midasr (version 0.5)

midas_r: 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 fmls 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 de

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 n

...

additional arguments supplied to optimisation function

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

Details

Given MIDAS regression:

$$y_t = \sum_{j=1}^p\alpha_jy_{t-j} +\sum_{i=0}^{k}\sum_{j=0}^{l_i}\beta_{j}^{(i)}x_{tm_i-j}^{(i)} + u_t,$$

estimate the parameters of the restriction

$$\beta_j^{(i)}=g^{(i)}(j,\lambda).$$

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 $x_\tau^{(i)}$ 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.

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)))