Public and private capital productivity puzzle: a nonparametric approach.

1. Introduction

Returns to public capital generated a great deal of controversy in the productivity literature. After Aschauer (1989, 1990) published a series of papers relating declining labor productivity to the decline in public investment, journals were flooded with papers from those who agreed (e.g., Munnell 1990) and those who disagreed (e.g., Holtz-Eakin 1994). Each side developed convincing arguments for why public capital was productive (having a positive effect on output) or unproductive (not having a direct effect on output--perhaps only increasing utility). Econometricians later entered the picture and stated that the positive coefficient associated with public capital was most likely attributable to model misspecification (e.g., ignoring state and time effects, etc.). Although there is no consensus with state-level data, the majority of empirical evidence supports the view that the marginal return from public capital is not significantly different from zero. Numerically, the estimates are found to be quite small and often negative, especially when either state or both state and time effects are controlled for (see Holtz-Eakin 1994; Baltagi and Pinnoi 1995; Garcia-Mila, McGuire, and Porter 1996, among others).

Almost all the studies use a Cobb-Douglas (CD) production function (with the exception of Lynde and Richmond 1992; Nadiri and Mamuneas 1994; Morrison and Schwartz 1996a, b; Batina 2001; Cohen and Morrison Paul 2004; who use flexible cost function specifications) to estimate the productivity of inputs. Because of the functional form, elasticity (marginal productivity) of each input can be simply measured from the estimated coefficient of each input. Thus, by construction, elasticities are exactly the same for all states and over all years. This is a very strong assumption. In this paper, we argue that the negative or insignificant returns on public capital might be attributable to an inappropriate functional form (e.g., failure to take the nonlinearity into account). To take account of possible nonlinearities without imposing a functional form, we use a nonparametric approach. This method requires no specific assumptions on the form of the underlying production function. Additionally, the returns from the factor inputs are observation-specific in the nonparametric approach. The statistical results reveal two important concepts. One is that the sizes of the estimated output elasticities (as well as marginal products) of private capital and labor are similar to those reported in other studies. Second, our results show that the return to public capital is positive and statistically significant.

The remainder of the paper is organized as follows: section 2 gives additional background on the productivity puzzle and describes the data; section 3 describes the generalized kernel estimation procedure, whereas the fourth section presents the results. Section 5 employs an alternative data set to check for robustness of the results, and the final section summarizes the conclusions.

2. The Productivity Puzzle

Two competing approaches are typically employed to measure productivity. In the primal approach, a parametric production function (e.g., CD) is often estimated. This approach is widely used because it requires information on only output and input quantities. On the other hand, the dual approach, in which mostly a parametric cost function is estimated, requires information on input prices along with the input and output quantities. The main advantage of using the cost function is that it explicitly takes into account possible endogeneity of inputs into the analysis. However, the cost function approach is less popular because of the fact that data on input prices are not easily available.

In this paper, we follow the primal approach (and hence leave the dual for future research) and estimate an aggregate production function

[y.sub.it] = f([x.sub.kit], [beta]) + [[epsilon].sub.it],

using state-level panel data (48 contiguous states observed for the period 1970-1986). (1) Output (y) is the gross state product for each state i in each time period t and [beta] is a vector of unknown parameters. As for the regressors (x), labor (L) is employment in nonagricultural payrolls, private capital stock (KP) is the Bureau of Economic Analysis' national stock estimates, and public capital (KG) aggregates highways and streets (KH), water and sewer facilities (KW), and other public buildings and structures (KO). Details on these variables can be found in Munnell (1990) as well as in Baltagi and Pinnoi (1995). Following Baltagi and Pinnoi (1995), we use the unemployment rate (u) to control for business-cycle effects. We use three different estimation methods, one is pooled ordinary least squares (OLS), one allows the existence of fixed state and time effects in the error term [[epsilon].sub.it] and is written as [[epsilon].sub.it] = [[mu].sub.it] + [[gamma].sub.t] + [v.sub.it], and finally, we consider a random-effects formulation in which the error components [[mu].sub.i] and [[gamma].sub.t] are assumed to be random variables.

The standard error for each estimate is given in italics. Controls for state and time effects included in both fixed- and random-effects models.

The results based on the CD production function (linear in logs)

[y.sub.it] = [K.summation over (k=1)][[beta].sub.k][x.sub.kit] + [[epsilon].sub.it],

are reported in Table 1. (2) The coefficients on KG (in the fixed- and random-effects models) are found to be quite small (-0.030 and 0.018, respectively) and statistically insignificant. On the contrary, coefficients on private capital are found to be positive (0.169 and 0.266, respectively) and statistically significant. Such a large difference in the returns between public and private capital is difficult to explain. Using the estimated elasticities (the coefficients on capital and labor in the CD production function), the marginal products (same as the value of marginal products since the inputs are measured in constant dollars) are also evaluated. For example, the average estimates (as shown in Table 2) for KP and KG (for the random-effects model) are found to be 0.263 and 0.042, respectively. If one views this as a problem of allocation of funds between private and public capital, returns from a dollar from public and private investment should be the same. Although an optimizing model would suggest equal returns to private and government capital, finding a political process to allocate government capital in such an optimal manner is difficult. Given this difficulty, it may be suggested that public capital would have a lower, but positive return (e.g., see Pinnoi 1994; Aschauer 2001). However, much of the recent literature does not find this phenomenon. Since the marginal product (NIP) using the CD production function of KG is much less than that of KP (both are measured in 1982 year dollars), there must be some explanation for such a massive overinvestment in public projects. More specifically, the MP for the random-effects model of KP is six times bigger than that of KG. Thus, to justify the results, the price of KP has to be six times the price of KG. Because of the positive externality in KG, one would expect that the effective price of KG would be lower. Furthermore, the MP of KG is found to be negative in the fixed-effects model (although not significantly different from zero), which indicates overinvestment in public capital, even if it is assumed to be costless. One explanation offered by Holtz-Eakin (1994, p. 12) is that "... government capital budgeting decisions focused at best on the consumption benefits accruing from public goods and services, and at worst on the pork-barrel punch they carried." While this might be true to some extent, the question is whether such a big difference in the returns can be explained. Thus, it seems natural to ask whether the results from the CD model can be trusted. In fact, if the true model is nonlinear and one ignores it, the resulting estimates of returns to inputs are likely to be inconsistent.

To avoid the model misspecification problem, we use a nonparametric kernel regression approach (we also try flexible parametric specifications). Nonparametric regression is simply a method where local averaging is used. Dependent values are estimated using predictor values that are close to a target value. As more distant observations are used for averaging, the curve will be a straight line, as in linear regression. On the other hand, if only the closest observations of the predictor values are used, the resulting curve becomes less smooth (there will be a more in-depth discussion on bandwidth selection in the next section). Specifically, we use the Li-Racine generalized kernel estimation procedure (which allows us to smooth both continuous and categorical variables) and estimate the returns to public capital, private capital, labor, and unemployment, while controlling for state and time effects.

3. Generalized Kernel Estimation

In this section, we describe Li-Racine generalized kernel estimation (see Li and Racine 2004; Racine and Li 2004), which we use to estimate marginal products and elasticities of the inputs in our models. First, consider the nonparametric regression model

[y.sub.i] = m([x.sub.i]) + [w.sub.i], i = 1, ..., NT, (1)

where m([x.sub.i]) is the unknown smooth production function with argument [x.sub.i] = [[x.sup.c.sub.i], [x.sup.u.sub.i], [x.sup.0.sub.i]], [x.sup.c.sub.i] is a vector of continuous regressors (private capital, public capital, labor, and the unemployment rate), [x.sup.u.sub.i] is a vector of regressors that assume unordered discrete values (state effects), [x.sup.o.sub.i] is a vector of regressors that assume ordered discrete values (time effects), w is an additive error, N is the number of cross-sectional units, and T is the number of periods in the sample (N = 48, T = 17). Taking a first-order Taylor expansion of m([x.sub.j]) at [x.sub.j], Equation 1 yields

[y.sub.i] [approximately equal to] m([x.sub.j]) + ([x.sup.c.sub.i] - [x.sup.c.sub.j]) [beta]([x.sub.j]) + [w.sub.i],

where [beta]([x.sub.j]) is defined as the partial derivative of m([x.sub.j]) with respect to [x.sup.c]. When y and x are both expressed in logarithmic form, [beta]([x.sub.j]) is interpreted as an elasticity.

The estimator of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [K.sub.[??]] = [[PI].sup.q.sub.s=1] [([[??].sup.s.sub.s]).sup.-1] [l.sup.c] ([x.sup.c.sub.si]- [x.sup.c.sub.sj]/[[??].sup.c.sub.s]) [[PI].sup.r.sub.s=1] [l.sup.u]([x.sup.u.sub.si], [x.sup.u.sub.sj] [[??].sup.u.sub.s] [[PI].sup.p.sub.s=1] [l.sup.o] ([x.sup.o.sub.si], [x.sup.o.sub.sj], [[??].sup.o.sub.s]). [K.sub.[lambda]] is the commonly used product kernel (see Pagan and Ullah 1999), [l.sup.C] is the standard normal kernel function with window width [[lambda].sup.c.sub.s] = [[lambda].sup.c.sub.s](NT) associated with the sth component of [x.sup.c], [l.sup.u] is a variation of Aitchison and Aitken's (1976) kernel function, which equals one if [x.sup.u.sub.si] = [x.sup.u.sub.sj], and [[lambda].sup.u.sub.s] otherwise, and [l.sup.o] is the Wang and van Ryzin (1981) kernel function, which equals one if [x.sup.o.sub.si] = [x.sup.o.sub.sj], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] otherwise. (3)

Estimation of the bandwidths ([[lambda].sup.c], [[lambda].sup.u], [[lambda].sup.o]) is perhaps the most salient factor when performing nonparametric estimation. For example, choosing a very small bandwidth means that there may not be enough points for smoothing, and thus we may get an undersmoothed estimate (low bias, high variance). On the other hand, a very large bandwidth may include too many points and thus produce an oversmoothed estimate (high bias, low variance). This trade-off is a well-known dilemma in applied nonparametric econometrics, and thus we usually resort to automatic determination procedures to estimate the bandwidths. Although many selection methods exist, we choose Hurvich, Simonoff, and Tsai's (1998) Expected Kullback Leibler (AI[C.sub.c]) criteria. This method chooses smoothing parameters using an improved version of a criterion based on the Akaike information criteria. AI[C.sub.c] has been shown to perform well in small samples and avoids the tendency to undersmooth, as often happens with other approaches such as least-squares cross-validation. (4) Specifically, the bandwidths are chosen to minimize

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [??]([x.sub.j]) = H[y.sub.j].

4. Results

To accommodate the possible nonlinearity parametrically (as well as obtain observation-specific estimates), we estimate the model using several versions of the translog production function (5)

[y.sub.it] = [K.summation over (k=1)] [[beta].sub.k] [x.sub.kit] + [1/2] [K.summation over (k=1)] [K.summation over (l=1)] [[beta].sub.kl] [x.sub.kit] [x.sub.lit] + [[epsilon].sub.it],

to which the CD model is the special case where [[beta].sub.kl] = 0 [for all] k, l. The results for the three specifications of the translog model (OLS, fixed effects, and random effects) are given in Tables 3 and 4. So far as private capital is concerned, these procedures give similar "reasonable" coefficient estimates. However, the results for the other three variables are somewhat alarming. Large negative and significant elasticities are given for public capital for at least a quarter of the observations. Also, elasticities of output with respect to labor are found to be greater than unity for about 25% of the observations. Finally, the translog model finds positive coefficients on unemployment for about half of the observations. Each of these phenomena seem economically unreasonable. The only specification for which the results look economically meaningful is the pooled OLS model. However, this specification has been heavily criticized for ignoring state and time effects. Thus, we suggest that the results in Tables 3 and 4 cast doubt on the usefulness of this flexible parametric procedure for this particular data set. It appears that the translog specification is not sufficiently flexible to capture the nonlinearities in the data. (6)

The results for the nonparametric model are reported in Tables 5 and 6. (7) Here we again report the mean and quartile values of the estimated elasticities (and marginal products, respectively) of public and private capital, labor, and the unemployment rate. It should be noted that, on average, we find similar results as in the CD case (by using nonparametric regression that captures nonlinearity in the functional form) in terms of private capital, labor, and unemployment. However, our results are significantly different in terms of the returns to public capital. Specifically, we find evidence of a significant positive return to public capital. We find that the mean and median elasticities associated with public capital (0.106 and 0.123) are positive and statistically different from zero. Although these elasticities are smaller than those of private capital, the difference is not all that large. To compare returns from private and public capital, perhaps it is better to examine their marginal products. As hypothesized, the MP of KG evaluated at the mean (0.251) is positive and significant, but still smaller than that of KP (0.306). If one argues that public capital generates a positive externality, and therefore its effective price is smaller than the actual price, then the overinvestment in public capital and smaller marginal product (at the mean) can be justified. Based on these findings, we come to the conclusion that the parametric models are too simple and fail to capture the nonlinearities inherent in the model. Thus, we conclude that model misspecification causes coefficients on public capital to be small (positive/negative) and statistically insignificant.

Although these results are striking, we feel it necessary to test for a known parametric specification. Here we employ the Hsiao, Li, and Racine (2006) specification test for mixed categorical and continuous data. This test will help us determine if the parametric functional forms applied are acceptable. Here we test the null hypothesis that the parametric model is correctly specified ([H.sub.0]: P[E(y|x) = f(x, [beta])] = 1) against the alternative that it is not ([H.sub.1]: P[E(y|x) = f(x, [beta])] < 1). First, we test the null hypothesis that the model is CD (the preferred fixed-effects model). The test firmly rejects the null hypothesis (p-value = 0.0000) that the underlying model is CD. Second, we tested the null hypothesis that the model is translog (the preferred fixed-effects model). Again, the test strongly rejects the null hypothesis (p-value = 0.0000).

In addition to the Hsiao-Li-Racine test, we also undertake several other diagnostic checks. First, we examine the mean squared error (MSE) from each regression. The MSE from the nonparametric model (0.0003) is more than three times smaller than that of the translog (0.0010) and over four times smaller than that of the CD model (0.0012). (8)

On an intuitive level, reliability of a model is often judged in terms of its closeness to some statistics that can be obtained without estimating any econometric model. One such statistic is average product, which can be computed simply from the observed data. If the predicted average products from an econometric model (parametric or nonparametric) closely resemble those obtained simply from the observed data, it might be argued that the model fits the data well. In this vein, we compare the prediction of average product based on the preferred parametric model (fixed-effects [time and state] translog model) (9) and the nonparametric model against the observed values that do not depend on any specific model. Specifically, we compute the average product of public capital, estimate its kernel density, and compare it to the estimated average products of public capital from the translog and nonparametric models. Figure 1 gives the three kernel density plots on one graph. It is obvious from first glance at the densities that the actual and nonparametric densities are nearly identical. The translog, which mimics the shape of the actual, is shifted to the left. To give a statistical measure of the closeness of the empirical distributions, we implemented the Li (1996) test to examine closeness/ difference between unknown distributions. Using this test, we fail to reject the null hypothesis that the actual and nonparametric kernel density estimates are different from one another (p-value = 0.9857), but are able to reject the nulls that the translog kernel density estimate is different from both the actual (p-value = 0.0000) and nonparametric (p-value = 0.0000) densities. (10)

[FIGURE 1 OMITTED]

As a final test, we choose subsets of the data to obtain out-of-sample forecasts. We find that the nonparametric model has significantly lower predicted MSE than the parametric models for this particular data set. As an example, we estimate the model using the first sixteen years of data for each state and forecast the remaining year of data. In this particular scenario, our predicted MSE for the nonparametric model (0.0010) is roughly fifty times smaller than both the CD (0.0570) and the translog (0.0489) models. (11)

Based on these tests, we conclude that the nonparametric model performs better than its parametric counterparts. We provide further evidence on this by examining an alternative data set.

5. An Alternative Data Set

In order to check our results for robustness, we have chosen to perform the results of this experiment on an alternative data set. Specifically, we test our procedures on the data used in Cohen mad Morrison Paul (2004). Although this is also a U.S. state-level data set, it is different from the one in the previous section in several ways. First, the data set is more recent and covers the period 1982-1996. Second, the cross sections are less aggregated, and the focus is on the manufacturing industry. Finally, this data set does not use unemployment as a regressor. If our result holds for this data, then it will show that our result is robust.

Specifically, we estimate the model

[y.sub.it] = f([x.sub.1it], [x.sub.2it], [3.sub.it], [beta]) + [[epsilon].sub.it],

where y is the aggregate output in state i at time t, [x.sub.1it], is the public infrastructure stock, [x.sub.2i], is fixed (private) capital, and [x.sub.3it] is labor (nonproduction and production).

The results for the estimation of this data set are given in Tables 7 through 9. The CD model (with fixed state and time effects), found to be preferred over the random-effects model using the Hausman test, again gives insignificant returns to public capital (Table 7). The same holds true for the fixed-effects translog model (which is preferred to the random-effects translog model using the Hausman test), the results of which are reported in Table 8. Finally, as with the previous data, the nonparametric estimates for public capital (Table 9) are positive and significant at both the mean and median.

For the sake of comparison, Figure 2 plots the estimated kernel densities of the average product of public capital for the actual data, and both the nonparametric and the preferred parametric model (fixed-effects [time and state] translog model). (12) Even more so than in Figure l, the nonparametric model outperforms the translog model. The difference between the estimated densities of the actual average product and the estimated average product using the nonparametric technique is extremely small. This picture is mimicked by the Li test, which fails to reject the null hypothesis that the two distributions are different from one another (p-value = 0.9975). The results using the translog model reinforce the importance of using the nonparametric technique. Whereas the shape of the density using the translog model is similar in Figure 1, in Figure 2 the shape of that density is far different from the actual data. Again, this is reflected in the Li test (p-value = 0.0000), which firmly rejects the null hypothesis. (13)

[FIGURE 2 OMITTED]

In summary, this section of the paper attempts to test the results of the previous section for robustness. Here we use an alternative data set based on U.S. manufacturing industry data from a more recent period to test our hypothesis. In addition, we do not include the unemployment rate in our models used in this section. Nonetheless, the results of this experiment are similar to the previous section. In both cases, the preferred fixed-effects translog model produces small and insignificant returns to public capital expenditure. However, similar to the previous case, the nonparametric model is able to uncover the positive and significant return to public capital. Based on these results, we conclude that our results using the nonparametric estimation procedure are robust.

6. Conclusion

In this paper, we show that the popular parametric specifications (the Cobb-Douglas, as well as the translog specifications) of the production function are not supported by the state-level panel data that are used to estimate returns on public and private capital. The parametric specifications are unable to capture the nonlinearity in the functional relationship underlying the production technology. Consequently, the parametric models are likely to give incorrect estimates of returns to inputs. To avoid model misspecification, we estimate the production technology using the Li-Racine generalized kernel estimation technique. These procedures are used to estimate the returns to private capital, employment, and public capital in (i) gross state product using a panel of 48 states for 17 years (19701986), and (ii) the manufacturing industry using a panel of 48 states for 15 years (1982-1996).

We find that the return to public capital is positive and significantly different from zero, even after controlling for state and time effects. Based on these findings, we come to the conclusion that the parametric models used in many previous studies were too simple and failed to capture the nonlinearities inherent in the production function. Thus, we feel that model misspecification is what caused the insignificant coefficients on public capital. We are not suggesting, however, that this is the end of the story. We have simply shown that past econometric evidence that public capital is not productive might not hold under tight scrutiny. At the very least, we suggest that this discussion should be reopened.

The authors would like to thank Christopher Hanes, Qi Li, Christopher Parmeter, Jeff Racine, Dek Terrell, Yue Xu, and two anonymous referees for useful comments on the subject matter of this paper. The research on this project has also benefited from participants of seminars at the State University of New York at Binghamton, the University of Calgary, the State University of New York at Albany, Texas Christian University, and Southern Methodist University, and participants at the 2005 winter meetings of the Econometric Society, Philadelphia, Pennsylvania. Finally, the authors would also like to thank Badi Baltagi and Catherine Morrison Paul for providing the data necessary for this project.

Received December 2004; accepted October 2005.

References

Aitchison, John, and Colin G.G. Aitken. 1976. Multivariate binary discrimination by kernel method. Biometrika 63:413-20.

Aschauer, David A. 1989. Is public expenditure productive? Journal of Monetary Economics 23:177-200.

Aschauer, David A. 1990. Why is infrastructure important? In Is there a shortfall in public capital investment? edited by Alicia H. Munnell. Boston, MA: Federal Reserve Bank of Boston, pp. 21-50. Aschauer, David A. 2001. Output and employment effects of public capital. Public Finance & Management 1:135-60.

Baltagi, Badi H., and Nat Pinnoi. 1995. Public capital stock and state productivity growth: Further evidence from an error components model. Empirical Economics 20:351-9.

Batina, Raymond G. 2001. The effects of public capital on the economy. Public Finance and Management 1:113-34.

Boarnet, Marlon G. 1998. Spillovers and the locational effects of public infrastructure. Journal of Regional Science 38:381-400.

Cohen, Jeffrey P., and Catherine J. Morrison Paul. 2003. Airport infrastructure spillovers in a network system. Journal of Urban Economics 54:459-73.

Cohen, Jeffrey P., and Catherine J. Morrison Paul. 2004. Public infrastructure investment, interstate spatial spillovers and manufacturing costs. Review of Economics and Statistics 86:551-60.

Garcia-Mila, Teresa, Therese J. McGuire, and Robert H. Porter. 1996. The effects of public capital in state-level production functions reconsidered. Review of Economics and Statistics 78:177-80.

Hall, Peter, Jeff Racine, and Qi Li. 2004. Cross-validation and the estimation of conditional probability densities. Journal of the American Statistical Association 99:1015-26.

Holtz-Eakin, Douglas. 1994. Public-sector capital and the productivity puzzle. Review of Economics and Statistics 76:12-21.

Holtz-Eakin, Douglas, and A. E. Schwartz. 1995. Spatial productivity spillovers from public infrastructure: Evidence from state highways. International Tax and Public Finance 2:459-68.

Hsiao, Cheng, Qi Li, and Jeff Racine. 2006. A consistent model specification test with mixed categorical and continuous data. Journal of Econometrics. In press.

Hurvich, Clifford M., Jeffrey S. Simonoff, and Chih-Ling Tsai. 1998. Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. Journal of the Royal Statistical Society. Series B 60:271-93.

Li, Qi. 1996. Nonparametric testing of closeness between two unknown distribution functions. Econometric Reviews 15:261-74.

Li, Qi, and Desheng Ouyang. 2005. Uniform convergence rate of kernel estimation with mixed categorical and continuous data. Economics Letters 86:291-6.

Li, Qi, and Jeff Racine. 2004. Cross-validated local linear nonparametric regression. Statistica Sinica 14:485-512.

Li, Qi, and Jeff Racine. 2006. Nonparametric econometrics: Theory and practice. Princeton, NJ: Princeton University Press. In press.

Lynde, Catherine, and James Richmond. 1992. The role of public capital in production. Review of Economics and Statistics 74:37-44.

Morrison, Catherine J., and Amy E. Schwartz. 1996a. Public infrastructure, private input demand, and economic performance in New England manufacturing. Journal of Business and Economic Statistics 14:91-101.

Morrison, Catherine J., and Amy E. Schwartz. 1996b. State infrastructure and productive performance. American Economic Review 86:1095-111.

Munnell, Alicia H. 1990. How does public infrastructure affect regional economic performance? New England Economic Review (September) 11-32.

Nadiri, M. Ishaq, and Theofanis P. Mamuneas. 1994. The effects of public infrastructure and R&D capital on the cost structure and performance of U.S. manufacturing industries. Review of Economics and Statistics 76:22-37.

Pagan, Adrian, and Aman Ullah. 1999. Nonparametric econometrics. Cambridge, UK: Cambridge University Press.

Pinnoi, Nat. 1994. Public infrastructure and private production measuring relative contributions. Journal of Economic Behavior and Organization 23:127-48.

Racine, Jeff. N, nonparametric software. Available http://www.economics.mcmaster.ca/racine/.

Racine, Jeff, and Qi Li. 2004. Nonparametric estimation of regression functions with both categorical and continuous data. Journal of Econometrics 119:99-130.

Rousseeuw, Peter J., and Bert C. van Zomeren. 1990. Unmasking multivariate outliers and leverage points. Journal of the American Statistical Association 85:633-9.

Wang, Min-Chiang, and John van Ryzin. 1981. A class of smooth estimators for discrete estimation. Biometrika 68:301-9.

(1) We decided not to use an updated version of the data to be able to directly compare our results to previous studies. However, in the fifth section of the paper we employ a similar but more recent data set to show robustness of the results with respect to the time period.

(2) The results in the table are slightly different from those in Baltagi and Pinnoi (1995). This is because we use a two-way and not a one-way error component model. However, it should be noted that when we employ the one-way error component model, we obtain identical results.

(3) See Hall, Racine, and Li (2004), Racine and Li (2004), Li and Racine (2004, 2006), and Li and Ouyang (2005) for further details.

(4) One possible cause for undersmoothing with the LSCV procedure is due to the presence of outliers in the data. The inclusion of outliers causes the cross-validation procedure to undersmooth kernel regression estimates (give too small values for the bandwidths in order to capture observations lying away from the underlying data-generating process). The end result is that the regression estimates have far too much variation even though they possess small sample bias. In an earlier version of this paper, we included an alternative approach to the AIC, and LSCV bandwidth selection criterion. Specifically, we proposed the robust cross-validation procedure (which gives similar results as AI[C.sub.c] for this particular data set). In simple terms, it consists of determining the outliers in a particular data set, putting them aside, and then running the (least-squares) cross-validation procedure. Once the window widths have been obtained, then we can use those on the full sample to obtain consistent estimates of the regression parameters.

The question is then, how do we find the outliers? We suggest using the widely cited robust distances method of Rousseeuw and van Zomeren (1990). This procedure amounts to encircling the data in a K + 1 dimensional sphere and identifying the leverage points (of which some are good and some are bad--the good ones help in obtaining correct estimates while the ad ones are true outliers) that lie outside the sphere.

(5) We also estimated the model using a random coefficients specification. The results are similar to the translog model and are thus omitted for sake of brevity.

(6) It should be noted here that there are many extensions to this flexible parametric model. For instance, there is a large literature on the issue of spillovers across jurisdictions or across states, as well as the issue of spatial autocorrelation (e.g., see Holtz-Eakin and Schwartz 1995; Boarnet 1998; Cohen and Morrison Paul 2003, 2004). Failure to account for this inherent dependence in the data can lead to biased estimates. Fortunately, Li-Racine generalized kernel estimation allows for weak dependence in the data (see Li and Racine 2006). Therefore, if spatial autocorrelation exists in our data set, this should not affect the estimates. Thus we leave the issue of spillovers within the nonparametric framework to future research.

(7) All bandwidths were estimated using N [c] (Racine).

(8) We also computed mean absolute error (MAE) and mean absolute percentage error (MAPE) for each model and found similar results.

(9) First, we performed an F-test to determine whether the translog or the CD functional form (with time and stateeffects) was appropriate for the data. The test rejected the CD specification (the F value was found to be 18.5307) at the 1% level of significance. Second, we used the Hausman test to determine whether the fixed- or random-effects translog model was appropriate. The Hansman test rejected the random-effects specification. Based on these tests, we conclude that thefixed effects (both time and state) translog is the preferred parametric model for the data.

(10) Similar results are obtained for private capital.

(11) We also computed predicted MAE and MAPE and found similar results. Further, these results are robust to alternative subsamples of the data.

(12) To find the preferred parametric functional form, we first tested between the CD fixed- and random-effects specifications. The same test was then done for the translog model. The Hausman test favored the fixed-effects specification in both cases. Finally, we tested between the fixed-effects CD and translog specifications. The test favored the fixed-effects translog model over its CD counterpart.

(13) As with the previous data set, we performed several other diagnostic checks. We found that the results were similar to those found with the previous data set, and so we omit them for the sake of brevity. These are available from the authors upon request.

Daniel J. Henderson * and Subal C. Kumbhakar ([dagger])

* Department of Economics, State University of New York at Binghamton, Binghamton, NY 13902-6000, USA; E-mail [e-mail]; corresponding author.

([dagger]) Department of Economics, State University of New York at Binghamton, Binghamton, NY 13902-6000, USA; E-mail [e-mail].

Table 1. Estimates of Output Elasticity: Cobb-Douglas Estimates

                 [beta](KG)   [beta](KP)   [beta](L)   [beta](u)

Pooled OLS          0.155       0.309        0.594      -0.007
                    0.017#      0.012#       0.014#      0.001#
Fixed effects      -0.030       0.169        0.769      -0.004
                    0.026#      0.027#       0.027#      0.001#
Random effects      0.018       0.266        0.745      -0.005
                    0.023#      0.020#       0.023#      0.001#

The standard error for each estimate is given in italics. Controls
for state and time effects included in both fixed- and
random-effects models.

Note: Standard error for each estimate is indicated with #.

Table 2. Estimates of Marginal Products: Cobb-Douglas Estimates

                 [beta](KG)   [beta](KP)   [beta](L)   [beta](u)

Pooled OLS
  Mean             0.362        0.306       20.536     -410.808
                   0.047#       0.013#       0.717#      72.182#
  [q.sub.1]        0.312        0.243       18.626     -458.691
                   0.040#       0.010#       0.651#      80.555#
  [q.sub.2]        0.361        0.299       19.859     -269.231
                   0.047#       0.013#       0.682#      47.249
  [q.sub.3]        0.407        0.352       21.493     -111.111
                   0.053#       0.015#       0.748#      19.639#
Fixed effects
  Mean            -0.071        0.167       26.600     -257.547
                   0.049#       0.013#       0.721#      72.105#
  [q.sub.1]       -0.079        0.133       24.125     -287.566
                   0.054#       0.011#       0.653#      80.448#
  [q.sub.2]       -0.070        0.164       25.722     -168.789
                   0.049#       0.013#       0.697#      47.199
  [q.sub.3]       -0.061        0.192       27.840      -69.659
                   0.042#       0.015#       0.754#      19.662#
Random effects
  Mean             0.042        0.263       25.756     -279.170
                   0.048#       0.012#       0.747#      73.017#
  [q.sub.1]        0.036        0.209       23.360     -311.710
                   0.041#       0.009#       0.677#      81.487#
  [q.sub.2]        0.042        0.258       24.906     -182.960
                   0.048#       0.012#       0.724#      47.675
  [q.sub.3]        0.047        0.302       26.957      -75.507
                   0.053#       0.014#       0.781#      19.011#

The standard error for each estimate is given in italics. Controls
for state and time effects included in both fixed- and
random-effects models.

Note: The standard error for each estimate is indicated with #.

Table 3. Estimates of Output Elasticity: Translog Estimates

                 [beta](KG)   [beta](KP)    [beta](L)   [beta](u)

Pooled OLS
  Mean             0.136         0.256        0.667      -0.004
                   0.036#        0.029#       0.021#      0.002#
  [q.sub.1]        0.077         0.142        0.581      -0.009
                   0.033#        0.018#       0.046#      0.002#
  [q.sub.2]        0.146         0.216        0.689      -0.004
                   0.036#        0.015#       0.046#      0.002#
  [q.sub.3]        0.204         0.347        0.760       0.001
                   0.035#        0.030#       0.046#      0.003#
Fixed effects
  Mean            -0.095         0.153        0.872      -0.001
                   0.048#        0.021#       0.036#      0.002#
  [q.sub.1]       -0.199         0.099        0.785      -0.005
                   0.034#        0.033#       0.027#      0.003#
  [q.sub.2]       -0.107         0.155        0.897       0.000
                   0.064#        0.025#       0.022#      0.002#
  [q.sub.3]       -0.034         0.213        0.999       0.003
                   0.019#        0.031#       0.047#      0.003#
Random effects
  Mean            -0.051         0.267        0.831      -0.004
                   0.035#        0.036#       0.032#      0.002#
  [q.sub.1]       -0.143         0.219        0.731      -0.008
                   0.052#        0.017#       0.037#      0.004#
  [q.sub.2]       -0.066         0.259        0.851      -0.003
                   0.034#        0.029#       0.036#      0.002#
  [q.sub.3]        0.024         0.312        0.963       0.000
                   0.048#        0.029#       0.047#      0.002#

The standard error for each estimate is given in italics. Controls
for state and time effects included in both fixed- and
random-effects models.

Note: The standard error for each estimate is indicated with #.

Table 4. Estimates of Marginal Products: Translog Estimates

                 [beta](KG)   [beta](KP)   [beta](L)   [beta](u)

Pooled OLS
  Mean             0.330        0.208       22.525     -178.055
                   0.105#       0.015#       1.642#     233.510#
  [q.sub.1]        0.160        0.155       19.991     -379.282
                   0.068#       0.037#       1.485#     446.740#
  [q.sub.2]        0.334        0.213       22.469     -120.632
                   0.077#       0.029#       1.057#     207.150#
  [q.sub.3]        0.512        0.275       24.861       11.869
                   0.058#       0.018#       1.386#     137.410#
Fixed effects
  Mean            -0.266        0.145       29.715       30.495
                   0.137#       0.017#       0.787#     343.231#
  [q.sub.1]       -0.505        0.095       25.814     -119.740
                   0.185#       0.016#       1.498#     141.834#
  [q.sub.2]       -0.240        0.142       29.663        1.762
                   0.121#       0.017#       0.858#      94.052#
  [q.sub.3]       -0.070        0.185       33.298      184.631
                   0.038#       0.013#       0.988#     453.172#
Random effects
  Mean            -0.159        0.252       28.041     -184.084
                   0.086#       0.021#       1.316#     175.262#
  [q.sub.1]       -0.357        0.213       24.887     -245.170
                   0.124#       0.018#       0.927#     161.793#
  [q.sub.2]       -0.150        0.247       28.677     -103.164
                   0.075#       0.019#       1.162#     276.157#
  [q.sub.3]        0.049        0.280       31.857      -14.088
                   0.073#       0.020#       1.223#     263.010#

The standard error for each estimate is given in italics. Controls
for state and time effects included in both fixed- and
random-effects models.

Note: The standard error for each estimate is indicated with #.

Table 5. Estimates of Output Elasticity: Nonparametric Model

            [beta](KG)   [beta](KP)   [beta](L)   [beta](u)

Mean           0.106        0.310       0.641      -0.008
               0.034#       0.028#      0.045#      0.001#
[q.sub.1]      0.017        0.259       0.562      -0.013
               0.032#       0.025#      0.031#      0.003#
[q.sub.2]      0.123        0.304       0.641      -0.008
               0.037#       0.021#      0.045#      0.001#
[q.sub.3]      0.207        0.358       0.730      -0.003
               0.036#       0.029#      0.043#      0.002#

The standard error for each estimate is given in italics.
Controls for time and state effects are included in the model.

Note: Standard error for each estimate is indicated with #.

Table 6. Estimates of Marginal Products: Nonparametric Model

            [beta](KG)   [beta](KP)   [beta](L)   [beta](u)

Mean          0.251        0.306       21.870     -431.147
              0.041#       0.013#      0.717#      49.781#
[q.sub.1]     0.037        0.231       19.198     -555.573
              0.033#       0.015#      0.697#      83.880#
[q.sub.2]     0.279        0.298       22.018     -217.337
              0.041#       0.013#      0.640#      23.176#
[q.sub.3]     0.474        0.375       24.808     -76.497
              0.045#       0.016#      0.779#      36.754#

The standard error for each estimate is given in italics. Controls
for time and state effects are included in the model.

Note: Standard error for each estimate is indicated with #.

Table 7. Cobb-Douglas Estimates (Alternative Data Set)

                    [beta](KG)   [beta](KP)   [beta](L)

Elasticities
  Fixed effects       0.021        0.477        0.678
                      0.084#       0.056#       0.049#
Marginal products
  Mean                0.074        1.564        3.545
                      0.080#       0.102#       0.116#
  [q.sub.1]           0.050        1.356        2.728
                      0.056#       0.088#       0.089#
  [q.sub.2]           0.072        1.543        3.316
                      0.081#       0.100#       0.109#
  [q.sub.3]           0.096        1.720        4.073
                      0.107#       0.112#       0.134#

The standard error for each estimate is given in italics. Controls for
time and state effects are included in the model.

Note: Standard error for each estimate is indicated with #.

Table 8. Translog Estimates (Alternative Data Set)

                    [beta](KG)   [beta](KP)   [beta](L)

Elasticities
  Mean                -0.051       0.443        0.797
                       0.027#      0.041#       0.981#
  [q.sub.1]           -0.176       0.338        0.718
                       0.062#      0.066#       0.866#
  [q.sub.2]           -0.039       0.422        0.806
                       0.089#      0.041#       0.996#
  [q.sub.3]            0.060       0.517        0.889
                       0.038#      0.031#       0.982#
Marginal products
  Mean                -0.090       1.428        4.277
                       0.067#      0.131#       5.489#
  [q.sub.1]           -0.414       1.035        3.051
                       0.129#      0.130#       3.679#
  [q.sub.2]           -0.109       1.417        4.029
                       0.099#      0.094#       4.529#
  [q.sub.3]            0.267       1.699        4.950
                       0.077#      0.123#       6.212#

The standard error for each estimate is given in italics. Controls for
time and state effects are included in the model.

Note: Standard error for each estimate is indicated with #.

Table 9. Nonparametric Estimates (Alternative Data Set)

                    [beta](KG)   [beta](KP)   [beta](L)
Elasticities
  Mean                0.234        0.560        0.696
                      0.097#       0.107#       0.118#
  [q.sub.1]           0.023        0.303        0.292
                      0.073#       0.117#       0.157#
  [q.sub.2]           0.228        0.465        0.581
                      0.071#       0.152#       0.160#
  [q.sub.3]           0.452        0.800        0.920
                      0.133#       0.140#       0.164#
Marginal products
  Mean                0.862        2.911        2.159
                      0.350#       0.220#       0.281#
  [q.sub.1]           0.058        1.386        0.949
                      0.137#       0.721#       0.258#
  [q.sub.2]           0.852        2.551        1.877
                      0.302#       0.194#       0.335#
  [q.sub.3]           1.626        4.167        3.006
                      0.236#       0.280#       0.582#

The standard error for each estimate is given in italics. Controls for
time and state effects are included in the model.

Note: Standard error for each estimate is indicated with #.