Package {fixedCV}

fixedCV: Fixed-b Critical Values for Robust Inference with Time Series Data

Description

Provides functions for computing fixed-b critical values and conducting robust inference procedures for time series data with unknown correlation structures. Implements long-run variance estimators using various kernel functions and lugsail transformations for improved finite-sample properties as described by Kurtz-Garcia and Flegal (2026) <doi:10.48550/arXiv.2606.17369>.

Details

The main functionality of fixedCV includes:

• Robust Linear Model Inference: Use robust_lm to conduct inference on lm objects with robust standard errors and test statistics that account for serial correlation.

• Long-Run Variance Estimation: Estimate the long-run variance matrix using various kernel functions (Bartlett, Parzen, Tukey-Hanning, Quadratic Spectral) and lugsail adjustments.

• Critical Value Computation: Obtain fixed-b critical values using analytical, simulated, or fitted methods with get_cv.

• Automatic Bandwidth Selection: Select optimal bandwidth parameters using get_b to control Type I error distortion.

Getting Started

For most applications, use robust_lm as the main entry point:

# Generate data
set.seed(123)
error <- arima.sim(model = list(ar = 0.7), n = 100)
x1 <- rbinom(100, 1, .5)
x2 <- rnorm(100)
y <- 1 + 2*x1 - 0.5*x2 + error
mydata <- data.frame(x1, x2, y)

# Fit a linear model
model <- lm(y ~ x1 + x2, data = mydata)

# Conduct robust inference
result <- robust_lm(model, the_kernel = "bartlett", lugsail = "Zero")

Available Kernels

The package supports four kernel functions:

• bartlett: Bartlett kernel (q=1)

• parzen: Parzen kernel (q=2)

• th: Tukey-Hanning kernel (q=2)

• qs: Quadratic Spectral kernel (q=2)

Lugsail Options

Three lugsail adjustment types are available:

• "Mother": No transformation (original kernel)

• "Zero": Conservative adjustment (r=2)

• "Over": Moderate adjustment (r=3)

Critical Value Methods

Critical values can be obtained via:

• "simulated": Pre-computed lookup tables

• "fitted": Polynomial approximations

• "analytical": Closed-form approximations

• "adaptive": Classical limiting theory

Author(s)

Maintainer: Rebecca Kurtz-Garcia rkurtzgarcia@smith.edu

Authors:

• Rebecca Kurtz-Garcia rkurtzgarcia@smith.edu

• Thomas Robacker trobacker@umass.edu

See Also

Useful links:

• Main function: robust_lm

• Critical values: get_cv

• Bandwidth selection: get_b

Stream Temperature data

Description

This data set contains daily average temperature readings in Celsius for a stream in Gill, Massachusetts. The original data was recorded via a submerged logger at 15 minutes intervals from August to December 2024 by the Organismic & Evolutionary Biology Department at the University of Massachusetts Amherst.

Usage

GillStream

Format

A data frame with 153 rows and 3 columns:

Date

The date the temperature was recorded.

AvgTemp

Temperature in Celsius

time_index

index values for the date

Fixed Critical Values using the Fitted Method

Description

A look up table containing the fitted coefficients for robust critical values. It also contains the adjusted R^2 values for the respective fit.

Usage

fitted_CV

Format

A data frame with the following columns:

• intercept:a parameter for the approximation

• beta1: a parameter for the approximation

• beta2: a parameter for the approximation

• beta3: a parameter for the approximation

• adj.r.sq: Adjusted R^2

• d: dimensions

• kernel: mother kernel selected

• lugsail: lugsail setting

• alpha: objective type 1 error rate

Details

This data set contains critical values that the get_cv and get_cv_fitted functions search for. For a given kernel, lugsail setting, dimension, and desired \alpha, the critical value is approximated via the formula, CV = \text{intercept} + \beta_1*b+ \beta_2*b^2 + \beta_3*b^3

Generate Fixed-b Critical Values

Description

Generates fixed-b critical values for robust hypothesis testing. Simulate from a fixed-b distribution to get custom critical values.

Usage

generate_cv(
  b,
  d = 2,
  alpha = 0.05,
  the_kernel = bartlett,
  lugsail_type = "Mother",
  q = 1,
  return_F_stats = FALSE,
  num_replicates = 50000,
  replicate_size = 1000
)

Arguments

Details

This function is used to generate custom simulated critical values. Both package kernels and user supplied kernels maybe supplied. It is not recommended to decrease num_replicates or replicate_size, especially for higher dimensions.

Value

If return_F_stats = FALSE (default) a matrix is returned where the rows correspond to the values for b and the columns correspond to the significance levels (alpha) supplied by the user. If return_F_stats = TRUE, then a list is returned. The second element in the list is a matrix of the F statistics generated to approximate the fixed-b distribution.

Examples

# Multiple b's and alpha's
set.seed(62)
CVs <- generate_cv(b = c(0.005, 0.006), d=1, alpha = c(0.05, 0.01),
num_replicates = 100)
CVs
# Custom kernel function
my_kernel <- function(x){
  k_x <- exp(-x^2/2)
  return(k_x)
}
CVs <- generate_cv(b = 0.005, d=1, alpha = 0.05, the_kernel = my_kernel,
 q = 2, num_replicates = 100)
CVs

Automatic Bandwidth Selection

Description

Selects optimal bandwidth for long-run variance estimation by controlling Type I error distortion. The bandwidth determines what proportion of autocovariances receive non-zero weight in the spectral variance estimator.

Usage

get_b(
  the_data,
  alpha = 0.05,
  the_kernel = "Bartlett",
  lugsail = "Mother",
  tau = NA,
  auto_adjust = TRUE
)

Arguments

Details

Returns a bandwidth optimized for hypothesis testing. If no hypothesis test is being conducted, using alpha = 0.05 as a guideline is recommended. The selected bandwidth is the smallest value whose Type I error rate falls within tau of alpha. If no bandwidth yields a Type I error rate within tau of alpha and auto_adjust = TRUE, then tau is increased by 25% iteratively until a bandwidth satisfies the criterion.

When lugsail = "Zero", the default setting tau = NA uses the tolerance recommended by Kurtz-Garcia and Flegal (2026). For other lugsail settings, larger tolerance levels are typically needed, hence when tau = NA, the default is set to alpha * 0.15.

Value

Numeric scalar, the selected optimal bandwidth value between 0 and 1.

References

Rebecca P. Kurtz-Garcia and James M. Flegal. Inference optimal long run variance estimation with lugsail kernels, 2026. URL https://arxiv.org/abs/2606.17369.

Examples

# Simulate AR(1) data
set.seed(123)
data <- arima.sim(list(ar = 0.7), n = 100)

# Get optimal bandwidth
get_b(data)

# With different kernel
get_b(data, the_kernel = "QS")

# With custom tolerance
get_b(data, tau = 0.01)

Get Fixed-b Critical Values

Description

Retrieves fixed-b critical values for robust hypothesis testing. Critical values can be computed using simulated lookup tables, fitted polynomial approximations, or analytical formulas.

Usage

get_cv(
  new_b,
  d = 1,
  alpha = 0.05,
  the_kernel = "Bartlett",
  lugsail = "Mother",
  method = "simulated"
)

Arguments

Value

Numeric vector of critical values corresponding to each bandwidth in new_b. When b = 0, returns chi-square critical values.

Examples

# Get critical value for single bandwidth
get_cv(0.1, d = 1, alpha = 0.05)

# Get critical values for multiple bandwidths
get_cv(c(0, 0.1, 0.2, 0.3), d = 1, alpha = 0.05)

# Using different methods
get_cv(0.1, method = "fitted")
get_cv(0.1, method = "analytical")

# For F-test with 3 degrees of freedom
get_cv(0.1, d = 3, alpha = 0.05)

Kernel Functions for Long-Run Variance Estimation

Description

Kernel weight functions used in spectral variance (long-run variance) estimation. These functions generate weights for autocovariances in robust inference procedures.

Usage

bartlett(x)

qs(x)

parzen(x)

th(x)

Arguments

Details

The kernel functions are:

• bartlett: Bartlett (Newey-West) kernel. q = 1 kernel.

• parzen: Parzen kernel. q = 2 kernel.

• th: Tukey-Hanning kernel. q = 2 kernel.

• qs: Quadratic Spectral kernel. q = 2 kernel.

Value

Numeric kernel weight value.

Examples

# Evaluate kernels at x = 0.5
bartlett(0.5)
parzen(0.5)
th(0.5)
qs(0.5)

# Plot kernel shapes
x <- seq(-1.5, 1.5, length.out = 200)
plot(x, sapply(x, bartlett), type = "l", ylab = "Weight", main = "Bartlett Kernel")

Robust Linear Model Inference with Fixed-b Critical Values

Description

Conducts robust inference for linear models using fixed-b asymptotics. This function computes robust standard errors, test statistics, and p-values for linear model coefficients, accounting for unknown serial correlation in the errors.

Usage

robust_lm(
  fit,
  the_kernel = "Bartlett",
  lugsail = "Mother",
  method = "simulated",
  tau = NA,
  alpha = 0.05,
  conf.int = FALSE
)

Arguments

Details

This function largely follows the robust inference procedure described in Kurtz-Garcia and Flegal (2026). Standard errors are estimated using a HAC estimator with a choice of kernel functions. Both classical adaptive limiting theory `method = "adaptive"` and fixed limiting theory are supported, with multiple approximations available for the fixed-limiting theory.

The method = "simulated" option is only available for problems with up to 12 dimensions. In general, method = "analytical" is recommended for F-tests and for non-univariate tests.

By default, the null hypothesis for the F-test is that all slope coefficients are zero. If the model contains no slope coefficients, the null hypothesis is that the intercept is zero.

Because obtaining quantiles for the fixed-limit distribution is computationally expensive, exact p-values are not reported. Instead, thresholds are given indicating if the p-value is above 0.10 (>= 0.10), between 0.10 and 0.05 (<0.10), between 0.05 and 0.025 (<0.05.), between 0.025 and 0.01 (<0.025.), and less than 0.01 (<0.01*). For the same reason, confidence intervals are only available for a limited set of confidence levels.

Value

A list containing:

References

Rebecca P. Kurtz-Garcia and James M. Flegal. Inference optimal long run variance estimation with lugsail kernels, 2026. URL https://arxiv.org/abs/2606.17369.

Examples

# Simulate AR(1) data
set.seed(123)
n <- 100
x <- arima.sim(list(ar = 0.5), n)
y <- 2 + 3*x + arima.sim(list(ar = 0.7), n)
fit <- lm(y ~ x)

# Robust inference with default settings
robust_lm(fit)

# With different kernel and method
robust_lm(fit, the_kernel = "QS", method = "fitted")

# With confidence intervals
robust_lm(fit, conf.int = TRUE)

Fixed Critical Values using the Simulated Method

Description

The title of each data set is the <Mother>_<Lugsail>_<alpha>_Master.

Usage

Bartlett_Mother_01_Master

Bartlett_Mother_025_Master

Bartlett_Mother_05_Master

Bartlett_Mother_10_Master

Bartlett_Zero_01_Master

Bartlett_Zero_025_Master

Bartlett_Zero_05_Master

Bartlett_Zero_10_Master

Bartlett_Over_01_Master

Bartlett_Over_025_Master

Bartlett_Over_05_Master

Bartlett_Over_10_Master

QS_Mother_01_Master

QS_Mother_025_Master

QS_Mother_05_Master

QS_Mother_10_Master

QS_Zero_01_Master

QS_Zero_025_Master

QS_Zero_05_Master

QS_Zero_10_Master

QS_Over_01_Master

QS_Over_025_Master

QS_Over_05_Master

QS_Over_10_Master

Parzen_Mother_01_Master

Parzen_Mother_025_Master

Parzen_Mother_05_Master

Parzen_Mother_10_Master

Parzen_Zero_01_Master

Parzen_Zero_025_Master

Parzen_Zero_05_Master

Parzen_Zero_10_Master

Parzen_Over_01_Master

Parzen_Over_025_Master

Parzen_Over_05_Master

Parzen_Over_10_Master

TH_Mother_01_Master

TH_Mother_025_Master

TH_Mother_05_Master

TH_Mother_10_Master

TH_Zero_01_Master

TH_Zero_025_Master

TH_Zero_05_Master

TH_Zero_10_Master

TH_Over_01_Master

TH_Over_025_Master

TH_Over_05_Master

TH_Over_10_Master

Format

A look up table containing the simulated robust critical values. The 'b' column contains the bandwidth, and subsequent columns are the dimensions

• b: bandwidth of the test statistics. The proportion of autocovariances given a non-zero weight.

• d: the dimension of the test statistic.

Details

The title of each data set is the <Mother>_<Lugsail>_<alpha>_Master. Supported mother kernels are Bartlett, Parzen, Tukey-Hanning (TH), and Quadratic Spectral (QS). Supported lugsail settings are mother, zero, and over. The available critical values are 0.01, 0.025, 0.05, and 0.10. Note that when b = 0 the robust critical value are equivalent to chi-square critical values.