Public and Private Capital Productivity Puzzle: A Nonparametric Approach

Transcription

1 Public and Private Capital Productivity Puzzle: A Nonparametric Approach Daniel J. Henderson and Subal C. Kumbhakar Department of Economics State University of New York at Binghamton June 1, 2005 Abstract Is public expenditure productive? Is there a shortfall or excess in public capital investment? We address these old issues in the light of new econometric tools. It is argued that the Cobb-Douglas specification that ignores non-linearity inherent in the functional relationship of the production technology causes incorrect estimates of input elasticities. To avoid possible model misspecification, we use Li-Racine Generalized Kernel Estimation. This procedure is used to estimate the returns to private capital, employment, and public capital in Gross State Product from a panel of 48 states for 17 years. In contrast to previous studies, we find that the return to public capital is positive and significantly different from zero. Keywords: Nonparametric, Productivity, Public Capital JEL Classification No.: C1,C4,H4,O4 Acknowledgements: 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 benefitted from participants of seminars at the State University of New York at Binghamton, the University of Calgary, and the State University of New York at Albany; and participants at the 2005 Winter Meetings of the Econometric Society, Philadelphia, PA. Finally, the authors would also like to thank Badi Baltagi and Catherine Morrison Paul for providing the data necessary for this project. Daniel J. Henderson, Department of Economics, State University of New York, Binghamton, NY 13902, (607) , Fax: (607) , djhender@binghamton.edu. Subal C. Kumbhakar, Department of Economics, State University of New York, Binghamton, NY 13902, (607) , Fax: (607) , kkar@binghamton.edu.

2 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 attributed 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 Baltagi and Pinnoi 1995, Garcia-Milà, McGuire and Porter 1996, Holtz-Eakin 1994, among others). Almost all the studies use a Cobb-Douglas (CD) production function (with the exception of Batina 2001, Cohen and Morrison Paul 2004, Lynde and Richmond 1992, Morrison and Schwartz 1996a, 1996b, and Nadiri and Mamuneas 1994 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 attributed to an inappropriate functional form (e.g., failure to take the non-linearity 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 size 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 1

3 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 concludes. 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 takes into account possible endogeneity of inputs explicitly 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 it = f(x kit,β)+ε it, (1) using state level panel data (48 contiguous states observed for the period ). 1 Output (y) is the gross state product for each state i in each time period t. As for the regressors (x), labor (L) is employment in non-agricultural 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 OLS, one allows the existence of fixed 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

4 state and time effects in the error term ε it and is written as ε it = µ i +γ t +v it,andfinally, we consider a random effects formulation in which the error components µ i and γ t are assumed to be random variables. The results based on the CD production function (linear in logs) y it = KX β k x kit + ε it, (2) k=1 are reported in Table 1. 2 The coefficients on KG (in the fixed and random effects models) arefoundbequitesmall( and respectively) and statistically insignificant. On the contrary, coefficients on private capital are found to be positive (0.169 and 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 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 Aschauer 2001, and Pinnoi 1994). However, much of the recent literature does not find this phenomenon. Since the marginal product (MP) 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 over-investment in public projects. More specifically, the MP for the random effects model of KP is 6 times bigger than that of KG. Thus, to justify the results, the price of KP has to be 6 times the price of KG. Because of the positive externality in KG one would expect that the effective price of KG is lower. Furthermore, the MP of KG is found to be negative in the fixed effects model (although not significantly different from zero), which indicates 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

5 over-investment 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). The idea of 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 and Racine and Li 2004) which will be used to estimate marginal products and elasticities of the inputs in our models. First, consider the nonparametric regression model y i = m(x i )+v i, i =1,...,NT (3) where m(x i ) is the unknown smooth production function with argument x i =[x c i,xu i,xo i ], x c i is a vector of continuous regressors (private capital, public capital, labor, and the unemployment rate), x u i is a vector of regressors that assume unordered discrete values (state effects), x o i is a vector of regressors that assume ordered discrete values (time effects), v is an additive error, N is the number of cross-sectional units, and T is the 4

6 number of periods in the sample (N =48, T =17). Taking a first-order Taylor expansion of m(x j ) at x j in (3) yields y i m(x j )+(x c i x c j)β(x j )+v i (4) where β(x j ) is defined as the partial derivative of m(x j ) with respect to x c.whenyand x are both expressed in logarithmic form, β(x j ) is interpreted as an elasticity. The estimator of δ(x j ) m(x j ) β(x j ) is given by b δ(xj ) = µ bm(xj ) = X bβ(x j ) i X µ Kb ³ 1 h x c i xc j i Kb h 1 ³ x c i xc j ³ x c i xc j ³ x c i xc j ³ x c i xc j 0 y i, (5) where Kb h = Π q b h 1 s w x c si xc r ³ sj s=1 Π l u x u hs s=1 si,xu sj, λ c ³ s pπ u l o x o s=1 si,xo sj, c λ o s. K h is the commonly used product kernel (see Pagan and Ullah 1999), where w is the standard normal kernel function with window width h s = h s (NT) associated with the s th component of x c. l u is a variation of Aitchison and Aitken s (1976) kernel function which equals one if x u si = xu sj and λu s otherwise, and l o is the Wang and Van Ryzin (1981) kernel function which equals one if x o si = xo sj and (λo s) xo si xo sj otherwise. 3 Estimation of the bandwidths (h, λ u,λ 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, choosing a very large bandwidth, we may include too many points and thus get 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 there exist many selection methods, we choose Hurvich et al. s (1998) Expected Kullback Leibler (AIC c ) criteria. This method chooses smoothing parameters using an improved version of a criterion based on the Akaike Information Criteria. AIC 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 3 See Hall, Racine and Li (2004), Li and Ouyang (2005), Li and Racine (2004, 2005) and Racine and Li (2004) for further details. 1 5

7 Cross-Validation. 4 Specifically, the bandwidths are chosen to minimize AIC c =log bσ tr(H)/N T 1 [tr(h)+2]/n T (6) where bσ 2 = = 1 XNT (y j bm j (x j )) 2 (7) NT j=1 µ 1 y 0 (I H) 0 (I H)y, (8) NT where bm j (x j ) is the commonly used leave-one-out estimator of m(x j ). 4 Results To accommodate the possible non-linearity parametrically (as well as obtain observation specific estimates)weestimatethemodelusingseveral versions of the translog production function 5 KX yit = β k x kit + 1 KX KX β 2 kl x kit x lit + ε it, (9) k=1 k=1 l=1 to which the CD model is the special case where β kl =0 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 private capital are insignificant at the mean and quartile values. Further, the results for the other three variables 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 fartoomuchvariationeventhoughtheypossesssmallsamplebias. Inanearlierversionofthispaperwe included an alternative approach to the AIC c and LSCV bandwidth selection criterion. Specifically we proposed the Robust Cross-Validation procedure (which gives similar results as AIC 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 bad ones are true outliers) which 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

8 are somewhat alarming. Large negative, although insignificant, 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 phenomenon seem economically unreasonable. The only specification for which the results look economically meaningful is the pooled OLS model. However, in addition to the fact that only the coefficients on labor are significant, 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 over-investment in public capital and smaller 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 Boarnet 1998, Cohen and Morrison Paul 2003, 2004 and Holtz-Eakin and Schwartz 1995). 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 2005). 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. 7

9 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 caused coefficients on public capital to be small (positive/negative) and statistically insignificant. Although the results above are striking, we feel it necessary to test for a known parametric specification. Here we employ the Hsiao-Li-Racine (2003) specification test for mixed categorical and continuous data. This test will help us determine if the parametric functional forms applied are acceptable. Here we are testing the null hypothesis that the parametric model is correctly specified (H o : P [E(y x) =f(x, β)] = 1) against the alternative that it is not (H 1 : P [E(y x) =f(x, β)] < 1). First we tested the null 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 that the model is translog (the preferred fixed effects model). Again we strongly reject the null hypothesis that the underlying model is translog (p-value= ). In addition to the Hsiao-Li-Racine test, we also performed several other diagnostic checks. First, we examined the MSE from each regression. The MSE from the nonparametric model (0.0003) was 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 compared 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 which do not depend on any specific model. Specifically, we computed the average product of 8 We also computed MAE and MAPE for each model and found similar results. 9 First, we performed a F-test to determine whether the translog or the CD functional form (with time and state effects) is appropriate for the data. The test rejected the CD specification at the 1% level of significance. Second, we used the Hausman test to determine whether the fixed or random-effects translog model is appropriate. The Hausman test rejected the random-effects specification. Based on these tests, we conclude that the fixed-effects (both time and state) translog is the preferred parametric model for the data. 8

10 public capital and estimated its kernel density and compared 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 As a final test we chose subsets of the data to obtain out-of-sample forecasts. We find that the nonparametric model has significantly lower predicted mean squared error than the parametric models for this particular data set. As an example, we estimated the model using the first sixteen years of data for each state and forecasted on the remaining year of data. In this particular scenario our predicted MSE for the nonparametric model (0.0010) was 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 tested our procedures on the data used in Cohen and 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 Second, the cross-sections are less aggregated and the focus is on the manufacturing industry. Finally, this data set does 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 sub-samples of the data. 9

11 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 it = f(x 1it,x 2it,x 3it,β)+ε it, (10) where y is the aggregate output in state i at time t, x 1it is the public infrastructure stock, x 2it is fixed (private) capital, and x 3it is labor (non-production and production). The results for the estimation of this data set are given in tables 7 through 9. The CD model (with fixedstateandtimeeffects), found to be preferred over the randomeffects 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) results 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 1, 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 that the two distributions are different from one another (p-value = ). The results using the translog model reinforce the importance of using the nonparametric technique. Whereas the shape of the density using the translog model was 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 = ), which firmly rejects the null hypothesis. 13 In summary, this section of the paper attempted to test the results of the previous 12 To find the preferred parametric functional form, we first tested between the CD fixed and random effects specifications. The same test is then done for the 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 F-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 find that the results are similar to those found with the previous data set and omit them for the sake of brevity. These are available from the authors upon request. 10

12 section for robustness. Here we used an alternative data set to test our hypothesis. It used U.S. manufacturing industry data from a more recent period. In addition, we did not include the unemployment rate in our models used in this section. Nonetheless, the results of this experiment produced results similar to the previous section. In both cases the preferred fixed effects translog model produced small and insignificant returns to public capital expenditure. However, similar to the previous case, the nonparametric model was 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 showed 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 non-linearity 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 ( ), (ii) the manufacturing industry using a panel of 48 states for 15 years ( ). 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 of the 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. In the very least, we suggest that this discussion should be reopened. 11

13 References [1] Aitchison, J., and C.B.B. Aitken (1976). Multivariate Binary Discrimination by Kernel Method, Biometrika 63, [2] Aschauer, D.A. (1989). Is Public Expenditure Productive? Journal of Monetary Economics 23, [3] Aschauer, D.A. (1990). Why Is Infrastructure Important? in A. H. Munnell, ed., Is There a Shortfall in Public Capital Investment? Boston, MA: Federal Reserve Bank of Boston, [4] Aschauer, D.A. (2001). Output and Employment Effects of Public Capital, Public Finance & Management 1, [5] Baltagi, B.H., and N. Pinnoi (1995). Public Capital Stock and State Productivity Growth: Further Evidence from an Error Components Model, Empirical Economics 20, [6] Batina, R.G. (2001). The Effects of Public Capital on the Economy, Public Finance and Management 1, [7] Boarnet, M.G. (1998). Spillovers and the Locational Effects of Public Infrastructure, Journal of Regional Science 38, [8] Cohen, J.P. and C.J. Morrison Paul (2003). Airport Infrastructure Spillovers in a Network System, Journal of Urban Economics 54, [9] Cohen, J, and C.J. Morrison Paul (2004). Public Infrastructure Investment, Interstate Spatial Spillovers and Manufacturing Costs, Review of Economics and Statistics 86, [10] Garcia-Milà, T., T.J. McGuire, and R.H. Porter (1996). The Effects of Public Capital in State-Level Production Functions Reconsidered, Review of Economics and Statistics 78, [11] Hall, P., J. Racine, and Q. Li (2004). Cross-Validation and the Estimation of Conditional Probability Densities, Journal of The American Statistical Association 99,

14 [12] Holtz-Eakin, D. (1994). Public-Sector Capital and the Productivity Puzzle, Review of Economics and Statistics 76, [13] Holtz-Eakin, D. and A.E. Schwartz (1995). Spatial Productivity Spillovers from Public Infrastructure: Evidence from State Highways, International Tax and Public Finance 2, [14] Hsiao, C., Q. Li., and J. Racine (2003). A Consistent Model Specification Test with Mixed Categorical and Continuous Data, manuscript, McMaster University. [15] Li, Q. (1996). Nonparametric Testing of Closeness between Two Unknown Distribution Functions, Econometric Reviews,15, [16] Li, Q., and D. Ouyang (2005). Uniform Convergence Rate of Kernel Estimation with Mixed Categorical and Continuous Data, Economics Letters, (forthcoming). [17] Li, Q., and J. Racine (2004). Cross-Validated Local Linear Nonparametric Regression, Statistica Sinica 14, [18] Li, Q., and J. Racine (2005). Nonparametric Econometrics: Theory and Practice, Princeton, Princeton University Press, (forthcoming). [19] Lynde, C. and J. Richmond (1992). The Role of Public Capital in Production, Review of Economics and Statistics 74, [20] Morrison, C.J. and A.E. Schwartz (1996a). Public Infrastructure, Private Input Demand, and Economic Performance in New England Manufacturing, Journal of Business and Economic Statistics 14, [21] Morrison, C.J. and A.E. Schwartz (1996b). State Infrastructure and Productive Performance, American Economic Review 86, [22] Munnell, A.H. (1990). How Does Public Infrastructure Affect Regional Economic Performance? New England Economic Review, [23] N c, Nonparametric software by Jeff Racine ( 13

15 [24] Nadiri, M.I. and T.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, [25] Pagan, A., and A. Ullah (1999). Nonparametric Econometrics, Cambridge, Cambridge University Press. [26] Pinnoi, N. (1994). Public Infrastructure and Private Production Measuring Relative Contributions, Journal of Economic Behavior and Organization 23, [27] Racine, J., and Q. Li (2004). Nonparametric Estimation of Regression Functions with Both Categorical and Continuous Data, Journal of Econometrics 119, [28] Rousseeuw, P.J., and B.C. van Zomeren (1990). Unmasking Multivariate Outliers and Leverage Points, Journal of the American Statistical Association 85, [29] Wang, M.C., and J. Van Ryzin (1981). A Class of Smooth Estimators for Discrete Estimation, Biometrika 68,

16 Table 1 - Estimates of Output Elasticity: Cobb-Douglas Estimates β(kg) β(kp) β(l) β(u) Pooled OLS Fixed Effects Random Effects Notes: The standard error for each estimate is given in italics. Controls for time effects included in both fixed and random effects models.

17 Table 2 - Estimates of Marginal Products: Cobb-Douglas β(kg) β(kp) β(l) β(u) Pooled OLS mean q q q Fixed Effects mean q q q Random Effects mean q q q Notes: See Table 1.

18 Table 3 - Estimates of Output Elasticity: Translog Estimates β(kg) β(kp) β(l) β(u) Pooled OLS mean q q q Fixed Effects mean q q q Random Effects mean q q q Notes: See Table 1.

19 Table 4 - Estimates of Marginal Products: Translog Estimates β(kg) β(kp) β(l) β(u) Pooled OLS mean q q q Fixed Effects mean q q q Random Effects mean q q q Notes: See Table 1.

20 Table 5 - Estimates of Output Elasticity: Nonparametric Model β(kg) β(kp) β(l) β(u) mean q q q Notes: The standard error for each estimate is given in italics. Controls for time and state effects are included in the model.

21 Table 6 - Estimates of Marginal Products: Nonparametric Model β(kg) β(kp) β(l) β(u) mean q q q Notes: See Table 5.

22 Table 7 -Cobb-Douglas Estimates (Alternative Data Set) β(kg) β(kp) β(l) Elasticities Fixed Effects Marginal Products mean q q q Notes: The standard error for each estimate is given in italics. Controls for time and state effects are included in the model.

23 Table 8 - Translog Estimates (Alternative Data Set) β(kg) β(kp) β(l) Elasticities mean q q q Marginal Products mean q q q Notes: See Table 7.

24 Table 9 - Nonparametric Estimates (Alternative Data Set) β(kg) β(kp) β(l) Elasticities mean q q q Marginal Products mean q q q Notes: The standard error for each estimate is given in parentheses. Controls for time and state effects are included in the model.

25

26