|mIIIII/EMPIRICAL
Empirical Economics (1995)20: 299-302
|~IIIII/ECONOltlKS
On the Econometric Estimation of a Constant Rate of
Depreciation
INGMAR R. PRUCHA
Department of Economics, University of Maryland, College Park MD 20742, USA
JEL Classification System-Numbers: C13, E22
1
Introduction ~
Presently most estimates of the stocks of physical or R&D capital that are
derived via the perpetual inventory method are n o t based on "econometric"
estimates of the depreciation rates, an the estimates of those rates have not
been subjected to econometric testing. However, in the last decade several authors have contributed to a literature that attempts to econometrically estimate the depreciation rates of the stocks of physical and R&D capital goods;
see, e.g., Epstein and Denny (1980), Kollintzas and Choi (1985), Bischoff and
Kokkelenberg (1987), Prucha and Nadiri (1993) and Nadiri and Prucha (1993).2
A difficulty in econometrically estimating the depreciation rates of the stocks
of physical and/or R&D capital is that by treating the depreciation rates as
unknown parameters or as a functions of unknown parameters, the stocks
become - since they are functions of the depreciation rates - unobservable.
Adopting the framework of the perpetual inventory method, we may view the
capital stocks as functions of current and past investments and the unknown
depreciation rates. In principle we can substitute those functions for the stocks
in, say, the production function or factor demand equations and estimate the
resulting relationships. However, there is a difficulty if we try to use standard
econometric packages for this task. Estimation routines as programmed in standard econometric packages typically do not permit the specification of functions where the argument list changes from sample period to sample period)
However, as discussed in more detail below, this is exactly the case if stocks
are generated by the perpetual inventory method from some initial stock. As a
consequence, the authors of the above cited papers had to write their own
I would like to thank Gary Anderson, Harry Kelejian, Sang-Loh Kim and Sally Srinivasan
for helpful discussions.
2 Estimates of depreciation rates obtained by classical econometric techniques may serve as a
check of estimates obtained by non-econometric techniques and/or as an alternative to the latter.
Of course, nonparametric estimates of depreciation rates may not be available for certain capital
goods. For example, most researchers simply assumed a depreciation rate of 10 or 15 percent for
R&D capital. Clearly, for those goods econometric estimates of the depreciation rate are of special
interest.
3 Cp., e.g., the FIML or LSQ routine in the TSP (Time Series Processor) estimation package.
0377- 7332/95/2/299-302 $2.50 9 1995 Physica-Verlag, Heidelberg
300
I.R. Prucha
estimation programs in, e.g., FORTRAN (or settle for an approach that does
not generate a fully consistent capital stock series). Clearly writing ones own
estimation program is at a minimum inconvenient and time consuming.
The purpose of this paper is to introduce a method that permits the use of
standard econometric packages to estimate the depreciation rates of the stocks
of physical and/or R&D capital for the case where those rates are constant. In
essence, as will be discussed in more detail below, this method involves - using
a dummy variable approach - a rewriting of the equations in such a fashion
that the argument list does not change from sample period to sample period.
As a consequence, those equations can then be estimated by standard econometric packages, and the researcher no longer has to write her/his own estimation routine in, e.g., FORTRAN.
2 Estimation Methodology
In the following we explain the estimation methodology by considering a simple example. Suppose a researcher would like to estimate the following production function
Y, = F(Lt, K,, O)
(1)
where Y, L t and K t denote, respectively, output, labor input and the capital
stock at the end of period t, and 0 represents a vector of unknown model
parameters. The capital stock accumulates according to the perpetual inventory equation:
K t = I t + (1 - 6 ) K t _ , ,
(2)
where It denotes gross investment in period t and 6 denotes the constant but
unknown depreciation rate. Repeated substitution yields
t-1
K t = ~ (1 -- 8)ilt_i + (1 -- 8)tKo = G,(It . . . . . I1, K o , t, 8) .
(3)
i=O
Suppose the depreciation rate 8 is known. We could then use (2) or (3) to
generate a series of capital stock data from past investments (if an estimate of
the initial stock K0 is available), and use those capital stock data to estimate
the production function parameters from (1). However, if the depreciation rate
8 is unknown, the capital stocks Kt are also unknown/unobserved and we cannot directly estimate the production function parameters from (1). Of course,
in this case we can still substitute the expression for Kt on the r.h.s, of (3) into
the production function (1) to obtain:
Yt = F(L,, Gt(I, . . . . . 11, K o , t, 8), O) = nt(Lt, It . . . . . I,, K o , t, 8, O) .
(4)
O n the Econometric Estimation of a Constant Rate of Depreciation
301
Note that L1 . . . . . L T and 11. . . . . I T are observed and thus, in principle, we
can estimate the production function parameters 0 and the depreciation rate 6
jointly by some standard estimation methods for nonlinear models. 4' s
As stated in the introduction, the difficulty in using standard econometric
packages to estimate an equation like (4) is that the argument list of the function Ht depends on t. However, estimation routines as programmed in standard econometric packages typically do not permit the specification of a function where the argument list changes with t. It is for this reason that, e.g.,
Epstein and Denny (1980), KoUintzas and Choi (1985), and Nadiri and Prucha
(1993) and Prucha and Nadiri (1993) had to write their own estimation program in FORTRAN.
We next show how K, can be artificially rewritten as a function of investments, the initial stock and the depreciation rate such that the argument list
does not depend on t. Takingj = t - i we can rewrite (3) as
g t= ~
j=l
(1 - 6)'-Jlj + (1 - 6 ) t g o .
(5)
Now suppose we define T new variables I/ . . . . . I~ as (t = 1. . . . . T, and j = 1,
.... T)
(6a)
I] = IjD] ,
where
. {~
D~ =
fort<j
for t > j
(6b)
We then see from the representation (5) that we can write
T
g r= ~
(1 - 6)*-'I~ + (1 - b ) t g o = G ( g . . . . . I / , t, g o, 6) .
(7)
i=1
Substitution of (7) into (1) yields
Yt = F ( L , , G(It T . . . . . I / , K o, t, 6), O) = H ( L , , I,T . . . . . I~, K o , t, 6, O) .
(8)
Note that the length of the argument list of G(') and H(.) does not vary with t.
Thus we can now estimate the parametes 0 and 6 using standard econometric
packages. Clearly, the basic idea of rewriting equation (3) as (7) is not specific
to our particular example and can be applied more generally.
Nadiri and Prucha (1993) estimate the depreciation rates of physical and
R&D capital for the U.S. manufacturing sector from a factor demand model
4
For a survey of estimation methods for dynamic nonlinear models see, e.g., Prtscher and
Prucha (1991a, b).
5
If an estimate of the initial stock K o is not available we m a y treat the initial stock as a further
u n k n o w n parameter. This treatment is justified since although the initial stock depends on earlier
investments and the depreciation rate 6, this dependence imposes no restrictions on the value of
the initial stock unless those earlier investments are observed.
302
I.R. Prucha
using their own F O R T R A N estimation program. T o test the feasibility of the
above a p p r o a c h we reestimated the model based on the above a p p r o a c h using
TSP. We successfully duplicated the original results.
3
Conclusion
In this note we showed how, using a d u m m y variables approach, it is possible
to estimate the depreciation rate of capital g o o d s jointly with other model parameters using standard econometric packages. While the a p p r o a c h suggested
in this paper is - once explained - simple, it seems that researchers have so far
not been aware of this approach. As a suggestion for future research, it seems
of interest to extend the a p p r o a c h to situations where the depreciation rates
are allowed to be variable and are modeled as, say, functions of prices and
output; cp., e.g., Epstein and D e n n y (1980), Kollintzas and Choi (1985), P r u c h a
and Nadiri (1993).
References
Bischoff CW, Kokkelenberg EC (1987) Capacity utilization and depreciation-in-use. Applied Economics 19: 995-1007
Epstein LG, Denny M (1980) Endogenous capital utilization in a short-run production model:
Theory and empirical application. Journal of Econometrics 12:189-207
Koltintzas T, Choi J-B (1985) A linear rational expectations equilibrium model of aggregate investment with endogenous capital utilization and maintenance. Mimeo
Kokkelenberg EC (1984) The specification and estimation of interrelated factor demands under
uncertainty. Journal of Economic Dynamics and Control 7:181-207
Nadiri MI, Prucha IR (1993) Estimation of the depreciation rate of physical and R&D capital in
the US total manufacturing sector. Department of Economics, University of Maryland, forthcoming in Economic Inquiry
P6tscher BM, Prucha IR (1991a) Basic structure of the asymptotic theory in dynamic nonlinear
econometric models, Part I: Consistency and Approximation Concepts. Econometric Reviews
10:125-216
Prtscher BM, Prucha IR (1991b) Basic structure of the asymptotic theory in dynamic nonlinear
econometric models, Part II: Asymptotic Normality. Econometric Reviews 10:253-325
Prucha IR, Nadiri IM (1993) Endogenous capital utilization and productivity measurement
in dynamic factor demand models: Theory and application to the US electrical machinery
industry, national bureau of economic research, Working Paper 3680, forthcoming in Journal of
Econometrics
First version received:April 1994
Final version received: December 1994