rumidas (version 0.1.3)

ugmfit: Methods for obtaining (and evaluating) a variety of GARCH-MIDAS-based models

Description

Estimates several GARCH-MIDAS-based models, according to two errors' conditional distributions: Normal and Student-t, and the presence of asymmetric terms in the short- and long-run components.

Usage

ugmfit(
  model,
  skew,
  distribution,
  daily_ret,
  mv_m,
  mv_m_2 = NULL,
  K,
  K_2 = NULL,
  lag_fun = "Beta",
  X = NULL,
  out_of_sample = NULL,
  vol_proxy = NULL,
  R = 100
)

Value

ugmfit returns an object of class 'rumidas'. The function summary.rumidas

can be used to print a summary of the results. Moreover, an object of class 'rumidas' is a list containing the following components:

• model: The model used for the estimation.

• rob_coef_mat: The matrix of estimated coefficients, with the QML standard errors. For details, see: Bollerslev_Wooldridge_1992;textualrumidas.

• obs: The number of daily observations used for the (in-sample) estimation.

• period: The period of the in-sample estimation.

• loglik: The value of the log-likelihood at the maximum.

• inf_criteria: The AIC and BIC information criteria.

• loss_in_s: The in-sample MSE and QLIKE averages, calculated considering the distance with respect to the volatility proxy (if provided) or the squared daily returns.

• est_in_s: The one-step-ahead volatility, for the in-sample period, that is: \(\sqrt{\hat{\tau}_t \times \hat{g}_{i,t} }\).

• est_lr_in_s: The one-step-ahead long-run volatility, for the in-sample period.

• loss_oos: The out-of-sample MSE and QLIKE averages, calculated considering the distance with respect to the volatility proxy (if provided) or the squared daily returns.

• est_oos: The one-step-ahead volatility, for the out-of-sample period, that is: \(\sqrt{\hat{\tau}_t \times \hat{g}_{i,t} }\).

• est_lr_oos: The one-step-ahead long-run volatility, for the out-of-sample period.

Arguments

model

Model to estimate. Valid choices are: "GM" for GARCH-MIDAS, "GM2M" for GARCH-MIDAS with two MIDAS variables, "GMX" for GARCH-MIDAS-X, "DAGM" for Double Asymmetric GARCH-MIDAS (DAGM), "DAGM2M" for DAGM with two MIDAS variables, and "DAGMX" for DAGM-X

skew

The skewness parameter to include in the short--run equation. Valid choices are: "YES" and "NO"

distribution

The conditional density to use for the innovations. At the moment, valid choices are "norm" and "std", for the Normal and Student-t distribution, respectively

daily_ret

Daily returns, which must be an "xts" objectparamter

mv_m

(first) MIDAS variable already transformed into a matrix, through mv_into_mat function

mv_m_2

optional second MIDAS variable already transformed into a matrix, through mv_into_mat function

K

Number of (lagged) realizations of the (first) MIDAS variable to consider

K_2

optional Number of (lagged) realizations of the (second) MIDAS variable to consider

lag_fun

optional Lag function to use. Valid choices are "Beta" (by default) and "Almon", for the Beta and Exponential Almon lag functions, respectively

X

optional Additional "X" variable, which must be an "xts" object. Moreover, "X" must be observed for the same days of daily_ret

out_of_sample

optional A positive integer indicating the number of periods before the last to keep for out of sample forecasting

vol_proxy

optional If present, the 'vol_proxy' is the volatility proxy used for the in-sample and out-of-sample (again, if present) evaluation. It could be the realized variance. If it left unspecified, vol_proxy is replaced by the squared daily returns

R

optional. A positive integer indicating the number of replications used to find the best starting values. Equal to 100 by default

Details

Function ugmfit implements the estimation and evaluation of the GARCH--MIDAS--based models, with and without the asymmetric term linked to negative lagged daily returns, according to two distributions for the error term. The general framework assumes that: $$r_{i,t}= \sqrt{\tau_t \times g_{i,t} } \epsilon_{i,t},$$ where

• \(r_{i,t}\) is the daily return for the \(i\)-th day (\(i = 1, \ldots, N_t\)) of the period \(t\) (for example, a week, a month or a quarter; \(t = 1 , \ldots, T\));

• \(\tau_{t}\) is the long-run component, varying each period \(t\);

• \(g_{i,t}\) is the short--run term, varying each day \(i\) of the period \(t\);

• \(\epsilon_{i,t}\) is an \(iid\) error term which has a zero mean and unit variance.

The short--run component of the GARCH--MIDAS (parameter "model" set to "GM"), DAGM (parameter "model" set to "DAGM"), and DAGM with two MIDAS variables (parameter "model" set to "DAGM2M") models is as follows. When the parameter "skew" is present (set to "YES"): $$g_{i,t} = \left(1-\alpha-\gamma/2-\beta\right) + \left(\alpha + \gamma \cdot I_{\left(r_{i-1,t} < 0 \right)}\right) \frac{\left(r_{i-1,t}\right)^2}{\tau_t} + \beta g_{i-1,t},$$ where \(I_{(\cdot)}\) is an indicator function. The short--run component of the GARCH--MIDAS--X (parameter "model" set to "GMX") and DAGM--X (parameter "model" set to "DAGMX"), when the parameter "skew" is set to "YES", is: $$g_{i,t} = \left(1-\alpha-\gamma/2-\beta\right) + \left(\alpha + \gamma \cdot I_{\left(r_{i-1,t} < 0 \right)}\right) \frac{\left(r_{i-1,t}\right)^2}{\tau_t} + \beta g_{i-1,t} + z \cdot \left(X_{i-1,t}- E(X_{i-1,t})\right).$$ When, for the models GARCH--MIDAS, GARCH--MIDAS--X, DAGM, and DAGM--X, the parameter "skew" is set to "NO", parameter \(\gamma\) disappears. For details on the GARCH--MIDAS--X and DAGM--X models, see amendola_candila_gallo_2020;textualrumidas. The long-run component of the GARCH-MIDAS and GARCH--MIDAS--X models is: $$\tau_{t} = \exp \left\{ m + \theta \sum_{j=1}^K \delta_{j}(\omega) X_{t-j}\right\},$$ where \(X_{t}\) is the MIDAS term and \(\delta_{j}(\omega)\) is the chosen weighting function, which can be the Beta (beta_function) or Exponential Almon lag (exp_almon) functions. The long-run component of the DAGM and DAGM--X models is: $$\tau_t = \exp \left(m + \theta^{+} \sum_{k=1}^K \delta_k(\omega)^{+} X_{t-k} I_{\left( X_{t-k} \geq 0 \right)} + \theta^{-} \sum_{k=1}^K \delta_k(\omega)^{-} X_{t-k} I_{\left( X_{t-k} < 0 \right)} \right).$$

References

See Also

mv_into_mat.

Examples

# \donttest{
# estimate a GARH-MIDAS model, without the skewness parameter
r_t<-sp500['2008']
mv_m<-mv_into_mat(r_t,diff(indpro),K=12,"monthly") 
fit<-ugmfit(model="GM",skew="NO",distribution="norm",r_t,mv_m,K=12)
fit
summary.rumidas(fit)
names(fit)

# to see the estimated coefficients with the QML standard errors:
fit$rob_coef_mat

# estimate a DAGM model, with the skewness parameter, 
# including the volatility proxy (realized variance), and
# leaving the last 100 observations for the out-of-sample evaluation
r_t<-sp500['2002/2020']
mv_m<-mv_into_mat(r_t,diff(indpro),K=12,"monthly") 
fit_2<-ugmfit(model="DAGM",skew="YES",distribution="norm",r_t,
mv_m,K=12,vol_proxy=rv5['2002/2020'],out_of_sample=100)
fit_2
summary.rumidas(fit_2)
# estimate a GM-X model, without the skewness parameter
r_t<-sp500['2010/2013']
X<-vix['2010/2013'] 
mv_m<-mv_into_mat(r_t,diff(indpro),K=36,"monthly") 
fit_3<-ugmfit(model="GMX",skew="NO",distribution="norm",r_t,mv_m,K=36,X=X)
summary.rumidas(fit_3)
# }