the sum of the variances of the 8 endogenous variables 

(row 10 indicated by s 10,j) accounted for by the variance of a structural 

disturbance of an endogenous or exogenous variable, which is not set to near 

zero as an identifying restriction but is estimated (columns 2-11 indicated by 

2 σq , ..., 2 σd ). The particular values of the variance decompositions in table 3 

depend on the particular values of the set and estimated structural 

parameters, which, for the given model structure, are nonuniquely determined 

by the data, but depend somewhat arbitrarily on identifying restrictions. 

Nevertheless, the variance decompositions in table 3 are broadly similar to 

alternative variance decompositions resulting from alternative identifying 

restrictions and resulting parameter estimates. 
17

Table 3 is summarized as follows. If, somewhat arbitrarily, a number in 

the table less than 5, between 5 and 10, and more than 10 is considered 

insignificant, moderately significant, and strongly significant, then, the 

numbers in columns 6 and 9, reflecting ζτt and ζplt, are insignificant; the 

numbers in columns 2 and 8, reflecting ζqt and ζprt, are at most moderately 

significant; and, one or more numbers in each of columns 3, 4, 5, 7, 10, and 

11, reflecting ζit, ζrt, ζkt, ζpit, ζpmt, and ζdt, are strongly significant. There 

are notable symmetries. For example, investment and research disturbances have 

symmetrical effects (columns 3-4) and labor and materials respond very 

similarly in most cases (rows 4-5). Columns 3-6 indicate that investment and 

capital disturbances account for somewhat more of variations of individual 

endogenous variables (rows 2-9) and account for 7.47 times more of overall 

variations of endogenous variables (row 10) than do research and technology 

disturbances and, accordingly, counter the "real business cycle" premise that 

technology disturbances are the primary source of variations in variables. As 

an accounting of output growth, row 2 indicates that 94.5% of the variations of 

output are accounted for in decreasing order by variations in disturbances of 

investment, price of investment, research, output demand, price of materials, 

price of research, and output. Current and previous tables 3 are consistent. 

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

The model was estimated as described in section 3, using standardized 

levels of the data described in section 3.2. As in Chen and Zadrozny (2005), 

the MDKF was applied to the estimated model and the 1949-2005 data, which 

produced 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 1949-2005. Then, elements 7 and 8 of z t|t ˆ were picked as 

the model-based estimates of capital and technology, k t|t ˆ and t|t ˆτ , and the 

square roots of diagonal elements 7 and 8 of the error covariance matrices 

were picked as their estimated standard errors. Figure 3 depicts the modelbased and standard estimates of capital and technology for U.S. total 

manufacturing industries for 1949-2005. Solid lines depict model-based 

estimates and their 2-standard-error confidence bounds. Standard errors of 

technology estimates for 1949-1952 are large due to initialization effects in 

the MDKF. For capital, dashed lines depict weighted sums of BLS stock estimates 

of equipment, structures, inventories, and land, based on nonstochastic PIEs 

(equations (2.8)-(2.9) without disturbances are nonstochastic PIEs of capital 
18

and technology). BLS also produces service-flow estimates of equipment, 

structures, inventories and land, but weighted sums of these estimates are very 

similar and are, thus, not depicted or considered further. For technology, 

dashed lines depict BLS estimates of total factor productivity (TFP) based on 

Solow residuals. The BLS capital stock and TFP estimates are graphed in figure 

3 as examples of standard capital and technology estimates. 

Because MLE is tractable only if the data are scaled similarly, the data 

were standardized prior to estimation, by subtracting sample means and dividing 

by sample standard deviations. Being based on standardized data, the modelbased capital and technology estimates and standard errors are implicitly, but 

not exactly, in standardized form. Standard (BLS) capital (stock) and 

technology (TFP) estimates are in different and essentially arbitrary units. To 

make the capital and technology estimates and their standard errors comparable, 

before graphing them, we standardized them. Also, to make the estimates and 

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 in fundamentally arbitrary units, vertical differences between them should 

not be interpreted in percentage terms. 

[Put figure 3 approximately here] 

Graphs of the model-based and standard capital and technology for 1949-

2005 in figure 3 are now summarized. (1) For 1949-2000, trends of model-based 

and standard capital and technology are broadly similar. (2) For 2000-2005, 

model-based and standard estimates diverge significantly: model-based capital 

continues its previous growth; standard capital levels off; model-based 

technology declines and levels off; and, standard technology continues its 

previous growth. (3) For 1949-2005, model-based capital is noisy and 

uncertain and model-based technology is smooth and certain. To the extent 

that current results duplicate previous ones (Chen and Zadrozny, 2005), they 

support the model-based method as a general method for estimating capital and 

technology. The results have the following implications and interpretations. 

The similar trends of the model-based and standard estimates for 1949-

2000 make them mutually reinforcing. Being produced by government agencies 

and commonly used, the standard estimates are usually considered the "truer" 

ones. The intention here is not to challenge this view but to consider 

alternative estimates of capital and technology based on an estimated dynamic 

structural economic model which has features that are now standard in dynamic 
19

economic modeling: the variables of primary interest are endogenous in the 

model, in this case, capital and technology; agents in the model solve an 

explicitly considered dynamic optimization problem; the resulting dynamics of 

the endogenous variables arise naturally from elementary structural 

components, in this case, adjustment costs from the CES-CET production 

function; and, 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 model-based capital has more, larger, and noisier short-run variations 

than model-based technology. Consequently, model-based capital appears more 

uncertain than model-based technology, a conclusion which is supported by the 

2-standard-error confidence bounds in figure 3 produced by the Kalman filter. 

Standard errors of model-based capital and technology are, respectively, about 

1.01 and .036, which means that capital confidence bounds are about 28 times 

wider than technology confidence bounds. Being in unit-free standardized 

form, model-based estimates and confidence bounds are comparable. In essence, 

the residual role of technology in standard estimates switches to capital in 

model-based estimates, which is more realistic because capital is the 

residual-income earner and knowledge, the basis of technology, is presumably 

mostly invariant to higher-frequency economic variations. 

An important question is whether investment and capital or research and 

technology better account for variations in endogenous variables, in 

particular, output (Gordon, 2000; Oliner and Sichel, 2000; Stiroh, 2001). 

Columns 3-6 of table 3 indicate that the estimated model's investment and 

capital disturbances account for somewhat more of variations of individual 

endogenous variables (rows 2-9) and account for 7.47 times more of overall 

variations of endogenous variables (row 10) than do research and technology 

disturbances. 

The time-series properties of standard capital estimates depend entirely 

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

PIE. For given time-series properties of investment, standard capital estimates 

are smoother and more trendlike when capital depreciates more slowly. Time-
20

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 (2.8) are 

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

where xˆ t|s denotes estimated or expected xs, conditional on data through period 

t. Thus, the time-series properties of model-based capital and technology 

estimates also partly depend on the time-series properties of estimated 

disturbances, t|kt ˆζ . 

To consider how much noise the relatively large estimated capital 

disturbance variance of .99 passes to the model-based capital estimates through 

equation (3.6), as in Chen and Zadrozny (2005), we recomputed the capital and 

technology estimates with the capital disturbance variance set to 1.0×10-6, the 

value of the estimated technology disturbance variance, and left the other 

parameters at their estimated values. The resulting graphs (not shown here; 

cf., Chen and Zadrozny, 2005, figure 4) of model-based capital and technology 

are very similar to those in figure 3, except that both capital and technology 

are as smooth as technology in figure 3. Thus, the large capital disturbance 

variance in (2.8) appears to make model-based capital estimates noisy. 

Estimated annual capital and technology depreciation rates of 1 - φk1 = 

.39 and 1 - φτ1 = .96 are very high (cf., Jorgenson and Stephenson, 1967). To 

check whether system-wide MLE somehow caused high estimated depreciation rates, 

we reestimated capital and technology equations (2.8)-(2.9) separately by 

nonlinear least squares, using the model-based and BLS capital-stock and 

technology estimates as data, and obtained very similar estimated depreciation 

rates (not shown here; cf., Chen and Zadrozny, 2005, tables 2 and 4). For given 

depreciation, investment, and research rates, the model-based capital and 

technology estimates and their estimated disturbances, t|kt ˆζ and t|t ˆτ ζ , are 

smoother and more trendlike when system characteristic roots are near one. A 

system characteristic root, λ, is a solution of the characteristic equation, 

|I13λ2

 - B1λ - B2| = 0, of equation (3.2), where |•| denotes a determinant and I13

denotes the 13×13 identity matrix. The system has 26 roots: 2 exogenous roots 

arising from demand-state process (2.3), 8 exogenous roots arising from inputprice processes (2.7), 6 exogenous roots arising from residual autocorrelation 
21

coefficients θ, and 10 endogenous roots depending on the solution of the 

dynamic optimization problem. An accounting based on the parameter estimates 

in tables 1 and 2 indicates 9 near-unit roots (within .02 of 1): 8 exogenous 

near-unit roots and 1 endogenous near-unit root. The estimate ˆρ = 275 in table 

2 implies that output transformation in production is near zero (CET = .004), 

so that adjustment costs are very high and an endogenous root is near one, 

which is confirmed by the root accounting above. Model-based capital and 

technology estimates should be smoother and more trendlike to the extent that 

estimated ρ is high, estimated adjustment costs are high, and an endogenous 

root is near one. However, an unexplained paradox remains why technology 

estimates are very smooth or unit-root-like while their primary determinant, 

technology equation (2.9), has a very un-unit-root-like high depreciation rate. 

4. Conclusion. 

The paper has described and applied an economic method for estimating 

unobserved stocks of production capital and technology or total factor 

productivity of U.S. total manufacturing industries for 1949-2005. The method 

was applied to the merged 1949-2001 SIC and 1987-2005 NAICS data. The method 

involves using the data to estimate a dynamic structural economic model and, 

then, using the data, the estimated model, and the Kalman filter to compute 

filtered estimates of capital and technology. The estimated model gives the 

data and the capital and technology estimates an economic-statistical 

rationale missing from standard estimates based entirely on Solow-residual 

and perpetual-inventory accounting. 

Despite an equivocal result of a likelihood-ratio test of overidentifying 

restrictions, we accept the estimated model as an economic-statistical 

accounting, surely for 1960-1990 SIC-NAICS data, more tentatively for 1991-2005 

SIC-NAICS data. The variance decompositions of the estimated model in table 3 

indicate that investment and capital disturbances account for somewhat more of 

individual variations of endogenous variables (rows 2-9) and account for 7.47 

times more of overall variations of endogenous variables (row 10) than do 

research and technology disturbances. The model-based capital and technology 

estimates in figure 3 indicate the following. For 1949-2000, trends of modelbased and standard estimates of capital and technology are broadly similar 

and, thus, reinforce each other. For 2000-2005, the estimates diverge 

significantly: model-based capital continues its previous growth; standard 

capital levels off; model-based technology declines and levels off; and, 
22

standard technology continues its previous growth. For 1949-2005, model-based 

capital is noisy and uncertain and model-based technology is smooth and 

certain. Estimated capital and technology depreciation rates are very high. 

Specifying capital and technology PIEs as more general rational distributed 

lags, instead of (2.8)-(2.9), could result in lower estimated capital and 

technology depreciation rates. 

If possible, we should distinguish between degrees of effective 

allocation and utilization rates of capital and technology, although, without 

further detailed modelling, capital and technology estimates must, as usual, 

be considered fully effective and fully utilized. Both model-based and 

standard estimation methods effectively treat investment and research as 

fully successful, regardless of any misallocations and market valuations, and 

do not adjust the estimates for utilization rates. Thus, an optimally 

allocated factory adds the same amount to a capital estimate as a 

misallocated one built using the same resources. However, adjusting the 

capital estimates for effectiveness and utilization rates (also, possibly the 

technology estimates) would require expanding the model significantly beyond 

its present form. 

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Statistics 49: 16-27. 
23

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Laub, A.J. (1979), "A Schur Method for Solving Algebraic Riccati Equations," 

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24

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

 and Inputs, 1949-2005 

The dates on the horizontal axes refer to 1949-2003. 

a: Price of Output

49 55 61 67 73 79 85 91 97 03

2.8

3.0

3.2

3.4

3.6

3.8

4.0

c: Price of Materials

49 55 61 67 73 79 85 91 97 03

0.6

0.7

0.8

0.9

1.0

1.1

1.2

e: Price of Labor

49 55 61 67 73 79 85 91 97 03

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

g: Price of Investment

49 55 61 67 73 79 85 91 97 03

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

i: Price of Research

49 55 61 67 73 79 85 91 97 03

0.16

0.32

0.48

0.64

0.80

0.96

1.12

b: Quantity of Output

49 55 61 67 73 79 85 91 97 03

0.16

0.32

0.48

0.64

0.80

0.96

1.12

d: Quantity of Materials

49 55 61 67 73 79 85 91 97 03

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

f: Quantity of Labor

49 55 61 67 73 79 85 91 97 03

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

h: Quantity of Investment

49 55 61 67 73 79 85 91 97 03

0.00

0.16

0.32

0.48

0.64

0.80

0.96

1.12

j: Quantity of Research

49 55 61 67 73 79 85 91 97 03

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25
25

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
26

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

Estimates for U.S. Total Manufacturing, 1949-2005 

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 estimates and technology estimates as total 

factor productivity, produced by BLS. The dates on the horizontal axes refer 

to 1949-2005.

 Model-Based and Standard Estimates of Capital Stock

49 53 57 61 65 69 73 77 81 85 89 93 97 01 05

0

1

2

3

4

5

6

Model-Based and Standard Estimates of Technology

49 53 57 61 65 69 73 77 81 85 89 93 97 01 05

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5
27

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

Variable Parameter Estimates Fit Statistics 

1,

ˆφ⋅ 2,ˆφ⋅ | λ | 2 R Q 

Pi 1.37 

(10.5) 

-.356 

(2.61) 

1.02 .988 6.76 

(.563) 

pr 

1.90 

(28.2) 

-.900 

(13.1) 

.999 .999 17.6 

(.024) 

Pl 1.88 

(25.6) 

-.874 

(11.7) 

1.03 .999 13.3 

(.101) 

pm 

1.16 

(8.34) 

-.319 

(2.32) 

.712 .812 4.40 

(.820) 

Columns 2-6 show estimates of 1, ˆφ⋅ and 2, ˆφ⋅ , with absolute t statistics in 

parentheses, implied maximum absolute characteristic roots (solutions of λ2

 - 

1, ˆφ⋅ λ - 2, ˆφ⋅ = 0), unadjusted R2

, and Ljung-Box Q statistics for testing absence 

of residual autocorrelations at lags 1 to 10, with marginal significance 

levels or p values in parentheses. 
28

Table 2: Maximum Likelihood Estimates of Non-Input-Price Structural 

Parameters 

Production Function Parameters 

β

ˆ = -5.48 (CES = -.182), ρ

ˆ = 275 (CET = .004) 

Output-Demand Curve Parameters 

ˆ

η = .932, 1d ˆφ = 1.18, 2d ˆφ = -.367 

Capital and Technology Equation Coefficients 

1k ˆφ = .610, 0i ˆφ = .789, 1 ˆφτ = .043, 0r ˆφ = .304 

Residual Autocorrelation Coefficients 

pq

ˆθ = .999, qˆθ = .675, θl

ˆ = .999, m

ˆθ = .999, iˆθ = .848, rˆθ = .942 

Structural Disturbance Standard Deviations 

q σˆ = .144, i σˆ = .246, r σˆ = .106, k σˆ = .995, στ ˆ = .001, d σˆ = .207 

Reduced-Form Equation Fit Statistics 

 2

pq R = .932, 2q R = .942, 2 Rl = .651, 2

i R = .938, 2 Rr = .990 

 Qpq = 2.52, Qq = 2.55, Ql = 13.1, Qi = 13.6, Qr = 12.9 

 (.989) (.990) (.218) (.194) (.230) 

The sample span is 1949-2005 (57 years). R2

 and Q statistics and p values in 

parentheses are as in table 1. 
29

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 6.6 17.7 16.3 3.4 .00 17.4 7.6 1.6 8.0 21.4

s10,q,j⋅ 7.2 19.2 17.7 3.7 .00 18.8 8.3 1.8 8.6 14.7

s10,l,j 2.2 3.7 3.3 80.3 .00 1.5 .82 .01 5.7 2.4 

s10,m,j 2.0 3.2 2.9 70.7 .00 1.3 .73 .88 16.1 2.1 

s10,i,j .00 54.3 3.5 .54 .00 16.4 6.8 1.7 5.8 11.0

s10,r,j .00 1.6 41.9 3.5 .00 19.5 9.2 1.9 8.2 14.2

s10,k,j .00 23.5 1.5 57.8 .00 6.8 2.5 .63 2.6 4.8 

s10,τ,j .00 1.6 41.9 3.4 .00 19.5 9.2 1.9 8.2 14.2

s 10,j 1.4 19.8 8.3 42.2 .00 9.7 4.1 .99 5.6 7.8 

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

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

structural-disturbance variances. For example, s10,pq,2 = 6.6 is the percentage 

of the variance of pqt accounted for by the variance ( 2 σq ) of the structural 

disturbance of output (ζqt). Row 10 gives the percentage decomposition of the 

sum of the variances of the 8 endogenous variables. Each row sums to 100.