1.1 Foundations of Time Collection Evaluation
The fashionable idea of time collection evaluation traces its origins to the work of Yuletide (1927), who launched the autoregressive mannequin, and Slutsky (1937), who demonstrated that shifting common processes may generate apparently cyclical conduct 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 at this time. The Field-Jenkins method prescribes a three-stage iterative process: (i) mannequin identification by examination of the autocorrelation perform (ACF) and partial autocorrelation perform (PACF); (ii) parameter estimation by way of most probability or conditional least squares; and (iii) diagnostic checking by residual evaluation.
Within the context of monetary return collection, Hamilton (1994) supplied a complete therapy of time collection strategies, whereas Tsay (2010) specialised the framework for monetary purposes. 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 generally known 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: Growth and Extensions
Engle’s (1982) ARCH mannequin was the primary to formally parameterize time-varying conditional variance as a perform 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 constructive 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 price and fairness return volatility and located that the easy GARCH(1,1) is remarkably tough 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 information, offering extra correct benchmarks for evaluating conditional variance fashions.
1.3 Time Collection Fashions in Algorithmic Buying and selling
The applying of time collection fashions to buying and selling technique improvement has a considerable historical past. Chan (2009) supplied 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 idea, an extension of univariate time collection strategies to multivariate settings. Extra lately, the mixing of machine studying with classical time collection strategies has attracted appreciable consideration (Dixon, Halperin, and Bilokon, 2020), although the basic 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-Threat fashions for place sizing in commodity buying and selling. Brownlees, Engle, and Kelly (2011) developed a sensible framework for volatility forecasting in threat administration purposes. Regardless of this physique of labor, a rigorous tutorial therapy that bridges the hole between GARCH idea 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 beneficial properties for mean-variance buyers, even after accounting for transaction prices. West, Edison, and Cho (1993) established the theoretical situations underneath 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 straight maps to financial loss capabilities. Our analysis builds upon this custom by embedding GARCH volatility forecasts inside a whole algorithmic buying and selling structure and evaluating financial efficiency by 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 set 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 purposes, we outline the log-return course of rₜ = ln(Pₜ/Pₜ₋₁), the place Pₜ denotes the asset value at time t. Below normal regularity situations, the return course of rₜ is often (a minimum of roughly) covariance stationary, even when the worth course of Pₜ is non-stationary (built-in of order one). This remark motivates the widespread apply of modeling returns quite 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 strategy 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 exterior the unit circle. Below stationarity, the method admits the infinite shifting 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 shifting common strategy of order q, denoted MA(q), if rₜ = c + εₜ + θ₁εₜ₋₁ + θ₂εₜ₋₂ + … + θₙεₜ₋ₙ, the place θ₁, …, θₙ are shifting common coefficients.
An MA(q) course of is at all times covariance stationary whatever the parameter values. Nonetheless, 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 exterior 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 shifting common strategy 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 everyday case (since returns are usually stationary), lowering the ARIMA specification to an ARMA mannequin. Nonetheless, the ARIMA framework is related when modeling built-in variables reminiscent of cumulative returns, asset costs, or rate of interest ranges that require differencing to attain stationarity.
2.4 The ARCH Mannequin
Engle (1982) launched the ARCH mannequin to formalize the remark that the conditional variance of monetary returns varies over time in a predictable method. Think about 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 certain 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 information.
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 because 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 idea of random recurrence equations (Bougerol and Picard, 1992). The GARCH(1,1) conditional variance might 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 concept of Bougerol and Picard, a singular 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) supplied 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 improve volatility greater than constructive 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 perform 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 perform: σ²ₜ = ω + αε²ₜ₋₁ + γε²ₜ₋₁ · I(εₜ₋₁ < 0) + βσ²ₜ₋₁, the place I(·) is the indicator perform. The parameter γ captures the differential influence of adverse shocks. The information influence curve for the GJR-GARCH mannequin is piecewise linear, with slope α for constructive 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 Pupil-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 Pupil-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 Probability 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 perform underneath 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). Below Assumptions A1–A5 (acknowledged 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, underneath distributional misspecification (e.g., assuming Gaussian improvements when the true distribution is Pupil-t), the knowledge matrix equality A = B doesn’t maintain, and the usual inverse-Hessian covariance estimator is inconsistent. The strong sandwich estimator stays constant underneath 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 apply, 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 Checks for Serial Correlation
The Ljung-Field Q-statistic (4) assessments 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. Below 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 underneath 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 Checks
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 accurately 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 price: 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.