1 Estimation of the Primal Production Function Lecture V I. Ordinary Least Squares A. The most straightforward concept in the estimation of production function is the application of ordinary least squares. 1. Taking the quadratic production function as a starting point 20 1 1 2 2 3 3 11 1 12 1 2 13 1 32222 2 23 2 3 33 3ya ax ax ax Ax Axx AxxAx Axx Axε=+ + + + + + ++++ a. Note that we have already applied symmetry on the quadratic. From an estimation perspective, since 12 21xxxx= any other approach would not work. b. Using data from Indiana and Illinois, we apply ordinary least squares to this specification to estimate Table 1. Estimates of the Quadratic Production Function. Parameter Estimate α0 100.28194 (6.81173)a α1 -0.00535 (0.07911) α2 0.23445** (0.10630) α3 0.05900 (0.07947) A11 0.00014 (0.00030) A12 -0.00089 (0.00082) A13 0.00073 (0.00047) A22 0.00019 (0.00038) A23 -0.00087* (0.00049) A33 -0.00007 (0.00031) ** Denotes statistical significance at the .05 confidence level. * Denotes statistical significance at the .10 confidence level. a Numbers in parenthesis denote standard deviations.AEB 6184 Production Economics Lecture V Professor Charles B. Moss Fall 2005 2c. Do these estimates make any sense? What is wrong? 2. Turning to the Cobb-Douglas form ()()()()()123 1 2 3ln ln ln ln lnyAxxx y A x x xαβγαβγ=⇒=+++ a. One alternative is then to run the regression ()()()()01 1 2 2 3 3ln ln ln lnyxxxααα α ε=+ + + + b. What are some of the problems with this specification? (1) First, the one problem is that there may be zero input levels. What is the production theoretic problem with zero input levels? What is the econometric problem with zero input levels? (2) Second, what is the assumption of the error term? c. Regression results Table 2. Estimates of the Cobb-Douglas Function Parameter Estimate α0 4.5858*** (0.0561)a α1 0.0126 (0.0118) α2 0.0168** (0.0073) α3 0.0132** (0.0063) ***Denotes statistical significance at the .01 confidence level. **Denotes statistical significance at the .05 confidence level. *Denotes statistical significance at the .10 confidence level. aNumbers in parenthesis denote standard deviations. 3. Estimating the Transcendental Production Function: a. The transcendental production function has many of the same problems as the Cobb-Douglas. Specifically, the production function can be written as: 333111 2 2212 3abxabx a bxyAxexexe= Thus, it is estimated as in the Cobb-Douglas case (it is actually a generalization of the Cobb-Douglas form) in logarithmic form: ()()()()01 1 112 2 223 3 33ln ln ln lnyaa x bxa x bxa x bxε=+ + + + + + + b. Again, what are the assumptions about zeros or the distribution of error terms. c. Regression results:AEB 6184 Production Economics Lecture V Professor Charles B. Moss Fall 2005 3Table 3. Estimated Parameters from the Transcendental Production Function Parameter Estimates a0 4.55203*** (0.05875)a a1 -0.00006 (0.01565) b1 0.00039 (0.00038) a2 0.01656* (0.00887) b2 -0.00004 (0.00045) a3 0.00251 (0.00897) b3 0.00080* (0.00047) ***Denotes statistical significance at the .01 confidence level. **Denotes statistical significance at the .05 confidence level. *Denotes statistical significance at the .10 confidence level. aNumbers in parenthesis denote standard deviations. 0.00190.00190.00200.00200.00210.00210.00220 50 100 150 200 250 300Pounds of NitrogenMarginal Physical ProductAEB 6184 Production Economics Lecture V Professor Charles B. Moss Fall 2005 40.0000.0020.0040.0060.0080.0100.0120.0140.0160.0180 20 40 60 80 100 120 140 160Pounds of PhosphorousMarginal Physical Product 0.00000.00050.00100.00150.00200.00250.00300 20 40 60 80 100 120 140 160 180 200Pounds of PotashMarginal Physical Product of PotashAEB 6184 Production Economics Lecture V Professor Charles B. Moss Fall 2005 5B. One alternative to the problem with the distribution functions for errors involves the estimation of production functions using maximum likelihood. These likelihood functions may explicitly incorporate such considerations as gamma distributions. II. Nonparametric Production Functions A. It is clear from our discussions on production functions that the choice of production function may have significant implications for the economic results from the model. 1. The Cobb-Douglas function has linear isoquants that has implications for the input demand functions. 2. While the Cobb-Douglas function has no stage III, the quadratic production function is practically guaranteed a stage III. B. Thus, one approach is to generate nonparametric functional forms. 1. These nonparametric functional forms are intended to impose allow for the maximum flexibility in the input-output map. 2. The approach is different that the nonparametric production function suggested by Varian. C. Two approaches: 1. Fourier Expansions () () ()()0011011cos sin2AJkjjjAfxxxx jkxjkxkkααααααααααββ β β γβ===′′ ′ ′=+ + Β+ + −′Β=−∑∑∑ 2. Nonparametric regressions a. A nonparametric regression is basically a moving weighted average where the weights of the moving average change for various input levels. () () ( )ˆ,,,yx yz f yzx dz∞−∞=δ∫ (1) In this case ()ˆyx is the estimated function value conditioned on the level of inputs x. (2) The value ()yz is the observed output level at observed input level z . (3) (),,,fyzxδ is a kernel function which weights the observations based on a distance from the point of approximation. b. In this application, I use a Gaussian kernel. ()()2121,,, exp22zxfyzx−−δ= δδπ c. The multivariate form of the Gaussian kernel function is expressed as {}{}()()()111211,,,, exp22fyz x A A z xA z x−−−′δ= δ − − −πAEB 6184 Production Economics Lecture V Professor Charles B. Moss Fall 2005 6d. Because of the discrete nature of the expansion, I transform the continuous distribution into a discrete Gaussian distribution {}{}(){}{}{}{}1,,,,,,,,,,,,iiNjjfyx xAwy x x Afyx xA=δδ=δ∑ e. The estimated value of the production function at point x can then be computed as () {}{}()1ˆ,, ,Niiiyx wy x xA y==δ∑ Figure 5. Production Function for Corn Varying both Nitrogen and Potash100200300Nitrogen50100150200250Potash100110120130140Corn100200300Nitrogen50100150200250PotashAEB 6184 Production Economics Lecture V