Lecture06-2011 - Outline Simultaneity and Estimation of the...

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Unformatted text preview: Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Simultaneity and Other Simple Problems : Lecture VI Charles B. Moss September 13, 2011 Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas 1 Simultaneity and Estimation of the Production Function Irving Hock - Econometrica Lawrence Klein - A Textbook of Econometrics 2 Two Models of Simultaneity Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation 3 Zeros in the Cobb-Douglas Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Irving Hock - Econometrica Lawrence Klein - A Textbook of Econometrics Simultaneity and Estimation of the Production Function The above discussion (and estimates) makes the experimental plot design assumption regarding the data. I essentially assumed that the data are being generated from some sort of experimental design so that the errors are truly random. If the data are actually the result of farm level decisions, the data are endogenous. Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Irving Hock - Econometrica Lawrence Klein - A Textbook of Econometrics Irving Hock - Econometrica Hock, Irving. 1958. Simultaneous Equation Bias in the Context of the Cobb-Douglas Production Function. Econometrica 26(4), 566-78. The basic firm-level model is that we have an empirical model under the assumption of: A Cobb-Douglas production function, and Competition. Q ￿ a X 0 = K0 Xq q ￿q =1 ￿ ∂ X0 X0 P0 = P 0 aq = Pq ∂ ￿q X Xq ￿ X0 P 0 Y0 aq = aq =1 Xq P q Yq Charles B. Moss (1) Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Irving Hock - Econometrica Lawrence Klein - A Textbook of Econometrics Lawrence Klein - A Textbook of Econometrics Klein demonstrates that the best linar unbiased estimate of aq is I ￿ ￿ Yqi ￿ I 1 ˆq = a i =1 Y 0i In this approach the “average” firm is defined to be the optimal firm. As an alternative ￿ ￿ X0 P 0 a0 = Rq Pq Xq (2) (3) where Rq is “some constant and the investigator wishes to test whether it is equal to one.” Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Irving Hock - Econometrica Lawrence Klein - A Textbook of Econometrics “The firm sets the value of the marginal product equal to the price augmented by the effect of any restrictions that exist.” “In this formulation, Rq can of course vary among firms; but, for a sample of firms, the investigator would be interested in testing whether the average Rq is equal to one.” Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation Two models of Simultaneity Model 1: Production disturbance not transmitted to the “independent” variables “If the disturbance in the production equation affects only the output and is not transmitted to the other variables in the system, then there is no simultaneous equation bias. Single equation estimates are consistent.” For example if inputs are fixed or are predetermined . Model 2: Production disturbance transmitted to the “independent” variables. “Simultaneous equation bias arises when disturbances in the production relations affect the observed values of all variables, and, as a result, single equation estimates are not consistent.” Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation Empirical Setup Starting in level space X 0 = K0 Q ￿ a Xq q U q =1 aq P 0 Xq = X0 Vq q = 1, 2, · · · Q Rq Pq Taking the logarithms of this expression x0 = k 0 + Q ￿ (4) a q xq + u q =1 xq = kq + x0 + vq q = 1, 2, · · · Q x0 = ln (X0 ) ￿xq = ln (Xq ) ￿ aq P 0 kq = ln Rq Pq Charles B. Moss (5) Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation Indirect Least Square Solution Kmenta, J. 1964. Some Properties of Alternative Estimates of the Cobb-Douglas Production Function. Econometrica 32(1/2), 183-188. Simplifying the general system of equations x0 i = k 0 + a 1 x1 i + a 2 x2 i + ν 0 i x1 i = k 1 + x0 i + ν 1 i x2 i = k 2 + x0 i + ν 2 i Charles B. Moss (6) Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation Transforming the estimation problem to x0 i = b 0 + b 1 ( x1 i − x 0 i ) + b 2 ( x 2 i − x0 i ) + e i (7) yields estiamtes of b1 and b2 that are consistent. Note by the definitions x1 i = k 1 + x 0 i + ν 1 i ⇒ x 1 i − x0 i = k 1 + ν 1 i x2 i = k 2 + x0 i + ν 2 i ⇒ x 2 i − x0 i = k 2 + ν 2 i Charles B. Moss (8) Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation Working through the mechanics x0 i = b 0 + b 1 x1 i − b 1 x0 i + b 2 x2 i − b 2 x0 i + e i (1 + b1 + b2 ) x0i = b0 + b1 x1i + b2 x2i + ei b0 b1 b2 1 x0 i = + x1 i + x2 i + ei 1 + b1 + b2 1 + b1 + b2 1 + b1 + b2 1 + b1 + b2 ˆ br ˜r = a r = 1, 2 ˆ ˆ 1 + b1 + b2 (9) Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation Seemingling Unrelated Regression Formulation Building on the simple Cobb-Douglas function αβγ π = pAx1 x2 x3 − w1 x1 − w2 x2 − w3 x3 ∂π Y Y w1 = p α − w1 = 0 ⇒ α = ⇒ αYp = x1 w1 ⇒ αR = S1 ∂ x1 x1 x1 p ∂π Y Y w2 = p β − w2 = 0 ⇒ β = ⇒ β Yp = x2 w2 ⇒ β R = S2 ∂ x2 x2 x2 p ∂π Y w3 Y = p γ − w3 = 0 ⇒ = ⇒ γ pY = x3 w3 ⇒ γ R = S3 ∂ x3 x3 x3 p (10) Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation Three related empirical equations ln (Y ) = a0 + a1 ln (x1 ) + a2 ln (x2 ) + ln (x3 ) + ν1 S1 = k 1 + a1 R + ν 2 S2 = k 2 + a2 R + ν 3 S3 = k 3 + a3 R + ν 4 (11) System of Equations ln (Y1 ) 10 S11 0 1 S21 = 0 0 S31 00 0 0 1 0 0 ln (x11 ) ln (x21 ) ln (x31 ) 0 R1 0 0 0 0 R1 0 1 0 0 Rq Charles B. Moss a0 k1 k2 k3 a1 a2 a3 (12) Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas ln (Y1 ) S 11 S21 S31 ln (Y2 ) S12 S22 S32 ln (Y3 ) S13 S23 S33 = 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 ln (x11 ) R1 0 0 ln (x12 ) R2 0 ln (x13 ) R3 0 0 Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation ln (x21 ) 0 R1 0 ln (x22 ) 0 0 ln (x23 ) 0 R2 0 ln (x31 ) 0 0 R1 ln (x32 ) 0 0 R2 ln (x33 ) 0 0 R3 ￿ ￿￿ ￿ ˆ y = x β￿ ν ⇒ β￿= x ￿ x x ￿ y + ￿ ￿ ˜ ˆ ˆ β = x ￿ V −1 x x ￿ V −1 y Charles B. Moss k0 k1 k2 k3 a1 a2 a3 + ν11 ν21 ν31 ν41 ν12 ν22 ν32 ν42 ν13 ν23 ν33 ν34 (13) (14) Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Empirical Setup Indirect Least Squares Solution Seemingly Unrelated Regression Formulation ν ν ν ν ν ν 1 11 12 13 11 21 31 ˆ ν21 ν22 ν23 ν12 ν22 ν32 V= 3 ν31 ν32 ν33 ν13 ν23 ν33 Charles B. Moss (15) Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas Zeros in the Cobb-Douglas Functional Form Moss, Charles B. 2000. Estimation of the Cobb-Douglas with Zero Input Levels: Bootstrapping and Substitution. Applied Economics Letters 7(10), 677-679. Zeros raise several difficulties in estimating the Cobb-Douglas production function. On the theoretical side, the presence of a zero-level input is that it violates weak necessity of inputs. On the empirical side, how do you take the natural logarithm of zero. Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas The existence of zeros is the result of measurement error. Agronomically, production of a crop is impossible without some level of each fertilizer. Thus, production occurs based on some true level of each nutrient available to the plant xi∗ = xi + ￿i (16) In fact we can think of the soil as a sponge that contains a variety of nutrients that we can augment by applying fertilizer. The actual level of fertilizer used by the crop could then be a function of what we add, the weather (i.e., if adequate moisture is not present the crop does not use the full potential), etc. Second, a production function that does not admit zero input could represent a misspecification. Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI Outline Simultaneity and Estimation of the Production Function Two Models of Simultaneity Zeros in the Cobb-Douglas In this paper, I consider two techniques for adjusting the zero observations. First, I redraw from the sample averaging the result until the pseudo sample contains no zeros. Second, I substitute a small non-zero number for those observations that contain zeros (i.e., 0.1, 0.01, 0.001). The goodness of fit for each procedure is then compared using a Strobel measure of information I= N ￿ i =1 ￿￿ si si ln ˜i s (17) Where si is the theoretically appropriate budget share for each input and ˜i is the budget share estimated using each s empirical approximation of zero. Charles B. Moss Simultaneity and Other Simple Problems : Lecture VI ...
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