How connectivity choices change substantive conclusions
All regression examples in this chapter use simulated data. Illustrative distances and trade figures are drawn from real-world approximations for concreteness, but no empirical claims depend on them. The goal is to show how matrix choices affect inference, not to make claims about real policy diffusion.
Interdependence is a core feature of social phenomena. States share borders, trade with each other, join international organizations, wage war, and exchange migrants. To model these interdependencies, I need to specify a connectivity or spatial weights matrix W that encodes who influences whom and by how much.
Despite its centrality, many researchers treat W as a technical preprocessing step, something to get through quickly before running the “real” model. This is a mistake. W is itself a core substantive assumption. It defines the structure of interdependence in the model. As Neumayer and Plümper (2016) put it, the spatial weights matrix embodies a theory about how influence flows between units, and getting that theory wrong will bias every estimate that depends on it.
The consequences are not hypothetical. Whitten, Williams, and Wimpy (2021) surveyed 94 published articles using spatial autoregressive models in political science and found that only 42% attempted to interpret spatial effects beyond the raw coefficients. The survey found no examples of studies computing the proper direct and indirect effect decompositions that depend on W. Drolc, Gandrud, and Williams (2021) showed that minor specification errors, including the wrong W, can push false discovery rates for policy diffusion to alarmingly high levels.
In this chapter, I walk through the five substantive decisions that define W, explain why each one matters for substantive conclusions, and demonstrate with simulated data how sensitive spatial estimates can be to these choices. The goal is not to provide a definitive recipe for building W, but to make the case that these decisions deserve the same careful theoretical justification as any other part of a model.
At its core, W is a table of exposure. Each row represents a receiving unit, each column a sending unit, and each cell records how much influence flows from sender to receiver.
\[ \mathbf{W} = [w_{ij}]_{N \times N}, \quad w_{i=j}=0 \tag{5.1}\]
The diagonal is zero by convention because units do not influence themselves through the spatial channel. Everything else (who is connected, how strongly, and at what scale) is a modeling choice. At a coarse level, these choices bundle together three distinct decisions, namely which pairs are connected at all (\(w_{ij} = 0\) or not), how strongly (binary or weighted), and how scaled (raw or row-standardized). In practice, as the rest of this chapter shows, this trichotomy unfolds into five concrete design decisions: The connectivity concept, scope, weight function, normalization, and directionality.
In the formal SAR model, the spatial autoregressive parameter is conventionally written \(\rho\). In simulation code and output tables throughout this chapter, I label the estimated value \(\hat{\rho}\) as “spatial feedback” for readability. Some software packages and textbooks use \(\lambda\) for the same parameter (or reserve \(\lambda\) for the spatial error model). I use \(\rho\) consistently in equations and refer to the same quantity in code output.
A concrete example makes this tangible. Consider three countries linked by geographical contiguity.
W_border <- matrix(
c(
0, 1, 1, # USA receives from: —, Canada, Mexico
1, 0, 0, # Canada receives from: USA, —, —
1, 0, 0 # Mexico receives from: USA, —, —
),
nrow = 3,
byrow = TRUE,
dimnames = list(
c("USA", "Canada", "Mexico"),
c("USA", "Canada", "Mexico")
)
)
W_border USA Canada Mexico
USA 0 1 1
Canada 1 0 0
Mexico 1 0 0The convention matters here. Rows are receivers, columns are senders. Row 1 (USA) has ones in the Canada and Mexico columns, meaning the USA receives influence from both. Row 2 (Canada) has a one only in the USA column, meaning Canada receives influence from the USA alone. This becomes concrete when I compute the spatial lag \(Wy\), which is the quantity that enters a spatial regression.
Suppose each country has a tax rate stored in a vector \(y\). The USA has a tax rate of 21, Canada has a tax rate of 15, and Mexico has a tax rate of 30. The matrix product \(Wy\) gives each country the weighted sum of its neighbors’ tax rates.
\[ Wy = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} 21 \\ 15 \\ 30 \end{bmatrix} = \begin{bmatrix} 0 \cdot 21 + 1 \cdot 15 + 1 \cdot 30 \\ 1 \cdot 21 + 0 \cdot 15 + 0 \cdot 30 \\ 1 \cdot 21 + 0 \cdot 15 + 0 \cdot 30 \end{bmatrix} = \begin{bmatrix} 45 \\ 21 \\ 21 \end{bmatrix} \]
Each row of the result is the spatial lag for one country. The USA’s lag is \(15 + 30 = 45\) (it receives from Canada and Mexico), Canada’s is \(21\) (it receives only from the USA), and Mexico’s is \(21\) (same). The column a country sits in determines where it sends influence, the row it sits in determines what it receives. In a spatial regression, this lag replaces the vague idea of “neighboring outcomes” with a precise, numerically defined quantity.
This simple matrix already encodes a substantive claim, namely that the USA is exposed to influence from two neighbors while Canada and Mexico each face influence from one. If I study tax policy diffusion among these three countries, W says that the USA’s policy is shaped by a weighted combination of Canadian and Mexican policies, but Canada is only affected by the USA. Whether that reflects reality depends entirely on whether contiguity is the right connectivity concept for the mechanism I have in mind.
The rest of this chapter unpacks the five design decisions embedded in every W, explains the substantive implications of each choice, and illustrates how they can alter conclusions.
The most consequential choice in building W is deciding what relationship it represents. This is not a technical question but a theoretical one. As Neumayer and Plümper (2016) argue, “Spatial dependence is clearly not caused by geography, proximity and contiguity itself. Rather, it is caused by connectivity, i.e. contact in its various forms, transactions, interactions and relations.”
Yet geographic proximity remains the default in most applied work. Researchers routinely define neighbors as units that share a border (contiguity) or fall within some distance threshold, regardless of whether their theory implies that geography is the relevant channel. Di Salvatore and Ruggeri (2021) make the important distinction between spatial clustering (which can arise from common exposure or similar environments) and spatial interdependence (which requires a causal transmission mechanism). Seeing neighboring countries adopt similar policies does not mean they influenced each other. They may simply be responding to similar economic pressures or other common shocks, producing spatial clustering through common exposure rather than causal interdependence. A useful rule of thumb is that similar outcomes among neighbors are not by themselves evidence of diffusion. Instead, they can often be explained by shared covariates (clustering) even when no causal transmission exists. The burden of proof falls on the researcher to demonstrate that a genuine transmission mechanism is at work.
To illustrate, consider studying how corporate tax rates spread across countries. Several connectivity concepts are plausible, and each implies a different theory of tax competition.
Each of these produces a different W with a different set of non-zero entries and a different pattern of influence. Taking trade as an example, the USA’s most important “neighbors” for tax competition would include China, the EU, and Mexico. Under contiguity, however, the USA’s neighbors are exclusively Canada and Mexico.
The choice matters empirically, not just conceptually. Tan, Kesina, and Elhorst (2025) showed that different regressors in the same model can have different spatial ranges of influence. Military expenditure, for instance, might diffuse through security alliances (tightly clustered) while economic conditions diffuse through trade networks (more dispersed). Using a single geographic W for both forces a common spatial structure that fits neither mechanism well.
Once I have chosen a connectivity concept, I must decide how broadly to define neighborhoods. Should each unit be influenced only by its first nearest contact, or by contacts two, three, or more steps away? This neighborhood scope decision directly controls how much of the network participates in the diffusion process.
The common options are the following.
The substantive implication is direct. Broader neighborhoods assume more far-reaching influence. A k = 3 nearest-neighbor matrix says each European country is meaningfully shaped by only its three closest peers. A k = 10 matrix says influence reaches much further. Neither is inherently correct, and the right scope depends on how far the theorized mechanism actually operates.
This is not an idle concern. Consider studying how European countries set their corporate tax rates. A narrow contiguity-based W would say the Netherlands is influenced primarily by its direct border neighbors (Belgium, Germany). A broader k = 6 definition would also include Luxembourg, France, the UK, and Denmark, countries whose tax incentive packages the Netherlands actively competes against when attracting multinational headquarters. The choice between these two definitions is a choice between modeling localized border-crossing competition and modeling a wider race-to-the-bottom dynamic where countries compete with a larger set of jurisdictions for mobile corporate investment.
Once I know who is connected and how far connections reach, I must decide whether all connections are equally strong or whether some are more important than others. This is the weight function choice.
The simplest approach is binary weighting, where every connected pair gets a weight of 1 and every unconnected pair gets 0. The USA/Canada/Mexico matrix from Section 5.2 is exactly this kind of matrix where each connected pair gets a 1, and that is all the information W encodes. Binary contiguity matrices are by far the most common in the social sciences. They are easy to construct and easy to interpret, but they make a strong assumption, namely that every neighbor matters equally, regardless of distance, size, or intensity of interaction.
Alternative weight functions reflect the intuition that “closer means more influential”.
To make the difference between these approaches concrete, consider how each would weight the USA’s relationships with Canada and Mexico in a study of fiscal policy competition.
| Weight function | \(w_{\text{USA,Canada}}\) | \(w_{\text{USA,Mexico}}\) | Interpretation |
|---|---|---|---|
| Binary contiguity | 1 | 1 | Both neighbors matter equally |
| Inverse distance (\(\alpha=1\)) | \(1/730 \approx 0.00137\) | \(1/3040 \approx 0.00033\) | Canada is closer (Ottawa-Washington < Mexico City-Washington), so it gets more weight |
| Trade volume (USD billions) | ~760 | ~840 | Mexico can receive more weight, reflecting stronger trade integration |
The table reveals a key insight. Different weight functions do not just scale the same relationships up or down. They can reorder which neighbors matter most. Under inverse distance, Canada outweighs Mexico because Ottawa is much closer to Washington than Mexico City is. Under trade weights, Mexico can outweigh Canada because US-Mexico trade volume can be larger in aggregate. These are different substantive claims about the channel through which influence flows.
More broadly, the choice of weight function interacts with the choice of measurement scale. The raw magnitude of a connectivity variable (trade volume in USD, migration in persons, distance in kilometers) may not map linearly onto the intensity of influence. A country that trades $20 billion with a neighbor is not necessarily influenced exactly twice as much as one trading $10 billion. Depending on the mechanism, a log transformation, a rank ordering, or a threshold might better capture the relevant variation.
Normalization is perhaps the most consequential “technical” decision because it is the one researchers most often make on autopilot. Row-standardization, which divides each row’s entries by the row sum so that all rows sum to 1, is the default in virtually every spatial econometrics software package. Formally, each entry \(w_{ij}\) of the raw matrix is replaced by
\[ w_{ij}^* = \frac{w_{ij}}{\sum_{k=1}^{N} w_{ik}} \]
so that \(\sum_j w_{ij}^* = 1\) for every row \(i\). But as Neumayer and Plümper (2016) argue, this imposes a strong and often unjustified assumption, namely that every unit faces the same total exposure to spatial influence, regardless of how many neighbors it has.
To understand what row-standardization does concretely, return to the USA/Canada/Mexico binary contiguity matrix from Section 5.2. The USA has 2 neighbors, while Canada and Mexico each have only 1. Applying the formula row by row gives
\[ W^* = \begin{bmatrix} 0 & 1/2 & 1/2 \\ 1/1 & 0 & 0 \\ 1/1 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0.5 & 0.5 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix} \]
In the raw matrix, the USA’s spatial lag is the sum of Canada’s and Mexico’s tax rates (\(15 + 30 = 45\)). After row-standardization it becomes their average (\((15 + 30)/2 = 22.5\)). Canada and Mexico, having only one neighbor each, are unaffected because dividing by 1 changes nothing. Row-standardization has thus equalized total exposure (every row now sums to 1) by shrinking the influence of each individual neighbor for well-connected units.
Is this reasonable? It depends on the mechanism.
Why does this matter statistically? Tiefelsdorf, Griffith, and Boots (1999) identified the mechanism. Row-standardization gives higher leverage to peripheral units with few connections, inflating their importance in the overall estimate. In the example above, Canada’s spatial lag is determined entirely by one observation (the USA). The USA’s lag, by contrast, averages over two countries, making it more stable. Scale this up to a real application with 30 European countries where Germany has 9 neighbors and Portugal has 1, and the leverage imbalance becomes substantial. This topology-induced heterogeneity means that normalization does not just rescale weights. It changes which units drive the results.
The core normalization decision is whether to row-standardize. All other common normalization schemes (global standardization, spectral normalization, min-max normalization) divide every entry by the same constant. Because they are scalar multiples of the raw matrix, the spatial autoregressive coefficient can in principle absorb the scaling factor, making the spatial multiplier \((I - \hat{\rho} W)^{-1}\) equivalent up to reparameterization. In practice, however, the choice among these schemes is not entirely innocuous. Different scalings change the admissible parameter bounds for \(\rho\) (which depend on the eigenvalues of W), can affect optimization and log-determinant numerics, and may matter more in complex multi-parameter models. Still, the substantive distinction is small compared to the choice between row-standardization and everything else.
Row-standardization is fundamentally different. It divides each row by its own sum, so units with many connections get their individual weights shrunk more than units with few connections. This changes the relative influence structure, not just the scale. The spatial lag becomes an average rather than a sum, and the estimated effects genuinely change.
The simulation below makes this concrete. I generate an asymmetric weighted matrix for 50 units where each unit has a random number of neighbors (1 to 12) with random connection strengths, mimicking a trade-flow network where both the number of partners and the intensity of exchange vary. I then compare the raw matrix against its row-standardized version.
# Build an asymmetric matrix with heterogeneous connectivity.
# Each unit gets a random number of neighbors (1 to 12) with
# random positive weights, mimicking a trade-flow network
# where both the number of partners and trade intensity vary.
set.seed(42)
W_raw <- matrix(0, n, n)
for (i in 1:n) {
n_neighbors <- sample(1:12, 1)
neighbors <- sample(setdiff(1:n, i), n_neighbors)
W_raw[i, neighbors] <- runif(n_neighbors, min = 0.2, max = 5)
}
diag(W_raw) <- 0
# Row-standardized: each row sums to 1
W_row_std <- row_standardize(W_raw)
norm_tbl <- compare_model_sensitivity(
formula = policy_adoption ~ policy_pressure + fiscal_capacity,
data = dat,
W_list = list(
raw = W_raw,
row_standardized = W_row_std
)
)
norm_display <- norm_tbl[, c("W", "spatial_feedback",
"indirect_policy_pressure",
"total_policy_pressure")]
names(norm_display) <- c("Normalization", "Spatial feedback (\u03c1)",
"Indirect effect", "Total effect")
num_cols <- vapply(norm_display, is.numeric, logical(1))
norm_display[num_cols] <- lapply(norm_display[num_cols], round, 3)
knitr::kable(norm_display, row.names = FALSE)| Normalization | Spatial feedback (ρ) | Indirect effect | Total effect |
|---|---|---|---|
| raw | -0.031 | -0.279 | 0.507 |
| row_standardized | -0.271 | -0.168 | 0.617 |
Note that the raw \(\hat{\rho}\) values are not directly comparable across the two rows because the matrices have different scales. The substantively meaningful comparison is the indirect and total effects, which are computed from the full spatial multiplier. The table shows that the same connectivity structure produces different effect estimates depending on whether total exposure is preserved or equalized. In the raw matrix, row sums vary widely because units differ in how many trade partners they have and how intense those relationships are. Row-standardization collapses all of that variation to 1, asserting that every unit faces the same total influence. Whether that assertion matches the theory is the question researchers should ask before accepting the software default.
Ask whether the theory implies that total exposure is homogeneous across units. If not, consider raw weights, spectral normalization, or a variance-stabilizing scheme. At minimum, check how sensitive results are to this choice.
Most spatial weights matrices in practice are symmetric, meaning that if A influences B, then B influences A with the same weight. This is natural for geographic contiguity (if two countries share a border, the border exists for both) but it is a strong assumption for most other connectivity concepts.
Influence is rarely reciprocal in equal measure. Consider the following examples.
W would miss this asymmetry.When I use a symmetric W, I assert that all of these directional stories are wrong, that influence is always reciprocal and equal. This is a substantive claim, not a modeling convenience. If my theory suggests directional influence (e.g., large economies shape small economies more than vice versa), then W should be asymmetric.
Some units may have no neighbors under the chosen connectivity definition. Hawaii and Alaska have no contiguous US state neighbors. Island nations have no border neighbors. In a trade-weighted matrix, a sanctioned or autarkic country might have negligible connections. These units become isolates and their rows and columns in W are all zeros.
Isolates create practical and interpretive problems. The spatial lag for an isolated unit is always zero, regardless of what happens elsewhere. In a spatial autoregressive model, this means isolated units contribute no information to estimating the spatial feedback parameter (\(\rho\)) and their outcomes are modeled as if no interdependence exists. If many units are isolates, the effective sample for estimating spatial parameters shrinks considerably.
More subtly, isolates can fragment the connectivity graph into disconnected components. When the network splits into separate clusters with no cross-links, diffusion effects cannot cross the gaps. It is worth always checking connectivity statistics before modeling. How many connected components exist? Are there unexpected isolates? A bimodal distribution of neighbor counts may signal that the connectivity definition does not fit the geography or the theory equally well across all units. The lesson is not that isolates should be avoided at all costs, but that their presence should be diagnosed and their consequences understood. If my connectivity definition creates many isolates, it may not be the right definition for the question at hand.
The five decisions discussed above (connectivity concept, scope, weight function, normalization, and directionality) interact to produce a specific W. To drive home how much these choices matter jointly, I now fit the same regression model to the same simulated data under four deliberately different matrices.
W_sym_local <- W_true # already constructed above
W_sym_dense <- matrix(0, n, n)
for (i in 1:n) {
for (j in 1:n) {
if (i != j && abs(i - j) <= 2) W_sym_dense[i, j] <- 1
}
}
W_sym_dense <- row_standardize(W_sym_dense)
W_asym_forward <- matrix(0, n, n)
for (i in 1:n) {
if (i > 1) W_asym_forward[i, i - 1] <- 1
if (i > 2) W_asym_forward[i, i - 2] <- 0.5
}
W_asym_forward <- row_standardize(W_asym_forward)
W_asym_hub <- matrix(0, n, n)
hubs <- c(5, 20, 35, 45)
for (i in 1:n) {
if (!i %in% hubs) {
W_asym_hub[i, hubs] <- c(1.0, 0.8, 0.6, 0.4)
}
}
W_asym_hub <- row_standardize(W_asym_hub)W_candidates <- list(
symmetric_local = W_sym_local,
symmetric_dense = W_sym_dense,
asymmetric_forward = W_asym_forward,
asymmetric_hub = W_asym_hub
)
sensitivity_tbl <- compare_model_sensitivity(
formula = policy_adoption ~ policy_pressure + fiscal_capacity,
data = dat, W_list = W_candidates
)
sens_display <- sensitivity_tbl[, c("W", "spatial_feedback",
"indirect_policy_pressure",
"total_policy_pressure")]
names(sens_display) <- c("Matrix", "Spatial feedback (\u03c1)",
"Indirect effect", "Total effect")
num_cols <- vapply(sens_display, is.numeric, logical(1))
sens_display[num_cols] <- lapply(sens_display[num_cols], round, 3)
knitr::kable(sens_display, row.names = FALSE)| Matrix | Spatial feedback (ρ) | Indirect effect | Total effect |
|---|---|---|---|
| symmetric_local | 0.481 | 0.502 | 1.237 |
| symmetric_dense | 0.539 | 0.614 | 1.260 |
| asymmetric_forward | 0.568 | 0.846 | 1.527 |
| asymmetric_hub | -0.416 | -0.292 | 0.470 |
The indirect effect column is the estimated spillover, the part of policy pressure’s impact that propagates through the network. If it changes sign, magnitude, or substantive importance across matrices, then the diffusion claim is only as credible as the justification for the chosen W.
Tan, Kesina, and Elhorst (2025) found exactly this pattern in a systematic analysis. Direct effects tend to be relatively robust to W specification, but indirect effects are highly sensitive. This is intuitive. The direct effect captures a unit’s own response, while the indirect effect routes through the entire connectivity structure. Change the structure, change the spillover.
This sensitivity is not a flaw in spatial models. It honestly reflects the fact that claims about interdependence are only as strong as the theory that defines interconnection. A study reporting large tax competition spillovers under one W without showing what happens under plausible alternatives is incomplete.
W?An important robustness check asks what happens if W has no theoretical basis at all. Below, I compare the theory-aligned matrix (which matches the true data-generating process) against a randomly generated matrix.
set.seed(99)
W_random <- matrix(sample(c(0, 1), n * n, replace = TRUE, prob = c(0.9, 0.1)), n, n)
diag(W_random) <- 0
W_random <- row_standardize(W_random)
misspec_tbl <- compare_model_sensitivity(
formula = policy_adoption ~ policy_pressure + fiscal_capacity,
data = dat,
W_list = list(theory_aligned = W_sym_local, random_matrix = W_random)
)
mis_display <- misspec_tbl[, c("W", "spatial_feedback",
"indirect_policy_pressure",
"total_policy_pressure")]
names(mis_display) <- c("Matrix", "Spatial feedback (\u03c1)",
"Indirect effect", "Total effect")
num_cols <- vapply(mis_display, is.numeric, logical(1))
mis_display[num_cols] <- lapply(mis_display[num_cols], round, 3)
knitr::kable(mis_display, row.names = FALSE)| Matrix | Spatial feedback (ρ) | Indirect effect | Total effect |
|---|---|---|---|
| theory_aligned | 0.481 | 0.502 | 1.237 |
| random_matrix | -0.321 | -0.187 | 0.581 |
A significant-looking \(\hat{\rho}\) under a random matrix should not be interpreted as evidence for spatial diffusion. Spatial models will often find some spatial pattern in correlated data even when the connectivity structure is wrong. This is a core insight from Drolc, Gandrud, and Williams (2021). Omitted spatially correlated covariates can create spurious diffusion findings regardless of which W is used, and false positive rates under misspecification are alarmingly high.
The implication is straightforward. If I cannot defend the W theoretically, the spatial parameter estimates are uninterpretable. No amount of statistical significance compensates for a W that does not match the mechanism. If I find strong “tax competition spillovers” using a random connectivity matrix, I have not discovered tax competition. I have discovered that spatially correlated omitted variables can masquerade as diffusion.
Even with the right W, interpreting spatial model results is harder than it looks. Whitten, Williams, and Wimpy (2021) call interpretation “the final spatial frontier” and document how political scientists routinely misread their own models.
The problem is that coefficients in a spatial autoregressive model do not have the same straightforward interpretation as OLS coefficients. In a standard SAR model (Equation 5.2),
\[ y = \rho W y + X\beta + \varepsilon \tag{5.2}\]
the coefficient \(\beta\) captures only the zero-order direct effect, the immediate impact of \(X\) on \(y\) before any spatial feedback occurs. But changes in \(X_i\) propagate through the network. They affect \(y_i\) directly, which affects \(y_i\)’s neighbors through \(Wy\), which feeds back to \(y_i\) and propagates to neighbors’ neighbors, and so on. The full effect is captured by the spatial multiplier (Equation 5.3),
\[ (I_N - \rho W)^{-1} X\beta \tag{5.3}\]
This multiplier expands as \(I + \rho W + \rho^2 W^2 + \rho^3 W^3 + \ldots\), showing that effects cascade through the network with diminishing intensity at each step. The total effect on unit \(i\) from a change in unit \(j\)’s covariate depends on \(j\)’s specific position in the network as defined by W. This means that different units generally have different total effects, and a single coefficient cannot summarize them.
The inverse \((I_N - \rho W)^{-1}\) exists only when \(\rho\) lies within bounds determined by the eigenvalues of W. For a symmetric row-standardized matrix, eigenvalues are real and bounded between \(-1\) and \(1\), giving the common sufficient condition \(|\rho| < 1\). For asymmetric matrices (including asymmetric row-standardized ones), eigenvalues can be complex, and the admissible range must be derived from the real parts of the eigenvalues or computed numerically via the log-determinant. For non-standardized matrices with real eigenvalues, the admissible range is \(1/\omega_{\min} < \rho < 1/\omega_{\max}\), where \(\omega_{\min}\) and \(\omega_{\max}\) are the smallest and largest eigenvalues of W. This is one reason normalization choices matter in practice even when they do not change estimated effects in principle. They change the parameter space over which the optimizer searches.
The correct approach, as Whitten, Williams, and Wimpy (2021) demonstrate, is to compute the matrix of partial derivatives and report summary measures, specifically the average direct effect (diagonal elements of the multiplier matrix), the average indirect effect (off-diagonal elements), and the average total effect (their sum). None of these equal the raw coefficient \(\beta\), though they reduce to \(\beta\) when \(\rho = 0\) (i.e., when there is no spatial dependence).
This matters for every decision discussed in this chapter because the multiplier \((I_N - \rho W)^{-1}\) depends on W. Changing W changes not only the coefficient estimates but the entire mapping from coefficients to substantive effects.
To make this concrete, return to the corporate tax competition setting. If I model how policy pressure affects tax policy adoption with a SAR model, the raw coefficient \(\hat{\beta}\) tells me the immediate effect of a one-unit increase in policy pressure on a country’s own policy adoption before any spatial feedback. But when that country adjusts its tax rate, its neighbors notice and may respond, and those responses feed back. The average direct effect includes this feedback echo. The average indirect effect captures how changing one country’s policy pressure affects other countries’ outcomes through the connectivity structure. Both quantities depend entirely on W, because W defines the paths along which competitive pressure propagates. The sensitivity analysis in the previous section showed just how much these indirect effects can shift when I change the connectivity assumptions.
In R, spatialreg::impacts() computes average direct, indirect, and total effects from a fitted SAR model.
# dat and W_sym_local are already defined above in this chapter.
listw_obj <- spdep::mat2listw(W_sym_local, style = "W")
sar_fit <- spatialreg::lagsarlm(
policy_adoption ~ policy_pressure + fiscal_capacity,
data = dat,
listw = listw_obj,
method = "eigen",
zero.policy = TRUE
)
imp <- spatialreg::impacts(sar_fit, listw = listw_obj, R = 2000)
imp_summary <- summary(imp, zstats = FALSE, short = TRUE)
bnames <- attr(imp_summary, "bnames")
target_idx <- grep("^policy_pressure", bnames)[1]
lib_direct <- as.numeric(imp_summary$res$direct[target_idx, 1])
lib_indirect <- as.numeric(imp_summary$res$indirect[target_idx, 1])
lib_total <- as.numeric(imp_summary$res$total[target_idx, 1])
get_sd <- function(sum_obj, idx) {
stat <- sum_obj$statistics
if (is.null(stat)) return(NA_real_)
sd_col <- grep("^SD$|std", colnames(stat), ignore.case = TRUE)
if (!length(sd_col)) sd_col <- 2
as.numeric(stat[idx, sd_col[1]])
}
get_ci <- function(sum_obj, idx) {
qmat <- sum_obj$quantiles
if (is.null(qmat)) return(c(NA_real_, NA_real_))
lo_col <- grep("^2\\.5%$|^2\\.5 %$|^2.5%$", colnames(qmat))
hi_col <- grep("^97\\.5%$|^97\\.5 %$|^97.5%$", colnames(qmat))
if (!length(lo_col) || !length(hi_col)) {
return(c(as.numeric(qmat[idx, 1]), as.numeric(qmat[idx, ncol(qmat)])))
}
c(as.numeric(qmat[idx, lo_col[1]]), as.numeric(qmat[idx, hi_col[1]]))
}
lib_direct_se <- get_sd(imp_summary$direct_sum, target_idx)
lib_indirect_se <- get_sd(imp_summary$indirect_sum, target_idx)
lib_total_se <- get_sd(imp_summary$total_sum, target_idx)
lib_direct_ci <- get_ci(imp_summary$direct_sum, target_idx)
lib_indirect_ci <- get_ci(imp_summary$indirect_sum, target_idx)
lib_total_ci <- get_ci(imp_summary$total_sum, target_idx)
impact_tbl <- data.frame(
Effect = c("Direct", "Indirect", "Total"),
Estimate = c(lib_direct, lib_indirect, lib_total),
SE = c(lib_direct_se, lib_indirect_se, lib_total_se),
CI_95 = c(
sprintf("[%.4f, %.4f]", lib_direct_ci[1], lib_direct_ci[2]),
sprintf("[%.4f, %.4f]", lib_indirect_ci[1], lib_indirect_ci[2]),
sprintf("[%.4f, %.4f]", lib_total_ci[1], lib_total_ci[2])
),
stringsAsFactors = FALSE
)
knitr::kable(
impact_tbl,
digits = 4,
col.names = c("Effect", "Estimate", "SE", "95% CI"),
caption = "Average impacts of policy pressure from spatialreg::impacts()"
)| Effect | Estimate | SE | 95% CI |
|---|---|---|---|
| Direct | 0.7345 | 0.1271 | [0.5070, 1.0010] |
| Indirect | 0.5022 | 0.1916 | [0.2433, 0.9940] |
| Total | 1.2367 | 0.2888 | [0.7957, 1.9289] |
The table shows the average direct, indirect, and total effects of policy pressure with simulation-based standard errors and confidence intervals.
W Choice and Model Choice Interact
This chapter focuses on the SAR (spatial autoregressive lag) model, but the choice of W is not separable from the choice of model specification. A SAR model assumes that influence flows through the dependent variable (\(\rho Wy\)). An SLX model places spatial lags on the covariates instead (\(WX\theta\)), assuming neighbors’ characteristics matter but not their outcomes. An SDM (spatial Durbin model) includes both \(\rho Wy\) and \(WX\theta\). An SEM (spatial error model) places spatial structure on the error term, capturing unmodeled spatial correlation without implying a diffusion mechanism. Each specification makes different demands on W and assigns it a different substantive role. If the theory implies that neighbors’ policies matter (competitive pressure), SAR or SDM is appropriate. If the theory implies that neighbors’ characteristics matter (contextual effects), SLX may suffice. The five decisions discussed in this chapter apply to all of these models, but their consequences for estimation and interpretation differ across model types.
The following checklist synthesizes the lessons of this chapter into a workflow for specifying W in any project.
W. Choose a matrix that is plausible under the same general theory but differs in scope, weight function, or connectivity concept. This is not optional but rather the minimum standard for credible spatial analysis.Some connectivity measures (trade volumes, migration flows, capital flows) can themselves be influenced by the outcome variable. If you use bilateral trade to define W and then model how trade affects tax policy, you have a circularity problem. The fact that geographic distance is exogenous does not mean it is theoretically appropriate, but it is one reason geographic matrices remain popular despite their theoretical limitations. When using endogenous connectivity measures, consider using a theoretically motivated instrument. In panel settings, a common partial remedy is to lag W in time (e.g., using trade volumes from the previous period), though this does not fully resolve endogeneity if the connectivity measure is serially correlated. More broadly, researchers working with panel data should consider whether W should vary over time at all. A time-invariant W assumes a fixed network structure, which may be appropriate for geography but implausible for trade or alliance ties that evolve.
When writing up spatial analyses, the following information should appear in the main text or a prominent appendix, not buried in a footnote.
The connectivity matrix is not a technical nuisance to be dispatched with software defaults. It is a substantive hypothesis about the structure of interdependence, and it shapes every spatial estimate reported. Whether I am modeling corporate tax competition among European countries, policy diffusion through trade networks, or security spillovers across alliance partners, each decision embedded in W (what connects units, how broadly, how strongly, at what scale, and in which direction) is a claim about how the social world works.
As the simulations throughout this chapter have shown, these choices change results. Row-standardization versus raw weights produces different indirect effects because it changes which units drive the estimates. Symmetric versus asymmetric matrices encode different theories about the direction of influence. Narrow versus broad neighborhood scope controls how far spillovers can reach. And a theoretically unjustified W can produce spurious evidence of diffusion that has nothing to do with the mechanism under study.
The minimum standard, following the arguments of Neumayer and Plümper (2016), Whitten, Williams, and Wimpy (2021), Drolc, Gandrud, and Williams (2021), and Tan, Kesina, and Elhorst (2025), is to report at least one theoretically plausible alternative matrix, show how conclusions change under it, and discuss what that sensitivity means for the strength of the claims.