For more than a quarter century since the early 1970s, the countries in East Asia grew at phenomenal rates, leading observers to dub the period the “Asian Miracle.” The rapid growth came to an abrupt end when the financial crisis hit in 1997, with many of the high-performing countries in the region falling into painful recessions and facing the distinct possibility that the miracle, if indeed there had been one, was over.
Clearly the Asian financial crisis was an unprecedented event and was unforeseen by virtually everyone. But what is more troubling is that a decline in growth in East Asian countries, even abstracting from the effects of the financial crisis, was already being predicted by some. Paul Krugman (1994) in particular, using the results of Young (1992, 1995), argued persuasively that the rapid growth of the East Asian economies over the past three decades had come primarily from capital accumulation, increasing labor force participation, and improving labor quality, rather than from improvements in productivity. As such, the rate of growth of these countries was bound to slow down eventually. Even though Krugman did not see the Asian financial crisis coming, he certainly saw an end to the so-called Asian miracle—presumably the crisis only made the end come faster.
The Young (1992) and Krugman (1994) papers set off a heated debate on the sources of economic growth in East Asia.1 One group, subscribing to the “accumulation view,” claimed that growth in East Asian countries was mainly driven by high rates of capital formation.2 The second group adhered to the “assimilation view,” arguing that the essential component of East Asian high growth was the acquisition and mastery of foreign technology.3 In other words, high growth resulted largely, although not exclusively, from gains in efficiency and productivity.
Whether the accumulation or assimilation view of growth is a more accurate characterization of the East Asian miracle has important implications for growth strategies. If the accumulation view is correct and growth is mainly based on capital formation, it will not be sustainable for long because the law of diminishing returns (to capital) will eventually prevail. As Krugman (1994) puts it rather dramatically, the East Asian economies with their high rates of investment would end up looking like the former Soviet Union! Following this logic, the future looks quite bleak for the East Asian countries even when they recover from the fallout of the financial crisis; growth rates in the future will be permanently below those experienced in earlier years. The practical implication for growth-enhancing strategies under the accumulation view is that to improve living standards requires investment, which has to be paid for in large part through foregone consumption.
On the other hand, the assimilation view would point to a more optimistic outcome. The proponents of this view would argue that, following the downturn resulting from the financial crisis, East Asian countries can get back to their precrisis long-run growth paths. And, if growth indeed originates from a narrowing of the “idea gap” as the assimilation view claims, no significant opportunity costs need to be incurred to incorporate ideas from abroad (Romer, 1993). Instead, ideas can be transmitted to the mutual benefit of producers and no sacrifice of current consumption for future growth is required.
Both groups can point to empirical evidence for a variety of countries that supports their respective cases.4 Most of the studies associated with the accumulation view use time-series data and follow the conventional growth-accounting method based on the Solow (1957) model. This growth-accounting method relies on the assumption of competitive factor markets, enabling one to replace output elasticities of capital and labor with the respective income shares of these factors. While the use of income shares may well be a reasonable approximation in industrial countries,5 this procedure is more questionable for developing countries, including the East Asian countries, where the capital and labor markets are unlikely to be perfectly competitive. The assimilation view on the other hand is generally supported by cross-country empirical growth analysis where the values of the output elasticities of capital and labor are estimated rather than imposed. These estimated elasticities are then used to calculate productivity changes. To do the cross-country regression analysis, however, requires assuming a particular form for the underlying aggregate production function, which may or may not be valid. Indeed, as Hulten (2000) points out, the original growth-accounting formulation due to Solow (1957) is completely nonparametric, and thus assuming any particular form for the production function is basically incorrect.
This paper proposes a new method of estimating the sources of economic growth and the growth of total factor productivity (TFP) using nonparametric derivative estimation techniques. This method requires no specific assumptions on the competitive state of factor markets or the form of the underlying aggregate production function. Applying this methodology to East Asian countries over the period 1960–95 yields estimates of output elasticities with respect to capital and labor, as well as TFP growth.6 Two main results emerged from the analysis. First, the estimated output elasticities of capital and labor tend to be quite different from their respective income shares, casting some doubt on the conventional growth-accounting model assumption of competitive factor markets. Second, the growth rates of TFP turn out in many cases to be similar to those obtained in other studies, yet in certain important cases are much higher, lending some support to the assimilation view of sources of economic growth.
I. Estimating Sources of Economic Growth
In the context of the neoclassical growth model, we start with an aggregate production function, which typically is specified as
where Y is output, K and L are capital and labor inputs, and t indicates time. The aggregate production function approach is an analytical simplification that makes it possible to summarize detailed information about the complex process of economic growth within a simple unified framework (for a review of the neoclassical growth model, see, for example, Barro and Sala-i-Martin, 1995). Differentiating the logarithm of equation (1) with respect to t, we obtain
Function (1) is often specified more explicitly in Hicks neutral form
where A(t) is called total factor productivity or TFP and measures the shift in the production function
The last term on the right-hand side of equation (2) is interpreted in equation (4) as the growth rate of TFP.7 Since equation (3) is the form typically assumed in the literature, we base our discussion on this specification throughout this paper. Equation (4) can be written as
where εK and εL stand for the elasticity of output with respect to capital and labor, respectively. Since the growth rates of GDP, capital, and labor are available in the national income accounts data of most countries, TFP growth rates are obtained by subtracting from GDP growth the sum of the growth rates of capital and labor with appropriate weights εK and εL. An obvious problem with this procedure is that εK and εL are unknown parameters depending on the functional form of F(K, L) and it is these parameters that are critical in calculating TFP growth.
The question then is how to estimate εK and εL. There are two approaches developed in the literature. The first approach assumes that the factor markets are perfectly competitive so that the necessary equilibrium conditions are given by equalities between the income shares of capital and labor in GDP (νK and νL) and the elasticities of output. The rental price of capital, r, and the wage rate, w, are then given by r = A·∂F/∂K and w = A·∂F/∂L so that εK ≡ (∂F/∂K)(K/F) = rK/Y = vK and εL ≡ (∂F/∂L)(L/F) = wL/Y = νL. In other words, εK and εL are equal to the income share of each factor (νK and νL). Under constant to returns to scale, νK + νL = εK + εL = 1. Thus, with this replacement, the growth rate of TFP may be calculated by simple subtraction. The result is what is known as the “Solow residual.”8 The second approach assumes a particular parametric form of equation (1) and estimates the production function by running a regression either in level or difference form. The output elasticities are constructed using the parameter estimates and TFP growth is again calculated as a residual.
Neither assumption, however, is particularly attractive when dealing with developing economies. For one thing, capital and labor markets in these economies are likely to be far from perfectly competitive. Furthermore, there is no guarantee that any particular functional form of the production function is appropriate for these economies (see Hulten, 2000). The simplest form used has been the Cobb-Douglas function, which involves estimating a single parameter and then using the constant returns to scale assumption, calculating the other elasticity.9 The parametric form that became popular in the 1970s is the translog function,10 which essentially attempts to estimate the second-order Taylor approximation of general function (3). However, a straightforward application of a translog production function often results in severe collinearity problems when using time-series data. Kim and Lau (1994) apply a common translog form to all the East Asian countries they studied, with some parameter variations allowed for each country.11 Furthermore, as pointed out by White (1980), least squares does not provide a proper approximation to the unknown function, and hence, the resulting estimates are often misleading.12
This paper proposes a third approach that has not been utilized in this context. For this approach, we do not need the assumption of perfectly competitive factor markets. Nor do we need to assume any particular functional form of the aggregate production function. All that is needed is some kind of smoothness of the production function. The proposed approach is based on nonparametric kernel derivative estimation techniques developed recently in the statistics and econometrics literature (see Härdle, 1990; and Pagan and Ullah, 1999). The logarithmic transformation of equations (1) and (3) with the addition of a stochastic term is
where a(t) ≡ lnA(t) is an unknown function of t, F*(x1, x2) = ln F (ex1, ex2) and u(t) is an error term satisfying E[u(t) | ln K(t), lnL(t), t] = 0. The idea behind the estimation procedure is as follows. Note that output elasticities εK and εL are simply the partial derivatives of the systematic part of equation (5) with respect to the first two arguments. Hence, application of nonparametric derivative estimation techniques yields the estimates
The nonparametric regression method has been usefully applied in many areas of economics.13 In particular, a semi-parametric regression model, in which the function is partly parametric (usually linear) and partly nonparametric, has been implemented by Engle, Granger, Rice, and Weiss (1986), Stock (1989), and Iwata, Murao, and Wang (2000), among others. Another popular variant is the nonparametric estimation of derivatives of a regression function. Examples of nonparametric estimation of derivatives include Rilstone (1988), who applied the techniques to examine the properties of a production function, and Rilstone (1991) to estimate average uncompensated price elasticities. Also, Lewbel (1993) provides nonparametric estimates of average compensated price elasticities, while Lewbel (1995) nonparametrically tests demand constraints and compares them with those yielded by the standard parametric test, assuming that the demand system has the quadratic, almost ideal, form. But, as far as we are aware, the nonparametric derivatives method has not been utilized to calculate the sources of growth or TFP growth.
II. Estimation Method
In order to estimate output elasticities εK and εL, first note that the conditional expectation of ln Y(t) in equation (5) is given by
which we call the mean function. This additive separability form imposes structure on the conditional mean, which can improve the efficiency of estimation of derivatives. Note that
are the partial derivatives of the mean function m(x1, x2, x3) with respect to the first two arguments, or the slopes of the regression curve in equation (5).
We now introduce the nonparametric estimation approach. The idea of the nonparametric regression is simply local averaging, that is, averaging the y values of observations having predictor values x = (x1, x2, x3) close to a target value. As one includes more distant observations for averaging, the resulting curve would be smoother and smoother until all observations are included for averaging, in which case the curve would be a straight line. This is the case of a linear regression. On the other hand, if only the closest observations are averaged, the resulting curve would become less smooth. How smooth the function should be is controlled by the parameter called bandwidth (h).
Below we essentially focus on estimation of the average TFP growth performance over the whole sample period. This focus is reasonable for two reasons. First, unlike the point estimation, the average estimator can attain the same speed of convergence as when the parametric model is used. Second, comparison with the previous estimates using the conventional method is straightforward.
To estimate the average TFP growth, we use three different methods. The first approach is to estimate the TFP growth rate at each observation point and then average them over the whole period. The second approach is to estimate directly the average derivatives and then get the corresponding TFP growth. The third approach is an extension of the first, assuming some kind of trend in the derivatives.14 We outline each in turn.
To see how to estimate the derivatives in the first method, it is helpful to consider nonparametric estimation of the mean function F* first.15 Let K(z) be the kernel function satisfying ∫K(z)dz = 1. The well-known Nadaraya-Watson kernel regression estimator of m(x) is given by
which is a type of estimator discussed by Chen and others (1996). Although this estimator itself is not operational because a(t) is unknown, the derivatives of F*, like εK and εL, are estimable in the following manner. Vinoid and Ullah (1988) show how an estimate of the derivative of F* is obtained by analytically differentiating
Kj(zi) = ∂K(zi)/∂zji and zji is the j-th element of zi for j = 1,2. This gives a point-wise derivative at each observation point. The growth rate of TFP is then calculated as a residual after substituting
by its sample counterpart
Then the average TFP growth is obtained from the Solow formula, using the average growth rates of output, capital, and labor.
The third approach is based on the idea of locally weighted regression. Basically the regular kernel estimator assumes that the conditional expectation is a constant in a neighborhood of the estimation point and this justifies the averaging. But we might expect that the function would be better approximated with a lower-order polynomial around that point. Instead of using a linear or quadratic relation as a global approximation (which yields a Cobb-Douglas and Translog function, respectively), the method uses them locally. It then reduces to a so-called local linear or quadratic estimator (Fan, 1992; Härdle and Linton, 2000).
In implementing the above procedures, we need to choose a kernel function and the bandwidth. The accuracy of kernel smoothers as estimators of m (the mean function) or derivatives of m is a function of the kernel K and the bandwidth h. In practice, the accuracy depends mainly on the bandwidth h (Härdle, 1990). Appendix III describes how the selection is made based on the cross-validation method.
The large-sample properties of the above estimators are well established. When the sample size is small, however, a large-sample approximation may be misleading. To find out how reliable the estimators are in a situation like ours, we conducted a small Monte Carlo experiment with the constant elasticity of substitution (CES) production function, which is reported in Appendix IV.18 The result indicates that even with a sample size as small as ours, the nonparametric derivative estimates look superior, in terms of bias, to the parametric estimates yielded by the Cobb-Douglas and Translog models. On the other hand (as expected), the variances of the nonparametric estimators are found to be larger than the parametric ones. The standard errors reported in the tables in the next section, therefore, should be interpreted with a degree of caution.19
III. Estimation Results
Using the procedures outlined in the previous section, we estimate TFP growth rates for nine East Asian countries: Hong Kong SAR, Indonesia, Korea, Malaysia, Philippines, Singapore, Taiwan Province of China, Thailand, and China. We use two data sets for this study: the data set based on the Penn World tables covering the period 1960–1990 (see Summers and Heston, 1991), and the data set constructed by Collins and Bosworth (1996), which covers the period 1960–1995. Each of the data sets is described in Appendix I. The data for Hong Kong SAR are only available in the Penn data set.
Before presenting our nonparametric estimates of TFP growth rates, we summarize the results obtained with the conventional methods in Table 1. Because the results using two data sets are similar, only the estimates with the Collins data set are reported here. The results from the Penn data set, as well as the estimates for subperiods—1960s, 1970s, 1980s, and 1990s—are available upon request. Table 1 reports the conventional growth-accounting estimates of TFP growth rates for the nine East Asian countries. The estimates in the column labeled “Young” are reported for Hong Kong SAR, Korea, Singapore, and Taiwan Province of China, using weighted labor shares calculated by Young (1995). In the column labeled “Collins” are the TFP growth estimates based on constant labor and capital shares equal to 0.65 and 0.35, respectively, which serve as a benchmark.20 As can be seen from Table 1, differences between the Young and the Collins estimates are quite small in three of the countries that are common to both studies. The exception is Singapore, where the difference is substantial. This result follows directly from the relatively small income share estimate of labor used by Young.21
|TFP Growth Estimate|
|Growth (in percent)|
|Hong Kong SAR||4.1||4.1||0.63||7.7||5.2||2.6|
|Taiwan Province of China||3.4||2.6||0.74||8.3||11.4||2.7|
We now present the results of nonparametric estimation. Among the three estimation methods, the results of the first two (the average of pointwise derivative estimates and the average derivative estimates) are reported in Tables 2 and 3, respectively. The third method (local linear and quadratic estimation) fails to yield reasonable point estimates in terms of the stability of elasticities over time (although their averages turn out quite close to the results of the first method). As such, we do not report these last results here. The reason for the failure is unclear, although it is possible that the local linear and quadratic estimation is sometimes sensitive to the size of the sample.
Table 2 presents nonparametric TFP growth estimates averaged over the whole time period,22 together with estimated output elasticities for nine East Asian countries.23 There is no important difference between the nonparametric estimates using the different data sets (not reported in the table). However, the nonparametric estimation methods yield quite different estimates of the TFP growth rate, as compared to the conventional methods. Except for the cases of Hong Kong SAR and China, our estimates of TFP growth over the period 1960–1995 are higher than the estimates obtained using the “Young” and “Collins” methods. Table 2 also reports the estimated output elasticities of capital and labor. All the estimates are highly significant based on the values of their asymptotic standard errors.
|TFP Growth||Elasticity||Estimates||Growth (in percent)|
|Hong Kong SAR||3.4||0.41||0.71||7.7||5.2||2.6|
|Taiwan Province of China||3.8||0.19||0.76||8.3||11.4||2.7|
|TFP Growth||Elasticity||Estimates||Growth (in percent)|
|Hong Kong SAR||3.7||0.50||0.52||7.7||5.2||2.6|
|Taiwan Province of China||4.1||0.23||0.59||8.3||11.4||2.7|
Table 3 presents nonparametric TFP growth estimates based on the average derivative method. The estimated TFP growth rates are a little larger (5–8 percent larger in most cases) than those estimated with the pointwise derivative method reported in Table 2, but the standard errors are almost twice as large as the first method. Therefore, in what follows, we focus on the results of the first method only, namely averaging local estimates of the derivatives, and refer to them simply as nonparametric estimates.
Table 4 provides a comparison of the conventional estimates and the nonparametric estimates for four East Asian countries: Hong Kong SAR, Korea, Singapore, and Taiwan Province of China. We find that the nonparametric estimates and the conventional estimates using Young’s weighted labor shares (“Nonparametric” and “Young” in Table 4) are quite similar to each other for Hong Kong SAR, Korea, and Taiwan Province of China, but very different for Singapore. These results indicate that the Young and Collins-Bosworth estimates are validated, except for Singapore. But TFP growth in Singapore turns out to be much higher than would be the case using the Young and Collins-Bosworth methods. The estimate here results from the fact that the labor elasticity (0.63) is higher than that used by Young and is in fact close to that of Collins and Bosworth (1996). However, the estimated capital elasticity (0.17) is considerably smaller than the income share of capital (0.35) assumed by Collins and Bosworth.
|TFP Growth Estimate (in percent)|
|Country||Young||Collins||Nonparametric||Elasticity||Income share||Elasticity||Income share|
|Hong Kong SAR||4.1||4.1||3.4||0.41||0.37||0.71||0.63|
|Taiwan Province of China||3.8||2.1||3.8||0.19||0.26||0.76||0.74|
A comparison of factor elasticities and factor shares reveals three interesting points. First, in most East Asian countries, the estimated capital elasticity is smaller than the capital share, while the estimated labor elasticity is larger than the labor share.24 Second, the difference between the labor share and the estimated labor elasticity is quite large in the East Asian countries. Third, the sum of the capital and labor elasticity is not far away from unity in most East Asian countries, seemingly verifying the constant returns to scale assumption.
The above results have an intuitive appeal. The first point above implies that capital is compensated higher than its marginal product, while labor is compensated less than its marginal product. This finding is in line with the typical government policy in many East Asian countries that taxes labor and subsidizes capital in order to attract foreign investment. It is interesting to observe that this pattern is not applicable to Hong Kong SAR, which is known as a free capitalist economy, as compared to government-led capitalist economies like Korea and Singapore. The second point is consistent with the view that the equality of the factor elasticity and the factor income share is unlikely to hold in developing countries. This casts doubt on the validity of the conventional procedure of the growth-accounting calculation. The third point suggests that the aggregate production functions of most East Asian economies exhibit constant returns to scale.25
It is fairly common to assume that the aggregate production function exhibits constant returns to scale. Therefore, we next estimate elasticities by explicitly imposing this restriction. More specifically, with constant returns to scale, equation (1) together with (3) is replaced by
while equation (5) is replaced by
The elasticities are then obtained as
Table 5 presents the results of this constrained nonparametric derivative estimation. Comparing the results in Table 5 and Table 2, we find that the constrained estimates of TFP growth are very close to the unconstrained ones. In particular, the constrained growth estimates for Hong Kong SAR, Korea, Taiwan Province of China, and China lie within one standard error from the unconstrained estimates. The estimates for Singapore and Thailand lie within two standard errors. The constrained and unconstrained estimates are significantly different from each other only in the cases of Indonesia, Malaysia, and the Philippines.
|TFP Growth||Elasticity||Estimates||Growth (in percent)|
|Hong Kong SAR||3.5||0.51||0.49||7.7||5.2||2.6|
|Taiwan Province of China||3.4||0.25||0.75||8.3||11.4||2.7|
Overall, our estimates are quite similar to Young’s modified estimates (Young (1995)) with one major exception. Our TFP growth estimate for Singapore (3.6–3.7 percent) turns out to be much larger than Young’s (0.3–0.5 percent). This is interesting because Hsieh (2002) calculates the dual measure of TFP growth and found the value to be similar to ours. Klenow and Rodriguez-Clare (1997) also report a similar number (3.3 percent) for Singapore.
This paper develops a new method of estimating the growth rate of TFP that does not require the strong assumptions that are needed for the conventional growth-accounting method. Our findings, based on the new estimation procedure, can be summarized as follows. First, we find that Hong Kong SAR, Korea, Singapore, and Taiwan Province of China all have very similar TFP growth of 3.4–3.9 percent over the period 1960–1995, which represents 44–47 percent of the output growth of each country during that period. On the other hand, capital growth contributes only 25–28 percent of output growth in these countries. These results provide little support for the strong version of the accumulation hypothesis.
Second, we find that the output elasticities of capital and labor are quite different from the income shares of those factors in the East Asian countries. The actual capital elasticity appears to be much smaller than the measured income share of capital, resulting in a misleadingly high contribution of capital growth to output growth in conventional growth-accounting exercises.
In conclusion, our findings appear to suggest an alternative view about East Asian economic growth that is somewhat different from either the strict “accumulation” or “assimilation” views. On one hand, as the “assimilation” view suggests, economic growth in East Asian countries appears to come from productivity improvements rather than only capital accumulation. On the other hand, in order to attract foreign investment through which new technology is transferred to the country’s economy, the government has to encourage a higher capital compensation and a lower labor compensation than what is economically justified. As a result, there is likely to have been excessive capital investment. Therefore, according to this scenario, unlike what the pure assimilation view would predict, there appear to be some opportunity costs associated with a narrowing of the “idea gap.” All in all, on the basis of the new estimates, we would argue that East Asian growth reflects a combination of the accumulation and assimilation views of economic growth.
Two data sets, referred to as “Penn” and “Collins,” were utilized. The Penn data set is based on the Penn World Tables Mark 5.6 (Summers and Heston, 1991). The Penn World Tables Mark 5.6 covers the period from 1950 to 1992. The output measure is gross domestic product (GDP) expressed in international prices. The labor data are extracted from the Penn World Tables. Although the capital data are available in the Penn World Tables, many East Asian countries have missing observations in the early years. We therefore obtained the capital data from the World Bank (available at http://www.worldbank.org/research/growth/ddhehdha.htm), which cover the period from 1950 to 1990 for most of the East Asian countries except Hong Kong SAR. The capital data for Hong Kong SAR are extracted from the Penn World Tables.
The Collins data set is developed by Collins and Bosworth (1996) (available at http://www.brookings.edu/es/research/project/develop/develop.htm), which covers the period from 1960 to 1996. The output measure is an index of real GDP based on World Bank data. The labor input measure is an index of total labor force based on OECD employment or International Labor Organization data. The capital stock is an index of capital stock based on World Bank data.
Two types of conventional estimates are calculated under the labels “Collins” and “Young,” depending on the source of labor share data. The Collins method uses the labor share set equal to 0.65 as in Collins and Bosworth (1996). The Young method uses the labor share estimates used in Young (1995). Young’s estimates are available for Hong Kong SAR, Korea, Singapore, and Taiwan Province of China up to 1990.
It can be shown that the limiting distribution of (
Therefore, the asymptotic α% pointwise error bands are given by
To implement the procedures described in Section II, we use the second-order Gaussian product kernel to reduce the bias
the derivative of which is given by
for j = 1,2,3, where ϕ(·) stands for the standard normal density function. The selection of bandwidth hj is made on the basis of the cross-validation method outlined below.
Taking into account
we select the bandwidth h so as to minimize the cross-validation function (CV):
In the above, the estimates (d/dt)â(t),
The bandwidths of (capital, labor, time) used for the estimation are (0.75, 0.45, 14.6) for Hong Kong, (1.78, 0.51, 19.5) for Indonesia, (2.42, 0.54, 19.5) for Korea, (1.87, 0.60, 19.5) for Malaysia, (1.18, 0.57, 19.5) for Philippines, (2.44, 0.67, 19.5) for Singapore, (2.29, 0.58, 19.5) for Taiwan, Province of China, (1.93, 0.56, 19.5) for Thailand, and (1.43, 0.42, 19.5) for China.
The purpose of this Monte Carlo experiment is to ascertain how well the nonparametric estimates of derivatives perform with small samples. In this experiment, the true form of the regression function is known, and the data are drawn from a known population. The original model and data set were initially constructed by White (1980) and used by Byron and Bera (1983) and Rilstone (1989). (The latter result is reproduced in Table 4.1 of Pagan and Ullah, 1999.) We modified the model by adding a time trend assuming the Hicks neutral form.
The true model is a stochastic CES production function
for t = 1,…,T,
where yt, xt1 and xt2 stand for the values of output, capital, and labor, respectively. The time trend of output p(t) is set equal to the third-degree polynomial, estimated from the data by regressing output on its first lag as well as the trend. The terms ln xi1 and ln xi2 are generated from an estimated second-order autoregression fitted to the data for log capital and labor and then transformed to have a mean equal to (0.5, 0.5) and each variance equal to 1/12, while ut, is generated independently from a normal distribution with mean zero and variance 0.01.
Our goal is to estimate derivatives or elasticities ∂ ln y / ∂ ln xj. In the literature on production economics, the parametric approximations often used for this purpose are
(1) CD: Cobb-Douglas function
(2) TL: Translog function
Table A1 presents three sets of estimates of the output elasticities bj(x1, x2) = ∂ ln y/∂ ln xj for j = 1, 2 at the mean point (ln x1, ln x2) = (0.5, 0.5), using OLS estimation assuming (i) the Cobb-Douglas (CD) and (ii) Translog (TL) forms of production function as well as (iii) non-parametric estimation (NP). The sample size is 200. It is clear that in terms of bias, the nonparametric estimates are much superior to the parametric counterparts. The standard errors for the Cobb-Douglas and Translog estimates, however, are somewhat smaller compared to the nonparametric standard errors. This may reflect the slow speed of convergence of nonparametric estimates.
Table A2 presents the same comparison using only 35 observations, which is equal to the sample size in our study. The result is strikingly similar to Table A1, except the standard errors are larger. The nonparametric estimates remain less biased for elasticities.
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