Conditional-Skew

This project fits autoregressive and regime-switching time-series models with non-Gaussian noise, starting from plain AR models and extending to GED, SGED, and level-dependent SGED specifications.

The current examples focus on VIX and SPY, with particular attention to whether the conditional distribution of noise changes as VIX changes, and whether latent Markov regimes improve fit relative to a single-regime AR model.

In terms of prior literature, there is extensive work on fitting GED innovations to financial returns and a smaller literature on fitting SGED distributions to returns or conditional-volatility models. By contrast, there appears to be comparatively little work on SGED models in which the scale, skewness, and tail-shape parameters are explicit functions of a lagged state variable such as the level or log-level of VIX. That is the main direction explored here.

A useful reference for the SGED itself is Panayiotis Theodossiou, Skewed Generalized Error Distribution of Financial Assets and Option Pricing

Files

From simplest to most complex:

• xxar.py
  Fits ordinary autoregressive models to each time series in a CSV file, prints autocorrelations, and reports residual diagnostics.

• ar_noise_report.py
  Shared reporting helpers for residual standard deviation, skew, excess kurtosis, and bin-by-bin summaries based on the previous level.

• xar_ged.py
  Fits AR models with GED innovations. This allows heavy tails but not skewness.

• ar_ged_model.py
  Core GED fitting utilities used by xar_ged.py.

• xar_sged.py
  Fits AR models with skewed GED innovations, adds standard errors, implied skew and excess kurtosis, and supports analytic or numerical Hessians.

• ar_sged_model.py
  Core SGED fitting code for constant-parameter models.

• xar_sged_level.py
  Fits AR models where SGED noise parameters vary with the previous level or log-level of the series. This is the main single-series script.

• ar_sged_level_model.py
  Core level-dependent SGED fitting utilities used by xar_sged_level.py.

• xmarkov_autoreg_n_regimes.py
  Fits Gaussian Markov-switching autoregressions using statsmodels over a range of regime counts, reports residual moment diagnostics, smoothed regime probabilities, probability-weighted regime levels, and compares AIC and BIC across k_regimes.

• xvix_spy_sged.py
  Two-series application for vix_spy.csv. The first column is treated as the driver series (VIX), while the second column (SPY) is analyzed through returns and returns divided by lagged VIX.

Model progression

The scripts reflect a progression in modeling assumptions:

1. xxar.py
   Mean dynamics only. Residuals are examined after fitting a standard AR model.

2. xar_ged.py
   Keeps the AR mean but replaces Gaussian noise with GED noise, allowing tail thickness to differ from normality.

3. xar_sged.py
   Adds skewness to the innovation distribution. This allows the noise to be both heavy-tailed and asymmetric.

4. xar_sged_level.py
   Allows the SGED parameters themselves to vary with the previous level. In practice this means volatility, skewness, and tail behavior can all change as the state variable changes.

5. xmarkov_autoreg_n_regimes.py
   Lets the number of Markov regimes vary and selects the preferred count by information criteria.

6. xvix_spy_sged.py
   Uses VIX as the state variable and studies both SPY returns and SPY returns normalized by lagged VIX.

Main result from xar_sged_level.py

The key result is that a constant-noise SGED model is not enough for log(VIX). A model in which the SGED noise parameters vary with the previous log(VIX) fits materially better.

Setup

• Data file: vix.csv
• Date range: 1990-01-02 to 2026-04-24
• Number of observations: 9145
• AR order selected: 1
• Trend specification: constant only (trend = "c")
• Driver for time-varying noise parameters: previous log(VIX)

Basic properties of the series

The autocorrelation function is extremely persistent:

• lag 1 ACF: 0.980
• lag 12 ACF: 0.863

So a persistent AR(1) structure is a reasonable baseline.

Constant-parameter SGED benchmark

The benchmark AR(1)-SGED fit is:

• BIC: -24735.690
• AR coefficients: [0.051, 0.977]
• beta = 1.111
• xi = 1.173
• scale = 0.054

Interpretation:

• beta < 2 implies heavier tails than Gaussian.
• beta = 1.111 is also fairly close to the Laplace case beta = 1, so the fitted innovation density is much closer to Laplace-like than Gaussian.
• xi > 1 implies positive skew in the fitted noise.
• The test for xi = 1 gives z = 11.712, p = 1.108e-31, so the fitted skew is decisively nonzero.

The implied innovation moments from this constant-SGED fit are:

• implied skew: 0.565
• implied excess kurtosis: 2.427

Level-dependent SGED model

The richer model lets the SGED parameters depend on a standardized previous-level driver z:

• log(scale_t) = a0 + a1*z
• log(beta_t) = b0 + b1*z
• log(xi_t) = c0 + c1*z

Estimated coefficients:

• AR coefficients: [0.032, 0.984]
• a0 = -2.892, a1 = 0.167
• b0 = 0.142, b1 = 0.018
• c0 = 0.167, c1 = -0.036

Interpretation:

• a1 > 0: the noise scale rises as VIX rises.
• b1 > 0: the GED shape parameter rises mildly with VIX.
• c1 < 0: the skew parameter falls as VIX rises.

How the implied noise distribution changes with VIX

Across previous-VIX quantiles:

q	VIX level	scale	beta	xi	implied skew	implied excess kurtosis
1%	10.160	0.042	1.116	1.258	0.779	2.591
5%	11.430	0.044	1.123	1.243	0.735	2.504
10%	12.140	0.045	1.127	1.235	0.713	2.461
25%	13.960	0.049	1.135	1.217	0.661	2.365
50%	17.620	0.054	1.150	1.187	0.574	2.214
75%	22.763	0.062	1.166	1.155	0.479	2.063
90%	28.597	0.069	1.180	1.127	0.395	1.942
95%	32.989	0.074	1.189	1.110	0.343	1.872
99%	46.766	0.088	1.212	1.070	0.218	1.721

The pattern is clear:

• higher VIX is associated with larger innovation scale
• higher VIX is associated with lower positive skew
• higher VIX is associated with lower implied excess kurtosis

So the conditional noise distribution changes meaningfully with the level of VIX, not just its variance.

Model comparison

The BIC comparison is decisive:

• constant-parameter SGED BIC: -24735.690
• level-dependent SGED BIC: -24925.226
• delta BIC (level - constant): -189.537

Since lower BIC is better, BIC strongly favors the level-dependent noise model.

Markov AR results from vix_markov.txt

The Markov-switching results give a different route to non-Gaussian behavior. Instead of making the innovation law depend explicitly on lagged VIX, they treat the series as switching across latent regimes with regime-specific intercepts and variances.

Setup

• Data file: vix.csv
• Date range: 1990-01-02 to 2026-04-24
• Series: log(VIX)
• AR order: 1
• Trend: constant
• Switching AR coefficient: False
• Switching trend: True
• Switching variance: True
• Regime counts compared: 1, 2, and 3

Information-criterion comparison

The model comparison in vix_markov.txt is:

# regimes	AIC	BIC
1	-23254.593	-23233.231
2	-25392.322	-25342.476
3	-25713.231	-25620.660

Both AIC and BIC choose 3 regimes.

So a latent-regime Gaussian model substantially improves fit relative to a plain AR(1), and a 3-state version improves further on the 2-state version.

Three-regime fit

For the preferred 3-regime model:

• log-likelihood: 12869.616
• AIC: -25713.231
• BIC: -25620.660
• AR coefficient: 0.9875

The average regime probabilities and probability-weighted levels are:

state	probability	avg level
2	0.5564	17.925
1	0.3525	20.340
0	0.0911	25.480

This suggests:

• one dominant lower-level regime
• one medium-probability mid-level regime
• one relatively rare high-level regime

The regimes therefore line up reasonably well with low-, medium-, and high-VIX states.

Residual moments

The residual moment diagnostics from vix_markov.txt are:

# regimes	sd	skew	excess kurtosis
1	0.068	1.057	6.806
2	0.061	0.976	8.080
3	0.051	1.346	14.106

Two points matter here:

1. These moments show that the pooled one-step residuals remain strongly non-Gaussian.
2. In a Markov-switching Gaussian model, pooled residuals need not look Gaussian even when the regime-conditional errors are Gaussian, because they are a mixture over latent states.

So these residual moments are useful descriptively, but they are not a clean specification test for the Markov model in the same way they are for a single-regime Gaussian AR fit.

The right interpretation is:

• the Gaussian AR(1) model is plainly inadequate
• latent Markov regimes improve fit a great deal
• the pooled forecast errors still show large positive skew and excess kurtosis
• this is consistent with either remaining misspecification or simply the fact that a latent-state Gaussian mixture is non-Gaussian when pooled over time

Markov AR versus level-dependent SGED

The two main approaches in the repository address non-Gaussian behavior differently:

• xar_sged_level.py uses an observed state variable, lagged VIX, to let the innovation law change deterministically with the previous level.
• xmarkov_autoreg_n_regimes.py uses latent regimes to let intercepts and variances change probabilistically over time.

So the Markov AR results do not replace the level-dependent SGED results. They provide a second lens on the same broad fact: log(VIX) is highly persistent, and its conditional distribution is not well described by a single Gaussian AR(1) innovation law.

Using the scripts

The scripts are simple command-line programs with parameters set near the top of each file.

Typical controls include:

• input file name
• AR lag range
• trend specification
• date range
• whether to use logs
• whether to show AIC and/or BIC
• number of previous-level bins
• whether to fit constant or level-dependent SGED models
• number of Markov regimes to fit or compare
• whether to export regime probabilities

For the two-series VIX/SPY script, there are also toggles to turn the SPY returns and SPY returns / VIX(t-1) sections on or off.

Summary

The main empirical message of this project is that for VIX, and for SPY returns conditioned on VIX, the distribution of innovations is not well described by a fixed Gaussian law. The codebase explores two complementary ways to capture that:

• observed-state models, where SGED parameters vary with lagged VIX
• latent-state models, where AR dynamics switch across hidden Markov regimes

Both approaches reject the idea that a single-regime Gaussian AR(1) is adequate for log(VIX).