Nonnegative Garrote as a Variable Selection Method in Panel Data

Nonnegative Garrote as a
Variable Selection Method
in Panel Data
Borislava Vrigazova
Sofia University “St. Kliment Ohridski”
Abstract: In this research, we broaden the advantages of nonnegative garrote as a feature selection method and empirically
show it provides comparable results to panel models. We compare nonnegative garrote to other variable selection methods like
ridge, lasso and adaptive lasso and analyze their performance on a dataset, which we have previously analyzed in another
research. We conclude by showing that the results from nonnegative garrote are comparable to the robustness checks applied to
the panel models to validate statistically significant variables. We conclude that nonnegative garrote is a robust variable
selection method for panel orthonormal data as it accounts for the fixed and random effects, which are present in panel datasets.
Keywords: nonnegative garrote, feature selection, panel models, fixed effects
A. Introduction
Large number of variables and observations characterizes big datasets. They are ‘big’ in dimension and length.
The challenge before researchers is to extract all valuable information from big data for statistical inference.
Researchers can derive valuable information by identifying the significant variables and/or reducing the size of the
sample. In data mining theory, these methods are known as dimensionality reduction.
Dimensionality reduction results in reducing the size of a dataset so that the researcher can extract all significant
information for research. It includes two techniques – feature selection and feature extraction. Feature extraction
transforms high-dimensional data into a space of fewer dimensions, which contain all significant characteristics of
data for statistical modelling. The data transformation may be linear, as in principal component analysis (PCA),
linear discriminant analysis (LDA) or nonlinear, as in non-linear LDA, kernel PCA. Feature selection, in contrast,
reduces the number of variables by choosing those, which are statistically significant. As feature selection excludes
variables from a statistical model, the reduction of dimensionality is performed by variable selection. What type of
dimensionality reduction technique will be used depends on the data type, researcher’s goals and the quality of data.
In this research, we make comparison between the performance of variable selection methods like ridge, lasso,
adaptive lasso and nonnegative garrote. Although these methods are popular in academic literature, we will test their
performance on panel data to prove empirically that nonnegative garrote outperforms other feature selection methods
in panel data. As the nonnegative garrote is the only variable selection method that accounts for fixed and random
regressors, we show empirically that nonnegative garrote provides robust variable selection in datasets without
multicollinearity. Thus, we broaden the set of advantages of nonnegative garrote. Next section reviews academic
literature. Section 3 and 4 present the theoretical and empirical framework. Section 5 concludes.
B. Literature overview
For many years, one of the most popular methods for variable selection has been the selection criteria of Mallows.
During the 1960s, the Mallows’ C was new and modern tool to select significant variables in regression problems.
As linear regression suggested a pool of 2k possible models and only a few would profit good enough fit of data,
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criterion for choice of the best model was necessary. Mallows [4] invented Mallows’ Cp as criterion for rejecting
unfit models to filter the ones, which provide best fit of data provided the best model had the smallest prediction
error. Gorman and Toman [3] explored further the advantages of the selection criterion using visualization tools. The
best fit between 2k models would select based on the combination of variables, which provided the smallest
prediction error. The Mallows’ criteria appeared to be simple for visual and numeric interpretation.
Due to computational exhaustion and other disadvantages of Mallows’ criterion [5], researchers continued to look
for other methods to select statistically significant variables in linear regression problems. The Russian
mathematician Tikhonov, who worked on mathematical applications in physics, invented ridge regression as a
solution to ill-posed problems. In 1962, Hoerl [2] applied ridge regression as a variable selection method in statistics
for the first time. Unlike, Mallows’ Cp criterion, ridge regression imposes penalty on regression coefficients. Some
of the coefficients shrink to a number very close to zero, which filters out the variables with insignificant impact on
the model. Hoerl and Kennard [6, 7, and 8] defined the ridge trace and its applications in orthogonal and
nonorthogonal problems. Unsolved was the problem about a way of choosing the optimal value of the ridge penalty
parameter. In 1979 generalized cross-validation (GCV) was proposed as a method for selecting the ridge parameter
[9]. In modern research articles, ridge regression is applied with GCV to various statistical problems. Although it
does not shrink regression coefficients to zero, it is used along with other variable selection methods for model
validation. The use of ridge regression is broader which justifies its application nowadays in data analysis [10].
Ridge regression as a feature selection method is often compared to lasso regression [11]. Lasso was devised in
1996 [11] as a method which filters significant variables. Similar to ridge, lasso imposes penalty to regression
coefficients but unlike ridge, shrinks some coefficients to zero. In 2006, Efron and Hastie [19] devised least angle
regression to perform shrinkage. The advantages of lasso range from fast computation to being easily used to provide
solution paths to other methods like lasso. Many software packages, for example, provide solutions for lasso
regression based on lars paths [20].
Lasso [16] is based on another shrinkage method introduced a year earlier – nonnegative garrote [12].
Nonnegative garrote has an advantage to lasso as it can provide robust results in small and big datasets. The
difference among ridge, lasso and nonnegative garrote appears to be in the type of constraint on parameters.
Depending on the constraint, the methods yield different results. Although ridge and lasso use cross validation to
select the value for the tuning parameter [9 and13], the nonnegative garrote can use both cross validation and
bootstrapping depending on the data type [12]. The two methods and their adaptations are widely used in various
topics [13 and 14].
Zou [15] criticized the lasso approach for yielding a non-robust solution in big datasets. He described its
disadvantages in details in [15] and proposed oracle method called adaptive lasso. Adaptive lasso overcomes some
disadvantages of lasso by weighing the variables in a dataset. He theoretically justified his hypothesis that adaptive
lasso performs more reliable variable selection than lasso. The properties of adaptive lasso allow variable selection in
big datasets with heteroscedasticity [17] and for choice of generalized empirical likelihood estimators [19].
In 2005, Zou and Hastie introduced a hybrid between ridge and lasso called elastic net regularization [21]. The
elastic net combines the penalty terms of lasso and ridge and outperforms lasso. The elastic net can be applied in
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metric learning [22] and portfolio optimization [23]. Similar to lasso [24], elastic net can be reduced to support
vector machine model for classification problems [25 and 26].
The evolution of variable selection methods has led to adaptations of these methods depending on the research
goal and the quality of dataset. In the next section, we will provide the theoretical framework behind ridge,
nonnegative garrote, lasso and adaptive lasso as they appear in the original research articles.
C. Methodological Framework
In this section, we will present the theoretical framework behind the shrinkage methods. They all fulfill the task of
minimizing a target given constraint on the coefficients. The shrinkage methods differ only by the type of the
constraint.
Ridge regression [11]:
(1)
As λ goes towards zero, the ridge coefficients coincide with the coefficients of the OLS regression. When λ goes
towards one, ridge coefficients shrink towards zero.
A special case of the estimator for an orthonormal matrix is
Equation 2 shows that with the shrinkage of coefficients, variance is minimized but bias is introduced. Ridge
regression rarely shrinks regression coefficients to zero. Instead, it shrinks OLS coefficients to values close to zero.
Lasso regression [11]:
(3)
The lasso, on the other hand, shrinks some coefficients to zero, thus eliminating the statistically insignificant
variables. The penalty is called L1 penalty. The penalty parameter λ determines the amount of
shrinkage. When λ = 0, no shrinkage is performed and the lasso parameters equal the estimates of the OLS
regression. With the increase of the value of λ, more parameters are excluded from the regression. When λ = ∞,
theoretically all coefficients are removed.
A disadvantage of the least absolute selection operator (lasso) is its inability to perform robust variable selection in
datasets, which are highly correlated. To solve this problem, Zou and Hastie [21] devised the elastic net, which is
another regularization method, which combines penalty terms from ridge and lasso. The avid reader can review their
article "Regularization and Variable Selection via the Elastic Net" for further details on elastic net.
Despite the advantage of lasso over ridge in terms of variables shrinkage, lasso is robust only in very big datasets.
It fails to provide robust feature selection in smaller datasets. Adaptive lasso was devised [15] to overcome this
disadvantage of lasso.
Adaptive lasso [15] :
(4)
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The adaptive lasso estimator has similar structure as lasso but the difference is the each coefficients has weight in
the penalty term. The tuning parameter behaves as λ in lasso. By weighing the coefficients, statisticians avoid
spurious elimination of variables. Adaptive lasso can result in robust feature selection even in smaller datasets.
Choosing the tuning parameter in feature selection method is another important issue. We have used the k-fold
cross validation as the underlying method for choosing the optimal value of the tuning parameter in ridge and lasso.
The k-fold CV being the most widely used. The k-fold cross validation draws k different samples from a dataset and
compares the models’ error. The goal is to be chosen the value of the tuning parameter, which minimizes the CV
error. The k-fold CV method is given by equation 5:
(5),
Where K is the number of samples drawn from the dataset and k varies from one to K.
K-fold CV can be used for determination of the tuning parameter in the nonnegative garrote (eq. 6). However,
nonnegative garrote uses k-fold CV when the assumption for the random independent variables is fulfilled. The
nonnegative garrote can select significant variables in small and big datasets.
Nonnegative garrote [14, 29]
(6)
Where dj >0 for all j and
),
being the least square estimate, λ is tuning parameter. The nonnegative garrote estimate of the regression
coefficient is given by
(7)
Under orthogonal designs the nonnegative garrote estimator can be expressed by
The shrinkage factor, as a result, will be close to one if the least square estimator is large. If the least square
estimator is small, the shrinkage factor can reduce to zero.
The problem about the optimal value of the tuning parameter in the nonnegative garrote case can be solved either
by k-fold cross validation or by little bootstrap procedure [1]. K – fold CV in nonnegative garrote, unlike lasso and
ridge, is performed if the independent variables are assumed to random and uncorrelated. Lasso and ridge lack such
an assumption. When nonnegative garrote assumes fixed independent variables, the little bootstrap procedure
described in [4] can select the tuning parameter. When the X variables are random, the selection of the best tuning
parameter can be performed by cross validation [1].
Nonnegative garrote differs from other variable selection methods like ridge and lasso in its ability to perform
robust variable selection for various assumptions for X variables (random or fixed) [1]. This property of nonnegative
garrote reminds of panel models where the panel OLS accounts for fixed or random effects. The equation of the
panel models is:
ititXit cfY   )( (9),
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Where itc denotes the types of effects and it marks the error term. Although academic literature has not
investigated the connection between panel models and nonnegative garrote as a panel variable selection method, we
believe there is such connection. More specifically, we have conducted a research of panel fixed effects model and
compared the results of variable selection via nonnegative garrote with little bootstrap to conclude that nonnegative
garrote for fixed X variables successfully performs feature selection in panels. We compared the results of the
nonnegative garrote procedure to lasso, ridge and adaptive lasso and discovered that they fail in panel datasets.
Although we base our conclusion on one dataset and many experiments must be conducted, we believe this
observation can be used to test the robustness of panel models and, on the other hand, it can be a time saving
instrument for initial investigation of panel models.
Scientists apply panel models to datasets, which contain both time series and cross-sectional observations. The
problem with panel datasets is the fact that the model should account for the effects of time and individual
characteristics. To do that, scientists perform tests (Hausman [30], Pesaran [31]) to examine whether time and/or
individual effects are present. They also investigate whether data are connected by factors that are characteristics of
the dataset (fixed effects) or randomness is the underlying process in dataset (random effects) [28]. As a result,
finding the right panel model is usually time-consuming process as it includes running a big amount of models and
testing them for robustness.
We believe the similar results between panel models and the nonnegative garrote can be attributed to the
similarities between panel fixed effects and little bootstrap in the X fixed case. In the X random case, the panel
random effects can describe a procedure similar to cross validation. Breiman [1] describes the little bootstrap
procedure as a method for choosing random samples with replacement from a dataset. The replacement corresponds
with fixing the X variables to be a particular quantity [1]. The panel fixed effects models also assume X variables to
be fixed quantities. Thus, the little bootstrap procedure generates random samples of fixed X and finds the tuning
parameter of nonnegative garrote, which results in the smallest MSE in the fixed case. Then, the nonnegative garrote
shrinks the coefficients of the fixed variables and selects only those, which are statistically significant. The fixed
effects panel model, on the other hand, is based on panel OLS method, which estimates the coefficients of X
variables with fixed quantities. The estimates of some coefficients can be statistically significant that leads to their
elimination. Despite the fact that nonnegative garrote with bootstrap and panel fixed effects models choose the
significant variables in different ways, the two of them address the problem of fixed quantities in X variables and
should result in similar statistical significance.
Similar parallel between random panel effects models and nonnegative garrote with cross validation can be done.
Random effects assume that X quantities are random. The panel model estimates the coefficients, based on a panel
OLS method with random independent variables. Some of the coefficients become statistically significant. Cross
validation in the nonnegative garrote, procedure chooses random samples from data, which, unlike the little
bootstrap, are independent. In this way, cross validation accounts for randomness in data. Once cross validation
estimates the MSE of many random independent samples, the tuning parameter for the nonnegative garrote can be
selected and shrinkage can be performed. The statistically insignificant variables are shrunk to zero.
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In the next section, we show our experiment on a panel dataset about property rights which contains fixed effects.
Although our research needs to be extended with bigger amount of datasets, we believe that our results contribute to
the practical advantages of the nonnegative garrote.
D. Results
We have carried out our experiment on a two datasets - property rights dataset, described in table 1 and a
scoreboard of indicators for financial crisis in the EU described in [30].
We have analyzed dataset 1 in a research for determinants of property rights in a panel of data [29]. Our analysis
shows that panel fixed effects models are present in the dataset. We have outlined the main determinants of property
rights (model 1) and tested whether shadow economy (model 4), income (model 3) and gender inequality (model 2)
affect the index of property rights. Using the panel OLS method, we have statistical significance of variables
presented in table 1:
Table 1: OLS Panel method results:
Model 1 Model 2 Model 3 Model 4
dbirth -0.06** -0.06** -0.06** -0.05*
(-28.66) (-28.67) (-27.47) (-25.22)
drate -0.10** -0.10** -0.10** -0.11**
(-26.17) (-26.16) (-25.81) (-27.16)
mortality -0.05*** -0.05*** -0.05*** -0.05***
(-70.89) (-72.08) (-72.58) (-70.78)
unempl -0.02** -0.02** -0.02** -0.02**
(-27.96) (-27.96) (-25.91) (-28.54)
urban 0.01*** 0.01*** 0.01*** 0.01***
(35.26) (34.75) (32.54) (35.84)
dmilitary 0.26* 0.27* 0.27* 0.26*
(24.76) (24.78) (25.06) (24.08)
dgender -0.00
(-0.05)
lgini -0.09
(-0.51)
dshadow 0.07***
(40.37)
R2
0.67 0.67 0.67 0.68
F-stat 135.59*** 115.93*** 116.55*** 118.45***
Data set 2000 -2014 2000 -2014 2000 -2014
2000 -
2014
Annual Annual Annual Annual
N 420 420 420 419
Source: authors’ calculations
According to the results of our experiment, income and gender inequality are statistically insignificant. The panel
models lack cross sectional dependencies and multicollinearity and robustness checks have been made [29].
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We have also performed variable selection on the transformed dataset (table 1) via lasso, ridge, adaptive lasso and
the nonnegative garrote for fixed X variables. Table two presents the results of feature selection methods and
compares them with panel models (1-4):
Table 2: Feature selection methods
Lasso Ridge Adaptive Lasso NNG Panel models significance
dbirth -0,05 -0,05 0 0,00 **/*
drate -0,11 -0,13 -0,03 -0,05 **
mortality -0,04 -0,04 -0,04 -0,48 ***
unempl -0,02 -0,02 -0,02 -0,22 **
dshadow 0,04 0,04 0,01 0,01 ***
dgender 0 0,01 0 0,00
infl 0 0 0 0,00
loginternet -0,02 -0,01 0 0,00
dexpect -0,05 -0,07 0 0,00
emissions 0,01 0,01 0,01 0,00
dhealth 0 0 0 0,00
urban 0,01 0,01 0,01 0,33 ***
lgini -0,07 -0,11 0 0,00
dmilitary 0,2 0 0,13 0,08 *
When we compare the results of the feature selection methods with the results of the panel models we see that
lasso identifies 11 non-zero variables against seven significant variables from the panel models. Zou and Hastie [15]
criticize lasso estimation for providing non-robust estimates in relatively small datasets and highly correlated
variables. Although the variables lack multicollinearity and autocorrelation, the number of observations in the dataset
is relatively small, so lasso fails to perform robust shrinkage of all insignificant variables. Another reason why lasso
fails in this dataset is the presence of panel fixed effects, which are absent in cross-sectional and time series datasets.
Lasso does not account for panel effects, fixed or random. Lasso was designed with the purpose to perform shrinkage
of coefficients to zero in large datasets, most often applied on cross sectional data [11]. As a result, fixed and random
effects in panels were not included in the theoretical framework of lasso and lasso fails to provide robust shrinkage in
panel data.
The disadvantages of lasso have encouraged Zou and Hastie [15] to propose a weighted version of lasso, which is
robust in the presence of multicollinearity and smaller number of observations. They called it adaptive lasso. In table
2, we see that the adaptive lasso outlines seven variables with non-zero coefficients and their amount coincides with
the amount of significant variables in panel models. However, a more detailed analysis shows that the non-zero
variables from the adaptive lasso are different from the statistically significant variables in the panel models.
According to the adaptive lasso, the birth rate is not significant and emissions of carbon dioxide are significant,
which differs from panel models. Similar to lasso, adaptive lasso does not account for panel effects. Ridge regression
[2] results in similar estimations like lasso with more nonzero coefficients as it does not shrink to zero. From all three
methods ridge is the least robust as it does not perform variable selection in panel data and brings no valuable
information about the dataset.
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Bearing these results in mind, we wanted to examine whether nonnegative garrote with little bootstrap can provide
competitive results to panel models. As Leo Breiman [1] has introduced the advantages of nonnegative garrote as a
feature selection method, we raised the question of whether it can be robust in panel data. Breiman [1] has proposed
nonnegative garrote as an alternative of best subset selection and ridge. Later, Tibshirani and Hastie [11] have
developed lasso based on the nonnegative garrote of Breiman. According to Breiman, nonnegative garrote performs
robust variable selection not only in bigger datasets like lasso, but also in smaller ones. However, it fails when the
dataset has multicollinearity and outliers. As our dataset lacks outliers and multicollinearity and has fixed effects, we
ran nonnegative garrote with little bootstrap on our dataset.
As the nonnegative garrote is the only variable selection method, which proposes two selection procedures for the
tuning parameter based on the assumptions for the X variables, it is reasonable to test it on panel datasets. As table
two shows, the nonnegative garrote selects the same variables as the panel models. Although the coefficients have
different values in nonnegative garrote and panels, the same variables appear to be statistically significant. According
to the nonnegative garrote, the birth rate has zero coefficient, so it is statistically insignificant but in models, 1-3 it is
significant at the 1% significance level and in model 4 the significance level is 10%, which shows the effects of birth
rate may be negligible. Thus, this result is comparable with the result of the nonnegative garrote.
As the fixed effects in panel data imply fixed X variables and the little bootstrap in nonnegative garrote uses
sampling with fixed variables, nonnegative garrote can be successfully applied as a panel feature selection method.
In this research, we have validated the results from the panel models with panel GMM and robust covariance matrix
(table 3). After we have confirmed the results, we have run nonnegative garrote. The results show that nonnegative
garrote with little bootstrap was another way to validate the results of our research. Not only the selected variables
via nonnegative garrote coincided with those from the panel models, but also with the results of the robustness
checks. Nonnegative garrote proved to be useful not only for smaller datasets in variable selection in cross sectional
data but also in panel data. As a result, nonnegative garrote with little bootstrap can be applied to fixed effects panel
models without multicollinearity and outliers. Nonnegative garrote with cross validation can be applied to random
effects panel models.
Table 3: Robustness checks
Model 5 Model 6 Model 7 Model 8
DBIRTH -0.06** -0.08*** -0.06** -0.05*
(-1.93) (-2.57) (-1.92) (-1.60)
DRATE -0.11*** -0.10*** -0.10*** -0.11***
(-2.35) (-2.53) (-2.35) (-2.41)
MORTALITY -0.05*** -0.05*** -0.05*** -0.05***
(-15.93) (-14.30) (-15.33) (-15.78)
UNEMPL -0.02*** -0.02*** -0.02*** -0.02***
(-7.08) (-7.56) (-6.57) (-7.07)
URBAN 0.01*** 0.01*** 0.01*** 0.01***
(10.21) (9.95) (10.01) (10.32)
DMILITARY 0.27*** 0.26*** 0.27*** 0.26***
(3.96) (3.83) (4.01) (3.86)
DGENDER 0.01
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(0.81)
LGINI -0.09
(-1.19)
DSHADOW 0.08***
(2.39)
R2
0.67 0.63 0.67 0.68
Instrument rank 22 9 23 23
J-stat 139.34*** 105.40*** 149.22*** 137.48***
Similar conclusion can be made from table 4 where we have analyzed the determinants of the change in the GDP
in the context of EU scoreboard indicators for financial crisis [29]. Table 4 presents the results from the panel models
with period and cross section fixed effects. Table 5 makes comparison between the panel model and the feature
selection methods, which we have applied to the property rights dataset.
Table 4: Panel two-way fixed effects model for scoreboard of indicators for financial crisis in the EU
Dependent Variable: GDPCHANGE
Variable Coefficient Std. Error t-Statistic Prob.
c 1.808557 0.736731 2.454842 0.0144 ***
dcpi 0.055932 0.020464 2.733148 0.0065 ***
dfincon -0.109693 0.040897 -2.682185 0.0075 ***
dgfcf 0.307502 0.053179 5.782384 0.0000 ***
dggdebt -0.010726 0.010249 -1.046570 0.2958
dnip -0.012993 0.005526 -2.351405 0.0191 ***
dpsdebt -0.001152 0.004332 -0.265959 0.7904
drd 0.058476 0.193896 0.301584 0.7631
dreer 0.005063 0.016126 0.313974 0.7537
dresidential 0.137797 0.145171 0.949207 0.3429
lactivity 0.188133 0.111589 1.685951 0.0924 *
ldcredit -0.056197 0.118423 -0.474543 0.6353
privatecf 0.021183 0.012114 1.748612 0.0809 *
riskpovst 0.033869 0.019763 1.713716 0.0871 *
totalfsliab 0.027452 0.007956 3.450620 0.0006 ***
ulc -0.019610 0.003790 -5.174630 0.0000 ***
youthunemp -0.031945 0.013864 -2.304101 0.0216 ***
R-squared 0.620469
F-statistic 14.04933
Prob(F-statistic) 0.000000
Table 5: Comparison feature selection / panel model for scoreboard of indicators
ridge lasso adaptive nng
panel
significance
dcpi 0,01671 0,02767 0,07399 0,12600 ***
dfincon -0,07761 -0,05544 0,00000 -1,82711 ***
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dgfcf 0,23082 0,44817 0,58651 3,18178 ***
dggdebt -0,01821 0,01809 0,00000 0,00000
dnip -0,00686 -0,00558 0,00000 0,00368 ***
dpsdebt 0,00131 0,00000 0,00000 0,00000
drd 0,01177 0,00000 0,00000 0,00000
dreer 0,01476 0,00000 0,00000 0,04267
dresidential 0,33380 0,28351 0,00000 0,00000
lactivity 0,02823 0,00000 0,00000 0,00515 *
ldcredit -0,11780 -0,07883 0,00000 0,00000
privatecf 0,02555 0,03043 0,02798 1,06233 *
riskpovst -0,01291 0,00000 0,00000 0,32034 *
totalfsliab 0,01771 0,03002 0,03608 1,28388 ***
ulc -0,00515 -0,00988 -0,01187 0,00000 ***
youthunemp -0,01350 -0,00493 0,00000 0,73674 ***
The column for panel significance in table 5 shows which variables are statistically significant according to the
panel two-way fixed effects model. When we compare ridge with the panel model, it becomes clear that ridge failed
to capture statistical significance, as all coefficients are different from zero. Ridge is not a reliable method for feature
selection in panel datasets with fixed effects.
Similarly, to the property rights dataset, lasso shrunk some variables to zero. However, lasso identified variables,
which are not significant in the panel model. For instance, the differenced government debt and the domestic credit
are not significant in the panel in contrast to lasso. Lasso returned the log of activity rate (lactivity) as insignificant
when, in fact, it is significant. Similar is the case with the risk of poverty (riskpovst). The lasso regression failed to
perform feature selection successfully.
Adaptive lasso in contrast to lasso and ridge resulted in feature selection similar to the panel model. Some
discrepancies can be observed, however. The adaptive lasso selected the differenced final consumption (dfincon), the
net international investment position (nip), the log of activity rate (lactivity), the risk of poverty and the youth
unemployment as insignificant when, in fact, they are significant. The accuracy of the adaptive lasso in the dataset is
questionable.
When we look at the nng column, we see that the nonnegative garrote outlined as significant variables almost the
same variables as the two-way panel model. The only discrepancy is in the dreer variable. The nonnegative garrote
considers the real effective exchange rate to have nonzero coefficient while the panel model excludes it as
insignificant variable. According to the nonnegative garrote, if the rate of increase in the real exchange rate becomes
faster, the GDP will increase by 0.04%, which may be a very small, almost unobservable increase. In contrast to the
other feature selection methods, the nonnegative garrote resulted in identifying all significant variables in the panel.
Similarly to the property rights dataset, the nonnegative garrote performed accurate feature selection in the panel
scoreboard indicators for financial crisis. Despite the fact that different indicators and dependencies characterize the
two datasets, the common line between them is the presence of fixed effects. The presence of fixed effects in panel
models is modelled by OLS with fixed effects. As the little bootstrap captures the fixed effects, the nonnegative
garrote successfully performs variable selection in panel data.
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E. Concluding remarks
Breiman [1] introduced the advantages of nonnegative garrote as a variable selection method. He proved its
computational advantage to subset selection methods and its ability to perform robust variable selection in small and
big datasets, unlike lasso and ridge. We broaden this set of advantages by showing that nonnegative garrote, unlike
other variable selection methods, accounts for fixed X variables by little bootstrap procedure in fixed effects panel
data and for random panel effects by cross validation. Although we lack sufficient empirical evidence, we believe
our finding has an important impact on how panel data can be preprocessed.
ACKNOWLEDGMENT
We thank prof. Ivan Ivanov1
for the valuable comments and recommendations.
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