What Changes When Causality Lives in Space?

John Snow: A classic spatial causal argument

Snow’s argument, in five questions

The same five questions you will ask in the practical, applied to 1854.

Practical

Remember the assumptions of causal inference

Spatial causal inference

Space puts the most pressure on exchangeability and no interference.

Three characteristics of geospatial analysis

Spatial spillovers stress no interference

A “control” cell next door is then already partly treated, the comparison is contaminated.

Spatial confounding stresses exchangeability

Scale impacting consistency / positivity

▶ Practice now: Part 0 & Step 1

First meet the data, then run the first regression. The grid joins three sources on a 1 km raster: RIVM modelled NH₃ concentrations (2018–2024), firm locations & sectors from Bureau van Dijk (Orbis), and CBS neighbourhood covariates (population density, urbanity).

Spatial models diagnose dependence: they don’t identify

In Lecture 1 we fought confounding by adding controls to the regression. Here we fight spatial dependence the same way, by adding spatial terms to the model. The general one is the Spatial Durbin (SDM):

\[y = \color{#0b789d}{\rho\, W y} + \color{#c44e52}{\beta X} + \color{#e08214}{\theta\,W X} + \varepsilon\]

• ρWy, my outcome tracks my neighbours’ outcome (outcome spillover)
• βX, the ordinary regression on my own covariates
• θWX, my neighbours’ covariates (their farms) reach me (covariate spillover)

The two famous models just keep part of this:

\[\text{SAR: } y = \color{#0b789d}{\rho W y} + \color{#c44e52}{\beta X} + \varepsilon \qquad\qquad \text{SEM: } y = \color{#c44e52}{\beta X} + u,\;\; u = \lambda W u + \varepsilon\]

SAR when the outcome itself spreads (prices, congestion, a gas), our choice here (NH₃ drifts). SEM when the leftover clustering comes from unmeasured factors that happen to cluster in space (not from the outcome spreading): it corrects the standard errors but leaves the coefficients’ meaning unchanged. SDM adds neighbours’ covariates (\(\theta WX\)) directly. We use SAR here.

Do spatial models clean spatial correlation?

A key problem: the weight matrix W

W encodes who is a neighbour of whom, an assumption you choose, not data.

Useful models of dependence, not designs for identification — unmeasured confounders are likely present.

Those models in python

# W = who-neighbours-whom (inverse-distance, 3 km)
w = DistanceBand(coords, threshold=3000,
                 binary=False, alpha=-3); w.transform = "R"
m_sar = ML_Lag(y, X, w=w)              # ρ W y
m_sem = ML_Error(y, X, w=w)            # λ in errors
m_sdm = ML_Lag(y, X, w=w, slx_lags=1)  # + W X θ

slx_lags controls how many orders of neighbours’ X enter:

• slx_lags=0 (SAR): \(y = \rho Wy + \beta X + \varepsilon\)
• slx_lags=1 (SDM): \(y = \rho Wy + \beta X + \theta_1\,WX + \varepsilon\)
• slx_lags=2: \(y = \rho Wy + \beta X + \theta_1\,WX + \theta_2\,W^2X + \varepsilon\)  (adds neighbours-of-neighbours)

Another key problem: interpreting coefficients

In SAR/SDM (\(\text{SAR: } y = \color{#0b789d}{\rho W y} + \color{#c44e52}{\beta X} + \varepsilon\)) the outcome sits on both sides: I affect my neighbours, and they affect me back. So a change in \(X\) doesn’t move just one cell, it ripples out and partly returns.

• the raw coefficient is not the effect
• spreg reports the impact decomposition:

\[\underbrace{\text{direct}}_{\text{own cell}} + \underbrace{\text{indirect}}_{\text{spillover + feedback}} = \text{total}\]

• the feedback sums to the spatial multiplier \(\dfrac{1}{1-\rho}\), a larger \(\rho\) inflates the total.

▶ Practice now: Step 2

Refit the same regression with spatial weights.

A clean residual map is spatially honest SE, not causal identification.

Geography hands us ways to rescue exchangeability

Space made exchangeability harder via confounding, no-interference harder via spillovers, consistency / positivity harder via scale.

But geography provides sources of exogenous variation:

Each of the next slides takes one design: which assumption it rescues, what comparison it supplies, what new bet we make.

Time-based design: rescues exchangeability through exogenous timing

E-ZPass (Currie & Walker, 2011). Toll plazas were automated on a fixed national schedule. Local congestion fell. Birth outcomes improved near the plazas, not far away.

Boundary design: rescues exchangeability through local continuity

Black (1999). Two houses 50 m apart, same street, opposite sides of a school attendance boundary. Same bakery, bus stop, neighbors. Different school.

Instrument design: rescues exchangeability through an exogenous shifter

Deryugina, Heutel, Miller, Molitor & Reif (2019, AER). They use daily changes in wind direction as an instrument for fine-particulate (PM₂.₅) exposure, and estimate its effect on mortality and health-care use among older Americans.

▶ Practice now: Step 3, Difference-in-Differences

The timing design, on the NH₃ grid, the comparison this lecture builds toward.

smf.ols("nh3 ~ treated * post", did).fit()

But DiD breaks under interference

DiD trusts that the controls are untreated. In space they often aren’t, NH₃ drifts across cell boundaries.

Cell B looks like a control on the map but is partly treated by Cell A’s farm. The no-interference assumption fails in space. We look at three ways to respond.

Response 1: model the spillover (spatial lag / SAR)

Keep the timing comparison by taking first-differences (\(\Delta Y_i = Y_{i,2024} - Y_{i,2020}\)), but correct for spillovers with SAR:

\[\Delta Y_i = \alpha + \color{#6a51a3}{\rho\,(W\Delta Y)_i} + \color{#0b789d}{\tau\, T_i} + \varepsilon_i\]

ML_Lag(dY, X[["treated"]], w=w_fd)

The cost of Response 1: it’s all about W

The spatial-lag (SAR) answer is only as good as the weight matrix you assumed.

• W is a choice, not data. A different neighbourhood or decay → a different \(\rho\) → a different total. No test tells you W is “right.”
• because \(\rho\) enters as \(1/(1-\rho)\), a high \(\rho\) amplifies the own effect: here ρ = 0.67 turns the τ ≈ 0.26 own gain into a ~0.79 total.
• the raw \(\tau\) is not the effect, you must read the direct/indirect/total decomposition.
• and it’s not only W. SAR pins all the spatial structure of ΔNH₃ on outcome feedback (ρWΔY). If some is common spatial shocks (weather, regional trends hitting neighbours alike), ρ soaks those up too and inflates the total: the same identification gap as SAR-vs-SEM.

Modelling spillovers trades “my controls are clean” for “I picked the right W.”

Response 2: redesign the comparison with distance rings

Instead of modelling the leak, change who counts as a control: keep treated cells, drop controls near a treated cell (partly exposed), and compare to far controls.

Two answers

Rings clean the control side; with clustered treatment the treated cells still receive drift, so 0.5 is an upper bound on the pure own-cell effect, cf. SAR direct 0.30.

A third route: make the spillover part of the exposure

Interference depends on the question you ask:

• Do this cell’s own farms cause NH₃? → neighbours’ drift contaminates the controls = interference (Responses 1 & 2)
• Does the NH₃ you actually breathe harm you? → drift is simply part of your exposure = no interference.

▶ Practice now: Step 4

Make the DiD spillover-aware, and compare the two responses.

ML_Lag(dY, X, w=w_fd)  # first-diff SAR (spatial lag)

Recap: two lectures, three ideas to keep

Be honest about your design: name treatment → estimand → comparison → assumption → failure