tvccointreg 0.1.1

Time-Varying Coefficient regression and Generalized Cointegration (Hall, Swamy & Tavlas)

tvccointreg

Time-Varying-Coefficient regression and Generalized Cointegration in Python.

📦 PyPI: https://pypi.org/project/tvccointreg/ — install with pip install tvccointreg 🌐 Documentation site: https://merwanroudane.github.io/tvccointreg/

tvccointreg implements the generalized cointegration framework of

Hall, S. G., Swamy, P. A. V. B., & Tavlas, G. S. (2015). A Note on Generalizing the Concept of Cointegration.

together with the surrounding Swamy time-varying-coefficient (TVC) literature (Swamy & Mehta 1975; Swamy, Tavlas, Hall & co-authors 2003–2014; Granger 2008).

Conventional cointegration (Engle–Granger 1987) is an inherently linear concept: it can only recover a structural relationship if that relationship happens to be linear in unit-root variables. Most economic theory, however, implies nonlinear relationships among variables that are nonstationary but not necessarily unit-root. tvccointreg lets you:

• estimate a model that is linear in variables but with time-varying coefficients — the Swamy–Mehta / Granger representation of any nonlinear relationship (eq. 7);
• decompose each coefficient into a bias-free structural component plus omitted-variable bias and measurement-error bias using coefficient drivers (eq. 8);
• test for generalized cointegration — i.e. whether the bias-free structural partial derivative is nonzero — with standard χ²/normal inference (no Dickey–Fuller tables);
• produce journal-quality tables (text / LaTeX booktabs / HTML) and publication-quality plots with the MATLAB Parula colormap by default.

---

Table of contents

• Installation
• The 60-second tour
• Concepts: how the paper maps to the code
• The estimator
• API reference
• Detailed syntax
• Visualizations
• Worked example: spurious vs. real
• Testing
• Citing
• References

---

Installation

From PyPI:

pip install tvccointreg

# with the optional ADF stationarity diagnostic
pip install "tvccointreg[adf]"

Or from source:

git clone https://github.com/merwanroudane/tvccointreg.git
cd tvccointreg
pip install -e ".[dev]"      # editable install with test extras

Requirements: numpy, scipy, pandas, matplotlib. statsmodels is optional (used only for the ADF stationarity diagnostic and some examples).

---

The 60-second tour

from tvccointreg import TVCModel, DriverSpec
from tvccointreg.datasets import simulate_nonlinear_cointegration

# 1) A nonlinear relationship between nonstationary variables.
sim = simulate_nonlinear_cointegration(T=300, seed=1, nonlinearity=0.4)

# 2) Partition the drivers into the three sets (Assumption 1 / eq. 8).
spec = DriverSpec(
    names=list(sim.drivers.columns),
    bias_free=["x_lag", "y_lag"],   # true coefficient variation
    omitted=["w"],                  # omitted-variable bias
    measurement=[],                 # measurement-error bias
)

# 3) Fit by iteratively rescaled GLS.
res = TVCModel(sim.y, sim.X, sim.drivers, driver_spec=spec).fit()

# 4) Journal table + generalized cointegration test.
print(res.summary())
res.coint_test()

==============================================================================
      Time-Varying-Coefficient Regression  /  Generalized Cointegration
==============================================================================
Dep. variable: y                       No. observations: 300
No. coefficients: 2                    No. drivers: 3
Estimator: Iteratively rescaled GLS    Covariance: GLS
R-squared: 0.8311                      Log-likelihood: -78.07
Converged: True                        Resid ADF p-value: 0.0000
==============================================================================
Variable    Coef (mean)  Bias-free    Std.Err          t    p-value    G-Coint
------------------------------------------------------------------------------
x                0.3283     0.3392     0.0698     4.8614     0.0000 ***     Yes
==============================================================================
                   Signif.: *** p<0.01  ** p<0.05  * p<0.10
       Bias-free = structural derivative; G-Coint via Wald test on it.
==============================================================================

On this DGP the recovered bias-free coefficient correlates 0.99 with the true time-varying derivative, and the residuals are stationary (ADF p ≈ 0) — exactly the standard-inference result the paper proves.

---

Concepts: how the paper maps to the code

Paper (Hall–Swamy–Tavlas 2015)	Symbol	In tvccointreg
TVC model, eq. (7)	y_t = γ_0t + γ_1t x_1t + … + γ_{K-1,t} x_{K-1,t}	TVCModel(y, X, drivers)
Coefficient-driver eq., eq. (8)	γ_jt = π_j0 + Σ_d π_jd z_dt + ε_jt	DriverSpec(...) + the drivers matrix
Three components of a coefficient	bias-free / omitted-variable / measurement-error	DriverSpec(bias_free=, omitted=, measurement=)
Concentrated model	y_t = w_t'π + u_t, u_t = Σ_j x_jt ε_jt	estimation.build_design, estimation.irgls
Variance structure	Var(u_t) = Σ_j σ_j² x_jt²	estimation.variance_components (Hildreth–Houck–Swamy)
Consistent estimator (§3.3)	iteratively rescaled GLS	TVCModel.fit(method="irgls")
Bias-free component	γ_jt^BF = π_j0 + Σ_{d∈BF} π_jd z_dt	res.bias_free_coefficients()
Generalized cointegration, eqs. (5)–(6)	∂y/∂x ≠ 0 ⇔ bias-free component ≠ 0	res.coint_test()
Standard inference (§3.3)	χ² Wald / normal, not Dickey–Fuller	Wald + delta-method t in cointegration.py
Drivers vs. instruments (Table 1)	drivers should be correlated with the misspecification	the omitted / measurement sets

The definition, in one line

y and x are generalized-cointegrated if, holding all other relevant preexisting conditions w constant, the bias-free part of ∂y/∂x is nonzero.

Cointegration is thus a property of the real-world structural relationship, not of any particular statistical model — and it survives nonlinearity and omitted regressors.

---

The estimator

Substituting the driver equation (8) into the TVC model (7) gives a model that is linear in the stacked parameter vector π:

y_t = w_t' π + u_t ,   w_t = ( x_0t·z_t , x_1t·z_t , … ) ,   u_t = Σ_j x_jt ε_jt
Var(u_t | x_t) = Σ_j σ_j² x_jt²          (Hildreth–Houck–Swamy structure)

tvccointreg then:

1. OLS start on the concentrated design W.
2. Variance components σ_j² from a non-negative regression of squared residuals on squared regressors (scipy.optimize.nnls).
3. GLS with weights 1/h_t, h_t = Σ_j σ_j² x_jt²; iterate to convergence (iteratively rescaled GLS).
4. Recover the time-varying coefficients γ_jt = z_t'π_j + ε̂_jt, where the random part is the best linear predictor ε̂_jt = σ_j² x_jt u_t / h_t.
5. Decompose γ_jt into bias-free / omitted / measurement parts using the driver-set masks, and test the bias-free block.

Inference uses the GLS covariance (W'Ω⁻¹W)⁻¹ (the paper's standard-inference result) or a heteroskedasticity-robust sandwich (cov_type="robust").

Note / honest caveat. As in the paper, validity hinges on Assumption 1 — that the chosen drivers genuinely span the bias components. This is an identifying assumption, not something the data can confirm; choose drivers that are plausibly correlated with the suspected misspecification (Table 1).

---

API reference

TVCModel(y, X, drivers, driver_spec=None, driver_sets=None, add_const=True, index=None)

Build a model. y (T,), X (T, K−1), drivers (T, m). A constant regressor and a constant driver are added automatically. Accepts NumPy arrays or pandas objects (column names are picked up automatically).

• .fit(method="irgls", max_iter=100, tol=1e-8, cov_type="gls", verbose=False) → TVCResults.

DriverSpec(names, bias_free=[], omitted=[], measurement=[])

Three-set partition of the drivers. Any driver not listed defaults to bias_free. .describe() prints the partition.

TVCResults

Method	Returns
`.summary(fmt="text"	"latex"
.coef_table()	per-regressor average coefficient frame
.coint_test(alpha=0.05, skip_const=True)	tidy DataFrame of test results (and .coint_results_)
.tv_coefficients(include_random=True)	T×K time-varying coefficients
.bias_free_coefficients()	T×K bias-free (structural) coefficients
.components(regressor)	bias-free / omitted / measurement / random / total
.coefficient_se()	pointwise standard errors
.diagnostics()	R², log-lik, convergence, residual ADF
.plot_coefficients(...)	TVC paths with CI bands
.plot_decomposition(regressor)	the three-component decomposition
.plot_fit()	actual vs fitted + residuals
.plot_coint_heatmap()	Parula heatmap of bias-free paths

Datasets

• simulate_nonlinear_cointegration(T, seed, nonlinearity, omitted, measurement_error)
• simulate_spurious(T, seed)

Colormaps

parula_colors(n), matlab_jet_colors(n), turbo_colors(n), bluered_colors(n), sinha_colors(n), resolve_colorscale("Parula", n), get_cmap("parula").

---

Detailed syntax

Using raw NumPy arrays and reading off a specific test:

import numpy as np
from tvccointreg import TVCModel, DriverSpec

res = TVCModel(y, X, Z,                       # y:(T,), X:(T,K-1), Z:(T,m)
               driver_sets={"bias_free": ["z1"],
                            "omitted":   ["z2"],
                            "measurement": ["z3"]}).fit()

ct = res.coint_test()
print(ct.loc["x1", "cointegrated"], ct.loc["x1", "p_value"])

Robust covariance and OLS (homoskedastic) comparison:

res_robust = TVCModel(y, X, Z, driver_spec=spec).fit(cov_type="robust")
res_ols    = TVCModel(y, X, Z, driver_spec=spec).fit(method="ols")

Exporting a LaTeX table for a paper:

with open("table1.tex", "w") as f:
    f.write(res.summary(fmt="latex"))

Pulling the time-varying coefficient path of one regressor:

gamma_x = res.tv_coefficients()["x"]          # pandas Series indexed by t
bf_x    = res.bias_free_coefficients()["x"]

---

Visualizations

All plots default to the MATLAB Parula colormap and return a matplotlib Figure.

plot_coefficients()	plot_decomposition("x")
plot_fit()	plot_coint_heatmap()

---

Worked example: spurious vs. real

examples/spurious_vs_real.py reproduces the central point of the paper:

================ SPURIOUS (independent random walks) ================
Naive OLS slope = 1.125  (p = 5.97e-46)   <- spuriously 'significant'
Generalized cointegration: NO (p = 0.708)

================ GENUINE generalized cointegration ================
Generalized cointegration: YES (p = 1.34e-35)

A naive OLS regression of one random walk on another is wildly "significant", yet the generalized-cointegration test on the bias-free component correctly finds no structural relationship.

---

Empirical application: the US consumption function (real data)

examples/real_data_consumption.py applies the method to real US quarterly macro data (statsmodels macrodata, 1959Q1–2009Q3, 202 obs after lagging). We model the long-run relationship between real personal consumption and real disposable income (both in logs, both strongly trending / I(1)):

import numpy as np, pandas as pd
from statsmodels.datasets import macrodata
from tvccointreg import TVCModel, DriverSpec

d = macrodata.load_pandas().data
d.index = pd.period_range("1959Q1", periods=len(d), freq="Q")
zscore = lambda s: (s - s.mean()) / s.std(ddof=0)

y = np.log(d["realcons"]).rename("log_cons")
X = np.log(d[["realdpi"]]).rename(columns={"realdpi": "log_inc"})
drivers = pd.DataFrame({
    "inc_lag":  zscore(np.log(d["realdpi"]).shift(1)),
    "trend":    zscore(pd.Series(np.arange(len(d)), index=d.index)),
    "realint":  zscore(d["realint"]),
    "unemp":    zscore(d["unemp"]),
    "cons_lag": zscore(np.log(d["realcons"]).shift(1)),
})
df = pd.concat([y, X, drivers], axis=1).dropna()

spec = DriverSpec(
    names=["inc_lag", "trend", "realint", "unemp", "cons_lag"],
    bias_free=["inc_lag", "trend"],   # true elasticity variation
    omitted=["realint", "unemp"],     # interest-rate / labour-market channels
    measurement=["cons_lag"],         # dynamics / measurement
)
res = TVCModel(df["log_cons"], df[["log_inc"]], df[spec.names],
               driver_spec=spec, index=df.index.astype(str)).fit()
print(res.summary())
print(res.coint_test())

Result (verbatim output):

==============================================================================
      Time-Varying-Coefficient Regression  /  Generalized Cointegration
==============================================================================
Dep. variable: log_cons                No. observations: 202
No. coefficients: 2                    No. drivers: 5
Estimator: Iteratively rescaled GLS    Covariance: GLS
R-squared: 0.9999                      Log-likelihood: 752.32
Converged: True                        Resid ADF p-value: 0.0000
==============================================================================
Variable    Coef (mean)  Bias-free    Std.Err          t    p-value    G-Coint
------------------------------------------------------------------------------
log_inc          0.3886     0.3886     0.0528     7.3626     0.0000 ***     Yes
==============================================================================

regressor	avg_bias_free	std_err	t_stat	wald	df	p_value	cointegrated
log_inc	0.3886	0.0528	7.3626	56.9504	3	0.0000	True

Reading the result.

• Generalized cointegration is confirmed between consumption and income (Wald = 56.95, 3 df, p ≈ 0): the bias-free structural elasticity is firmly nonzero, so the long-run relationship is genuine, not spurious.
• Standard inference is valid here: the composite residuals are stationary (ADF p = 1.9 × 10⁻⁷), exactly the condition under which Hall–Swamy–Tavlas show the TVC test uses ordinary χ²/normal critical values rather than Dickey–Fuller ones.
• The bias-free income elasticity drifts down over the sample, from 0.43 (1959Q2) to 0.34 (2009Q3) — the direct income channel after the dynamics (cons_lag) and omitted business-cycle conditions (realint, unemp) are stripped out into the bias terms. A declining direct sensitivity of consumption to current income over five decades is consistent with greater consumption smoothing / financial deepening.

Bias-free income elasticity over time	Coefficient decomposition

Run it yourself:

python examples/real_data_consumption.py

Multiple regressors

The framework handles several regressors at once — each gets its own time-varying coefficient and its own cointegration verdict. examples/consumption_multivariate.py estimates a two-regressor consumption function (disposable income and the real interest rate) on the same data:

Regressor	Bias-free coef	t	Wald	p-value	Cointegrated
log_realDPI (income)	0.6666 ***	10.13	111.40	<0.0001	Yes
realint (real rate)	−0.0007 **	−2.26	8.37	0.039	Yes

Positive income elasticity (~0.67) and a negative real-rate effect (intertemporal substitution); R² = 0.9998, residuals stationary (ADF p = 3.5×10⁻⁹). See the rendered results on the documentation site.

---

Testing

pip install -e ".[dev]"
pytest -q

The suite checks coefficient recovery, that the three components sum exactly to the total coefficient, detection of genuine generalized cointegration, rejection of spurious relationships, and the Parula palette/colorscale helpers.

---

Publishing (maintainer notes)

The package is PyPI-ready (python -m build + twine check both pass).

First release — manual upload with an API token

1. Create accounts on TestPyPI and PyPI, and generate an API token for each (Account settings → API tokens).
2. Build and check:

   python -m build
   python -m twine check dist/*
   

3. Dry-run on TestPyPI first, then install from there to confirm:

   python -m twine upload --repository testpypi dist/*
   pip install --index-url https://test.pypi.org/simple/ \
       --extra-index-url https://pypi.org/simple/ tvccointreg
   

4. Upload to the real PyPI:

   python -m twine upload dist/*
   

   When prompted, use __token__ as the username and the API token (starting with pypi-) as the password.

Subsequent releases — automated via Trusted Publishing (recommended)

.github/workflows/publish.yml publishes automatically when you publish a GitHub Release. One-time setup: on PyPI add a Trusted Publisher (Project → Publishing) for repository merwanroudane/tvccointreg, workflow publish.yml, environment pypi. After that, no tokens or secrets are needed — just bump version in pyproject.toml and __version__, tag, and publish a Release.

Remember: a version number can only be uploaded to PyPI once. Bump the version for every release.

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Citing

If you use this package, please cite both the package and the underlying paper.

@software{roudane_tvccointreg,
  author  = {Merwan Roudane},
  title   = {tvccointreg: Time-Varying-Coefficient Regression and
             Generalized Cointegration in Python},
  year    = {2026},
  url     = {https://github.com/merwanroudane/tvccointreg}
}

@incollection{hall_swamy_tavlas_2015,
  author  = {Hall, Stephen G. and Swamy, P. A. V. B. and Tavlas, George S.},
  title   = {A Note on Generalizing the Concept of Cointegration},
  year    = {2015}
}

---

References

• Hall, S. G., Swamy, P. A. V. B., & Tavlas, G. S. (2015). A Note on Generalizing the Concept of Cointegration.
• Hall, S. G., Swamy, P. A. V. B., & Tavlas, G. S. (2014). Time Varying Coefficient Models; A Proposal for Selecting the Coefficient Driver Sets. University of Leicester WP 14/18.
• Swamy, P. A. V. B., & Mehta, J. S. (1975). Bayesian and non-Bayesian Analysis of Switching Regressions and of Random Coefficient Regression Models. JASA.
• Swamy, P. A. V. B., Tavlas, G. S., Hall, S. G., et al. (2010). Nonparametric Nonstationary Regression.
• Granger, C. W. J. (2008). Non-linear Models: Where Do We Go Next — Time-Varying Parameter Models? Studies in Nonlinear Dynamics & Econometrics.
• Hildreth, C., & Houck, J. P. (1968). Some Estimators for a Linear Model with Random Coefficients. JASA.
• Engle, R. F., & Granger, C. W. J. (1987). Co-integration and Error Correction. Econometrica.
• Cramér, H. (1946). Mathematical Methods of Statistics.

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Author

Dr Merwan Roudane · [e-mail] · github.com/merwanroudane

Released under the MIT License.