estimation of production functions: fixed effects in panel data
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Estimation of Production Functions: Fixed Effects in Panel Data. Lecture VIII. Analysis of Covariance. Looking at a representative regression model - PowerPoint PPT PresentationTRANSCRIPT
Analysis of Covariance
Looking at a representative regression model
It is well known that ordinary least squares (OLS) regressions of y on x and z are best linear unbiased estimators (BLUE) of α, β, and γ
* 1,
1,it it it ity x z u i N
t T
However, the results are corrupted if we do not observe z. Specifically if the covariance of x and z are correlated, then OLS estimates of the β are biased.
However, if repeated observations of a group of individuals are available (i.e., panel or longitudinal data) they may us to get rid of the effect of z.
For example if zit = zi (or the unobserved variable is the same for each individual across time), the effect of the unobserved variables can be removed by first-differencing the dependent and independent variables
, 1 , 1 , 1 , 1it i t it i t it i t it i ty y x x z z u u
Similarly if zit = zt (or the unobserved variables are the same for every individual at a any point in time) we can derive a consistent estimator by subtracting the mean of the dependent and independent variables for each individual
it i it i it i it iy y x x z z u u
OLS estimators then provide unbiased and consistent estimates of β.
Unfortunately, if we have a cross-sectional dataset (i.e., T = 1) or a single time-series (i.e., N = 1) these transformations cannot be used.
Next, starting from the pooled estimates
Case I: Heterogeneous intercepts (αi ≠ α) and a homogeneous slope (βi = β).
*it i it ity x u
*it it ity x u
Empirical Procedure
From the general model, we pose three different hypotheses: H1: Regression slope coefficients are
identical and the intercepts are not. H2: Regression intercepts are the same
and the slope coefficients are not. H3: Both slopes and the intercepts are the
same.
Estimation of different slopes and intercepts
1 1
1 1T T
i it i itt t
y y x xT T
, ,
, ,1 1
2
,1
ˆ ˆˆ 1,XX i XY i i i i i
T T
XX i it i it i XY i it i it it t
T
YY i it it
W W y x i N
W x x x x W x x y y
W y y
Covariance Matrices
X'X Nitrogen Phosphorous Potash X'Y beta alpha
Illinois
Nitrogen 1.2823 0.7194 1.5488 0.7415 0.7985 3.7917
Phosphorous 0.7160 0.6410 1.0156 0.2204-0.9813
Potash 1.5427 1.0174 2.0326 0.7894 0.2734
Indiana
Nitrogen 1.0346 0.2489 0.7220 0.6577 0.4386 3.6162
Phosphorous 0.2348 0.3717 0.2320 -0.0913-0.8905
Potash 0.7268 0.2448 0.6072 0.4587 0.5894
Pooled
Nitrogen 2.3168 0.9683 2.2708 1.3992 0.5924 3.9789
Phosphorous 0.9508 1.0128 1.2475 0.1291-0.9335 3.8851
Potash 2.2695 1.2622 2.6398 1.2481 0.4098
Estimation of different intercepts with the same slope
1 *
, ,1 1
,1
ˆ ˆ 1,W XX XY i i W i
N N
XX XX i XY XY ii i
N
YY YY ii
W W y x i N
W W W W
W W
12 YY XY XX XYS W W W W
Estimation of homogeneous slopes and intercepts
1 *
1 1
1 1
2
1 1
1 1 1 1
ˆ ˆˆ
1 1
XX XY
N T
XX it iti t
N T
XY it iti t
N T
YY iti t
N T N T
it iti t i t
T T y x
T x x x x
T x x y y
T y y
y y x xNT NT
Testing first for pooling both the slope and intercept terms:
* * *3 1 2
1 2
3 1
31
:
1 1~ 1 1 , 1
1
N
N
H
S SN K
F F N K NT N KSNT N K
If this hypothesis is rejected, we then test for homogeneity of the slopes, but heterogeneity of the constants
1 1 2
2 1
11
:
1~ 1 , 1
1
NH
S SN K
F F N K NT N KSNT N K
1 1 1 2 1 1
2 1 2 2 2 2
1 2
1
1 1 2
2
1 1 1
0 0
i i i Ki
i i i Kii i
it iT iT KiT
T
i T i i i iN
i i i T i j
y x x x
y x x xy x
y x x x
e M e
u M u u u u
E u E u u I E u u i j
Given this formulation, we know the OLS estimation of
The OLS estimation of α and β are obtained by minimizing
*it i it ity x u
* *
1 1
N N
i i i i i i i ii i
S u u y e x y e x
*
1 1
1 1 1 1
ˆ 1,
1 1
ˆ
i i i
T T
i it i iti t
N T N T
CV it i it i it i it ii t i t
y x i N
y y x xT T
x x x x x x y y
Sweeping the data
1TQ I eeT
1 1 1 11 4 4 4 41 0 0 0 1 1 1 11 1 1 110 1 0 0 1 1 1 1 4 4 4 41
40 0 1 0 1 1 1 1 1 1 1 114 4 4 40 0 0 1 1 1 1 1 1 1 1 114 4 4 4