1.1 Foundations of Time Collection Evaluation
The fashionable concept of time collection evaluation traces its origins to the work of Xmas (1927), who launched the autoregressive mannequin, and Slutsky (1937), who demonstrated that transferring common processes might generate apparently cyclical habits from random shocks. The synthesis of those concepts by Field and Jenkins (1970) into the ARIMA modeling framework established the usual methodology for time collection identification, estimation, and forecasting that is still extensively used as we speak. The Field-Jenkins method prescribes a three-stage iterative process: (i) mannequin identification by way of examination of the autocorrelation operate (ACF) and partial autocorrelation operate (PACF); (ii) parameter estimation by way of most probability or conditional least squares; and (iii) diagnostic checking by way of residual evaluation.
Within the context of economic return collection, Hamilton (1994) offered a complete remedy of time collection strategies, whereas Tsay (2010) specialised the framework for monetary functions. A important perception from this literature is that monetary returns sometimes exhibit weak serial dependence in ranges however sturdy dependence in squared returns—a phenomenon referred to as volatility clustering. This remark motivated the event of conditional heteroskedasticity fashions, as conventional ARMA fashions assume a continuing unconditional variance.
1.2 ARCH and GARCH Fashions: Improvement and Extensions
Engle’s (1982) ARCH mannequin was the primary to formally parameterize time-varying conditional variance as a operate of previous squared improvements. The GARCH(1,1) generalization by Bollerslev (1986) launched lagged conditional variance phrases, yielding a extra parsimonious specification that captured long-memory results in volatility. Nelson (1991) proposed the Exponential GARCH (EGARCH) mannequin to deal with the leverage impact [1] —the uneven volatility response to optimistic and adverse return shocks—whereas Glosten, Jagannathan, and Runkle (1993) developed the GJR-GARCH threshold mannequin for a similar goal. Zakoian (1994) launched the Threshold GARCH (TGARCH) mannequin with an alternate uneven specification.
The empirical adequacy of GARCH fashions has been extensively studied. Hansen and Lunde (2005) in contrast 330 ARCH-type fashions for trade charge and fairness return volatility and located that the straightforward GARCH(1,1) is remarkably troublesome to outperform [2] . Andersen and Bollerslev (1998) reconciled the obvious poor efficiency of GARCH fashions in predicting realized volatility by demonstrating that a lot of the forecast analysis literature used noisy proxies for the true latent variance course of. Subsequent work by Andersen, Bollerslev, Diebold, and Labys (2003) launched realized volatility estimators based mostly on high-frequency knowledge, offering extra correct benchmarks for evaluating conditional variance fashions.
1.3 Time Collection Fashions in Algorithmic Buying and selling
The appliance of time collection fashions to buying and selling technique growth has a considerable historical past. Chan (2009) offered a practitioner-oriented overview of statistical arbitrage methods based mostly on mean-reverting time collection processes. Avellaneda and Lee (2010) developed pairs buying and selling methods grounded in cointegration concept, an extension of univariate time collection strategies to multivariate settings. Extra just lately, the mixing of machine studying with classical time collection strategies has attracted appreciable consideration (Dixon, Halperin, and Bilokon, 2020), although the elemental constructing blocks stay the ARMA and GARCH specs.
A number of research have particularly examined GARCH-based buying and selling methods. Engle and Sokalska (2012) proposed intraday volatility-based buying and selling guidelines. Alexander and Lazar (2006) evaluated GARCH-based Worth-at-Danger fashions for place sizing in commodity buying and selling. Brownlees, Engle, and Kelly (2011) developed a sensible framework for volatility forecasting in threat administration functions. Regardless of this physique of labor, a rigorous educational remedy that bridges the hole between GARCH concept and full-stack algorithmic buying and selling system design stays conspicuously absent from the literature.
1.4 Volatility Forecasting and Financial Worth
The query of whether or not superior volatility forecasts translate into financial worth has been addressed from a number of angles. Fleming, Kirby, and Ostdiek (2001, 2003) demonstrated that volatility timing methods based mostly on conditional variance forecasts can generate substantial financial features for mean-variance buyers, even after accounting for transaction prices. West, Edison, and Cho (1993) established the theoretical circumstances below which improved volatility forecasts yield larger anticipated utility for portfolio buyers. Engle and Colacito (2006) developed a model-confidence-set method to volatility forecast analysis that instantly maps to financial loss capabilities. Our analysis builds upon this custom by embedding GARCH volatility forecasts inside an entire algorithmic buying and selling structure and evaluating financial efficiency by way of strategy-level metrics together with Sharpe ratio, most drawdown, and Calmar ratio.
2.1 Stochastic Processes and Stationarity
Let (Ω, ℱ, P) denote a likelihood house. A discrete-time stochastic course of {Xₜ}ₜ∈ℤ is a group of random variables outlined on this house, listed by integer-valued time. We undertake the next definition of covariance stationarity.
Definition 3.1 (Covariance Stationarity). A stochastic course of {Xₜ} is covariance stationary (weakly stationary) if (i) E[Xₜ] = μ for all t; (ii) Var(Xₜ) = γ(0) < ∞ for all t; and (iii) Cov(Xₜ, Xₜ₋ₖ) = γ(ok) relies upon solely on the lag ok and never on t.
For monetary functions, we outline the log-return course of rₜ = ln(Pₜ/Pₜ₋₁), the place Pₜ denotes the asset value at time t. Underneath normal regularity circumstances, the return course of rₜ is usually (no less than roughly) covariance stationary, even when the worth course of Pₜ is non-stationary (built-in of order one). This remark motivates the frequent observe of modeling returns slightly than value ranges.
2.2 Autoregressive and Shifting Common Processes
Definition 3.2 (AR(p) Course of). A stochastic course of {rₜ} follows an autoregressive means of order p, denoted AR(p), if rₜ = c + ϕ₁rₜ₋₁ + ϕ₂rₜ₋₂ + … + ϕₚrₜ₋ₚ + εₜ, the place c is a continuing, ϕ₁, …, ϕₚ are autoregressive coefficients, and {εₜ} is a white noise course of with E[εₜ] = 0 and Var(εₜ) = σ².
The AR(p) course of is covariance stationary if and provided that all roots of the attribute polynomial Φ(z) = 1 − ϕ₁z − ϕ₂z² − … − ϕₚzᵖ lie outdoors the unit circle. Underneath stationarity, the method admits the infinite transferring common (Wold) illustration rₜ = μ + ∑ⱼ₌₀^∞ ψⱼεₜ₋ⱼ, the place the coefficients ψⱼ are completely summable.
Definition 3.3 (MA(q) Course of). A stochastic course of {rₜ} follows a transferring common means of order q, denoted MA(q), if rₜ = c + εₜ + θ₁εₜ₋₁ + θ₂εₜ₋₂ + … + θₙεₜ₋ₙ, the place θ₁, …, θₙ are transferring common coefficients.
An MA(q) course of is at all times covariance stationary whatever the parameter values. Nevertheless, for the method to be invertible—a requirement for distinctive identification—all roots of the MA attribute polynomial Θ(z) = 1 + θ₁z + θ₂z² + … + θₙz⁾ should lie outdoors the unit circle.
2.3 ARIMA Processes
Definition 3.4 (ARIMA(p,d,q) Course of). A stochastic course of {Xₜ} follows an autoregressive built-in transferring common means of orders (p, d, q) if the d-th distinction (1 − L)ᵈ Xₜ is a stationary and invertible ARMA(p,q) course of, the place L denotes the lag operator.
For monetary return collection, d = 0 is the standard case (since returns are typically stationary), lowering the ARIMA specification to an ARMA mannequin. Nevertheless, the ARIMA framework is related when modeling built-in variables equivalent to cumulative returns, asset costs, or rate of interest ranges that require differencing to realize stationarity.
2.4 The ARCH Mannequin
Engle (1982) launched the ARCH mannequin to formalize the remark that the conditional variance of economic returns varies over time in a predictable method. Contemplate the return course of decomposition rₜ = μₜ + εₜ, the place μₜ = E[rₜ | ℱₜ₋₁] is the conditional imply and εₜ is the innovation.
Definition 3.5 (ARCH(m) Course of). The innovation course of εₜ follows an ARCH(m) course of if εₜ = σₜ zₜ, the place zₜ ~ i.i.d.(0,1), and the conditional variance is given by σ²ₜ = α₀ + α₁ε²ₜ₋₁ + α₂ε²ₜ₋₂ + … + αₘε²ₜ₋ₘ, the place α₀ > 0 and αᵢ ≥ 0 for i = 1, …, m.
The positivity constraints on the αᵢ parameters be sure that the conditional variance is non-negative. The unconditional variance of εₜ is given by Var(εₜ) = α₀ / (1 − ∑ᵢ₌₁ᵐ αᵢ), which exists and is finite if and provided that ∑ᵢ₌₁ᵐ αᵢ < 1. The kurtosis of ARCH improvements exceeds 3 (the Gaussian worth), thereby producing the fat-tailed distributions noticed in empirical return knowledge.
2.5 The GARCH Mannequin
Definition 3.6 (GARCH(p,q) Course of). The innovation course of εₜ follows a GARCH(p,q) course of if εₜ = σₜ zₜ, the place zₜ ~ i.i.d.(0,1), and the conditional variance satisfies σ²ₜ = ω + ∑ᵢ₌₁ᵖ αᵢε²ₜ₋ᵢ + ∑ⱼ₌₁ᵐ βⱼσ²ₜ₋ⱼ, the place ω > 0, αᵢ ≥ 0, and βⱼ ≥ 0.
The GARCH(1,1) specification, σ²ₜ = ω + αε²ₜ₋₁ + βσ²ₜ₋₁, is the workhorse mannequin in monetary econometrics on account of its parsimony and empirical adequacy. The persistence of volatility shocks is ruled by the sum α + β, the place values near unity point out excessive persistence (the Built-in GARCH or IGARCH case). The unconditional variance equals ω/(1 − α − β) when α + β < 1.
Theorem 3.1 (Stationarity of GARCH(1,1)). The GARCH(1,1) course of εₜ = σₜ zₜ with σ²ₜ = ω + αε²ₜ₋₁ + βσ²ₜ₋₁ is strictly stationary and ergodic if and provided that E[ln(αz²ₜ + β)] < 0. The method is covariance stationary if and provided that α + β < 1.
Proof. The strict stationarity situation follows from the speculation of random recurrence equations (Bougerol and Picard, 1992). The GARCH(1,1) conditional variance could be written as σ²ₜ = ω + (αz²ₜ₋₁ + β)σ²ₜ₋₁, which is a stochastic recurrence of the shape Yₜ = Aₜ Yₜ₋₁ + Bₜ with Aₜ = αz²ₜ + β and Bₜ = ω. By the theory of Bougerol and Picard, a novel strictly stationary resolution exists if and provided that the highest Lyapunov exponent is adverse, i.e., E[ln Aₜ] = E[ln(αz²ₜ + β)] < 0. The covariance stationarity situation follows by taking unconditional expectations: E[σ²ₜ] = ω + (α + β)E[σ²ₜ₋₁], yielding E[σ²ₜ] = ω/(1 − α − β) offered α + β < 1. □
2.6 Uneven GARCH Extensions
2.6.1 The EGARCH Mannequin
The Exponential GARCH mannequin of Nelson (1991) specifies the logarithm of the conditional variance as ln(σ²ₜ) = ω + α(|zₜ₋₁| − E[|zₜ₋₁|]) + γzₜ₋₁ + β ln(σ²ₜ₋₁). The leverage impact is captured by the parameter γ: when γ < 0, adverse return shocks enhance volatility greater than optimistic shocks of equal magnitude. A key benefit of the EGARCH specification is that no non-negativity constraints are required on the parameters, for the reason that exponential operate ensures σ²ₜ > 0 robotically.
2.6.2 The GJR-GARCH Mannequin
The GJR-GARCH mannequin (Glosten, Jagannathan, and Runkle, 1993) augments the usual GARCH specification with an indicator operate: σ²ₜ = ω + αε²ₜ₋₁ + γε²ₜ₋₁ · I(εₜ₋₁ < 0) + βσ²ₜ₋₁, the place I(·) is the indicator operate. The parameter γ captures the differential affect of adverse shocks. The information affect curve for the GJR-GARCH mannequin is piecewise linear, with slope α for optimistic improvements and slope α + γ for adverse improvements.
2.7 Innovation Distributions
Whereas the usual GARCH specification assumes Gaussian improvements zₜ ~ N(0,1), empirical proof strongly favors fat-tailed distributions. The Scholar-t distribution with ν levels of freedom offers a pure extension, with the density f(z; ν) = [Γ((ν+1)/2) / (√(π(ν−2)) Γ(ν/2))] · (1 + z²/(ν−2))⁻⁽ᵛ⁺¹⁾ᐟ², outlined for ν > 2. As ν → ∞, the Scholar-t converges to the Gaussian. An alternate is the Generalized Error Distribution (GED), which nests the Gaussian (ν = 2) and permits for each thinner and thicker tails.
3. Estimation Methodology and Diagnostic Testing
3.1 Most Chance Estimation
Let θ = (ω, α₁, …, αₙ, β₁, …, βₘ)ᵀ denote the parameter vector of a GARCH(p,q) mannequin. Given a pattern of T return observations {r₁, …, rₜ}, the conditional log-likelihood operate below Gaussian improvements is ℓ(θ) = −(T/2)ln(2π) − (1/2)∑ₜ₌₁ᵀ [ln(σ²ₜ) + ε²ₜ/σ²ₜ], the place εₜ = rₜ − μₜ and σ²ₜ is specified by the GARCH equation. The utmost probability estimator (MLE) θ̂ = argmaxₓ ℓ(θ) is obtained by way of numerical optimization, sometimes utilizing the BFGS or Marquardt algorithms [3] .
Theorem 4.1 (Consistency and Asymptotic Normality of GARCH MLE). Underneath Assumptions A1–A5 (said in Appendix A), the QMLE θ̂ₜ satisfies: (i) θ̂ₜ → θ₀ in likelihood as T → ∞ (consistency); and (ii) √T(θ̂ₜ − θ₀) →ᵈ N(0, V) the place V = A⁻¹BA⁻¹ is the sandwich covariance matrix, A = E[∂²ℓₜ/∂θ∂θᵀ] is the Hessian, and B = E[(∂ℓₜ/∂θ)(∂ℓₜ/∂θ)ᵀ] is the outer product of gradients.
The sandwich type of the asymptotic covariance matrix V is crucial as a result of, below distributional misspecification (e.g., assuming Gaussian improvements when the true distribution is Scholar-t), the knowledge matrix equality A = B doesn’t maintain, and the usual inverse-Hessian covariance estimator is inconsistent. The sturdy sandwich estimator stays constant below quasi-maximum probability estimation, offering legitimate inference even when the innovation distribution is misspecified.
3.2 Mannequin Choice Standards
We make use of three data standards for mannequin choice: the Akaike Data Criterion AIC = −2ℓ(θ̂) + 2k, the Bayesian Data Criterion BIC = −2ℓ(θ̂) + ok·ln(T), and the Hannan-Quinn Criterion HQC = −2ℓ(θ̂) + 2k·ln(ln(T)), the place ok is the variety of estimated parameters. The BIC is constant (selects the true mannequin with likelihood approaching one as T → ∞), whereas the AIC tends to overfit in finite samples. In observe, we report all three standards and choose the specification that achieves the perfect out-of-sample forecasting efficiency.
3.3 Diagnostic Testing
3.3.1 Exams for Serial Correlation
The Ljung-Field Q-statistic [4] checks for residual serial correlation: Q(m) = T(T+2) ∑ₖ₌₁ᵐ ρ̂²ₖ/(T−ok), the place ρ̂ₖ is the pattern autocorrelation of standardized residuals at lag ok. Underneath the null speculation of no serial correlation, Q(m) ~ χ²(m) asymptotically. We apply this take a look at to each the standardized residuals ẑₜ = ε̂ₜ/σ̂ₜ and the squared standardized residuals ẑ²ₜ.
3.3.2 ARCH-LM Check
Engle’s (1982) Lagrange Multiplier take a look at for ARCH results [5] regresses ẑ²ₜ on a continuing and m lags: ẑ²ₜ = α₀ + α₁ẑ²ₜ₋₁ + … + αₘẑ²ₜ₋ₘ + uₜ. The take a look at statistic T·R² from this auxiliary regression follows a χ²(m) distribution below the null speculation of no remaining ARCH results. Rejection of the null on uncooked return residuals motivates GARCH modeling; failure to reject on GARCH-filtered residuals validates the volatility specification.
3.3.3 Distributional Exams
We assess the distributional assumptions on the standardized residuals utilizing the Jarque-Bera take a look at for normality, the Kolmogorov-Smirnov take a look at for goodness-of-fit to the hypothesized innovation distribution, and likelihood integral remodel (PIT) histograms. If the mannequin is appropriately specified, the PIT values uₜ = F(ẑₜ; θ̂) ought to be roughly uniformly distributed on [0,1].
3.4 Volatility Forecasting
For a GARCH(1,1) mannequin, the h-step-ahead conditional variance forecast from time T is given by the recursion σ²ₜ₊ₕ|ₜ = ω + (α + β)σ²ₜ₊ₕ₋₁|ₜ for h ≥ 2, with preliminary situation σ²ₜ₊₁|ₜ = ω + αε²ₜ + βσ²ₜ. The forecast converges to the unconditional variance at a geometrical charge: σ²ₜ₊ₕ|ₜ = σ̄² + (α + β)ʰ⁻¹(σ²ₜ₊₁|ₜ − σ̄²), the place σ̄² = ω/(1 − α − β). This mean-reversion property is central to the buying and selling technique: when the present conditional variance is elevated relative to the long-run degree, we anticipate a subsequent decline in volatility, and vice versa.
