Hedonic Price Models with Omitted
Variables and Measurement Errors
A Constrained Autoregression Structural Equation Modeling Approach
with Application to Urban Indonesia
Yusep Suparman (Universitas Padjadjaran)
Henk Folmer (Groningen University)
Johan HL Oud (Radboud University Nijmegen)
Typical Problems in HP Studies
(i)Under-specification: variables that actually
belong to the true population model are
missing.
• Upward bias of the coefficients of the
observed variables if the omitted variables are
positively correlated with the observed
variables and have a positive impact on the
dependent variable
Problem (ii)
(ii) Measurement errors in explanatory variable.
Leads to attenuation of the estimator of that
variable and arbitrary biases of the
estimators of the coefficients of the other
variables.
Constrained Autoregression-Structural
Equation Modeling (ASEM)
• Omitted variables
Let the price of house i at time t  pit  be determined
by a+b systematic house characteristics q1it ,, qa b it
according to the linear function
a b
(1)
pit   0t    jt q jit   it
j 1
with  0t the intercept,  jt the marginal price for
the-jth characteristic, and  it an iid error term for
which the zero conditional mean assumption
holds.
If b characteristics which are correlated with the a
characteristics are omitted from (1), the estimator
of the model made up of a characteristics
a
pit   0t    jt q jit   it
j 1
will usually be biased ( omitted variables bias)
(2)
• Standard panel data approaches
Based on the assumption that the omitted
variables are constant overtime the
approaches are
– Differencing → OLS to the differences
– Fixed effect
• Time-varying omitted variables
Let
 t   0t 
a b

j  a 1
jt
q jt   t
(3)
i.e. the sum of the intercept, plus the sum of the a
omitted variables, and the error term for which
the zero conditional mean assumption applies i.e.
it is not correlated with the a observed systematic
characteristics
• Constrained Autoregression
Let  0t include the expected value of the house
characteristics captured by  t . Accordingly, the
expected value of (3) is:
E t    0t 
a b




E
q


 jt jt 0t
j  a 1
(4)
• Assumption
Let
 t*   t   0t
(5)
Combining (5) and (1) gives
a
pt   0t    jt q jt   t
j 1
Or
(6)
a
 t  pt   0t    jt q jt
j 1
(7)
• Approximation
We approximate the model of the time-varying
omitted house characteristics (7) by the following
first order autoregression
 t   0t  1t t1   t
with  t an iid error term.
(8)
• Subtitution
Substituting the right hand of (7) into the left side of
(8) and its lag into the right hand side for the
(T+1)-waves (t=0,1,…,T) of observations gives:
p




pt   0t    jt q jt   0t  1t  pt 1   0t 1    jt 1q jt 1    t
j 1
j 1


a
(9)
• Constrained Autoregression
Rearranging (9) we obtain the following constrained
autoregressive price model


a
pt   0t   0t  1t 0t 1  1t pt 1    jt q jt
j 1
a
for t=1,2,…,T.
 1t   jt 1q jt 1  t
j 1
(10)
By combining the three intercept components in
(10) into a single parameter  0t which reduces
(10) to
a
a
j 1
j 1
pt   0t  1t pt 1    jt q jt  1t   jt 1q jt 1   ti
(11)
for t=1,2,…,T.
• Measurment Error
Standard approach:
– Instrumental variables
• Hard to obtain adequate instruments
• Non testable assumptions
• SEM
with twelve parameter matrices and vectors
η  α  Γξ  Βη  ζ
with covζ  Ψ
(12)
and
x  τ x  Λ xξ  δ
with covδ  Θ
(13)
with covε   Θ 
(14)
y  τ y  Λyη  ε
• Decomposition
The measurement models decompose the variance
of an observed variable into the variance
explained by the latent variable and the variance
of the corresponding measurement error. Hence
the parameters of the structural model are free
from measurement errors and hence not
attenuated. In addition multicollinearity is
mitigated by subsuming highly correlated
variables under one and the same latent variable
in the structural model
An ASEM Housing HP for Urban Indonesia:
Measurement model parameter estimates
Variable
log (Median Household
Monthly Expenditure)
Measurement
error variance
Reliability
0.03
0.97
(39.51 )
log (Floor area)
0.01
0.99
(1.26)
House condition
0.41
(7.62)
0.98
Parameter Estimates of Housing HP
for Urban Indonesia
Model
Variable
ASEM
0.26
(0.03)
AUT
0.28
(0.02)
SEM
n.a.
n.a.
FE
n.a.
n.a.
log(Median
Household Monthly
Expenditure)
log(Floor Area)
1.13
(0.06)
0.80
(0.05)
1.32
(0.05)
0.84
(0.04)
0.09
(0.03)
0.11
(0.03)
0.12
(0.03)
0.11
(0.03)
House condition
0.32
(0.02)
0.26
(0.01)
0.33
(0.01)
0.27
(0.01)
Constant
-2.79
(0.12)
-2.29
(0.09)
-3.85
(0.07)
-3.09
(0.09)
1997’s R^2
0.70
0.64
0.70
0.72
2000’s R^2
0.76
0.73
0.74
0.78
RMSEA
0.06
0.16
0.08
0.56
Lagged log(Rent
Appraisal)
Relative Estimates Differences with ASEM Estimates
Variable
AUT
log(Median Household
Monthly Expenditure)
Model
SEM
FE
-29.20
16.81
-25.66
22.22
33.33
22.22
-18.75
3.12
-15.62
log(Floor Area)
House condition
Conclusion
• ASEM allows handling of time-variant missing
variables and thus supplements standard
econometric procedures applied to timeinvariant missing variables
• Omitted variables and measurement errors in
explanatory variables should be handled
simultaneously, as done by ASEM