Wavelet Quantile Regression — A comprehensive Python toolkit for quantile-on-quantile regression, wavelet quantile regression, nonparametric quantile causality, and wavelet quantile mediation/moderation with publication-quality visualizations.

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Project description

WQR — Wavelet Quantile Regression

A comprehensive, high-performance Python library for wavelet-based quantile regression econometrics — providing eight cutting-edge methodologies with publication-quality MATLAB-style visualizations, significance testing, and journal-ready LaTeX table export.

Migrated from R/MATLAB. This package unifies and extends the R CRAN packages QuantileOnQuantile, wqc, nonParQuantileCausality, and several MATLAB toolboxes into a single, optimized Python framework.

Author

Dr. Merwan Roudane
📧 merwanroudane920@gmail.com
🔗 GitHub

Features
Installation
Quick Start
Modules
Visualization Gallery
Tables & LaTeX Export
API Reference
Dependencies
Citation
References
License

Features

Econometric Methods

#	Module	Method	Key Function	Reference
1	`qq_regression`	Quantile-on-Quantile Regression	`wqr.qq_regression()`	Sim & Zhou (2015)
2	`wavelet_qr`	Wavelet Quantile Regression (WQR)	`wqr.wavelet_qr()`	Adebayo & Ozkan (2023)
3	`multivariate_wqr`	Multivariate WQR (MWQR)	`wqr.multivariate_wqr()`	Adebayo et al. (2025)
4	`wavelet_qqr`	Wavelet QQR with P-values (WQQR)	`wqr.wavelet_qqr()`	Adebayo, Özkan et al. (2025)
5	`causality`	Nonparametric Quantile Causality	`wqr.np_quantile_causality()`	Balcilar et al. (2016)
6	`causality`	Wavelet NP Quantile Causality (WNQC)	`wqr.wavelet_np_causality()`	Balcilar et al. (2016)
7	`mediation`	Wavelet Quantile Mediation & Moderation	`wqr.wavelet_mediation()`	Adebayo (2025)
8	`correlation`	Wavelet Quantile Correlation (WQC)	`wqr.wavelet_quantile_correlation()`	Roudane (2024)
9	`quantile_density`	Wavelet Quantile Density Estimation	`wqr.wavelet_quantile_density()`	Chesneau, Dewan & Doosti

Visualization Types

Plot	Function	Description
🏔️ 3D Surface	`plot_qq_3d()`	MATLAB-style Jet colorscale 3D surface
🗺️ Heatmap	`plot_qq_heatmap()`	Coefficient heatmap with significance stars (``, ``, `**`)
📊 Contour	`plot_qq_contour()`	Filled contours with isolines
📈 Causality	`plot_causality()`	Test statistic vs critical value lines
🔗 Correlation	`plot_correlation_heatmap()`	Quantile-by-quantile correlation matrix
📐 P-value Map	`plot_wqqr_pvalue_heatmap()`	Binary significance grid
🔀 Mediation	`plot_mediation_panel()`	5-panel direct/indirect/interaction heatmaps
📉 Density	`plot_quantile_density()`	4-panel wavelet density estimation
📊 WQR Heatmap	`plot_wqr_heatmap()`	Band × quantile heatmap with stars
📈 WQQR Surface	`plot_wqqr_surface()`	3D surface of WQQR coefficients
🔄 WQR vs WQQR	`plot_wqr_vs_wqqr()`	Line comparison plot
🌊 Wavelet Causality	`plot_wavelet_causality()`	Heatmap across wavelet scales
🔗 WQC Heatmap	`plot_wqc_heatmap()`	Wavelet quantile correlation with CI borders

Additional Features

📋 LaTeX Tables — Journal-ready tables with significance stars and proper formatting
📊 Summary Statistics — Descriptive statistics tables (mean, std, skewness, kurtosis)
💾 CSV/DataFrame Export — All results exportable to CSV and pandas DataFrames
🎨 Custom Colormaps — MATLAB Jet, Blue-Red, Green-Orange-Red, Green-Yellow-Red, and all matplotlib colormaps
⚡ MODWT Decomposition — Maximal Overlap Discrete Wavelet Transform with MRA
📐 Band Aggregation — Automatic Short/Medium/Long frequency band aggregation

Installation

pip install wqr

Install with optional interactive plotting support:

pip install wqr[plotly]    # Plotly + Kaleido for interactive HTML
pip install wqr[full]      # All optional: plotly, seaborn, openpyxl

Or install from source:

git clone https://github.com/merwanroudane/wqrr.git
cd wqrr
pip install -e .

Quick Start

import numpy as np
import wqr

# Generate sample data
np.random.seed(42)
n = 300
x = np.random.normal(size=n)
y = 0.5 * x + np.random.normal(scale=0.5, size=n)

# ── Quantile-on-Quantile Regression ──
result = wqr.qq_regression(y, x)
result.summary()

# Publication-quality visualizations
fig, ax = wqr.plot_qq_3d(result, colormap="jet")
fig, ax = wqr.plot_qq_heatmap(result, colormap="jet")
fig, ax = wqr.plot_qq_contour(result)

# Export LaTeX table
latex = result.export_latex()
print(latex)

Modules

1. Quantile-on-Quantile Regression (`qq_regression`)

Implements the methodology of Sim & Zhou (2015) to examine the dependence between the quantiles of two variables. For each x-quantile τ, subsets data where x ≤ Qₓ(τ) and runs quantile regression of y on x at each y-quantile θ.

result = wqr.qq_regression(
    y,                              # Dependent variable
    x,                              # Independent variable
    y_quantiles=np.arange(0.05, 1.0, 0.05),  # Y quantile grid
    x_quantiles=np.arange(0.05, 1.0, 0.05),  # X quantile grid
    min_obs=10,                     # Minimum observations for QR
    se_method="boot",               # SE method: 'boot', 'iid', 'robust'
    n_boot=200,                     # Bootstrap replications
    verbose=True
)

# Access results
result.summary()                     # Print comprehensive summary
mat = result.to_matrix("coefficient")  # (19×19) coefficient matrix
mat = result.to_matrix("p_value")      # P-value matrix
mat = result.to_matrix("r_squared")    # R² matrix
df = result.to_dataframe()             # Full results DataFrame
result.export_csv("results.csv")       # Export to CSV
latex = result.export_latex()          # LaTeX table

# Visualize
fig, ax = wqr.plot_qq_3d(result, colormap="jet", elev=25, azim=-135)
fig, ax = wqr.plot_qq_heatmap(result, show_significance=True)
fig, ax = wqr.plot_qq_contour(result, levels=20)

QQResult attributes:

Attribute	Type	Description
`results`	`pd.DataFrame`	Full results (y_quantile, x_quantile, coefficient, std_error, t_value, p_value, r_squared)
`y_quantiles`	`np.ndarray`	Y quantile grid
`x_quantiles`	`np.ndarray`	X quantile grid
`n_obs`	`int`	Number of observations

2. Wavelet Quantile Regression (`wavelet_qr`)

Decomposes time series using MODWT (Maximal Overlap Discrete Wavelet Transform) then performs quantile regression on each wavelet detail level. Supports aggregation into Short/Medium/Long frequency bands. Based on Adebayo & Ozkan (2023).

result = wqr.wavelet_qr(
    y,                       # Dependent variable
    x,                       # Independent variable
    quantiles=np.arange(0.05, 1.0, 0.05),
    wavelet="la8",           # Wavelet family (R-style: la8, d4, haar, etc.)
    J=5,                     # Decomposition levels
    bands=True,              # Aggregate to Short/Medium/Long
    n_boot=500,
    verbose=True
)

result.summary()
coef_matrix = result.to_matrix()            # (quantiles × bands) matrix
sig = result.significance_matrix(alpha=0.05)  # Binary significance

fig, ax = wqr.plot_wqr_heatmap(result.coefficients, colormap="green_orange_red")

WQRResult attributes:

Attribute	Type	Description
`coefficients`	`pd.DataFrame`	Columns: quantile, level, beta, se, p_value
`quantiles`	`np.ndarray`	Quantile grid
`levels`	`list`	Band/level names (e.g., ['Short', 'Medium', 'Long'])
`wavelet`	`str`	Wavelet name
`J`	`int`	Decomposition levels

3. Multivariate Wavelet Quantile Regression (`multivariate_wqr`)

Extends WQR to multiple independent variables, estimating the effect of each regressor at each quantile across wavelet frequency bands. Based on Adebayo et al. (2025).

result = wqr.multivariate_wqr(
    y,                                  # Dependent variable
    X_dict={"GDP": gdp, "CO2": co2, "Energy": energy},  # Multiple regressors
    quantiles=np.array([0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95]),
    wavelet="la8",
    J=6,
    dep_name="Temperature",
    verbose=True
)

result.summary()

# Per-variable analysis
gdp_df = result.get_variable("GDP")
gdp_mat = result.to_matrix("GDP")
stars = result.significance_matrix("GDP")

# Visualize each variable
fig, ax = wqr.plot_wqr_heatmap(gdp_df, col_col="band", title="GDP → Temperature")

MWQRResult attributes:

Attribute	Type	Description
`coefficients`	`pd.DataFrame`	Columns: quantile, band, variable, beta, se, p_value
`dep_name`	`str`	Dependent variable name
`indep_names`	`list`	Independent variable names
`bands`	`list`	Frequency bands

4. Wavelet QQR with P-values (`wavelet_qqr`)

Combines MODWT wavelet decomposition with kernel-weighted local polynomial quantile regression to produce a full QQ coefficient grid with pointwise p-values. Based on Adebayo, Özkan, Uzun Ozsahin, Eweade & Gyamfi (2025).

result = wqr.wavelet_qqr(
    y, x,
    quantile_step=0.05,      # → 19 quantiles (0.05 to 0.95)
    wavelet="la8",
    J=5,
    band="long",              # 'short', 'medium', 'long', 'all', or 'D1', 'D2', ...
    bandwidth=1.0,            # Gaussian kernel bandwidth
    verbose=True
)

result.summary()
avg_coef = result.avg_qqr_coef()  # Average WQQR at each y-quantile

# 3D surface
fig, ax = wqr.plot_wqqr_surface(result, colormap="jet")

# P-value significance map
fig, ax = wqr.plot_wqqr_pvalue_heatmap(result, alpha=0.05)

# Compare WQR vs WQQR
fig, ax = wqr.plot_wqr_vs_wqqr(result)

WQQRResult attributes:

Attribute	Type	Description
`coef_matrix`	`np.ndarray`	(19×19) coefficient grid
`pval_matrix`	`np.ndarray`	(19×19) p-value grid
`qr_coef`	`np.ndarray`	Standard QR slope coefficients
`qr_pval`	`np.ndarray`	Standard QR p-values
`quantiles`	`np.ndarray`	Quantile grid
`band_name`	`str`	Selected frequency band

5. Nonparametric Quantile Causality (`np_quantile_causality`)

Implements the Balcilar–Jeong–Nishiyama nonparametric quantile Granger causality test. Tests whether x_{t-1} Granger-causes y_t in quantile τ, using local linear quantile regression and the Song, Whang & Shin (2012) test statistic.

result = wqr.np_quantile_causality(
    x,                        # Candidate cause
    y,                        # Effect variable
    test_type="mean",         # 'mean' (1st moment) or 'variance' (2nd moment)
    q=np.arange(0.05, 1.0, 0.05),
    bandwidth=None            # None → Silverman plug-in bandwidth
)

result.summary()
sig = result.significant(alpha=0.05)    # Boolean array
df = result.to_dataframe()              # Full results DataFrame

fig, ax = wqr.plot_causality(result)

CausalityResult attributes:

Attribute	Type	Description
`statistic`	`np.ndarray`	Test statistic at each quantile
`quantiles`	`np.ndarray`	Quantile grid
`bandwidth`	`float`	Kernel bandwidth used
`test_type`	`str`	'mean' or 'variance'
`cv_5pct`	`float`	5% critical value (1.96)

6. Wavelet Nonparametric Quantile Causality (`wavelet_np_causality`)

MODWT decomposition followed by the nonparametric quantile causality test at each wavelet level or aggregated band.

result = wqr.wavelet_np_causality(
    x, y,
    test_type="mean",
    wavelet="la8",
    J=5,
    bands=True,               # Aggregate to Short/Medium/Long
    verbose=True
)

result.summary()
stat_matrix = result.to_matrix()           # (quantiles × bands) DataFrame
stars = result.significance_matrix(cv=1.96)  # Stars matrix

fig, ax = wqr.plot_wavelet_causality(result)

WaveletCausalityResult attributes:

Attribute	Type	Description
`results`	`dict`	`{band_name: CausalityResult}`
`quantiles`	`np.ndarray`	Quantile grid
`test_type`	`str`	'mean' or 'variance'
`wavelet`	`str`	Wavelet family

7. Wavelet Quantile Mediation & Moderation (`wavelet_mediation`)

Implements five estimation paths in a wavelet-decomposed setting based on Adebayo (2025):

Path	Regression	Interpretation
Direct	Y ~ X	Direct effect
Interaction	Y ~ X + Z + X·Z	Moderation (X·Z coefficient)
Path a	Z ~ X	Effect of X on mediator
Path b	Y ~ X + Z	Effect of mediator on Y
Indirect	a(τ) × b(τ)	Mediation (product of coefficients)

result = wqr.wavelet_mediation(
    y,                         # Dependent (Y)
    x,                         # Main independent (X)
    z,                         # Mediator/Moderator (Z)
    quantiles=np.array([0.01, 0.05, 0.10, 0.20, 0.30, 0.40,
                        0.50, 0.60, 0.70, 0.80, 0.90, 0.95, 0.99]),
    wavelet="la8",
    dep_name="Emissions",
    main_name="Renewable",
    mod_name="ESG",
    verbose=True
)

result.summary()

# Five-panel visualization
fig, axes = wqr.plot_mediation_panel(result, colormap="green_yellow_red")

MediationResult attributes:

Attribute	Type	Description
`direct`	`pd.DataFrame`	Direct effect (beta, se, p_value by quantile × band)
`interaction`	`pd.DataFrame`	Moderation interaction coefficient
`path_a`	`pd.DataFrame`	X → Z mediation path
`path_b`	`pd.DataFrame`	Z → Y mediation path
`indirect`	`pd.DataFrame`	Indirect effect a×b

8. Wavelet Quantile Correlation (`wavelet_quantile_correlation`)

Computes quantile correlations between wavelet-decomposed time series at each wavelet detail level, with Monte Carlo bootstrap confidence intervals. Converted from the R CRAN package wqc (Roudane).

result = wqr.wavelet_quantile_correlation(
    x, y,
    quantiles=np.array([0.05, 0.25, 0.50, 0.75, 0.95]),
    wavelet="la8",
    J=8,
    n_sim=1000,               # Monte Carlo simulations for CI
    verbose=True
)

result.summary()
mat = result.to_matrix()               # (levels × quantiles) matrix
sig = result.significant_cells()       # Cells outside 95% CI

fig, ax = wqr.plot_wqc_heatmap(result)

WQCResult attributes:

Attribute	Type	Description
`results`	`pd.DataFrame`	Level, Quantile, Estimated_QC, CI_Lower, CI_Upper
`quantiles`	`np.ndarray`	Quantile grid
`J`	`int`	Number of levels
`n_sim`	`int`	Monte Carlo replications

9. Wavelet Quantile Density Estimation (`wavelet_quantile_density`)

Implements the Daubechies-Lagarias wavelet-based estimator for the quantile density function q(p) = 1/f(F⁻¹(p)), with linear wavelet estimation, hard thresholding, and local linear smoothing. Converted from MATLAB by Chesneau, Dewan & Doosti.

from scipy.stats import norm

# Generate sample
y = norm.rvs(size=500, random_state=42)

result = wqr.wavelet_quantile_density(
    y,
    j0=5,                     # Coarsest decomposition level
    bandwidth=0.15,            # Local linear smoothing bandwidth
    wavelet="coif1",           # PyWavelets wavelet name
    gld_params=None            # Optional: (λ1, λ2, λ3, λ4) for true GLD quantile density
)

result.summary()

# Four-panel visualization
fig, axes = wqr.plot_quantile_density(result)

QuantileDensityResult attributes:

Attribute	Type	Description
`grid`	`np.ndarray`	Probability grid p ∈ (0,1)
`true_qd`	`np.ndarray` or `None`	True quantile density (if GLD params given)
`linear_estimate`	`np.ndarray`	Linear wavelet estimate
`thresholded_estimate`	`np.ndarray`	After hard thresholding
`smoothed_estimate`	`np.ndarray`	After local linear smoothing

Visualization Gallery

All plots are publication-quality (300 DPI) with configurable colormaps, labels, and save paths.

Available Colormaps

Name	Style	Best For
`"jet"`	MATLAB Jet (rainbow)	3D surfaces, QQ regression
`"blue_red"`	Diverging blue→white→red	Coefficient heatmaps
`"green_orange_red"`	Sequential green→orange→red	WQR band heatmaps
`"green_yellow_red"`	Sequential green→yellow→red	Mediation panels
`"viridis"`	Perceptually uniform	Correlation heatmaps
`"RdBu_r"`	Red-Blue diverging	Correlation matrices
`"coolwarm"`	Cool-warm diverging	General purpose
`"plasma"` / `"inferno"`	Sequential	Density plots

Common Plot Parameters

All plot functions accept:

figsize=(width, height) — Figure size in inches
save_path="figure.pdf" — Auto-save (supports PDF, PNG, EPS, SVG)
colormap="jet" — Colormap name
title="Custom Title" — Override default title
Returns (fig, ax) tuple for further customization

Tables & LaTeX Export

Formatted Results Table

# From any DataFrame with beta and p_value columns
table = wqr.results_table(
    result.coefficients,
    value_col="beta",
    pval_col="p_value",
    row_col="quantile",
    col_col="level",
    digits=4
)
print(table)

LaTeX Export

latex = wqr.export_latex(
    result.coefficients,
    caption="Wavelet Quantile Regression Results",
    label="tab:wqr",
    digits=4
)
print(latex)

Output:

\begin{table}[htbp]
\centering
\small
\caption{Wavelet Quantile Regression Results}
\label{tab:wqr}
\begin{tabular}{lccc}
\hline\hline
$\tau$ & Long & Medium & Short \\
\hline
0.05 & 0.1234*** & -0.0567 & 0.0891** \\
...
\hline\hline
\multicolumn{4}{l}{\footnotesize Note: ***, **, * denote significance at 1\%, 5\%, 10\% levels.} \\
\end{tabular}
\end{table}

Summary Statistics

stats = wqr.summary_statistics(
    {"GDP": gdp, "CO2": co2, "Energy": energy}
)
print(stats)

API Reference

Core Functions

Function	Returns	Description
`wqr.qq_regression(y, x, ...)`	`QQResult`	Quantile-on-Quantile regression
`wqr.wavelet_qr(y, x, ...)`	`WQRResult`	Wavelet quantile regression
`wqr.multivariate_wqr(y, X_dict, ...)`	`MWQRResult`	Multivariate wavelet QR
`wqr.wavelet_qqr(y, x, ...)`	`WQQRResult`	Wavelet QQR with p-values
`wqr.np_quantile_causality(x, y, ...)`	`CausalityResult`	Nonparametric quantile causality
`wqr.wavelet_np_causality(x, y, ...)`	`WaveletCausalityResult`	Wavelet NP quantile causality
`wqr.wavelet_mediation(y, x, z, ...)`	`MediationResult`	Mediation & moderation
`wqr.wavelet_quantile_correlation(x, y, ...)`	`WQCResult`	Wavelet quantile correlation
`wqr.wavelet_quantile_density(y, ...)`	`QuantileDensityResult`	Quantile density estimation

Plotting Functions

Function	Description
`wqr.plot_qq_3d(result, ...)`	3D surface of QQ coefficients
`wqr.plot_qq_heatmap(result, ...)`	Heatmap with significance stars
`wqr.plot_qq_contour(result, ...)`	Filled contour plot
`wqr.plot_wqr_heatmap(df, ...)`	WQR/MWQR band heatmap
`wqr.plot_wqqr_surface(result, ...)`	WQQR 3D surface
`wqr.plot_wqqr_pvalue_heatmap(result, ...)`	Binary significance map
`wqr.plot_wqr_vs_wqqr(result, ...)`	WQR vs WQQR comparison
`wqr.plot_causality(result, ...)`	Test statistic vs CV lines
`wqr.plot_wavelet_causality(result, ...)`	Wavelet causality heatmap
`wqr.plot_correlation_heatmap(y, x, ...)`	Quantile correlation matrix
`wqr.plot_wqc_heatmap(result, ...)`	WQC heatmap with CI borders
`wqr.plot_quantile_density(result, ...)`	4-panel density estimation
`wqr.plot_mediation_panel(result, ...)`	5-panel mediation/moderation

Table Functions

Function	Description
`wqr.results_table(df, ...)`	Formatted table with significance stars
`wqr.export_latex(df, ...)`	Journal-ready LaTeX table
`wqr.summary_statistics(data, ...)`	Descriptive statistics (mean, std, skew, kurt)

Dependencies

Required

Package	Version	Purpose
`numpy`	≥ 1.20	Array computations
`pandas`	≥ 1.3	DataFrames & table export
`scipy`	≥ 1.7	Statistical distributions, kernel functions
`statsmodels`	≥ 0.13	Quantile regression engine
`matplotlib`	≥ 3.5	Publication-quality static plots
`PyWavelets`	≥ 1.1	MODWT wavelet decomposition

Optional

Package	Install	Purpose
`plotly` + `kaleido`	`pip install wqr[plotly]`	Interactive HTML plots
`seaborn` + `openpyxl`	`pip install wqr[full]`	Enhanced styling & Excel export

Citation

If you use wqr in academic research, please cite:

@software{roudane2025wqr,
  author    = {Roudane, Merwan},
  title     = {WQR: Wavelet Quantile Regression for Python},
  year      = {2025},
  publisher = {PyPI},
  url       = {https://pypi.org/project/wqr/},
  version   = {1.0.0}
}

References

Sim, N. & Zhou, H. (2015). Oil prices, US stock return, and the dependence between their quantiles. Journal of Banking & Finance, 55, 1–12. doi:10.1016/j.jbankfin.2015.01.013
Balcilar, M., Gupta, R. & Pierdzioch, C. (2016). Does uncertainty move the gold price? New evidence from a nonparametric causality-in-quantiles test. Resources Policy, 49, 74–80. doi:10.1016/j.resourpol.2016.04.004
Balcilar, M., Bekiros, S. & Gupta, R. (2016). The role of news-based uncertainty indices in predicting oil markets. Open Economies Review, 27(2), 229–250.
Song, K., Whang, Y.-J. & Shin, Y. (2012). Testing distributional treatment effects. Econometric Reviews.
Adebayo, T. S. & Ozkan, O. (2023). Investigating the influence of socioeconomic conditions on the environment using wavelet quantile regression. Journal of Cleaner Production. doi:10.1016/j.jclepro.2023.140321
Adebayo, T. S. et al. (2025). Unpacking policy ambiguities in residential and commercial renewable energy adoption: A novel multivariate wavelet quantile regression analysis. Applied Economics. doi:10.1080/00036846.2025.2590632
Adebayo, T. S., Özkan, O., Uzun Ozsahin, D., Eweade, B. S. & Gyamfi, B. A. (2025). Environmental Sciences Europe. doi:10.1186/s12302-025-01059-z
Adebayo, T. S. (2025). Can ESG Uncertainty Alter the Emissions Impact of Renewable Energy Consumption? Statistical Journal of the IAOS. doi:10.1177/18747655261445375
Chesneau, C., Dewan, I. & Doosti, H. Nonparametric estimation of the quantile density function by wavelet methods. (MATLAB implementation)
Roudane, M. (2024). QuantileOnQuantile: R package for quantile-on-quantile regression. CRAN.
Roudane, M. (2024). wqc: R package for wavelet quantile correlation. CRAN.

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WQR — Wavelet Quantile Regression

Author

Table of Contents

Features

Econometric Methods

Visualization Types

Additional Features

Installation

Quick Start

Modules

1. Quantile-on-Quantile Regression (qq_regression)

2. Wavelet Quantile Regression (wavelet_qr)

3. Multivariate Wavelet Quantile Regression (multivariate_wqr)

4. Wavelet QQR with P-values (wavelet_qqr)

5. Nonparametric Quantile Causality (np_quantile_causality)

6. Wavelet Nonparametric Quantile Causality (wavelet_np_causality)

7. Wavelet Quantile Mediation & Moderation (wavelet_mediation)

8. Wavelet Quantile Correlation (wavelet_quantile_correlation)

9. Wavelet Quantile Density Estimation (wavelet_quantile_density)

Visualization Gallery

Available Colormaps

Common Plot Parameters

Tables & LaTeX Export

Formatted Results Table

LaTeX Export

Summary Statistics

API Reference

Core Functions

Plotting Functions

Table Functions

Dependencies

Required

Optional

Citation

References

License

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1. Quantile-on-Quantile Regression (`qq_regression`)

2. Wavelet Quantile Regression (`wavelet_qr`)

3. Multivariate Wavelet Quantile Regression (`multivariate_wqr`)

4. Wavelet QQR with P-values (`wavelet_qqr`)

5. Nonparametric Quantile Causality (`np_quantile_causality`)

6. Wavelet Nonparametric Quantile Causality (`wavelet_np_causality`)

7. Wavelet Quantile Mediation & Moderation (`wavelet_mediation`)

8. Wavelet Quantile Correlation (`wavelet_quantile_correlation`)

9. Wavelet Quantile Density Estimation (`wavelet_quantile_density`)