Reference guide

The ARCHModels package for Julia. For documentation, see https://s-broda.github.io/ARCHModels.jl/dev.

BG96

Data from Bollerslev and Ghysels (JBES 1996).

DOW29

Stock returns, in procent, from 03/19/2008 through 04/11/2019, for tickers AAPL, IBM, XOM, KO, MSFT, INTC, MRK, PG, VZ, WBA, V, JNJ, PFE, CSCO, TRV, WMT, MMM, UTX, UNH, NKE, HD, BA, AXP, MCD, CAT, GS, JPM, CVX, DIS.

ARCH{q, T<:AbstractFloat} <: UnivariateVolatilitySpec{T}

ARCH{q}(coefs) -> UnivariateVolatilitySpec

Construct an ARCH specification with the given parameters.

Example:

julia> ARCH{2}([1., .3, .4])
TGARCH{0, 0, 2} specification.

──────────────────────────
               ω   α₁   α₂
──────────────────────────
Parameters:  1.0  0.3  0.4
──────────────────────────

ARCHLMTest(am::UnivariateARCHModel, p=max(o, p, q, ...))

Conduct Engle's (1982) LM test for autoregressive conditional heteroskedasticity with p lags in the test regression.

ARCHLMTest <: HypothesisTest

Engle's (1982) LM test for autoregressive conditional heteroskedasticity.

ARCHLMTest(u::Vector, p::Integer)

Conduct Engle's (1982) LM test for autoregressive conditional heteroskedasticity with p lags in the test regression.

ARCHModel <: StatisticalModel

ARMA{p, q, T} <: MeanSpec{T}

An ARMA(p, q) mean specification.

ARMA{p, q}(coefs::Vector)

Create an ARMA(p, q) model.

CCC{VS<:UnivariateVolatilitySpec, T<:AbstractFloat, d} <: MultivariateVolatilitySpec{T, d}

DCC(Qbar, coefs, univariatespecs; method=:largescale)

Construct a CCC specification with the given parameters. coefs must be passed as a length-zero Vector of the same element type as Qbar.

DCC{p, q, VS<:UnivariateVolatilitySpec, T<:AbstractFloat, d} <: MultivariateVolatilitySpec{T, d}

DCC{p, q}(Qbar, coefs, univariatespecs; method=:largescale)

Construct a DCC(p, q) specification with the given parameters.

DQTest <: HypothesisTest

Engle and Manganelli's (2004) out-of-sample dynamic quantile test.

DQTest(data, vars, level, p=1)

Conduct Engle and Manganelli's (2004) out-of-sample dynamic quantile test with p lags in the test regression. vars shoud be a vector of out-of-sample Value at Risk predictions at level level.

EGARCH{o, p, q, T<:AbstractFloat} <: UnivariateVolatilitySpec{T}

EGARCH{o, p, q}(coefs) -> UnivariateVolatilitySpec

Construct an EGARCH specification with the given parameters.

Example:

julia> EGARCH{1, 1, 1}([-0.1, .1, .9, .04])
EGARCH{1, 1, 1} specification.

─────────────────────────────────
                ω   γ₁   β₁    α₁
─────────────────────────────────
Parameters:  -0.1  0.1  0.9  0.04
─────────────────────────────────

GARCH{p, q, T<:AbstractFloat} <: UnivariateVolatilitySpec{T}

GARCH{p, q}(coefs) -> UnivariateVolatilitySpec

Construct a GARCH specification with the given parameters.

Example:

julia> GARCH{2, 1}([1., .3, .4, .05 ])
GARCH{2, 1} specification.

────────────────────────────────
               ω   β₁   β₂    α₁
────────────────────────────────
Parameters:  1.0  0.3  0.4  0.05
────────────────────────────────

Intercept{T} <: MeanSpec{T}

A mean specification with just an intercept.

Intercept(mu)

Create an instance of Intercept. mu can be passed as a scalar or vector.

MeanSpec{T}

Abstract supertype that mean specifications inherit from.

MultivariateARCHModel{T<:AbstractFloat,
     				  d,
					  VS<:MultivariateVolatilitySpec{T, d},
					  SD<:MultivariateStandardizedDistribution{T, d},
					  MS<:MeanSpec{T}
					 } <: ARCHModel

MultivariateARCHModel(spec::MultivariateVolatilitySpec, data::Matrix;
      			  	  dist=MultivariateStdNormal,
				  	  meanspec::[NoIntercept{T}() for _ in 1:d]
	  			  	  fitted::Bool=false
				  	  )

Create a MultivariateARCHModel.

MultivariateStandardizedDistribution{T, d} <: Distribution{Multivariate, Continuous}

Abstract supertype that multivariate standardized distributions inherit from.

MultivariateStdNormal{T, d} <: MultivariateStandardizedDistribution{T, d}

The multivariate standard normal distribution.

MultivariateVolatilitySpec{T, d} <: VolatilitySpec{T}

Abstract supertype that multivariate volatility specifications inherit from.

NoIntercept{T} <: MeanSpec{T}

A mean specification without an intercept (i.e., the mean is zero).

NoIntercept(T::Type=Float64)
NoIntercept{T}()
NoIntercept(v::Vector)

Create an instance of NoIntercept.

Regression{k, T} <: MeanSpec{T}

A linear regression as mean specification.

Regression(coefs::Vector, X::Matrix; coefnames=[β₀, β₁, …])
Regression(X::Matrix; coefnames=[β₀, β₁, …])
Regression{T}(X::Matrix; coefnames=[β₀, β₁, …])

Create a regression model.

Standardized{D<:ContinuousUnivariateDistribution, T}  <: StandardizedDistribution{T}

A wrapper type for standardizing a distribution from Distributions.jl.

StandardizedDistribution{T} <: Distributions.Distribution{Univariate, Continuous}

Abstract supertype that standardized distributions inherit from.

StdGED{T} <: StandardizedDistribution{T}

The standardized (mean zero, variance one) generalized error distribution.

StdGED(p)

Create a standardized generalized error distribution parameter p. p can be passed as a scalar or vector.

StdNormal(T::Type=Float64)
StdNormal(v::Vector)
StdNormal{T}()

Construct an instance of StdNormal.

StdNormal{T} <: StandardizedDistribution{T}

The standard Normal distribution.

StdSkewT{T} <: StandardizedDistribution{T}

Hansen's standardized (mean zero, variance one) Skewed Student's t distribution.

StdSkewT(v,λ)

Create a standardized skewed t distribution with v degrees of freedom and λ shape parameter. ν,λ` can be passed as scalars or vectors.

StdT{T} <: StandardizedDistribution{T}

The standardized (mean zero, variance one) Student's t distribution.

StdT(ν)

Create a standardized t distribution with ν degrees of freedom. ν` can be passed as a scalar or vector.

TGARCH{o, p, q, T<:AbstractFloat} <: UnivariateVolatilitySpec{T}

TGARCH{o, p, q}(coefs) -> UnivariateVolatilitySpec

Construct a TGARCH specification with the given parameters.

Example:

julia> TGARCH{1, 1, 1}([1., .04, .9, .01])
TGARCH{1, 1, 1} specification.

─────────────────────────────────
               ω    γ₁   β₁    α₁
─────────────────────────────────
Parameters:  1.0  0.04  0.9  0.01
─────────────────────────────────

UnivariateARCHModel{T<:AbstractFloat,
          		    VS<:UnivariateVolatilitySpec,
          			SD<:StandardizedDistribution{T},
          			MS<:MeanSpec{T}
          			} <: ARCHModel

UnivariateARCHModel(spec::UnivariateVolatilitySpec, data::Vector; dist=StdNormal(),
          			meanspec=NoIntercept(), fitted=false
          			)

Create a UnivariateARCHModel.

Example:

julia> UnivariateARCHModel(GARCH{1, 1}([1., .9, .05]), randn(10))

GARCH{1, 1} model with Gaussian errors, T=10.


─────────────────────────────────────────
                             ω   β₁    α₁
─────────────────────────────────────────
Volatility parameters:     1.0  0.9  0.05
─────────────────────────────────────────

UnivariateVolatilitySpec{T} <: VolatilitySpec{T} end

Abstract supertype that univariate volatility specifications inherit from.

VolatilitySpec{T}

Abstract supertype of UnivariateVolatilitySpec{T} and MultivariateVolatilitySpec{T} .

VaRs(am::UnivariateARCHModel, level=0.01)

Return the in-sample Value at Risk implied by am.

correlations(am::MultivariateARCHModel)

Return the estimated conditional correlation matrices.

covariances(am::MultivariateARCHModel)

Return the estimated conditional covariance matrices.

means(am::UnivariateARCHModel)

Return the conditional means of the model.

selectmodel(::Type{<:ARMA}, data; kwargs...)  -> UnivariateARCHModel

Fit a number of ARMA{p, q} models to data and return that which minimizes the BIC.

Keyword arguments:

• dist=StdNormal: the error distribution.
• minlags=1: minimum lag length to try in each parameter of VS.
• maxlags=3: maximum lag length to try in each parameter of VS.
• criterion=bic: function that takes a UnivariateARCHModel and returns the criterion to minimize.
• show_trace=false: print criterion to screen for each estimated model.
• algorithm=BFGS(), autodiff=:forward, kwargs...: passed on to the optimizer.

selectmodel(::Type{VS}, data; kwargs...) -> UnivariateARCHModel

Fit the volatility specification VS with varying lag lengths and return that which minimizes the BIC.

Keyword arguments:

• dist=StdNormal: the error distribution.
• meanspec=Intercept: the mean specification, either as a type or instance of that type.
• minlags=1: minimum lag length to try in each parameter of VS.
• maxlags=3: maximum lag length to try in each parameter of VS.
• criterion=bic: function that takes a UnivariateARCHModel and returns the criterion to minimize.
• show_trace=false: print criterion to screen for each estimated model.
• algorithm=BFGS(), autodiff=:forward, kwargs...: passed on to the optimizer.

Example

julia> selectmodel(EGARCH, BG96)

EGARCH{1, 1, 2} model with Gaussian errors, T=1974.

Mean equation parameters:
───────────────────────────────────────────────
      Estimate   Std.Error    z value  Pr(>|z|)
───────────────────────────────────────────────
μ  -0.00900018  0.00943948  -0.953461    0.3404
───────────────────────────────────────────────

Volatility parameters:
──────────────────────────────────────────────
      Estimate  Std.Error    z value  Pr(>|z|)
──────────────────────────────────────────────
ω   -0.0544398  0.0592073  -0.919478    0.3578
γ₁  -0.0243368  0.0270414  -0.899985    0.3681
β₁   0.960301   0.0388183  24.7384      <1e-99
α₁   0.405788   0.067466    6.0147      <1e-08
α₂  -0.207357   0.114161   -1.81636     0.0693
──────────────────────────────────────────────

simulate(am::ARCHModel; warmup=100, rng=Random.GLOBAL_RNG)
simulate(am::ARCHModel, T; warmup=100, rng=Random.GLOBAL_RNG)
simulate(spec::UnivariateVolatilitySpec, T; warmup=100, dist=StdNormal(), meanspec=NoIntercept(), rng=Random.GLOBAL_RNG)

Simulate a length-T time series from a UnivariateARCHModel. simulate(spec::MultivariateVolatilitySpec, T; warmup=100, dist=MultivariateStdNormal(), meanspec=[NoIntercept() for i = 1:d], rng=Random.GLOBAL_RNG) Simulate a length-T time series from a MultivariateARCHModel.

simulate!(am::ARCHModel; warmup=100, rng=Random.GLOBAL_RNG)

Simulate an ARCHModel, modifying am in place.

volatilities(am::UnivariateARCHModel)

Return the conditional volatilities.

fit!(am::ARCHModel; algorithm=BFGS(), autodiff=:forward, kwargs...)

Fit the uni- or multivariate ARCHModel specified by am, modifying am in place. Keyword arguments are passed on to the optimizer.

fit(am::ARCHModel; algorithm=BFGS(), autodiff=:forward, kwargs...)

Fit the uni- or multivariate ARCHModel specified by am and return the result in a new instance of ARCHModel. Keyword arguments are passed on to the optimizer.

fit(t::Type{<:ARMA}, data; kwargs...) -> UnivariateARCHModel

Fit an ARMA{p, q} model to data.

fit(VS::Type{<:UnivariateVolatilitySpec}, data; dist=StdNormal, meanspec=Intercept,
    algorithm=BFGS(), autodiff=:forward, kwargs...)

Fit the ARCH model specified by VS to data. data can be a vector or a GLM.LinearModel (or GLM.TableRegressionModel).

Keyword arguments:

• dist=StdNormal: the error distribution.
• meanspec=Intercept: the mean specification, either as a type or instance of that type.
• algorithm=BFGS(), autodiff=:forward, kwargs...: passed on to the optimizer.

Example: EGARCH{1, 1, 1} model without intercept, Student's t errors.

julia> fit(EGARCH{1, 1, 1}, BG96; meanspec=NoIntercept, dist=StdT)

EGARCH{1, 1, 1} model with Student's t errors, T=1974.


Volatility parameters:
──────────────────────────────────────────────
      Estimate  Std.Error    z value  Pr(>|z|)
──────────────────────────────────────────────
ω   -0.0162014  0.0186806  -0.867286    0.3858
γ₁  -0.0378454  0.018024   -2.09972     0.0358
β₁   0.977687   0.012558   77.8538      <1e-99
α₁   0.255804   0.0625497   4.08961     <1e-04
──────────────────────────────────────────────

Distribution parameters:
─────────────────────────────────────────
   Estimate  Std.Error  z value  Pr(>|z|)
─────────────────────────────────────────
ν   4.12423    0.40059  10.2954    <1e-24
─────────────────────────────────────────

fit(::Type{SD}, data; algorithm=BFGS(), kwargs...)

Fit a standardized distribution to the data, using the MLE. Keyword arguments are passed on to the optimizer.

fit(DCCspec::Type{<:DCC{p, q, VS<:UnivariateVolatilitySpec}}, data::Matrix;
    method=:largescale,  dist=MultivariateStdNormal, meanspec=Intercept,
    algorithm=BFGS(), autodiff=:forward, kwargs...)

Fit the DCC model specified by DCCspec to data. If p and q or VS are unspecified, then these default to 1, 1, and GARCH{1, 1}.

Keyword arguments:

• method: one of :largescale or twostep
• dist: the error distribution.
• meanspec: the mean specification, as a type.
• algorithm, autodiff, kwargs, ...: passed on to the optimizer.

Example: DCC{1, 1, GARCH{1, 1}} model:

julia> fit(DCC, DOW29)

29-dimensional DCC{1, 1} - GARCH{1, 1} - Intercept{Float64} specification, T=2785.

DCC parameters, estimated by largescale procedure:
────────────────────
       β₁         α₁
────────────────────
  0.88762  0.0568001
────────────────────

Calculating standard errors is expensive. To show them, use
`show(IOContext(stdout, :se=>true), <model>)`

predict(am::MultivariateARCHModel, what=:covariance)

Form a 1-step ahead prediction from am. what controls which object is predicted. The choices are :covariance (the default) or :correlation.

predict(am::UnivariateARCHModel, what=:volatility, horizon=1; level=0.01)

Form a horizon-step ahead prediction from am. what controls which object is predicted. The choices are :volatility (the default), :variance, :return, and :VaR. The VaR level can be controlled with the keyword argument level.

Not all prediction targets / volatility specifications support multi-step predictions.

residuals(am::MultivariateARCHModel; standardized = true, decorrelated = true)

Return the residuals.

residuals(am::UnivariateARCHModel; standardized=true)

Return the residuals of the model. Pass standardized=false for the non-devolatized residuals.