In this study, we examine dynamic relationships in the house price movements in central and surrounding regions in Taipei. We apply several dynamic spatial-panel data methods to estimate how national and local conditions affect prices in housing sub-markets. We start with outlining the standard panel models following with a detailed discussion of the spatial models.

### 3.1 Standard panel data models

A typical bias in estimations with the panel data is the presence of unobserved heterogeneity, which may correlate with the independent variables and the residual term. For demonstration, the standard panel data model can be represented as:

{y}_{\mathit{it}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\mathit{\alpha}+\phantom{\rule{0.25em}{0ex}}\beta {x}_{\mathit{kit}}+\phantom{\rule{0.25em}{0ex}}{\u03f5}_{\mathit{it}}

(1)

where *y*_{
it
} indicates the house price in the *i*^{th} housing market in year *t. α*_{
i
} denotes the time-invariant individual effect allowing region-specific characteristics such as location, local conditions and economic structure. *x*_{
kit
} is the *k*^{th} factor in the *i*^{th} housing market in *t*, and *β* is the coefficient vector; *ε*_{
it
} is the error term. If *α* is constant, the model will be a conventional linear regression model and OLS can serve as an appropriate method to estimate the parameters. On the other hand, if correlation exists between *α* and independent variables, it can lead to the typical problem of omitted variable bias in panel data. The panel data models provide efficient tools to address unobserved heterogeneity by controlling for the fixed effects. Therefore, with potential region-specific effects across housing markets, a two-way error components model incorporating both the region-specific and the time-specific fixed effects can be employed to examine the regional housing markets. The region-specific fixed effect implies that individual factors may vary across regions but are time-invariant, and more importantly, it could have long-term effects on the housing markets. Time-specific fixed effect specification indicates that specific period of time would cause short-term disequilibrium in the housing markets but these effects may not vary across regions. Fixed effect model can be seen as a Least Square Dummy Variable (LSDV) formulation because it uses dummy variables to estimate the unobserved heterogeneity, which is tantamount to the mean-differencing approach. The model can be expressed as:

{y}_{\mathit{it}}=\mathit{\alpha}+\phantom{\rule{0.25em}{0ex}}\beta {x}_{\mathit{kit}}+\phantom{\rule{0.25em}{0ex}}{\mu}_{i}+\phantom{\rule{0.25em}{0ex}}{\u03f5}_{\mathit{it}}

(2)

where *μ*_{
i
} is the region-specific constant term that is time-invariant. The model can be augmented to a two-way fixed effects model if we add the time effects *λ*_{
t
} in the equation as well:

{y}_{\mathit{it}}=\mathit{\alpha}+\phantom{\rule{0.25em}{0ex}}\beta {x}_{\mathit{kit}}+\phantom{\rule{0.25em}{0ex}}{\mu}_{i}+{\lambda}_{t}+\phantom{\rule{0.25em}{0ex}}{\u03f5}_{\mathit{it}}

(3)

Moreover, if the residuals *ε*_{
it
} exhibit temporal autocorrelation or the dependent variable *y*_{
it
} shows high persistency, dynamic panel model would work better because it allows feedbacks from current or past shocks. Therefore, equation (3) can be written as in a dynamic set-up:

{y}_{\mathit{it}}=\phantom{\rule{0.25em}{0ex}}\mathit{\alpha}+\phantom{\rule{0.25em}{0ex}}\beta {x}_{\mathit{kit}}+\eta {y}_{\mathit{it}-s}+\phantom{\rule{0.25em}{0ex}}{\mu}_{i}+{\lambda}_{t}+\phantom{\rule{0.25em}{0ex}}{\u03f5}_{\mathit{it}}

(4)

However, the estimator is inconsistent and biased in dynamic models by using LSDV method due to existence of correlations between lagged values of independent variables and residual terms (Roodman 2009). The bias would turn out to be worse when the autoregressive coefficient is high or the number of time periods is short. Therefore, we turn to dynamic panel modelling with controls for spatial correlation.

### 3.2 Dynamic panel-spatial model

In this paper, we are interested in exploring the presence of spatial patterns or correlations. Therefore, we include a spatially lagged dependent variable to capture the spatial correlation between regions. It implies that the value of the dependent variable is jointly determined by the neighbouring units and local characteristics (Elhorst 2010). Then, the one-lagged spatial panel model can be expressed as:

{y}_{\mathit{it}}=\mathit{\varphi}{y}_{\mathit{it}-1}+\phantom{\rule{0.25em}{0ex}}\beta {x}_{\mathit{kit}}+\rho \mathbf{W}{y}_{\mathit{jt}}+\phantom{\rule{0.25em}{0ex}}{\mu}_{i}+{\lambda}_{t}+\phantom{\rule{0.25em}{0ex}}{\u03f5}_{\mathit{it}}

(5)

where *ρ* is the coefficient of spatial autoregressive term and *Wy*_{
jt
} is called a spatial lag as a weighted average of observations on the variable over neighbouring units. *y*_{it−1} is the lag of the dependent variable, ϕ the autoregressive time dependence parameter and *w*_{
ij
} is the *N × N* spatial weight matrix. The spatial matrix *W* is pre-determined by contiguity, where the value of the spatial correlation is 1 if the region *i* and region *j* are neighbours, otherwise the value is 0. The spatial matrix is normalised with each row summing up to unity. The stability condition is: (|*ρ| + |* ϕ*| < 1*).

Due to the correlation between the spatial regressor *w*_{
ij
}*y*_{
jt
} and the error term, the estimation of standard fixed effects models could be inconsistent. There are several approaches suggested in the literature with varied levels of merits and demerits (for example, see Kuethe and Pede 2011; Beenstock and Felsentein 2007 for Vector Auto-regression approaches). There are two major methods - maximum likelihood (MLE) and instrumental variables or generalised method of moments (IV/GMM) approaches - that are used to deal with the spatial interactions. However, considering the complex moment conditions in GMM and a lack of a direct GMM estimator for the spatial dynamic-panel model, an instrumental variable approach within a two-stage estimation process has also been suggested in the literature (Brady 2011).

However, the simple SAR formulation in equation (5) has limitations due to the absence of effective control for potential space-time covariance. Ignoring the space-time covariance may lead to violation of the stability condition (|*ρ| + |* ϕ*| <* 1). Therefore, based on works by Anselin (2001), Yu et al. (2008) and Parent and LeSage (2012), Debarsy et al. (2012) present a more general dynamic *spatial lag* panel model that allows for time and spatial dependence both as well as a cross-product term reflecting spatial dependence at a one-period time lag. They also add spatially lagged exogenous variables to the set of covariates, leading to a dynamic *Spatial Durbin Model* (SDM).

\begin{array}{l}{y}_{\mathit{it}}=\mathit{\varphi}{y}_{\mathit{it}-1}+\phantom{\rule{0.25em}{0ex}}\beta {x}_{\mathit{kit}}+\rho \mathbf{W}{y}_{\mathit{jt}}+\theta \mathbf{W}{y}_{\mathit{jt}-1}\\ \phantom{\rule{2.5em}{0ex}}+\phantom{\rule{0.25em}{0ex}}\gamma \mathbf{W}{x}_{\mathit{jt}}+\phantom{\rule{0.25em}{0ex}}{\mu}_{i}+{\lambda}_{t}+\phantom{\rule{0.25em}{0ex}}{\u03f5}_{\mathit{it}}\end{array}

(6)

where *ρ* the spatial dependence parameter, ϕ the autoregressive time dependence parameter, and *θ* the spatio-temporal diffusion parameter. *ϵ*_{
it
} is assumed *i.i.d*. across *i* and *t* with zero mean and constant variance. The stability condition is: (|*ρ*| *+* |ϕ| *+* |*θ*| *< 1*). However, this stability condition may be too restrictive in many cases (Elhorst 2012). However, a less restrictive condition may also be applied counting the negative values.

Furthermore, the direct (or, own), indirect (or, spillover) and total effects can be estimated from spatial models by computing partial derivatives of the impact from changes to the variable. In the most general *Spatial Auto-Regressive* (SAR) model, the equation (6) could be rewritten in vector form in one housing characteristic. Thus the derivative of *Y* with respect to *x*_{
k
}′ is as:

\frac{\partial \mathrm{Y}}{\partial {\mathrm{x}}_{k}^{\prime}}=\phantom{\rule{0.25em}{0ex}}\left[\begin{array}{ccc}\hfill \frac{\partial {\mathrm{Y}}_{1}}{\partial {x}_{1k}}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial {\mathrm{Y}}_{1}}{\partial {x}_{\mathit{nk}}}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill \frac{\partial {\mathrm{Y}}_{n}}{\partial {x}_{1k}}\hfill & \hfill \cdots \hfill & \hfill \frac{\partial {\mathrm{Y}}_{n}}{\partial {x}_{\mathit{nk}}}\hfill \end{array}\right]

(7)

The marginal effect is derived as (see Elhorst 2014):

\begin{array}{c}\hfill \frac{\partial \mathrm{Y}}{\partial {\mathrm{x}}_{f}^{\prime}}={\left(I-\mathrm{\rho}\phantom{\rule{0.25em}{0ex}}W\right)}^{-1\phantom{\rule{0.25em}{0ex}}}\phantom{\rule{0.5em}{0ex}}\left[\begin{array}{ccc}\hfill {\beta}_{k}\hfill & \hfill \cdots \hfill & \hfill {w}_{1n}{\gamma}_{k}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {w}_{n1}{\gamma}_{k}\hfill & \hfill \cdots \hfill & \hfill {\beta}_{k}\hfill \end{array}\right]\hfill \\ \phantom{\rule{2em}{0ex}}={\left(I-\mathrm{\rho}\phantom{\rule{0.25em}{0ex}}W\right)}^{-1\phantom{\rule{0.25em}{0ex}}}\left[{\beta}_{k}\phantom{\rule{0.25em}{0ex}}I+\phantom{\rule{0.25em}{0ex}}\gamma \phantom{\rule{0.25em}{0ex}}W\right]\hfill \end{array}

(8)

*β*_{
k
} is the marginal implicit price, but the marginal implicit price of the SDM is [*β*_{
k
} *I* + *γ W*](*I* − ρ *W*)^{− 1}. The house price in location *i* could be affected by both of a marginal change of one housing characteristic in location *i* and marginal changes of housing in the other locations. The former is called the direct or own effect and the later an indirect or spillover effect. When both *ρ* and *γ* are equal to zero, the indirect effects do not exist. The indirect effects also known as spillover effects due to from an observation’s neighbourhood set, but the effect of *x*_{
jk
} on *y*_{
j
} is also zero if the element *w*_{
ij
} of the spatial weights matrix is zero (Elhorst 2012). According to LeSage and Pace (2009), the direct effect could be estimated by the average of the diagonal elements, and the indirect effect measured by the average of the row sums of non-diagonal elements of the matrix.

In our estimation framework, we employ several specifications: (1) Brady (2011) SAR model; (2) Debarsy et al. (2012) SDM model without time effects; (3) Debarsy et al. (2012) SDM model with time effects; and (4) Debarsy et al. (2012) SDM model with time effects to calculate direct, indirect and total effects.