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 March 30, 2004 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 speci cation that ignores non-linearity inherent in the functional relationship of the production technology causes incorrect estimates of input elasticities. To avoid possible model misspeci cation, we use Li-Racine Gerneralized Kernel Estimation and propose the Robust Cross- Validation procedure in order to combat outliers a ecting estimated bandwidths. These procedures are 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 nd that the return to public capital is positive and signi cantly di erent from zero. Keywords : Nonparametric, Cross-Validation, Productivity, Public Capital JEL Classi cation No.: C1, C4, H4, O4 Acknowledgements: The authors would like to thank Christopher Hanes for useful comments on the subject matter of this paper. We would also like to thank participants of the Economics Department Seminar Series at the State University of New York at Binghamton. Contact information: 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 ßooded 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 misspeciþcation (viz., ignoring state and time effects, possible endogeneity of inputs, etc.). Although there is no consensus, the majority of empirical evidence supports the view that the marginal return from public capital is not signiþcantly 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, Holtz-Eakin 1994, Garcia-Milà, McGuire, and Porter 1996, among others). Almost all the studies used a Cobb-Douglas (CD) production function (with the exception of Lynde and Richmond 1992, Morrison and Schwartz 1996a, 1996b and Nadiri and Mamuneas 1994 who used ßexible cost function speciþcations) to estimate the productivity of inputs. Because of the functional form, productivity of each input (elasticity) 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 insigniþcant returns on public capital might be attributed to the choice of wrong functional form (viz., failure to take the non-linearity into account). To take the possible nonlinearity without imposing a functional form, we use a nonparametric approach. The main advantage of the nonparametric approach is that the returns from factor inputs are observation speciþc. In doingsowecancontrolforbothstateandtimeeffects. Our results show that the return of public capital is positive and statistically signiþcant. 1

3 2 The Productivity Puzzle Two competing approaches are usually used to measure productivity. In the primal approach, a parametric production function (e.g., a Cobb-Douglas) is mostly estimated. This approach is most 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 the endogeneity of inputs explicitly into the analysis. However, this 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 estimate an aggregate production function using the state level panel data (48 contiguous states observed for the period ). Output (y) are the gross state products, 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 variable can be found in Munnell (1990) as well as in Baltagi and Pinnoi (1995). Following Baltagi and Pinnoi (1995) we used the unemployment rate (u) to control for business cycle effects. The results based on the CD production function (after controlling for state and time effects) are exactly the same as in Baltagi and Pinnoi (1995) and are not reported here. The coefficients on KG (in the Þxed and random effects models) are found be quite small (-0.03 and in the Þxed and random effects models, respectively) and statistically insigniþcant. On the contrary, coefficientsonprivate capital arefoundtobe positive (0.29 and 0.31) and statistically signiþcant. 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 evaluated at the mean of the data. These estimates (associated with the random effects model) for KP and KG are found to be and 0.009, respectively. If one views this as a problem of allocation of funds between private and public capital, returns from a 2

4 dollar from public and private investment should be the same. Since the marginal product (MP) of KG is much less than that of KP (both are measured in 1982 year dollars), there must be some explanations for such a massive over-investment in public projects. The MP of KG is found to be negative in some studies (although not signiþcantly different from zero), which indicates over-investment in public capital even if it is assumed to be costless. One explanation offered by Eakin-Holtz (1994) is that... government capital budgeting decisions focused at best on the consumption beneþts accruing from public goods and services, and at worst on the pork-barrel punch they carried (p. 12). While thismightbetruetosomeextent,onecanalwaysaskwhethertheresultsfromthe CD model (linear in logs) 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 misspeciþcation problem, we use the Li-Racine Generalized Kernel Estimation procedure (see the appendix for details) and estimate the returns to private capital, employment, and public capital while controlling for the state and time effects. 3 Generalized Kernel Estimation We use Li-Racine Generalized Kernel Estimation (see Li and Racine 2003 and Racine and Li 2004) to estimate the production function, which is written as y i = m(x i )+ε i, i =1, 2,..., NT, where m is the unknown smooth production function. The arguments of m( ) are x i = (x c i,xu i,xo i ), where xc i is a continuous random vector of dimension q, xu i is a r 1 vector of regressors that assume unordered discrete values (e.g., individuals), and x o i is a p 1 vector of regressors that assume ordered discrete values (e.g., time). By taking a Þrstorder Taylor expansion of (the above model) with respect to x j we obtain y i m(x j )+(x c i x c j)β(x j )+ε i, where β(x) is deþned to be the partial derivative of m(x) with respect to x c. We then obtain a leave-one-out local linear kernel estimator of δ(x) m(x) β(x) using the formula given in the appendix. Since selection of the window width is the most important factor in obtaining nonparametric estimates, we discuss it at some length. Many methods have been developed 3

5 to select the most appropriate bandwidth. One of the most popular methods is that of Least-Squares Cross-Validation. The bandwidths are chosen using the popular leaveone-out estimator by minimizing the Least-Squares Cross-Validation function. As often is the case, these methods depend on particular regularity conditions holding. One of those conditions is that outliers are not present 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. To address this problem, we propose the Robust Cross-Validation procedure. 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 Þnd the outliers. In a regression with one input, we can simply look at the scatter plot. However, in most regressions, we have multiple inputs. Rousseeuw and Leroy (1987) argue that it is not appropriate to look at each variable separately or even at all plots of the variables. Typically, we look at the least squares residuals to determine outliers. This popular method, as shown by Rousseeuw and Leroy (1987), can sometimes cause the researcher to throw out good observations while leaving in the outliers. Instead, we suggest using the Robust Distances method of Rousseeuw and van Zomeren (1990). This procedure amounts to encircling the data in a q +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. The Robust Distances estimate for each observation is RD i =[(x T (x))c(x) 1 (x T(x)) 0 ] 1/2 where T (x) and C(x) are robust estimates of the mean values of the x variables and the covariance matrix respectively. The robust estimates are calculated according to the Fast Minimum Covariance Determinant (FAST-MCD) estimator outlined in Rousseeuw and Van Driessen (1999). FAST-MCD is a computationally faster procedure than its 4

6 predecessor, the Minimum Covariance Determinant estimator. This estimator (shown to be asymptotically normal) searches for a subset containing half of the data, for which the covariance matrix has the smallest determinant. 4 Results A necessary Þrst step in any nonparametric application is to test for a known parametric speciþcation. Here we employ the Hsiao-Li-Racine (2003) SpeciÞcation Test for Mixed Categorical and Continuous Data. This test will help us determine if the CD functional form is acceptable. The null hypothesis is that the parametric model is correctly speciþed (H o : P [E(y x) =x 0 β]=1) against the alternative that it is not (H 1 : P [E(y x) =x 0 β] < 1). The test statistic, which is based on I def = E [ue(u x)f(x)] where u = y x 0 β, approaches (in distribution) the standard normal under the null. The computed test statistic for our data is with a p-value of Thus,weÞrmly reject the null hypothesis that the underlying model is CD. Given that bandwidth selection is an important issue in nonparametric regression, we estimate the production function with bandwidths chosen from three different methods. First, following many applied researchers, we set the scale factors c = 1.06 (note that h = cσ x NT 1 4+q, where σx is the standard deviation of a particular regressor). This is regarded as the benchmark in density estimation (although it has no direct bearing on nonparametric regressions). Second, we choose bandwidths from the Least-Squares Cross-Validation procedure. Finally, we use the Robust Cross-Validation procedure (to minimizes the impact of outliers) in selecting bandwidths. When simply setting c =1.06 we Þnd that our results look reasonable. We Þnd that the returns to each of the variables (on average) are similar to previous studies (excluding the return to aggregated public capital). However, getting economically reasonable results are not sufficient. Is this a case of a lucky shot in the dark, or well behaved biased estimates? Being that this scale factor is based on density estimation and not on regressions, we need to see if these results are robust to alternative methods. Cross-Validation procedures have been highly useful in applied nonparametric econometrics. The problem is that if outliers are present in an otherwise well behaved data generating process, then the Cross-Validation procedure will undersmooth the regression coefficients. This happens to our data when we employ the Naive Cross-Validation 5

7 procedure. The presence of a number of outliers give estimates of input elasticities (the betahats) that are economically unreasonable. We believe that some of these strange elasticity estimates are due to the presence of outliers in the data. Because of this, we employ the Robust Cross-Validation technique and hope for a reduction in the variance of the estimated elasticities. As expected, now the estimated elasticities lie within a reasonable range. The results are reported in Table 1 where we report the mean and the quartile values of estimated elasticities of private and public capital, labor and the unemployment rate. We Þnd that mean (median) returns to public capital (0.16 and 0.17) are positive and statistically different from zero. 1 The same holds for the quartile 1, quartile 2 and quartile 3 values. Although these elasticities are smaller than those of private capital, the difference is not all that high. Furthermore, variability of the input elasticities (KG, KP, andl), measured by inter-quartile range are much smaller (0.22, 0.10, and 0.18, respectively), compared to the other two cases. It is worth mentioning here that the MP of KG evaluated at the mean is 0.37 which is higher than that of KP (0.27). This clearly shows that the MP of KG is not only positive but for some states the MP of KG is higher than that of KP thereby suggesting a possible under-investment in public capital. 2 To put our results in perspective with the other studies using either the same or similar data, we summarize some of them in Table 2. It is surprising that our results are in line with Munnell (1990) and Eisner (1991) who were criticized for not taking state and time effects into accounts in their regressions. We Þnd similar results (by using nonparametric regression that captures nonlinearity in the functional form) even after controlling for state and time effects. Based on these Þndings we come to the conclusion that the CD model (which assumes a constant input elasticity for all data points) is too simple and failed to capture the nonlinearities inherent in the model. Thus, we conclude that model misspeciþcation caused coefficients on public capital to be small (positive/negative) and statistically insigniþcant. 3 1 Results for the disaggregated model are similar and are available from the authors upon request. 2 It should be noted that this appears to rise because of the ratio of y/k relative to that of y/g (e.g. MP(G) = β(g) y/g). This switch does not occur in the parametric version because the estimated coefficient associated with public capital is so small. 3 Finally, we chose subsets of the data to obtain out-of-sample forecasts. We Þnd that the nonparametric model has signiþcantly lower predicted mean squarred error than the Cobb-Douglas model for this particular data set. 6

8 5 Conclusion In this paper we showed that the popular Cobb-Dougals (CD) speciþcation of the production function (linear in log) is not supported by the state level panel data that are used to estimate returns on public and private capital. The CD speciþcation ignores non-linearity in the functional relationship underlying the production technology. Consequently, the CD model is likely to give incorrect estimates of returns to inputs. To avoid model misspeciþcation, we estimate the production technology using the Li-Racine Gerneralized Kernel Estimation technique. Furthermore, we propose the Robust Cross- Validation procedure in order to combat outliers affecting estimated bandwidths. These procedures are used to estimate the returns to private capital, employment, and public capital in Gross State Product using a panel of 48 states for 17 years. We Þnd that the return to public capital is positive and signiþcantly different from zero, even after controlling for state and time effects. Based on these Þndings, we come to the conclusion that the CD model (which assumes a constant input elasticity for all data points) used in many of the previous studies is too simple and failed to capture the nonlinearities inherent in the production function. Thus, we feel that model misspeciþcation is what caused the insigniþcant coefficients on public capital. We are not suggesting, however, that this is the end to the story. We have simply shown that past econometric evidence is not sufficient to condem the idea that public capital is productive. In the very least, we suggest that this discussion should be reopened. 7

9 Table 1 - Estimates of Output Elasticity 4 KG KP L u mean (0.05) (0.05) (0.06) (0.007) q (0.05) (0.06) (0.07) (0.003) q (0.06) (0.05) (0.07) (0.006) q (0.07) (0.05) (0.05) (0.008) Table 2 - Estimates of the Output Elasticity of Private and Public Capital Author KP KG Munnell (1990) Eisner (1991) Holz-Eakin (1994) Baltagi and Pinnoi (1995) Henderson and Kumbhakar The standard error for each estimate is given in parentheses. 8

10 References [1] Aschauer, D. A. (1989). Is Public Expenditure Productive? Journal of Monetary Economics 23, [2] 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, [3] Aitchison, J., and C.B.B. Aitken (1976). Multivariate Binary Discrimination by Kernel Method, Biometrika 63, [4] Baltagi, B. H., and N. Pinnoi (1995). Public Capital Stock and State Productivity Growth: Further Evidence from an Error Components Model, Empirical Economics 20, [5] Eisner, R. (1991). Infrastructure and Regional Economic Performance: Comment, New England Economic Review, [6] Garcia-Milà, T., T. J. McGuire, and R. H. Porter (1996). The Effects of Public Capital in State-Level Production Functions Reconsidered, The Review of Economics and Statistics 78, [7] Holtz-Eakin, D. (1994). Public-Sector Capital and the Productivity Puzzle, The Review of Economics and Statistics 76, [8] Hsiao, C., Q. Li., and J. Racine (2003). A Consistent Model SpeciÞcation Test with Mixed Categorical and Continuous Data, manuscript, Syracuse University. [9] Li, Q., and J. Racine (2003). Cross-Validated Local Linear Nonparametric Regression, Statistica Sinica, forthcoming. [10] Lynde, C. and J. Richmond (1992). The Role of Public Capital in Production, The Review of Economics and Statistics 74, [11] 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,

11 [12] Morrison, C. J. and A. E. Schwartz (1996b). State Infrastructure and Productive Performance, The American Economic Review 86, [13] Munnell, A. H. (1990). How Does Public Infrastructure Affect Regional Economic Performance? New England Economic Review, [14] N c, Nonparametric software by Jeff Racine ( [15] 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, The Review of Economics and Statistics 76, [16] Pagan, A., and A. Ullah (1999). Nonparametric Econometrics, Cambridge, Cambridge University Press. [17] Racine, J., and Q. Li (2004). Nonparametric Estimation of Regression Functions with Both Categorical and Continuous Data, Journal of Econometrics, 119, [18] Rousseeuw, P. J., and A. M. Leroy (1987). Robust Regression and Outlier Detection, New York, Wiley. [19] Rousseeuw, P. J., and K. Van Driessen (1999). A Fast Algorithm for the Minimum Covariance Determinant Estimator, Technometrics 41, [20] Rousseeuw, P. J., and B. C. van Zomeren (1990). Unmasking Multivariate Outliers and Leverage Points, Journal of the American Statistical Association 85, [21] Wang, M.C., and J. Van Ryzin (1981). A Class of Smooth Estimators for Discrete Estimation, Biometrika 68,

12 Appendix: Li-Racine Generalized Kernel Estimation In this section we describe Li-Racine Generalized Kernel Estimation (see Li and Racine 2003 and Racine and Li 2004) which will be used in order to estimate the returns to our four inputs. First, consider the nonparametric regression model y i = m(x i )+ε i, i =1, 2,..., NT, where m is an unknown smooth function. x i =(x c i,xu i,xo i ), where xc i is a continuous random vector of dimension q, x u i is a r 1 vector of regressors that assume unordered discrete values (e.g. individuals), and x o i is a p 1 vector of regressors that assume ordered discrete values (e.g. time). Next, by taking the Þrst order Taylor expansion of (the above model) with respect to x j we obtain y i m(x j )+(x c i x c j)β(x j )+ε i, where β(x) is deþned to be the partial derivative of m(x) with respect to x c. A leaveone-out local linear kernel estimator of δ(x) m(x) β(x) is given by b δ j (x j ) = µ bm j (x j ) = X bβ j (x j ) i6=j X µ K ³ 1 h i6=j K h y i, 5 1 ³ ³ ³ ³ where K h = Π q h 1 s w x c si xc r sj s=1 h s Π l ³x u u s=1 si,xu sj,λu s pπ l ³x o o s=1 si,xo sj,λo s. K h is the commonly used product kernel function (see Pagan and Ullah 1999) where w is the standard normal kernel function with window width h s = h s (NT) associated with the sth component of x c. l u is a variation of Aitchison and Aitken s (1976) kernel function which takes the value 1 if x u si = xu sj and λu s otherwise, and l o is the Wang and Van Ryzin (1981) kernel function which takes the value 1 if x o si = xo sj and (λo s) xo si xo sj otherwise. Next, we use the leave-one-out estimator in order to choose (h, λ u,λ o ) which minimize the least-squares cross-validation function given by CV (h, λ u,λ o )= 1 XNT [y j bm j (x j )] 2. NT j=

13 Finally, we use ( b h, λ cu, c λ o ) (the values that minimize the cross-validation function) to estimate δ(x) where µ b bm(x) δ(x) = = X bβ(x) i X µ Kb ³ 1 h i Kb h y i, 1 ³ ³ ³ ³ where Kb h = Π q b h 1 s w x c si xc r ³ sj Π l u x u s=1 ĥ s s=1 si,xu sj, λ c ³ s pπ u l o x o s=1 si,xo sj, c λ o s