# Spatial Vector Error Correction

• Michael Beenstock
• Daniel Felsenstein
Chapter

## Abstract

We begin with a theoretical discussion on the spatiotemporal dynamics induced by spatial VECMs (SpVECM). Subsequently, a hierarchy of empirical error correction models for house prices and housing construction in Israel is reported based on the empirical results for spatial panel cointegration reported in Chap. 8. Empirical illustrations for single equation error correction are presented and discussed. This is followed by empirical illustration of error correction involving two equations (house prices and housing construction) in which error correction occurs within and between equations, as in a SpVECM.

## 9.1 Introduction

In Chap. we focused on the estimation of cointegrating vectors using nonstationary spatial panel data. In this chapter we show how these cointegrating vectors may be used to estimate spatial vector error correction models (SpVECM) as defined in Chap. . Whereas cointegrating vectors are concerned with long-run equilibrium relations between spatial panel data, SpVECMs are concerned with their spatiotemporal convergence to these long-term relations. In non-spatial VECMs involving difference stationary variables (Chap. ), error correction models for individual variables such as y depend on the lagged disequilibria for y as well as the lagged disequilibria for other variables, such as x. In SpVECMs the error correction model for y depends on these lagged disequilibria, as well as their spatial lags.

The cointegrating vectors for y and x are represented by Eqs. (9.1a and 9.1b):
$${y}_{it}={\alpha}_{1i}+{\beta}_1{x}_{it}+{\gamma}_1{z}_{it}+{\uplambda}_1{\tilde{y}}_{it}+{\delta}_1{\tilde{x}}_{it}+{u}_{it}$$
(9.1a)
$${x}_{it}={\alpha}_{2i}+{\beta}_2{y}_{it}+{\gamma}_2{w}_{it}+{\delta}_2{\tilde{y}}_{it}+{\uplambda}_2{\tilde{x}}_{it}+{v}_{it}$$
(9.1b)
where the α’s denote spatial fixed effects, spatial lag coefficients are denoted by λ’s, spatial Durbin lags are denoted by δ’s, and the disequilibrium error components are denoted by u and v, which are stationary by definition of cointegration. Notice that Eqs. (9.1a and 9.1b) are identified because z is specified in the former but not the latter, and w is specified in the latter but not the former.
In Chap. we introduced the concept of spatial error correction, and spatial vector error correction. For convenience, we recall the latter here. The first-order SpVECM associated with Eqs. (9.1a and 9.1b) is:
$$\Delta {y}_{it}={\mu}_{1i}+{\pi}_{11}\Delta {y}_{it-1}+{\pi}_{12}\Delta {x}_{it-1}+{\uplambda}_3\Delta {\tilde{y}}_{it}+{\delta}_3\Delta {\tilde{x}}_{it}-{\xi}_{11}{\widehat{u}}_{it-1}+{\xi}_{12}{\widehat{v}}_{it-1}+{\zeta}_{11}{\widehat{\tilde{u}}}_{it-1}+{\zeta}_{12}{\widehat{\tilde{v}}}_{it-1}+{\varepsilon}_{1 it}$$
(9.2a)
$$\Delta {x}_{it}={\mu}_{2i}+{\pi}_{22}\Delta {y}_{it-1}+{\pi}_{21}\Delta {x}_{it-1}+{\delta}_4\Delta {\tilde{y}}_{it}+{\lambda}_4\Delta {\tilde{x}}_{it}-{\xi}_{22}{\widehat{u}}_{it-1}+{\xi}_{21}{\widehat{v}}_{it-1}+{\zeta}_{22}{\widehat{\tilde{u}}}_{it-1}+{\zeta}_{21}{\widehat{\tilde{v}}}_{it-1}+{\varepsilon}_{2 it}$$
(9.2b)
Where the μ’s are spatial fixed effects, the π’s are VAR coefficients, the λ’s are spatial lag coefficients, the δ’s are spatial Durbin lag coefficients, the ξ’s are the error correction coefficients, the ζ’s are spatial error correction coefficients, and the ε’s are iid random variables. In non-spatial panel data the λ’s, δ’s, and ζ’s are zero in Eqs. (9.1a, 9.1b, 9.2a and 9.2b). Hence, in bivariate SpVECMs there are 12 spatial effects. For simplicity, terms in Δz, Δw and their spatial lags have been omitted from Eqs. (9.2a and 9.2b) as have terms in $$\Delta {\tilde{y}}_{it-1}\ and\ \Delta {\tilde{x}}_{it-1}$$. The expected signs of the error correction coefficients are positive for ξ11 and ξ12, and positive for ξ22 and ξ21. The expected signs of the spatial error correction coefficients are positive for ζ11 and ζ21 and positive for ζ12 and ζ22.

All the variables in Eqs. (9.2a and 9.2b) must be stationary if the variables used to estimate Eqs. (9.1a and 9.1b) are difference stationary. Therefore, the principles of unit root econometrics, which apply to Eqs. (9.1a and 9.1b) do not apply to Eqs. (9.2a and 9.2b). Instead, standard econometric principles apply to Eqs. (9.2a and 9.2b), e.g. the parameter estimates have standard distributions so that t-tests etc. may be used, and the parameter estimates are consistent but no longer super-consistent. The main econometric concern with respect to Eqs. (9.2a and 9.2b) is that the error terms (ε) should be serially uncorrelated, otherwise lagged dependent variables will not be weakly exogenous for the πs. Another econometric concern is that contemporaneous SAR coefficients (λ3, λ4, δ3 and δ4) are not identified by OLS.

In SpVECMs as in VECMs error correction is interdependent. Hence, the disequilibrium in xit−1, measured by vit−1, is specified in the error correction model for yit in Eq. (9.2a) alongside the disequilibrium in yit−1, measured by uit−1. In Eq. (9.2b) error correction in xit depends on both vit−1 and uit−1. Whereas error correction is implied by cointegration theory, matters are different for vector error correction. If y and x are completely independent vector error correction would not make sense. For example, in the SGE model for Israel reported in Table , it might be reasonable to expect that vector error correction applies within markets, such as housing, rather than between markets, such as housing and labor markets. In the empirical application below, we test for vector error correction between house prices and housing starts because these variables belong to the same market. It would have been less sensible to test for vector error correction between housing starts and wages. In the final analysis, vector error correction is an empirical matter, except for variables that are completely independent a priori.

## 9.2 Stability of Error Correction Models

If the variables in Eqs. (9.1a and 9.1b) are cointegrated, by definition it must be the case that the SpVECM in Eqs. (9.2a and 9.2b) implies that y and x eventually converge on their long-run equilibria determined by Eqs. (9.1a and 9.1b). This means that the roots or eigenvalues of SpVECMs must be less than 1. The number of roots in SpVECMs tends to be large since there are MNP roots, where M is the number of state variables, N is the number of spatial units, and P is the lag order of the SpVECM. In Eqs. (9.2a and 9.2b) P = 2 because they are first order difference equations in changes in yit and xit and consequently second order difference equations in yit and xit. Therefore, in Eqs. (9.1a, 9.1b, 9.2a and 9.2b) where M = 2 and P = 2 there are 4N roots. In data sets where N exceeds 100, the number of roots may run into thousands. In what follows we try to expose the basic issues involved by focusing on simple but nonetheless illuminating cases. We begin with time series data before turning to spatial panel data.

### Bivariate Time Series: Vector Error Correction

To fix ideas suppose y and x are time series data rather than panel data, to which we shall return. Suppose for simplicity that the VECM is symmetric and does not involve VAR components involving lags of Δy and Δx:
$$\Delta {u}_t=-{\xi}_1{u}_{t-1}+{\xi}_2{v}_{t-1}+{\varepsilon}_{1t}$$
(9.3a)
$$\Delta {v}_t={\xi}_2{u}_{t-1}-{\xi}_1{v}_{t-1}+{\varepsilon}_{2t}$$
(9.3b)
where u and v are determined in the time series counterparts to Eqs. (9.1a and 9.1b):
$${y}_t={\beta}_1{x}_t+{\gamma}_1{\mathrm{z}}_t+{u}_t$$
(9.3c)
$${x}_t={\beta}_2{y}_t+{\gamma}_2{w}_t+{v}_t$$
(9.3d)
Equations (9.3a and 9.3b) may be rewritten as:
$$\left[\begin{array}{cc}1+\left({\xi}_1-1\right)L& -{\xi}_2L\\ {}-{\xi}_2L& 1+\left({\xi}_1-1\right)L\end{array}\right]\left[\begin{array}{c}{u}_t\\ {}{v}_t\end{array}\right]=\left[\begin{array}{c}{\varepsilon}_{1t}\\ {}{\varepsilon}_{2t}\end{array}\right]$$
(9.3e)
for which the characteristic equation is quadratic:
$${\omega}^2-2\left(1-\xi \right)\omega +{\left(1-{\xi}_1\right)}^2-{\xi}_2^2=0$$
(9.3f)
The roots are:
$${\omega}_1=1-{\xi}_1+{\xi}_2$$
(9.3g)
$${\omega}_2=1-{\xi}_1-{\xi}_2$$
(9.3h)
Since the ξs are fractions the roots are less than 1 in absolute value. The general solutions for u is:
$${u}_t=\frac{1}{\omega_1-{\omega}_2}{\sum}_{\tau =0}^{\infty}\left({\omega}_1^{1+\tau }-{\omega}_2^{1+\tau}\right)\left[{\varepsilon}_{1t-\tau }-\left(1-{\xi}_1\right){\varepsilon}_{1t-\tau -1}+{\xi}_2{\varepsilon}_{2t-\tau -1}\right]+{A}_1{\omega}_1^t+{A}_2{\omega}_2^t$$
(9.3i)
where the As are arbitrary constants determined by initial conditions. Since the ωs are fractions their effect on u vanish over time. The conditional expected value of u from Eq. (9.3i) differs from zero. However its unconditional expected value is zero because the εs have zero expected value. A similar result applies to the solution for v. Therefore, the parameters of the VECM are expected to induce y and x to converge on their equilibrium values determined in Eqs. (9.3c and 9.3d).

### Panel Data: Spatial Error Correction with 2 Spatial Units and 1 Variable

Equations (9.3a9.3i) may also be used to illustrate the properties of spatial error correction. Suppose there are two spatial units and yt denotes the time series of a variable of interest in unit 1 while xt denotes the time series of the same variable in unit 2. Therefore, β1 in Eq. (9.3c) and β2 in Eq. (9.3d) constitute spatial lags in global cointegrated vectors, ξ1 in Eqs. (9.3a and 9.3b) are homogeneous error correction coefficients, and ξ2 in Eq. (9.3a) is the spatial error correction coefficient for unit 1, and ξ2 in Eq. (9.3b) is the spatial error correction coefficient for unit 2.

In general the number of roots or eigenvalues (ω) equals NMP where M is the number of variables in the SpVECM, N is the number of spatial units, and P is the temporal lag order. In the 1 × 2 case (when M = 1 and N = 2) the characteristic Eq. (9.3g) is quadratic and has an analytical solution. Therefore, even in the 2 × 2 case, the condition for convergent roots cannot be derived analytically because there are at least four eigenvalues. Equations (9.2a and 9.2b) refer to an SpVECM in which M = 2, there are N spatial units and P = 1, hence there are 2N eigenvalues. In the SGE model (Table ) M = 7 and N = 9 in which case there would be 63 eigenvalues in its first order SpVECM.

### Spatial Error Correction: N Spatial Units and M = 2 Variables

In what follows we restrict M = 2 and P = 1 but N is not restricted. Let ut and vt now denote N-vectors of error terms generated by cointegrating vectors for the two variables, such as y and x. The SpECMs for these variables are:
$${u}_t=\left(1-{\xi}_u\right){u}_{t-1}+{\zeta}_uW{u}_{t-1}+{\varepsilon}_{ut}$$
(9.4a)
$${v}_t=\left(1-{\xi}_v\right){v}_{t-1}+{\zeta}_vW{v}_{t-1}+{\varepsilon}_{vt}$$
(9.4b)
where, as before, ξ denotes error correction coefficients, ζ denotes spatial error correction coefficients, and ε denotes iid ECM error terms. The solution to Eq. (9.4a) is:
$${u}_t=A{\varepsilon}_{ut}$$
(9.4c)
$$A={\left[{\mathrm{I}}_N-\left[\left(1-{\xi}_u\right){I}_N+{\zeta}_uW\right]L\right]}^{-1}={\mathrm{I}}_N+{\sum}_{\tau =1}^{\infty }\ {\left[\left(1-{\xi}_u\Big){I}_N-{\zeta}_uW\right)\right]}^{\tau }{L}^{\tau }$$
(9.4d)

The N eigenvalues will be less than 1 if 1 − ξu + ζu < 1. The solution to Eq. (9.4b) is similar. Because the eigenvalues are less than 1 the unconditional expected values for u and v are zero because A is convergent.

The SpVECM for these variables is:
$${u}_t=\left(1-{\xi}_{uu}\right){u}_{t-1}+{\zeta}_{uu}W{u}_{t-1}+{\xi}_{uv}{v}_{t-1}+{\zeta}_{uv}W{v}_{t-1}+{\varepsilon}_{ut}$$
(9.4e)
$${v}_t=\left(1-{\xi}_{vv}\right){v}_{t-1}+{\zeta}_{vv}W{v}_{t-1}+{\xi}_{vu}{u}_{t-1}+{\zeta}_{vu}W{u}_{t-1}+{\varepsilon}_{vt}$$
(9.4f)
where ξuu and ξvv denote own or within error correction coefficients, ξuv and ξvu denote cross or between error correction coefficients, ζuu and ζvv denote own or within spatial error correction coefficients, and ζuv and ζvu denote cross or between spatial error correction coefficients. Whereas in SpECMs u and v may be solved separately because they are independent, in SpVECMs u and v are dependent and must be solved jointly. For example, the solution for ut is:
$${u}_t=A\left\{{\varepsilon}_{ut}-\left[\left(1-{\xi}_{vv}\right){\mathrm{I}}_N+{\zeta}_{vu}W\right]{\varepsilon}_{ut-1}+\left[{\xi}_{uv}{\mathrm{I}}_N+{\zeta}_{uv}W\right]{\varepsilon}_{vt-1}\right\}$$
(9.4g)
$$A={\left[{\mathrm{I}}_N-\left({A}_1L-{A}_2{L}^2\right)\right]}^{-1}={\mathrm{I}}_N+{\sum}_{\tau =1}^{\infty }{\left[{A}_1L-{A}_2{L}^2\right]}^{\tau }$$
(9.4h)
$${A}_1=\left({\xi}_{vv}+{\xi}_{uu}\right){I}_N+\left({\zeta}_{vu}+{\zeta}_{uu}\right)W$$
$${A}_2=-\left({\xi}_{vu}{\xi}_{uv}-{\xi}_{vv}\right){\mathrm{I}}_N+\left({\zeta}_{vu}-{\xi}_{uv}{\zeta}_{vu}-{\zeta}_{uv}{\xi}_{vu}\right)W-{\zeta}_{uv}{\zeta}_{vu}{W}^2$$
where A is an N × N matrix. Notice that ut depends on temporal and spatial lags of εv as well as εu. If W is row summed to 1, sufficient conditions for convergence are A1–A2 = ξuu + ζuu + ξvuξuv + ξuvζvu + ζuv ζvu < 1.

### Final Forms of ARSAR Models

In the empirical illustrations below we estimate various error correction models, such as Eqs. (9.4a and 9.4b) for SpECMs, or Eqs. (9.4e and 9.4f) for SpVECMs. Since u and v are generated by cointegrating vectors for y = Zy + u and x = Zx + v, where the Zs are exogenous variables and include iid innovations, we are naturally interested in the spatial and temporal dynamic solutions for y and x in terms of Zy and Zx. These solutions inherit the characteristics of the SpECM or SpVECM from which they are derived. In Eqs. (9.4a9.4h) the error correction models are AR(1)SAR(1) because they embody first order temporal and spatial dynamics. Hence, the first order ARSAR models for y and x are assumed to be:
$${y}_t=\pi {y}_{t-1}+\left(1-\pi \right)W{y}_{t-1}+\beta {x}_{t-1}+\gamma W{x}_{t-1}+{Z}_{yt}$$
(9.5a)
$${x}_t=\delta {x}_{t-1}+\varphi W{x}_{t-1}+\theta {y}_{t-1}+\eta W{y}_{t-1}+{Z}_{xt}$$
(9.5b)
to ensure that feedbacks between y and x are not explosive β + γ and θ + η are less than 1 in absolute value. In Eq. (9.5a) the AR(1) and SAR(1) coefficients (π and 1 − π) sum to 1 inducing spatio-temporal nonstationarity in y. Nevertheless, if x is stationary, y might be stationary because it depends on temporal and spatial lags of x, which is stationary provided δ + φ < 1. To show this we derive the final forms for y and x.
The final forms for y and x generated by Eqs. (9.5a and 9.5b) are:
$${y}_t=\left(\pi +\delta \right){y}_{t-1}+\left(\theta \beta -\pi \delta \right){y}_{t-2}+\left(\varphi +1-\pi \right){\tilde{y}}_{t-1}+\left[-\pi \varphi -\delta \left(1-\pi \right)+\gamma \theta +\beta \eta \right]{\tilde{y}}_{t-2}+\left[-\varphi \left(1-\pi \right)+\gamma \eta \right]{\tilde{\tilde{y}}}_{t-2}+{Z}_{xt-1}+\gamma {Z}_{xt-1}+{Z}_{yt}-\delta {Z}_{yt-1}-\varphi {\tilde{Z}}_{yt-1}$$
(9.5c)
$${x}_t=\left(\pi +\delta \right){x}_{t-1}+\left(\theta \beta -\pi \delta \right){x}_{t-2}+\left(\varphi +1-\pi \right){\tilde{x}}_{t-1}+\left[-\pi \varphi -\delta \left(1-\pi \right)+\gamma \theta +\beta \eta \right]{\tilde{x}}_{t-2}+\left[-\varphi \left(1-\pi \right)+\gamma \eta \right]{\tilde{\overset{\check{} }{x}}}_{t-2}+{Z}_{xt}-\pi {Z}_{xt-1}-\left(1-\pi \right){\tilde{Z}}_{xt-1}+\theta {Z}_{yt-1}+\eta {\tilde{Z}}_{yt-1}$$
(9.5d)

As expected, the final forms are second order ARSAR models. The final form ARSAR coefficients sum to (β + γ)(θ + η) < 1. Therefore, despite the presence of a spatio-temporal unit root in Eq. (9.6a), both y and x are stationary. Matters would be different if (β + γ)(θ + η) ≥ 1, or if δ + φ = 1. In summary, a variable such as y that has a spatiotemporal unit root in isolation, may nevertheless be stationary if it depends on a variable such as x, which in isolation is stationary. If despite this y is nonstationary, so must x be nonstationary if it depends on y.

### Spatial Error Correction and Spatial Cointegration

As mentioned in Chap. , Yu et al. (2012) compared alternative estimators of spatial error correction models where it is known that the nonstationary variables in the model are panel cointegrated. They did not provide critical values for statistical tests of the null hypothesis that these variables are not panel cointegrated. Nor was this their intention. Nevertheless, Elhorst et al. (2013) and Ciccarelli and Elhorst (2018) use the QMLE estimator suggested by Yu et al. to test hypotheses about financial liberalization and cigarette consumption under the assumption that the spatial panel data concerned are in fact cointegrated. However, it is first necessary to establish that the variables are indeed spatially panel cointegrated e.g. along the lines proposed in Chap. . Only then would there be justification in using QMLE to estimate the spatial error correction model.

This two-step process dates back to Engle and Granger (1987) who suggested that in the first stage it is necessary to test whether the nonstationary variables in the model are cointegrated. If they are, they must be related through error correction in the second stage.

## 9.3 Empirical Illustration of Spatial Error Correction

To set the scene, we begin by illustrating the estimation of a spatial error correction model (SpECM) for annual house prices in Israel in which the cointegrating vector is represented by Eq. (9.6a) taken from Beenstock and Felsenstein (2010) in which N = 9 and T = 17:
$$\begin{array}{ll}\mathit{\ln}{P}_{it}=& {\alpha}_1+1.137\mathit{\ln}{POP}_{it}+0.036\mathit{\ln}{Y}_{it}-0.317\mathit{\ln}{H}_{it}\\ {}& -0.073 lnP\overset{\sim }{O}{P}_{it}+0.184\mathit{\ln}{\tilde{P}}_{it}+{v}_{it}\end{array}}$$
(9.6a)
$$\mathrm{GADF}=-3.11\ \mathrm{SpGrho}=0.17$$
where P denotes house prices, POP denotes population, Y denotes income and H denotes housing stock (square meters). Equation (9.6a) includes a spatial lagged dependent variable and a spatial Durbin lag in population. Since the group ADF statistic is less than its critical values of −2.82 (Pedroni 1999) and SpGrho is less than its critical value of 0.79, u is stationary. According to Eq. (9.6a) regional house prices vary directly in the long run with demand (population and income), vary inversely with supply (housing stock), vary directly with house prices nearby and vary inversely with population nearby.
The estimated residuals (v) from Eq. (9.6a) are used to estimate the SpECM for regional house prices (Table 9.1). The SpECM includes the lagged first difference of the variables in Eq. (9.6a). These include lags of the spatially lagged variables as well as spatial lags of the estimated residuals.
Table 9.1

Spatial error correction model for regional house prices (dependent variable: ΔlnPit)

Coefficient

t-statistic

Intercept

−0.0005

−0.05

ΔlnPit−1

0.1732

3.1

Δln$$\overset{\sim }{P}$$it−1

0.1006

4.567

ΔlnHit−1

0.6759

4.521

ΔlnPOPit−1

−0.3926

−3.591

Δln$${\overset{\sim }{Y}}_{it}$$

0.0762

1.882

$${\widehat{u}}_{\mathrm{it}-1}$$

−0.7047

−8.622

$${\overset{\sim }{\widehat{u}}}_{\mathrm{it}-1}$$

−0.6348

−4.63

0.511

Standard error

1.035

DW

2.021

Method of estimation: Panel SUR with common effects. Source: Beenstock and Felsenstein (2010)

In Table 9.1 both error correction terms are negative and statistically significant, indicating that house prices are both spatially and locally cointegrated. Indeed, the sizes of their coefficients indicate that about 70% of the local error is corrected within a year and 63% of the neighboring error spills over onto the local region. The latter also means that if house prices were too high in neighboring regions this exerts downward pressure on local house prices, i.e. there is spatial spillover in error correction, just as there might be with any other variable.

Table 9.1 also incorporates temporally-lagged spatial lags for the first differences in house prices in the autoregressive component of the model. Had this difference been contemporaneous rather than lagged one period, the estimated parameters of the SpECM would not have been consistent, in which case estimation by ML or IV would have been necessary. The same would have applied had the SpECM residuals been autocorrelated, in which event they would have been correlated with the lagged difference in the spatial lag of house prices. However, the panel Durbin Watson statistic indicates that the SpECM residuals are not autocorrelated. Therefore, the estimate of the SAR coefficient (0.1006) is consistent. This means that the current rate of change in local house prices depends on the lagged rate of change in house prices in neighboring regions, as well as the rate of change of lagged house prices in the locality. The spatial lag coefficient is 0.1 in Table 9.1, whereas the coefficient on the lagged dependent variable is 0.1732.

Substituting Eq. (9.6a) into Table 9.1 for $${\widehat{u}}_{it-1}$$ and $${\overset{\sim }{\widehat{u}}}_{it-1}$$ produces the following ARSAR (autoregressive and spatial autoregressive) model in the logarithm of house prices:
$$\mathit{\ln}{P}_t=\kern-0.30em \left(0.4685{I}_N-0.4045W+0.1169{W}^2\right)\mathit{\ln}{P}_{t-1}-\left(0.1732{I}_N+0.1006W\right)\mathit{\ln}{P}_{t-2}+{X}_t$$
(9.6b)
where P is an N-vector of house prices, and Xt is an N-vector of all the other variables in Eq. (9.6a), such as POPt−1 and in the SpECM such as ΔPOPt−1. The temporal and spatial dynamics are second order because Table 9.1 is a second order difference equation, which involves W and W2. There are 2N = 18 roots to Eq. (9.6b), which is too many to consider analytically. However, we may gain some insight by obtaining the conditional roots for unit i and by setting N = 2.
The solution in Eq. (9.6b) for house prices in unit i is:
$$\mathit{\ln}{P}_{it}=0.4685\mathit{\ln}{P}_{it-1}-0.1732\mathit{\ln}{P}_{it-2}-0.4045\mathit{\ln}{\tilde{P}}_{it-1}-0.1006\mathit{\ln}{\tilde{P}}_{it-2}+0.1169\mathit{\ln}{\tilde{\tilde{P}}}_{it-1}+{X}_{it}$$
(9.6c)
House prices in unit i are a second order AR and SAR process and depend on house prices in second order neighbors. Conditional on spatial lagged house prices the two roots of Eq. (9.6c) are less than 1 but complex:
$$\omega =0.4162\left(0.5629\pm i0.8265\right)$$
(9.6d)

The roots are not real because, as mentioned, the ECM model contains a lagged endogenous variable (ΔlnPit−1). However, they are less than 1 because the modulus is a fraction (0.4162).

Setting N = 2 in Eq. (9.6b) would generate the following fourth order characteristic equation:
$${\omega}^4-0.937{\omega}^3+0.4023{\omega}^2-0.1623\omega +0.03=0$$
(9.6e)
for which the roots are: ω1 = 0.5681, ω2 and ω3 = 0.032 ± i0.415, and ω4 = 0.3049. The complex roots arise for the same reason as in Eq. (9.6d). Equation (9.6e) implies that the AR coefficients weaken in absolute size with their lag order. The AR(1) coefficient is 0.937, the AR(2) coefficient is −0.4023, the AR(3) coefficient is 0.1623 and the AR(4) coefficient is 0.03. This is why the roots are than 1.
Since N = 9 there are 36 roots, which are too many to consider here. Nevertheless, they share the features of Eq. (9.6e). The general solution to Eq. (9.6b) is:
$$\mathit{\ln}{P}_t=\left[{I}_N+{\sum}_{\tau =1}^{\infty }{\left({\Omega}_1L-{\Omega}_2{L}^2\right)}^{\tau}\right]{X}_t+{\sum}_i^{18}{A}_i{\omega}_i^t$$
(9.6f)
where
$${\Omega}_1=0.4685{\mathrm{I}}_{\mathrm{N}}-0.4045\mathrm{W}+0.1169{\mathrm{W}}^2$$
$${\Omega}_2=0.1732{\mathrm{I}}_{\mathrm{N}}+0.1006\mathrm{W}$$
and the As are arbitrary constants determined by initial conditions. Since the roots (ω) are (positive) fractions the final term in Eq. (9.6f) tends to zero with time. Equation (9.6f) generates spatio-temporal impulse responses. For example, the response of Pit to an impulse in Xi after two periods is:
$$\frac{\partial \mathit{\ln}{P}_{it}}{\partial {X}_{it-2}}=1+{\psi}_{ii}$$
(9.6g)
where Ψ = $${\Omega}_1^2-{\Omega}_2$$ embodies a quartic in W. The direct effect is 1 and the indirect effect is ψii. Note that ψii incorporates up to fourth order spatial effects. The impulse response between spatial units is ψij since there is no direct effect. Table 9.1 implies that the direct elasticity of house prices with respect to population is 0.1732 after 1 year. The indirect effect will increase this because the SAR coefficient 0.1006. According to Eq. (9.6a) the long term elasticity is 1.137. The intermediate impulse elasticities are generated by Eq. (9.6f) since population is a component of X.

Spatial lags for other variables such as income also feature in Table 9.1. Indeed, whereas there is no local income effect in Table 9.1 there is a small but statistically significant spatially lagged effect of 0.076. The short run effects of the housing stock and population on house prices in the ECM have opposite signs to their long-run counterparts in the cointegrating vector. The long run effect of housing stock on house prices is negative, but the short term effect is positive. This means that shocks to the housing stock initially increase house prices, but eventually lower them, and because the roots are complex, house prices overshoot their long run value before settling down. The same applies to the dynamic effect of population shocks on house prices, except in the opposite direction.

Finally, we note that because according to the panel DW statistic the SpECM residuals are serially uncorrelated, $$\Delta \mathit{\ln}{\tilde{P}}_{it-1}$$ is weakly exogenous for its SAR coefficient (0.1006). This means that since this variable has been lagged one period its SAR coefficient is estimated consistently without recourse to ML or IV. Matters would have been different if the SpECM residuals were serially correlated.

## 9.4 Empirical Example of Spatial Vector Error Correction

There is a natural hierarchy to error correction models, which has four tiers:
1. 1.

In the basic error correction model (ECM) the dynamics of y in Eq. (9.2a) depends on ut−1 and the dynamic of x in Eq. (9.2b) depends on vt−1.

2. 2.

In spatial error correction models (SpECM) the dynamics of y depends on ut−1 and its spatial lag $${\tilde{u}}_{t-1}$$, and the dynamics of x depends on vt−1 and its spatial lag $${\tilde{v}}_{t-1}$$. In SpECMs there is spatial spillover in error correction.

3. 3.

In vector error correction models (VECM) the dynamics of y and x depend on both ut−1 and vt−1. In VECMs there is mutual dependence in error correction.

4. 4.

In spatial vector error correction models (SpVECM) the dynamics of y and x depend on ut−1, vt−1, $${\tilde{u}}_{t-1}$$ and $${\tilde{v}}_{t-1}$$.

In spatial error correction models the ζ coefficients in Eqs. (9.2a and 9.2b) are assumed to be zero. This means that error correction is induced by “own” or within residuals generated by cointegrating vectors as measures of disequilibrium, and does not depend on measures of disequilibrium regarding other variables. Thus, in Eq. (9.2a) Δyt depends on ut−1 and $${\tilde{u}}_{t-1}$$ and in Eq. (9.2b) Δxt depends on vt−1 and $${\tilde{v}}_{t-1}$$. By contrast, in spatial vector error correction models, the ζ coefficients are not zero so that error correction occurs within and between variables. Hence, Δyt and Δxt depend on ut−1 and $${\tilde{u}}_{t-1}$$ and vt−1 and $${\tilde{v}}_{t-1}$$.

By way of illustration we take two cointegrating vectors from Table . The first is represented by Eq. I for housing starts (measured in square meters), and the second is represented by Eq. III for house prices (at constant prices). Recall, that Eq. I represents the supply of housing, and Eq. III is in inverted demand schedule for housing. These cointegrating vectors refer to the long run relationships between the nonstationary panel data in these models, which include spatial lagged dependent variables as well as spatial Durbin lagged variables. The panel residuals for housing starts (u) and house prices (v) are plotted in Figs. 9.1 and 9.2. Fig. 9.1 Cointegrating residuals: housing starts (u) Fig. 9.2 Cointegrating residuals: house prices (v)

The residuals are mean-zero, autocorrelated and mean reverting as expected. In Fig. 9.2 the residuals (v) are more correlated than in Fig. 9.1 for u. In Fig. 9.1 the residuals for Haifa are more volatile than the rest.

We use the general-to-specific (GTS) methodology (Hendry 1995) to estimate the error correction model. GTS starts with an unrestricted error correction model in which first differences of all the variables in the cointegrating vector are specified, as well as the lagged cointegrating residuals uit−1 and vit−1. GTS is a backward stepwise procedure in which variables are omitted provided their omission does not induce autocorrelation and a deterioration of goodness-of fit in terms of equation standard error or other criteria such as AIC and BIC. To check for path dependence variables omitted at an earlier stage are subsequently respecified. The restricted model should have superior goodness-of-fit to the unrestricted model, and its residuals should be serially independent, or at least no worse than in the unrestricted model.

We present a hierarchy of results. We begin with error correction and move onto spatial error correction. Then we consider vector error correction before we move onto spatial vector error correction.

### Error Correction Models

Table 9.2 reports spatial error correction models for housing starts and house prices. Since the data in ECMs are stationary, SAR coefficients such as λ3 in Eq. (9.2a) and λ4 in Eq. (9.2b) must be estimated by maximum likelihood (or IV). Unlike their counterparts, λ1 and λ2 in Eqs. (9.1a and 9.1b), which were estimated by least squares and are super-consistent, least squares estimates of λ3 and λ4 would not have been consistent. In the case of housing starts (Table 9.2, model 1) the ECM omits the spatial lagged dependent variable because it was not statistically significant (SAR = 0.024 with t-statistic 0.25). The error correction coefficient is −0.474 (t statistic = −6.617), implying that almost half of the error correction for housing starts occurs within a year. Apart from this, the only other variable in model 1 is the lagged change in house prices, suggesting that the short term price elasticity of supply in terms of housing starts is almost two, which is much greater than the long term elasticity reported in the cointegrating vector for housing starts (Table , Eq. I).
Table 9.2

Error correction models

Model

1 Housing starts

2 House prices

3 House prices

Coeff

t-stat

Coeff

t-stat

Coeff

t-stat

ΔlnPit−1

1.879

3.871

0.320

5.00

−0.441

6.939

ΔlnPOPit−1

0.252

1.125

0.077

0.407

ΔlnYit−1

−0.186

−2.29

0.116

0.349

$$\Delta \mathit{\ln}{\tilde{P}}_{it}$$

−0.155

1.544

uit−1

−0.474

−6.617

vit−1

−0.286

−6.318

−0.426

−0.69

R2

0.061

0.044

0.136

se

0.258

0.057

DW

2.23

1.93

LM

13.64

9.92

30.72

BP

31.16

42.96

108.49

CD

−1.72

−0.07

−1.70

Notes: Estimation period: 1988–2014. Estimated with fixed spatial effects. Models 1 and 2 are estimated by EGLS (SUR). Model 3 is estimated by maximum likelihood. The dependent variables are ΔlnSit for housing starts and ΔlnPit for house prices. POP denotes population, Y denotes real wages, u denotes the residuals in Fig. 9.1 and v the residuals in Fig. 9.2. LM is the Lagrange multiplier panel test statistic for second order autocorrelation in the residuals (distributed $${\chi}_2^2$$), BP is the Breusch–Pagan statistic for cross-section dependence in the residuals (distributed $${\chi}_{\mathit{\frac{1}{2}}N\left(N-1\right)}^2$$) and CD tests for strong cross-section dependence in the residuals (distributed N(0,1))

The panel Durbin–Watson statistic indicated that the residuals of model 1 are not serially autocorrelated, in which case the lagged change in log house prices are weakly exogenous. This means that the lagged change in log house prices have a causal effect on subsequent housing starts. By contrast, the LM statistic indicates that the residuals are serially correlated. The DW statistic is biased towards 2 when the covariates include lagged endogenous variables. However, there are no lagged endogenous variables in model 1. Table 9.2 and subsequent tables omit contemporaneous first differences because variables such ΔlnPit are jointly determined with ΔlnSit. Therefore, to test whether current house price changes causally affect changes in housing starts would require estimation by IV.

The error correction coefficient in model 2 for house prices (−0.286) is smaller than its counterpart for housing starts. Hence house prices adjust considerably more slowly than housing starts. Although according to model 1 there is no inertia in housing starts, the autoregressive coefficient in model 2 indicates first order inertia in house prices. Apart from this the ECM for house prices suggests that population growth raises house prices, but the opposite applies to income growth. This means that whereas income raises house prices in the long run, the opposite applies in the short run.

Model 3, estimated by ML includes a contemporaneous term in the spatial lagged endogenous variable. Since the SAR coefficient is negative, it implies negative spatial spillover in house prices. In contrast, the spatial spillover in the cointegrating vector for house prices was positive (Table , Eq. III). The error correction coefficient increases to −0.426 (becomes more negative) from −0.236, but surprisingly it is not statistically significant. Since cointegration and error correction are mirror images of each other, it is not clear why error correction is statistically significant in model 2, but not in model 3.

Finally, the BP and CD statistics test for cross-section dependence in the residuals of the error correction models. Since cross-section dependence and its econometric implications are the focus of Chap. , we defer detailed discussion to the next chapter. However, these statistics are reported her for reference. In the meanwhile, we note that since the critical value of chi square for BP is about 50 and the critical value for CD is −1.96 the BP statistic is not significant in models 1 and 2, and the CD statistic is not significant at conventional levels in all models.

In Table 9.6 we report the counterparts to Eq. (9.5b) for the ARSAR coefficients of the error correction models reported in Tables 9.2, 9.3, 9.4 and 9.5 and the cointegrating vectors reported in Table . The first column refers to lnPt which is AR(2) and SAR(1) and does not directly depend on S since this variable does not feature in the ECM for house prices or the cointegrating vector for house prices. Since housing stocks feature in the cointegrating vector, housing starts affect house prices indirectly, but we ignore this for our present illustrative purposes. The second column refers to lnSt, which is AR(1) and SAR(1), and depends on lags and spatial lags of lnP. Since the ARSAR coefficients sum to fractions, 0.84 in the case of lnP and 0.901 in the case of lnS, the 4N = 36 roots are fractions but some will be complex especially since the AR2 coefficient for lnP is negative.
Table 9.3

Spatial error correction models

Model

1 Housing starts

2 House prices

3 House prices

Coeff

t-stat

Coeff

t-stat

Coeff

t-stat

ΔlnPit−1

1.780

3.539

0.295

4.687

−0.380

6.039

ΔlnPOPit−1

0.292

1.325

0.015

0.083

ΔlnYit−1

−0.132

−1.551

0.136

0.426

$$\Delta \mathit{\ln}{\tilde{P}}_{it}$$

−0.169

1.712

uit−1

−0.470

−6.548

$${\overset{\sim }{u}}_{it-1}$$

−0.233

0.753

vit−1

−0.223

−4.621

−0.032

−0.054

$${\overset{\sim }{v}}_{it-1}$$

0.371

3.229

6.090

4.044

R2

0.062

0.066

0.109

se

0.258

0.056

DW

2.23

1.99

LM

13.91

14.91

24.53

BP

31.63

47.60

95.47

CD

−1.18

−1.41

−1.70

See notes to Table 9.2

Table 9.4

Vector error correction models

Model

1 Housing starts

2 House prices

3 House prices

Coeff

t-stat

Coeff

t-stat

Coeff

t-stat

ΔlnPit−1

2.009

4.053

0.317

4.952

−0.440

6.914

ΔlnPOPit−1

0.235

1.045

0.077

0.405

ΔlnYit−1

−0.185

−2.184

0.116

0.351

$$\Delta \mathit{\ln}{\tilde{P}}_{it}$$

−0.159

−1.580

uit−1

−0.474

−6.630

−0.285

−6.307

0.005

0.020

vit−1

−0.245

−1.253

0.019

1.188

−0.426

−0.688

R2

0.062

0.037

se

0.258

0.057

DW

2.23

1.92

LM

14.02

8.32

BP

32.23

42.58

108.54

CD

−1.43

−0.049

−1.685

See notes to Table 9.2

Table 9.5

Spatial vector error correction models

Model

1 Housing starts

2 House prices

3 House prices

Coeff

t-stat

Coeff

t-stat

Coeff

t-stat

ΔlnPit−1

1.601

3.047

0.274

4.362

−0.386

6.114

ΔlnPOPit−1

0.481

2.046

0.016

0.089

ΔlnYit−1

−0.066

−0.753

0.130

0.409

$$\Delta \mathit{\ln}{\tilde{P}}_{it}$$

−0.174

1.763

uit−1

−0.488

−6.830

0.019

1.196

0.098

0.409

$${\overset{\sim }{u}}_{it-1}$$

−0.197

−0.640

−0.172

−2.33

0.552

1.00

vit−1

−0.026

−0.120

−0.236

−4.911

0.060

0.098

$${\overset{\sim }{v}}_{it-1}$$

1.185

2.239

0.352

3.081

6.219

4.122

R2

0.058

0.0511

0.109

se

0.255

0.055

DW

2.20

1.99

LM

13.08

11.50

24.46

BP

35.56

44.15

92.77

CD

−1.23

−1.23

−1.76

See notes to Table 9.2

### Spatial Error Correction

As in Table 9.1, Table 9.3 reports spatial error correction models (SpECM). Model 1 in Table 9.3 is the same as model 1 in Table 9.2 except it includes a spatial Durbin lag for uit−1. The coefficient on the spatial error correction coefficient is −0.233, but it is not statistically significant. In model 2 both error correction coefficients are statistically significant. The spatial error correction coefficient is 0.371 and positive. The own error correction coefficient (ξ) implies that if house prices were too high in the previous period, they decrease subsequently. The spatial error correction coefficient implies that if house prices were too high elsewhere in the previous period, this disequilibrium spills-over onto house prices. As in Table 9.2 the panel DW statistic suggests that the residuals are not serially correlated, however, the LM statistic continues to contradict this.

Columns 3 and 4 of Table 9.6 report the ARSAR coefficients generated by models 2 and 1 in Table 9.3. In the case of house prices these coefficients are AR(2) and SAR(2) while for housing starts they are AR(1) and SAR(2). The sum of the ARSAR coefficients for lnP is close to 1 (1.083) and the sum for lnS is 0.897. The former implies that the SpECM is not convergent and is inconsistent with the result that variables in Eq. III in Table are cointegrated. Matters would have been different had the relation between housing starts and housing stocks been fully articulated (as in Table ) since house prices vary inversely with housing stocks. Because this indirect effect has been omitted for present illustrative purposes, the ARSAR coefficients for lnP sum to slightly above 1 in Tables 9.3, 9.4 and 9.5. The final form ARSAR coefficients sum to slightly less than 1. Our purpose here is simply to illustrate spatial error correction when there are only two state variables, housing starts (S) and house prices (P). Had we included the nexus between starts, completions and housing stocks, transparency would have suffered.
Table 9.6

ARSAR coefficients

ECM

SpECM

VECM

SpVECM

lnPt

lnSt

lnPt

lnSt

lnPt

lnSt

lnPt

lnSt

Pt−1

1.034

2.082

1.072

1.981

1.495

1.967

1.023

1.784

Pt−2

−0.320

−1.879

−0.295

−1.780

−0.317

−2.009

−0.274

−1.601

$${\overset{\sim }{P}}_{t-1}$$

0.126

−0.230

0.469

−0.129

−0.163

−0.123

0.538

−1.326

$${\tilde{\tilde{P}}}_{t-1}$$

0

0

−0.163

−0.113

0

0

−0.238

−0.616

$${\overline{P}}_{t-1}$$

0

0.197

0

0.292

0

0

0.064

0.284

St−1

0

0.526

0

0.530

−0.285

0.374

0.019

0.512

$${\tilde{S}}_{t-1}$$

0

0.375

0

0.183

0.225

0

0

0.189

$${\tilde{\tilde{S}}}_{t-1}$$

0

0

0

0.184

0

0

0

0.157

Sum

0.84

0.901

1.083

0.897

1.015

0.374

1.04

0.858

Final Form

0.984

1.410

1.011

0.967

Notes: “Sum” refers to sums of ARSAR coefficients. “Final form” refers to the sums of ARSAR coefficients using Eq. (9.5c)

### Vector Error Correction

In Tables 9.1, 9.2 and 9.3 error correction applies within equations but not between them. In VECM models error correction applies within and between equations. Hence, house price dynamics depend on the disequilibrium in housing starts, and the dynamics of housing starts depend on the disequilibrium in house prices. In VECMs there is no spatial error correction. Results are reported in Table 9.4.

In model 1 there is negative error correction from u and v for housing starts. The former means that housing starts decrease if in the previous period they were too high (u > 0), which makes economic sense. The latter means that housing starts decrease if in the previous period house prices were too high (v > 0), which makes economic sense if building contractors expect house prices to fall. The former effect is clearly statistically significant, but the latter effect is not significant at conventional levels. In model 2 there is negative error correction from u and positive error correction from v. The former means that house prices decrease if starts were too high in the previous period, which makes economic sense. The latter means that if house prices were too high in the previous period, they grow even higher, which does not make economic sense. Moreover, the former effect is statistically significant, and the latter effect is not.

The ARSAR coefficients sum to 1.015 for house prices and to 0.734 for housing starts (Table 9.6). The final form ARSAR coefficients sum to 1.41. The VECM for lnP is AR(2) and SAR(1) and for lnS it is AR(1) and SAR(0). Whereas the ECMs in Table 9.2 and the SpECMs in Table 9.3 have recursive structures because starts depend on house prices, but house prices do not depend directly on starts, matters are different in the VECMs. Table 9.6 shows that the VECMs are not recursive because house prices depend on lagged starts with a coefficient of −0.285 as well as their spatial lag with a coefficient of 0.225. Since these coefficients roughly offset each other the sum of the final form ARSAR coefficients sum to 1.011.

Model 3 in Table 9.4 is estimated by ML because it includes a contemporaneous spatial lagged dependent variable. The spatial lag coefficient remains similar to what it was in Tables 9.2 and 9.3, and the error correction coefficients cease to be statistically significant. However, the spatial lag coefficient continues to be not statistically significant at conventional levels.

### Spatial Vector Error Correction

Table 9.5 presents illustrative results for spatial vector error correction models in which error correction is both spatial and interdependent. For example, in model 1 housing starts decrease if they were too high in the previous year; the error correction coefficient is −0. 488 and is statistically significant. There is a negative spatial lag (−0.197) suggesting that housing starts decrease if they were too high elsewhere, however, this effect is not statistically significant. Nor is the error correction effect of excess house prices (−0.026). On the other hand, there is strong and significant spatial error correction, implying that housing starts increase if house prices were too high elsewhere.

In the SpVECM for house prices three of the four error correction coefficients are statistically significant (model 2). House prices decrease when house prices were too high locally, and they increase if they were too high elsewhere. They decrease if housing starts were too high elsewhere. Surprisingly, house prices depend on local error correction. The ARSAR coefficients in Table 9.6 sum to 1.049 for lnP and to −0.858 for ln S. Their final form counterparts sum to 0.967. The SpVECM in Table 9.6 incorporates six spatial effects; two for house prices (first and second order spatial lags for house prices) and four for housing starts (first and second order spatial lags for house prices and starts).

Our purpose here is not to choose the best, or indeed any, of the error correction models. Rather, our intension is simply to illustrate the hierarchy of error correction models. There is no reason why the best model should not be hybrid. For example, the error correction for one variable might be SpECM while for another variable it might be VECM. Nevertheless, error correction models with final form ARSAR coefficients summing to 1 and more are less admissible because they do not converge to equilibrium. In Table 9.6 this rules out SpECM, VECM and SpVECM.

We have already noted that this artefact happens because for illustrative purposes we chose to focus on only two variables, house prices and housing starts. A complete analysis would have involved specifying the relationship between housing starts and housing stocks, which would have involved error correction models for housing completions. In this case the ARSAR coefficients would have summed to less than 1 because housing starts, housing completions, house prices and housing stocks are spatially coinregrated.

Chapter includes the first episode of a plot that has four parts. The first episode involves spatiotemporal unit root tests for spatial panel data in Israel. The second episode in Chap. involves spatial panel cointegration tests among these variables. The third episode, presented in this chapter, involves spatial error correction between these variables. The final episode is in the next chapter, where we ask whether the spatial panel data in the series are related through common factors rather just spatial dependence alone.

## References

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## Authors and Affiliations

• Michael Beenstock
• 1
• Daniel Felsenstein
• 2
1. 1.Department of EconomicsHebrew University of JerusalemJerusalemIsrael
2. 2.Department of GeographyHebrew University of JerusalemJerusalemIsrael