Computers and productivity: Are aggregation effects important?

McGuckin, Robert H. ; Stiroh, Kevin J.

KEVIN J. STIROH (*)

This article examines the empirical implications of aggregation bias when measuring the productive impact of computers. To isolate "aggregation in variables" and "aggregation in relations" problems, we compare production function estimates across specifications, econometric estimators, and data levels. The results show both sources of bias are important, especially when moving from sectors to the economy level, and when the elasticity of all types of non-computer capital are restricted to be equal. The elasticity of computers is surprisingly stable between industry and sector regressions and does not appear biased by incorporating a restrictive measure of non-computer capital. The data consistently show that computers have a large impact on output. (JEL 03)

I. INTRODUCTION

Economic research relies heavily on aggregated data, and there are good reasons for this. Aggregation combines details to give a clearer picture of performance. It also simplifies analysis because it is much easier to work with a small set of aggregate variables than to try to model the myriad of details that lie beneath. A third reason is that aggregation enables survey organizations to collect data in broad categories, which limits processing and collection costs. Finally, aggregation protects confidential microdata, although this is becoming less important as new institutions are developed to maintain secrecy while still allowing access by researchers.

Aggregation, however, also has important problems and the possibility of systematic bias from the use of aggregate data has long been recognized. In an early paper, Theil (1954) formally examined the possibilities for biased estimates of economic relationships when aggregates were used in place of information on individuals or firms. Similarly, the long debate over the "existence" of an aggregate production function or an aggregate measure of capital is essentially about the difficulties when aggregate data and relationships are used to represent the underlying microeconomic production relationships. (1) Although much has been learned about how aggregation should be done and what conditions are required for meaningful aggregates, for example, Diewert (1976), aggregation remains an important issue.

The recent availability of microdata, for example, has led to a host of studies documenting widespread differences in the characteristics of economic units within seemingly homogeneous groups or industries, for example, Baily et al. (1992), Jensen and McGuckin (1996), and Haltiwanger (1997). Arguably, the common practice of using a "representative firm" model to justify use of aggregates is not appropriate in these situations and recent empirical work supports this in a wide variety of situations, for example, productivity growth, investment dynamics, and employment movements as surveyed in Haltiwanger (1997). As a concrete example of how aggregation may be misleading, McGuckin and Nguyen (1999) find an impact of ownership change on productivity in plant-level data, but the relationship becomes obscured when firm-level observations are used.

This article examines the empirical implications of using different types of aggregate data in a specific context-measuring the productive impact of computers. This application is of particular interest in light of the widespread discussion and controversy concerning the so-called computer productivity paradox. This puzzle can be traced to the observation by Robert Solow (1987) that, although computers were becoming ubiquitous, aggregate productivity remained sluggish. One potential explanation, presented by Oliner and Sichel (1994) and Sichel (1997), emphasized that computers represent a relatively small portion of economy-wide capital. Because computer capital was small, it is not surprising that it did not have a large impact on aggregate productivity or growth in earlier years. (2)

Though certainly true in an aggregate growth accounting framework, this doesn't explain the disparity of findings at lower levels of aggregation that analyzed data for same period of the 1980s and early 1990s. Berndt and Morrison (1995), for example, found that computers were having little impact on productivity in the 1980s, while Morrison (1997) reports evidence of overinvestment in high-tech capital. In contrast, Brynjolfsson and Hitt (1995), Lehr and Lichtenberg (1999), Lichtenberg (1995), Siegel (1997), and Steindel (1992) found that computers had a significant impact on productivity and output.

One factor that may be contributing to this divergence is the different degree of aggregation across studies. The purpose of this article, therefore, is to use a common econometric approach and data set to isolate the impact of aggregation effects. By comparing econometric results using the same specification at three distinct levels of aggregation--the private business economy, ten major sector groups, and 55 detailed industries-we can quantify the importance of bias from "aggregation of relations" problems.(3) Similarly, by comparing estimates from different decompositions of capital, we can quantify the importance of bias from "aggregation of variables" problems.

We begin by documenting widespread differences in computer intensity across business sectors and component industries. We then use a continuous measure of computer intensity--measured either as the stock of computer capital or the share of equipment capital in the form of computers--in various production function specifications. Comparison of the estimated elasticity from the different regressions provides insights about the relationship between computers, productivity, and output and the importance of aggregation effects.

Our results suggest that both sources of bias are important in general, although the estimated computer elasticity is surprisingly robust across different levels of data and alternative specifications. The level of aggregation has a direct impact on the estimated elasticity of other types of capital, most notably structures, while the decomposition of noncomputer capital matters because the marginal products of different types of capital are clearly not equal.

In terms of computers, the estimated elasticity is quite large and stable, typically in the range of 0.15, across both the industry and the sector regressions and does not appear to be biased by the use of a restrictive measure of noncomputer capital. Because computers experience rapid depreciation and large capital losses due to obsolescence, they must earn a large gross rate of return to cover these costs; this explains the relatively large output elasticity for computers. We conclude that despite the presence of aggregation bias, previous estimates of a large elasticity of computers appear reasonable and are entirely consistent with economic theory.

II. EMPIRICAL APPROACH

Research on computers and productivity has generally followed two empirical traditions. Jorgenson and Stiroh (1995; 1999; 2000b), Sichel (1999), Haimowitz (1998), Stiroh (1998), and Oliner and Sichel (1994, 2000) use growth accounting techniques to compare the growth rate of output to the share-weighted growth rates of inputs and estimate the contribution of each input to economic growth. In contrast, Gera et al. (1999), Lehr and Lichtenberg (1999), Morrison (1997), Siegel (1997), Berndt and Morrison (1995), Brynjolffson and Hitt (1995), Lichtenberg (1995), and Steindel (1992) estimate production or cost functions for firms or industries that explicitly include some measure of computer capital or computer-related labor. (4)

The empirical work typically begins with a production function, such as

(1) [Y.sub.i,t] = [A.sub.t] * f([K.sup.computer.sub.i,t], [K.sup.other.sub.i,t] [L.sub.i,t], i),

where [Y.sub.i,t] is output, [K.sup.computer.sub.i,t] is capital in the form of computers, [K.sup.other.sub.i,t] is other capital, and [L.sub.i,t], is labor for firm, industry, or sector i at time t. [A.sub.t] represents total factor productivity or disembodied technical change that depends solely on time.

The econometric studies take some version of equation (1), assume a functional form, transform the relationship into a regression, and estimate that regression with the data at hand. Our empirical work follows this tradition and estimates various specifications of equation (1) at different levels of aggregation using several different econometric estimators to quantify the importance of two different types of aggregation errors. (5)

Aggregation Issues

Broadly speaking, aggregation bias is a special case of specification error, but ascertaining the bias in any particular application is not straightforward. Maddala (1977) divides the topic of aggregation bias into two broad sets of questions--problems in the "aggregation of variables" and problems in the "aggregation of relations." We examine both issues and, although we do not formally derive or model these aggregation biases here, it is useful to put our comparisons in this context. See Maddala (1977) for details.

Aggregation of variables includes the whole area of index number construction in which an index variable is created to represent the movement of a set of prices or quantities over time. This area has a rich history, and we focus on one example relating to incorrectly specifying the inputs to a production function.

To fix ideas, consider the example in Lichtenberg (1990) where the true structural relationship is y = [[beta].sub.1][X.sub.1] + [[beta].sub.2][X.sub.2] + u, but the econometrician incorrectly estimates y = [beta]([X.sub.1] + [X.sub.2]) + [epsilon]. Lichtenberg shows that the probability limit of the estimate of [beta] is a weighted average of [[beta].sub.1] and [[beta].sub.2], but the weights need not be between zero and one, so perverse effects are possible. In his application, Lichtenberg (1990) examines the consequences of using an aggregate measure of research and development (R&D) in place of a disaggregated specification that explicitly includes federal government R&D and private R&D. Because the productivity impact of federal R&D was less than private R&D, the estimated impact of aggregate R&D understated the true impact. (6) Aizcorbe (1990) makes an important contribution to this literature by developing tests of the validity of particular aggregates in the context of generalized production functions that are discussed below.

Aggregation of relations includes many issues that deal with the interaction between micro- and macrorelationships. The seminal work of Theil (1954) studied one specific case dealing with the conditions needed for the preservation of the parameters of microrelations when estimation is carried out using macrovariables. (7) We examine a very specific form of this question and compare differences in the coefficients of the same production function that are estimated at various levels of aggregation. Morrison-Paul and Siegel (1999) undertake a similar exercise in the context of measuring economies of scale in manufacturing and find that the aggregation bias from moving between four-digit, two-digit, and total manufacturing data is not substantive.

In the case of computers and productivity, we speculate that both types of aggregation issues are important. Regarding aggregation of variables, the marginal productivity of different forms of capital can be quite varied, so aggregating heterogeneous types of capital may be an important source of bias. For example, econometric work that simply includes an aggregate measure of "noncomputer capital" as in equation (1) may generate the exact bias described by Lichtenberg (1990). Regarding aggregation of relations, both production structure and computer intensity vary widely across industries and combining these disparate industries into aggregates may lead to biased coefficient estimates. Thus the level of aggregation may have a direct impact on the estimated parameters of a production function and create a misleading picture of the productive role of computers.

Econometric Framework

The remainder of this section outlines three related empirical specifications that can be used to examine how production function coefficients vary when they are estimated with different data. In particular, we focus on how the estimated elasticity of computer capital changes across different decompositions of capital (aggregation of variables) and across different levels of aggregation (aggregation of relations).

Simple Production Function. In the empirical literature on the role of computers, it is standard practice to decompose capital into two parts-computer capital and noncomputer capital as in equation (1)-and econometrically estimate a production function. A common Cobb-Douglas specification, for example, yields the following model:

(2) Y = [e.sup.A(t)] * [e.sup.[phi](i)] * [L.sup.[[beta].sub.0]] * [K.sup.[[beta].sub.1].sub.c] * [K.sup.[[beta].sub.2].sub.n] * [e.sup.[epsilon]],

where L is labor, [K.sub.c] is computer capital, [K.sub.n] is noncomputer capital, A(t) and [phi](i) are general time and industry variables that simply shift the production function but do not interact with the inputs, and [epsilon] is a random error term. (8)

To estimate equation (2), take logs and include time and industry effects as dummy variables, T and I, to allow flexibility in A(t) and [phi](i), respectively. This implies the following simple production function regression:

(3) ln Y = [alpha] + [[beta].sub.0] ln L + [[beta].sub.1] ln [K.sub.c] +[[beta].sub.2] ln [K.sub.n] + [[theta].sub.t]T + [[theta].sub.i]I + [epsilon].

This regression provides the first way to estimate the impact of computers on output and productivity. Specifications of this type are common in the literature, for example, Brynjolfsson and Hitt (1995), Lichtenberg (1995), and Steindel (1992). (9)

We emphasize that although equation (3) isolates computers from other forms of capital, there are still problems relating to aggregation of variables. [K.sub.n] contains many heterogeneous assets across many vintages, for example, cars, structures, and machine tools, which likely do not have the same elasticity. Moreover, [K.sub.c] is an aggregate that includes mainframes, personal computers, displays, printers, storage devices, and other peripheral equipment, and these components may have different production characteristics that may make aggregation inappropriate. We do not focus on bias at this level to provide comparability with earlier studies and to maintain tractability of our results. (10)

Extended Production Function. The implicit assumption in the simple production function of equation (2) is that all forms of non-computer capital are perfect substitutes and have the same output elasticity. To examine the impact of this assumption, we consider a more general production function that explicitly includes four types of capital--computer capital, [K.sub.c], other high-tech equipment, [K.sub.h], other equipment, [K.sub.o], and structures, [K.sub.s]-- and allows the output elasticity to vary across each asset class. As shown in Lichtenberg (1990), incorrectly imposing a common elasticity can lead to biased estimates due to the aggregation of variables problem.

The extended production function, again in Cobb-Douglas form, is

(4) Y = [e.sup.A(t)] * [e.sup.[phi](i)] * [L.sup.[[beta].sub.0]] * [K.sup.[[beta].sub.1].sub.c]

* [K.sup.[[beta].sub.2].sub.h] * [K.sup.[[beta].sub.3].sub.o] * [K.sup.[[beta].sub.4].sub.s] * [e.sup.[epsilon]],

which implies the following extended production function regression:

(5) ln Y = [alpha] + [[beta].sub.0] ln L + [[beta].sub.1] ln [K.sub.c]

+ [[beta].sub.2] ln [K.sub.h] + [[beta].sub.3] ln [K.sub.o] + [[beta].sub.4] ln [K.sub.s]

+ [[theta].sub.t]T + [[theta].sub.i]I + [epsilon].

The regression in equation (5) provides a second way to evaluate the impact of computers. By allowing a more general specification that removes a particular form of aggregation error, these regressions may provide better estimates. Moreover, by comparing the estimated elasticities from equation (3) to the estimated elasticities from equation (5), we can directly assess the practical importance of the aggregation of variables bias.

Alternative Production Function. The two approaches described above estimate the impact of computers by explicitly decomposing capital into different types. As an alternative, we also examine a specification similar to Berndt and Morrison (1995) and Lehr and Lichtenberg (1999). This approach uses capital shares to identify differences in the productive impact of equipment capital in general and computer capital in particular.

Consider the slightly modified Cobb-Douglas production function

(6) Y = A * [L.sup.[[beta].sub.0]] [([K.sup.*]).sup.[[beta].sub.1]],

where [K.sup.*] is "effective" capital that is measured as

(7) [K.sup.*] = K * [([K.sub.e]/K).sup.[delta]] * [([K.sub.c]/[K.sub.e]).sup.[gamma]],

so that

(8) ln [K.sup.*] = ln K + [delta] * ln([K.sub.e]/K) + [gamma] * ln([K.sub.c]/[K.sub.e]),

where [K.sub.e] is total equipment capital and [K.sub.c] is computer capital.

Aizcorbe (1990) provides a formal justification and defense of this type of approach. She shows that, under reasonable conditions, a general production function Y = f([X.sub.1], [X.sub.2], ... [X.sub.K]) can be restated as Y = f(X, [M.sub.1], [M.sub.2], ... , [M.sub.K-1]) where X is some aggregate of the individual inputs and [M.sub.i] = [M.sub.i]([X.sub.i], [X.sub.K]) is a "mix function" that relates the two arguments. To test if the aggregate X is valid, one can test if [[partial]]Y/[[partial]][M.sub.i] = 0, [for all]i.

In this framework, [delta] and [gamma] measure the compositional effects associated with different types of capital. That is, [delta] > 0 implies that a larger proportion of capital in the form of equipment increases the amount of effective capital relative to the measured aggregate. Likewise, [gamma] > 0 implies that a larger proportion of equipment in the form of computers increases the effective amount of capital.

Note that this specification only captures composition effects because individual capital stock series are calculated with quality-adjusted price indexes for computers to account for improvement embodied in more recent vintages and all series are aggregated using a Divisia index. Thus vintage differences and traditional index number problems are eliminated.

Combining equations (6) and (8) yields the following alternative production function regression:

(9) In Y = [alpha] + [[beta].sub.0] ln L

+ [[beta].sub.1] ln K + [[beta].sub.1][delta]ln([K.sub.e]/K)

+ [[beta].sub.1][gamma]([K.sub.c]/[K.sub.e]) + [[theta].sub.t]T + [[theta].sub.i]I + [epsilon].

Again, the interpretation of [delta] and [gamma] is clear. If equipment and computer capital do not have any differential impact and the composition of the capital stock does not matter, then [delta] = [gamma] = 0. If this is true, then [[beta].sub.1][delta] = [[beta].sub.1][gamma] = 0 in equation (9). Conversely, if composition effects do matter, then the estimated coefficients on the shares will be statistically significant different from zero (assuming the [[beta].sub.1] [not equal to] 0 as implied by standard production theory). Of course, the coefficients on the two shares need not be the same and none, either one, or both of the share effects could matter.

The regressions in equations (3), (5), and (9) provide the means for assessing the practical important of biases from aggregation of variables and aggregation of relations. By comparing estimates from the simple production function to the extended or alternative production functions, we can assess the bias created from a restrictive measure of capital (aggregation of variables). Likewise, by estimating each regression at differ levels of aggregation, for example, industry, sector, and private business economy, we can assess the bias from incorrectly imposing the same relationship (aggregation of relations).

III. DATA ISSUES

Data comes from the Bureau of Economic Analysis (BEA) and include gross product originating (GPO) by industry and capital stock by industry and asset.

BEA GPO

GPO represents each industry's contribution to gross domestic product as calculated by BEA. These data, also called value-added data, equal gross output less intermediate inputs and thus equal payments to labor and capital. The GPO data include current and chain-weighted constant dollar data for 62 detailed private industries. The current dollar GPO is from 1948-1996, and the constant dollar GPO is only from 1977-1996. Data on full-time equivalent employees is available for the same industries from 1947-1996. (11) Details are provided by Lum and Yuskavage (1997), Lum and Moyer (1998), and Yuskavage (1996).

BEA Tangible Wealth Survey

Investment and capital stock are estimated by the BEA (1998) as part of their tangible wealth study. These data include current dollar net capital stocks and corresponding chain-weighted quantity indexes for 62 private industries and 57 assets from 1947-1996. Details on the estimation and data sources can be found in Katz and Herman (1997) and these data correspond to those reported in the September 1997 Survey of Current Business.

Creating Consistent Data

The data used in this article represent consolidated data based on 1987 SIC codes. To focus on the private business economy, we excluded government enterprises, general government, real estate, and private households from our econometric analysis. Aggregation of output and capital stocks was done as a Divisia quantity index, which has the desired exact aggregation properties, whereas labor series are simple sums. This procedure resulted in 55 detailed industries that comprise ten major sectors with data on GPO, labor, and capital stock by asset.

To measure the composition of the capital stock, we created several aggregates from the detailed capital stock series. Computers include mainframes, personal computers, direct access storage devices, printers, terminals, tape drives, and other storage devices. Other high-tech equipment includes communications equipment, instruments, and photocopy equipment. Other equipment includes all other producers' durable equipment. Structures include all nonresidential structures. Thus our capital measure excludes residential structures, land, and inventories and includes only fixed, reproducible tangible assets owned by the business sector.

Descriptive Statistics

Table 1 shows the evolution of computer capital for major sectors and the detailed industries from 1970-1996. These data show the rapid accumulation of computers throughout the economy as documented in the aggregate work of Jorgenson and Stiroh (2000b) and Oliner and Sichel (2000). (12)

More important for our purposes, there is wide variation across major sectors and industries. The private business sector shows a nominal computer share in the capital stock of 1.8% in 1996, for example, and only 0.002% of farm capital is computers but more than 20% of business services capital is in the form of computers. Even within major sectors like services or manufacturing, there is substantial variation in computer shares of the total capital stock.

Table 1 also shows wide variation in the accumulation rates of computers across major sectors, ranging from 8.34% in mining to 28.84% in wholesale trade in the 1990s. These growth rates far exceed the growth in other forms of capital, typically by a factor of ten. Again, variation widens significantly at lower levels of aggregation and remains large within major sectors. Because capital stocks are calculated using the same methodology and the same underlying deflators, this reflects enormous differences in investment patterns across industries.

Table 2 presents the distribution of capital by type--computers, other high-tech, other equipment, and structures--within major sectors and detailed industries. As expected, there is wide variation in all forms of capital because different industries have fundamentally different production techniques. This suggests that the simple production function approach may be quite misleading because it does not account for the wide heterogeneity in capital.

Table 3 presents the distribution of capital in a different way by reporting the distribution of total computer and high-tech capital across major sectors. As reported in Triplett (1999) and Stiroh (1998), computers are highly concentrated in service-related sectors with wholesale trade, retail trade, finance insurance and real estate, and services owning over $120 billion of computer equipment, which accounts for over 78% of the U.S. business total. Manufacturing, on the other hand, holds only $26 billion, or 17% of the total.

IV. PRODUCTION FUNCTION ESTIMATES

Our empirical results focus on three regressions--equations (3), (5), and (9)--that are estimated at different levels of aggregation with different econometric methods. The structure of aggregation coincides with Tables 1 and 2 and includes three nested levels. "Aggregate" is the private business economy, (13) which consists of ten major "sectors," which in turn consist of 55 detailed "industries" at roughly the two-digit SIC level. All aggregation is done with a Divisia index so the 55 detailed industries sum to the ten major sectors and the ten major sectors sum to the aggregate in current dollars. (14)

We estimate the production functions in several different ways. We first perform ordinary least squares (OLS) on the aggregate, sector, and industry data. These regressions include year dummy variables for the sector and industry regressions and a linear time trend in the aggregate regression. We then estimate a traditional fixed effect (FE) specification that allows each industry or sector to have a unique intercept to account for unobserved heterogeneity. Because the aggregate data is a single series and cannot be estimated using the panel methods, only an OLS estimate is reported.

Although regressions of these types are quite common in the literature, there are important econometric concerns. There is an endogeneity problem because output and inputs, particularly variable inputs such as labor, are likely to be chosen simultaneously. There is also an omitted variable problem because we cannot observe all factors that determine output or productivity, for example, technology, R&D, efficiency, and input quality. Inclusion of FEs in a panel framework can control for unobservable factors that are constant over time, but to the extent that unobservable factors vary and are correlated with particular inputs, those coefficients will be biased. Finally, there are multicollinearity issues because all inputs are likely to be correlated, particularly at higher levels of aggregation with less cross-sectional variation. (15)

To control for both the unobserved heterogeneity and simultaneity problems, we also employ more sophisticated econometric tools developed by Arellano and Bover (1995) and Blundell and Bond (1998), and applied to production function estimates by Blundell and Bond (1999). Their system generalized method of moments (SYS-GMM) estimator utilizes a combination of regressions in levels and first-differences with lagged first-differences as instruments for the equations in levels and lagged levels as instruments for equations in first-differences. Simulation results in Blundell and Bond (1998) show this estimator offers efficiency gains relative to the basic first-differenced GMM estimator. (16)

A second data limitation forces us to use a measure of capital stock rather than the preferred flow of capital services. This difference has been recognized at least since Solow (1957) and has been an important part of the growth accounting literature. Jorgenson and Stiroh (2000b) provide details on the conceptual and empirical distinction.

A final issue is our use of GPO as the output concept. Ideally, we would prefer to use a measure of gross output, which includes the value of intermediate inputs, but we did not have the corresponding data for intermediate inputs. Such is data is available for manufacturing industries, for instance, National Bureau of Economic Research Grey-Bartelsman database, or for relatively high levels of aggregation, for example, Jorgenson and Stiroh (2000a), but they do not provide the comprehensive coverage of computer-intensive industries or the nested Levels of disaggregation required for this exercise. (17) With these caveats in mind, we proceed to the empirical results.

Simple Production Function

Table 4 reports estimates of the simple production function in equation (3) for different levels of aggregation and econometric methods. For the most part, the sector and industry results are reasonable with a large, positive, and significant coefficient on labor in the 0.5 range and capital coefficients that are typically statistically significant. (18) Consistent with Blundell and Bond (1999), the FE (within) estimator produces coefficients that appear to be biased downward, while the SYS-GMM estimators are more reasonable. In both the sector and industry SYS-GMM regression, the coefficient on computer capital is large and statistically significant.

These estimates of the computer elasticity are typically around 0.18 in the OLS and SYS-GMM models, somewhat larger than earlier estimates in Brynjolfsson and Hitt (1995) and Lichtenberg (1995). Those papers estimate a similar regression with firm-level data for earlier periods and report a statistically significant elasticity for computer capital in the range of 0.05-0.12. The larger impact in our data likely reflects the growing importance of computers in our sample relative to their earlier samples.

The large change in coefficients when sector or industry dummy variables are included in the FE estimates suggests that deviations in output over time from the mean for a particular sector or industry are not highly correlated with deviations in computer capital. Similarly, the elasticity of noncomputer capital falls and is actually significantly negative at the industry level. This is unexpected, although low and insignificant capital coefficients in FE regressions are common in empirical work and were an important motivation behind the more sophisticated GMM approaches. Griliches and Mairesse (1998), particularly page 178, discuss this phenomenon and suggest that the loss in variance of the right-hand-side variables is responsible as other errors like measurement and random noise dominate the remaining information.

In general, there appear to be small differences in the estimated coefficients when the industry- and sector-level regressions are compared. Consistent with Morrison-Paul and Siegel (1999), bias from aggregation of relations appears small. When the aggregate regression is compared, however, there are large changes in the estimated coefficients, suggesting the aggregation of relations problem becomes large at the aggregate level.

Extended Production Function

Table 5 reports estimates of the extended production function regressions in equation (5). The results, again except for the aggregate OLS regression, are generally well behaved with a coefficient on labor typically in the 0.5 range and mostly reasonable capital coefficients. Again, the SYS-GMM estimates appear the most reasonable, and the FE results appear biased downward.

In terms of comparison across levels of aggregation and biases from aggregation of relations, the computer coefficient is again quite stable across the industry and sector regressions using all estimators. The other capital coefficients show more variability. The estimated elasticity of structures, for example, increases from 0.048 in the sector regression to 0.194 in the industry SYS-GMM regression.

To assess the aggregation in variables bias, we compare the results in Table 5 to those in Table 4 and find evidence of an important problem. That is, the estimated elasticities on the different types of noncomputer capital are quite different and it appears inappropriate to impose a common elasticity on all types of noncomputer capital. (19) The point estimates of the computer elasticities, however, typically remain in the range of 0.15. This suggests that any bias introduced by incorrect restrictions on other forms of capital does not substantially change the estimated elasticity of computers.

Alternative Production Function

Table 6 presents estimates of equation (9), which includes aggregate capital and two share variables, as opposed to each type of capital separately. As discussed above, Aizcorbe (1990) shows this to be an equivalent representation of the production function if aggregation across variables is valid. Again, we report estimates from the OLS, FE, and SYS-GMM estimators.

The results are consistent with the earlier findings and mostly appear reasonable. Labor elasticities are in the 0.5 range, the capital elasticity is between 0.3-0.4 in the OLS and SYS-GMM estimates, and the FE estimates appear biased downward. These estimates are broadly consistent with expected income shares. As in the earlier specifications, the capital elasticity drops in the FE regression and the aggregate regression appears the least reasonable with a large, negative elasticity on capital and a labor elasticity that appears too large.

When comparing the sector and industry regressions, some estimated coefficients vary a great deal, but others do not. For example, labor elasticity in the FE regressian was estimated at 0.04 at the sector level and 0.46 at the industry level. In contrast, the computer share coefficient varies little across levels of aggregation but falls dramatically in the FE regression. This suggests that between-industry rather than within-industry variation is the primary source of output variation associated with computers.

These results imply that the composition of the capital stock matters with regard to computers, but not necessarily with regard to equipment in general. That is, a higher share of equipment in the form of computers is typically associated with higher output, while a larger share of capital in the form of equipment is not. More formally, in the sector and industry regressions, [delta] in equation (9) is typically small and insignificantly different from zero, while [gamma] is typically positive and often significant. The implied estimates and [delta] and [gamma] and the associated p-value are reported in Table 6. (20)

Lehr and Lichtenberg (1999) estimate a similar specification and find results that are broadly consistent. Their coefficient estimates are not directly comparable, however, because they do not include the equipment to capital ratio in their regression and estimate an approximation without taking logs of the computer share. Despite these differences, they also find that the share of computers is positively and significantly related to output across a panel of firms in the late 1980s and early 1990s.

It is important to recognize that our evidence that computers appear very productive does not necessarily imply that computers earn excess returns. In a neoclassical framework, for example, an asset's output elasticity equals its nominal income share. This income share is typically derived from a user-cost approach that includes tax factors, depreciation, capital gains/losses, and the acquisition price of the asset. In the case of computer equipment, rapid obsolescence and massive price declines yield a high user-cost, which makes a high marginal product necessary simply to make the computer a worthwhile investment.

Lichtenberg (1995) explicitly tests the hypothesis that computer equipment has excess returns by comparing the ratio of marginal products to user-costs for different types of capital and finds evidence of excess returns to computers. We do not perform this type of test, but one can get a rough sense by comparing the average income share reported in Jorgenson and Stiroh (2000b) and Oliner and Sichel (2000) to our estimated elasticity. Both studies use aggregate data for the United States and report relatively small income shares for computer hardware and software: about 2.5% in Jorgenson and Stiroh for 1990-96 and 3.4% in Oliner and Sichel for 1991-95. Both shares are considerably below our estimated coefficient, although the results are not directly comparable because we do not have detailed capital service data for all assets, we use a different vintage of capital stock data, and the output concepts are not identical.

Robustness Checks

One concern in estimating these types of regressions is measurement error. The recent divergence in productivity growth between manufacturing and nonmanufacturing industries, for example, has led some to believe that measurement problems cause output, and therefore productivity, to be understated in certain industries. Dean (1999, 24), for example, concludes, "there are important measurement problems in some service activities."

These difficulties may reflect inadequate data, conceptual problems in defining service sector output, or an inability to accurately decompose nominal changes into prices and quantities. In the specific context of measuring the impact of computers, Siegel (1997) finds that computers lead to both quality change and productivity growth, after accounting for potential measurement errors. Using a different framework, McGuckin and Stiroh (2001) conclude that measurement error associated with computer investment may be contributing to an underestimate of aggregate productivity growth.

To examine whether our results are robust to such potential problems, we split the industry data into manufacturing and nonmanufacturing industries. Table 7 reports OLS results for the simple production function and SYS-GMM estimators for all three specifications. In all cases, the coefficient related to computer capital is larger in the manufacturing regression, but typically less precisely estimated. This is similar to McGuckin and Stiroh (2001), but contrasts Brynjolfsson and Hitt (1995) who report a larger estimated coefficient on computers in the service sector compared to manufacturing. If one believes that manufacturing output is better measured than nonmanufacturing, these results suggest that the estimates for all industries and sectors may be understating the productive impact of computers. The pattern of coefficients, however, is similar to that found earlier, suggesting that the qualitative results are not being driven by service sector mismeasurement.

V. CONCLUSIONS

The purpose of this article is to examine the empirical importance of aggregation effects in the context of estimating production functions that include computer capital as a distinct asset. Drawing together the results from alternative specifications that were estimated at different levels of aggregation, several conclusions stand out.

It is clear that the economy-wide specification provides very unstable results and gives a misleading picture of computer productivity. In contrast, both the sector and the industry estimates show a very stable estimated elasticity for computers, typically around 0.15, across specifications in OLS and SYS-GMM regressions. Estimates are typically smaller in FE regressions, consistent with previous econometric work that documents a downward bias in this type of estimate. These findings also echo results of our earlier work in McGuckin and Stiroh (1998), which found that because computer use was concentrated in a small number of industries, the impact was obscured at higher levels of analysis.

The stability of the estimated impact of computers between the industry and sector regressions and across specifications was somewhat surprising but also reassuring. Because these estimates are robust to the specification and aggregation level, it suggests that computers are having a real impact on output. This does not mean, however, that aggregation of relations problems are not important because the estimated elasticities of the noncomputer aggregate, particularly structures, varied substantially between the industry and sector regressions. Thus, empirical work must still be careful when choosing the level of aggregation.

The bias from aggregation in variables was also generally found to be important. There is wide variation in the productivity of different types of capital, and it is inappropriate to include a single capital index in many cases. When included individually, structures showed a large and significant elasticity, and the impact from other forms of noncomputer equipment was harder to estimate. The estimated computer elasticity, however, did not change when more disaggregated capital variables were included, which suggests that the bias was not transmitted to all coefficients.

These results show a large and robust elasticity of computer capital across a wide range of specifications, estimation techniques, and aggregation levels. Despite the clear presence of other aggregation problems, our results suggest that computers are quite productive. This is entirely consistent with economic theory; computers must earn a high gross rate of return to offset the rapid depreciation and capital losses. These results support the growing body of work at both the micro and the aggregate level that shows computers are indeed an important source of growth and productivity.

ABBREVIATIONS

BEA: Bureau of Economic Analysis

FE: Fixed Effect

GPO: Gross Product Originating

OLS: Ordinary Least Squares

R&D: Research and Development

SYS-GMM: System Generalized Method of Moments

(*.) The authors thank Ana Aizcorbe, Charles Waite, participants of the Innovations in Economic Measurement session at the 1999 Joint Statistical Meetings, and two anonymous referees for helpful comments, as well as Michael Fort and Jennifer Poole for excellent research assistance. This article represents the views of the authors only and does not necessarily reflect those of the Federal Reserve Bank of New York, the Federal Reserve System, or their staffs.

McGuckin: Director of Economic Research, The Conference Board, New York, NY 10045. Phone 1-212-339-0303, E-mail robert_mcguckin@conferenceboard.org

Stiroh: Senior Economist, Banking Studies Function, Federal Reserve Bank of New York, New York, NY 10022. Phone 1-212-720-6633, Fax 1-212-720-8363, E-mail kevin.stiroh@ny.frb.org

(1.) Fisher (1992) provides details on both capital and production function aggregation, and Jorgenson (1990) discusses the stringent assumptions needed to move from a sectoral production function to an aggregate production function.

(2.) More recent estimates for the late 1990s in Jorgenson and Stiroh (2000b) and Oliner and Sichel (2000) show that information technology played an important role in the U.S. productivity revival after 1995.

(3.) Note, however, that data limitations prevent us from extending this comparison to include analysis of aggregation effects that move from the firm or plant level to the industry level. Microdata is available for manufacturing industries from the Longitudinal Research Database but is not available outside of manufacturing.

(4.) Brynjolfsson and Yang (1996) survey this field.

(5.) Some authors, for example, Lichtenberg (1995), also decompose labor into a information technology and noninformation technology portion. He finds evidence of excess return to labor associated with information technology.

(6.) The true coefficient of the aggregate variable is a weighted sum (equal to 1.0) of the true coefficients for the individual impacts with the bias depending on the ratio of the variances of each component of aggregate R&D and the correlation between the individual components. See Lichtenberg (1990) for details.

(7.) There is a technically similar class of problems that Madalla (1977) includes in aggregation of relations. These problems question whether aggregate or micro-based relationships give thc "best" prediction or forecast when both types of variables are available. We do not investigate this.

(8.) Time and industry subscripts have been dropped from the inputs and outputs for ease of exposition.

(9.) Under the assumption of constant returns to scale, it is straightforward to transform this output regression into a labor productivity regression, and this has been done in several of the studies cited. We estimated these types of labor productivity regressions and found similar results, so we focus on the output-level regressions.

(10.) There are also important differences across vintages for each detailed asset, for example, faster processor speed in more recent personal computers. The Bureau of Economic Analysis data, however, account for these vintage differences through constant-quality price deflators that implicitly makes investment across vintages perfect substitutes and allows aggregation across vintages.

(11.) As a consistency check, we compared alternative labor series available from the Bureau of Labor Statistics and hours by industry data from Gullickson and Harper (1999) and found very high correlations.

(12.) These studies use a later vintage of capital data and are thus not directly comparable to these industry estimates.

(13.) Real estate and private households are not included.

(14.) The series do not sum exactly in constant dollars, which is an artifact of Divisia aggregation. A similar property exists in the new chain-weighted indexes in the national accounts. See Landefeld and Parker (1997) for details.

(15.) Griliches and Mairesse (1998) provide a detailed review of the econometric difficulties associated with production function estimation and common approaches designed to mitigate the problems.

(16.) The SYS-GMM estimator was generated using the DPD98 Gauss software described by Arellano and Bond (1998). All reported SYS-GMM estimates are from the one-step GMM estimator, with standard errors corrected for heteroskedasticity. To avoid overfitting biases that are possible in our relatively small cross-sectional samples, we report estimates with two lags for the instruments in the sector regressions and three lags in the industry regressions.

(17.) We did combine our data with the real gross output series available from the Bureau of Labor Statistics Employment Projections division and estimated the regressions with gross output as the dependent variables and the same explanatory variables. Though not the proper specification, the results were similar to those reported.

(18.) Not surprisingly, the estimates from the aggregate regression are the least reasonable with a labor coefficient of 1.074 and a negative (though not significant) coefficient on noncomputer capital.

(19.) Econometric tests of the null hypothesis of a common coefficient on all types of noncomputer capital are reported at the bottom of Table 5.

(20.) The p-value is associated with the null hypothesis that [delta] or [gamma] is equal to zero.

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TABLE 1

Change in Computer Intensity by Major Sectors and Detailed Industries,
1970-1996

 Nominal Computer
 Capital Share
Industry 1970 1980

Agriculture, forestry, and 0.000 0.011
 fishing
 Farms 0.000 0.002
 Agricultural services, 0.000 0.131
 forestry and fishing
Mining 0.082 0.169
 Metal mining 0.000 0.039
 Coal mining 0.000 0.004
 Oil and gas extraction 0.104 0.203
 Nonmetallic minerals, 0.000 0.008
 except fuels
Construction 0.416 0.147
Manufacturing 0.905 0.926
 Durable manufacturing 1.146 1.346
 Lumber and wood products 0.311 0.869
 Furniture and fixtures 0.000 0.966
 Stone, clay, and glass 0.144 1.688
 products
 Primary metal industries 0.420 0.283
 Fabricated metal products 0.811 0.464
 Industrial machinery 4.566 3.664
 and equipment
 Electronic and other 0.525 2.130
 electric equipment
 Transportation equipment 0.779 0.862
 Instruments and related 1.751 1.324
 products
 Miscellaneous manufacturing 0.751 0.685
 industries
 Nondurable manufacturing 0.620 0.452
 Food and kindred products 0.580 0.360
 Tobacco products 0.000 0.527
 Textile mill products 0.812 0.262
 Apparel and other 2.354 1.191
 textile products
 Paper and allied products 0.181 0.392
 Printing and publishing 1.436 1.388
 Chemicals and allied products 0.465 0.289
 Petroleum and coal products 0.453 0.385
 Rubber and miscellaneous 1.123 0.664
 plastic products
 Leather and leather products 0.495 0.851
Transportation, comm., and 0.068 0.106
 utilities
 Railroad 0.012 0.013
 transportation
 Local and interurban 0.155 0.112
 passenger transit
 Trucking and 0.108 0.047
 warehousing
 Water transportation 0.149 0.057
 Transportation by air 0.430 0.438
 Pipelines, except 0.054 0.050
 natural gas
 Transportation services 0.214 0.163
 Communications 0.141 0.153
 Utilities 0.027 0.112
Wholesale trade 2.427 3.113
Retail trade 0.406 0.592
Finance, insurance, and 0.977 1.322
 real estate
 Depository institutions 3.664 4.439
 Nondepositoiy; 4.388 3.576
 holding and
 investment offices
 Security and 8.345 9.442
 commodity brokers
 Insurance carriers 7.732 6.381
 Insurance agents, 11.521 8.572
 brokers, and services
 Real estate 0.174 0.164
Services 1.361 1.582
 Hotels and other 0.031 0.236
 lodging places
 Personal services 0.000 0.619
 Business services 4.408 12.458
 Auto repair, services, 0.106 0.970
 and garages
 Miscellaneous repair 0.000 0.986
 shops
 Motion pictures 0.000 1.554
 Amusement and 0.000 0.481
 recreation services
 Health services 0.610 0.600
 Legal services 6.877 2.199
 Educational services 11.940 2.450
 Other 3.636 2.446
Private business sector 0.550 0.678

 Nominal Computer
 Capital Share
Industry 1990 1996

Agriculture, forestry, and 0.020 0.016
 fishing
 Farms 0.004 0.002
 Agricultural services, 0.156 0.099
 forestry and fishing
Mining 0.233 0.140
 Metal mining 0.041 0.020
 Coal mining 0.016 0.030
 Oil and gas extraction 0.274 0.158
 Nonmetallic minerals, 0.122 0.241
 except fuels
Construction 0.238 0.113
Manufacturing 1.758 1.747
 Durable manufacturing 2.291 2.097
 Lumber and wood products 0.494 0.899
 Furniture and fixtures 1.893 1.570
 Stone, clay, and glass 2.353 1.354
 products
 Primary metal industries 0.474 0.492
 Fabricated metal products 0.986 1.037
 Industrial machinery 4.557 3.255
 and equipment
 Electronic and other 3.200 3.585
 electric equipment
 Transportation equipment 1.962 1.801
 Instruments and related 3.742 3.348
 products
 Miscellaneous manufacturing 1.519 2.034
 industries
 Nondurable manufacturing 1.170 1.382
 Food and kindred products 0.842 1.069
 Tobacco products 1.128 1.318
 Textile mill products 0.517 0.640
 Apparel and other 0.864 1.914
 textile products
 Paper and allied products 0.982 0.685
 Printing and publishing 4.469 3.943
 Chemicals and allied products 0.854 1.531
 Petroleum and coal products 0.659 0.857
 Rubber and miscellaneous 1.120 1.376
 plastic products
 Leather and leather products 1.344 1.482
Transportation, comm., and 0.332 0.300
 utilities
 Railroad 0.023 0.009
 transportation
 Local and interurban 0.170 0.030
 passenger transit
 Trucking and 0.060 0.051
 warehousing
 Water transportation 0.098 0.100
 Transportation by air 0.869 0.404
 Pipelines, except 0.026 0.198
 natural gas
 Transportation services 1.261 0.316
 Communications 0.441 0.575
 Utilities 0.359 0.283
Wholesale trade 4.636 6.948
Retail trade 1.705 1.853
Finance, insurance, and 2.277 2.285
 real estate
 Depository institutions 5.314 3.273
 Nondepositoiy; 5.626 6.981
 holding and
 investment offices
 Security and 7.411 8.712
 commodity brokers
 Insurance carriers 7.979 5.170
 Insurance agents, 5.062 4.295
 brokers, and services
 Real estate 0.452 0.928
Services 4.083 4.966
 Hotels and other 0.338 0.167
 lodging places
 Personal services 2.979 1.099
 Business services 20.406 36.25
 Auto repair, services, 3.870 0.907
 and garages
 Miscellaneous repair 2.660 2.454
 shops
 Motion pictures 3.069 1.868
 Amusement and 0.841 0.856
 recreation services
 Health services 1.423 1.924
 Legal services 7.869 5.852
 Educational services 1.166 0.685
 Other 4.936 5.849
Private business sector 1.575 1.847

 Real Computer Capital
 Growth Rates
Industry 1970-80 1980-90

Agriculture, forestry, and 22.77 20.11
 fishing
 Farms 0.00 20.26
 Agricultural services, 21.70 20.15
 forestry and fishing
Mining 36.62 18.83
 Metal mining 22.54 18.14
 Coal mining 0.00 14.48
 Oil and gas extraction 36.99 18.72
 Nonmetallic minerals, 0.00 30.34
 except fuels
Construction 16.67 19.52
Manufacturing 27.61 23.58
 Durable manufacturing 28.94 22.40
 Lumber and wood products 26.83 10.01
 Furniture and fixtures 29.95 24.98
 Stone, clay, and glass 37.05 19.78
 products
 Primary metal industries 21.18 19.78
 Fabricated metal products 23.54 24.50
 Industrial machinery 27.92 20.49
 and equipment
 Electronic and other 40.23 23.68
 electric equipment
 Transportation equipment 28.49 25.04
 Instruments and related 27.39 30.47
 products
 Miscellaneous manufacturing 15.15 23.79
 industries
 Nondurable manufacturing 24.08 26.76
 Food and kindred products 25.21 25.20
 Tobacco products 12.70 24.80
 Textile mill products 13.54 21.98
 Apparel and other 17.40 11.85
 textile products
 Paper and allied products 25.76 28.05
 Printing and publishing 29.11 31.89
 Chemicals and allied products 25.78 26.82
 Petroleum and coal products 25.54 22.66
 Rubber and miscellaneous 21.85 23.84
 plastic products
 Leather and leather products 10.88 17.30
Transportation, comm., and 24.59 27.77
 utilities
 Railroad 17.37 20.26
 transportation
 Local and interurban 9.51 18.44
 passenger transit
 Trucking and 9.97 19.90
 warehousing
 Water transportation 5.84 18.15
 Transportation by air 25.16 26.31
 Pipelines, except 11.16 2.13
 natural gas
 Transportation services 13.49 36.78
 Communications 29.79 28.12
 Utilities 28.52 28.54
Wholesale trade 34.17 25.01
Retail trade 31.57 29.08
Finance, insurance, and 32.10 27.07
 real estate
 Depository institutions 35.51 24.94
 Nondepositoiy; 29.13 28.43
 holding and
 investment offices
 Security and 28.97 20.70
 commodity brokers
 Insurance carriers 31.18 32.49
 Insurance agents, 29.11 10.41
 brokers, and services
 Real estate 26.50 29.89
Services 28.44 28.29
 Hotels and other 29.10 23.07
 lodging places
 Personal services 37.52 32.84
 Business services 27.63 29.46
 Auto repair, services, 35.51 34.09
 and garages
 Miscellaneous repair 29.54 27.42
 shops
 Motion pictures 47.04 28.99
 Amusement and 35.62 21.08
 recreation services
 Health services 24.75 30.94
 Legal services 19.08 35.82
 Educational services 18.50 13.05
 Other 20.92 25.46
Private business sector 29.97 26.17

 Real Computer Growth in
 Capital
 Growth Rates Total Capital
Industry 1990-96 1970-96 1970-96

Agriculture, forestry, and 15.36 19.93 0.96
 fishing
 Farms 8.13 15.71 0.59
 Agricultural services, 16.50 19.83 4.65
 forestry and fishing
Mining 8.34 23.25 1.77
 Metal mining 8.46 16.70 1.50
 Coal mining 29.37 17.83 2.98
 Oil and gas extraction 7.16 23.08 1.67
 Nonmetallic minerals, 26.41 28.86 1.47
 except fuels
Construction 7.77 15.71 0.90
Manufacturing 19.77 24.25 2.27
 Durable manufacturing 18.17 23.94 2.21
 Lumber and wood products 27.94 20.62 1.44
 Furniture and fixtures 16.59 24.54 2.54
 Stone, clay, and glass 9.79 24.12 0.77
 products
 Primary metal industries 18.49 20.02 -0.15
 Fabricated metal products 20.28 23.16 2.15
 Industrial machinery 14.06 21.86 3.50
 and equipment
 Electronic and other 23.88 30.09 4.96
 electric equipment
 Transportation equipment 17.90 24.72 2.11
 Instruments and related 18.03 26.41 4.69
 products
 Miscellaneous manufacturing 24.43 20.61 1.95
 industries
 Nondurable manufacturing 22.74 24.80 2.35
 Food and kindred products 23.99 24.93 1.86
 Tobacco products 19.14 19.35 3.68
 Textile mill products 21.61 18.65 0.15
 Apparel and other 31.10 18.43 2.09
 textile products
 Paper and allied products 13.70 23.86 2.61
 Printing and publishing 17.12 27.41 3.39
 Chemicals and allied products 29.78 27.10 2.93
 Petroleum and coal products 24.72 24.24 1.76
 Rubber and miscellaneous 25.75 23.52 3.38
 plastic products
 Leather and leather products 19.06 15.24 -0.64
Transportation, comm., and 18.55 24.42 2.07
 utilities
 Railroad 4.92 15.61 -0.83
 transportation
 Local and interurban -10.77 8.27 -1.24
 passenger transit
 Trucking and 20.18 16.15 3.20
 warehousing
 Water transportation 18.44 13.48 0.26
 Transportation by air 10.36 22.19 3.24
 Pipelines, except 51.93 17.09 1.09
 natural gas
 Transportation services 2.88 20.00 2.24
 Communications 24.73 27.98 4.54
 Utilities 15.96 25.63 2.40
Wholesale trade 28.84 29.42 6.38
Retail trade 23.48 28.75 3.61
Finance, insurance, and 21.33 27.68 4.39
 real estate
 Depository institutions 14.01 26.48 7.40
 Nondepositoiy; 26.67 28.29 7.48
 holding and
 investment offices
 Security and 21.00 23.95 6.77
 commodity brokers
 Insurance carriers 19.42 28.97 11.17
 Insurance agents, 16.10 18.91 3.24
 brokers, and services
 Real estate 31.00 28.84 3.09
Services 25.22 27.64 4.41
 Hotels and other 8.98 22.14 2.74
 lodging places
 Personal services 6.03 27.70 1.77
 Business services 31.37 5.34
 Auto repair, services, 3.18 27.50 5.22
 and garages
 Miscellaneous repair 20.22 26.33 3.16
 shops
 Motion pictures 19.37 31.97 6.06
 Amusement and 23.56 26.15 2.37
 recreation services
 Health services 27.38 27.74 5.67
 Legal services 12.89 24.09 4.74
 Educational services 15.80 15.78 8.22
 Other 23.96 23.37 4.97
Private business sector 22.96 26.89 2.90

Source: BEA (1998) and authors' calculations.

Note: Shares and growth rates are percentages.
TABLE 2

Distribution of Capital within Major Sectors and Detailed Industries,
1996

 Shares (%)
 Other Other
Industry Computers High Tech Equipment

Agriculture, forestry and fishing 0.02 0.87 41.76
 Farms 0.00 0.04 38.22
 Agricultural services, forestry,
 and fishing 0.10 5.95 63.35
Mining 0.14 1.89 16.88
 Metal mining 0.02 0.99 19.78
 Coal mining 0.03 0.42 33.89
 Oil and gas extraction 0.16 2.17 13.22
 Nonmetallic minerals, except fuel 0.24 1.34 43.03
Construction 0.11 0.81 59.31
Manufacturing 1.75 6.30 51.47
 Durable manufacturing 2.10 5.98 51.71
 Lumber and wood products 0.90 2.05 47.03
 Furniture and fixtures 1.57 1.81 36.76
 Stone, clay, and glass products 1.35 4.94 51.33
 Primary metal industries 0.49 2.76 58.89
 Fabricated metal products 1.04 2.23 62.95
 Industrial machinery and
 equipment 3.26 6.61 51.76
 Electronic and other electric
 equipment 3.59 12.87 45.54
 Transportation equipment 1.80 3.23 54.43
 Instruments and related product 3.35 13.23 33.44
 Miscellaneous manufacturing
 industries 2.03 3.28 43.11
 Nondurable manufacturing 1.38 6.64 51.19
 Food and kindred products 1.07 4.64 49.47
 Tobacco products 1.32 3.47 46.96
 Textile mill products 0.64 2.95 52.71
 Apparel and other textile
 products 1.91 2.81 39.85
 Paper and allied products 0.68 4.69 69.13
 Printing and publishing 3.94 9.66 42.69
 Chemicals and allied products 1.53 11.13 48.87
 Petroleum and coal products 0.86
 Rubber and miscellaneous 5.16 41.35
 plastics products 1.38 2.07 61.77
 Leather and leather products 1.48 0.84 31.12
 Transportation, comm., and
 utilities 0.30 12.42 23.81
 Railroad transportation 0.01 1.77 11.93
 Local and interurban passenger
 transit 0.03 6.56 21.39
 Trucking and warehousing 0.05 3.61 72.73
 Water transportation 0.10 4.04 76.74
 Transportation by air 0.40 7.94 72.31
 Pipelines, except natural gas 0.20 1.71 8.68
 Transportation services 0.32 12.16 71.34
 Communications 0.58 36.83 7.44
 Utilities 0.28 5.04 23.51
Wholesale trade 6.95 10.34 33.21
Retail trade 1.85 1.85 22.68
Finance, insurance, and real estate 2.28 5.11 16.95
 Depository institutions 3.27 8.14 30.79
 Nondepository; holding and
 investment offices 6.98 11.52 55.51
 Security and commodity brokers 8.71 5.62 13.89
 Insurance carriers 5.17 8.79 27.36
 Insurance agents, brokers, and
 services 4.29 6.79 29.91
 Real estate 0.93 2.87 6.55
Services 4.97 8.83 30.19
 Hotels and other lodging places 0.17 1.28 9.20
 Personal services 1.10 6.31 28.96
 Business services 20.41 14.20 37.01
 Auto repair, services, and garages 0.91 2.87 83.21
 Miscellaneous repair shops 2.45 2.13 59.37
 Motion pictures 1.87 23.21 22.24
 Amusement and recreation services 0.86 1.29 25.84
 Health services 1.92 14.30 12.64
 Legal services 5.85 11.48 26.79
 Educational services 0.69 1.46 6.93
 Other 5.85 12.56 18.01
Private business sector 1.85 7.32 28.87

 Shares (%) Total Value of
 Capital Stock
Industry Structures (Millions US$)

Agriculture, forestry and fishing 57.38 366,430
 Farms 61.69 315,425
 Agricultural services, forestry,
 and fishing 30.53 51,276
Mining 81.05 436,641
 Metal mining 79.20 35,165
 Coal mining 65.67 36,171
 Oil and gas extraction 84.45 344,343
 Nonmetallic minerals, except fuel 55.42 20,745
Construction 39.80 88,188
Manufacturing 40.49 1,480,426
 Durable manufacturing 40.23 759,300
 Lumber and wood products 50.05 29,331
 Furniture and fixtures 59.87 13,316
 Stone, clay, and glass products 42.41 43,403
 Primary metal industries 37.80 127,819
 Fabricated metal products 33.76 81,906
 Industrial machinery and
 equipment 38.33 128,598
 Electronic and other electric
 equipment 38.01 128,802
 Transportation equipment 40.55 139,315
 Instruments and related product 49.98 53,024
 Miscellaneous manufacturing
 industries 51.59 14,001
 Nondurable manufacturing 40.77 721,018
 Food and kindred products 44.86 145,868
 Tobacco products 48.27 9,182
 Textile mill products 43.74 37,655
 Apparel and other textile
 products 55.41 13,332
 Paper and allied products 25.47 98,503
 Printing and publishing 43.72 60,044
 Chemicals and allied products 38.47 206,387
 Petroleum and coal products
 Rubber and miscellaneous 52.64 92,578
 plastics products 34.81 54,711
 Leather and leather products 66.51 2,629
 Transportation, comm., and
 utilities 63.51 2,301,741
 Railroad transportation 86.24 360,656
 Local and interurban passenger
 transit 72.00 19,912
 Trucking and warehousing 23.59 108,208
 Water transportation 19.10 36,002
 Transportation by air 19.36 110,141
 Pipelines, except natural gas 89.38 48,361
 Transportation services 16.20 42,660
 Communications 55.17 562,226
 Utilities 71.15 1,014,044
Wholesale trade 49.47 402,858
Retail trade 73.67 540,565
Finance, insurance, and real estate 75.62 1,960,245
 Depository institutions 57.81 370,374
 Nondepository; holding and
 investment offices 26.02 151,641
 Security and commodity brokers 71.78 11,542
 Insurance carriers 58.67 180,296
 Insurance agents, brokers, and
 services 59.01 6,449
 Real estate 89.64 1,239,803
Services 56.06 754,670
 Hotels and other lodging places 89.37 127,093
 Personal services 63.63 26,266
 Business services 28.34 126,464
 Auto repair, services, and garages 13.01 114,535
 Miscellaneous repair shops 36.00 11,987
 Motion pictures 52.73 29,366
 Amusement and recreation services 71.96 48,536
 Health services 71.16 153,883
 Legal services 55.87 18,922
 Educational services 91.00 18,221
 Other 63.60 79,647
Private business sector 61.99 8,328,842

Source: BEA (1998) and authors' calculations.
TABLE 3

Distribution of Computer and High-Tech Capital across Major Sectors,
1996

 Computer Capital
Industry Nominal Value Percent of Total

Agriculture, forestry and fishing 58 0.04
Mining 612 0.40
Construction 100 0.06
Durable manufacturing 15,919 10.35
Non durable manufacturing 9,965 6.48
Transportation, comm., and 6,905 4.49
 utilities
Wholesale trade 27,991 18.20
Retail trade 10,018 6.51
Finance, insurance and real estate 44,789 29.11
Services 37,475 24.36
Private business sector 153,833 100.00

 High-Tech Capital
Industry Nominal Value

Agriculture, forestry and fishing 3,243
Mining 8,874
Construction 811
Durable manufacturing 61,278
Non durable manufacturing 57,829
Transportation, comm., and 292,832
 utilities
Wholesale trade 69,677
Retail trade 20,009
Finance, insurance and real estate 144,988
Services 104,160
Private business sector 763,700

 High-Tech Capital
Industry Percent of Total

Agriculture, forestry and fishing 0.42
Mining 1.16
Construction 0.11
Durable manufacturing 8.02
Non durable manufacturing 7.57
Transportation, comm., and 38.35
 utilities
Wholesale trade 9.12
Retail trade 2.62
Finance, insurance and real estate 18.99
Services 13.64
Private business sector 100.01

Source: BEA (1998) and authors' calculations.

Notes: Shares are percentages and values are millions of current
dollars. High-tech capital includes computers, scientific instruments,
photocopy equipment, and communications equipment.
TABLE 4

Simple Production Function Regressions, 1980-96

 OLS FE
 Aggregate Sector Industry Sector

ln(L) 1.074 (***) 0.511 (***) 0.493 (***) 0.051
 (0.104) (0.025) (0.020) (0.075)
ln([K.sub.c]) 0.052 (**) 0.188 (***) 0.178 (***) 0.046
 (0.018) (0.016) (0.010) (0.038)
ln([K.sub.n]) -0.227 0.154 (***) 0.256 (***) 0.030
 (0.291) (0.033) (0.015) (0.097)
Year 0.003
 (0.003)
[R.sup.2] 0.99 0.89 0.85 0.87
No. of obs. 17 170 935 170

 FE SYS-GMM
 Industry Sector Industry

ln(L) 0.468 (***) 0.554 (***) 0.340 (***)
 (0.062) (0.053) (0.114)
ln([K.sub.c]) -0.018 0.171 (***) 0.305 (***)
 (0.018) (0.064) (0.084)
ln([K.sub.n]) -0.122 (**) 0.123 -0.020
 (0.056) (0.120) (0.081)
Year

[R.sup.2] 0.50
No. of obs. 935 160 880

Notes: All regressions include year dummy variables, except for
aggregate OLS. Robust standard errors are in parentheses. [R.sup.2] is
adjusted-[R.sup.2] for OLS and within-[R.sup.2] for FE. Dependent
variable is log of real value-added; L is labor; [K.sub.c] is computer
capital; [K.sub.n] is noncomputer capital.

(***), (**), and (*)denote statistical significance at the 1%, 5%, and
10% level, respectively.
TABLE 5

Extended Production Function Regressions, 1980-96

 OLS
 Aggregate Sector Industry

ln(L) 1.299 (***) 0.533 (***) 0.505 (***)
 (0.126) (0.028) (0.021)
ln([K.sub.c]) 0.073 (*) 0.161 (***) 0.142 (***)
 (0.034) (0.024) (0.013)
ln([K.sub.h]) -0.028 0.057 0.070 (***)
 (0.388) (0.044) (0.012)
ln([K.sub.o]) -0.752 (***) -0.050 -0.015
 (0.184) (0.051) (0.013)
ln([K.sub.s]) -0.441 0.096 (*) 0.210 (***)
 (0.609) (0.056) (0.017)
Year 0.011 (**)
 (0.004)
[R.sup.2] 0.99 0.88 0.86
No. of obs. 17 170 935
p-value 0.013 0.116 0.000

 FE SYS-GMM
 Sector Industry Sector

ln(L) -0.001 0.465 (***) 0.557 (***)
 (0.089) (0.060) (0.072)
ln([K.sub.c]) 0.069 (*) 0.001 0.132 (*)
 (0.037) (0.019) (0.073)
ln([K.sub.h]) 0.122 (***) 0.071 (***) 0.123
 (0.041) (0.023) (0.098)
ln([K.sub.o]) -0.190 (**) -0.223 (***) -0.105
 (0.088) (0.060) (0.105)
ln([K.sub.s]) 0.246 (**) 0.059 0.048
 (0.115) (0.070) (0.203)
Year

[R.sup.2] 0.89 0.52
No. of obs. 170 935 160
p-value 0.019 0.000

 SYS-GMM
 Industry

ln(L) 0.485 (***
 (0.102)
ln([K.sub.c]) 0.177 (**)
 (0.076)
ln([K.sub.h]) 0.105 (*)
 (0.061)
ln([K.sub.o]) -0.170 (**)
 (0.078)
ln([K.sub.s]) 0.194 (**)
 (0.075)
Year

[R.sup.2]
No. of obs. 880
p-value

Notes: All regressions include year dummy variables, except for
aggregate OLS. Robust standard errors are in parentheses. [R.sup.2] is
adjusted-[R.sup.2] for OLS and within-[R.sup.2] for FE. Dependent
variable is log of real value-added; L is labor; [K.sub.c] is computer
captial; [K.sub.h] is other high-tech capital; [K.sub.o] is other
equipment capital; and [K.sub.s] is structures. p-value is associated
with null hypothesis of equal elasticities for all types of noncomputer
capital.

(***), (**) and (*)denote statistical significance at the 1%, 5%, and
10% level, respectively.
TABLE 6

Alternative Production Function Regressions, 1980-96

 OLS
 Aggregate Sectors Industry

ln(L) 1.342 (***) 0.539 (***) 0.512 (***)
 (0.094) (0.032) (0.021)
ln(K) -1.056 (***) 0.318 (***) 0.417 (***)
 (0.235) (0.031) (0.013)
ln([K.sub.e]/K) -0.503 (***) 0.061 0.006
 (0.134) (0.082) (0.032)
ln([K.sub.c]/[K.sub.e]) 0.092 (***) 0.181 (***) 0.182 (***)
 (0.015) (0.016) (0.010)
year 0.015 (***)
 (0.004)
Implied [delta] 0.476 0.191 0.015
p-value 0.001 0.454 0.841
Implied [gamma] -0.087 0.568 0.437
p-value 0.000 0.000 0.000
[R.sup.2] 0.99 0.89 0.86
No. of obs. 17 170 935

 FE SYS-GMM
 Sectors Industry Sectors

ln(L) 0.040 0.463 (***) 0.577 (***)
 (0.078) (0.061) (0.102)
ln(K) 0.217 (**) -0.035 0.344 (***)
 (0.098) (0.057) (0.116)
ln([K.sub.e]/K) -0.256 (**) -0.313 (***) -0.007
 (0.103) (0.086) (0.273)
ln([K.sub.c]/[K.sub.e]) 0.066 (*) 0.003 0.141 (**)
 (0.039) (0.020) (0.058)
year

Implied [delta] -1.179 8.921 -0.020
p-value 0.009 0.570
Implied [gamma] 0.303 -0.078 0.410
p-value 0.128 0.901
[R.sup.2] 0.88 0.51
No. of obs. 170 935 160

 SYS-GMM
 Industry

ln(L) 0.484 (***)
 (0.086)
ln(K) 0.281 (***)
 (0.085)
ln([K.sub.e]/K) -0.063
 (0.121)
ln([K.sub.c]/[K.sub.e]) 0.231 (***)
 (0.056)
year

Implied [delta] -0.224
p-value
Implied [gamma] 0.821
p-value
[R.sup.2]
No. of obs. 880

Notes: All regressions include year dummy variables, except for
aggregate OLS. Robust standard errors are in parentheses. [R.sup.2] is
adjusted-[R.sup.2] for OLS and within-[R.sup.2] for FE. Dependent
variable is log of real value-added; L is labor; [K.sub.c] is total
capital; [K.sub.e] is equipment capital; [K.sub.c] is computer capital.
Implied [delta] and implied [gamma] from equation (9), and p-value is
associated with null hypothesis that the implied coefficient equals
zero.

(***), (**), and (*) denote statistical significance at the 1%, 5%, and
10% level, respectively.
TABLE 7

Comparison of Manufacturing and Nonmanufacturing Results, 1980-96

 Simple OLS Simple SYS-GMM
 Mfg Non-Mfg Mfg

ln(L) 0.128 0.565 (***) 0.178
 (0.205) (0.058) (0.195)
ln(K)

ln[(K.sub.c)] 0.262 (***) 0.164 (***) 0.458 (***)
 (0.112) (0.032) (0.232)
ln[(K.sub.n)] 0.299 (***) 0.261 (***) 0.038
 (0.108) (0.066) (0.231)
ln[(K.sub.h)]

ln[(K.sub.o)]

ln[(K.sub.s)]

ln[(K.sub.e/K)]

ln[(K.sub.c/K.sub.e)]

No. of obs. 340 595 320

 Simple SYS-GMM Extended SYS-GMM
 Non-Mfg Mfg Non-Mfg

ln(L) 0.412 (***) 0.197 0.582 (***)
 (0.085) (0.155) (0.085)
ln(K)

ln[(K.sub.c)] 0.228 (***) 0.191 0.143 (***)
 (0.066) (0.126) (0.061)
ln[(K.sub.n)] 0.098
 (0.082)
ln[(K.sub.h)] 0.365 (***) 0.054
 (0.140) (0.069)
ln[(K.sub.o)] 0.147 -0.135 (*)
 (0.209) (0.073)
ln[(K.sub.s)] -0.398 0.231 (***)
 (0.328) (0.066)
ln[(K.sub.e/K)]

ln[(K.sub.c/K.sub.e)]

No. of obs. 560 320 560

 Alternative SYS-GMM
 Mfg Non-Mfg

ln(L) 0.127 0.602 (***)
 (0.152) (0.089)
ln(K) 0.384 (***) 0.265 (***)
 (0.082) (0.085)
ln[(K.sub.c)]

ln[(K.sub.n)]

ln[(K.sub.h)]

ln[(K.sub.o)]

ln[(K.sub.s)]

ln[(K.sub.e/K)] 1.512 (***) -0.266 (**)
 (0.528) (0.132)
ln[(K.sub.c/K.sub.e)] 0.358 (**) 0.169 (***)
 (0.159) (0.046)
No. of obs. 320 560

Notes: All regressions includes year dummy variables. Robust standard
errors are in parentheses. Variables are defined in Tables 4-6.

(***), (**), and (*) denote statistical significance at the 1%, 5%, and
10% level, respectively.