Spatial Dynamic Panels with Common Factors

Credit risk in US banking, the spxtivdfreg way

0.394spatial spillover psi

7.765long-run total effect

33.5%variance from factors

Carlos Mendez

Nagoya University (GSID)

June 11, 2026

The Tension

Act I

When the crisis hit, credit risk spread two ways — and standard tools saw only one

When the 2007–2009 crisis struck, non-performing loans surged across the whole US banking system at once.

Two channels did it: spatial spillovers between banks, and common factors — macro shocks that hit every bank at once.

Stata’s xsmle and spxtregress model only the first.

This is the central tension. Bank risk does not stay contained. It travels along lending relationships (spatial) AND it responds to aggregate shocks like interest-rate moves and the housing collapse (common factors). The standard spatial-panel toolkit in Stata can do spatial lags but cannot accommodate unobserved common factors — that gap is what the whole deck is about.

One equation must absorb four kinds of endogeneity at once

\[NPL_{it} = \psi \sum_{j=1}^{N} w_{ij}\, NPL_{jt} + \rho\, NPL_{i,t-1} + x_{it}\beta + \alpha_i + \lambda_i' f_t + \varepsilon_{it}\]

Spatial lag \(\psi W\,NPL\), temporal lag \(\rho\,NPL_{i,t-1}\), endogenous covariates \(x_{it}\beta\), bank effect \(\alpha_i\), and the latent factors \(\lambda_i' f_t\) — every term but \(\alpha_i\) is a source of endogeneity.

Walk it left to right. The spatial lag is endogenous (i depends on j and j on i — simultaneity). The temporal lag correlates with the fixed effect. INEFF is an endogenous regressor. And the interactive fixed effect λ_i’ f_t enters both the regressors and the error, inducing cross-sectional dependence and omitted-variable bias. Four problems, one estimator.

Where we’re going

• The lab: 350 US banks, 36 quarters, an economic-distance network
• The estimator: defactor first, then run IV
• The full model: a significant spatial lag and one-third of the variance from factors
• Three stress tests: drop the factors, drop the spatial lag, free the slopes

The Investigation

Act II

The lab: 350 US banks across the entire crisis, 12,250 observations

• Outcome — NPL, non-performing loans over total loans (%)
• Span — 2006:Q1 to 2014:Q4, 36 quarters straddling the GFC
• Network — a \(350 \times 350\) weight matrix \(W\), not geographic but economic-distance (Spearman rank correlations of bank debt ratios)

Strongly balanced: every bank in every quarter. The matrix is row-standardized with about 18 neighbours per bank, so the spatial lag is a weighted average of interconnected banks.

The economic-distance W is the key modelling choice. Two banks are neighbours if their debt-ratio rank correlation exceeds the 95th percentile — so two banks on opposite coasts can be “neighbours” if they lend alike. 6,300 nonzero entries / 350 banks ≈ 18 neighbours. Effective sample is 12,250 = 350 × 35 after the lag.

Defactored IV is a two-step trick: subtract the rainstorm, then read each pond

Step 1 — Defactor

• Extract latent common factors \(\lambda_i' f_t\) by principal components
• Remove them from data before estimating

Step 2 — IV / GMM

• Run instrumental variables on the defactored data
• Standard 2SLS asymptotics now hold

Removing the factors first kills the cross-sectional dependence — no incidental-parameters bias, no Lee–Yu correction.

The analogy from the post: a global rainstorm hits every pond at once. All ponds rise together — not because they’re connected, but because of the same outside force. Subtract that rain first; the remaining variation is the local circuitry the spatial/temporal model wants to learn. Because the factors come out before the IV step, classic 2SLS asymptotics apply.

INEFF is endogenous, so INTEREST does the instrumenting

spxtivdfreg NPL INEFF CAR SIZE BUFFER PROFIT QUALITY LIQUIDITY, ///
    absorb(ID) splag tlags(1) spmatrix("W.csv", import) ///
    iv(INTEREST CAR SIZE BUFFER PROFIT QUALITY LIQUIDITY, splags lag(1)) std

splag adds the spatial lag, tlags(1) the temporal lag, iv(...) instruments INEFF with INTEREST and lags, std standardizes before extracting factors.

One command carries the whole model. INEFF (operational inefficiency) is endogenous — high NPL lowers measured efficiency (reverse causality) and unobserved management quality drives both. INTEREST (interest expenses / deposits) is the excluded instrument, alongside lagged exogenous regressors. The std option standardizes covariates so the principal-components factor step is numerically stable across very different scales.

The estimator finds the factors itself: 2 in the regressors, 1 in the errors

• \(r_x = 2\) common factors in the regressors
• \(r_u = 1\) common factor in the error term
• Together they capture Fed policy, the housing collapse, the interbank freeze

Heterogeneous loadings \(\lambda_i\) let each bank react to the same aggregate shock with its own intensity — something two-way fixed effects cannot do.

This is the payoff of interactive fixed effects over plain time dummies. Time dummies would only work if a shock hit all 350 banks equally. The λ_i’ f_t structure lets a big, systemically-exposed bank load heavily on the recession factor while a small local bank barely moves. The estimator chooses the number of factors automatically here.

With factors in, the spatial lag is real: a 0.39 contemporaneous spillover

0.394

\(\psi\), spatial autoregressive parameter (z = 4.65, p < 0.001) — neighbours’ NPL up 1 pp raises this bank’s NPL by 0.39 pp

Financial contagion through interconnected lending: when one bank’s borrowers default, it cascades to economically linked banks. ψ = 0.394 is strongly significant. Pair this with ρ = 0.290 (z = 5.33) — about 29% of last quarter’s NPL persists to this quarter via workouts, foreclosures, write-offs.

LIQUIDITY dominates the covariates, and the J-test blesses the instruments

Term	Coef.	z	Sig.?
\(\psi\) (W·NPL)	0.394	4.65	yes
\(\rho\) (L1.NPL)	0.290	5.33	yes
LIQUIDITY	2.452	9.09	yes
INEFF	0.447	4.28	yes
BUFFER	−0.055	−4.59	yes

Hansen J: \(\chi^2(19) = 18.83\), \(p = 0.468\) — fails to reject, so the instruments survive overidentification.

LIQUIDITY has the biggest effect: a 1-pp rise in the loan-to-deposit ratio adds 2.45 pp of NPL — banks that lend aggressively relative to deposits carry more risk. INEFF confirms the “bad management” hypothesis. BUFFER is protective: better-capitalized banks have lower credit risk. And the Hansen J passing (p = 0.468) tells us the IV strategy — INTEREST plus lagged exogenous regressors — is appropriate.

One-third of the residual variance is pure macro shock

33.5%

\(\rho_{factor} = \sigma_f^2 / (\sigma_f^2 + \sigma_e^2)\) with \(\sigma_f = 0.642\), \(\sigma_e = 0.904\) — the share a no-factor model would bake into biased coefficients

This is the quantitative case for modelling factors explicitly. A full third of what’s left after the regressors is common, serially correlated macro variation. Sweep it into the error and it contaminates everything that’s correlated with it — which is the whole next act.

The Resolution

Act III

Drop the factors and temporal persistence doubles while the J-test rejects

Term	With factors	Without factors
\(\psi\) (spatial)	0.394***	0.288***
\(\rho\) (temporal)	0.290***	0.594***
LIQUIDITY	2.452***	0.843***
Factors \((r_x, r_u)\)	2, 1	0, 0
J-test \(p\)	0.468	0.000

Excluded factors are serially correlated, so \(\rho\) swells to absorb them — and the instruments are no longer valid.

The differences are striking and systematic. ρ doubles from 0.290 to 0.594 because the serially-correlated common factors, once excluded, get soaked up by the temporal lag — the model confuses macro persistence with bank-level risk persistence. LIQUIDITY collapses by two-thirds. And crucially the J-test now REJECTS (p < 0.001): without factors, the exclusion restriction breaks, so the model is misspecified. That’s a formal statistical justification for keeping the factors.

The no-factor model also masks a real effect: SIZE goes insignificant

With factors

• SIZE = 0.223, significant (z = 2.36)
• larger banks load more on macro shocks

Without factors

• SIZE = 0.089, not significant
• exposure mistaken for noise

Omitting factors can hide a genuine relationship, not just inflate spurious ones.

This is the subtle direction of the bias. Big banks have larger factor loadings — they’re more exposed to systematic macro shocks. Pull the factors out and that exposure is misattributed to noise, so SIZE drops to insignificance. Factor omission distorts coefficients in BOTH directions: inflating some (ρ), masking others (SIZE).

The spatial lag earns its place: dropping it inflates the covariates

Term	Full model	Without spatial lag
\(\psi\) (W·NPL)	0.394***	—
\(\rho\) (L1.NPL)	0.290***	0.323***
INEFF	0.447***	0.638***
SIZE	0.223**	0.346***
J-test \(p\)	0.468	0.226

Both pass the J-test — the choice is economic, and theory says banks are interconnected.

Remove the spatial lag and the leftover covariates must explain the cross-sectional co-movement on their own: INEFF jumps 43%, SIZE 55%, and ρ ticks up to absorb the missing spatial dynamics. Both specs have valid instruments (p = 0.468 vs 0.226), so the J-test can’t decide between them. We prefer the full model because financial theory predicts interdependence and ψ = 0.394 is highly significant.

A permanent liquidity shock costs three times what the coefficient shows

7.765

Long-run total effect of LIQUIDITY vs a short-run coefficient of 2.452 — amplified over time and across the network

The short-run coefficient is the first ripple; the long-run total is the whole wake. A permanent 1-pp rise in the loan-to-deposit ratio across all banks eventually lifts system-wide NPL by nearly 7.8 pp. The contemporaneous reading badly understates the true cost of excessive lending.

Long-run impact = short-run effect, amplified twice

\[\text{Total LR effect} = \frac{\beta}{(1-\rho)(1-\psi)} = \frac{2.452}{(1-0.290)(1-0.394)} \approx 7.77\]

The temporal multiplier \(1/(1-\rho) = 1.41\) compounds the shock over time; the spatial multiplier \(1/(1-\psi) = 1.65\) spreads it across the network.

Two multipliers stack. The temporal multiplier 1/(1−ρ) = 1.41 captures the shock feeding back through lagged NPL. The spatial multiplier 1/(1−ψ) = 1.65 captures it spreading through the ~18-neighbour network. For nearly every covariate the long-run INDIRECT (spillover) effect is as large as or larger than the direct effect — for LIQUIDITY, 4.22 indirect vs 3.55 direct.

The strongest objection — and the answer

Objection. The mean-group estimator drives the spatial lag to insignificance (\(\psi = 0.032\), \(p = 0.536\)). So maybe the spillovers were never real.

Response. Imposing homogeneous slopes on heterogeneous banks can manufacture spurious spatial dependence — but MG is only \(\sqrt{N}\)-consistent with just 35 periods, so a few outlier banks can swamp the average. Read it as a caution, not a refutation: \(\rho\) stays stable at 0.301, and theory still predicts interconnection.

Steelman, don’t strawman. The MG result is genuinely uncomfortable: free every bank’s slopes and the average ψ collapses to 0.032. That can happen because pooled homogeneity creates spurious spatial dependence when slopes truly differ. But MG trades efficiency for flexibility — √N vs √NT consistency — and with T = 35 a handful of extreme bank-specific estimates can dominate. The temporal persistence ρ ≈ 0.30 is rock-stable across all four specifications, which argues the dynamics are real.

Four specifications, one verdict: factors stay, spatial lag stays

	Full	No factors	No spatial	MG
\(\psi\)	0.394***	0.288***	—	0.032
\(\rho\)	0.290***	0.594***	0.323***	0.301***
LIQUIDITY	2.452***	0.843***	2.534***	6.330***
J-test \(p\)	0.468	0.000	0.226	—

The J-test decides factors; theory and significance decide the spatial lag; MG is a robustness check, read with care.

Decision rule from the post: run with and without factors first. The J-test rejects without (p < 0.001), so factors are mandatory. With factors, ψ is highly significant, so keep the spatial lag — that’s the full model. MG is a final robustness probe that flags possible slope heterogeneity but shouldn’t overturn the preferred specification given its small-T fragility.

Model the common factors, or the spatial story you tell will be the wrong one.

The single takeaway. Unobserved macro shocks aren’t a nuisance to sweep into the error — they’re 33.5% of the residual variance, and ignoring them doubles ρ, halves LIQUIDITY, and invalidates the instruments. Defactor first, then let IV recover the genuine spatial (0.394) and temporal (0.290) dynamics — whose long-run amplification (7.765 for LIQUIDITY) is the real policy story.