GARCH-from-scratch

Pure-Python (numpy/pandas) simulation and basic theory checks for:

• symmetric GARCH(1,1) with Student t innovations
• GJR-GARCH(1,1) (a.k.a. GJR / threshold GARCH) with Student t innovations

The scripts compare empirical autocorrelations from simulation to closed-form theoretical autocorrelations of squared returns. The GJR script also compares the leverage cross-correlation corr(ε_t, ε_{t+k}^2) to its theoretical geometric decay (the level uses a simulation estimate of E[h^(3/2)]).

All formulas in the code assume symmetric innovations with E[z^2]=1.

Files

• garch.py

  • core simulation functions
    • simulate_garch_t(...)
    • simulate_gjr_garch_t(...)
  • helper stats
    • acf(...)
    • crosscorr(...)
    • m4_student_t_standardized(...)
    • abs_moment_student_t_standardized(...)
  • theory
    • theoretical_acf_sq_returns_garch(...)
    • theoretical_acf_sq_returns_gjr(...)
    • theoretical_crosscorr_return_future_sq(...) (GJR leverage cross-corr)

• xgarch.py

  • main program for symmetric GARCH(1,1)
  • prints a table comparing theoretical vs empirical ACF of squared returns

• xgarch_gjr.py

  • main program for GJR-GARCH(1,1)
  • prints a table comparing theoretical vs empirical ACF of squared returns
  • prints a table comparing theoretical vs empirical corr(ε_t, ε_{t+k}^2)

Quick start

Install dependencies:

pip install numpy pandas

Run symmetric GARCH(1,1):

python xgarch.py

Run GJR-GARCH(1,1):

python xgarch_gjr.py

Both scripts run multiple simulations with different seeds and print:

• the ACF comparison table(s) as pandas DataFrames
• an RMSE across lags as a quick "match quality" metric

Model definitions

Standardized Student t innovations

The code uses numpy's standard_t(df=ν) and rescales it so Var(z)=1:

• if x ~ t_ν, then Var(x) = ν/(ν-2) for ν>2
• set z = x * sqrt((ν-2)/ν) so Var(z)=1

This is what the simulators use in:

• simulate_garch_t
• simulate_gjr_garch_t

Symmetric GARCH(1,1)

ε_t = sqrt(h_t) * z_t

h_t = ω + α * ε_{t-1}^2 + β * h_{t-1}

Stationarity (finite unconditional variance) requires:

κ = α + β < 1

GJR-GARCH(1,1)

ε_t = sqrt(h_t) * z_t

h_t = ω + α * ε_{t-1}^2 + γ * ε_{t-1}^2 * 1{ε_{t-1}<0} + β * h_{t-1}

For symmetric z_t, P(ε_{t-1}<0)=1/2, so stationarity (finite unconditional variance) requires:

κ = α + β + 0.5*γ < 1

Theoretical ACF of squared returns

Let s_t = ε_t^2. Under symmetric innovations with E[z^2]=1 and finite fourth moment, the autocorrelation of s_t is geometric:

ρ_s(k) = Corr(s_{t+k}, s_t) = ρ_s(1) * κ^(k-1), k>=1

The code computes ρ_s(1) in closed form by solving for E[h] and E[h^2].

Fourth-moment condition

To have finite Var(ε_t^2) (and therefore a well-defined ACF of squared returns), you also need a finite fourth moment and a contraction condition for E[h^2]. The code checks this via η < 1.

For standardized Student t, you need ν > 4 and:

m4 = E[z^4] = 3*(ν-2)/(ν-4)

Symmetric GARCH(1,1): η

η = β^2 + 2αβ + α^2*m4

GJR-GARCH(1,1): η

Define:

a2 = α^2 + αγ + 0.5γ^2

η = β^2 + 2β(α + 0.5γ) + a2m4

Leverage cross-correlation for GJR

The GJR script also reports:

corr(ε_t, ε_{t+k}^2), k>=1

For symmetric z_t, this cross-correlation decays geometrically at the same rate κ. The level depends on E[h^(3/2)], which is not computed in closed form in this repo; xgarch_gjr.py estimates it from the simulated path and plugs it into theoretical_crosscorr_return_future_sq.

For symmetric GARCH(1,1), the corresponding leverage cross-correlations are zero by symmetry, so xgarch.py does not compute them.

Customizing runs

Edit the constants near the top of main() in xgarch.py or xgarch_gjr.py:

• ω, α, β, ν (and γ for GJR)
• nobs number of kept observations
• burn burn-in length
• nlags number of ACF lags to compare
• nsim number of simulations
• seed schedule (currently seed0 + 1000000*i)

Notes

• This repo is intended for learning, sanity checks, and small experiments.
• Numerical safeguards:
  • floor prevents h_t from going non-positive due to rounding.
• If you choose parameters with κ >= 1, the process has no finite unconditional variance; the theory checks in theoretical_* will raise.