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Simultaneity and Estimation of the Production Function
Two Models of Simultaneity
Zeros in the CobbDouglas 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 CobbDouglas 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 CobbDouglas 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 CobbDouglas 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 CobbDouglas Irving Hock  Econometrica
Lawrence Klein  A Textbook of Econometrics Irving Hock  Econometrica
Hock, Irving. 1958. Simultaneous Equation Bias in the
Context of the CobbDouglas Production Function.
Econometrica 26(4), 56678.
The basic ﬁrmlevel model is that we have an empirical model
under the assumption of:
A CobbDouglas 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 CobbDouglas 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 CobbDouglas 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 CobbDouglas 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 CobbDouglas 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 CobbDouglas Empirical Setup
Indirect Least Squares Solution
Seemingly Unrelated Regression Formulation Indirect Least Square Solution Kmenta, J. 1964. Some Properties of Alternative Estimates of
the CobbDouglas Production Function. Econometrica
32(1/2), 183188.
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 CobbDouglas 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 CobbDouglas 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 CobbDouglas Empirical Setup
Indirect Least Squares Solution
Seemingly Unrelated Regression Formulation Seemingling Unrelated Regression Formulation Building on the simple CobbDouglas 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 CobbDouglas 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 CobbDouglas 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 CobbDouglas 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 CobbDouglas Zeros in the CobbDouglas Functional Form Moss, Charles B. 2000. Estimation of the CobbDouglas with
Zero Input Levels: Bootstrapping and Substitution. Applied
Economics Letters 7(10), 677679.
Zeros raise several diﬃculties in estimating the CobbDouglas
production function.
On the theoretical side, the presence of a zerolevel 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 CobbDouglas 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 CobbDouglas 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 nonzero 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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