# FCVARlikeGrid: Grid Search to Maximize Likelihood Function In FCVAR: Estimation and Inference for the Fractionally Cointegrated VAR

 FCVARlikeGrid R Documentation

## Grid Search to Maximize Likelihood Function

### Description

`FCVARlikeGrid` performs a grid-search optimization by calculating the likelihood function on a grid of candidate parameter values. This function evaluates the likelihood over a grid of values for `c(d,b)` (or `phi`). It can be used when parameter estimates are sensitive to starting values to give an approximation of the global maximum that can then be used as the starting value in the numerical optimization in `FCVARestn`. `plot.FCVAR_grid` plots the likelihood function from `FCVARlikeGrid`.

### Usage

```FCVARlikeGrid(x, k, r, opt)
```

### Arguments

 `x` A matrix of variables to be included in the system. `k` The number of lags in the system. `r` The cointegrating rank. `opt` An S3 object of class `FCVAR_opt` that stores the chosen estimation options, generated from `FCVARoptions()`.

### Value

An S3 object of type `FCVAR_grid` containing the optimization results, including the following parameters:

`params`

A vector `params` of `d` and `b` (and `mu` if level parameter is selected) corresponding to a maximum over the grid of `c(d,b)` or `phi`.

`dbHatStar`

A vector of `d` and `b` corresponding to a maximum over the grid of `c(d,b)` or `phi`.

`muHatStar`

A vector of the optimal `mu` if level parameter is selected.

`Grid2d`

An indicator for whether or not the optimization is conducted over a 2-dimensional parameter space, i.e. if there is no equality restriction on `d` and `b`.

`dGrid`

A vector of the grid points in the parameter `d`, after any transformations for restrictions, if any.

`bGrid`

A vector of the grid points in the parameter `b`, after any transformations for restrictions, if any.

`dGrid_orig`

A vector of the grid points in the parameter `d`, in units of the fractional integration parameter.

`bGrid_orig`

A vector of the grid points in the parameter `b`, in units of the fractional integration parameter.

`like`

The maximum value of the likelihood function over the chosen grid.

`k`

The number of lags in the system.

`r`

The cointegrating rank.

`opt`

An S3 object of class `FCVAR_opt` that stores the chosen estimation options, generated from `FCVARoptions()`.

### Note

If `opt\$LocalMax == 0`, `FCVARlikeGrid` returns the parameter values corresponding to the global maximum of the likelihood on the grid. If `opt\$LocalMax == 1`, `FCVARlikeGrid` returns the parameter values for the local maximum corresponding to the highest value of `b`. This alleviates the identification problem mentioned in Johansen and Nielsen (2010, section 2.3).

### References

Johansen, S. and M. Ø. Nielsen (2010). "Likelihood inference for a nonstationary fractional autoregressive model," Journal of Econometrics 158, 51-66.

`FCVARoptions` to set default estimation options. `plot.FCVAR_grid` plots the likelihood function from `FCVARlikeGrid`.

Other FCVAR auxiliary functions: `FCVARforecast()`, `FCVARsimBS()`, `FCVARsim()`, `FracDiff()`, `plot.FCVAR_grid()`

### Examples

```# Restrict equality of fractional parameters.

opt <- FCVARoptions()
opt\$dbStep1D     <- 0.2 # Coarser grid for plotting example.
opt\$dbMin        <- c(0.01, 0.01) # Set lower bound for d,b.
opt\$dbMax        <- c(2.00, 2.00) # Set upper bound for d,b.
opt\$constrained  <- 0 # impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
opt\$restrictDB   <- 1 # impose restriction d=b ? 1 <- yes, 0 <- no.
opt\$progress     <- 2 # Show progress report on each value of b.
x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
likeGrid_params <- FCVARlikeGrid(x, k = 2, r = 1, opt)
plot(likeGrid_params)

# Linear restriction on fractional parameters.

opt <- FCVARoptions()
opt\$dbStep1D     <- 0.2 # Coarser grid for plotting example.
opt\$dbMin        <- c(0.01, 0.01) # Set lower bound for d,b.
opt\$dbMax        <- c(2.00, 2.00) # Set upper bound for d,b.
opt\$constrained  <- 0 # impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
opt\$restrictDB   <- 0 # impose restriction d=b ? 1 <- yes, 0 <- no.
# Impose linear restriction on d and b:
opt\$R_psi        <- matrix(c(2, -1), nrow = 1, ncol = 2)
opt\$r_psi        <- 0.5
opt\$progress     <- 2 # Show progress report on each value of b.
x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
likeGrid_params <- FCVARlikeGrid(x, k = 2, r = 1, opt)
plot(likeGrid_params)

# Constrained 2-dimensional optimization.
# Impose restriction dbMax >= d >= b >= dbMin.

opt <- FCVARoptions()
opt\$dbStep1D     <- 0.2 # Coarser grid for plotting example.
opt\$dbStep2D     <- 0.2 # Coarser grid for plotting example.
opt\$dbMin        <- c(0.01, 0.01) # Set lower bound for d,b.
opt\$dbMax        <- c(2.00, 2.00) # Set upper bound for d,b.
opt\$constrained  <- 1 # impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
opt\$restrictDB   <- 0 # impose restriction d=b ? 1 <- yes, 0 <- no.
opt\$progress     <- 2 # Show progress report on each value of b.
x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
likeGrid_params <- FCVARlikeGrid(x, k = 2, r = 1, opt)

# Unconstrained 2-dimensional optimization.

opt <- FCVARoptions()
opt\$dbStep1D     <- 0.1 # Coarser grid for plotting example.
opt\$dbStep2D     <- 0.2 # Coarser grid for plotting example.
opt\$dbMin        <- c(0.01, 0.01) # Set lower bound for d,b.
opt\$dbMax        <- c(2.00, 2.00) # Set upper bound for d,b.
opt\$constrained  <- 0 # impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
opt\$restrictDB   <- 0 # impose restriction d=b ? 1 <- yes, 0 <- no.
opt\$progress     <- 2 # Show progress report on each value of b.
x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
likeGrid_params <- FCVARlikeGrid(x, k = 2, r = 1, opt)

```

FCVAR documentation built on May 5, 2022, 9:06 a.m.