### Simulate Copula-GARCH Model

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Simulates a Copula-GARCH model with specified simulation parameters, starting from a sample.

```r
cgarch_sim <- cgarchsim(
  cgarch_fit,
  n.sim = 500,
  m.sim = 100,
  startMethod = "sample"
)
```

--------------------------------

### Get Time-Varying Copula Correlations

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Extracts the time-varying copula correlations from a fitted Copula-GARCH model.

```r
R_copula <- rcor(cgarch_fit)
```

--------------------------------

### Simulate GO-GARCH Model

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Simulates a GO-GARCH model with specified parameters. Requires a fitted GO-GARCH object.

```r
gogarch_sim <- gogarchsim(
  object = gogarch_fit,
  n.sim = 250,
  n.start = 50,
  m.sim = 500,
  startMethod = "sample",
  rseed = 123
)
```

--------------------------------

### Simulate VAR Model

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Simulates a VAR model using provided coefficients, data, and residuals. Requires pre-return data and simulated residuals.

```r
var_sim <- varxsim(
  X = returns,
  Bcoef = var_fit$Bcoef,
  p = 2,
  n.sim = 500,
  n.start = 50,
  prereturns = tail(returns, 2),
  resids = matrix(rnorm(500 * 5), ncol = 5),
  mexsimdata = NULL
)
```

--------------------------------

### GO-GARCH Forecast and Simulation

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Produces forecasts and simulations from GO-GARCH models, including higher-order moment support.

```r
library(rmgarch)

data(dji30ret)
returns <- dji30ret[, 1:5]

# Fit GO-GARCH with out-of-sample
gogarch_spec <- gogarchspec(
  mean.model = list(model = "constant"),
  variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
  distribution.model = "manig",
  ica = "fastica"
)

gogarch_fit <- gogarchfit(gogarch_spec, data = returns, out.sample = 50)

# Forecast
gogarch_fcst <- gogarchforecast(
  fit = gogarch_fit,
  n.ahead = 10,
  n.roll = 49  # Rolling forecasts using out-of-sample
)

# Extract forecasts
fitted(gogarch_fcst)
sigma(gogarch_fcst, factors = FALSE)  # Asset volatilities (not factor)

# Forecast correlations
rcor(gogarch_fcst)
```

--------------------------------

### Calculate GO-GARCH Portfolio Distribution using FFT

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Calculates the exact portfolio distribution for GO-GARCH models with non-Gaussian distributions using Fast Fourier Transform (FFT) convolution.

```r
library(rmgarch)

data(dji30ret)
returns <- dji30ret[, 1:5]
```

--------------------------------

### DCC-GARCH Simulation

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Simulates paths from a fitted DCC-GARCH model for Monte Carlo analysis and scenario generation.

```r
library(rmgarch)
library(rugarch)

data(dji30retw)
returns <- dji30retw[, 1:5]

# Fit DCC model
uspec <- ugarchspec(
  mean.model = list(armaOrder = c(1, 1)),
  variance.model = list(garchOrder = c(1, 1), model = "sGARCH"),
  distribution.model = "norm"
)

dcc_spec <- dccspec(
  uspec = multispec(replicate(5, uspec)),
  dccOrder = c(1, 1),
  distribution = "mvnorm"
)

dcc_fit <- dccfit(dcc_spec, data = returns)

# Simulate from fitted model
dcc_sim <- dccsim(
  fitORspec = dcc_fit,
  n.sim = 500,          # Simulation horizon
  n.start = 100,        # Burn-in period
  m.sim = 1000,         # Number of Monte Carlo simulations
  startMethod = "sample",  # Use sample ending values to start
  rseed = 42            # For reproducibility
)

# Extract simulated series
sim_returns <- fitted(dcc_sim, sim = 1)  # First simulation path

# Get simulated correlations
sim_R <- rcor(dcc_sim, sim = 1)

# Get simulated covariances
sim_H <- rcov(dcc_sim, sim = 1)

# Access all simulation paths for VaR calculation
all_sims <- sapply(1:100, function(i) fitted(dcc_sim, sim = i)[500, 1])
VaR_95 <- quantile(all_sims, 0.05)
```

--------------------------------

### Backtest Portfolio VaR

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Performs a backtest on calculated Value-at-Risk (VaR) against actual portfolio returns at a 1% significance level.

```r
actual_returns <- as.numeric(returns %*% weights)
VaRTest(0.01, actual = actual_returns, VaR = VaR_1pct)
```

--------------------------------

### Calculate Portfolio VaR (Normal)

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Calculates the Value-at-Risk at the 1% level for a portfolio assuming a multivariate normal distribution, using calculated portfolio margins.

```r
VaR_1pct <- qdist("norm", 0.01, port_margin[, 1], port_margin[, 2], 0, 0, 0)
```

--------------------------------

### GO-GARCH Specification and Fit

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Implements factor-based multivariate volatility modeling using Independent Component Analysis.

```r
library(rmgarch)

data(dji30ret)
returns <- dji30ret[, 1:5]

# GO-GARCH specification with affine NIG distribution
gogarch_spec <- gogarchspec(
  mean.model = list(
    model = "VAR",  # Options: "constant", "AR", "VAR"
    lag = 2,
    robust = FALSE
  ),
  variance.model = list(
    model = "sGARCH",
    garchOrder = c(1, 1),
    variance.targeting = FALSE
  ),
  distribution.model = "manig",  # Multivariate affine NIG
  ica = "fastica"  # ICA algorithm: "fastica" or "radical"
)

# Fit GO-GARCH model
gogarch_fit <- gogarchfit(
  spec = gogarch_spec,
  data = returns,
  out.sample = 50,
  solver = "solnp",
  gfun = "tanh"  # ICA nonlinearity function
)

# View results
print(gogarch_fit)

# Extract mixing matrix A (maps factors to returns)
A <- as.matrix(gogarch_fit, which = "A")

# Extract whitening matrix K
K <- as.matrix(gogarch_fit, which = "K")

# Get dynamic covariance and correlation
H <- rcov(gogarch_fit)
R <- rcor(gogarch_fit)

# Higher-order co-moments (unique to GO-GARCH)
coskew <- rcoskew(gogarch_fit, from = 1, to = 100)  # Coskewness tensor
cokurt <- rcokurt(gogarch_fit, from = 1, to = 100)  # Cokurtosis tensor
```

--------------------------------

### Forecast with VAR Model

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Generates VAR forecasts for a specified number of steps ahead, rolling the forecast sample.

```r
var_fcst <- varxforecast(
  X = returns,
  Bcoef = var_fit$Bcoef,
  p = 2,
  out.sample = 50,
  n.ahead = 10,
  n.roll = 0,
  mregfor = NULL
)
```

--------------------------------

### Generate VAR-based scenarios

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Creates scenarios using a Vector Autoregression model for multivariate data.

```r
var_scenarios <- fscenario(
  Data = returns,
  sim = 1000,
  roll = 0,
  model = "var",
  var.model = list(lag = 2, robust = FALSE),
  cov.method = "ML"
)
```

--------------------------------

### Calculate Portfolio Moments from Forecast

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Calculates portfolio moments (mean, sigma, skewness, kurtosis) from a GO-GARCH forecast object using specified weights.

```r
weights <- rep(1/5, 5)
port_moments <- gportmoments(gogarch_fcst, weights = weights)
```

--------------------------------

### Fit DCC-GARCH Model in R

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Estimates a DCC-GARCH model using a two-stage approach. Can utilize pre-fitted univariate models for efficiency. Use for estimating time-varying correlations.

```r
library(rmgarch)
library(rugarch)
library(parallel)

data(dji30retw)
returns <- dji30retw[, 1:5]

# Define univariate specification
uspec <- ugarchspec(
  mean.model = list(armaOrder = c(1, 1)),
  variance.model = list(garchOrder = c(1, 1), model = "gjrGARCH"),
  distribution.model = "norm"
)

# Create multispec and fit univariate models first (optional but useful)
mspec <- multispec(replicate(5, uspec))
multfit <- multifit(mspec, returns)

# Create DCC specification
dcc_spec <- dccspec(
  uspec = mspec,
  dccOrder = c(1, 1),
  distribution = "mvnorm"
)

# Fit DCC model (passing pre-fitted univariate models)
dcc_fit <- dccfit(
  spec = dcc_spec,
  data = returns,
  out.sample = 50,  # Keep 50 observations for out-of-sample testing
  solver = "solnp",
  fit.control = list(eval.se = TRUE, stationarity = TRUE),
  fit = multfit  # Pass pre-fitted univariate models
)

# View model summary
print(dcc_fit)

# Extract DCC parameters
coef(dcc_fit, type = "dcc")
# Returns: dcca1, dccb1

# Extract log-likelihood
likelihood(dcc_fit)

# Get dynamic correlation matrices
R <- rcor(dcc_fit)  # Returns 3D array [assets x assets x time]
print(R[1, 2, 1:5])  # Correlation between asset 1 and 2, first 5 periods

# Get dynamic covariance matrices
H <- rcov(dcc_fit)  # Returns 3D array [assets x assets x time]

# Extract fitted values and residuals
mu <- fitted(dcc_fit)      # Conditional means
sigma <- sigma(dcc_fit)    # Conditional volatilities
resid <- residuals(dcc_fit) # Residuals
```

--------------------------------

### Moment-based Forecast Generation

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Generates forecast covariance and higher co-moment matrices for DCC and GO-GARCH models.

```r
library(rmgarch)

data(dji30ret)
returns <- dji30ret[, 1:5]

# DCC-based moment forecasts
dcc_spec <- dccspec(
  uspec = multispec(replicate(5, ugarchspec(
    mean.model = list(armaOrder = c(1, 0)),
    variance.model = list(garchOrder = c(1, 1)),
    distribution.model = "norm"
  ))),
  dccOrder = c(1, 1),
  distribution = "mvnorm"
)

moments_dcc <- fmoments(
  spec = dcc_spec,
  Data = returns,
  n.ahead = 5,
  roll = 10,  # Rolling estimation
  solver = "solnp"
)

# Access forecast moments
# moments_dcc contains forecast covariance matrices

# GO-GARCH-based moment forecasts (includes higher moments)
gogarch_spec <- gogarchspec(
  mean.model = list(model = "constant"),
  variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
  distribution.model = "manig",
  ica = "fastica"
)

moments_gogarch <- fmoments(
  spec = gogarch_spec,
  Data = returns,
  n.ahead = 1,
  roll = 20,
  solver = "solnp"
)
```

--------------------------------

### Scenario Generation

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Generates Monte Carlo scenarios for risk management using various multivariate models.

```r
library(rmgarch)

data(dji30ret)
returns <- dji30ret[, 1:5]

# DCC-based scenario generation
dcc_spec <- dccspec(
  uspec = multispec(replicate(5, ugarchspec(
    mean.model = list(armaOrder = c(1, 0)),
    variance.model = list(garchOrder = c(1, 1)),
    distribution.model = "norm"
  ))),
  dccOrder = c(1, 1),
  distribution = "mvnorm"
)

scenarios <- fscenario(
  Data = returns,
  sim = 1000,           # Number of scenarios
  roll = 10,            # Rolling scenarios for backtesting
  model = "dcc",        # Options: "gogarch", "dcc", "cgarch", "var", "mdist"
  spec = dcc_spec,
  solver = "solnp",
  rseed = 42
)

# Extract simulated scenarios
sim_returns <- fitted(scenarios)  # Simulated return scenarios

# Access forecast conditional mean and covariance
scenarios@scenario$forecastMu
scenarios@scenario$forecastCov
```

--------------------------------

### Fit GO-GARCH with NIG distribution

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Configures and fits a GO-GARCH model using the NIG distribution and FastICA for independent component analysis.

```r
gogarch_spec <- gogarchspec(
  mean.model = list(model = "AR", lag = 2),
  variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
  distribution.model = "manig",
  ica = "fastica"
)

gogarch_fit <- gogarchfit(gogarch_spec, data = returns, gfun = "tanh")

# Portfolio weights
weights <- rep(1/5, 5)

# Geometric (analytical) portfolio moments
geom_moments <- gportmoments(gogarch_fit, weights = weights)
# Returns: mu, sigma, skewness, kurtosis for each time point

# FFT-based convolution for exact portfolio density
conv <- convolution(
  gogarch_fit,
  weights = weights,
  fft.step = 0.001,
  fft.by = 0.0001,
  fft.support = c(-1, 1),
  use.ff = TRUE,  # Use ff package for large arrays
  support.method = "adaptive"
)

# Semi-analytic (FFT) portfolio moments
fft_moments <- nportmoments(conv)

# Get portfolio density function at specific time
df <- dfft(conv, index = 100)
x_vals <- seq(-0.1, 0.1, by = 0.001)
density_vals <- df(x_vals)

# Get portfolio quantile function for VaR
qf <- qfft(conv, index = 100)
VaR_1pct <- qf(0.01)
VaR_5pct <- qf(0.05)

# Sample from portfolio distribution using inverse transform
set.seed(42)
u <- runif(10000)
portfolio_samples <- qf(u)
```

--------------------------------

### Fit Copula-GARCH Model

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Fits a Copula-GARCH model to the provided return data using the 'solnp' solver and evaluating standard errors.

```r
cgarch_fit <- cgarchfit(
  spec = cgarch_spec,
  data = returns,
  solver = "solnp",
  fit.control = list(eval.se = TRUE)
)
```

--------------------------------

### Specify Copula-GARCH Model

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Specifies a Copula-GARCH model with time-varying correlation using a Student copula and DCC for dependence.

```r
cgarch_spec <- cgarchspec(
  uspec = multispec(replicate(5, uspec)),
  VAR = FALSE,
  dccOrder = c(1, 1),
  asymmetric = TRUE,  # Use aDCC for copula correlation
  distribution.model = list(
    copula = "mvt",           # Student copula
    method = "ML",            # Maximum Likelihood estimation
    time.varying = TRUE,      # Dynamic copula
    transformation = "parametric"  # Options: parametric, empirical, spd
  )
)
```

--------------------------------

### Calculate Weighted Portfolio Margins (Student-t)

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Calculates portfolio mean and volatility using specified weights, conditional means, and conditional covariances from a DCC fit with a Student-t distribution.

```r
port_margin_t <- wmargin(
  distribution = "mvt",
  weights = weights,
  mean = fitted(dcc_fit_t),
  Sigma = rcov(dcc_fit_t),
  shape = coef(dcc_fit_t, "dcc")["mshape"]
)
```

--------------------------------

### Create DCC-GARCH Specification in R

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Defines a DCC-GARCH specification object. Supports symmetric, asymmetric, and Flexible DCC models with different multivariate distributions. Use for setting up DCC models.

```r
library(rmgarch)
library(rugarch)

# Load example data (Dow Jones 30 weekly returns)
data(dji30retw)
returns <- dji30retw[, 1:5]

# Create univariate GARCH specification for each asset
uspec <- ugarchspec(
  mean.model = list(armaOrder = c(1, 1)),
  variance.model = list(garchOrder = c(1, 1), model = "sGARCH"),
  distribution.model = "norm"
)

# Create DCC specification with multivariate normal distribution
dcc_spec <- dccspec(
  uspec = multispec(replicate(5, uspec)),
  dccOrder = c(1, 1),
  model = "DCC",
  distribution = "mvnorm"
)

# Create asymmetric DCC specification with Student-t distribution
adcc_spec <- dccspec(
  uspec = multispec(replicate(5, uspec)),
  dccOrder = c(1, 1),
  model = "aDCC",
  distribution = "mvt"
)

# Create DCC with VAR for conditional mean
var_dcc_spec <- dccspec(
  uspec = multispec(replicate(5, uspec)),
  VAR = TRUE,
  lag = 2,
  dccOrder = c(1, 1),
  distribution = "mvnorm"
)
```

--------------------------------

### Calculate Weighted Portfolio Margins (Normal)

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Calculates portfolio mean and volatility using specified weights, conditional means, and conditional covariances from a DCC fit with a normal distribution.

```r
weights <- rep(1/10, 10)

port_margin <- wmargin(
  distribution = "mvnorm",
  weights = weights,
  mean = fitted(dcc_fit),      # Conditional means
  Sigma = rcov(dcc_fit),       # Conditional covariances
  shape = NA,
  skew = NA
)
```

--------------------------------

### Extracting Forecast Metrics

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Methods for retrieving forecast means, correlations, and covariances from a forecast object.

```r
fitted(forecast_ahead)  # Returns n.ahead x n.assets x (n.roll+1) array

# Extract forecast correlations
R_fcst <- rcor(forecast_ahead)  # List of correlation matrices

# For rolling forecasts, access each roll
sapply(rcor(forecast_roll), function(x) x[1, 2, 1])  # Pairwise correlation forecasts

# Extract forecast covariances
H_fcst <- rcov(forecast_ahead)
```

--------------------------------

### Forecast with DCC-GARCH Model in R

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Generates multi-step ahead or rolling forecasts from a fitted DCC-GARCH model. Supports unconditional n-ahead and rolling 1-ahead forecasts. Use for predicting future correlations.

```r
library(rmgarch)
library(rugarch)

data(dji30retw)
returns <- dji30retw[, 1:5]

# Fit DCC model with out-of-sample data
uspec <- ugarchspec(
  mean.model = list(armaOrder = c(1, 1)),
  variance.model = list(garchOrder = c(1, 1), model = "sGARCH"),
  distribution.model = "norm"
)

dcc_spec <- dccspec(
  uspec = multispec(replicate(5, uspec)),
  dccOrder = c(1, 1),
  distribution = "mvnorm"
)

dcc_fit <- dccfit(dcc_spec, data = returns, out.sample = 100)

# Multi-step ahead unconditional forecast
forecast_ahead <- dccforecast(
  fit = dcc_fit,
  n.ahead = 50,  # 50-step ahead forecast
  n.roll = 0     # No rolling
)

# Rolling 1-step ahead forecast using out-of-sample data
forecast_roll <- dccforecast(
  fit = dcc_fit,
  n.ahead = 1,
  n.roll = 99  # 100 rolling forecasts (1 base + 99 additional)
)
```

--------------------------------

### Extract Copula-GARCH Results

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Prints the fitted Copula-GARCH model object and extracts its coefficients.

```r
print(cgarch_fit)
cof(cgarch_fit)
```

--------------------------------

### DCC-GARCH Rolling Estimation

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Performs rolling window estimation and forecasting for backtesting with automatic refitting.

```r
library(rmgarch)
library(rugarch)

data(dji30ret)
returns <- dji30ret[, 1:5]

# Define specification
uspec <- ugarchspec(
  mean.model = list(armaOrder = c(1, 0)),
  variance.model = list(model = "gjrGARCH"),
  distribution.model = "norm"
)

dcc_spec <- dccspec(
  uspec = multispec(replicate(5, uspec)),
  dccOrder = c(1, 1),
  distribution = "mvlaplace"
)

# Rolling estimation with refitting every 25 observations
dcc_roll <- dccroll(
  spec = dcc_spec,
  data = returns,
  n.ahead = 1,
  forecast.length = 500,  # Total forecast period
  refit.every = 25,       # Refit every 25 observations
  refit.window = "moving", # Moving window (vs "recursive")
  solver = "solnp",
  fit.control = list(eval.se = FALSE)
)

# Extract rolling forecasts
sigma_roll <- sigma(dcc_roll)
rcor_roll <- rcor(dcc_roll)
rcov_roll <- rcov(dcc_roll)

# Get likelihood across rolls
likelihood(dcc_roll)

# Plot rolling correlation
plot(dcc_roll, which = 4, series = c(1, 2, 3))
```

--------------------------------

### Perform FastICA Independent Component Analysis

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Implements the FastICA algorithm to extract independent components from mixed signals.

```r
library(rmgarch)

# Create mixed signals
set.seed(100)
S <- matrix(runif(10000), 5000, 2)  # Independent sources
A <- matrix(c(1, 1, -1, 2), 2, 2, byrow = TRUE)  # Mixing matrix
X <- S %*% t(A)  # Mixed signals

# Apply FastICA
ica_result <- fastica(
  X = X,
  approach = "symmetric",  # Options: "symmetric", "deflation"
  n.comp = 2,
  demean = TRUE,
  pca.cov = "ML",  # Options: "ML", "LW", "ROB", "EWMA"
  gfun = "tanh",   # Options: "pow3", "tanh", "gauss", "skew"
  epsilon = 1e-4,
  maxiter1 = 1000,
  rseed = 42,
  trace = TRUE
)

# Results
ica_result$A  # Estimated mixing matrix
ica_result$W  # Unmixing matrix
ica_result$S  # Estimated independent components
ica_result$whiteningMatrix
ica_result$dewhiteningMatrix

# Verify: S_hat %*% t(A_hat) should approximate original X
reconstructed <- ica_result$S %*% t(ica_result$A)
all.equal(head(X), head(reconstructed), tolerance = 1e-10)
```

--------------------------------

### Fit DCC Model with Student-t Distribution

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Fits a Dynamic Conditional Correlation (DCC) model with univariate GJR-GARCH specifications and a multivariate Student-t distribution.

```r
dcc_spec_t <- dccspec(
  uspec = multispec(replicate(10, uspec)),
  dccOrder = c(1, 1),
  distribution = "mvt"
)
dcc_fit_t <- dccfit(dcc_spec_t, data = returns)
```

--------------------------------

### Specify Univariate GARCH Model

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Defines a univariate GARCH model specification with ARMA(1,1) mean, GJR-GARCH(1,1) variance, and Student-t distribution.

```r
uspec <- ugarchspec(
  mean.model = list(armaOrder = c(1, 1)),
  variance.model = list(garchOrder = c(1, 1), model = "gjrGARCH"),
  distribution.model = "std"  # Student-t margins
)
```

--------------------------------

### Fit VAR(2) Model

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Estimates a Vector Autoregressive model of order 2 with a constant term, using standard estimation.

```r
var_fit <- varxfit(
  X = returns,
  p = 2,                    # VAR lag order
  constant = TRUE,          # Include constant
  robust = FALSE,           # Use robust estimation
  postpad = "constant"      # Padding method for residuals
)
```

--------------------------------

### Extract VAR Model Results

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Extracts various components from a fitted VAR model, including coefficients, fitted values, residuals, standard errors, and t-statistics.

```r
var_fit$Bcoef      # Coefficient matrix
var_fit$xfitted    # Fitted values (conditional mean)
var_fit$xresiduals # Residuals
var_fit$se         # Standard errors
var_fit$tstat      # t-statistics
```

--------------------------------

### Fit DCC Model with Normal Distribution

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Fits a Dynamic Conditional Correlation (DCC) model with univariate GJR-GARCH specifications and a multivariate normal distribution.

```r
uspec <- ugarchspec(
  mean.model = list(armaOrder = c(1, 1)),
  variance.model = list(garchOrder = c(1, 1), model = "gjrGARCH"),
  distribution.model = "norm"
)

dcc_spec <- dccspec(
  uspec = multispec(replicate(10, uspec)),
  dccOrder = c(1, 1),
  distribution = "mvnorm"
)

dcc_fit <- dccfit(dcc_spec, data = returns)
```

--------------------------------

### Correlation Distance Visualization

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Computes and plots rolling distance measures between correlation matrices to identify regime shifts.

```r
library(rmgarch)

data(dji30retw)
returns <- dji30retw[, 1:10]

# Fit DCC model
uspec <- ugarchspec(
  mean.model = list(armaOrder = c(1, 0)),
  variance.model = list(garchOrder = c(1, 1)),
  distribution.model = "norm"
)

dcc_spec <- dccspec(
  uspec = multispec(replicate(10, uspec)),
  dccOrder = c(1, 1),
  distribution = "mvnorm"
)

dcc_fit <- dccfit(dcc_spec, data = returns)
R <- rcor(dcc_fit)

# Calculate correlation distance with various measures
dist_matrix <- cordist(
  R = R,
  distance = "cmd",  # Options: "ma", "ms", "meda", "meds", "eigen", "cmd"
  n = 25,            # Distance between correlation snapshots
  plot = TRUE,
  dates = as.POSIXct(rownames(returns)),
  title = "Correlation Matrix Distance Over Time"
)
```

--------------------------------

### Filter New Data with VAR Coefficients

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Applies existing VAR coefficients to new data to filter conditional means and residuals.

```r
new_data <- returns[1:500, ]
var_filt <- varxfilter(
  X = new_data,
  p = 2,
  Bcoef = var_fit$Bcoef,
  postpad = "constant"
)
```

--------------------------------

### DCC Test for Dynamic Correlation

Source: https://context7.com/alexiosg/rmgarch/llms.txt

Performs the Engle-Sheppard test to detect dynamic conditional correlation against a null hypothesis of constant correlation.

```r
library(rmgarch)

data(dji30retw)
returns <- dji30retw[, 1:5]

# Test for dynamic correlation
dcc_test <- DCCtest(
  Data = returns,
  garchOrder = c(1, 1),  # First-stage GARCH order
  n.lags = 5,            # Number of lags to test
  solver = "solnp"
)

# Results
print(dcc_test)
# $H0: "No dynamic conditional correlation"
# $TestStatistic: Chi-square test statistic
# $pvalue: p-value for the test

# Low p-value indicates evidence of dynamic correlation
if (dcc_test$pvalue < 0.05) {
  print("Evidence of dynamic correlation - use DCC model")
} else {
  print("No evidence of dynamic correlation - CCC may suffice")
}
```

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