Hedonic price models with omitted variables and measurement errors: a constrained autoregression–structural equation modeling approach with application to urban Indonesia

Abstract

Omitted variables and measurement errors in explanatory variables frequently occur in hedonic price models. Ignoring these problems leads to biased estimators. In this paper, we develop a constrained autoregression–structural equation model (ASEM) to handle both types of problems. Standard panel data models to handle omitted variables bias are based on the assumption that the omitted variables are time-invariant. ASEM allows handling of both time-varying and time-invariant omitted variables by constrained autoregression. In the case of measurement error, standard approaches require additional external information which is usually difficult to obtain. ASEM exploits the fact that panel data are repeatedly measured which allows decomposing the variance of a variable into the true variance and the variance due to measurement error. We apply ASEM to estimate a hedonic housing model for urban Indonesia. To get insight into the consequences of measurement error and omitted variables, we compare the ASEM estimates with the outcomes of (1) a standard SEM, which does not account for omitted variables, (2) a constrained autoregression model, which does not account for measurement error, and (3) a fixed effects hedonic model, which ignores measurement error and time-varying omitted variables. The differences between the ASEM estimates and the outcomes of the three alternative approaches are substantial.

Appendices

Appendix 1: Model specifications

1.1 FE

In SEM notation, the FE HP housing model reads as follows. For each wave, log(Rent Appraisal) is the only endogenous variable, while all the house characteristics are exogenous variables. Time-invariant unobserved heterogeneity is represented by the latent exogenous variable $\left( {\xi_{10} } \right)$ which is correlated with the other exogenous variables and has a fixed unit regression coefficient in the three waves. The model does not account for measurement error, and hence, the relationships between the observed and the latent variables are identity relationships. The measurement models thus read:

${\varvec{\Uplambda}}_{y} = {\mathbf{I}}_{{\left( {3 \times 3} \right)}} ,\quad {\varvec{\Uplambda}}_{x} = \left[ {{\mathbf{I}}_{{\left( {9 \times 9} \right)}} {\mathbf{0}}_{\left( 9 \right)} } \right],\quad {\varvec{\tau}}_{y} = {\mathbf{0}}_{\left( 3 \right)} ,\quad {\varvec{\tau}}_{x} = {\mathbf{0}}_{\left( 9 \right)} ,\quad {\varvec{\Uptheta}}_{\varepsilon } = {\mathbf{0}}_{{\left( {3 \times 3} \right)}}$ and ${\varvec{\Uptheta}}_{\delta } = {\mathbf{0}}_{{\left( {9 \times 9} \right)}}$.

The structural model parameter matrices are ${\varvec{\alpha}}^{\prime } = \left[ {\begin{array}{*{20}c} {\alpha_{1} } & {\alpha_{2} } & {\alpha_{3} } \\ \end{array} } \right]$, ${\mathbf{B}} = {\mathbf{0}}_{{\left( {3 \times 3} \right)}}$, ${\varvec{\Upgamma}} = \left[ {\begin{array}{*{20}c} {\gamma_{11} } & {\gamma_{12} } & {\gamma_{13} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\gamma_{24} } & {\gamma_{25} } & {\gamma_{26} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\gamma_{37} } & {\gamma_{38} } & {\gamma_{39} } \\ \end{array} \, \begin{array}{*{20}c} 1 \\ 1 \\ 1 \\ \end{array} } \right]$, ${\varvec{\Uppsi}} = diag\left[ {\begin{array}{*{20}c} {\psi_{11} } & {\psi_{22} } & {\psi_{33} } \\ \end{array} } \right]$, ${\varvec{\kappa}}^{\prime } = \left[ {\begin{array}{*{20}c} {\kappa_{1} } & \cdots & {\kappa_{9} } & 0 \\ \end{array} } \right]$, and $\Upphi = \left[ {\begin{array}{*{20}c} {\phi_{11} } & {} & {} & {} \\ {\phi_{21} } & {\phi_{22} } & {} & {} \\ \vdots & \vdots & \ddots & {} \\ {\phi_{10,1} } & {\phi_{10,2} } & \cdots & {\phi_{10,10} } \\ \end{array} } \right]$.

Under Constraint 2, γ _1k  = 0.7892γ _2l and γ _2l  = 1.1606γ _3m for $\left( {k,l,m} \right) = \left\{ {\left( {1,4,7} \right),\left( {2,5,8} \right),\left( {3,6,9} \right)} \right\}$, while under the time-invariant coefficients assumption γ _1k  = γ _2l and γ _2l  = γ _3m for $\left( {k,l,m} \right) = \left\{ {\left( {1,4,7} \right),\left( {2,5,8} \right),\left( {3,6,9} \right)} \right\}$.

1.2 SEM

The SEM HP structural model consists of the standard multiple regression model (2) in terms of latent variables, supplemented with the auxiliary autoregression models of the house characteristics for identification of the measurement error variances. The exogenous variables in this model are the house characteristics in wave-0. The exogenous and endogenous observed and latent variables are ${\mathbf{x}}^{\prime } = \left[ {\begin{array}{*{20}c} {x_{1} } & {x_{2} } & {x_{3} } \\ \end{array} } \right]$, ${\mathbf{y}}^{\prime } = \left[ {\begin{array}{*{20}c} {y_{1} } & \cdots & {y_{3} } \\ \end{array} } \right]$, ${\varvec{\upxi}}^{\prime } = \left[ {\begin{array}{*{20}c} {\xi_{1} } & {\xi_{2} } & {\xi_{3} } \\ \end{array} } \right]$, ${\varvec{\upeta}}^{\prime } = \left[ {\begin{array}{*{20}c} {\eta_{1} } & \cdots & {\eta_{9} } \\ \end{array} } \right]$. Note that the observed log(Rent Appraisal) variables are y ₁, y ₂, and y ₆, because there is no lagged dependent variable in the structural model. The structural parameter matrices are ${\mathbf{B}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {\beta_{23} } & {\beta_{24} } & {\beta_{25} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\beta_{67} } & {\beta_{68} } & {\beta_{69} } \\ 0 & 0 & {\beta_{73} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\beta_{84} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\beta_{95} } & 0 & 0 & 0 & 0 \\ \end{array} } \right]$ ${\varvec{\Upgamma}} = \left[ {\begin{array}{*{20}c} {\gamma_{11} } & {\gamma_{12} } & {\gamma_{13} } \\ 0 & 0 & 0 \\ {\gamma_{31} } & 0 & 0 \\ 0 & {\gamma_{42} } & 0 \\ 0 & 0 & {\gamma_{53} } \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right]$, ${\varvec{\alpha}} = \left[ {\begin{array}{*{20}c} {\alpha_{1} } \\ \vdots \\ {\alpha_{9} } \\ \end{array} } \right],$ ${\varvec{\kappa}} = \left[ {\begin{array}{*{20}c} {\kappa_{1} } \\ {\kappa_{2} } \\ {\kappa_{3} } \\ \end{array} } \right]$ ${\varvec{\Uppsi}} = diag\left[ {\begin{array}{*{20}c} {\phi_{11} } \\ \vdots \\ {\phi_{99} } \\ \end{array} } \right]$, and ${\varvec{\Upphi}} = \left[ {\begin{array}{*{20}c} {\phi_{11} } & {} & {} \\ {\phi_{21} } & {\phi_{22} } & {} \\ {\phi_{31} } & {\phi_{32} } & {\phi_{33} } \\ \end{array} } \right]$.

Under Constraint 2, γ _1k  = 0.7892β _2l and β _2l  = 1.1604β _6m for $\left( {k,l,m} \right) = \left\{ {\left( {1,3,7} \right),\left( {2,4,8} \right),\left( {3,5,9} \right)} \right\}$, while under time-invariant assumption γ _1k  = β _2l and β _2l  = β _6m for $\left( {k,l,m} \right) = \left\{ {\left( {1,3,7} \right),\left( {2,4,8} \right),\left( {3,5,9} \right)} \right\}$.

The parameter matrices in the measurement models are ${\varvec{\Uplambda}}_{y} = {\mathbf{I}}_{{\left( {9 \times 9} \right)}} ,$ ${\varvec{\Uplambda}}_{x} = {\mathbf{I}}_{{\left( {3 \times 3} \right)}} ,$ ${\varvec{\tau}}_{y} = {\mathbf{0}}_{\left( 9 \right)}$, ${\varvec{\tau}}_{x} = {\mathbf{0}}_{\left( 3 \right)}$, ${\varvec{\Uptheta}}_{\varepsilon } = diag\left[ {\begin{array}{*{20}c} 0 & 0 & {\theta_{33}^{\varepsilon } } & {\theta_{44}^{\varepsilon } } & {\theta_{55}^{\varepsilon } } & 0 & {\theta_{77}^{\varepsilon } } & {\theta_{88}^{\varepsilon } } & {\theta_{99}^{\varepsilon } } \\ \end{array} } \right]$, and ${\varvec{\Uptheta}}_{\delta } = diag\left[ {\begin{array}{*{20}c} {\theta_{11}^{\delta } } & {\theta_{11}^{\delta } } & {\theta_{11}^{\delta } } \\ \end{array} } \right].$ Constraint 1 is $\theta_{kk}^{\delta } = \theta_{ll}^{\varepsilon } = \theta_{mm}^{\varepsilon }$ $\left( {k,l,m} \right) = \left\{ {\left( {1,3,7} \right),\left( {2,4,8} \right),\left( {3,5,9} \right)} \right\}$.

1.3 AUT

The AUT model is (11). The endogenous variables in the model are log(Rent Appraisal) in wave-1 and wave-2, while all other variables are exogenous. The exogenous and endogenous observed and latent vectors are ${\mathbf{x}}^{\prime } = \left[ {\begin{array}{*{20}c} {x_{1} } & \cdots & {x_{10} } \\ \end{array} } \right]$, ${\mathbf{y}}^{\prime } = \left[ {\begin{array}{*{20}c} {y_{1} } & {y_{2} } \\ \end{array} } \right]$, ${\varvec{\xi}}^{\prime } = \left[ {\begin{array}{*{20}c} {\xi_{1} } & \cdots & {\xi_{10} } \\ \end{array} } \right]$, and ${\varvec{\eta}}^{\prime } = \left[ {\begin{array}{*{20}c} {\eta_{1} } & {\eta_{2} } \\ \end{array} } \right]$. Because of the absence of latent variables, the measurement models are ${\varvec{\Uplambda}}_{y} = {\mathbf{I}}_{{\left( {2 \times 2} \right)}}$, ${\varvec{\Uplambda}}_{x} = {\mathbf{I}}_{{\left( {10 \times 10} \right)}}$, ${\varvec{\tau}}_{y} = {\mathbf{0}}_{\left( 2 \right)}$, ${\varvec{\tau}}_{x} = {\mathbf{0}}_{{\left( {10} \right)}}$, ${\varvec{\Uptheta}}_{\varepsilon } = {\mathbf{0}}_{{\left( {2 \times 2} \right)}}$, and ${\varvec{\Uptheta}}_{\delta } = {\mathbf{0}}_{{\left( {10 \times 10} \right)}}$.

The structural model matrices are ${\varvec{\alpha}} = \left[ {\begin{array}{*{20}c} {\alpha_{1} } \\ {\alpha_{1} } \\ \end{array} } \right]$, ${\mathbf{B}} = \left[ {\begin{array}{*{20}c} 0 & 0 \\ {\beta_{21} } & 0 \\ \end{array} } \right]$, ${\varvec{\Upgamma}} = \left[ {\begin{array}{*{20}c} {\gamma_{11} } & {\gamma_{12} } & {\gamma_{13} } & {\gamma_{14} } & {\gamma_{15} } & {\gamma_{16} } & {\gamma_{17} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\gamma_{25} } & {\gamma_{26} } & {\gamma_{27} } & {\gamma_{28} } & {\gamma_{29} } & {\gamma_{210} } \\ \end{array} } \right]$, ${\varvec{\Uppsi}} = diag\left[ {\begin{array}{*{20}c} {\psi_{11} } & {\psi_{22} } \\ \end{array} } \right]$, ${\mathbf{\kappa^{\prime}}} = \left[ {\begin{array}{*{20}c} {\kappa_{11} } & \cdots & {\kappa_{1010} } \\ \end{array} } \right]$, and ${\varvec{\Upphi}} = \left[ {\begin{array}{*{20}c} {\phi_{11} } & {} & {} & {} \\ {\phi_{21} } & {\phi_{22} } & {} & {} \\ \vdots & \vdots & \ddots & {} \\ {\phi_{10,1} } & {\phi_{10,2} } & \cdots & {\phi_{10,10} } \\ \end{array} } \right]$.

Under Constraint 2, $\gamma_{1k} = 1.1604\gamma_{2l}$ for $\left( {k,l} \right) = \left\{ {\left( {5,8} \right),\left( {6,9} \right),\left( {7,10} \right)} \right\}$, γ _1k  = −γ ₁₁(0.7892γ _1l ) and γ _2l  = − β ₂₁ γ _1l for $\left( {k,l} \right) = \left\{ {\left( {2,5} \right),\left( {3,6} \right),\left( {4,7} \right)} \right\}$. Under the time-invariant coefficients, assumption γ _1k  = γ _2l for (k, l) = {(5, 8), (6, 9),(7, 10)}, γ _1k  = − γ ₁₁ γ _1l and γ _2l  = − β ₂₁ γ _1l for $\left( {k,l} \right) = \left\{ {\left( {2,5} \right),\left( {3,6} \right),\left( {4,7} \right)} \right\}$.

Appendix 2: Time-invariant coefficients models

See Table 4.

Table 4 ASEM, AUT, SEM, and FE under the assumption of time-invariant coefficients