midas_r: Restricted MIDAS regression

In mpiktas/midasr: Mixed Data Sampling Regression

Restricted MIDAS regression

Description

Estimate restricted MIDAS regression using non-linear least squares.

Usage

midas_r(
  formula,
  data,
  start,
  Ofunction = "optim",
  weight_gradients = NULL,
  ...
)

Arguments

formula	formula for restricted MIDAS regression or midas_r object. Formula must include mls function
data	a named list containing data with mixed frequencies
start	the starting values for optimisation. Must be a list with named elements.
Ofunction	the list with information which R function to use for optimisation. The list must have element named Ofunction which contains character string of chosen R function. Other elements of the list are the arguments passed to this function. The default optimisation function is optim with argument method="BFGS". Other supported functions are nls
weight_gradients	a named list containing gradient functions of weights. The weight gradient function must return the matrix with dimensions d_k \times q, where d_k and q are the number of coefficients in unrestricted and restricted regressions correspondingly. The names of the list should coincide with the names of weights used in formula. The default value is NULL, which means that the numeric approximation of weight function gradient is calculated. If the argument is not NULL, but the name of the weight used in formula is not present, it is assumed that there exists an R function which has the name of the weight function appended with _gradient.
...	additional arguments supplied to optimisation function

Details

Given MIDAS regression:

y_t = ∑_{j=1}^pα_jy_{t-j} +∑_{i=0}^{k}∑_{j=0}^{l_i}β_{j}^{(i)}x_{tm_i-j}^{(i)} + u_t,

estimate the parameters of the restriction

β_j^{(i)}=g^{(i)}(j,λ).

Such model is a generalisation of so called ADL-MIDAS regression. It is not required that all the coefficients should be restricted, i.e the function g^{(i)} might be an identity function. Model with no restrictions is called U-MIDAS model. The regressors x_τ^{(i)} must be of higher (or of the same) frequency as the dependent variable y_t.

MIDAS-AR* (a model with a common factor, see (Clements and Galvao, 2008)) can be estimated by specifying additional argument, see an example.

The restriction function must return the restricted coefficients of the MIDAS regression.

Value

a midas_r object which is the list with the following elements:

coefficients	the estimates of parameters of restrictions
midas_coefficients	the estimates of MIDAS coefficients of MIDAS regression
model	model data
unrestricted	unrestricted regression estimated using midas_u
term_info	the named list. Each element is a list with the information about the term, such as its frequency, function for weights, gradient function of weights, etc.
fn0	optimisation function for non-linear least squares problem solved in restricted MIDAS regression
rhs	the function which evaluates the right-hand side of the MIDAS regression
gen_midas_coef	the function which generates the MIDAS coefficients of MIDAS regression
opt	the output of optimisation procedure
argmap_opt	the list containing the name of optimisation function together with arguments for optimisation function
start_opt	the starting values used in optimisation
start_list	the starting values as a list
call	the call to the function
terms	terms object
gradient	gradient of NLS objective function
hessian	hessian of NLS objective function
gradD	gradient function of MIDAS weight functions
Zenv	the environment in which data is placed
use_gradient	TRUE if user supplied gradient is used, FALSE otherwise
nobs	the number of effective observations
convergence	the convergence message
fitted.values	the fitted values of MIDAS regression
residuals	the residuals of MIDAS regression

Author(s)

Virmantas Kvedaras, Vaidotas Zemlys

References

Clements, M. and Galvao, A., Macroeconomic Forecasting With Mixed-Frequency Data: Forecasting Output Growth in the United States, Journal of Business and Economic Statistics, Vol.26 (No.4), (2008) 546-554

Examples

##The parameter function
theta_h0 <- function(p, dk, ...) {
   i <- (1:dk-1)/100
   pol <- p[3]*i + p[4]*i^2
   (p[1] + p[2]*i)*exp(pol)
}

##Generate coefficients
theta0 <- theta_h0(c(-0.1,10,-10,-10),4*12)

##Plot the coefficients
plot(theta0)

##Generate the predictor variable
xx <- ts(arima.sim(model = list(ar = 0.6), 600 * 12), frequency = 12)

##Simulate the response variable
y <- midas_sim(500, xx, theta0)

x <- window(xx, start=start(y))

##Fit restricted model
mr <- midas_r(y~fmls(x,4*12-1,12,theta_h0)-1,
              list(y=y,x=x),
              start=list(x=c(-0.1,10,-10,-10)))

##Include intercept and trend in regression
mr_it <- midas_r(y~fmls(x,4*12-1,12,theta_h0)+trend,
                 list(data.frame(y=y,trend=1:500),x=x),
                 start=list(x=c(-0.1,10,-10,-10)))

data("USrealgdp")
data("USunempr")

y.ar <- diff(log(USrealgdp))
xx <- window(diff(USunempr), start = 1949)
trend <- 1:length(y.ar)

##Fit AR(1) model
mr_ar <- midas_r(y.ar ~ trend + mls(y.ar, 1, 1) +
                 fmls(xx, 11, 12, nealmon),
                 start = list(xx = rep(0, 3)))

##First order MIDAS-AR* restricted model
mr_arstar <-  midas_r(y.ar ~ trend + mls(y.ar, 1, 1, "*")
                     + fmls(xx, 11, 12, nealmon),
                     start = list(xx = rep(0, 3)))

mpiktas/midasr documentation built on Aug. 24, 2022, 2:32 p.m.