Package {aersn}

aersn: Affine-Equivariant Adjusted-Range Self-Normalization

Description

The package implements affine-equivariant adjusted-range self-normalization for inference on a fixed-dimensional parameter of a dependent time series. The construction takes the centered partial-sum path of estimated influence contributions, forms the convex hull of all its increments (the increment hull), and normalizes the scaled estimation error by the gauge of that hull. The statistic has a pivotal limit that depends only on the dimension of the parameter of interest, so no long-run covariance matrix, bandwidth, kernel or block length has to be chosen.

What the package computes

Let \hat\theta_n \in R^q estimate \theta_0 and let \hat\psi_{n,t} \in R^q, t = 1, \ldots, n, be estimated influence contributions with \sqrt n(\hat\theta_n - \theta_0) = n^{-1/2}\sum_t \psi_t + o_p(1). The centered path on the observation grid is

\hat G_n(k/n) = n^{-1/2}\{\sum_{t \le k}\hat\psi_{n,t} - \tau(k/n)\sum_{t \le n}\hat\psi_{n,t}\}, \quad k = 0, \ldots, n,

with \tau(r) = r under calendar-time centering. Its increment hull is K(\hat G_n) = \mathrm{conv}\{\hat G_n(r) - \hat G_n(s)\}, whose support function in direction u equals the adjusted range (maximum minus minimum) of the projected path u^\top \hat G_n. The test statistic for H_0: \theta_0 = v is the gauge T_n(v) = \gamma_{K(\hat G_n)}\{\sqrt n(\hat\theta_n - v)\}, evaluated by a linear program over the n + 1 path values. The confidence region is \hat\theta_n + (c/\sqrt n) K(\hat G_n), where c is a quantile of the reference law \gamma_{K(B_q)}(Z_q) formed from a standard Brownian bridge B_q and an independent standard normal vector Z_q.

Comparator methods

Five further methods are available on the same fit, the same influence contributions and the same level, through the method argument of aersn_test(), confint.aersn(), aersn_contrast() and aersn_region(), and side by side through aersn_compare(): componentwise adjusted ranges after lag-zero LDL partial prewhitening (aersn_ldl_normalizer()), quadratic self-normalization (aersn_shao_normalizer()), kernel long-run covariance estimation (aersn_hac_lrv()), Bartlett fixed-b (aersn_fixed_b_normalizer()) and the equal-weighted cosine method (aersn_ewc_lrv()). Their statistics are on different scales and are never compared as numbers; the comparable quantities are p-values, decisions and interval widths.

Main functions

• aersn() builds the inference object from an estimate and its influence contributions; aersn_mean(), aersn_lm(), aersn_gmm() and aersn_mle() compute the contributions for common estimators and call the same core construction.

• aersn_test(), confint.aersn(), aersn_contrast(), aersn_region() and aersn_contains() provide tests, simultaneous intervals, and confidence-region membership.

• aersn_reference() simulates or evaluates reference distributions; aersn_scalar_cdf() and aersn_scalar_quantile() give the closed-form scalar law.

• aersn_path(), aersn_hull(), aersn_gauge() and aersn_support() expose the geometric core, and aersn_normalizer() the normalizer of any method.

• aersn_compare() applies several methods to one estimate and reports the tuning values each of them used.

Validity conditions

Inference is asymptotically valid under the conditions of the manuscript: (i) the partial-sum process of the influence contributions satisfies a functional central limit theorem with a nonsingular scale matrix; (ii) the estimator is asymptotically linear in those contributions; and (iii) the feasible increment hull converges in Hausdorff distance to the hull of the infeasible centered path. The package cannot verify these conditions for arbitrary estimators; see the vignette Supplied influence contributions and the documentation of each model interface for what must hold.

Author(s)

Maintainer: Jiajing Sun jiajing.sun@gmail.com

Authors:

• Jiajing Sun jiajing.sun@gmail.com

• Yongmiao Hong

• Zhuo Lin

• Oliver Linton

• Whitney K. Newey

References

Hong, Y., Lin, Z., Linton, O., Newey, W. K. and Sun, J. (2026). Affine-equivariant adjusted-range self-normalization. Cambridge Working Papers in Economics No. 2678; Janeway Institute Working Paper No. 2637. Publication page.

Hong, Y., Linton, O., McCabe, B., Sun, J. and Wang, S. (2024). Kolmogorov-Smirnov type testing for structural breaks: a new adjusted-range based self-normalization approach. Journal of Econometrics, 238(2), 105603. doi:10.1016/j.jeconom.2023.105603

Shao, X. (2010). A self-normalized approach to confidence interval construction in time series. Journal of the Royal Statistical Society: Series B, 72(3), 343-366. doi:10.1111/j.1467-9868.2009.00737.x

See Also

Working paper and publication details.

Adjusted-range self-normalized inference from an estimate and its influence contributions

Description

The central constructor of the package. Given a parameter estimate and the observation-level influence contributions of its asymptotically linear representation, it builds the centered influence path, the increment-hull self-normalizer and the diagnostics needed for tests (aersn_test()), simultaneous confidence intervals (confint.aersn(), aersn_contrast()) and confidence regions (aersn_region()). The model interfaces aersn_mean(), aersn_lm(), aersn_gmm() and aersn_mle() compute the contributions and call this function.

Usage

aersn(
  estimate,
  psi,
  n = NROW(psi),
  profile = NULL,
  names = NULL,
  model = NULL,
  call = NULL
)

Arguments

Details

Conventions. Parameters are ordered as in estimate; column j of psi is the contribution to parameter j. The representation assumed is manuscript equation (asymptotic-linear-representation),

\sqrt n(\hat\theta_n - \theta_0) = n^{-1/2}\sum_{t=1}^n \psi_t + o_p(1),

with n = nrow(psi) observations in time order. The centered path is formed by aersn_path(): partial sums of the contributions scaled by n^{-1/2} and centered by \tau(k/n) times the full-sample sum, with \tau(r) = r under calendar-time centering. The statistic for a candidate value v is T_n(v) = \gamma_{K(\hat G_n)}\{\sqrt n(\hat\theta_n - v)\} and the level-(1-\alpha) confidence region is \hat\theta_n + (c_{q,1-\alpha}^{(n)}/\sqrt n) K(\hat G_n), where c_{q,1-\alpha}^{(n)} is the matched-grid critical value from aersn_reference().

Nuisance parameters and target transformations. When the parameter of interest is a function h(\beta) of a larger estimated vector, the contributions must already incorporate the first-order effect of estimating the nuisance coordinates: for smooth GMM, \hat\psi_{n,t} = -\hat{\dot h}_n \hat M_n m_t(\hat\beta_n), and for conditional likelihood \hat\psi_{n,t} = \hat{\dot h}_n \hat J_n^{-1} s_t(\hat\beta_n) (manuscript Appendix A). Discarding nuisance coordinates from raw moment contributions or scores does not give the correct path; use aersn_gmm(), aersn_mle() or aersn_target().

What the package does not check. Validity requires (i) a functional central limit theorem for the partial sums of \psi_t with a nonsingular scale matrix, (ii) the asymptotically linear representation above, and (iii) Hausdorff convergence of the feasible increment hull to that of the infeasible centered path (manuscript Assumption 1). Uniform convergence of the feasible path is sufficient for (iii). These conditions hold for the built-in interfaces under the stated assumptions; for user-supplied contributions they are the user's responsibility. Under strong persistence the finite-sample null rejection rate can exceed the nominal level (manuscript Section 5).

Value

An object of class "aersn": a list with elements estimate, n, q, names, psi, path (an aersn_path()), hull (an aersn_hull()), centering, nodes, model and call.

References

Hong, Lin, Linton, Newey and Sun (2026), Sections 2-3 and Appendix A.

See Also

aersn_test(), confint.aersn(), aersn_contrast(), aersn_region(), aersn_target(), aersn_reference().

Examples

## Synthetic example: mean of a bivariate AR(1) series
set.seed(10)
n <- 200; e <- matrix(rnorm(2 * n), n, 2); Y <- e
for (t in 2:n) Y[t, ] <- 0.4 * Y[t - 1, ] + e[t, ]
fit <- aersn(colMeans(Y), sweep(Y, 2, colMeans(Y)), names = c("m1", "m2"))
fit
aersn_test(fit, null = c(0, 0), draws = 500)          # small draws: example only
confint(fit, draws = 500)

Clear the session cache of reference distributions

Description

Clear the session cache of reference distributions

Usage

aersn_clear_cache()

Value

Invisibly, the number of cached objects removed.

Compare inference methods on one estimate

Description

Applies several methods to the same estimate, the same influence contributions, the same null value or linear restriction and the same confidence level, and collects the results in one table. Each row reports the statistic, its critical value, a p-value, the reference law and the tuning values actually used, so that differences between methods can be attributed to the normalizer rather than to a difference in setup.

Usage

aersn_compare(
  object,
  null = NULL,
  contrast = NULL,
  level = 0.95,
  methods = aersn_methods(),
  settings = list(),
  draws = NULL,
  seed = 1L,
  cores = 1L,
  verbose = FALSE
)

Arguments

Details

Statistics are not comparable across rows as numbers. The increment-hull statistic is a gauge, homogeneous of degree one in the estimation error, while the other five are Wald statistics, homogeneous of degree two, and each is referred to its own law. The comparable quantities are the decisions, the p-values and the interval widths, which the reject, p_value and half_width columns report. The half_width column is the half-width of the confidence interval for the first contrast, or for the first coordinate when no contrast is given, at the stated level.

Methods that fail on the data, for example because a normalizer matrix is numerically singular, are reported with NA statistics and the reason in the status column rather than being dropped from the table.

Value

A data frame of class "aersn_comparison" with one row per method and columns method, label, statistic, statistic_scale, critical_value, p_value, p_mcse, reject, half_width, reference_family, reference, tuning and status. The estimate, null value and level are kept as attributes.

See Also

aersn_test(), aersn_normalizer().

Examples

set.seed(20)
n <- 300
e <- matrix(rnorm(2 * n), n, 2)
Y <- e
for (t in 2:n) Y[t, ] <- 0.4 * Y[t - 1, ] + e[t, ]
fit <- aersn_mean(Y, names = c("m1", "m2"))
aersn_compare(fit, null = c(0, 0), draws = 400, seed = 1)

Membership test for a confidence region

Description

Tests whether candidate parameter vectors belong to a confidence region, equivalently whether the point-null test does not reject them.

Usage

aersn_contains(region, v, tol = 1e-09)

Arguments

Value

A logical vector with attribute "gauge" holding the gauge of each candidate relative to the region (at most one inside).

Simultaneous intervals for linear contrasts

Description

Confidence intervals for the contrasts a_k^\top\theta, rows of a contrast matrix, obtained from the support function of the confidence region: a_k^\top\hat\theta_n \pm (c/\sqrt n)\,h_{K(\hat G_n)}(a_k), where h_K(a) is the adjusted range of the projected path a^\top\hat G_n (manuscript equation (projection-interval)).

Usage

aersn_contrast(
  object,
  contrasts,
  level = 0.95,
  type = c("simultaneous", "joint", "marginal"),
  method = "hull",
  ...,
  nu = NULL,
  reference = NULL,
  draws = NULL,
  seed = 1L,
  cores = 1L
)

Arguments

Value

A data frame with columns estimate, lower, upper, one row per contrast, with attributes type, level, critical.value, dimension and reference. For marginal inference, tuning_by_contrast records the tuning selected separately for each contrast.

Examples

set.seed(8)
fit <- aersn_mean(matrix(rnorm(300), 100, 3), names = c("a", "b", "c"))
A <- rbind("a - b" = c(1, -1, 0), "mean" = c(1, 1, 1) / 3)
aersn_contrast(fit, A, draws = 500)

Equal-weighted cosine (EWC) covariance estimate

Description

Estimates the long-run covariance matrix by averaging the outer products of the projections of the influence contributions on a cosine basis, and returns it as an aersn_normalizer(). EWC is a heteroskedasticity- and autocorrelation-robust (HAR) method in the fixed-smoothing class: the reference law holds the number of basis terms fixed as the sample grows, retaining the randomness of the covariance estimate through a scaled F distribution. The default number of terms increases with sample size; at each sample size the corresponding scaled F reference is used.

Usage

aersn_ewc_lrv(object, nu = NULL, center = TRUE)

Arguments

Details

With \nu basis terms the type II cosine functions are

\phi_j(t) = \sqrt{2/n}\,\cos\{\pi j (t - 1/2)/n\}, \qquad j = 1, \ldots, \nu,\ t = 1, \ldots, n,

which are orthonormal on the observation grid and exactly orthogonal to the constant, so demeaning the contributions does not change the estimate. The projections and the estimate are

\Lambda_j = \sum_{t=1}^n \phi_j(t)\, e_t, \qquad \hat\Omega_{\mathrm{EWC}} = \frac 1\nu \sum_{j=1}^\nu \Lambda_j \Lambda_j^\top .

The statistic for H_0: \theta_0 = v is z^\top\hat\Omega_{\mathrm{EWC}}^{-1}z with z = \sqrt n(\hat\theta_n - v).

Reference law. For a target of dimension m the statistic is referred to a scaled F law: \{(\nu - m + 1)/(\nu m)\}\,T is compared with F_{m,\,\nu - m + 1}, equivalently the critical value is \nu m/(\nu - m + 1) times the F_{m,\nu-m+1} quantile. This is the Hotelling form, and it requires \nu \ge m. For a linear restriction the dimension m of the restriction is used, not the full parameter dimension. The law is the fixed-smoothing asymptotic reference for dependent data; it is not an exact finite-sample distribution.

Number of terms. The default is \nu = \lfloor 0.4\, n^{2/3}\rfloor, the rule used in the manuscript's comparison, following the recommendation of Lazarus, Lewis, Stock and Watson (2018). Admissible values satisfy q \le \nu \le n - 1; the upper limit is the number of cosine terms that remain orthonormal on the grid. Larger \nu reduces the variance of the estimate and moves the reference law towards chi-squared, at the cost of a larger bias under strong dependence.

Value

An aersn_normalizer() object of class "aersn_normalizer_ewc", whose tuning records nu, the rule used and the centering flag.

References

Lazarus, E., Lewis, D. J., Stock, J. H. and Watson, M. W. (2018). HAR inference: recommendations for practice. Journal of Business and Economic Statistics, 36(4), 541-559.

See Also

aersn_normalizer(), aersn_hac_lrv(), aersn_fixed_b_normalizer(), aersn_compare().

Examples

set.seed(6)
fit <- aersn_mean(matrix(rnorm(600), 300, 2))
nz <- aersn_ewc_lrv(fit)
nz$tuning$nu
aersn_test(fit, method = "ewc")
aersn_test(fit, method = "ewc", nu = 20)

Bartlett fixed-b normalizer

Description

Estimates the long-run covariance matrix with the Bartlett kernel and a bandwidth proportional to the sample size, M = bn, and returns it as an aersn_normalizer(). Because the bandwidth fraction b is held fixed as the sample grows, the estimate does not converge to the long-run covariance matrix; the randomness that remains in the limit is carried by a nonstandard reference law that depends on b.

Usage

aersn_fixed_b_normalizer(object, b = 0.5)

Arguments

Details

The estimate is computed from the centered influence path through the partial-sum identity

\hat\Omega_b = \frac{2}{b_{\mathrm{grid}}}\cdot \frac 1n\sum_{k=1}^{n} \hat G_{n,k}\hat G_{n,k}^\top - \frac{1}{b_{\mathrm{grid}}}\cdot\frac 1n\sum_{k=0}^{n-m} \bigl(\hat G_{n,k}\hat G_{n,k+m}^\top + \hat G_{n,k+m}\hat G_{n,k}^\top\bigr),

which is algebraically the Bartlett estimator with bandwidth m = \mathrm{round}(bn) and weights 1-\ell/m for \ell < m, applied to the demeaned contributions. In aersn_hac_lrv() this corresponds to lag = m - 1.

Requested and realized b. The bandwidth in grid intervals must be an integer, so the realized fraction is b_{\mathrm{grid}} = m/n, which differs from the requested b by at most 1/(2n). The realized value is what enters the estimate, is recorded in tuning$b_grid, and is the value used to key and simulate the reference law. A request with \mathrm{round}(bn) = 0 is rejected rather than silently replaced.

Relation to quadratic self-normalization. At b = 1 we have m = n, the lagged term involves only the two endpoints of the path, which are zero, and the identity reduces to \hat\Omega_1 = 2V^{\mathrm Q}_n exactly in finite samples, where V^{\mathrm Q}_n is the aersn_shao_normalizer() matrix. The Wald statistic is therefore exactly half the quadratic self-normalized statistic, and the fixed-b reference law at b = 1 is exactly half the quadratic reference law, so the two tests reject on the same samples. The package tests verify both the matrix identity and the equality of the resulting decisions using common reference draws.

Other kernels. Only the Bartlett kernel is supported here, because the package supplies a matching fixed-b reference law only for that kernel. A Parzen or quadratic spectral estimate with a fixed bandwidth fraction would need its own reference law; use aersn_hac_lrv() with a numeric bandwidth if a consistent chi-squared reference is intended instead.

Value

An aersn_normalizer() object of class "aersn_normalizer_fixedb", whose tuning records the requested b, the realized b_grid, and the integer bandwidth m.

References

Kiefer, N. M. and Vogelsang, T. J. (2005). A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory, 21(6), 1130-1164.

See Also

aersn_normalizer(), aersn_shao_normalizer(), aersn_hac_lrv(), aersn_compare().

Examples

set.seed(5)
fit <- aersn_mean(matrix(rnorm(400), 200, 2))
nz <- aersn_fixed_b_normalizer(fit, b = 0.5)
nz$tuning$b_grid
## b = 1 gives exactly twice the quadratic self-normalizer
max(abs(aersn_fixed_b_normalizer(fit, b = 1)$matrix -
          2 * aersn_shao_normalizer(fit)$matrix))

Gauge of an increment hull, or the adjusted-range statistic

Description

The gauge \gamma_K(x) = \inf\{\lambda > 0: x \in \lambda K\} is the factor by which the increment hull must be enlarged to contain x. It is evaluated by the linear program of manuscript equation (gauge-lp), which uses only the n + 1 path values,

\gamma_K(x) = \max_{u, \ell}\ u^\top x \quad\text{subject to}\quad \ell \le u^\top \hat G_{n,j} \le \ell + 1,\ j = 0, \ldots, n.

For q = 1 the gauge is |x| divided by the adjusted range of the path (exact scalar reduction, equation (scalar-reduction)).

Usage

aersn_gauge(x, v, ...)

## S3 method for class 'aersn_hull'
aersn_gauge(x, v, details = FALSE, ...)

## S3 method for class 'aersn'
aersn_gauge(x, v, details = FALSE, ...)

## S3 method for class 'aersn_region'
aersn_gauge(x, v, details = FALSE, ...)

Arguments

Details

Applied to an aersn() object with a candidate parameter value v, the function returns the test statistic T_n(v) = \gamma_{K(\hat G_n)}\{\sqrt n(\hat\theta_n - v)\}. Applied to an aersn_region() object it returns the gauge relative to the region, so that values at most one indicate membership.

Value

A numeric vector of gauge values, or a list when details = TRUE.

Examples

set.seed(4)
Y <- matrix(rnorm(200), 100, 2)
fit <- aersn_mean(Y)
aersn_gauge(fit, c(0, 0))                    # statistic for H0: mean = 0
aersn_gauge(fit, c(0, 0), details = TRUE)$direction

Smooth generalized method of moments

Description

Adjusted-range inference for a target \theta = h(\beta) of a GMM estimator \hat\beta_n with moment contributions m_t(\beta) \in R^{d_m}, d_m \ge p. With \hat D = n^{-1}\sum_t \partial m_t(\hat\beta_n)/\partial\beta^\top, weighting matrix \hat W and \hat M = (\hat D^\top \hat W \hat D)^{-1}\hat D^\top \hat W, the influence contributions are

\hat\psi_{n,t} = -\hat{\dot h}_n \hat M m_t(\hat\beta_n),

manuscript Proposition "Feasible influence paths for smooth GMM". The matrix \hat M uses the Jacobian with respect to all coordinates of \beta, so the joint estimation of nuisance coefficients is retained in the contributions of the target.

Usage

aersn_gmm(
  moments,
  jacobian,
  estimate,
  weight = NULL,
  target = NULL,
  profile = NULL,
  names = NULL
)

Arguments

Details

The estimator must (approximately) solve the first-order condition \hat D^\top \hat W \bar m_n(\hat\beta_n) = o_p(n^{-1/2}); the function reports each component of this scaled condition divided by the sample standard deviation of the corresponding component of \hat D^\top\hat W m_t. It warns when the largest absolute standardized residual exceeds .Machine$double.eps^0.25 (about 1.2e-4). This numerical convergence check is not a statistical test or a verification of the asymptotic assumptions. The sufficient conditions are manuscript Assumption "Sufficient conditions for smooth GMM": stationarity and ergodicity of the moment process, \sqrt n-consistency, full column rank of D_m, consistent Jacobian and weighting matrix, smoothness of the moments, and a joint functional central limit theorem for the moments and their Jacobian. Under these conditions the feasible path is uniformly equivalent to the path based on the true influence function. This includes OLS, IV, and stacked local-projection systems; it does not cover non-smooth moments (for example quantile regression) or estimators with nonstandard rates.

Value

An aersn() object for the target, with model$foc holding the scaled first-order condition \sqrt n \hat D^\top\hat W\bar m_n. model$foc_standardized and model$foc_tolerance record the standardized residuals and the numerical warning threshold.

Examples

## Instrumental-variables example (synthetic): y = 1 + 0.5 x + u,
## x endogenous, instrument z.
set.seed(12)
n <- 300; z <- rnorm(n); v <- rnorm(n)
x <- z + v; u <- 0.5 * v + rnorm(n); y <- 1 + 0.5 * x + u
Zm <- cbind(1, z); Xm <- cbind(1, x)
beta <- solve(crossprod(Zm, Xm), crossprod(Zm, y))   # exactly identified IV
moments <- Zm * as.numeric(y - Xm %*% beta)          # z_t (y_t - x_t' beta)
D <- -crossprod(Zm, Xm) / n
fit <- aersn_gmm(moments, D, beta, target = 2, names = "slope")
aersn_test(fit, null = 0.5, draws = 2000)

HAC long-run covariance estimate

Description

Estimates the long-run covariance matrix of the influence contributions of an aersn() fit by kernel smoothing of the sample autocovariances, and returns it as an aersn_normalizer() so that tests, intervals and regions can be built from it exactly as for the other methods.

Usage

aersn_hac_lrv(
  object,
  kernel = c("Bartlett", "Parzen", "Quadratic Spectral"),
  bandwidth = "short",
  lag = NULL,
  prewhite = FALSE,
  adjust = FALSE,
  center = TRUE
)

Arguments

Details

Write e_t for the influence contributions, demeaned when center = TRUE. With kernel weights w_\ell = w(\ell/h) and bandwidth h,

\hat\Omega = \hat\Gamma_0 + \sum_{\ell = 1}^{n-1} w_\ell (\hat\Gamma_\ell + \hat\Gamma_\ell^\top), \qquad \hat\Gamma_\ell = \frac 1n \sum_{t=1}^{n-\ell} e_t e_{t+\ell}^\top .

The statistic for H_0: \theta_0 = v is z^\top\hat\Omega^{-1}z with z = \sqrt n(\hat\theta_n - v), and the reference law is \chi^2_q (or \chi^2_m for an m-dimensional linear restriction). All n-1 lags are used; the Bartlett and Parzen weights vanish beyond the bandwidth, while the quadratic spectral kernel has unbounded support and is not truncated.

Kernels. "Bartlett" gives w(x) = 1 - |x| for |x| \le 1; "Parzen" gives 1 - 6x^2 + 6|x|^3 for |x| \le 1/2 and 2(1 - |x|)^3 for 1/2 < |x| \le 1; "Quadratic Spectral" gives 3\{\sin(y)/y - \cos(y)\}/y^2 with y = 6\pi x/5. These match sandwich::kweights().

Bandwidth. Distinguish three quantities that are easy to confuse. The real-valued bandwidth h enters through w(\ell/h). A lag truncation L for the Bartlett kernel corresponds to h = L + 1, since w_\ell = 1 - \ell/(L+1) is then zero at \ell = L+1. The fixed-b fraction b of aersn_fixed_b_normalizer() is a different parameterisation again. bandwidth accepts:

• a positive number, used directly as h;

• "short": L = \lfloor 4(n/100)^{2/9}\rfloor, h = L + 1, the rule used for the Newey-West comparison in the manuscript;

• "long": L = \lceil 1.3\sqrt n\,\rceil, h = L + 1;

• "andrews": the plug-in bandwidth of Andrews (1991) for the chosen kernel, from sandwich::bwAndrews() with equal weights on the coordinates. For the Parzen and quadratic spectral kernels the returned bandwidths satisfy h_{\mathrm{Parzen}} = h_{\mathrm{QS}}\times 2.6614/1.3221 exactly, the ratio of Andrews' kernel constants; the manuscript's replication code uses that identity and the package checks it in its tests.

• "newey-west": the lag-selection rule of Newey and West (1994), from sandwich::bwNeweyWest(), applied as L = \lfloor bw \rfloor and h = L + 1. This is the convention of sandwich::NeweyWest(), which applies Bartlett weights; it is therefore supported for the Bartlett kernel only, and other kernels raise an error rather than silently using a different rule.

Alternatively, lag sets the Bartlett or Parzen lag truncation directly, giving h = L + 1; it is not available for the quadratic spectral kernel, whose weights do not vanish at any finite lag.

Prewhitening. prewhite = TRUE applies the Andrews-Monahan procedure: a VAR(1) is fitted to the contributions by least squares, the kernel estimate is formed from its residuals, and the result is recolored as (I - \hat A)^{-1}\hat\Omega_e (I - \hat A)^{-\top}. Bandwidth selection then uses the residuals, and the scaling divisor stays n although there are n - 1 residuals, which is the convention of sandwich::vcovHAC(). This is VAR prewhitening of the HAC estimator and is unrelated to the lag-zero LDL partial prewhitening of aersn_ldl_normalizer(). The default FALSE reproduces the manuscript's comparison.

Finite-sample adjustment. adjust = TRUE multiplies the estimate by n/(n-q), where q is the target dimension, not the total number of fitted coefficients. For a selected target this can differ from sandwich::meatHAC() applied to the full model. The default FALSE reproduces the manuscript's comparison. The influence contributions already absorb the estimation of nuisance parameters, so this multiplier is a finite-sample convention rather than a correction derived for this framework.

Value

An aersn_normalizer() object of class "aersn_normalizer_hac". tuning records the kernel, the bandwidth rule, the realized numeric bandwidth, the implied Bartlett lag truncation where that is meaningful, and the prewhitening, adjustment and centering flags.

References

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817-858.

Newey, W. K. and West, K. D. (1994). Automatic lag selection in covariance matrix estimation. Review of Economic Studies, 61(4), 631-653.

Andrews, D. W. K. and Monahan, J. C. (1992). An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica, 60(4), 953-966.

See Also

aersn_normalizer(), aersn_fixed_b_normalizer(), aersn_ewc_lrv(), aersn_compare().

Examples

set.seed(4)
n <- 300
e <- matrix(rnorm(2 * n), n, 2)
Y <- e
for (t in 2:n) Y[t, ] <- 0.5 * Y[t - 1, ] + e[t, ]
fit <- aersn_mean(Y)
aersn_hac_lrv(fit, kernel = "Bartlett", bandwidth = "short")
aersn_hac_lrv(fit, kernel = "Quadratic Spectral", bandwidth = "andrews")
aersn_test(fit, method = "hac", kernel = "Parzen", bandwidth = "andrews")

Increment-hull self-normalizer

Description

Constructs the affine-equivariant normalizing set of the manuscript: the convex hull of all increments of the centered influence path,

K(\hat G_n) = \mathrm{conv}\{\hat G_n(r) - \hat G_n(s): 0 \le r, s \le 1\},

manuscript equation (increment-hull). The set is compact, convex and centrally symmetric. Its support function in direction u is the adjusted range of the projected path, h_K(u) = \max_j u^\top \hat G_{n,j} - \min_j u^\top \hat G_{n,j} (equation (support-range)), and its gauge has the dual representation \gamma_K(x) = \sup_{u \ne 0} |u^\top x| / h_K(u) (equation (gauge-dual)). The hull is never stored as a list of pairwise increments; it is represented by the path itself, and the gauge is evaluated by the linear program of equation (gauge-lp).

Usage

aersn_hull(path, n_directions = 1000L)

Arguments

Details

The object records diagnostics recommended in the Supplement (Section "Multivariate critical values and affine transformations"): the singular values of the ⁠(n + 1) x q⁠ path matrix, its numerical rank after dividing each nonconstant coordinate by its adjusted range (threshold max(n + 1, q) * s_1 * eps on this scaled path), the condition number s_1 / s_q, and the minimum, maximum and spread of the projected adjusted range over the coordinate axes and a fixed set of n_directions prespecified unit directions. If the numerical rank is below q the hull is not full dimensional and joint statistics are not evaluated (Algorithm 1, step 2). Scaling prevents units alone from causing numerical rank deficiency; unscaled diagnostics remain available as singular_values, raw_rank and condition. The scanned minimum range is an upper bound on the all-direction minimum.

Value

An object of class "aersn_hull" with elements path, G, n, q, singular_values, rank, full_rank, condition, coordinate_ranges, min_projected_range, max_projected_range, projected_range_spread, and n_directions.

References

Hong, Lin, Linton, Newey and Sun (2026), Sections 2.2-2.3 and Supplement Section "Multivariate critical values and affine transformations".

See Also

aersn_gauge(), aersn_support(), aersn_path().

Examples

set.seed(3)
Y <- matrix(rnorm(300), 100, 3)
hull <- aersn_hull(aersn_path(sweep(Y, 2, colMeans(Y))))
hull
aersn_support(hull, c(1, 0, 0))             # adjusted range of column 1
aersn_gauge(hull, c(0.5, -0.2, 0.1))

Hull normalizer

Description

Wraps the increment hull of an aersn() fit as an aersn_normalizer() object, so that the affine-equivariant adjusted-range method can be handled by the same inference and comparison functions as the other methods. The hull itself is computed by aersn_hull() when the fit is created; this function does not recompute it.

Usage

aersn_hull_normalizer(object)

Arguments

Value

An "aersn_normalizer" object with family = "geometric".

See Also

aersn_normalizer(), aersn_hull().

Examples

set.seed(1)
fit <- aersn_mean(matrix(rnorm(200), 100, 2))
aersn_hull_normalizer(fit)

LDL factorization of a symmetric positive-definite matrix

Description

Factorizes S = L D L^\top with L unit lower triangular and D diagonal with positive entries. The factorization is computed from the Cholesky factor as L = C \mathrm{diag}(C)^{-1} and D = \mathrm{diag}(C)^2, where C is the lower Cholesky factor, so that C = L D^{1/2}.

Usage

aersn_ldl(S, what = "matrix")

Arguments

Details

No ridge term, pseudo-inverse or nearest-positive-definite projection is applied. If the matrix is not numerically positive definite the function stops and reports the smallest eigenvalue, because a silently modified factorization would change the statistic without changing its name.

Value

A list with L (unit lower triangular), D (the diagonal of D, a vector), chol (the lower Cholesky factor C) and reconstruction_error, the maximum absolute deviation of L D L^\top from the symmetrized input.

Examples

S <- crossprod(matrix(c(2, 1, 0, 1, 3, 1, 0, 1, 4), 3, 3))
f <- aersn_ldl(S)
f$L
max(abs(f$L %*% diag(f$D) %*% t(f$L) - S))
max(abs(f$chol - f$L %*% diag(sqrt(f$D))))

Componentwise adjusted-range normalizer after LDL partial prewhitening

Description

Implements the componentwise adjusted-range self-normalizer of Hong, Linton, McCabe, Sun and Wang (2024), as described in the Supplement of the package's main reference. The influence contributions are transformed by the inverse of the unit lower triangular factor of their sample lag-zero covariance, which removes contemporaneous cross dependence, and each transformed coordinate is normalized by its own adjusted range. The joint statistic is the sum of the squared componentwise ratios.

Usage

aersn_ldl_normalizer(object, order = NULL)

Arguments

Details

Write \hat\Gamma_0 for the sample lag-zero covariance of the influence contributions, factorized as \hat\Gamma_0 = \hat L \hat D \hat L^\top with \hat L unit lower triangular (aersn_ldl()). With z = \sqrt n(\hat\theta_n - v), transformed error \tilde z = \hat L^{-1} z and transformed path \tilde G_n = \hat L^{-1}\hat G_n, the statistic is

T^{\mathrm{comp}}_n(v) = \sum_{j=1}^q \left(\frac{|\tilde z_j|}{\max_k \tilde G_{n,k,j} - \min_k \tilde G_{n,k,j}}\right)^2 ,

the range being taken over the observation grid including the origin. This is the Wald form z^\top V^{-1} z with V = \hat L \,\mathrm{diag}(R_j^2)\, \hat L^\top, where R_j is the adjusted range of transformed coordinate j, so the confidence region is an ellipsoid in the original parameter coordinates and linear contrasts are obtained from V in the usual way.

Cholesky and LDL give the same statistic. Writing the lower Cholesky factor as \hat C = \hat L \hat D^{1/2}, the transformed error becomes \hat D^{-1/2}\tilde z and transformed coordinate j of the path is divided by the same \hat D_{jj}^{1/2}, so each ratio is unchanged and the diagonal scale cancels exactly. The function returns both factors and reports the difference between the two statistics as a diagnostic.

Conditions. The reference law used for this statistic treats the q componentwise ratios as independent. Diagonalizing the lag-zero covariance does not in general diagonalize the limiting long-run covariance: the independent-component law applies when \hat L^{-1}\Omega \hat L^{-\top} is diagonal in the limit, where \Omega is the long-run covariance of the influence contributions. The prewhitening is partial because serial and lead-lag dependence remain. The statistic depends on the order of the coordinates, and it is not affine equivariant: a rotation or shear of the parameters changes its value. Use method = "hull" when equivariance under general nonsingular reparameterization is required.

Value

An aersn_normalizer() object of class "aersn_normalizer_ldl". In addition to the common fields it carries lag_zero (the sample lag-zero covariance in factorization order), ldl (the factorization in that same order), transform (the matrix applying the coordinate permutation followed by \hat L^{-1} to the error and path in their original column order), transformed_path, component_ranges and, in diagnostics, the reconstruction error, the maximum absolute off-diagonal element of the transformed lag-zero covariance, and the Cholesky agreement check.

References

Hong, Y., Linton, O., McCabe, B., Sun, J. and Wang, S. (2024). Kolmogorov-Smirnov type testing for structural breaks: a new adjusted-range based self-normalization approach. Journal of Econometrics, 238(2), 105603. doi:10.1016/j.jeconom.2023.105603

See Also

aersn_normalizer(), aersn_ldl(), aersn_compare().

Examples

set.seed(2)
fit <- aersn_mean(matrix(rnorm(600), 200, 3))
nz <- aersn_ldl_normalizer(fit)
nz$ldl$L
nz$component_ranges
aersn_test(fit, method = "ldl", draws = 400, seed = 1)

Linear regression by ordinary least squares

Description

Adjusted-range inference for regression coefficients. OLS is the exactly identified GMM estimator with moments m_t(b) = z_t(y_t - z_t^\top b); its influence contributions are \hat\psi_{n,t} = \hat{\dot h}_n(\hat Q^{-1} z_t \hat u_t) with \hat Q = n^{-1}\sum_t z_t z_t^\top and residuals \hat u_t (manuscript Appendix B, OLS example). The function forms the moments and Jacobian from a fitted stats::lm() object and calls aersn_gmm().

Usage

aersn_lm(object, ..., target = NULL, profile = NULL, names = NULL)

Arguments

Details

Weighted least squares, generalized linear models, multiple-response models and models with aliased (NA) coefficients are not supported. Observations removed by na.action (including na.exclude) are dropped before the path is formed; the function warns because the remaining rows are then treated as consecutive in time. Validity requires the conditions of aersn_gmm() for the moments z_t u_t, in particular a functional central limit theorem for their partial sums and positive-definite E(z_t z_t^\top).

Value

An aersn() object.

Examples

set.seed(13)
n <- 250
x <- arima.sim(list(ar = 0.5), n)
u <- arima.sim(list(ar = 0.5), n)
y <- 1 + 0.8 * x + u
fit <- aersn_lm(y ~ x, target = "x")
aersn_test(fit, null = 0.8, draws = 2000)
fit2 <- aersn_lm(y ~ x)                           # both coefficients
confint(fit2, draws = 1000)

Sample mean of a scalar or vector time series

Description

Adjusted-range inference for the mean. The estimate is the sample mean and the influence contributions are the demeaned observations, \hat\psi_{n,t} = Y_t - \bar Y_n (manuscript Section 2.1). The centered path is the scaled cumulative sum of the demeaned series.

Usage

aersn_mean(Y, profile = NULL, names = NULL)

Arguments

Details

Validity requires the functional central limit theorem for the partial sums of Y_t - \theta_0 with nonsingular long-run covariance (manuscript Assumption 1(i)); the representation is exact and the feasible path equals the infeasible centered path, so Assumption 1(ii)-(iii) hold automatically.

Value

An aersn() object.

Examples

set.seed(11)
x <- arima.sim(list(ar = 0.5), 300)
fit <- aersn_mean(x)
aersn_test(fit, 0, reference = "continuous")
confint(fit, draws = 5000)

Method identifiers

Description

The character identifiers accepted by the method argument of aersn_test(), confint.aersn(), aersn_contrast(), aersn_region() and aersn_compare(), in the order used by comparison tables.

Usage

aersn_methods()

Details

• "hull": the affine-equivariant adjusted-range increment hull, the method of the package's main reference and the default everywhere.

• "ldl": componentwise adjusted ranges after lag-zero partial prewhitening by an LDL factorization (aersn_ldl_normalizer()).

• "shao": quadratic self-normalization (aersn_shao_normalizer()).

• "hac": kernel long-run covariance estimation (aersn_hac_lrv()).

• "fixedb": Bartlett fixed-b (aersn_fixed_b_normalizer()).

• "ewc": equal-weighted cosine (aersn_ewc_lrv()).

"hull", "ldl" and "shao" are self-normalized: the normalizer is a random object with a nondegenerate limit and the reference law is nonstandard. "hac" uses a consistent covariance estimate with a chi-squared reference. "fixedb" and "ewc" use inconsistent covariance estimates with fixed-smoothing reference laws.

Value

A character vector of method identifiers.

Examples

aersn_methods()

Conditional likelihood scores

Description

Adjusted-range inference for a target \theta = h(\beta) of a conditional maximum-likelihood (or M-) estimator from its fitted scores and normalized negative Hessian. The influence contributions are

\hat\psi_{n,t} = \hat{\dot h}_n \hat J_n^{-1} s_t(\hat\beta_n), \qquad \hat J_n = n^{-1}J_{n,n}(\hat\beta_n),

manuscript Appendix A.2 (equation (fitted-score-clock) and the text following it). Because the fitted scores sum to zero, the calendar-time and profile-centered paths coincide; a profile changes only the reference grid used for matched-grid quantiles.

Usage

aersn_mle(
  scores,
  neg_hessian,
  estimate,
  target = NULL,
  profile = "calendar",
  names = NULL
)

Arguments

Details

With profile = "opg" the variance-accumulation profile is estimated from cumulative score outer products by aersn_opg_profile(). This is justified when the scores form a martingale-difference array and the cumulative conditional score variance and cumulative expected negative Hessian accumulate the same fraction \tau(r) of their full-sample matrices (Supplement, Propositions "Multivariate fitted scores under a common variance-accumulation profile" and "Feasible variance-accumulation profile from score outer products"). Under correct specification and information equality the two profiles agree. For serially correlated scores the outer-product profile is not valid and calendar-time centering should be used. The proportionality measure returned in model$proportionality describes departures from a common scalar profile.

Value

An aersn() object.

Examples

## Gaussian linear model with known variance: the score interface
## reproduces the OLS influence contributions.
set.seed(14)
n <- 200; x <- rnorm(n); y <- 0.5 + x + rnorm(n)
X <- cbind(1, x); b <- solve(crossprod(X), crossprod(X, y))
s <- X * as.numeric(y - X %*% b)          # scores with sigma^2 = 1
J <- crossprod(X) / n
fit <- aersn_mle(s, J, b, target = 2, names = "slope")
confint(fit, draws = 4000)
fit_opg <- aersn_mle(s, J, b, target = 2, profile = "opg")
fit_opg$model$proportionality

Construct the normalizer used by an inference method

Description

Builds the object that a method uses to normalize the estimation error: the increment hull for "hull", or a q by q positive-definite matrix for the other five methods. The inference functions call this internally; it is exported so that a normalizer can be inspected, reused across several tests, or built once and passed to aersn_test() and friends. A prebuilt normalizer is tied to its influence contributions and profile. For a different data set or a lower-dimensional target, pass a method name and tuning arguments to construct the normalizer for that target.

Usage

aersn_normalizer(object, method = "hull", ...)

Arguments

Details

Method-specific tuning arguments are passed through ... and validated by the individual constructors:

• "hull": none.

• "ldl": order.

• "shao": integration.

• "hac": kernel, bandwidth, lag, prewhite, adjust, center.

• "fixedb": b.

• "ewc": nu, center.

Passing an unknown tuning argument is an error rather than a silent default, so that a misspelled kernel or bandwidth rule cannot be ignored.

Value

An object of class "aersn_normalizer". Common fields are method, label, family ("geometric" or "quadratic"), scale ("gauge" or "wald"), reference_family, q, n, names, matrix (the long-run covariance or quadratic normalizer, NULL for the hull), hull (for the hull method), tuning (the realized tuning values) and diagnostics.

See Also

aersn_ldl_normalizer(), aersn_shao_normalizer(), aersn_hac_lrv(), aersn_fixed_b_normalizer(), aersn_ewc_lrv(), aersn_compare().

Examples

set.seed(1)
fit <- aersn_mean(matrix(rnorm(300), 150, 2))
aersn_normalizer(fit, "shao")
aersn_normalizer(fit, "hac", kernel = "Parzen", bandwidth = "andrews")

Feasible variance-accumulation profile from score outer products

Description

Estimates the variance-accumulation profile \tau from fitted conditional-likelihood scores by the normalized trace of cumulative outer products, manuscript equation (opg-trace-clock):

\hat\tau_n(k/n) = p^{-1}\,\mathrm{tr}\big(\hat\Sigma_{s,n}^{-1/2} \hat\Sigma_{s,k}\hat\Sigma_{s,n}^{-1/2}\big), \qquad \hat\Sigma_{s,k} = n^{-1}\sum_{t \le k} s_t s_t^\top .

The estimate is nondecreasing, lies in [0, 1], and has endpoints zero and one whenever \hat\Sigma_{s,n} is positive definite.

Usage

aersn_opg_profile(scores)

Arguments

Details

The estimator is justified (Supplement, Proposition "Feasible variance-accumulation profile from score outer products") for martingale-difference conditional scores whose cumulative conditional variance and cumulative expected negative Hessian accumulate the same fraction \tau(r) of their full-sample matrices. For serially correlated influence contributions, lag-zero outer products do not estimate cumulative long-run covariance, and this profile is not valid. The returned proportionality measure \mathcal E_n = \max_k p^{-1/2}\|\hat\Xi_{s,k} - \hat\tau_n(k/n)I_p\|_F (manuscript equation (clock-proportionality-measure)) is descriptive: large values indicate that the accumulation is not proportional across score components, so a common scalar profile is doubtful.

Value

A numeric vector of length n + 1 (the nodes, with attribute "centering" = "profile"), with attributes "proportionality" (\mathcal E_n) and "score_sum" (the scaled full-sample score n^{-1/2}\sum_t s_t, which should be close to zero for fitted scores).

See Also

aersn_mle(), aersn_profile().

Examples

set.seed(2)
s <- matrix(rnorm(400), 200, 2)      # synthetic scores
s <- sweep(s, 2, colMeans(s))         # fitted scores sum to zero
tau <- aersn_opg_profile(s)
range(diff(tau)); attr(tau, "proportionality")

Parametric reference law for a Wald statistic

Description

Builds an aersn_reference() object for the two comparator methods whose reference law is available in closed form: the chi-squared law used with a consistent HAC covariance estimate, and the scaled F law used with the equal-weighted cosine estimate.

Usage

aersn_parametric_reference(law = c("chisq", "ewc_F"), df, nu = NULL)

Arguments

Details

For law = "chisq" the statistic is referred to \chi^2_{df}. For law = "ewc_F" the critical value is \nu\,df/(\nu - df + 1) times the F_{df,\ \nu - df + 1} quantile, equivalently \{(\nu - df + 1)/(\nu\,df)\}\,T \sim F_{df,\nu-df+1}. Both are asymptotic reference laws for dependent observations, not exact finite-sample distributions.

Value

An object of class "aersn_reference" with type = "parametric".

See Also

aersn_reference(), aersn_hac_lrv(), aersn_ewc_lrv().

Examples

ref <- aersn_parametric_reference("chisq", df = 3)
aersn_critical_value(ref, 0.95)
aersn_parametric_reference("ewc_F", df = 2, nu = 20)

Centered influence path

Description

Computes the centered partial-sum path of estimated influence contributions on the observation grid, including the initial and final zero points. This is the object whose increment hull normalizes the estimation error.

Usage

aersn_path(psi, profile = NULL, names = NULL)

Arguments

Details

With contributions \hat\psi_{n,t} \in R^q in the rows of psi and profile nodes \tau_k = \tau(k/n), the path is

\hat G_n(k/n) = n^{-1/2}\Big\{\sum_{t=1}^k \hat\psi_{n,t} - \tau_k \sum_{t=1}^n \hat\psi_{n,t}\Big\}, \qquad k = 0, \ldots, n,

manuscript equations (feasible-path) and (feasible-clock-path). Under calendar-time centering \tau_k = k/n. The empty sum at k = 0 is zero, so \hat G_n(0) = \hat G_n(1) = 0. The scaling by n^{-1/2} uses n = nrow(psi), which must be the number of contributions in the asymptotically linear representation.

Centering subtracts \tau_k times the full-sample sum, not the sample mean of the contributions at each date; the two coincide only under calendar-time centering. If the contributions sum to zero (fitted scores or least-squares moments), the second term vanishes and the profile does not change the path; it still changes the reference grid used by aersn_reference().

Value

An object of class "aersn_path": a list with the ⁠(n + 1) x q⁠ matrix G of path values, n, q, the profile nodes, the centering label, and total, the full-sample sum of the contributions.

See Also

aersn_hull(), aersn().

Examples

set.seed(1)
Y <- matrix(rnorm(60), 30, 2)
path <- aersn_path(sweep(Y, 2, colMeans(Y)))
path$G[c(1, 31), ]   # zero at both ends
plot(path)

Variance-accumulation profile on the observation grid

Description

Validates, or constructs from a function, the nodes \tau(k/n), k = 0, \ldots, n, of a variance-accumulation profile used for profile centering of the influence path. The profile must be continuous and nondecreasing from \tau(0) = 0 to \tau(1) = 1; the package only checks these properties on the grid.

Usage

aersn_profile(profile, n, tol = 1e-10)

Arguments

Details

Under calendar-time centering the nodes are k/n. The manuscript (Section 4, "Nonuniform Variance Accumulation") replaces k/n by \tau(k/n) when the cumulative long-run covariance satisfies the fixed-shape condition C(r) = \tau(r)\,C(1). Two additional conditions are required for a feasible profile-centered path (manuscript Assumption "Feasible scalar time change"):

• the estimated profile belongs to \{(t_0, \ldots, t_n): 0 = t_0 \le \cdots \le t_n = 1\} and is uniformly consistent for \tau; and

• the increment hull of the feasible profile-centered path converges in Hausdorff distance to the hull of the infeasible one.

Consistency of the profile alone does not deliver the second condition: for the sample mean, the fitted contributions sum to zero, so substituting any profile leaves the calendar-centered path unchanged, and the resulting path differs from the required one by \{\tau(r) - r\}S_n(1) (Supplement, Section "Cumulative information and profile-based reindexing"). The manuscript establishes feasibility for conditional-likelihood scores whose cumulative score variance and expected negative Hessian accumulate the same fraction of their full-sample matrices; see aersn_mle() and aersn_opg_profile(). The package does not estimate a valid common profile for arbitrary dependent data.

Value

A numeric vector of length n + 1 with attribute "centering" equal to "calendar" or "profile".

Examples

aersn_profile(NULL, 4)
aersn_profile(function(r) r^3, 4)
aersn_profile(c(0, 0.1, 0.3, 0.7, 1), 4)

Two-dimensional projections and slices of a confidence region

Description

aersn_projection() returns the polygon of the orthogonal projection of the region onto two coordinates. Because projection is linear, the projection of \hat\theta_n + s K(\hat G_n) is the same construction applied to the two-dimensional sub-path, so it is exact and has simultaneous coverage. aersn_slice() returns the boundary of the intersection of the region with the plane through the center in which all other coordinates are held at their estimates; each boundary point is \hat\theta_n + s\,d/\gamma_K(d) for a direction d in the plane, evaluated by the gauge linear program. A slice is a cross-section, not a confidence set for the two coordinates.

Usage

aersn_projection(region, coords = c(1L, 2L))

aersn_slice(region, coords = c(1L, 2L), n_angles = 180L)

Arguments

Value

A numeric matrix with two columns (polygon vertices in order).

Reference distribution of a self-normalized statistic

Description

Returns the reference law of the statistic T_q = \gamma_{K(B_q)}(Z_q), where B_q is a standard q-dimensional Brownian bridge and Z_q is an independent standard normal vector (manuscript equation (brownian-reference)). With grid = "matched" the bridge is evaluated on the same grid as the sample path (n intervals, or the supplied profile nodes), which gives the matched-grid reference statistic T_q^{(n)} whose quantiles the manuscript uses as critical values. With grid = "continuous" and q = 1 the closed-form law of aersn_scalar_cdf() is used.

Usage

aersn_reference(
  x,
  n = NULL,
  draws = NULL,
  seed = 1L,
  grid = c("matched", "continuous"),
  nodes = NULL,
  batches = 20L,
  cores = 1L,
  cache = TRUE,
  verbose = FALSE,
  statistic = "hull_gauge",
  args = list()
)

Arguments

Details

Each Monte Carlo draw simulates Brownian increments with variances equal to the node spacings, forms the bridge W_q(\tau_j) - \tau_j W_q(1) at the nodes, and evaluates the gauge of the piecewise-linear bridge at an independent standard normal vector with the same linear program used for the sample statistic (aersn_gauge()). For nested grids the finite-grid statistic decreases to the continuous-path statistic (Supplement, Lemma "Observed grids and continuous paths"), so matched-grid quantiles exceed continuous-path quantiles. Corollary "Validity of matched-grid critical values" in the Supplement covers uniform and nonuniform deterministic grids and, when the grid is the estimated profile, conditional quantiles given the sample.

Draws are generated in fixed-size chunks with chunk-specific seeds derived from seed, using the Mersenne-Twister generator with inversion for normal variates; the user's random-number state is saved and restored. Results are therefore reproducible for a given seed, draws, q and grid, independently of cores. Monte Carlo standard errors of quantiles are estimated from batches interleaved batches of draws. Objects are cached in the session by a key that contains every setting that changes the law (statistic, q, grid nodes, draws, seed, chunk size, and the linear-program settings), together with batches. The grid must contain at least q + 1 positive increments. If any draw fails numerically, simulation stops without returning or caching a reference; failed draws are never discarded to estimate quantiles. Matching the grid approximates the reference law and does not make inference finite-sample exact for general dependent observations.

Computational cost. Scalar matched-grid simulation stores several matrices with n columns and up to 2000 rows per worker. Peak memory therefore grows with n and min(draws, 2000); decreasing draws while keeping it above 2000 mainly reduces runtime, not peak memory. For orientation, a run with n = 20000, 50000 draws and one worker used about 3 GB of memory and one minute on an arm64 Mac; requirements vary with the platform. Fewer than 2000 draws reduces peak memory but increases Monte Carlo uncertainty. For q = 1, grid = "continuous" avoids simulation when a continuous-path approximation is appropriate; it is a different reference law, not the matched-grid calculation.

Value

An object of class "aersn_reference". For Monte Carlo references the element draws holds the simulated statistics in generation order (and signed the signed ratios when q = 1). Use quantile(), aersn_critical_value(), aersn_pvalue() and aersn_mcse() to query it.

See Also

aersn_scalar_cdf(), aersn_registry.

Examples

ref1 <- aersn_reference(1, grid = "continuous")
quantile(ref1, c(0.90, 0.95, 0.99))
ref2 <- aersn_reference(2, n = 50, draws = 400, seed = 1)
ref2
aersn_pvalue(ref2, 2.4)

Joint confidence region

Description

The level-(1-\alpha) confidence region \mathcal C_n = \hat\theta_n + (c^{(n)}_{q,1-\alpha}/\sqrt n) K(\hat G_n) (manuscript equation (confidence-region)). The region is a translated and scaled copy of the increment hull; it is convex, centrally symmetric about the estimate, generally not an ellipsoid, and affine equivariant. It is represented by its center, scale, and hull; membership is evaluated through the gauge and projections through the support function. Explicit vertices are available for q = 2 (aersn_vertices()), and two-dimensional projections or slices for plotting in any dimension (plot.aersn_region()).

Usage

aersn_region(
  object,
  level = 0.95,
  method = "hull",
  ...,
  nu = NULL,
  reference = NULL,
  draws = NULL,
  seed = 1L,
  cores = 1L
)

Arguments

Value

An object of class "aersn_region" with elements center, scale (c/\sqrt n), critical.value, level, hull, q, n, names and reference.

See Also

aersn_contains(), aersn_support(), aersn_gauge(), aersn_vertices(), plot.aersn_region().

Examples

set.seed(9)
fit <- aersn_mean(matrix(rnorm(400), 200, 2))
reg <- aersn_region(fit, draws = 1000)
reg
aersn_contains(reg, rbind(c(0, 0), c(0.5, 0.5)))
plot(reg)

Verified matched-grid critical values from the manuscript replication package

Description

Tabulated matched-grid quantiles of the increment-hull gauge statistic T_q^{(n)} taken from the replication package of Hong, Lin, Linton, Newey and Sun (2026), version of 8 September 2026. They are included as regression targets for the package's own simulations and as a fast tabulated reference (aersn_test() with reference = "registry").

Usage

aersn_registry

Format

A data frame with one row per (dimension, grid size, probability):

q

parameter dimension (1, 2, 3, 5, 10)

n

number of grid intervals (200, 500, 738, 1000)

prob

quantile level (0.90, 0.95, 0.975, 0.99)

cv

matched-grid quantile (type-8 sample quantile of the draws)

mcse

Monte Carlo standard error from 100 interleaved batches

draws

number of Monte Carlo draws (2,000,000 for q = 1, 30,000 otherwise)

batches

number of batches used for mcse

quantile_type

quantile type (8)

seed

master seed recorded in the production run manifest

source

file of the replication package the row was taken from

Details

The q = 1 rows were produced by scalar_matched_range_quadratic_cv.R (endpoint an independent standard normal; bridge the equispaced linear interpolant; range including the zero at the origin). The q >= 2 rows were produced by multivariate_matched_cv.R with the same dual linear program used by aersn_gauge(). The production run manifests record master seeds 20260715 and 20260717 and the replication counts stored in the draws column (runs completed 15 July 2026). These tabulated simulation results are distributed under this package's MIT license.

Source

Replication package, files core_methods/inputs/matched_scalar_critical_values.csv and core_methods/inputs/multivariate_matched_cv_registry.csv.

Examples

subset(aersn_registry, prob == 0.95)

Closed-form scalar adjusted-range reference law

Description

Distribution function, quantile function and density of the continuous-path scalar reference law. Let Z \sim N(0, 1) and let R be the range (supremum minus infimum) of an independent standard Brownian bridge on [0, 1]. The signed ratio M = Z / R is the limit of the signed scalar statistic, and |M| is the limit of the two-sided statistic T_n(\theta_0) for q = 1 (manuscript Theorem 1 with q = 1, equation (scalar-reduction)).

Usage

aersn_scalar_cdf(c, method = c("auto", "bessel", "series"), tol = 1e-13)

aersn_scalar_quantile(p, tol = 1e-12)

aersn_scalar_density(u, method = c("auto", "bessel", "series"), tol = 1e-11)

Arguments

Details

The Supplement (Proposition "Density and distribution") gives

f_M(u) = \tfrac12 - 12u^2\sum_{k \ge 1}\frac{k^2}{(u^2 + 4k^2)^{5/2}}, \qquad \Pr(|M| \le c) = c - 2c^3\sum_{k \ge 1}(c^2 + 4k^2)^{-3/2},

with truncation errors at most c^3/(8N^2) for the distribution function and 3U^2/(16N^2) for the density on |u| \le U after N terms. Poisson summation gives the exponentially convergent tail representation

\Pr(|M| > c) = 2\pi c^2\sum_{k \ge 1} k\,K_1(\pi k c),

where K_1 is the modified Bessel function of the second kind (Supplement, equation (bessel-tail)), and differentiating it gives f_M(c) = \pi^2 c^2\sum_k k^2 K_0(\pi k c) - \pi c\sum_k k K_1(\pi k c). The "auto" method uses the Bessel representations except very close to zero, where the direct series is used; the two representations are cross-checked in the package tests.

The quantile function inverts the distribution function of |M| with stats::uniroot(). Continuous-path critical values at two-sided levels 10, 5 and 1 percent are 1.397390, 1.705776 and 2.367378 (Supplement Table "Continuous-path scalar critical values").

These are continuous-path laws. For a sample of size n, the manuscript uses matched-grid quantiles, which are larger; see aersn_reference() with grid = "matched".

Value

Numeric vectors: probabilities \Pr(|M| \le c), quantiles of |M|, or density values of M.

See Also

aersn_reference().

Examples

aersn_scalar_cdf(c(1.397390, 1.705776, 2.367378))
aersn_scalar_quantile(c(0.90, 0.95, 0.99))
integrate(aersn_scalar_density, -Inf, Inf)$value   # integrates to one

Quadratic self-normalizer

Description

Builds the quadratic self-normalizer of Shao (2010) from the centered influence path of an aersn() fit. The normalizer is the discretized integral of the outer product of the path, and the statistic is the Wald form with that matrix. Like the increment hull it needs no bandwidth, kernel or block length, and its reference law is nonstandard.

Usage

aersn_shao_normalizer(object, integration = c("auto", "calendar", "profile"))

Arguments

Details

With the centered path \hat G_n(k/n), k = 0, \ldots, n, the calendar-time normalizer is

V^{\mathrm Q}_n = \frac 1n \sum_{k=1}^{n} \hat G_n(k/n)\,\hat G_n(k/n)^\top ,

the origin contributing zero. The statistic for H_0: \theta_0 = v is z^\top (V^{\mathrm Q}_n)^{-1} z with z = \sqrt n(\hat\theta_n - v), whose limit is B_q(1)^\top \{\int_0^1 \mathbb B_q \mathbb B_q^\top\}^{-1} B_q(1). For q = 1 the two-sided statistic is the square of |z| / \{n^{-1}\sum_k \hat G_n(k/n)^2\}^{1/2}.

Integration rule. Under nonuniform variance accumulation the manuscript integrates with respect to the variance-accumulation profile rather than calendar time. With integration = "profile" the normalizer is

V^{\mathrm Q,\tau}_n = \sum_{k=1}^{n} \hat G^\tau_n(k/n)\, \hat G^\tau_n(k/n)^\top \{\hat\tau_n(k/n) - \hat\tau_n((k-1)/n)\},

the Riemann-Stieltjes sum with the estimated profile increments as weights. With a linear profile the two rules coincide exactly. The reference law is simulated with the same rule and the same nodes, so a calendar-time statistic is never combined with a profile-weighted reference. The adjusted-range methods do not need this correction, because projected ranges are invariant under a monotone change of time.

Value

An aersn_normalizer() object of class "aersn_normalizer_shao", carrying the matrix V^{\mathrm Q}_n and the integration weights actually used.

References

Shao, X. (2010). A self-normalized approach to confidence interval construction in time series. Journal of the Royal Statistical Society: Series B, 72(3), 343-366. doi:10.1111/j.1467-9868.2009.00737.x

See Also

aersn_normalizer(), aersn_compare().

Examples

set.seed(3)
fit <- aersn_mean(matrix(rnorm(400), 200, 2))
nz <- aersn_shao_normalizer(fit)
nz$matrix
aersn_test(fit, method = "shao", draws = 400, seed = 1)

Support function of an increment hull or confidence region

Description

For an aersn_hull() object, returns the support function h_K(u) = \sup_{z \in K} u^\top z, which equals the adjusted range of the projected path u^\top \hat G_n (manuscript equation (support-range)). For an aersn_region() object, returns the support function of the confidence region u^\top\hat\theta_n + (c/\sqrt n)\,h_K(u), which is the upper end of the simultaneous projection interval for the contrast u^\top\theta.

Usage

aersn_support(x, u, ...)

## S3 method for class 'aersn_hull'
aersn_support(x, u, ...)

## S3 method for class 'aersn_region'
aersn_support(x, u, ...)

Arguments

Value

A numeric vector with one value per direction.

Build a reference object from tabulated quantiles

Description

Wraps tabulated critical values (for example a row set of aersn_registry) as an "aersn_reference" of type "table". Such a reference supports aersn_critical_value() at the tabulated probabilities only, and aersn_pvalue() returns the bracket of tabulated tail probabilities that contains the p-value.

Usage

aersn_table_reference(q, n, probs, values, mcse = NULL, source = "user table")

Arguments

Value

An "aersn_reference" object of type "table".

Examples

reg <- subset(aersn_registry, q == 2 & n == 200)
ref <- aersn_table_reference(2, 200, reg$prob, reg$cv, reg$mcse,
                             source = "aersn_registry")
aersn_critical_value(ref, 0.95)

Transform the target parameter

Description

Maps an aersn() object for a parameter vector \beta to one for a target \theta = h(\beta) by transforming the estimate and the influence contributions with the Jacobian of h: \hat\theta_n = h(\hat\beta_n) and \hat\psi^\theta_{n,t} = \hat{\dot h}_n \hat\psi^\beta_{n,t}. For a linear transformation A\beta, the estimate is A\hat\beta_n and the contributions are A\hat\psi^\beta_{n,t}; the increment hull transforms accordingly (K(Ag) = AK(g)). Inference from the returned object uses the target dimension's reference law. When dimension is reduced, its region generally differs from the projection of the original region, which retains the original critical value. Use aersn_contrast() with type = "simultaneous" for that projection.

Usage

aersn_target(object, transform, jacobian = NULL, names = NULL)

Arguments

Details

The transformation must have full row rank m \le q; otherwise the transformed increment hull cannot be full dimensional. The delta method underlying the function-valued case requires h to be continuously differentiable at the true parameter with a full-row-rank derivative (manuscript Proposition "Feasible influence paths for smooth GMM").

Value

An "aersn" object for the target, with the same sample size and centering.

Examples

set.seed(5)
fit <- aersn_mean(matrix(rnorm(300), 100, 3))
diff12 <- aersn_target(fit, matrix(c(1, -1, 0), 1, 3), names = "m1 - m2")
confint(diff12, draws = 500)

Point-null test with the increment-hull gauge

Description

Tests H_0: \theta_0 = v (or H_0: A\theta_0 = v for a full-row-rank contrast matrix A) using the statistic T_n(v) = \gamma_{K(\hat G_n)}\{\sqrt n(\hat\theta_n - v)\} and the reference law of aersn_reference(). The null is rejected at level \alpha = 1 - level when T_n(v) exceeds the (1-\alpha) quantile of the reference law.

Usage

aersn_test(
  object,
  null = NULL,
  contrast = NULL,
  level = 0.95,
  alternative = c("two.sided", "greater", "less"),
  method = "hull",
  ...,
  nu = NULL,
  reference = NULL,
  draws = NULL,
  seed = 1L,
  cores = 1L
)

Arguments

Details

A linear restriction A\theta_0 = v is tested as a point null for the transformed target A\theta (see aersn_target()); nuisance coordinates are handled through the influence contributions, not by profiling. Unrestricted composite hypotheses are not supported. For q = 1 one-sided alternatives use the signed ratio \sqrt n(\hat\theta_n - v) divided by the adjusted range, whose limit is M = Z/R (Supplement, Section "Scalar specialization").

The p-value is the reference tail probability at the observed statistic. For a Monte Carlo two-sided reference it is the fraction of draws at or above the statistic. One-sided scalar tails use the symmetry of the law, consistently with the critical values. The result includes Monte Carlo uncertainty; see aersn_pvalue(). For the closed-form scalar law no Monte Carlo approximation is used. This is a limiting reference law, not a finite-sample exact test for arbitrary dependent observations.

Value

An object of class c("aersn_test", "htest") with the statistic, critical value, p-value, level, estimate, null value, the maximizing direction of the dual program, and the reference summary.

Examples

set.seed(6)
fit <- aersn_mean(rnorm(150, mean = 0.2))
aersn_test(fit, null = 0, reference = "continuous")
aersn_test(fit, null = 0, draws = 2000)

Vertices of a two-dimensional confidence region

Description

For q = 2, the increment hull is a polygon whose vertices are among the pairwise differences of the vertices of the convex hull of the path values (K(g) = P(g) - P(g), Supplement Proposition "Characterization of the bridge-increment hull"). The region's vertices are the center plus the scaled hull vertices, in counter-clockwise order.

Usage

aersn_vertices(region)

Arguments

Value

A numeric matrix with two columns.

Confidence intervals from the increment hull

Description

Confidence intervals for coordinates of the parameter vector. Three constructions are distinguished:

• "simultaneous": projections of the joint q-dimensional confidence region onto the coordinate axes, manuscript equation (projection-interval), \hat\theta_j \pm (c_{q}/\sqrt n)\{\max_k \hat G_{n,k,j} - \min_k \hat G_{n,k,j}\}, using the q-dimensional critical value. These intervals hold simultaneously for every linear projection of the region.

• "joint": projections of the joint region of the selected coordinates only (dimension m = length(parm)), using c_m. Coverage is simultaneous over linear combinations of the selected coordinates.

• "marginal": separately constructed scalar intervals, each applying the univariate adjusted-range statistic to the coordinate path with c_1. They cover each coordinate individually and do not provide simultaneous coverage (manuscript Theorem 1(iii), last paragraph).

Usage

## S3 method for class 'aersn'
confint(
  object,
  parm = NULL,
  level = 0.95,
  type = c("simultaneous", "joint", "marginal"),
  method = "hull",
  ...,
  nu = NULL,
  reference = NULL,
  draws = NULL,
  seed = 1L,
  cores = 1L
)

Arguments

Value

A matrix with columns lower and upper and one row per coordinate, of class "aersn_confint", with attributes type, level, critical.value, dimension (the dimension of the reference law used) and reference. For marginal inference, tuning_by_contrast records the tuning selected separately for each coordinate.

Examples

set.seed(7)
fit <- aersn_mean(matrix(rnorm(400), 200, 2))
confint(fit, draws = 1000)                      # simultaneous
confint(fit, type = "marginal", draws = 4000)   # separate scalar intervals

Plot a confidence region

Description

For q = 1 the interval is drawn; for q = 2 the region polygon; for q >= 3 a matrix of two-dimensional views, either projections of the joint region (type = "projection", simultaneous coverage) or slices through the estimate (type = "slice", cross-sections). The estimate is marked, and optionally a null value.

Usage

## S3 method for class 'aersn_region'
plot(
  x,
  type = c("projection", "slice"),
  null = NULL,
  coords = NULL,
  col = "steelblue",
  ...
)

Arguments

Value

The region, invisibly.

Query a reference distribution

Description

Quantiles, critical values, p-values and Monte Carlo standard errors of an aersn_reference() object.

Usage

## S3 method for class 'aersn_reference'
quantile(x, probs = c(0.9, 0.95, 0.99), type = 8, ...)

aersn_critical_value(ref, level = 0.95)

aersn_mcse(ref, probs = c(0.9, 0.95, 0.99), type = 8)

aersn_pvalue(ref, statistic, alternative = c("two.sided", "greater", "less"))

Arguments

Value

quantile() and aersn_critical_value() return numeric vectors. aersn_mcse() returns the Monte Carlo standard error of each quantile estimated from interleaved batches (NA for analytic references). aersn_pvalue() returns a list with elements p.value, mcse, resolution (one over the number of draws), and for table references the bracket lower/upper. Monte Carlo results also contain conf.int, a 95 percent binomial interval for the reference tail probability, and conf.level. One-sided scalar tails use symmetry: half the absolute-tail probability on one side and its complement on the other; their resolution is half that of the two-sided tail. A zero estimated tail probability is not an upper bound on the true probability. Quantile standard errors are NA when fewer than two batches or two draws per batch are available. Type-8 quantile interpolation and empirical tail counting can give decisions differing within one Monte Carlo resolution unit at a boundary.

Examples

ref <- aersn_reference(1, n = 100, draws = 4000, seed = 2)
quantile(ref, 0.95); aersn_mcse(ref, 0.95)
aersn_pvalue(ref, 1.9)