ERSA Training Workshop
Lecture 4: Estimation of Production
Functions with Micro Data
Måns Söderbom
Thursday 15 January 2009
1
Introduction
Earlier in this course you will have seen how panel data methods can be used to
estimate the parameters of a production function, which may take the following
form:
yjt = k kjt + l ljt + ! j + ujt ;
where y; k; l denote output (or value-added), capital, labour, respectively, j; t
denote …rm and time (panel data), respectively, ! j is a …rm-speci…c unobserved
e¤ect, ujt is a time varying residual, and k ; l are unknown parameters.
The main reasons for using a panel estimator in this context are as follows:
The researcher might suspect there is time-invariant unobserved heterogeneity across …rms in underlying productivity - controlling for ’…xed
e¤ects’, either by means of di¤erencing, by going within, is meant to take
care of this.
The researcher might suspect that the time varying component of the
residual is serially correlated in levels - pseudo-di¤erencing the levels
equation which results in a dynamic model is meant to take care of this.
Recall: if
ujt = uj;t 1 + ejt;
we have
yjt =
k kjt
+ l ljt + ! j + ujt
yjt =
k kjt
+ l ljt + ! j + uj;t 1 + ejt ;
and since, by de…nition,
uj;t 1 = yj;t 1
k kj;t 1
l lj;t 1
!j ;
the production function can be written as a dynamic equation (with common factor restrictions):
yjt = yj;t 1 + k kjt
k kj;t 1 + l ljt
l lj;t 1 + ! j (1
) + ejt:
The researcher might suspect that the time varying component of the
residual is correlated with the factor inputs (e.g. capital, labour) using instrumental variables is meant to take care of this.
The panel data methods discussed earlier in this course are obviously very
general, and not speci…c to production functions. In this lecture we discuss in
more detail the econometrics of estimating production functions using panel
data, typically at the …rm level.
2
Why are we interested?
In so far as there is one thing on which economists appear to be able to agree it is
the desirability of higher productivity. The production function is an important
tool that can be used to analyze various aspects of productivity. Here are some
research questions/issues that can be addressed using a production function
approach:
Scale and productivity. In most datasets on enterprises in Sub-Saharan
Africa, labour productivity (usually de…ned as value-added per worker) is
much higher large than small …rms (see e.g. the survey paper by Bigsten
& Söderbom, WBRO, 2006). Is this because large …rms have more capital
per worker, or because there are increasing returns to scale? If we believe
the production function above is correctly speci…ed, we can answer this
question by estimating k and l .
Suppose we convince ourselves there are increasing returns to scale, i.e.
k + l > 1. One implication would be that if a …xed set of inputs (at the
national level) gets allocated to a small number of large …rms this results in
more aggregate output than if allocated to a large number of small …rms.
This may be important for policy.
In contrast, if we convince ourselves returns to scale of constant, + = 1,
a reallocation of resources between …rms of di¤ering size may not impact
on aggregate output (e.g. two small …rms will produce as much output
as one large …rm using the same amount of inputs as the two small ones
between them).
In fact, the evidence on returns to scale in developing countries is most
consistent with constant returns to scale (see e.g. Söderbom and Francis
Teal, 2004, for evidence on enterprises in Ghana). That is, while there
are many small …rms in developing countries, this does not imply foregone
scale economies. You will have seen that Blundell-Bond obtain the same
result for …rms in the US.
Although I won’t be taking about farms in this lecture, production functions are commonly used in agricultural economics too. For example, a
common view is that small farms are more productive than large farms;
however the empirical evidence on the matter is somewhat mixed (e.g.
Lamb, 2001, refutes this notion, concluding that large farms are as productive as small ones) .
Rates of technological change.
Lamb, R. L. “Inverse productivity: land quality, labor markets and measurement error”Journal
of Development Economics, 2003, 71: 71-95.
Rates of return on, for example, R&D or exporting (’learning-by-exporting’)
The contribution of various forms of inputs to output (e.g..skilled & unskilled labour).
3
Production functions & the basic endogeneity
issue
We focus on the simple 2-factor Cobb-Douglas production function:
Yj = Aj Kj k Lj l ;
or, in natural logarithms,
y j = 0 + k k j + l lj + j ;
where
ln Aj = 0 + j
is log TFP. 0 is a constant, interpretable as the mean of log TFP, while
j measures the deviation in productivity from the mean, for …rm j . TFP is
typically assumed unobserved (at least partially).
Suppose we have micro data on output, capital and labour. How can the
parameters of this equation be estimated?
As you know, for OLS to consistently estimate the -parameters, the error term must have zero mean and be uncorrelated with the explanatory
variables:
E
j
= 0;
Cov kj ; j
= 0;
(1)
Cov lj ; j
= 0
(2)
The zero mean assumption is innocuous, as the intercept
up a non-zero mean in j .
0
would pick
The crucial assumption is zero covariance. Is this likely to hold in the
present context?
No - because it seems quite possible that the …rm’s capital and labour
decisions are in‡uenced by factors that are observed to the …rm’s manager
but unobserved to the econometrician, i.e. by j . This would set up a
correlation between the regressors and the residuals, rendering the OLS
estimates biased and inconsistent.
3.1
Illustration
Assumptions:
Firms operate in perfectly competitive input and output markets (so that
input and output prices are not a¤ected by the actions of …rm j );
Capital is a …xed input (decided upon one period in advance, say) rented
at rate r;
Firms observe j before hiring labour (at rate W ), and labour is a ’‡exible
input’that can be altered without dynamic implications.
The …rm’s pro…t is given by
j
= pYj
j
= p Aj Kj k Lj l
W Lj
rKj
wLj
rKj ;
where p is the output price. Assuming the …rm maximizes pro…ts, it will choose
labour such the following …rst-order condition is ful…lled:
k
l
l pAj Kj Lj
1
= W;
which implies
Lj =
l pAj
W
!
1
1
l
1
Kj
k
l;
or, in logs,
lj =
1
1
l
h
ln l + ln p
i
ln W + ln 0 + j + k kj :
Clearly in this case lj depends on unobserved TFP (which is the interpretation assigned to the residual j ) and so estimating the production
function
y i = 0 + k k j + l lj + j :
by means of OLS will give biased and inconsistent results.
Note that, since the …rst-order condition for labour implies a positive correlation between lj and j , we would expect the OLS estimate of l to be
upward biased.
3.2
Other endogeneity issues
Attrition. Suppose the probability of exit is a negative function of the
value of the …rm, and suppose the value of the …rm depends on unobserved
productivity and the level of capital stock installed:
Pr exitj;t+1 = 1j j ; kj
= Pr Vj
= f
j ; kj
<
j ; kj ;
where f1 < 0; f2 < 0. That is, the typical …rm that would exit would be
one with a low level of productivity and a low level of capital (this would
be a low value …rm).
Think about what this means for the correlation between unobserved productivity and observed capital in the "selected sample", i.e. in the sample
of survivors.
Firms with a lot of capital are likely to survive even if they have low
productivity, because they have high values.’
However …rms with little capital will only survive if they have high levels
of productivity.
Hence, in the sample of survivors there will be a negative correlation
between kj and unobserved productivity j .
Thus, if we estimate the production function
y j = 0 + k k j + l lj + j ;
this mechanism would tend to yield a downward bias in the coe¢ cient on
kj .
Measurement errors. In general, we expect measurement errors in inputs
to lead to downward bias (attenuation bias) in the estimated coe¢ cients.
Recall the attenuation bias formula:
yit = xit + vit;
where xit is the true but unobserved value of the explanatory variable, and
vit is a non-autocorrelated, homoskedastic error term with zero mean. We
observe an imperfect measure of xit , namely xit such that
xit = xit + uit;
where uit is a random measurement error uncorrelated with xit. Our
estimable equation is
yit = xit + (vit
uit) ;
so the regressor xit is correlated with the error term (vit
uit). It can
be shown that this will lead to a downward bias in the OLS estimate of
- that is, estimated is lower than true . To give you an idea of what
the bias looks like, consider the following formula showing the bias caused
by measurement errors:
p lim ^
OLS
=
2
x
2
x
!
;
+ 2u
where 2x is the variance of the true, unobserved explanatory variable, and
2 is the variance of the measurement error. The operator p lim can be
u
thought of as showing the value of estimated in a large sample. Loosely
speaking, this is what we can expect to get if there are measurement
errors in the explanatory variable. Clearly the higher the variance of the
measurement error, the more severe is the bias.
What happens if we take …rst di¤erences? Clearly,
p lim ^
FD
2
dx
=
2
dx
+
2
de
!
;
where d indicates that the variance refers to the di¤erenced variable. Assumed that the variance is constant over time and that the mean of z is
zero, it can be shown that
p lim
^F D
=
0
B
@
2
x
+
2
x
2 (1
e (1
e)
x )
1
C
A
where e is the serial correlation of the measurement errors and x is the
serial correlation of the true values of the regressors.
Now compare the following expressions:
OLS
^
p lim
=
p lim
^F D
=
Which one has the most severe bias?
2
x
2
0 x
B
@
2
x
!
;
+ 2e
+
2
x
2 (1
e (1
e)
x )
1
C
A:
The bias of the FD estimator will be more severe than that of the levels esti(1
)
mator if (1 e ) > 1, i.e. if x > e.
x
This is an important result. In most applications we assume that the serial
correlation of the measurement errors typically is quite small or zero, while the
serial correlation of the true unobserved explanatory variable is positive. In this
case …rst di¤erencing the data is bound to exacerbate the measurement error
bias, and OLS estimation of the levels equation would be preferable to the FD
model.
In practice, estimating the coe¢ cient on the capital stock whilst controlling
for …xed e¤ects has proved di¢ cult - see Söderbom and Teal, 2004, for
details.
4
Traditional solutions to the endogeneity problem
The two traditional solutions to endogeneity problems are instrumental variables and …xed e¤ects. We are now going to write the production function
as
yjt = k kjt + l ljt + ! jt + jt;
i.e. we have added time subscripts re‡ecting the panel dimension in the data;
and we have decomposed the residual into two components, ! jt + jt
! jt represents the part of TFP observable to the …rm but not to the
econometrician - hence this is the source of endogeneity problems. You
can think of ! jt as a measure of the managerial quality of the …rm. From
now on, we will refer to ! jt as ’unobserved productivity’.
on the other hand is assumed not to impact on the …rm’s input decisions. You can think of jt as representing measurement errors in output, for example (other interpretations are possible too; see Section 2.2 in
ABBP). What’s important is that jt is not a source of endogeneity bias.
jt
4.1
Instrumental Variables
Our problem: We want to estimate
yjt = k kjt + l ljt + ! jt + jt;
but we cannot use OLS, since
Cov ljt; ! jt 6= 0:
(It is likely, of course, that capital is endogenous too, but we abstract from that
possibility for the moment.)
Suppose an instrument zjt is available, that ful…lls the following conditions:
1. The instrument is valid (or exogenous):
cov zjt; ! jt = 0:
This is an exclusion restriction - zjt is excluded from the structural
equation (the production function).
2. The instrument is informative (or relevant). This means that the instrument zjt must be correlated with the endogenous regressor (labour in the
current example), conditional on all exogenous variables in the model (i.e.
capital, if this is thought exogenous). That is, if we assume there is a
linear relationship between ljt and zjt and kjt;
ljt = 0 + 1kjt + 1zjt + rjt;
(3)
where rjt is mean zero and uncorrelated with the variables on the righthand side, we require 1 6= 0.
Many economists take the view that, for instrumental variable estimation to be
convincing, the instruments used must be motivated by theory. Recall the …rstorder condition for labour derived above - with my slightly modi…ed notation
we get
ljt =
1
1
l
h
ln l + ln p
i
ln W + k kjt + ! jt :
This suggests the wage rate W might be a useful instrument:
– Our theory says it is (negatively) correlated with labour.
– The wage rate also must be uncorrelated with ! jt. This may not be
an entirely innocuous assumption to make. While the wage rate does
not directly enter the production function, wages might be correlated
with unobserved productivity for other reasons - e.g. if more productive
…rms have stronger market power in input markets - in which case the
wage will not be a valid instrument.
It also follows from the …rst-order condition above that the output price
is a potential instrument - however, that has been used less often in the
literature. Why might we be concerned about using the output price is an
instrument?
A similar way of reasoning can be applied for capital, if that is thought
endogenous (i.e. use the cost of capital as an instrument).
Five reasons why the IV approach based on prices as instruments has
not been very successful
1. Market power. Wages and capital prices (and output prices) could well
be correlated with unobserved productivity if input (output) markets are
not perfectly competitive: e.g. high unobserved productivity gives the …rm
market power and so enables it to in‡uence the price.
2. Wages and unobserved worker quality. When labour costs are reported
in …rm-level datasets, they typically come in the form of average wage per
worker, and you may well be concerned that the average wage in the …rm is
correlated with unobserved quality of the workforce. Since the unobserved
quality of the workforce likely impacts on unobserved productivity, this
would imply the average wage is an invalid instrument.
3. Law of one price. If, as is typically the case, one wants to include time
dummies in the production function, there must be variation in input prices
across …rms at a given point in time for these to be useful instruments.
If input markets are essentially national in scope, this seems unlikely. (If
average wages indeed vary across …rms in most datasets, you suspect this
is at least partly picking up unobserved worker quality).
4. Endogenous unobserved productivity. Suppose unobserved productivity
! jt actually depends on input choices - e.g. investment in modern technology raises productivity. In that case it will be hard to argue that input
prices are valid instruments, since these surely will impact on investment.
5. Attrition. A di¤erent kind of endogeneity problem sometimes discussed
in the literature is posed by endogenous attrition, i.e. that the …rm’s exit
decision depends on unobserved productivity as well as input prices (after
all, these jointly determine the pro…tability of the …rm). In such a case
we will have a Heckman type selection problem, in which all variables
determining the exit decision will go into the residual of the production
function in the selected sample. Clearly input prices cannot be used as
instruments in this case.
The common theme across these reasons is that prices are unlikely to be valid
instruments.
4.2
Fixed E¤ects
A second traditional solution to the endogeneity problem is …xed e¤ects estimation, which as you know requires panel data. One key assumption underlying
this approach is that unobserved productivity is constant over time,
! jt = ! j
but varies across …rms. We would now write the production function as
yjt = k kjt + l ljt + ! j + jt ;
and use perhaps the within estimator (’…xed e¤ects’ estimator) or the …rstdi¤erenced estimator to estimate the parameters - in the latter case for example
we would thus estimate
yjt
yj;t 1 = k kjt
kj;t 1 + l ljt
lj;t 1 +
jt
j;t 1 ;
using OLS (probably with …rm-clustered standard errors since the di¤erenced
residual is likely serially correlated).
Notice that the source of endogeneity bias has been eliminated, thus e¤ectively
solving the endogeneity problem (subject of course to strict exogeneity; see
earlier lectures in this course).
Three reasons why the …xed e¤ects approach has not been very successful
1. Time invariant unobserved productivity. The assumption that unobserved productivity is …xed over time is quite restrictive, especially in longer
panels.
2. Di¤erencing may exacerbate measurement error bias. When there
are measurement errors in inputs, the …xed e¤ects estimator may well be
more severely biased than the OLS estimator. Discuss.
3. Poor performance in practice. Fixed e¤ects estimates of the capital
coe¢ cient are often implausibly low, and estimated returns to scale is
often (severely) decreasing ( k + l << 1).
[EXAMPLE 1. To be discussed in class]
5
The Olley and Pakes (1996) approach
The Olley & Pakes (1996; henceforth OP) use a di¤erent approach to solve the
endogeneity problems discussed above. Similar to the IV approach, OP derive
their solution from the input demand equations, however OP do not require
factor prices to be observed. In what follows I will discuss a simpli…ed version
of the OP model.
The production function:
yjt = 0 + k kjt + l ljt + ! jt + jt :
(the original OP model also allows for an e¤ect of …rm age, but I ignore
that here).
Summary of key assumptions:
Labour is a ‡exible input chosen in period t, after observing productivity
! jt.
Capital is a "quasi-…xed" input chosen in period t 1 and subject to strictly
convex adjustment costs. Capital evolves according to the equation
Kjt = (1
) Kj;t 1 + Ij;t 1;
where Ij;t 1 denotes investment.
Unobserved productivity ! it is assumed to follow a …rst order Markov
process,
p ! j;t+1jf! j gt =0; Ijt = p ! j;t+1j! jt ;
where Ijt is the …rm’s information set in period t. This means that, given
the present information, future states are independent of the past states lags of the productivity variable do not provide additional information as
to what might happen to productivity in the future. Examples:
– Linear process
! jt = ! j;t 1 + jt:
– Nonlinear process
! jt = 1! j;t 1 + 2! 3j;t 1 + jt:
– Nonparametric process
! jt = f ! j;t 1 + jt:
Recall the linear process was adopted by Blundell and Bond in their analysis of
production functions based on US data.
The pro…t in period t is de…ned as
t
= pKjtk Ljtl exp
0
+ ! jt
WjtLjt
pI Ijt
G(Ijt; Kjt);
where p is the output price, pI is the price of one unit of capital, and
G(Ijt; Kjt) is the adjustment cost for capital. Note: labour is not a
function of jt since we’re assuming this term is just random noise (output
measurement error).
Since labour is assumed to be a ‡exible input, the static …rst-order condition applies:
k
l
l pKjt Ljt
1
exp
0
+ ! jt = Wjt;
0
l p exp
Ljt = @
0
+ ! jt
Wjt
1
1
1
l
A
1
Kjt
k
l:
Using this expression for labour in the pro…t function above, we can rewrite
pro…ts as
t
= (1
1
l) l
pI Ijt
l
1
l
p exp
0
+ ! jt
1
l
l
Wjt
l 1
1
Kjt
k
l
G(Ijt; Kjt);
or, in more reader-friendly notation,
1
t
= ' Wjt; ! jt Kjt
k
l
pI Ijt
G(Ijt; Kjt):
You see how the labour variable has "disappeared" - replaced by the variables and parameters determining Ljt as implied by the …rst-order condition for labour. Using a notation more similar to that in OP, we might
therefore write pro…ts as
t
=
kjt; ! jt
c Ijt
where jt is sales minus labour costs, and c Ijt is the cost of investment,
including strictly convex adjustment costs, for example
G(Ijt; Kjt) =
where
2
Ijt
Kjt
!2
Kjt:
is a parameter measuring the marginal adjustment cost of capital.
The …rm’s objective
The …nal important assumption underlying the OP framework concerns the
behaviour of the …rm.
It is assumed that the …rm chooses investment and employment to maximize the present value of current and expected future net revenues. We
have already seen how labour is "optimized out" at each period, which
means we can write the value of the …rm as a function of capital and
productivity only:
V (kjt; ! jt) = max Et
It
1
X
s=t
(s t)
h
kjs; ! js
c Ijs
i
;
where Et denotes expectation given the information available in period t,
and is a discount factor. The choice variable (or control variable) here
is investment in period t:
Note: the fact that labour is not visible in this equation does not mean
labour is irrelevant. Labour is not visible here because we have implicitly
replaced it by the variables and parameters determining labour as implied
by the …rst-order condition. Indeed, estimating the coe¢ cient on labour
in the production function is a central objective in the analysis.
Alternatively, we can write the value of the …rm recursively as a (stochastic)
Bellman equation:
V (kjt; ! jt) = max
It
kjt; ! jt
c Ijt + Et[V (kj;t+1; ! j;t+1)] (4)
The …rm’s investment demand
Key for the OP approach is the …rm’s investment. In a model of the form
outlined above, optimal investment in period t will depend on
– the existing capital stock; and
– expectations about the future pro…tability of capital.
The …rst-order Markov assumption implies that expected productivity in
the future depends on current, but not past, productivity.
OP hence write down an investment demand function of the following
form:
Ijt = It kjt; ! jt :
This function needs to be strictly increasing in unobserved productivity
for the OP procedure to work - a …rm with a high value of ! jt will invest
strictly more than a …rm with a low value of ! jt, conditional on kjt.
[EXAMPLE 2. From Bond, Söderbom and Wu, 2008. To be discussed in
class]
Controlling for the endogeneity of input choice We are now ready to
discuss the estimation strategy proposed by OP. Notice that this is motivated
by the theory discussed above.
The key "trick" in OP. Recall that investment is assumed to be a strictly
monotonic in ! jt. This implies that the investment demand function
Ijt = Ijt kjt; ! jt
can be inverted so that productivity is expressed as a function of investment and capital:
! jt = ht kjt; Ijt :
Intuitively, capital kjt and investment Ijt "tells" us what ! jt must be.
Now return (to the production function:
yjt = k kjt + l ljt + ! jt + jt :
Recall that unobserved productivity ! jt is a source of endogeneity bias.
We now use ! jt = ht kjt; Ijt and rewrite the production function as
yjt = k kjt + l ljt + ht kjt; Ijt + jt:
By including the function ht kjt; Ijt as an additional term on the righthand side, we have e¤ectively "controlled" for unobserved productivity.
Building on this, OP proposed a two stage procedure to estimate the
parameters l and k This works as follows.
First stage: De…ne
t kjt ; Ijt
= k kjt + ht kjt; Ijt ;
and rewrite the production function
yjt = k kjt + l ljt + ! jt + jt :
as
yjt = l ljt + t kjt; Ijt + jt:
In general, the function t is not linear. OP propose either approximating
t using a polynomial, e.g.
t kjt ; Ijt = 0 + 1 Ijt + 2 kjt + 3 Ijt
2 +
2
kjt + 4Ijt
5 kjt ;
or using kernel methods (nonparametric). In any case, what is clear now is
that, provided we control for t kjt; Ijt , we may be able to identify the
labour coe¢ cient l in the …rst stage. Indeed, if we use the polynomial
above, all we have to do is to estimate the following regression
yjt = 0 + l ljt + 1Ijt + 2kjt + 3 Ijt
2 +
2
kjt + 4Ijt
5 kjt + jt
using OLS.
[EXAMPLE 3: Applying the …rst-stage OP procedure to the Blundell-Bond
data. To be discussed in class.]
Second stage: We have now estimated l . In the second stage we shall
estimate the capital coe¢ cient k - this cannot be estimated in the …rst
stage. Note that the …rst-stage estimation will give us an estimate of the
function t, e.g.
^ t kjt; Ijt = ^ 0 + ^ 1Ijt + ^ 2kjt + ^ 3 Ijt
2 + ^ k2 ;
kjt + ^ 4Ijt
5 jt
if we are using the polynomial above.
It follows that
^ t kjt; Ijt = ^ jt
!
^ jt = h
k kjt :
Now, recall that unobserved productivity follows a …rst-order Markov process;
this means we can decompose ! jt as follows:
! jt = Et 1 ! jt + jt
! jt = g ! j;t 1 + jt;
where jt is the innovation (shock) to productivity. If productivity follows
a linear autoregressive process, for example, we would have
! jt = ! j;t 1 + jt;
c.f Blundell-Bond.
The production function, again:
yjt = k kjt + l ljt + ! jt + jt;
which given the insights above can be written
yjt
l ljt
= k kjt + g ! j;t 1 + jt + jt;
or
yjt
l ljt
= k kjt + g ^ j;t 1
0
k kj;t 1
+ jt + jt:
(5)
Now, because capital is chosen one period in advance, the residual jt + jt
will be uncorrelated with all the right-hand side variables (remember we
have already estimated l , which is why I have moved l ljt to the left-hand
side here).
Depending on how ‡exible you want to be, (5) can be estimated using
either OLS (if g is linear); NLLS (if g is a polynomial); or kernel methods
(if g is treated nonparametrically).
[EXAMPLE 4: Applying the second-stage OP procedure to the BlundellBond data. To be discussed in class.]
5.0.1
Discussion
Scalar unobservable assumption must hold, otherwise can’t invert.
– Suppose there are two stochastic components of unobserved productivity, so that
Ijt = It kjt; ! 1jt; ! 2jt ;
and suppse ! 1jt; ! 2jt follow di¤erent stochastic processes; for example,
let’s suppose ! 1jt is highly persistent whereas ! 2jt exhibits only moderate serial correlation.
– In such a case, ! 1jt and ! 2jt will impact di¤erently on investment. For
example, conditional on capital,
! 1jt =
>0
! 2jt = 0;
will give a stronger investment response than
! 1jt = 0
! 2jt =
> 0;
because the …rm understands that expected future pro…ts are higher in
the former case than in the latter case (since ! 1jt more persistent).
– Because ! 1jt; ! 2jt are both unobserved, we are stuck. The investment
demand equation cannot be inverted; put di¤erently, we can’t infer
from capital and investment the values of ! 1jt; ! 2jt separately. The OP
approach won’t work.
Zero investment levels potentially problematic Recall that investment
needs to be a strictly monotonic function of (scalar) unobserved productivity.
The presence of lots of zero investments in the data strongly indicates that this
is not the case - surely it’s wildly unrealistic to assume that all …rms that invest
nothing have precisely the same level of unobserved productivity (conditional
on capital).
If investment is irreversible, for example, the investment demand function will
not be a monotonic function of productivity, and there will be lots of investment
zeros in the data (see the graph taken from Bond, Söderbom and Wu, 2008).
Levinsohn & Petrin (2003) proposed using raw materials as a proxy for
unobserved productivity in such a case. Raw materials is rarely if ever zero
in datasets and so strict monotonicity might hold. Below I will discuss a
generalized approach in this vein proposed by Ackerberg, Caves and Frazer
(2006), so I will not discuss the Levinsohn-Petrin estimator here.
Alternatively, we can retain the OP approach provided we simply drop all
observations for which investment is equal to zero. Provided the model is
correctly speci…ed, such a procedure would control for unobserved productivity and yield consistent estimates.
Labour really ‡exible? The OP approach just described is really only appropriate if labour is a ‡exible input. If not, e.g. because …rms can’t easily hire
and …re workers from one day to another, then the investment demand function
speci…ed as part of the OP approach,
Ijt = It kjt; ! jt :
would no longer be correct - investment would depend on capital and unobserved productivity, but it would also depend on labour:
Ijt = It kjt; ! jt; ljt :
Inverting out ! jt would leave you with a function of the following form
~ t kjt; Ijt; ljt ;
! jt = h
and so it would clearly not be possible to identify anything in the …rst stage:
~ t kjt; Ijt; ljt + jt:
yjt = k kjt + l ljt + h
Collinearity and other issues
Ackerberg, Caves and Frazer (2006) noted that the parameter l on the
‡exible labour input is not identi…ed by estimating the …rst stage unless
in a pretty special case involving either serially uncorrelated wages or serially correlated optimization errors. More generally, parameters on ‡exible inputs in Cobb-Douglas production functions are not identi…ed from
cross-section variation if all …rms face common input prices and inputs are
optimally chosen (Bond & Söderbom, work in progress).
To see this, consider the f.o.c. for labour again:
ljt =
1
1
l
h
ln l + ln p
i
ln W + k kjt + ! jt :
Identi…cation of L in OP stage 1,
yjt = k kjt + l ljt + ! jt + jt :
requires variation across …rms in lit at given levels of kit and ! it (or, under
the assumptions of OP, at given levels of kit and ht kjt; Ijt ):
t kjt ; Ijt
= k kjt + ht kjt; Ijt ;
Yet the structure of the conditional labour demand function indicates that
variation across …rms in lit is fully explained by kit and ! it, if the real
wage is common to all …rms and the labour input is optimally chosen.
In general, identi…cation of parameters on ‡exible inputs from cross-section
variation thus requires either variation across …rms in the real price of those
inputs, or some form of optimization error in the choice of those inputs.
As discussed in Ackerberg, Caves and Frazer (2006), identi…cation of L
using the …rst stage of the Olley-Pakes estimation procedure further requires that any variation across …rms in the real wage, or any optimization
error in the choice of labour, must be serially uncorrelated. The reason
is that either persistent variation in real wages or persistent optimization
error in the choice of labour would a¤ect the decision rule for capital, i.e.
Ijt = It kjt; ! jt :
would no longer be correct - instead,
Ijt = It kjt; ! jt; Wjt :
implying that the unobserved level of log TFP could no longer be adequately proxied using a function of investment and capital alone.
For the same reason, consistent estimation of L from the …rst stage also
requires that there must be no variation across …rms in the cost of capital,
and no optimization error in the investment decision.
Thus, if inputs are optimally chosen, the only form of input price variation
that allows identi…cation of L using the …rst stage of the estimator proposed by Olley and Pakes (1996) is the presence of serially uncorrelated
variation across …rms in the real wage.
5.1
The Ackerberg, Caves and Frazer (2006) approach
Ackerberg, Caves and Frazer (2006; henceforth ACF) suggest an alternative
estimation approach that avoids some of the problems discussed above
(e.g. the potential collinearity problems), and that will work under less
restrictive assumptions than those underlying the OP model.
Just like OP (and LP), the ACF estimator is a two-step estimator. The
main di¤erence between the ACF approach and the OP (and LP) approach
is that, with the ACF approach, no coe¢ cients of interest will be estimated
in the …rst stage of estimation. Instead, all input coe¢ cients are estimated
in the second stage. As we shall see, the …rst stage is still important,
however.
A useful starting point for the ACF approach is a three-factor CobbDouglas output production function,
qjt = k kjt + l ljt + mmjt + !
~ jt;
where qjt is output, and I have added log raw materials, denoted mit, to
the basic speci…cation used above (a constant is subsumed in !
~ jt). Raw
materials is assumed a perfectly ‡exible input, and so the following static
…rst order condition applies:
pQjt
= pm;
m
Mjt
where Qjt is the level of output, Mjt is the level of raw materials and pm
is the unit price of raw materials, assumed constant in the cross-section.
Note that raw materials will be proportional to output. Using this in the
output production, we can obtain a value-added production function as
follows:
yjt = k kjt + l ljt + ! jt:
Supposing that there are measurement errors in value-added, we modify
this accordingly:
yjt = k kjt + l ljt + ! jt + jt;
where jt denotes the measurement error as usual (note: jt doesn’t have
to be measurement error, it can be a "real" shock to output, provided it
does not impact on any of the factor inputs - see ABBP for a discussion).
So the raw materials variable has disappeared from the scene. Why did
we introduce it then? The answer is that the raw materials variable will
play a role similar to that played by investment in the OP model - i.e. as
a proxy for unobserved productivity.
Having set the scene, we are now ready to consider the ACF approach. Key
assumptions are as follows
Materials is a ‡exible input chosen in period t, after observing productivity
! jt.
Capital is a "quasi-…xed" input chosen in period t
strictly convex adjustment costs.
1 and subject to
Labour is chosen before material inputs, but after capital has been chosen.
in period t. Suppose labour is chosen at time t 0:5.
Unobserved productivity ! it is assumed to follow a …rst order Markov
process between the subperiods t 1; t 0:5, and t:
p ! jtjIj;t 0:5 = p ! jtj! j;t 0:5 ;
and
p ! j;t 0:5jIj;t 1 = p ! j;t 0:5j! j;t 1 :
Capital for period t production is decided in view of ! j;t 1 and the …rm’s
capital in t 1.
Labour for period t production is decided in view of ! j;t 0:5 and the
…rm’s capital in t (which is already known at this point). This may be
quite realistic: labour decisions need to be made in advance, since new
workers need to be trained or worker to be laid o¤ will have to be given
some period of notice.
Materials for period t production is decided in view of ! jt and the …rm’s
capital and labour in period t, both of which are known at this point:
mjt = ft ! jt; kjt; ljt :
Key "trick" in ACF. Under the assumption that materials is a strictly
monotonic (increasing, to be consistent with the theory) function of ! jt,
conditional on capital and labour, we can invert this function for ! jt, along
the same lines as in OP:
! jt = ft 1 mjt; kjt; ljt ;
and so we rewrite the value-added function
yjt = k kjt + l ljt + ! jt + jt;
as
yjt = k kjt + l ljt + ft 1 mjt; kjt; ljt + jt:
Note the close similarity with OP: by including the function ft 1 mjt; kjt; ljt
as an additional term on the right-hand side, we have e¤ectively "controlled" for unobserved productivity. The remaining residual jt is innocuous since it has no impact on factor inputs (e.g. because it’s simply
measurement error in output).
The snag here is that no parameter of interest can be identi…ed based on
this speci…cation. But don’t let that distract you. The key goal in the …rst
stage is to get rid of the jt term - why this is desirable will be clearer
later.
First stage. Regress log value added on a polynomial function of capital,
labour and raw materials, e.g.
yjt =
+ 1tkjt + 2tljt + 3tmjt +
2 +
2
2
+ 4tkjt
5tljt + 6tmjt
+ 7tkjtmjt + 8tljtmjt + 9tkjtljt
+ jt;
0t
using OLS. The estimated -parameters are not the parameters of interest.
Now de…ne
= k kjt + l ljt + ft 1 mjt; kjt; ljt ;
t =
k kjt + l ljt + ! jt
which represents log value added, net of the term jt. Having estimated
the …rst stage regression, we can thus estimate t simply by requesting
the predicted values.
t
The next task is to decompose unobserved productivity:
! jt = Et 1 ! jt + jt;
or
! jt = E ! jtj! j;t 1 + jt;
(remember …rst order Markov property), where jt is independent of all
information known in period t 1.
Given the timing assumption that capital was decided in period t
must then be that
E
h
jt kjt
i
= 0;
1, it
(6)
which you recognize is an orthogonality condition that can be used to
estimate the parameters of interest.
Labour on the other hand is chosen in t 0:5, and so at that time part of
jt has been observed - hence, lit is not uncorrelated with jt :
E
h
i
jt ljt 6= 0:
However, lagged labour, li;t 1, was chosen in period t 0:5 1, and so
at that point nothing was known about the innovation to productivity in
period t:
E
h
jt lj;t 1
i
(7)
= 0:
The two moments (6) and (7) can therefore be used to identify
l . This is what happens in the second stage.
k
and
Second stage. We have the following population moments:
E
E
h
h
jt kjt
jt lj;t 1
i
i
= 0
= 0;
Provided we have a random sample, we can appeal to the analogy principle and replace population moments by sample moments. We can then
obtain consistent estimates of k and l by minimizing the criterion function
2
4
T X
N
X
t=1 i=1
jt
"
kjt
#30
5
2
C 4
lj;t 1
(1 x 2)
N
T X
X
t=1 i=1
jt
"
(2 x 2)
kjt
lj;t 1
#3
5
(2 x 1)
with respect to the parameters k ; l . Since the model is exactly identi…ed,
the choice of C is irrelevant - the minimum will always occur at zero.
The computation of jt. We saw above that
t
= k kjt + l ljt + ! jt;
hence
! jt =
t
k kjt
l ljt :
Also, remember that we have an estimate of t from the …rst stage; thus,
conditional on the parameters k ; l we can compute ! jt.
Moreover, remember that
! jt = E ! jtj! j;t 1 + jt;
which we write as a nonparametric regression:
! jt = ' ! j;t 1 + jt:
This suggests the following estimation recipe, for the second stage:
G
;
1. Guess k ; l - denote these by G
k
l
2. Compute
^
!G
jt = t
Gk
k jt
Gl
l jt
(remember ^ t is …xed since estimated in the …rst stage).
G
3. Regress ! G
jt on ! j;t 1 using some suitable technique - e.g. linear regression (perhaps allowing for a polynomial), or nonparametric techniques.
Compute the productivity innovation, based on the results:
G
jt
= !G
jt
'
^ !G
j;t 1 :
4. Compute the criterion function
2
4
T X
N
X
t=1 i=1
jt
"
kjt
lj;t 1
#30
5
2
C 4
N
T X
X
t=1 i=1
jt
"
kjt
lj;t 1
#3
5
5. Check if this looks like the global minimum; if it does, then STOP (you
G and go
have obtained your estimates); if not, judiciously change G
;
k
l
back to step 2 above.
Of course, you’d use some pre-programmed minimization routine to do this.
To get standard errors, the easiest procedure is probably to rely on bootstrapping (you should include the …rst stage as well).
Relation between ACF and DPD approach
ACF discuss how their estimator compares with the type of estimator
("DPD") used by Blundell and Bond. The identify distinct advantages
and disadvantages of both approaches.
The main advantage of ACF:
– Unobserved productivity can follow an arbitrary …rst order Markov
process. That is, ACF can accommodate a nonparametric process,
such as ! jt = f ! j;t 1 + jt. This is not possible with the DPD
approach. ACF can do this because they recover the unobserved productivity term ! jt. 1st stage estimation is important in this context,
as this procedure eliminates the "irrelevant" part of the residual (e.g.
output measurement error).
The main advantage of DPD:
– Easy to allow for …rm …xed e¤ects - i.e. unobserved time invariant
heterogeneity across …rms. Recall we have suggested earlier in this
course that this is arguably the main advantage of having panel data.
This is not possible in the ACF approach, if one were modelling the
dynamics of ! jt nonparametrically.
However, the estimators can be made very similar to each other - if we were
using a linear AR1 model for productivity, so that ! jt = ! j;t 1 + jt,
then the second stage of ACF would literally be the dynamic COMFAC
model adopted by Blundell & Bond.
ERSA Training Workshop
Måns Söderbom
University of Gothenburg, Sweden
[email protected]
January 2009
Estimation of Production Functions with Micro Data
Example 1
OLS and FE results based on firm-level panel data on Ghanaian manufacturing firms
. xi: reg lvad lk lemp i.year, cluster(firm)
i.year
_Iyear_1992-2000
(naturally coded; _Iyear_1992 omitted)
Linear regression
Number of obs
F( 10,
246)
Prob > F
R-squared
Root MSE
=
=
=
=
=
1438
143.11
0.0000
0.7385
1.1832
(Std. Err. adjusted for 247 clusters in firm)
-----------------------------------------------------------------------------|
Robust
lvad |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------lk |
.2926854
.034091
8.59
0.000
.225538
.3598328
lemp |
.7983413
.0644447
12.39
0.000
.6714075
.9252751
_Iyear_1993 |
.0893364
.0910193
0.98
0.327
-.0899401
.268613
_Iyear_1994 |
.1384356
.104344
1.33
0.186
-.0670859
.3439571
_Iyear_1995 |
.5282265
.0917705
5.76
0.000
.3474704
.7089825
_Iyear_1996 |
.3443942
.1054644
3.27
0.001
.1366657
.5521226
_Iyear_1997 | -.0112335
.1326423
-0.08
0.933
-.272493
.250026
_Iyear_1998 |
.0657469
.1221722
0.54
0.591
-.17489
.3063838
_Iyear_1999 |
.2862481
.1229291
2.33
0.021
.0441203
.528376
_Iyear_2000 |
.2987483
.1240461
2.41
0.017
.0544203
.5430764
_cons |
4.662286
.2365325
19.71
0.000
4.196399
5.128173
-----------------------------------------------------------------------------. test lk+lemp=1
( 1)
lk + lemp = 1
F(
1,
246) =
Prob > F =
4.93
0.0274
1
. xi: xtreg lvad lk lemp i.year, fe cluster(firm)
i.year
_Iyear_1992-2000
(naturally coded; _Iyear_1992 omitted)
Fixed-effects (within) regression
Group variable: firm
Number of obs
Number of groups
=
=
1438
247
R-sq:
Obs per group: min =
avg =
max =
1
5.8
9
within = 0.0938
between = 0.8142
overall = 0.7074
corr(u_i, Xb)
F(10,246)
Prob > F
= 0.7317
=
=
10.87
0.0000
(Std. Err. adjusted for 247 clusters in firm)
-----------------------------------------------------------------------------|
Robust
lvad |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------lk |
.0481399
.1981945
0.24
0.808
-.3422347
.4385146
lemp |
.5240789
.1008461
5.20
0.000
.325447
.7227108
_Iyear_1993 |
.1918263
.0920392
2.08
0.038
.0105409
.3731117
_Iyear_1994 |
.2632723
.108263
2.43
0.016
.0500317
.4765129
_Iyear_1995 |
.5801393
.101955
5.69
0.000
.3793231
.7809555
_Iyear_1996 |
.4213335
.1173489
3.59
0.000
.1901968
.6524702
_Iyear_1997 |
.0633159
.1362419
0.46
0.643
-.2050336
.3316654
_Iyear_1998 |
.1064584
.1312125
0.81
0.418
-.1519848
.3649016
_Iyear_1999 |
.3280545
.1301013
2.52
0.012
.0717999
.5843091
_Iyear_2000 |
.3077651
.1381064
2.23
0.027
.0357432
.579787
_cons |
8.0316
1.989564
4.04
0.000
4.112848
11.95035
-------------+---------------------------------------------------------------sigma_u | 1.4748081
sigma_e | .87081827
rho |
.7414847
(fraction of variance due to u_i)
-----------------------------------------------------------------------------. test lk+lemp=1
( 1)
lk + lemp = 1
F(
1,
246) =
Prob > F =
4.51
0.0348
2
Figure 1: Investment Policy for Quadratic Adjustment Costs Only
Quadratic Adjustment Costs Only
1
0.8
0.6
0.4
I/K
t
0.2
t
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
const1*Zt/Kt-1
Figure 2: Investment Policy for Partial Irreversibility Only
Partial Irreversibility Only
1
0.8
0.6
0.4
t
I/K
t
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
const1*Zt/Kt-1
Figure 3: Investment Policy for Fixed Adjustment Costs Only
Fixed Adjustment Costs Only
1
0.8
0.6
0.4
t
I/K
t
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
const1*Zt/Kt-1
59
0.4
0.6
0.8
1
Example 3: OP estimate of the labour coefficient – BB data
use usbal89.dta, clear
tsset id year
ge ik=d.k
ge ikk=ik*k
ge ik2=ik^2
ge k2=k^2
ge
ge
ge
ge
ik3=ik^3
k3=k^3
ik2k=ik2*k
ikk2=ik*k2
a) OLS, no OP correction for endogeneity
. xi: reg y k n i.year, robust cluster(id)
i.year
_Iyear_1982-1989
(naturally coded; _Iyear_1982 omitted)
Linear regression
Number of obs
F( 9,
508)
Prob > F
R-squared
Root MSE
=
4072
= 2507.63
= 0.0000
= 0.9693
= .35256
(Std. Err. adjusted for 509 clusters in id)
-----------------------------------------------------------------------------|
Robust
y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------k |
.4322828
.0274846
15.73
0.000
.3782853
.4862803
n |
.5578836
.0308763
18.07
0.000
.4972227
.6185445
_Iyear_1983 | -.0568626
.0083657
-6.80
0.000
-.0732982
-.0404269
_Iyear_1984 |
-.050041
.0110933
-4.51
0.000
-.0718355
-.0282465
_Iyear_1985 | -.0875714
.0135255
-6.47
0.000
-.1141442
-.0609987
_Iyear_1986 |
-.092866
.016461
-5.64
0.000
-.125206
-.0605259
_Iyear_1987 | -.0580931
.0174944
-3.32
0.001
-.0924634
-.0237228
_Iyear_1988 | -.0211632
.0185846
-1.14
0.255
-.0576754
.015349
_Iyear_1989 | -.0382923
.020265
-1.89
0.059
-.0781058
.0015213
_cons |
3.046843
.0915369
33.29
0.000
2.867005
3.22668
------------------------------------------------------------------------------
3
b) OP with linear terms only, interacted with time
. xi: reg y i.year*k i.year|ik n , robust cluster(id)
i.year
_Iyear_1982-1989
(naturally coded; _Iyear_1982 omitted)
i.year*k
_IyeaXk_#
(coded as above)
i.year|ik
_IyeaXik_#
(coded as above)
Linear regression
Number of obs
F( 21,
508)
Prob > F
R-squared
Root MSE
=
3563
= 1127.34
= 0.0000
= 0.9692
= .35206
(Std. Err. adjusted for 509 clusters in id)
-----------------------------------------------------------------------------|
Robust
y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------_Iyear_1983 | -.0117025
.0397574
-0.29
0.769
-.0898116
.0664066
_Iyear_1984 | -.0003504
.0386859
-0.01
0.993
-.0763545
.0756537
_Iyear_1985 |
.0087549
.0338052
0.26
0.796
-.0576604
.0751701
_Iyear_1986 | -.0039311
.0271325
-0.14
0.885
-.0572368
.0493746
_Iyear_1987 | -.0064952
.0213817
-0.30
0.761
-.0485027
.0355122
_Iyear_1988 | (dropped)
_Iyear_1989 | -.0085923
.0218335
-0.39
0.694
-.0514875
.0343028
k |
.4305507
.0278018
15.49
0.000
.37593
.4851714
_IyeaXk_1983 |
.0003229
.0039407
0.08
0.935
-.0074191
.0080649
_IyeaXk_1984 | (dropped)
_IyeaXk_1985 | -.0045234
.0034495
-1.31
0.190
-.0113004
.0022536
_IyeaXk_1986 | -.0071593
.0049993
-1.43
0.153
-.0169812
.0026626
_IyeaXk_1987 |
.0011513
.0052254
0.22
0.826
-.0091146
.0114173
_IyeaXk_1988 |
.0069001
.0056709
1.22
0.224
-.0042412
.0180414
_IyeaXk_1989 |
.0056234
.0059086
0.95
0.342
-.0059849
.0172316
ik | -.1679462
.1046535
-1.60
0.109
-.3735531
.0376607
_IyeaXi~1983 |
.2370482
.1352143
1.75
0.080
-.0285999
.5026963
_IyeaXi~1984 |
.2041958
.1285209
1.59
0.113
-.048302
.4566937
_IyeaXi~1985 | (dropped)
_IyeaXi~1986 |
.2056813
.1355581
1.52
0.130
-.0606422
.4720048
_IyeaXi~1987 |
.1325166
.1263455
1.05
0.295
-.1157073
.3807406
_IyeaXi~1988 |
.1283877
.148174
0.87
0.387
-.1627217
.419497
_IyeaXi~1989 |
.124821
.1406473
0.89
0.375
-.1515009
.401143
n |
.5609091
.0303342
18.49
0.000
.5013132
.620505
_cons |
2.996726
.0999766
29.97
0.000
2.800308
3.193145
-----------------------------------------------------------------------------.
4
c) OP, 2nd order polynomial terms, interacted with time
. xi: reg y i.year*k i.year|ik i.year|ikk i.year|ik2
i.year|k3 i.year|ik2k i.year|ikk2 n , robust cluster(id)
i.year
i.year*k
i.year|ik
i.year|ikk
i.year|ik2
i.year|k2
i.year|ik3
i.year|k3
i.year|ik2k
i.year|ikk2
Linear regression
_Iyear_1982-1989
_IyeaXk_#
_IyeaXik_#
_IyeaXikk_#
_IyeaXik2_#
_IyeaXk2_#
_IyeaXik3_#
_IyeaXk3_#
_IyeaXik2a#
_IyeaXikka#
i.year|k2
i.year|ik3
(naturally coded; _Iyear_1982 omitted)
(coded as above)
(coded as above)
(coded as above)
(coded as above)
(coded as above)
(coded as above)
(coded as above)
(coded as above)
(coded as above)
Number of obs
F( 70,
508)
Prob > F
R-squared
Root MSE
=
=
=
=
=
3563
628.25
0.0000
0.9708
.34482
(Std. Err. adjusted for 509 clusters in id)
-----------------------------------------------------------------------------|
Robust
y |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------_Iyear_1983 | -.1269729
.1447766
-0.88
0.381
-.4114075
.1574616
_Iyear_1984 |
.0142485
.1417061
0.10
0.920
-.2641536
.2926507
_Iyear_1985 |
-.125383
.1042455
-1.20
0.230
-.3301883
.0794223
_Iyear_1986 | -.0997699
.1192445
-0.84
0.403
-.3340429
.1345031
_Iyear_1987 | (dropped)
_Iyear_1988 |
.1148153
.1008125
1.14
0.255
-.0832454
.312876
_Iyear_1989 |
.0660589
.1243963
0.53
0.596
-.1783356
.3104534
k | (dropped)
_IyeaXk_1983 |
.4731363
.0809219
5.85
0.000
.3141534
.6321191
_IyeaXk_1984 |
.4266294
.0900538
4.74
0.000
.2497057
.6035532
_IyeaXk_1985 |
.4882909
.0942897
5.18
0.000
.3030452
.6735366
_IyeaXk_1986 |
.4260074
.0949073
4.49
0.000
.2395482
.6124665
_IyeaXk_1987 |
.3817328
.0918138
4.16
0.000
.2013512
.5621144
_IyeaXk_1988 |
.3188845
.0938174
3.40
0.001
.1345666
.5032024
_IyeaXk_1989 |
.341171
.101227
3.37
0.001
.1422958
.5400461
ik |
.4519549
.6751802
0.67
0.504
-.8745344
1.778444
_IyeaXik_1~3 |
.190244
.7111746
0.27
0.789
-1.206961
1.587449
_IyeaXik_1~4 | -.7517249
.8771012
-0.86
0.392
-2.474917
.9714673
_IyeaXik_1~5 | (dropped)
_IyeaXik_1~6 | -.5032231
.8679689
-0.58
0.562
-2.208474
1.202027
_IyeaXik_1~7 | -1.070335
.8999296
-1.19
0.235
-2.838377
.6977074
_IyeaXik_1~8 | -.5777575
.8575433
-0.67
0.501
-2.262525
1.10701
_IyeaXik_1~9 | -.7952576
.8732094
-0.91
0.363
-2.510804
.9202887
ikk |
.2549711
.2312303
1.10
0.271
-.1993143
.7092564
_IyeaXikk_~3 | -.6242853
.3032464
-2.06
0.040
-1.220057
-.0285138
_IyeaXikk_~4 | -.1361169
.3207742
-0.42
0.671
-.7663243
.4940905
_IyeaXikk_~5 | -.5145717
.3433347
-1.50
0.135
-1.189102
.159959
_IyeaXikk_~6 | -.0358771
.3145985
-0.11
0.909
-.6539514
.5821971
_IyeaXikk_~7 |
.1029058
.3823818
0.27
0.788
-.6483386
.8541503
_IyeaXikk_~8 | -.1844971
.2655423
-0.69
0.488
-.7061934
.3371993
_IyeaXikk_~9 | (dropped)
5
ik2 |
.0286008
.6650747
0.04
0.966
-1.278035
1.335236
_IyeaXik2_~3 |
.1570187
.715695
0.22
0.826
-1.249068
1.563105
_IyeaXik2_~4 | -.1403274
.7453135
-0.19
0.851
-1.604604
1.323949
_IyeaXik2_~5 | (dropped)
_IyeaXik2_~6 |
.7096964
.7661279
0.93
0.355
-.7954727
2.214866
_IyeaXik2_~7 |
.4335649
.8305117
0.52
0.602
-1.198095
2.065225
_IyeaXik2_~8 | -.0142493
.8681223
-0.02
0.987
-1.719801
1.691303
_IyeaXik2_~9 |
1.113024
.8462974
1.32
0.189
-.5496499
2.775698
k2 | -.0228113
.016048
-1.42
0.156
-.0543398
.0087173
_IyeaXk2_1~3 | -.0049018
.0093979
-0.52
0.602
-.0233655
.0135618
_IyeaXk2_1~4 | (dropped)
_IyeaXk2_1~5 | -.0073795
.0095139
-0.78
0.438
-.0260709
.0113119
_IyeaXk2_1~6 |
.0075806
.0131017
0.58
0.563
-.0181596
.0333208
_IyeaXk2_1~7 |
.0157766
.0113246
1.39
0.164
-.0064723
.0380255
_IyeaXk2_1~8 |
.0279305
.0149167
1.87
0.062
-.0013755
.0572365
_IyeaXk2_1~9 |
.022291
.0181463
1.23
0.220
-.01336
.0579421
ik3 | -.6825286
.2868345
-2.38
0.018
-1.246056
-.1190008
_IyeaXik3_~3 |
.5448936
.2999866
1.82
0.070
-.0444736
1.134261
_IyeaXik3_~4 |
.7374314
.3611197
2.04
0.042
.0279594
1.446903
_IyeaXik3_~5 |
.5114732
.3572593
1.43
0.153
-.1904143
1.213361
_IyeaXik3_~6 | (dropped)
_IyeaXik3_~7 |
.3755959
.3004034
1.25
0.212
-.21459
.9657818
_IyeaXik3_~8 |
.9349795
.3288258
2.84
0.005
.2889535
1.581006
_IyeaXik3_~9 |
.051696
.305618
0.17
0.866
-.5487349
.6521268
k3 |
.0022769
.0009018
2.52
0.012
.0005053
.0040486
_IyeaXk3_1~3 | (dropped)
_IyeaXk3_1~4 |
-.00012
.0004725
-0.25
0.800
-.0010482
.0008083
_IyeaXk3_1~5 |
.0001134
.0006357
0.18
0.858
-.0011356
.0013624
_IyeaXk3_1~6 | -.0008324
.0007789
-1.07
0.286
-.0023627
.0006979
_IyeaXk3_1~7 | -.0012529
.0007453
-1.68
0.093
-.0027172
.0002114
_IyeaXk3_1~8 | -.0019506
.0009004
-2.17
0.031
-.0037196
-.0001815
_IyeaXk3_1~9 | -.0015104
.0010739
-1.41
0.160
-.0036202
.0005994
ik2k |
.2063111
.1328624
1.55
0.121
-.0547163
.4673386
_IyeaXik2a~3 | -.2057004
.1397434
-1.47
0.142
-.4802466
.0688457
_IyeaXik2a~4 |
-.23376
.1737731
-1.35
0.179
-.5751625
.1076424
_IyeaXik2a~5 | -.1579847
.1739745
-0.91
0.364
-.4997828
.1838134
_IyeaXik2a~6 |
-.408215
.1566099
-2.61
0.009
-.7158978
-.1005321
_IyeaXik2a~7 | -.3061767
.1599863
-1.91
0.056
-.6204929
.0081394
_IyeaXik2a~8 | (dropped)
_IyeaXik2a~9 | -.3404561
.13897
-2.45
0.015
-.6134827
-.0674294
ikk2 |
.0513622
.0248273
2.07
0.039
.0025853
.1001391
_IyeaXikka~3 | (dropped)
_IyeaXikka~4 | -.0542164
.0271281
-2.00
0.046
-.1075135
-.0009194
_IyeaXikka~5 | -.0298157
.0324681
-0.92
0.359
-.0936039
.0339725
_IyeaXikka~6 | -.0789603
.0330882
-2.39
0.017
-.1439668
-.0139538
_IyeaXikka~7 | -.0868584
.0349478
-2.49
0.013
-.1555184
-.0181985
_IyeaXikka~8 | -.0577613
.0301529
-1.92
0.056
-.117001
.0014783
_IyeaXikka~9 | -.0777462
.0320057
-2.43
0.015
-.1406262
-.0148663
n |
.5939467
.0296368
20.04
0.000
.535721
.6521725
_cons |
3.194397
.1735615
18.40
0.000
2.85341
3.535384
------------------------------------------------------------------------------
6
Example 4: OP estimate of the capital coefficient – BB data
Based on 2nd order polynomial in 1st stage (see (c) in Example 3)
/* 2nd order polynomial terms , interacted with time */
xi: reg y i.year*k i.year|ik i.year|ikk i.year|ik2 i.year|k2 i.year|ik3
i.year|k3 i.year|ik2k i.year|ikk2 n , robust cluster(id)
(output omitted)
predict predy, xb
ge phihat=predy-_b[n]*n
ge phihat_1=l.phihat
keep if e(sample)==1
drop if phihat_1==. | k_1==.
ge y_bn=y-_b[n]*n
. nl (y_bn = {b0} + {bk} * k + {bm}*(phihat_1 - {bk} * k_1) + {bm2}*(phihat_1 {bk} * k_1)^2), initial(b0 3 bk 0.3 bm 0.1 bm2 0)
(obs = 3054)
Iteration
Iteration
Iteration
Iteration
Iteration
Iteration
Iteration
Iteration
0:
1:
2:
3:
4:
5:
6:
7:
residual
residual
residual
residual
residual
residual
residual
residual
SS
SS
SS
SS
SS
SS
SS
SS
=
=
=
=
=
=
=
=
404.649
364.3411
359.907
359.6591
359.6583
359.6583
359.6583
359.6583
Source |
SS
df
MS
-------------+-----------------------------Model | 2474.18542
3 824.728475
Residual | 359.658315 3050 .117920759
-------------+-----------------------------Total | 2833.84374 3053 .928216095
Number of obs
R-squared
Adj R-squared
Root MSE
Res. dev.
=
=
=
=
=
3054
0.8731
0.8730
.3433959
2134.209
-----------------------------------------------------------------------------y_bn |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------/b0 |
12.81522
2.699174
4.75
0.000
7.522835
18.1076
/bk |
.3875877
.0072252
53.64
0.000
.3734209
.4017544
/bm | -6.683843
1.653108
-4.04
0.000
-9.925161
-3.442526
/bm2 |
1.148054
.2538301
4.52
0.000
.6503589
1.645749
-----------------------------------------------------------------------------Parameter b0 taken as constant term in model & ANOVA table
7