Measuring the Purple Line’s Effect on Residential Rents

Spatial Econometrics and Causal Inference in Urban Policy Analysis

Eli Knaap

eknaap@sdsu.edu

San Diego State University

February 13, 2024

TL;DR

PL offers a unique opportunity to explore spatial model specifications in a causal inference context
‘Causal’ hedonic models need spatial econometrics
The spatial weights matrix (\(W\)/Graph) matters a lot, especially for microdata
Spatial models with Fixed Effect (within) designs have a different counterfactual than DID (between) designs
- within estimator is more appropriate than the between estimator for transit studies (but more difficult data)
Contrary to prior research the Purple Line probably caused rents to decrease slightly during the announcement phase, and the spillover effect likely extends beyond the station areas

The Purple Line

A 16-mile light rail line that will extend from Bethesda in Montgomery County to New Carrollton in Prince George’s County. It will provide a direct connection to the Metrorail Red, Green and Orange Lines; at Bethesda, Silver Spring, College Park, and New Carrollton. The Purple Line will also connect to MARC, Amtrak, and local bus services.

The Montgomery County Rental Survey

The annual Rental Facility Occupancy Survey included single unit properties (single family homes, townhomes and condos). Inquiries are addressed by the Department of Housing and Community Affairs (DHCA), Licensing and Registration Office (L&R). The survey is required based upon Montgomery County Code 29-51, and says all landlords of rental housing units must participate in the survey, tracking vacancies; turnover rates; average rents; and amenities.

MontgometryCountyMD.org

Rent by # Bedrooms

Causal Inference in Hedonic Models

Does the Announcement of New Transit Cause an Increase in Land Rents/Prices?

Does space matter?
How should a causal hedonic model include spatial effects?
- DID vs FE designs
- SAR, SEM, SDM, SDEM
How should co-located units consider spatial weights?
- How much does it matter?

Does the Announcement of New Transit Cause an Increase in Land Rents/Prices?

Here, the treatment is defined by space (inside/outside the station area). Existing work uses classic Difference in Differences or spatial extensions to Difference-in-Differences (SAR-DID, SEM-DID, SDEM-DID).

DID is often the only choice for estimating land price responses to policy, because there is not enough within-unit variation in housing transactions.

Peng, Knaap, and Finio (2023)
Bardaka, Delgado, and Florax (2018)
McMillen (2018)

Dubé, Legros, and Devaux (2018)
Dubé et al. (2014)
Diao, Leonard, and Sing (2017)

When a treatment is defined by space, does DID generate a valid counterfactual?

How Should A Causal Hedonic Model Include Spatial Effects? (1)

A Theory of Spillover in Hedonic Models

“The value of a house at any location is dependent on its counterparts at nearby locations in addition to its structural and neighborhood attributes. The hypothesized spatial dependence among residential structures is determined by \(W\) which is specified in an a priori fashion. The coefficient \(\rho\) measures the absolute price impact of nearby houses on the price of a particular house.”

— Can (1992)

How Should A Causal Hedonic Model Include Spatial Effects? (2)

A Theory of Spillover in Hedonic Models

“As argued in Can (1990), this conceptualization corresponds more closely with the actual workings of the real estate institution in urban housing markets. A realtor will appraise a house given the price history of houses in the immediate vicinity in addition to other substantive characteristics. At the same time, home owners will initiate or forego certain improvements based on the anticipated return on their investment considering housing prices in the immediate area.”

— Can (1992)

How Should A Causal Hedonic Model Include Spatial Effects? (3)

A Problem for Causal Inference

With spatially-interactive observations, the cardinal rule of regression–the Stable Unit Treatment Value Assumption (SUTVA) (i.e. the independence of observations)–is violated (Delgado and Florax 2015).

Classic identification techniques will fail unless spatial effects are included, but different specifications imply different counterfactuals.

…And require additional subjective decisions among competing model specifications and \(W\) specifications

How should co-located units consider spatial weights?

Co-Located Observations

Buildings by Unit Count

How should co-located units consider spatial weights?

Distance-Based vs K-Nearest Neighbor Weights

Spatial Econometric Models

LeSage and Pace (2014)

“Global spillovers arise when changes in a characteristic of one region impact outcomes in more than just immediately neighboring regions. Global impacts fall on neighbors to the neighbors, neighbors to the neighbors to the neighbors, and so on…

Local spillovers represent a situation where the impacts fall only on nearby or immediate neighbors, dying out before they impact regions that are neighbors to the neighbors.

Another distinction between local and global spillovers is that feedback effects arise in the case of global spillovers, but not in the case of local spillovers.”

Haleck, Vega & Elhorst Typology

LeSage (2014, 22)

“Practitioners cannot simply estimate an SDM and SDEM model and draw meaningful conclusions about the correct model specification from point estimates of \(\rho\) or \(\lambda\)… the SDM specification implies global spatial spillovers, while the SDEM is consistent with local spillovers. These have very different policy implications in applied work”

Quasi-Experiments in Space (1)

Difference in Differences: the between estimator

The 2x2 DD design has a treatment group \(k\) and untreated group \(U\). There is a pre-period for the treatment group, \(pre(k)\); a post-period for the treatment group, \(post(k)\); a pre-treatment period for the untreated group, \(pre(U)\); and a post-period for the untreated group, \(post(U)\) So: \[ \widehat{\delta}^{2\times 2}_{kU} = \bigg(E\big[Y_k | Post\big] - E\big[Y_k | Pre\big]\bigg)- \bigg(E\big[Y_U | Post\big] - E\big[Y_U | Pre\big]\bigg) \]

— Cunningham (2018)

The point being that DID relies on between group variation, which depends on: the parallel trends assumption, the equivalence between groups, or random assignment

DID: Identifying Assumption

Parallel trends for treatments defined by space:

The units located here, are on average, no different from the units located there
Face validity? Are these groups equivalent? What about spatial heterogeneity?
Are places near the metro the same as places far from the metro?

A Pharmacological Metaphor:

A drug trial where treatment and control group are perfectly matched on covariates, but the groups are stratified by age.

Are 50-year olds a valid reference group for 20-year olds?

Unit-Level Fixed Effects: the within estimator

The repeat-sales model is the gold standard in hedonic modeling because a house cannot change locations. Any change in price is attributable to something that happened to that house

Unit-level fixed effects do not solve the issue of spatial autocorrelation
But they do avoid the issue of cross-regime comparisons
But housing is a durable good, transacted infrequently.
- very few hedonic models can use unit-level fixed effects to study location shocks

Spatial Panel Econometrics for TWFE

Consider a treatment \(D\), that takes the value of unity if a unit is located within a half-mile of a Purple Line station and zero otherwise. Data are a fully-balanced panel with rents measured in 2015 and 2018

\[ D = \begin{cases} 1 & \text{if distance} \between i, \text{PL} \leq 800m \\ 0 & \text{otherwise.} \end{cases} \]

where \(\text{PL}\) is the nearest Purple Line Station

A Spatial Durbin Model with TWFE

\[ Y_{it} = \rho W_{ij}y_{jt} + \delta D + \theta WD + \tau_t + \mu_i + \epsilon_{it} \]

\(Y_{it}\) is the rent for unit \(i\) at time \(t\) ,
\(W_{ij}y_{jt}\) is the spatial lag of price,
\(D\) is a binary treatment indicator
\(WD\) is a spatial lag of the treatment variable capturing local spillovers

\(\tau_t\) is a fixed effect for time
\(\mu_i\) is a unit-level fixed effect
\(\epsilon_{it}\) is a random error term.

Spatial 2x2 Designs

SAR

\[ Y_{it} = \rho W_{ij}y_{jt} +\delta D + \tau_t + \mu_i + \epsilon_{it} \]

SDM

\[ Y_{it} = \rho W_{ij}y_{jt} + \delta D + \theta WD + \tau_t + \mu_i + \epsilon_{it} \]

SEM

\[ \begin{gathered} Y_{it} = \delta D + \tau_t + \mu_i + u \\ u = \lambda Wu + \epsilon_{it} \end{gathered} \]

SDEM

\[ \begin{gathered} Y_{it} = \delta D + \theta WD + \tau_t + \mu_i + u \\ u = \lambda Wu + \epsilon_{it} \end{gathered} \]

SAR and SDM models include global spillovers which require computation of direct and indirect marginal effects. Importantly, indirect effects could accrue outside the station area (and feedback within the station area)

Results

Contrary to the findings of Peng, Knaap, and Finio (2023), the Purple Line did not cause rents to increase during the announcement phase.
Aspatial models show overwhelming evidence of spatial structure in the residuals
Results for spatial models depend on \(W\), however:
- PL effect is insignificant or negative
- significant spatial effects throughout
- spillover or diffusion?

Two BR

Model	W	Direct	Indirect	\(\theta\)
SEM	KNN	Insignificant	NA	NA
SDEM	KNN	Insignificant	NA	Insignificant
SAR	KNN	Insignificant		NA
SDM	KNN	Insignificant		Insignificant

SEM	Distance	Negative (.05)	NA	NA
SDEM	Distance	Negative (.001)	NA	Negative (.001)
SAR	Distance	Negative (.001)	Negative	NA
SDM	Distance	Negative (.001)	Negative	Negative (.001)

Three BR

Model	W	Direct	Indirect	\(\theta\)
SEM	KNN	Insignificant	NA	NA
SDEM	KNN	Insignificant	NA	Insignificant
SAR	KNN	Insignificant		NA
SDM	KNN	Insignificant	Negative	Insignificant

SEM	Distance	Insignificant	NA	NA
SDEM	Distance	Insignificant	NA	Negative
SAR	Distance	Insignificant		NA
SDM	Distance	Insignificant		Insignificant

The PL Effect

” First, why would the rents of multifamily units within a half mile from the planned stations increase more than the control group even before service begins? And second, why would this rise occur only for two-, three-, and four-bedroom units?… Before service begins because there would be no transportation cost savings or gains in household utility before service begins. One could even argue that disruption brought by construction would cause utility and willingness to pay to fall in the pre-service period”

— Peng, Knaap, and Finio (2023) p.17

Like Dubé, Legros, and Devaux (2018), I find that Peng et al’s suspicions are probably accurate

Thanks

References

Bardaka, Eleni, Michael S. Delgado, and Raymond J. G. M. Florax. 2018. “Causal Identification of Transit-Induced Gentrification and Spatial Spillover Effects: The Case of the Denver Light Rail.” Journal of Transport Geography 71 (June): 15–31. https://doi.org/10.1016/j.jtrangeo.2018.06.025.

Can, Ayse. 1992. “Specification and Estimation of Hedonic Housing Price Models.” Regional Science and Urban Economics 22 (3): 453–74. https://doi.org/10.1016/0166-0462(92)90039-4.

Cunningham, Scott. 2018. Causal Inference: The Mixtape (V1.7). http://www.apache.org/licenses/.

Delgado, Michael S, and Raymond J G M Florax. 2015. “Difference-In-Differences Techniques for Spatial Data: Local Autocorrelation and Spatial Interaction.” Economics Letters 137: 123–26. https://doi.org/10.1016/j.econlet.2015.10.035.

Diao, Mi, Delon Leonard, and Tien Foo Sing. 2017. “Spatial-Difference-in-Differences Models for Impact of New Mass Rapid Transit Line on Private Housing Values.” Regional Science and Urban Economics 67 (May): 64–77. https://doi.org/10.1016/j.regsciurbeco.2017.08.006.

Dubé, Jean, Diègo Legros, and Nicolas Devaux. 2018. “From Bus to Tramway: Is There an Economic Impact of Substituting a Rapid Mass Transit System? An Empirical Investigation Accounting for Anticipation Effect.” Transportation Research Part A: Policy and Practice 110: 73–87. https://doi.org/10.1016/j.tra.2018.02.007.

Dubé, Jean, Diègo Legros, Marius Thériault, and François Des Rosiers. 2014. “A Spatial Difference-in-Differences Estimator to Evaluate the Effect of Change in Public Mass Transit Systems on House Prices.” Transportation Research Part B: Methodological 64: 24–40. https://doi.org/10.1016/j.trb.2014.02.007.

LeSage, James P. 2014. “What Regional Scientists Need to Know About Spatial Econometrics.” SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2420725.

LeSage, James P., and R. Kelley Pace. 2014. “Interpreting Spatial Econometric Models.” In Handbook of Regional Science, 1535–52. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-23430-9_91.

McMillen, Daniel. 2018. “Spatial Variation in House Prices and the Opening of Chicago’s Orange Line.” Lincoln Institute for Land Policy. https://www.lincolninst.edu/sites/default/files/sources/events/mcmillen_spatial_variation_in_house_prices_and_the_opening_of_chicagos_orange_line.pdf.

Peng, Qiong, Gerrit-Jan Knaap, and Nicholas Finio. 2023. “Do Multifamily Unit Rents Increase in Response to Light Rail in the Pre-service Period?” International Regional Science Review, 016001762311625. https://doi.org/10.1177/01600176231162563.