ECO 310: Empirical Industrial Organization
Lecture 8 - Production Functions (II)
Dimitri Dimitropoulos
Fall 2014
UToronto
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References
I
ABBP Section 2
I
Olley and Pakes (1996). "The Dynamics of Productivity in the Telecommunications Equipment Industry." Econometrica, Vol. 64(6), pp.
1263-1297.
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Olley and Pakes (1996, Ecta)
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Olley and Pakes (1996) - Motivation
I
Olley and Pakes (OP) take a very di¤erent approach to solving the
simultaneity problem inherent in production function estimation
I
OP use a control function method – instead of instrumenting the
endogenous regressors, they include additional regressors that capture the
endogenous part of the error term
I
Olley and Pakes begins with two premises:
I
Since we are using panel data, and observe the transition of each …rm from
year to year, we should model the stochastic process for this.
I
Since the endogeniety problem stems from a …rm’s input demands, we should
add structure to these input demands
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Olley and Pakes (1996) - Motivation
I
Suppose we have a panel of J …rms observed over T years.
I
For each …rm j at time t, we observe (the logs of) output, labor and capital
fqjt ; ljt ; kjt : j = 1; 2; :::J and t = 1; 2; :::T g
I
Consider the following model of simultaneous equations
(PF)
qjt =
(LD)
ljt = fL (wjt ; rt ; kjt )
(KD)
ijt = fK (rt ; kjt ; !jt )
0
+
L
ljt +
K
kjt + !jt + "jt
I
(LD) & (KD) rep. …rms’optimal decision rules for labor & cap. investment
I
We explicitly account for (LD) and (KD) as it is through these choices that
the endogeneity problem makes itself present
I
However, we make no assumptions about the functional forms of fL or fK
I
We’ll rely on assumps. about the covariance structure of ljt ; kjt ; and !jt
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Assumptions
I
Olley and Pakes assume:
I
Assumption 1. Common Capital Markets
I
Firms purchase capital in a common, competitive capital market
I
Since the capital market is competitive, no individual …rm has any in‡uence on
the equilibrium price of capital.
I
And, since …rms opperate in a common capital market, the price of capital is
the same for all …rms.
I
As such, there is no cross-section variation in capital prices
rjt = rt for all j ,t
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Assumptions
I
Olley and Pakes assume:
I
Assumption 2. Capital is Dynamic
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There is a time-to-build assumption made about capital.
I
A …rm’s decision is about its level of investment (not its capital per se)
I
Speci…cally, capital is accumulated by through a dynamic investment process
k jt = (1
)k jt
1
+ ijt
1
where
ijt
is the rate of depreciation
1 is the …rm’s capital investment in period t
1
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However, while investment ijt is chosen in period t; it doesn’t become
productive capital until period t + 1
I
Thus, in period t; the quantiy of capital for the …rm is …xed.
I
Instead, taking as given the price of capital rt ; its current level of capital k jt ;
and its productivity shock !jt ; the capital decison problem of the …rm in
period t is to choose its investment ijt into capital for next period
ijt = fk (rt ; k jt ; !jt )
where fk ( ) is the …rm’s optimal decision rule for capital investment
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Assumptions
I
Olley and Pakes assume:
I
Assumption 2*. Labor is Non-Dynamic
I
Labor is assumed to be a completely variable input in production.
I
A …rm is free to choose whatever quantity of labor ljt it wishes in the current
period
I
Taking as given the price of labor w jt ; its current level of capital k jt ; and its
productivity shock !jt ; the labor decison problem of the …rm in period t is to
choose its quantity of labor for current period production
ljt = fL (w jt ; rt ; k jt )
where where fL ( ) is the …rm’s optimal decision rule for labor
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Note: this assumption is not necessary for the Olley-Pakes estimator
I
In fact, Olley and Pakes do not actually use this assumption.
I
Rather, we include this assumption for completeness
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Assumptions
I
Olley and Pakes assume:
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Assumption 3. Investment is invertible in !jt
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Firms with higher productivity are expected to invest more into capital
accumulation.
I
Thus, the …rm’s optimal decision rule for capital investment fk (k jt ; rt ; !jt ) is
strictly increasing in !jt
I
This means that the investment capital investment function fk (k jt ; rt ; !jt ) is
invertible in its !jt argument
ijt = fk (rt ; k jt ; !jt )
!
!jt = fk
1
(rt ; k jt ; ijt )
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Assumptions
I
Olley and Pakes assume:
I
Assumption 4. Productivity Shock follows a 1st Order Markov Process
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The evolution of the productivity shock !jt productivity shock from year to
year is determined by an exogneous stochastic process.
I
Speci…cally, !jt follows a 1st Order Markov Process
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Thus, the current value of the productivity shock !jt is soley dependent on its
one-period lagged value
!jt = !jt 1 + jt
where
is a parameter with j j < 1
!jt
jt
1
is the persistence in !jt between t
is the innovation in the !jt between t
1 and t
1 and t
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The Model
I
Suppose we have a panel of J …rms observed over T years.
I
For each …rm j at time t , we observe (the logs of) output, labor and capital
fqjt ; ljt ; kjt : j = 1; 2; :::J and t = 1; 2; :::T g
I
Consider the following model of simultaneous equations
(PF)
qjt =
(L)
ljt = fL (wjt ; rt ; kjt )
(K)
ijt = fK (rt ; kjt ; !jt )
0
+
L
ljt +
K
kjt + !jt + "jt
where (L) and (K) represent …rms’optimal decision rules for labor & capital
I
We explicitly account for (L) and (K) as it is through these choices that the
endogeneity problem makes itself present
I
However, we make no assumption about the functional forms of fL or fK
I
Rather, we will rely on the covariance structure assumptions in (A1) - (A4)
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The Model
I
In summary, our model of simultaneous equations
(PF)
qjt =
(LD)
ljt = fL (wjt ; rt ; kjt )
(KD)
ijt = fK (rt ; kjt ; !jt )
0
+
L
ljt +
K kjt
+ !jt + "jt
when (LD) & (KD) rep. …rms’optimal decision rules for labor & cap. investment
I
We make no assumptions about the functional forms of fL or fK
I
Instead, we assume
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(A1) rt is common across all …rms
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(A2) kjt is decided at t
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(A3) ijt = fK (rt ; kjt ; !jt ) is invertible in !jt
I
(A4) !jt = !jt
1
+
1 with kjt = (1
)kj ;t
1
+ ij ;t
1
jt
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Estimation
I
The Olley Pakes estimator proceeds in two stages
I
Stage 1
I
I
I
The …rst stage estimates L using a "control function" approach
This stage relies on assumptions (A1) and (A3)
Stage 2
I
I
The second stage estimates K ; taken as given the estimates from Stage 1.
This stage relies on assumptions (A2) and (A4).
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Estimation - Stage 1
I
The …rst stage estimates
I
L
using assumptions (A1) and (A3)
Since, by (A3), ijt = fK (rt ; kjt ; !jt ) is invertible in !jt ; hence we can write
!jt = fK 1 (rt ; kjt ; ijt )
I
We can use this to substitute for !jt into (PF)
qjt =
0
+
L
ljt +
K
kjt + fK 1 (rt ; kjt ; ijt ) + "jt
or
qjt =
where (rt ; kjt ; ijt )
0
+
K
L
ljt + (rt ; kjt ; ijt ) + "jt
kjt + fK 1 (rt ; kjt ; ijt )
I
This last equation is a partially linear regression model
I
The parameter L and the function ( ) can be estimated using
semiparametric methods
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Estimation - Stage 1
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We face two di¤erent problem in estimating the function (rt ; kjt ; ijt )
1. We do not observe the price of capital rt
I
2.
depends on rt ; which is not a variable observed in our data.
I
This is why we need assumption (A1)
I
Since rt is common across …rms, we can treat it as a year …xed e¤ect.
I
That is, we can control for the unobserved rt by a set of year-dummies
I
We write
t (kjt ; ijt )
t (k jt ; ijt )
to make clear the dependence of
on time
has unknown functional form
I
Without a function form assumption about fK ; the function
I
Nevertheless, we can appeal to Taylor’s Theorem, which tells us that we can
approximate any function using a high-order polynomial.
I
For example, a 2nd-Order of
t
=
0
+
1
k jt +
is unknown
t (k jt ; ijt )
i +
2 jt
3
k jt ijt +
4
k jt2 +
i 2 + yeart
5 jt
where yeart is a set of year dummies
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Estimation - Stage 1
I
Thus, the OP …rst stage regession
qjt =
L
ljt + (
0
+
1
kjt +
i +
2 jt
3
kjt ijt +
4
kjt2 +
i 2 + yeart ) + "jt
5 jt
I
This method is a control function method.
I
Instead of instrumenting the endogenous regressors, we include additional
regressors that capture the endogenous part of the error term (i.e., proxy for the
productivity shock).
I
By including a ‡exible function in (kjt ; ijt ), we are controlling for !jt
I
Thus standard estimation methods (e.g. OLS) yield us a consistent estimates of
L
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Estimation - Stage 1
I
Thus, the OP …rst stage regession
qjt =
I
L
ljt + (
+
0
1
kjt +
i +
2 jt
3
kjt ijt +
4
kjt2 +
i 2 + yeart ) + "jt
5 jt
The …rst stage provides us with:
I
an estimate of
L
I
an estimate of
t (kjt ; ijt )
I
I
as the coe¢ cient on ljt
for each …rm j in each period t
in particular, if we construct the …tted values b
q jt
then an estimate of t (kjt ; ijt )
bjt = b
q jt
b ljt
L
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Estimation - Stage 2
I
The second stage estimates
I
K
using assumptions (A2) and (A4)
Recall, we de…ned
t (kjt ; ijt )
I
But fK 1 (rt ; kjt ; ijt ) = !jt
I
Thus
0
jt
I
=
+
0
K
kjt + fK 1 (rt ; kjt ; ijt )
+
kjt + !jt
Now, by (A4),
!jt = !jt
I
K
1
+
jt
Using this to substitute in for !jt ; we have
jt
=
0
+
K
kjt + !jt
1
+
jt
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Estimation - Stage 2
I
The second stage estimates
I
In principal, the period t equation
jt
I
I
I
using assumptions (A2) and (A4)
K
=
0
+
K
kjt + !jt
1
+
jt
is a regression model through which we can estimate K
However, in this equation !jt 1 is an unobservable variable
However, a similar logic applies to jt in every period.
In particular, in period t 1, we also have
jt
1
=
!j ;t
1
=
0
+
K
kj ;t
1
+ !j ;t
1
kj ;t
1
or
I
j ;t
1
0
K
Using this last expression to substitute in for !jt
jt
=
0
+
K
kjt + (
jt
1
0
1
into the period t equation
K
kjt
1)
+
jt
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Estimation - Stage 2
I
Thus, the OP second stage regession
jt
I
= (1
)
0
+
K
kjt +
j ;t 1
K
kj ;t
1
+
jt
Note:
1. We do not observe the jt ’s. But we do have consistent estimates of their
values from Stage 1. So, our regression model
b = (1
jt
)
0
+
K
kjt + bj ;t
1
K
kj ;t
1
+
jt
2. The parameters of this model enter non-linearly. So, we estimate this model
by Non-Linear Least Squares
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Aside - Nonlinear Least Squares
I
Consider the Nonlinear Regression Model
yi = f (xi ; ) + "i
where
I
I
I
xi is a vector of explanatory variables for the outcome yi
is a vector of parameters
f (xi ; ) is a function in which the parameters enter nonlinearly.
I
We have data fyi ; xi : i = 1; :::; ng; and wish to estimate the
I
Of course, we do not observe the error term "i
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But, if we had an estimate of ; we could constuct an estimate of "i
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The residuals
e i = yi
parameters
f (xi ; b)
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Aside - Nonlinear Least Squares
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The Nonlinear Least Squares (NLLS) estimator of the parameters is
vector tha minimizes the sum of of square residuals
X
2
min
[yi f (xi ; b)]
b
i
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Generally, there is no anlytical solution for the NLLS estimator
I
Instead, we ask computer software to minimize the sum of squared residuals
by searching numerically for the NLLS estimator.
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Aside - Nonlinear Least Squares
I
Example 1.
I
Consider the Nonlinear Regression Model
qi =
0
+
1
li +
2
ki +
1
2
l i k i + "i
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We can estimate this model in STATA using Nonlinear Least Squares
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The STATA syntax is
nl ( q = fb0g + fb1g l + fb2g k + fb1g fb2g l
I
k )
Notes
I
The STATA keyword is nl
I
We type out our model explicitly, and enclose it in round brackets ( )
I
Parameters are indicated using brace brakets { }
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Variables are indicated using roman letters
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Aside - Nonlinear Least Squares
I
Example 2.
I
To STATA syntax to estimate the OP second stage regression is
nl (
t
= (1
fr 0g)*fb0g + fbkg*k + fr 0g*
where the one period lagged values of
j ;t
1
by sort …rmid : generate
t
1
+ frog*fbkg*kt
and kj ;t
1
were generated by
t
1
= [_n-1]
by sort …rmid : generate kt
1
= k[_n-1]
1
)
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Estimation - OP Algorithm
I
Is summary, the Olley and Pakes Alorithm
1. Stage 1. Estimate
qjt =
I
I
L
ljt + (
L
0
using the model
+
1
kjt +
i +
2 jt
3
kjt ijt +
4
kjt2 +
i 2 + yeart ) + "jt
5 jt
Regress q jt on ljt ; k jt ; ijt ; k jt ijt ; k jt2 ; ijt2 ; and a set of year e¤ects.
L
is estimated as the coe¢ cient by ljt :
2. Construct estimates of the
jt
bjt = q
bjt
b ljt
L
bjt are the …tted values from Step 1.
where the q
3. Stage 2. Estimate
I
I
K
b = (1
jt
using the model
)
0
+
K
kjt + bj ;t
Regress (nonlinearly) bjt on k jt ; bjt
K
1
; k jt
1
K
kj ;t
1
+
jt
1
is estimated as the coe¢ cient on k jt
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Empirical Application
I
During the 1970s and early 1980s, technological change and deregulation
caused a major restructuring of the telecommunication equipment industry.
I
Olley and Pakes apply their estimation algorithm to estimate the prameters
of the production function for …rms in the telecommunications equipment
industry.
I
They use their estimates to analyze changes that occurred in the
distribution of plant-level performance between 1974 and 1987
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Empirical Application - Results
Estimates of Production Function Parameters - Table VI
OLS
Fixed E¤ects
Olley Pakes
(3)
(4)
(8)
Labor
0.693
(0.019)
0.629
(0.026)
0.608
(0.027)
Capital
0.304
(0.018)
0.150
(0.026)
0.342
(0.035)
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