Abstract
This paper studies the spatial dependence of residential resale housing returns in ten major Canadian Census Metropolitan areas (or CMAs) from 1992Q4 to 2012Q4 and makes the following methodological contributions. Firstly, in the context of a spatial dynamic panel data model we use grid search to derive the appropriate spatial weight matrix W among different possible specifications. We select the compound W with the minimum root mean squared error formed from geographical distances and the ten CMAs’ gross domestic product. Secondly, contrary to common practice in the literature, we decompose the impacts of explanatory variables into direct and indirect impacts and proceed to derive and plot the impulse response functions of housing returns to external shocks. The empirical results suggest that Canadian residential housing markets exhibit statistically significant spatial dependence and spatial autocorrelation and that both geographical distances and economic closeness are the dominant channels of spatial interaction. Furthermore, the special feature of the Canadian housing market is that the responses to the shocks do not spread widely across regions and that they fade fast over time.
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Notes
 1.
Canada Mortgage and Housing Corporation (or CMHC) Canadian Housing Observer 2012.
 2.
Another available housing price measure is the Multiple Listing Services (MLS\(^{{\circledR }}\)) conducted by Canadian Real Estate Association (or CREA). However, the MLS\(^{{\circledR }}\) index are not quality adjusted. Holios and Pesando (1992) discussed some shortcomings of MLS\(^{{\circledR }}\) index, including mix of units sold varies over time, so that the ’quality’ of dwelling units sold is not held constant; the shortrun signals regarding price movements often diverge; and the MLS index suggests a more rapid rate of price escalation.
 3.
Detailed definition of bungalow and twostory executive can be found in Glossary of Housing Types of Royal LePage House Price Survey.
 4.
See Appendix A for the detailed list of selected 10 CMAs and their geographical locations. A CMA is a larger area than a city as it also includes municipalities adjacent to the city urban core. Geographic definitions used by Royal LePage is slightly differ from those used by Statistics Canada. However, it is the closest data available.
 5.
More specifically, we use annual CMAlevel household income divided by the annual national level household income, and we assume this ratio remain constant in a given year. Then, we use this ratio multiplied by the quarterly national level income to get the quarterly CMAlevel household income.
 6.
In most applied spatial econometrics, the same spatial weight matrix is used in both spatial lags of dependent variable and disturbances. We tried to estimate our model with different weight matrices in spatial lags and spatial errors, yet the results did not change much. This indicates that spatial dependence of housing returns and spatial autocorrelation in disturbances comes from a similar channel.
 7.
We thank the anonymous referee for clarifying this point.
 8.
We also tried to construct economic weight matrix using the negative exponential function. However, in those cases we ran into computational problems and as such we only use the inverse distance weighting function to construct the economic weight.
 9.
The variable mov_{i,j} includes all people lived in different address 5 years before the census year, regardless of where did they moved. While the other variable in Eq. 6mig_{i,j} only counts the number of people moved between CMA i and j specifically, and it does not include the population growth from immigrants and external migrants. These two variables are different, and we think they are both important to be used to explain the spatial dependence of housing returns.
 10.
The absolute value of eigenvalues of the matrix \(\hat {\lambda }W\) are 0.7984, 0.2289, 0.1600, 0.1548, 0.1180, 0.1010, 0.0984, 0.0712, 0.05338, and 0.0147. The absolute value of eigenvalues of the matrix \(\hat {\rho }W\) are 0.9916, 0.2843, 0.1987, 0.1922, 0.1465, 0.1254, 0.1222, 0.0885, 0.0663, and 0.0183. All absolute eigenvalues are less than 1, which ensures our spatial system is stationary. Equation 21 in Appendix C shows the overall impacts of spatial interactive relations on the housing returns.
 11.
It is worth noting that the weight matrices used in spatial models are typically assumed to be exogenous. However, the exogeneity assumption may not be reasonable. Kelejian and Piras (2014) discuss an estimation method with an endogenous weight matrix. Similarly, in Case et al. (1993), many of the weight matrices used to capture economic similarities are likely to be endogenous. In our case we also use weight matrices based on economic factors, yet the finally selected W in Eq. 12 formed by geographical distances and the real GDP levels is taken to be exogenous as geographical factors and GDP levels are taken as given. In that sense, the issue of endogeneity that we face may not be as severe. Given the complexity of our approach for arriving at the final choice of the weight matrix based on a data driven grid search method, we leave any issue of endogeneity for future research.
 12.
The magnitude of one standard deviation of population growth in each CMA is listed in last column of Table 7.
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Acknowledgments
This paper is part of the Ph.D. thesis of the first author. It has been presented at the 2nd Annual Doctoral Workshop in Applied Econometrics, the 41th Annual Conference of the Eastern Economics Association, and the 49th Annual Conference of the Canadian Economics Association. We wish to thank an anonymous referee for very useful comments that helped with interpretation of our results. We also want to thank Dr. Martin Burda, Xuefeng Pan, Dr. Christos Ntantamis and participants in the conferences, for their helpful comments on earlier drafts of this paper. We wish also thank Dr. Paul Anglin and Dr. Min Seong Kim for their valuable comments. The second author would like to thank for the financial support from the Social Science and Humanities Research Council of Canada Insightful Grant 43520160340.
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Appendices
Appendix
A. List of CMAs
B. Detailed Data Sources

Building Permits: CANSIM Table 0260006.

Housing Starts: CANSIM Table 0270052.

Completed: CANSIM Table 0270060.

Construction Union Wage Index (UWI): Table 3270045. The UWI data is not available for Regina from 1992Q2 to 2007 Q1, so the Prairie Region data is used for that period instead.

NHPI: CANSIM Table 3270046

Rent index: CANSIM Table 3260020.

Population and unemployment rate: CANSIM Table 2820090 for year 1992Q21995Q4 and Table 2820109 for year 1996Q12013Q1. Data in January and February 1996 is not available, so the data in March 1996 is used for 1996Q1 instead of quarterly average.

Household income: quarterly and annually national household income are from CANSIM Table 3800083, and annual CMAlevel household income is from CANSIM Table 2020605, which is real median aftertax household income. The income data is not available for Regina from year 1992 to 1997, so the household income for province Saskatchewan is used at that period.
C. The IRF of the Shock to Housing Returns
C.1 The Derivation of the IRF of the Shock to Housing Returns
An impulse response function traces out the magnitude and duration of a variable in response to a shock. We apply this technique to our dynamic spatial panel data model to better understand the effect of shocks on housing returns over time and across spaces. From model (1)–(2), we have
To simplify the notation, let I_{N} − λW_{N} = A, and I_{N} − ρW_{N} = B. Rearranging (20), we have
The instant impulse response to a onetime shock in ε_{t} of magnitude η at time t is
At period t + 1,
If we substitute Eq. 21 into Eq. 23, we get
We assume the shock to ε_{t} does not pass on to the the variable X_{s}, ε_{s} and v_{s} for s ≥ t, then we hold X_{s} and v_{s}unchanged for all period s ≥ t. Therefore, we could clearly observe how the housing return in response to such a shock. Then, one time shock in ε at time t results
The corresponding impulse response function in period t + 1 is
Repeating this procedure we have
The corresponding impulse response functions are
for h = 0, 1, 2,...T.
In order to construct the 95% confidence interval for each impulse response, we need to compute the standard error for the IRF_{h}. The delta method is used to compute the standard error. We take the derivative of the transformed function with respect to the parameter, and multiply it by the asymptotic variance of the untransformed parameter. To be specific,
where Σ is the asymptotic variancecovariance matrix of the estimators of γ, λ and ρ. We then construct the 95% confidence interval
where the \(\hat {IRF_{h}}\) is the estimate of IRF_{h}. The detailed derivatives of IRF_{h} can be found below.
C.2 Derivatives of IRF_{h} of the shock to housing returns
In order to construct the 95% confidence interval, we need to compute the standard errors for the estimates of IRF_{h}, then apply the delta method. The derivatives of IRF_{h} with respect to parameter (γ,λ,ρ) in each period, from h = 0 to h = 6, are presented below.
In period t, h = 0, we impose a shock and denote it by η, then
The derivative of IRF_{0} with respect to \(\left (\gamma ,\lambda ,\rho \right )\) is
where “ ⊗” refers to the Kronecker product, and \(vec\left (W_{N}\right )\) is the vectorization of matrix W_{N} and transforms W_{N} into a column vector.
In period t + h, where h ≥ 1
For h ≥ 1, the derivative of IRF_{h} with respect to \(\left (\gamma ,\lambda ,\rho \right )\) is
D. The IRF to a Population Growth Shock
D.1 The Derivation of IRF to the Population Growth Shock
We first need to compute the impulse response functions of a onetime shock to the explanatory variables X_{t−1}. From Eq. 1, we have
where I_{N} − λW_{N} = A. We denote an N × 1 vector ψ to be the shock on population growth at time t − 1. β_{p} is the coefficient in front of the population growth. Then, the corresponding impulse response functions are
Equation 34 is the response in quarter t, and it is also referred to as impulse response at horizon zero. We also apply the delta method and Eq. 28 to construct the 95% confidence interval for each IRF’s. The detail of derivatives that required to construct the error bound is given below.
D. 2 The Derivatives of IRF to a Population Growth Shock
In this section, we include the detailed derivatives that required by Eq. 27 to construct the 95% confidence interval of the impulse response functions. This section is similar to the previous section Appendix C.
We calculate the derivatives of IRF with respect to parameters space γ, λ and β_{p}, where A = I_{N} − λW_{N}, β_{p} is the parameter in front of the population growth, and ψ is an N × 1 vector denotes the shock to the population growth at horizon zero.
When h = 0, IRF_{0} = A^{−1}ψβ_{p}, the derivative of IRF_{0} with respect to \(\left (\gamma ,\lambda ,\beta _{p}\right )\) is
In period t + h, where h ≥ 1
For h ≥ 1, the derivative of IRF_{h} with respect to \(\left (\gamma ,\lambda ,\beta _{p}\right )\) is
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Zhang, Y., Sun, Y. & Stengos, T. Spatial Dependence in the Residential Canadian Housing Market. J Real Estate Finan Econ 58, 223–263 (2019). https://doi.org/10.1007/s1114601796232
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Keywords
 Canadian residential resale housing returns
 Impulse response functions
 Spatial dependence
 Spatial dynamic panel data models
 Spatial weight matrix