The R package specs implements the single-equation
penalized error-correction selector proposed in Smeekes and Wijler
(2018a) as an automated approach towards sparse single-equation
cointegration modelling. In addition, the package contains the dataset
used in Smeekes and Wijler (2018a) to predict Dutch unemployment rates
based on Google Trends.
The development version of the specs package can be
installed from GitHub using
# install.package("devtools")
devtools::install_github("wijler/specs")After installation, the package can be loaded in the standard way:
library(specs)Loading specs provides access to the dataset
Unempl_GT which contains the monthly unemployment rates
(x1000) in the Netherlands and a set of 87 related Google Trends
aggregated to a monthly frequency. The data covers the period from 1
January 2014 to December 2017 and is used in the empirical application
in Smeekes and Wijler (2018a).
The package specs enables sparse and automated
estimation of single-equation cointegration models. To estimate a
conditional error-correction model (CECM) on the untransformed levels of
a collection of time series, use the function specs.
Conversion of the data matrix in levels to a CECM representation is
performed automatically within the function. For example,
#library(specs)
unemployment <- Unempl_GT[,1] #Extract the Dutch unemployment levels (x1,000)
GT <- Unempl_GT[,2:11] #Select the first ten Google Trends
my_specs <- specs(unemployment,GT,p=1) #Estimate specs
my_coefs <- my_specs$gammas #store the coefficients
y_d <- my_specs$y_d #Transformed dependent variable
z_l <- my_specs$v #Transformed independent variablesestimates a regularized CECM on Dutch unemployment rates and ten
Google Trends by appropriately differencing and/or lagging the levels of
the time series, see Smeekes and Wijler (2018a, eq. 7) for the full
model specification. my_coefs contains 1,000 column, each
column being the solution for a unique combination of the group and
individual penalty (see the penalty section for more
details).
The short-run dynamics in a CECM are modeled via the inclusion of
lagged differences of the data. By default, specs includes
once lagged differences, but the user is free to specify the desired
number via the input p. Continuing the above example,
my_specs <- specs(unemployment,GT,p=0) #Estimate specs without any lagged differencesThe inclusion of deterministic terms is often desired for correct
model specification. When it is believed that a constant and/or trend
should be included, it is advisable to estimate the model without
imposing regularization on these deterministic components. Accordingly,
the input
deterministics = c("constant","trend","both","none") allows
the user to choose the deterministic specification without penalizing
the deterministic terms. For example,
my_specs <- specs(unemployment,GT,deterministics = "both") #Estimate specs with a constant and trend included
my_coefs <- my_specs$gammas #Store the new coefficients
my_deterministics <- my_specs$thetas #Store the coefficients of the deterministic componentThere are cases in which it may not be of interest to explicitly
model cointegration. In this instance, the lagged levels can be omitted
from the model by setting ADL = TRUE. This estimates a
penalized autoregressive distributed lag (ADL) model on the differenced
data:
my_adl <- specs(unemployment,GT,ADL=TRUE) #Estimate and ADL model
my_coefs <- my_specs$gammas #Store the coefficients (smaller matrix than before)The matrix of coefficients stored in my_coefs does not
contain the contribution of z_l anymore and, consequently,
has a smaller row dimension than before. my_coefs now
contains only 100 columns, as there is no more group penalty included.
The reduced number of penalties to estimate a solution far and some
algorithmic change (see Algorithm and Implementation)
speed up the estimation procedure considerably.
Alternatively, the user may choose to pre-transform the data into the
form of a CECM/ADL, for example to save on computation time in rolling
window forecast exercises. In this case, the function
specs_tr can be used instead. This function operates
entirely analogous to specs, with the exception of
requiring a differenced dependent variable (y_d), the
lagged levels of the time series (z_l) and the required
differences of the data (w) as inputs. Since w
is directly provided to the function, the option to set the lag length
p is omitted. When ADL=TRUE, the user may omit
z_l as an input. For example:
z <- cbind(unemployment,GT)
y_d <- diff(unemployment) #Difference the dependent variable
z_l <- GT[-nrow(GT),] #Lagged levels of the data
w <- diff(GT) #Contemporaneous differences (corresponding to p=0)
my_specs <- specs_tr(y_d,z_l,w) #Estimate a CECM on pre-transformed data
my_adl <- specs_tr(y_d,NULL,w,ADL=TRUE) #Estimate an ADL model on pre-transformed dataFinally, a word on the naming of objects. Within this package the
coefficients corresponding to z_l are referred to as
delta, those corresponding to w as
pi, and the numeric object that stacks both
delta and pi is referred to as
gamma. Therefore, the solutions of specs are
referred to as gammas. The deterministic terms are passed
to the function outputs as D, with there coefficients being
referred to as thetas. As seen in the above examples, when
ADL=TRUE, delta is omitted from the numeric
object gamma in the output. The naming of objects here is
congruent with Smeekes and Wijler (2018a), which may serve as a helpful
guideline for implementation.
Penalty
The functions specs, or equivalently
specs_tr, estimate the model by variants of penalized
regression, customized to the error-correction framework. In its most
general form, specs penalizes each individual coefficient via
lambda_i, and adds a group penalty on delta
via lambda_g. Unless sequences of positive numbers for
lambda_i and/or lambda_g are supplied,
sequences are generated automatically within the function. A grid of 100
values is generated for lambda_i and a grid of 10 values
for lambda_g. The largest value in each grid corresponds to
the smallest value that sets all the coefficients that it penalizes
equal to zero (with the other penalty set equal to zero). The smallest
penalty in the grid is chosen as 1e-4 times the largest value in that
grid. As an important special case, the user may set
lambda_g = 0, in which case the function will estimate the
model by (weighted) L1-penalized regression,
i.e. the (adaptive) lasso:
my_specs <- specs(unemployment,GT,lambda_g=0) #Estimate a CECM without group penaltyIn practice, one typically requires a single choice of penalties that
provides the optimal solution for the model building exercise at hand.
To facilitate selection of such an optimal penalty, the functions
specs_opt and specs_tr_opt, being the
equivalents of specs and specs_tr,
respectively, come with the added functionality of automated selection
of optimal values for lambda_i and lambda_g.
The selection criteria can be set via the input rule, with
the possible choices being BIC, AIC or time
series cross-validation (TSCV). The first two are
information criteria in which the degrees of freedom is approximated by
the number of non-zero coefficients for a particular solution, whereas
the latter is a form of cross-validation that respects the time series
structure of the data. The implementation details for TSCV can be found
in Smeekes and Wijler (2018b, p. 411). The full matrix of solutions, as
well as the solution corresponding to the optimal penalty choice are
included in the output.
my_specs <- specs_opt(unemployment,GT,rule="BIC") #Estimate a CECM with the optimal penalty chosen by BIC
coefs_opt <- my_specs$gamma_opt #Extract the optimal coefficients
my_adl <- specs_opt(unemployment,GT,rule="AIC",ADL=TRUE) #Estimate an ADL model with the optimal penalty chosen by BIC
my_specs <- specs_opt(unemployment,GT,rule="TSCV",CV_cutoff=4/5) #Estimate a CECM with the optimal penalty chosen by TSCV
#Training sample is 4/5 of the total sample
my_specs <- specs_tr_opt(y_d,z_l,w,rule="BIC") #Estimate a CECM based on pre-transformed data, penalty chosen by BICWeights
Finally, specs can be estimated with the use of adaptively weighted
penalization. Automatically generated weighting schemes are available
via the input weights = c("ridge","ols","none"). The
default option, "ridge" constructs the weights via the use
of initial estimates obtained by ridge regression. In detail, the
weights for δi and
πj are constructed as
|δ̂i| − kδ
and
|π̂j| − kπ,
respectively. The penalty parameter for the ridge regression is
automatically chosen by TSCV. Alternative options are to automatically
generate weights via initial ols estimates
(weights = "ols") or to refrain from adaptive weighting
altogether (weights = "none"). Alternatively, it is also
possible to supply a sequence of positive weights directly. Finally, the
values for kδ and
kπ can be chosen by the user via the
equivalently named input options k_delta and
k_pi. The optimal values for these parameters are
case-dependent, although some theoretical guidance is provided in table
1 of Smeekes and Wijler (2018a).
my_specs <- specs(unemployment,GT,weights="ols",k_delta=2,k_pi=1) #Estimate specs with OLS and variable weight exponentsAlgorithm and Implementation
The package specs combines accelerated generalized
gradient descent for the estimation of δ with coordinatewise
descent for the estimation of π. Since δ is penalized
by both an L1- and L2-penalty,
its estimation fits into the framework of the so-called sparse group
lasso, for which numerous computational procedures have been proposed.
This package adopts the algorithm of Simon et al. (2013), as the use of
accelerated gradient descent via Nesterov updates greatly improves
computational time. However, since π is regularized via an
L1-penalty, and is separable from the penalty on
δ, the optimal solution for π is calculated via the
coordinate-wise descent procedure proposed in Friedman et al. (2009).
Essentially, specs iterates between optimizing for
δ and π, where within each iteration one of the
aforementioned two algorithms is repeated until numerical convergence.
All calculations are performed in C++, with the help of the Rcpp and
Armadillo packages.