Robust Inference

AbstractRobustVCOV

A robust covariance specification, passed to robust_vcov, robust_coeftable or spatial_effects.

Subtypes: DriscollKraayVCOV and ConleySpatialHAC.

BartlettKernel()

Triangular kernel, max(0, 1 - |x|). The Newey-West default and the default throughout this package: weights decline linearly to zero at the bandwidth, which guarantees a positive semi-definite covariance.

See also KernelType.

ConleySpatialHAC(coords, dist_cutoff; max_time_lag = 0, metric = EuclideanDistance(),
                 space_kernel = BartlettKernel(), time_kernel = BartlettKernel(),
                 psd_fix = true)

Conley spatial HAC covariance: correlation decaying with geographic distance.

Pairs of units closer than dist_cutoff contribute to the covariance with a kernel weight based on their distance; pairs beyond it contribute nothing. Unlike DriscollKraayVCOV this needs coordinates, but in exchange it works with few time periods and with pure cross-sections.

Arguments

• coords: N x 2 matrix of positions, positional. With HaversineDistance the columns are latitude then longitude in degrees.
• dist_cutoff: distance beyond which units are treated as independent, positional, in the units of metric (kilometres for Haversine).

Keywords

• max_time_lag: periods of serial correlation to allow; 0 for a cross-section.
• metric: a DistanceMetric.
• space_kernel, time_kernel: KernelTypes for the spatial and temporal dimensions.
• psd_fix: project the result to the nearest positive semi-definite matrix if the kernel weighting leaves it indefinite. Keep this on unless you are diagnosing the covariance itself.

Examples

m = fit_spatial(SAR(), W, X, y)
robust_coeftable(m, ConleySpatialHAC(coords, 3.0))

ConleySpatialHAC(coords, 150.0; metric = HaversineDistance(), max_time_lag = 2)

References

Conley, T. (1999). GMM estimation with cross sectional dependence. Journal of Econometrics, 92(1).

DistanceMetric

How distance between two units is measured when building spatial HAC weights. Subtypes: EuclideanDistance, HaversineDistance.

DriscollKraayVCOV(; max_lag = nothing, kernel = BartlettKernel(), small_sample_correct = true)

Driscoll-Kraay spatially-correlated-consistent covariance for panel data.

Averages the scores across units within each period and then applies a Newey-West correction over periods. Because the cross-sectional dependence is handled by that averaging, it is robust to arbitrary spatial correlation — no distance, coordinates or cutoff are needed — as well as to heteroskedasticity and serial correlation.

The price is that it leans on having a reasonable number of time periods; with very short panels the time-series correction is poorly estimated, and ConleySpatialHAC is the better choice when you have coordinates and few periods.

Keywords

• max_lag: lag truncation. nothing picks a rule-of-thumb bandwidth from the number of periods.
• kernel: a KernelType.
• small_sample_correct: apply the finite-sample degrees-of-freedom scaling.

Examples

m = fit_spatial_panel(SAR(), W, X, y, N, T; fe = TwoWayFE())
robust_coeftable(m, DriscollKraayVCOV())
robust_coeftable(m, DriscollKraayVCOV(max_lag = 5, kernel = ParzenKernel()))

References

Driscoll, J. & Kraay, A. (1998). Review of Economics and Statistics, 80(4).

EuclideanDistance()

Straight-line distance in the units of the coordinates, hypot(dx, dy).

The default. Appropriate for projected coordinates; for raw latitude and longitude use HaversineDistance instead, since a degree of longitude is not a fixed distance.

HaversineDistance(; radius_km = 6371.0088)

Great-circle distance in kilometres between points given as (latitude, longitude) in degrees.

Use this whenever coordinates are unprojected geographic degrees. The default radius is the Earth's mean radius.

Examples

spec = ConleySpatialHAC(coords, 150.0; metric = HaversineDistance())

KernelType

Kernel used to down-weight distant observations in a HAC covariance.

Subtypes: BartlettKernel, ParzenKernel, QuadraticSpectralKernel, UniformKernel. The choice affects the weight profile but rarely the conclusion; what matters more is the bandwidth (max_lag or dist_cutoff).

ParzenKernel()

Parzen kernel: smoother decay than BartlettKernel, also guaranteeing positive semi-definiteness. Slightly more efficient when the true autocorrelation decays smoothly.

See also KernelType.

QuadraticSpectralKernel()

Quadratic spectral kernel. Asymptotically optimal in the rate sense (Andrews, 1991) and has no hard truncation point: weights decay smoothly without ever reaching exactly zero, so every lag contributes something.

See also KernelType.

RobustInferenceTable

Coefficient table pairing robust with model-based standard errors, returned by robust_coeftable. Pretty-prints via show.

Fields

• estimator_name: which robust estimator was used.
• coefnames, coef: parameter names and estimates.
• se_robust, se_baseline: robust and model-based standard errors. The ratio between them is the diagnostic worth reading.
• z_robust: coef ./ se_robust.
• vcov_robust: the full robust covariance.

UniformKernel()

Equal weight on every lag inside the bandwidth and zero outside — the truncated, unweighted estimator.

See also KernelType.

vcov_conley(Z, residuals, N, T_per, spec; bread_inv = nothing)

Conley spatial HAC covariance from a score matrix directly.

The low-level routine behind robust_vcov; see vcov_driscoll_kraay for why the fitted-model path is usually the one you want. spec is a ConleySpatialHAC, which carries the coordinates, cutoff, metric and kernels.

vcov_driscoll_kraay(Z, residuals, N, T_per; bread_inv = nothing, spec = DriscollKraayVCOV())

Driscoll-Kraay covariance from a score matrix directly.

The low-level routine behind robust_vcov. Prefer that on a fitted model, which builds Z and bread_inv correctly for you — in particular it maps maximum-likelihood scores through the Jacobian of the spatial transformation, which a raw call here will not do.

Arguments

• Z: (N*T_per) x K regressor or score matrix, stacked with units varying fastest within each period.
• residuals: matching residual vector.
• N, T_per: units and periods.

Keywords

• bread_inv: the (X'X)^{-1}-analogue for the sandwich. nothing uses the OLS form.
• spec: a DriscollKraayVCOV carrying the kernel and bandwidth.