arfimaOLS: Arfima-OLS for repeated cross-sectional data and/or pooled...
In ArfimaMLM: Arfima-MLM Estimation For Repeated Cross-Sectional Data

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/arfimaOLS.R

Estimates Arfima-OLS model for repeated cross-sectional data or pooled cross-sectional time-series data. For the variables specified by the user, the function automatically implements the aggregation and fractional differencing of time/level variables as well as the necessary procedures to remove deterministic components from the dependent as well as the major independent variables.

arfimaOLS(formula, data, timevar
          , d = "Hurst", arma = NULL
          , ecmformula = NULL, decm = "Hurst"
          , drop = 5, report.data = TRUE, ...)

`formula`	An object of the class “`formula`” that specifies the linear model to be estimated (see `lm` for details), typically with the response on the left of a ~ operator and the terms, separated by + operators, on the right. See details below for further information about selecting variables for automatic aggregation, fractional differencing, and the removal of deterministic components.
`data`	Data frame containing the original variables named in `formula`.
`timevar`	Name of the variable indicating different timepoints in `data`.
`d`	Call for a specific estimation method for the fractional differencing parameter in the `fractal`-package (``Hurst'') or in the `fracdiff`-package (``ML'', ``GPH'', or ``Sperio''). Default estimation procedure is by estimating the `Hurst` exponent. If the user wants to specify the methods for each variable individually, `d` can be a list containing a call for every individual variable. Furthermore, the list can contain numeric values for differencing parameters which were estimated externally (e.g. 1 for simple differencing, also see example for further details). A variable will not be differenced if `d` is specified as 0.
`arma`	List of variables for which AR and MA parameters are to be estimated (after fractional differencing) as well as a vector containing the respective orders of the model to fit. `order[1]` corresponds to the AR part and `order[2]` to the MA part, similar to the model specification in `arima` (just excluding the d parameter here). For variables specified in `arma`, the function will use the residuals of the ARMA model for the subsequent model estimation in order to remove their deterministic components. All variables included in the `arma` list have to included in either `varlist.fd`, `varlist.ydif`, or `varlist.xdif`. It is also possible to keep some of the AR/MA parameters fixed at zero (e.g. if the model is only supposed to estimate AR[1] and AR[3] parameters, but not AR[2]). In order to specify such a model, replace the vector containing the orders of the model with a list containing two vectors indicating each individual AR or MA parameter to be estimated. Please see the examples for clarification.
`ecmformula`	Specification of the cointegration regression to receive the residuals for the error correction mechanism (ecm) included in `formula`: linear formula object with the response on the left of a ~ operator and the independent variables, separated by + operators, on the right. Note that the variable names included here cannot be the original variable names included in `data`, but rather has to be extended by adding “`.mean`” to the original names, since the ecm is always based on the level/time aggreegates (see example).
`decm`	Call for estimation method for the fractional differencing parameter (see `d` for details). Can be either ``Hurst'' ``ML'', ``GPH'', or ``Sperio''. Default is ``Hurst''.
`drop`	Number of time points from the beginning of the series dropped from analysis. Default is 5.
`report.data`	Logical. `arfimaOLS` returns the transformed dataset used to estimate the final model as part of the results. Default is `TRUE`.
`...`	Further arguments passed to the estimation procedures used within the function (e.g. for `lm`).

The original variable names of data used in formula can be extended by adding three different suffixes: .fd, .xdif, and .ydif. These suffixes select variables for transformations according to the Arfima-OLS framework before estimating the actual model.

The suffix .fd allows the user to select variables which are supposed to be transformed to a fractionally differenced level-variable (by aggregating individuals over each time point prior to fractionally differencing the series), or variables which are already included as a level-variable in the original dataset and are just supposed to be fractionally differenced before the multilevel model is estimated.

If the suffix .xdif is added to an independent variable, the variable is simply filtered through the timepoint averages:

x.star[it]=x[it]-X[t]

If the suffix .ydif is added to the dependent variable (e.g. y[it]), the function will remove the temporal deterministic component from the individual level variable, such that it only consists of within-timepoint, as well as non-temporally autocorrelated between-timepoint variation:

y.star[it]=y[it]-(Y[t]-Δ[df]Y[t])
If formula contains ecm as one of the independent variables, and ecmformula is correctly specified, the function will include the lag of the fractionally differenced residuals of the cointegration regression as an error correction mechanism in the multilevel model. The ECM does not have to be estimated prior to calling the function.
In order to prevent errors in the estimation procedure, none of the original variable names in data should include “.ydif”, “.xdif” or “.fd”.

The function returns a list of the class 'arfimaOLS' with the following items:

`result`	Output of the linear model as specified in `formula`.
`d`	Matrix of fractional differencing parameters estimated for the level variables (`.fd` and `.ydif`) as well as the estimation method for each variable. Returns the specified value for d if it was specified in the initial call of the function.
`arma`	List of `arima` results for each variable specified in the model call. Contains AR/MA estimates as well as the model residuals.
`ecm`	Output of the cointegration regression (returned if `ecmformula` is specified). The lagged residuals of the cointegration regression are included in the multilevel model if `ecm` is included in `formula`.
`data.mean`	Data frame of variable means declared in `formula` as `.fd`, `.xdif` or `.ydif` (as well as `.mean` in `ecmformula`) for each time point specified by the level variable in `timevar`.
`data.fd`	Data frame of fractionally differenced level variables for each time point specified in `timevar`, which were declared as `.fd` or `.ydif` in `formula`. If `arma` was additionally specified for a variable, it contains the residuals of the ARMA model fitted after (fractionally) differencing.
`data.merged`	Merged data frame used to estimate the OLS model consisting of the original data, `data.mean`, `data.fd`, as well as the variables specified as `.xdif` and `.ydif` in `formula`

Patrick Kraft

Lebo, M. and Weber, C. 2015. “An Effective Approach to the Repeated Cross Sectional Design.” American Journal of Political Science 59(1): 242-258.

lm, fracdiff, hurstSpec, arfimaPrep, fd, and ArfimaMLM for a package overview.

require(fractal)
require(fracdiff)

### set basic parameters for simulation
t = 100 # number of time points
n = 500 # number of observations within time point
N = t*n # total number of observations

### generate fractional ARIMA Time Series for y_t, x1_t, z1_t, z2_t
set.seed(123)
y_t <- fracdiff.sim(t, d=0.4, mu=10)$series
x1_t <- fracdiff.sim(t, d=0.3, mu=5)$series
z1_t <- fracdiff.sim(t, d=0.1, mu=2)$series
z2_t <- fracdiff.sim(t, d=0.25, mu=3)$series

### simulate data
data <- NULL; data$time <- rep(seq(1:t),each=n)
data <- data.frame(data)
data$x1 <- rnorm(N,rep(x1_t,each=n),2)
data$x2 <- rnorm(N,0,40)
data$z1 <- rnorm(N,rep(z1_t,each=n),3)
data$z2 <- rep(z2_t,each=n)
b1 <- 0.2+rep(rnorm(t,0,0.1),each=n)
data$y <- (b1*data$x1-0.05*data$x2+0.3*rep(z1_t,each=n)
            +0*data$z2+rnorm(N,rep(y_t,each=n),1))


### estimate models

# basic example
m1 <- arfimaOLS(y.ydif ~ x1.xdif + x2 + z1.fd + z2.fd
                , data = data, timevar = "time")
                
# model including error correction mechanism
# change estimation method for differencing parameter for all variables
m2 <- arfimaOLS(y.ydif ~ x1.xdif + x2 + z1.fd + z2.fd + ecm
                , data = data, timevar = "time", d="ML"
                , ecmformula = y.mean ~ x1.mean
                , decm="Sperio")
                
# vary estimation method for differencing parameter between variables
# specify AR/MA models
m3 <- arfimaOLS(y.ydif ~ x1.xdif + x2 + z1.fd + z2.fd
                , data = data, timevar = "time"
                , d=list(y="ML", z1="Sperio", z2=0.25)
                , arma=list(y=c(1,0),z2=c(0,1)))                
                
# specify AR/MA models while holding AR[2] fixed for y
m4 <- arfimaOLS(y.ydif ~ x1.xdif + x2 + z1.fd + z2.fd
                , data = data, timevar = "time"
                , arma=list(y=list(c(1,3),0),z2=c(0,1)))                 

m1
summary(m2)
summary(m3$result)
m4$arma

Loading required package: lme4
Loading required package: Matrix
Loading required package: fractal
Loading required package: splus2R
Loading required package: ifultools
Loading required package: fracdiff

###################################
Fractional Differencing Parameters: 

   Method   Estimate
y   Hurst 0.41713434
z1  Hurst 0.01679005
z2  Hurst 0.10887414

##################
Results OLS Model: 

Call:
lm(formula = formula, data = new$data.merged)

Coefficients:
(Intercept)      x1.xdif           x2        z1.fd        z2.fd  
    0.05505      0.20334     -0.04985      0.20352     -0.07058  


###################################
Summary Error Correction Mechanism: 

Call:
lm(formula = ecmformula, data = data.mean)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.2427 -0.7130  0.0182  0.7618  2.6427 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  11.1164     0.5400  20.584   <2e-16 ***
x1.mean       0.1630     0.1194   1.364    0.176    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.13 on 98 degrees of freedom
Multiple R-squared:  0.01864,	Adjusted R-squared:  0.008626 
F-statistic: 1.861 on 1 and 98 DF,  p-value: 0.1756


###################################
Fractional Differencing Parameters: 

    Method   Estimate
y       ML 0.27174837
z1      ML 0.09300433
z2      ML 0.12142851
ecm Sperio 0.36545132


##################
Summary OLS Model: 

Call:
lm(formula = formula, data = new$data.merged)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.8499 -0.9610  0.0289  0.9987  5.6594 

Coefficients:
              Estimate Std. Error  t value Pr(>|t|)    
(Intercept)  0.0590680  0.0066877    8.832   <2e-16 ***
x1.xdif      0.2033355  0.0033374   60.926   <2e-16 ***
x2          -0.0498911  0.0001671 -298.572   <2e-16 ***
z1.fd        0.2226591  0.0071783   31.019   <2e-16 ***
z2.fd       -0.1202803  0.0065465  -18.373   <2e-16 ***
ecm         -0.0664588  0.0063169  -10.521   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.455 on 47494 degrees of freedom
Multiple R-squared:  0.6649,	Adjusted R-squared:  0.6648 
F-statistic: 1.885e+04 on 5 and 47494 DF,  p-value: < 2.2e-16


Call:
lm(formula = formula, data = new$data.merged)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.8569 -0.9601  0.0273  0.9961  5.6257 

Coefficients:
              Estimate Std. Error  t value Pr(>|t|)    
(Intercept)  0.0585643  0.0066831    8.763   <2e-16 ***
x1.xdif      0.2033354  0.0033381   60.913   <2e-16 ***
x2          -0.0498788  0.0001671 -298.453   <2e-16 ***
z1.fd        0.2282936  0.0071695   31.842   <2e-16 ***
z2.fd       -0.1073094  0.0064299  -16.689   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.455 on 47495 degrees of freedom
Multiple R-squared:  0.6647,	Adjusted R-squared:  0.6647 
F-statistic: 2.354e+04 on 4 and 47495 DF,  p-value: < 2.2e-16

$y

Call:
arima(x = data.fd[, names(arma)[i]], order = c(max(arma[[i]][[1]]), 0, max(arma[[i]][[2]])), 
    include.mean = FALSE, transform.pars = FALSE, fixed = fixed)

Coefficients:
          ar1  ar2     ar3
      -0.1323    0  0.1575
s.e.   0.0978    0  0.0972

sigma^2 estimated as 1.057:  log likelihood = -144.69,  aic = 295.39

$z2

Call:
arima(x = data.fd[, names(arma)[i]], order = c(arma[[names(arma)[i]]][1], 0, 
    arma[[names(arma)[i]]][2]), include.mean = FALSE)

Coefficients:
         ma1
      0.0470
s.e.  0.0973

sigma^2 estimated as 1.048:  log likelihood = -144.24,  aic = 292.47