# Measuring and Interpreting Urban Externalities in Real-Estate Data: A Spatio-Temporal Difference-in-Differences (STDID) Estimator

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## Abstract

**:**

## 1. Introduction

## 2. Measuring Urban Externalities with the Hedonic Pricing Model

#### 2.1. The Hedonic Pricing Model

**y**= α

**ι**+

**Xβ**+

**ε**

**y**of dimension (N × 1), the list of independent variables are stacked in a matrix

**X**of dimension (N × K),

**ι**is a vector of one, and

**ε**a vector of errors, both of dimension (N × 1). The vector of parameters

**β**is designed to evaluate the implicit price of each amenity and is of dimension (K × 1), while α is a scalar parameter.

**y**= ρ

**Wy**+ α

**ι**+

**Xβ**+

**ε**

**y**= α

**ι**+

**Xβ**+

**WXθ**+

**ε**

**y**= α

**ι**+

**Xβ**+

**u**

**u**= λ

**Wu**+

**ε**

**W**is an (exogenous) spatial weights matrix of dimension (N × N), expressing the possible spatial relations (or connections) among the observations, the parameters ρ and λ, and the vector of parameters

**θ**of dimension (P × 1), that allows us to capture the spatial effects (with P ≤ K), while

**ε**is the vector of error terms, which is assumed to be homoskedastic and independent. The autoregressive parameter ρ capture the global spatial effect, while the vector of parameters

**θ**capture the local spatial effect [16,17,37].

**y**= λ

**Wy**+ α

**ι**+

**Xβ**+

**WX**(λ

**β**) +

**ε**).

**y**= ρ

**Wy**+ α

**ι**+

**Xβ**+

**WXθ**+

**ε**

_{k}(Equation (6)).

**y**/∂

**x**

_{k}= (

**I**β

_{k}+

**W**θ

_{k})

**I**is the identity matrix of dimension (N × N). With the weights matrix

**W**row-standardized, the interpretation can be simplified to include the local spatial (externalities) effect (θ

_{k}). In such a case, the marginal effect is decomposed into a direct effect (β

_{k}) and the total effect (β

_{k}+ θ

_{k}).

**y**reduced form equation with respect to a change in variable

**x**

_{k}(Equation (7)). In the SAR specification θ

_{k}= 0 and the second term on the right-hand side can be simplified.

**y**/∂

**x**

_{k}= (

**I**− ρ

**W**)

^{−1}(

**I**β

_{k}+

**W**θ

_{k})

**I**− ρ

**W**)

^{−1}is a spatial multiplier that can be expressed as the sum of powers of the weights matrix

**W**(Equation (8)—[16,39]) (According to the infinite series properties).

**I**− ρ

**W**)

^{−1}=

**I**+ ρ

**W**+ (ρ

^{2}

**W**

^{2}+ ρ

^{3}

**W**

^{3}+ ρ

^{4}

**W**

^{4}+ …) =

**S**(

**W**)

**I**β

_{k}(or by [

**I**β

_{k}+

**W**θ

_{k}] for the SDM); (ii) an induced effect, as measured by ρ

**W**β

_{k}(or ρ

**W**[

**I**β

_{k}+

**W**θ

_{k}])); (Or by ρβ

_{k}or ρ(β

_{k}+ θ

_{k}) when the weights matrix

**W**is row-standardized) and (iii) a spillover effect, as measured by the rest of the terms appearing in parentheses on the right-hand side of Equation (8) multiplied by

**I**β

_{k}(or (

**I**β

_{k}+

**W**θ

_{k})) [39]. On the other hand, others argue that the marginal effect should be decompose into: (i) the direct effect, as measured by the mean elements appearing on the diagonal of the matrix

**S**(

**W**)

**I**β

_{k}for the SAR model or

**S**(

**W**)(

**I**β

_{k}+

**W**θ

_{k}) for the SDM specification; (ii) the indirect effect, as measured by the mean of the elements appearing on the off-diagonal; and (iii) the sum of both effect to yield the total marginal effect [5].

^{−1}(see [16,39,40,41,42]). In such a case, the total marginal effect can be expressed as β

_{k}/(1 − ρ) for the SAR model and (β

_{k}+ θ

_{k})/(1 − ρ) in the SDM model [43]. Using such a simplification to express the total marginal effect, [28] have suggested that β

_{k}, for the SAR model, or (β

_{k}+ θ

_{k}) for the SDM model can be interpreted as direct effects. The difference between the total marginal effect and the direct marginal effect enables us to ascertain the indirect marginal effect. This interpretation greatly simplifies the calculation, while returning an interpretation akin to the one usually made within a multiple regression framework. However, it does not allow us to directly retrieve the significance of such marginal effects, except for the direct marginal effects.

**x**

_{k}, results from a technological change and not from a pecuniary change. Thus, there is a challenge to distinguish between technological and pecuniary changes. However, this challenge can be partly solved when working with cross-sectional data that also include a time factor, such as real-estate transactions.

#### 2.2. Spatial Data Pooled over Time and Weights Matrices

_{t}= 1, 2, …, N

_{t}), while individual observations are rarely repeated over time. In such a case, the total sample size is equal to N

_{T}(N

_{T}= Σ

_{t}N

_{t}for t = 1, 2, …, τ). The visual reading of such data can be seen as a collection of spatial layers of information pooled over time when data are aggregated into discrete time periods [25]. Consequently, the dimensions of the matrices and vectors depend on the nature of the data under consideration, while the validity of the estimation of the parameters relies on the observance of the usual assumptions made about the error terms.

**T**, with a general element t

_{ij}taking a value of 1 if observations i and j are collected in the same time period (or interval) and 0 otherwise is built [23,43,44]. This temporal weights matrix is then after multiplied, element-by-element, with a general spatial weights matrix,

**W**, to obtain a general weights matrix that enables us to isolate multidirectional spatial effect:

**S**=

**W**$\u2a00$

**T**, where $\u2a00$ is a Hadamard product. The individual elements of the spatial weights matrix are defined as usual, i.e., as a function of the distance separating two observations i and j such that w

_{ij}= f(d

_{ij}), where f(·) could be any decreasing function, assigning a higher value to closer observations and a lower value to more remote observations.

**T**a general weights matrix indicating the temporal distance among observations i and j if observation j has been collected before i, then a general weights matrix

**W**=

**W**$\u2a00$

**T**, can be used to isolate a spatial unidirectional effect [24,28] (As previously argued by [43,44], the form of both temporal weights matrices,

**T**and

**T**, can be simplified if the observations are chronologically ordered beforehand. In such a case, the temporal weights matrix takes on a structure divided into three separate parts: (i) the upper-triangular part is defined by elements all equal to zero (0), indicating that future observations cannot exert any influence on present or past observations; (ii) a block-diagonal structure that isolates observations occurring in the same time period, or in the same time window; and (iii) a lower-triangular part that expresses the possible relations between previous observations and current observations).

**Wy**

_{t−1}, that, when

**W**is row-standardized, expresses the mean value of

**y**for observations collected previously within a spatial delimitation. One interesting feature of this new variable is that it can be seen as strictly exogenous (Since the observations are not the same in each time period, it is necessary to use a different notation than the spatial panel case where the observations are repeated in each time period and the same weights matrix is used).Thus, the standard linear regression model can incorporate the variable

**Wy**

_{t−1}in the specification and the model can still be estimated using ordinary least squares (OLS) or generalized least squares (GLS) [17,34]. By denoting ψ the additional coefficient related to the variable

**Wy**

_{t−1}, and by using the weights matrix

**S**based on the Hadamard product to isolate spatial multidirectional effects, one obtains a more complex set of (spatio-temporal) models (Equations (9)–(12), see [23]).

**y**

_{t}= ρ

**Sy**

_{t}+ ψ

**Wy**

_{t−1}+ α

**ι**+

**X**

_{t}

**β**+

**ε**

_{t}

**y**

_{t}= α

**ι**+ ψ

**Wy**

_{t−1}+

**X**

_{t}

**β**+

**SX**

_{t}

**θ**+

**ε**

_{t}

**y**

_{t}= α

**ι**+ ψ

**Wy**

_{t−1}+

**X**

_{t}

**β**+

**u**

_{t}

**u**

_{t}= λ

**Su**

_{t}+

**ε**

_{t}

**y**

_{t}= ρ

**Sy**

_{t}+ ψ

**Wy**

_{t−1}+ α

**ι**+

**X**

_{t}

**β**+

**SX**

_{t}

**θ**+

**ε**

_{t}

_{k}in time t does not influence the variable y

_{t−1}and

**W**y

_{t−1}is exogenous from the time period t (In such a case, the data generating process can be expressed as

**y**

_{t}= (

**I**− ρ

**S**)

^{−1}[ψ

**Wy**

_{t−1}+ α

**ι**+

**X**

_{t}

**β**+

**SX**

_{t}

**θ**] and the partial derivative of the function

**y**

_{t}with respect to

**x**

_{kt}become ∂

**y**

_{t}/∂

**x**

_{kt}= (

**I**− ρ

**S**)

^{−1}(

**I**β

_{k}+

**S**θ

_{k})).

**y**

_{t+1}/∂

**x**

_{kt}= ψ × (

**I**− ρ

**S**)

^{−1}(

**I**β

_{k}+

**S**θ

_{k})

**y**

_{t+s}/∂

**x**

_{kt}= (1 − ψ)

^{−1}× (

**I**− ρ

**S**)

^{−1}(

**I**β

_{k}+

**S**θ

_{k})

## 3. Difference-in-Differences (DID) Estimator

#### 3.1. Repeated-Sales (RS) Approach, or DID Estimator

**y**

_{s}= α

_{s}

**ι**+

**X**

_{s}

**β**+

**u**

_{s}

**y**

_{r}= α

_{r}

**ι**+

**X**

_{r}

**β**+

**u**

_{r}

**y**

_{r}−

**y**

_{s}) = (α

_{r}− α

_{s})

**ι**+ (

**X**

_{r}−

**X**

_{s})

**β**+ (

**u**

_{r}−

**u**

_{s})

Δ

**y**

_{t}= (α

_{r}− α

_{s})

**ι**+ Δ

**X**

_{t}

**β**+

**ε**

_{t}

_{t}. In such a case, the difference in the amenities of the goods (

**X**

_{r}−

**X**

_{s}) is assumed to be zero, while the exponential of the coefficients α

_{t}allows us to ascertain a global price index, with the first time period as the reference (price index = 1).

**X**over time on price (or price growth) (This approach can also be extended to the case where no particular amenities change over time, but where the implicit prices of the amenities are assumed to evolve over time by denoting the implicit price by

**β**

_{t}instead of

**β**in Equations (13) and (14)). This approach is used to measure the impact of changes in environmental (and exogenous) amenities on price or price growth [29,30]. Such an approach allows us to adequately control for most criticisms levelled at the hedonic pricing model.

_{T}instead of N

_{T}, where N

_{T}>> n

_{T}and where the total number of observations per period is given by n

_{t}.

#### 3.2. DID, SDID, STDID and Marginal Effects

**y**

_{t}, as a function of its own characteristics,

**X**

_{t}, but also as a function of the other sale prices occurring in the same time period around the good i,

**Sy**

_{t}, and as a function of the characteristics of the goods sold within the same time period,

**SX**

_{t}(Equation (18)).

**y**

_{t}= ρ

**Sy**

_{t}+ α

_{t}

**ι**+

**X**

_{t}

**β**+

**SX**

_{t}

**θ**+

**u**

_{t}

**S**is a row-standardized weights matrix expressing the spatial relations for observations recorded in the same time period.

**y**

_{t}= (α

_{r}− α

_{s})

**ι**+ ρ

**S**Δ

**y**

_{t}+ Δ

**X**

_{t}

**β**+

**S**Δ

**X**

_{t}

**θ**+

**ε**

_{t}

**y**

_{t}/∂Δ

**x**

_{kt}= (β

_{k}+ θ

_{k})/(1 − ρ)

**Wy**

_{t−1}(Equation (21)).

**y**

_{t}= ρ

**Sy**

_{t}+ ψ

**Wy**

_{t−1}+ α

_{t}

**ι**+

**X**

_{t}

**β**+

**SX**

_{t}

**θ**+

**u**

_{t}

**W**is a row-standardized weights matrix isolating the spatial unidirectional relations, from observations collected from the previous time period.

**y**

_{t}= (α

_{r}− α

_{s})

**ι**+ ρ

**S**Δ

**y**

_{t}+ ψ

**W**Δ

**y**

_{t−1}+ Δ

**X**

_{t}

**β**+

**S**Δ

**X**

_{t}

**θ**+

**ε**

_{t}

**y**

_{t+1}/∂Δ

**x**

_{kt}= ψ(β

_{k}+ θ

_{k})/(1 − ρ)

**y**

_{t+∞}/∂Δ

**x**

_{kt}= (β

_{k}+ θ

_{k})/[(1 − ρ)(1 − ψ)]

_{ikt}→ ∆y

_{it}) (Figure 1). This is denoted by the direct (or first-round) effect. Thus, the first observation to record a change in the value of y

_{it}is the observation that first records an exogenous change in x

_{kt}. The variation in the value of y

_{it}will necessarily imply changes in values of the dependent variables for the other observations spatially close to the observations that have recorded a change in x, and so on (∆y

_{it}→

**S**∆y

_{it}→ ∆y

_{jt}). Thus, the second wave of change implies the indirect effects. The change in the values of y

_{it}, and by extension of

**S**∆y

_{it}, will necessarily impact the values of y

_{it+1}, and

**S**∆y

_{it+1}, and so on. These additional effects are spatially localized dynamic effects, generating short- and long-term direct effects.

_{k}); (ii) direct spatial effect (β

_{k}+ θ

_{k}) if the SDM or STDM specification is used); (iii) a total spatial effect ((β

_{k}+ θ

_{k})/(1 − ρ)); and (iv) a total short-term (ψ × [(β

_{k}+ θ

_{k})/(1 − ρ)]) and total long-term marginal effect ((β

_{k}+ θ

_{k})/[(1 − ρ) × (1 − ψ)]) (Based on Equations (13) and (14)).

## 4. Empirical Investigation

`Stata`software and the

`spreg`command. More detail on the estimation procedure can be found in appendix of the book of [24]). The case study examines two separate commuter rail trains (CRT) in the Montreal (QC, Canada) metropolitan area. The first CRT serves the north shore part of the metropolitan area and was inaugurated in May 1997, when the first station opened in the municipality of Blainville and Sainte-Thérèse (the green line—Figure 2). Two more stations opened after that. The first was in Rosmère, a few months after the inauguration (1 January 1998), while the next stations only opened in January 2007 in St-Jérôme, the municipality located in the northern part of the metropolitan area. The impact of this new mass transit system has already been studied by [20]. The second CRT serves the eastern part of Montreal and has six stations. The service was officially introduced in 2000 when the first three stations opened, respectively Saint-Bruno (February 2000), and Saint-Lambert and McMasterville (May 2000). Two years after, the station of Mont-Saint-Hilaire was inaugurated (September 2002), while the last two stations opened in late 2003: Saint-Basile-le-Grand and Saint-Hubert (the purple line—Figure 2).

**SX**will necessarily be closely correlated to those in the matrix

**X**, introducing a collinearity problem. However, the price determination process can clearly depict a spatial component that can be controlled for using the STAR model (when

**θ**=

**0**in Equation (21)), which allows us to decompose the spatial effect into two separate spatial components.

#### 4.1. Descriptive Statistics

#### 4.2. Estimation Results

#### 4.2.1. The South-East CRT Line

^{2}is higher in the STDID model, but the gain is marginal. The Akaike information criterion (AIC) and the Schwartz information criterion (BIC) both indicate the significant gain of considering the SDID and STDID estimator instead of the usual DID approach. This is also confirmed by comparing the log-likelihood statistics. A log-likelihood ratio (LR) test indicates that both specifications are statistically preferable to the DID model (LR = 142.62 for the SDID and LR = 159.92 for the STDID). This is also supported by the fact that the autoregressive coefficients are statistically significant, as proposed by the t-test. The comparison between the SDID and the STDID is less clear, but the LR test indicates that the STDID model is preferable (LR = 17.20). Thus, statistically, the STDID model performs better globally.

#### 4.2.2. The North CRT Line

^{2}is higher in the STDID model, but the gain is once again marginal, while the AIC and BIC statistics are both lower in the STDID specification. The log-likelihood ratio (LR) test also points in favor of the superiority of the SDID and the STDID models over the DID model (72.10 for the SDID and 83.21 for the STDID). The comparison between the SDID and the STDID is less clear, but the LR test indicates that the STDID model is preferable (LR = 10.20). Thus, statistically, the STDID model gives, once again, a better global performance.

#### 4.2.3. Calculating and Interpreting the Marginal Effect

_{CBD}= 5), but not within walking distance of the station, the DID (in such as case, ∂

**y**

_{t}/∂

**d**

_{0–2t}= β

_{0–2}+ d

_{CBD}× β

_{0–2;CBD}) suggests a direct price increase of about 14.45% (with ∂

**y**

_{t}/∂

**d**

_{0–2t}= β

_{0–2}+ d

_{CBD}× β

_{0–2; CBD}= 0.1774 + (5 × (−0.0066)) versus 14.71% for the SDID and 14.72% for the STDID. Moreover, the spatial spillover effect implies that the total marginal effect is evaluated at 15.58% for the SDID (∂

**y**

_{t}/∂

**d**

_{0-2t}= (β

_{0–2}+ d

_{CBD}×β

_{0–2CBD}) × (1 − ρ)

^{−1}) and 15.40% for the STDID, while the total and final impact for the STDID (∂

**y**

_{t+∞}/∂

**d**

_{0–2t}= (β

_{0–2}+ d

_{CBD}×β

_{0–2CBD}) × [(1 − ρ) × (1 − ψ)]

^{−1}) is estimated at 15.74% (Table 5). Thus, the total effect is higher by more than 1 percentage point in the STDID than in the DID model.

#### 4.3. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic representation of the decomposition of the marginal effect according to the type of model selected.

**Figure 2.**Map of the CRT system in the Montréal suburbs; Source: Agence Métropolitaine de Transport (AMT).

**Figure 3.**Temporal distribution of sales and resales over time for both samples, 1992–2009; Legend: grey→south shore, black→north shore; solid line→sale, dashed line→resale.

**Figure 4.**Number of housing experiencing a change in accessibility to the nearest CRT station. Legend: grey→south shore, black→north shore; solid line→sale, dashed line→resale. Notes: For the x-axis, the sign [ on the left signify that the value in included, while the sign [ on the right means that the value is not included.

**Figure 5.**Distribution of the sale and resale price according to the logarithmic transformation among samples: (

**a**) Distribution of (first) sale price for the south-east corridor; (

**b**) Distribution of (second) resale price for the south-east corridor; (

**c**) Difference between resale and sale price for the south-east corridor; (

**d**) Distribution of (first) sale price for the north corridor; (

**e**) Distribution of (second) resale price for the north corridor; (

**f**) Difference between resale and sale price for the north corridor.

**Table 1.**Temporal distribution of the transactions along the south-east (purple) CRT line, 1995–2009.

Sale | |||||||||||||||

Resale | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 |

1995 | |||||||||||||||

1996 | 57 | 11 | |||||||||||||

1997 | 117 | 68 | 13 | ||||||||||||

1998 | 126 | 141 | 60 | 15 | |||||||||||

1999 | 151 | 188 | 140 | 73 | 16 | ||||||||||

2000 | 149 | 200 | 193 | 151 | 87 | 15 | |||||||||

2001 | 160 | 212 | 226 | 198 | 204 | 143 | 22 | ||||||||

2002 | 136 | 204 | 188 | 220 | 252 | 209 | 161 | 28 | |||||||

2003 | 100 | 150 | 176 | 193 | 197 | 219 | 266 | 202 | 30 | ||||||

2004 | 90 | 132 | 159 | 138 | 193 | 234 | 288 | 297 | 216 | 43 | |||||

2005 | 82 | 123 | 139 | 147 | 178 | 190 | 257 | 298 | 257 | 217 | 42 | ||||

2006 | 64 | 118 | 120 | 104 | 131 | 137 | 216 | 308 | 268 | 286 | 159 | 29 | |||

2007 | 61 | 113 | 114 | 103 | 117 | 151 | 198 | 254 | 276 | 290 | 307 | 175 | 40 | ||

2008 | 70 | 75 | 96 | 103 | 120 | 162 | 149 | 192 | 227 | 248 | 297 | 230 | 146 | 31 | |

2009 | 58 | 67 | 87 | 93 | 95 | 139 | 175 | 206 | 194 | 235 | 246 | 257 | 229 | 148 | 19 |

Sale | ||||||||||||||||||

Resale | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 |

1992 | ||||||||||||||||||

1993 | 5 | 0 | ||||||||||||||||

1994 | 21 | 34 | 7 | |||||||||||||||

1995 | 32 | 56 | 29 | 1 | ||||||||||||||

1996 | 58 | 89 | 64 | 39 | 5 | |||||||||||||

1997 | 47 | 94 | 93 | 78 | 46 | 8 | ||||||||||||

1998 | 39 | 81 | 104 | 93 | 90 | 59 | 7 | |||||||||||

1999 | 39 | 79 | 119 | 89 | 138 | 111 | 69 | 11 | ||||||||||

2000 | 47 | 63 | 91 | 101 | 149 | 146 | 119 | 95 | 17 | |||||||||

2001 | 38 | 84 | 102 | 93 | 141 | 173 | 171 | 209 | 109 | 17 | ||||||||

2002 | 31 | 87 | 112 | 105 | 147 | 181 | 197 | 235 | 212 | 199 | 32 | |||||||

2003 | 43 | 64 | 81 | 74 | 125 | 114 | 163 | 198 | 217 | 264 | 216 | 48 | ||||||

2004 | 27 | 52 | 66 | 67 | 100 | 127 | 140 | 201 | 228 | 276 | 324 | 218 | 43 | |||||

2005 | 16 | 40 | 56 | 68 | 95 | 98 | 143 | 138 | 183 | 269 | 318 | 309 | 258 | 30 | ||||

2006 | 18 | 42 | 50 | 56 | 74 | 86 | 75 | 125 | 128 | 189 | 242 | 313 | 319 | 209 | 23 | |||

2007 | 20 | 40 | 48 | 51 | 91 | 78 | 112 | 109 | 168 | 188 | 247 | 284 | 362 | 369 | 228 | 40 | ||

2008 | 16 | 28 | 32 | 45 | 60 | 68 | 74 | 94 | 110 | 156 | 194 | 212 | 275 | 283 | 305 | 165 | 24 | |

2009 | 10 | 29 | 39 | 52 | 54 | 68 | 71 | 104 | 87 | 157 | 171 | 201 | 263 | 305 | 310 | 269 | 134 | 25 |

∆ Walking Distance | South | North |
---|---|---|

(0–500) m | 12 | 22 |

(50–1000) m | 122 | 78 |

(1000–1500) m | 424 | 198 |

No improvement | 17,762 | 19,510 |

Total | 18,320 | 19,808 |

Models | DID Equation (17) | SDID Equation (19) | STDID Equation (22) | |||
---|---|---|---|---|---|---|

Variables | Coeff. | Sign. | Coeff. | Sign. | Coeff. | Sign. |

∆ Sale situation | ||||||

Forclosure | −0.1468 | *** | −0.1473 | *** | −0.1474 | *** |

Without legal warranty | −0.0792 | *** | −0.0778 | *** | −0.0776 | *** |

Succession | −0.0898 | *** | −0.0824 | *** | −0.0816 | *** |

Transfer | −0.0178 | *** | −0.0189 | *** | −0.0190 | *** |

∆ Walking distance | ||||||

(0–500) m | 0.0573 | 0.0570 | 0.0578 | |||

(500–1000) m | 0.0408 | * | 0.0379 | * | 0.0380 | * |

(1000–1500) m | 0.0080 | 0.0076 | 0.0075 | |||

∆ Driving distance | ||||||

(0–2) min. | 0.1743 | *** | 0.1773 | *** | 0.1774 | *** |

(2–4) min. | 0.1199 | *** | 0.1179 | *** | 0.1177 | *** |

(4–6) min. | 0.0528 | *** | 0.0517 | *** | 0.0519 | *** |

(6–8) min. | −0.0148 | −0.0149 | −0.0145 | |||

(8–10) min. | 0.0427 | *** | 0.0410 | ** | 0.0408 | ** |

(10–12) min. | 0.0378 | ** | 0.0363 | 0.0361 | ||

(12–14) min. | 0.0490 | 0.0514 | 0.0519 | |||

(0–2) min. × distance to CBD | −0.0060 | *** | −0.0060 | *** | −0.0060 | *** |

(2–4) min. × distance to CBD | −0.0034 | *** | −0.0033 | *** | −0.0033 | *** |

(4–6) min. × distance to CBD | −0.0007 | * | −0.0007 | −0.0007 | ||

(6–8) min. × distance to CBD | 0.0020 | *** | 0.0020 | *** | 0.0020 | *** |

(8–10) min. × distance to CBD | −0.0012 | −0.0011 | −0.0011 | |||

(10–12) min. × distance to CBD | −0.0011 | −0.0011 | −0.0010 | |||

(12–14) min. × distance to CBD | −0.0016 | −0.0017 | −0.0017 | |||

Temporal dummies variables | Yes | Yes | Yes | |||

Dynamic spatial effect (ψ) | -- | -- | 0.0215 | *** | ||

Multidirectional spatial effect (ρ) | -- | 0.0562 | *** | 0.0443 | *** | |

R^{2} | 0.7371 | 0.7372 | 0.7373 | |||

LL | 9540.08 | 9611.39 | 9619.98 | |||

AIC | −18,920 | −19,058 | −19,074 | |||

BIC | −18,294 | −18,417 | −18,425 | |||

N_{T} | 18,320 | 18,320 | 18,320 |

Models | DID Equation (17) | SDID Equation (19) | STDID Equation (22) | |||
---|---|---|---|---|---|---|

Variables | Coeff. | Sign. | Coeff. | Sign. | Coeff. | Sign. |

∆ Sale situation | ||||||

Forclosure | −0.1160 | *** | −0.1162 | *** | −0.1163 | *** |

Without legal warranty | −0.0727 | *** | −0.0726 | *** | −0.0727 | *** |

Succession | −0.0945 | *** | −0.0900 | *** | −0.0891 | *** |

Transfer | −0.0177 | *** | −0.0185 | *** | −0.0186 | *** |

∆ Walking distance | ||||||

(0–500) m | 0.0009 | 0.0024 | 0.0027 | |||

(500–1000) m | −0.0182 | −0.0165 | −0.0170 | |||

(1000–1500) m | −0.0035 | −0.0010 | −0.0009 | |||

∆ Driving distance | ||||||

(0–2) min. | 0.1643 | ** | 0.1589 | *** | 0.1611 | *** |

(2–4) min. | 0.0650 | * | 0.0633 | * | 0.0633 | * |

(4–6) min. | 0.0431 | 0.0410 | 0.0390 | |||

(6–8) min. | 0.0585 | 0.0553 | 0.0547 | |||

(8–10) min. | 0.0228 | 0.0236 | 0.0230 | |||

(10–12) min. | −0.0566 | −0.0562 | −0.0564 | |||

(12–14) min. | 0.0018 | 0.0013 | 0.0016 | |||

(0–2) min. × distance to CBD | −0.0026 | * | −0.0025 | ** | −0.0026 | ** |

(2–4) min. × distance to CBD | −0.0012 | −0.0012 | −0.0012 | |||

(4–6) min. × distance to CBD | −0.0007 | −0.0007 | −0.0006 | |||

(6–8) min. × distance to CBD | −0.0009 | −0.0009 | −0.0009 | |||

(8–10) min. × distance to CBD | −0.0004 | −0.0004 | −0.0004 | |||

(10–12) min. × distance to CBD | 0,0016 | 0.0016 | * | 0.0016 | * | |

(12–14) min. × distance to CBD | −0,0001 | −0.0001 | −0.0001 | |||

Temporal dummies variables | Yes | Yes | Yes | |||

Dynamic spatial effect (ψ) | -- | -- | 0.0139 | ** | ||

Multidirectional spatial effect (ρ) | -- | 0.0342 | *** | 0.0273 | *** | |

R^{2} | 0.7875 | 0.7875 | 0.7876 | |||

LL | 13,126.78 | 13,162.83 | 13,167.93 | |||

AIC | −26,072 | −26,140 | −26,148 | |||

BIC | −25,353 | −25,406 | −25,406 | |||

N_{T} | 19,808 | 19,808 | 19,808 |

Distances | [0–2[ min. ‡ | [2–4[ min. | [4–6[ min. | [6–8[ min. | [8–10[ min. |
---|---|---|---|---|---|

St-Lambert | |||||

DID | 0.1445 | 0.1031 | 0.0492 | −0.0046 | 0.0368 |

SDID | 0.1558 | 0.1073 | 0.0509 | −0.0051 | 0.0377 |

STDID | 0.1574 | 0.1081 | 0.0517 | −0.0048 | 0.0379 |

St-Hubert | |||||

DID | 0.0850 | 0.0695 | 0.0419 | 0.0158 | 0.0249 |

SDID | 0.0919 | 0.0720 | 0.0433 | 0.0162 | 0.0263 |

STDID | 0.0929 | 0.0727 | 0.0439 | 0.0166 | 0.0265 |

St-Bruno | |||||

DID | 0.0553 | 0.0528 | 0.0383 | 0.0260 | 0.0190 |

SDID | 0.0599 | 0.0544 | 0.0395 | 0.0269 | 0.0206 |

STDID | 0.0607 | 0.0550 | 0.0400 | 0.0274 | 0.0208 |

St-Basile-le-Grand | |||||

DID | −0.0042 | 0.0192 | 0.0310 | 0.0464 | 0.0072 |

SDID | −0.0041 | 0.0191 | 0.0318 | 0.0482 | 0.0091 |

STDID | −0.0039 | 0.0196 | 0.0323 | 0.0488 | 0.0094 |

McMasterville | |||||

DID | −0.0340 | 0.0024 | 0.0274 | 0.0566 | 0.0013 |

SDID | −0.0361 | 0.0015 | 0.0280 | 0.0589 | 0.0034 |

STDID | −0.0361 | 0.0019 | 0.0284 | 0.0595 | 0.0037 |

Mont-Saint-Hilaire | |||||

DID | −0.0637 | −0.0144 | 0.0237 | 0.0668 | −0.0047 |

SDID | −0.0681 | −0.0162 | 0.0242 | 0.0695 | −0.0023 |

STDID | −0.0684 | −0.0158 | 0.0245 | 0.0702 | −0.0020 |

Distances | [0–2[ min. ‡ | [2–4[ min. | [4–6[ min. | [6–8[ min. | [8–10[ min. |
---|---|---|---|---|---|

Rosemère | |||||

DID | 0.0861 | 0.0283 | 0.0213 | 0.0302 | 0.0107 |

SDID | 0.0864 | 0.0283 | 0.0213 | 0.0304 | 0.0106 |

STDID | 0.0874 | 0.0285 | 0.0211 | 0.0302 | 0.0105 |

Ste-Thérèse | |||||

DID | 0.0731 | 0.0222 | 0.0176 | 0.0255 | 0.0086 |

SDID | 0.0734 | 0.0221 | 0.0178 | 0.0259 | 0.0083 |

STDID | 0.0740 | 0.0223 | 0.0178 | 0.0258 | 0.0082 |

Blainville | |||||

DID | 0.0601 | 0.0161 | 0.0140 | 0.0208 | 0.0066 |

SDID | 0.0604 | 0.0159 | 0.0142 | 0.0215 | 0.0060 |

STDID | 0.0606 | 0.0160 | 0.0146 | 0.0213 | 0.0060 |

St-Jérôme | |||||

DID | 0.0210 | −0.0022 | 0.0031 | 0.0067 | 0.0006 |

SDID | 0.0213 | −0.0027 | 0.0037 | 0.0080 | −0.0009 |

STDID | 0.0203 | −0.0027 | 0.0048 | 0.0079 | −0.0008 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Dubé, J.; Legros, D.; Thériault, M.; Des Rosiers, F.
Measuring and Interpreting Urban Externalities in Real-Estate Data: A Spatio-Temporal Difference-in-Differences (STDID) Estimator. *Buildings* **2017**, *7*, 51.
https://doi.org/10.3390/buildings7020051

**AMA Style**

Dubé J, Legros D, Thériault M, Des Rosiers F.
Measuring and Interpreting Urban Externalities in Real-Estate Data: A Spatio-Temporal Difference-in-Differences (STDID) Estimator. *Buildings*. 2017; 7(2):51.
https://doi.org/10.3390/buildings7020051

**Chicago/Turabian Style**

Dubé, Jean, Diègo Legros, Marius Thériault, and François Des Rosiers.
2017. "Measuring and Interpreting Urban Externalities in Real-Estate Data: A Spatio-Temporal Difference-in-Differences (STDID) Estimator" *Buildings* 7, no. 2: 51.
https://doi.org/10.3390/buildings7020051