What a local coefficient means

Video lesson · 01:50 Read a local coefficient surface correctly

Vera begins with the difference between a global slope and a local coefficient, showing that a GWR surface reports conditional local association rather than an automatically valid local causal effect. The lesson establishes the reading frame before the note addresses bandwidth, edge samples, and multiple testing.

The lesson establishes the interpretive boundary of local coefficients; stability, collinearity, and inference remain in the full note.

A global regression gives you one slope: “an extra minute of walk to the station is worth −x% of price,” everywhere, identically. Geographically weighted regression (GWR) drops the “everywhere.” It fits a separate, distance-weighted regression at every location and returns a coefficient surface — one $\beta$ per place:

$$y_i = \beta_0(u_i, v_i) + \sum_k \beta_k(u_i, v_i)x_{ki} + \varepsilon_i$$

where (u_i, v_i) are coordinates. Observations near location i get more weight through a kernel; the bandwidth — how fast that weight decays, usually a number of nearest neighbours chosen to minimise AICc or a cross-validation score — sets how local the fit is. A small bandwidth gives a noisy, very local surface; a large one smooths toward the global model.

So what is a single local coefficient? $\beta_k(u_i, v_i)$ is the estimated marginal effect of $x_k$ in the neighbourhood of location $i$, holding the other variables fixed — the implicit local valuation of that attribute for a typical case near $i$. Reading the surface:

• A strong, consistent value across a cluster of locations → that attribute is strongly valued (or penalised) there.
• Near-zero across a region → not capitalised there (maybe abundant supply, maybe buyers don’t prioritise it).
• A sign reversal across space → theoretically interesting, and a flag to investigate: a genuine local externality, or an artefact of the data.

The gain is real — letting coefficients vary usually lifts fit substantially over a global model, which is itself evidence that the relationship is spatially heterogeneous rather than constant. But three traps come with it:

1. Description, not cause. Local coefficients summarise local data patterns. Spatial sorting and omitted, spatially-varying factors leak straight into the surface. It maps where a relationship differs; it does not explain why, and it is not a causal estimate.
2. Local collinearity. Nearby observations tend to resemble one another on all variables at once, so local estimates can be far less stable than global ones — collinearity that was mild globally can be severe locally.
3. Coefficients-as-data. It is tempting to carry the surface into a second stage and regress the local coefficients on something else. But estimated coefficients carry first-stage uncertainty; treat them as fixed data and your second-stage standard errors are biased downward. If you must, propagate the uncertainty (e.g. weight by inverse variance) and don’t trust naïve SEs.

A classical GWR also uses one bandwidth for every variable. If different processes operate at different spatial scales, multiscale GWR (MGWR) lets each covariate carry its own bandwidth — a natural next step when a single bandwidth is clearly too blunt.

The one-line version: a local coefficient is an honest answer to “how does this relationship look here?” — and a dishonest answer to any question with the word “cause” in it.