Content uploaded by Robert Nyabwanga
Author content
All content in this area was uploaded by Robert Nyabwanga on Mar 20, 2024
Content may be subject to copyright.
Electronic Journal of Applied Statistical Analysis
Vol. 17, Issue 01, March 2024, 153-171
DOI: 10.1285/i20705948v17n1p153
Hybrid Methodology for Sparse
Selection of Generalized Estimating
Equations Model for the Drivers of Firm
Value
Robert Nyamao Nyabwanga 1
*
a
aKisii University, Mathematics and Actuarial Science Department, 408-40200, Kisii, Kenya
15 March 2024
The study proposes a two-step hybrid methodology for sparse generalized
estimation equations modeling of the drivers of shareholder value creation.
Through the methodology, the validity of the Gordon constant growth model
is established and other non-dividend factors’ contribution to shareholder
value creation is assessed. The two-step hybrid method involves picking
out the right intra-subject correlation matrix and set of regressors using
EAIC and QIC respectively (EAIC-QIC) and then obtaining the penalized
GEE estimators of the selected model. Penalization is useful in removing
redundant regressors from the final model. The performance of the proposed
method was compared to that of exclusively using QIC method in selecting
both the correlation matrix and set of regressors. The study results showed
that, whereas EAIC preferred the parsimonious order one auto-aggressive
{AR(1)}structure for the data, QIC preferred the unstructured matrix which
estimates the highest number of correlation parameters. Using the AR(1)
structure and Algorithm 2, the GEE model chosen had higher efficiency
compared to when QIC is used to select both the correlation matrix and
regressors. Based on the results, the study concludes that adopting hybrid
methods enhances efficiency of GEE estimators. On firm value, the study
concludes that besides the elements in the Gordon-Constant growth model,
the financial health of a firm is a vital indicator of value creation ability by
firms.
*
Corresponding authors: nyamaonyabwanga@gmail.com
©
Universit`a del Salento
ISSN: 2070-5948
http://siba-ese.unisalento.it/index.php/ejasa/index
154 Nyabwanga
Keywords: Generalized Estimating Equations, Model Selection, Correla-
tion Structure, Sparsity, Penalized GEE, Shareholder Value Creation.
1 Introduction
The Constant-Growth Model (Gordon, , 1959) which was built on the assumption that
the company is a going concern and will grow forever, paying dividends at a continually
increasing rate, has been the fundamental model in predicting value creation by firms
for their shareholders in decades. Based on the model, value creation is a function of
dividends and the discount rate:
P0=D1
r−g=D0(1 + g)
r−g(1)
where, P0=the present-day market value of the anticipated flows of dividends per
share(D1); D0=Current Earning Per Share; g = growth in earnings and r is the equity
cost of capital. According to model 1, dividend per share and economic profitability
proxied by income per share are the key determinants of the present market value of a
share. This leads to the model:
Mv=α0+α1D+α2Y(Gordon,, 1959).(2)
Where Mv= the year-end price, D = the year’s dividend per share, and Y = the year’s
income per share.
Given that growth in earnings (g) is a function of the return on equity (ROE) and
retention ratio (γ) and further that earnings per share is a function of ROE and the
book value of equity shares (Bv), model (1) can be modified to take the form;
Mv
Bv
=ROE −g
r−g(3)
Considering the ratio the current year’s dividend and the book value of equity (d) and
equation (3) it can be observed that ROE adjusted for the cost of equity capital (ROE-r)
which is a measure of economic profitability (Ep), growth rate of earnings and dividend
policy (d) are the key drivers of firm value hence Equation (2) can be refined to take the
form: Mv
Bv
=α0+α1Ep+α2g+α3EP∗g+α4d+ϵ(4)
Model (4) implies that the key drivers of shareholder value creation are growth in earn-
ings (g), economic profitability proxied by earnings per share (γ) and dividend policy.
The consideration of dividend policy proxied by the dividend pay-out ratio as a driver
for firm value was also considered by Hansda, et al. (2020) and Agung et al. (2021a).
Hansda, et al. (2020) using dynamic panel regression with two-step system Generalised
Method of Moments (GMM) established that dividend policy had no significant effect on
firm value while Agung et al. (2021a) just like Bataha et al. (2023) who both used multiple
linear regression method, established that dividend policy had a significant positive effect
Electronic Journal of Applied Statistical Analysis 155
on firm value which were in line with the signalling theory. These contrasting findings
limits the development of theory on the relationship between dividend policy and firm
value. The disagreements have been attributed to changes in study contexts, research
periods, research design and methods of analysis. For instance the methods employed
by Hansda, et al. (2020), Bataha et al. (2023) and Agung et al. (2021a)) resulted to
differing results.
Further to the factors considered in Equation (4), the study considers the Modigliani
and Miller (MM) hypothesis which under some specified assumptions established that
debt policy had no effect on firm value and include debt policy as a factor that can influ-
ence value creation by firms. Chen and Chen, (2011) considered debt policy represented
by leverage (proxied by the debt-equity ratio) and established a negative relationship
with firm value, a result that was in contrast to the MM hypothesis.
Likewise, the study considered accounting profitability measured by return on assets
(ROA) ratio, firm size, financial distress measured by the Atlman Z score and working
capital policy (WCP). Pandey, (2005) established that accounting profitability was an
important value driver. In relation to firm size, Chen and Chen, (2011) found out that
firm value had a positive relationship with profitability while Mule et al. (2015) showed
that firm size had no significant relationship with firm value. Honjo and Harada, (2006)
considered the number of board members (Bsize), working capital policy and financial
health as indicators of firm value. Just like Honjo and Harada, (2006), Nguyen and
Faff, (2007) established that there was no significant relationship between firm value
and board size. The study therefore considers the expanded model in Equation (5)
Mv
Bv
=α0+α1g+α2Ep+α3Ep∗g+α4d+α5ROA +α6Lev +α7F siz e
+α8Bsize +α9Z+α10W C P +ϵ(5)
The standard multiple regression model has often been used in establishing the rela-
tionship between the various factors and firm value (Agung et al., 2021a; Bataha et al.,
2023). This method does not take into consideration the intra-class correlation and as-
sumes that observations within and between clusters are independent. As observed by
Gordon, (1959), a cross-section of studies that used regressions analysis yielded high
correlation but with differing regression coefficients and corresponding standard errors
amongst samples from different industries. This has led to questions being raised on the
economic significance of the results despite the fact that the variation in price among
shares is of paramount importance in guiding investors’ choice of investment possibilities
and in directing corporate financial policy formulation.
This study sought to extend the use of the method of generalized estimating equations
(GEE) to corporate finance so as to consider each firm as a cluster from which a number
of measurements are taken and have a defined working correlation structure that defines
the within-cluster correlation. The method will also allow for the choice of both an
appropriate link function that depend on the distribution of the response variable and
an appropriate set of covariates for the mean structure (Carey and Wang , 2011). GEE
is a population average method developed based on the quasi-likelihood theory hence
does not need one to specify the distribution of the response variable but only the mean
156 Nyabwanga
and variance functions of the response observations (Carey and Wang , 2011). Further,
as observed by Cui and Qian (2007), even with a mis-specified working correlation
structure, GEE analysis still yields consistent regression coefficient estimators.
Pan (2001) developed and championed for the routine use of Quasi-likelihood Infor-
mation Criteria (QIC) for the selection of both the correlation matrix and best subset
of explanatory variables. He commended the QIC criteria for its good performance in
variable selection. QIC has however been established to more often select a wrong cor-
relation structure leading to less efficient GEE estimators to the extent of 40%. The
finding on correlation matrix selection performance by Pan (2001) were supported by
other findings which established that QIC was weak in picking out the true correlation
matrix for repeated measurements (Nyabwanga et al., 2019a; Wang et al. , 2012; Cui
and Qian , 2007). The wanting performance of QIC has over the years led to several
modifications by scholars such as Hin and Wang, (2009) who developed the Correla-
tion Information Critieria (CIC), Chen and Lazar (2012) who developed EAIC and
EBIC among others in efforts to increase chances of selecting the correct matrix hence
enhance efficiency of the estimates. On their part Oyebayo and Mohdi, (2019) cham-
pioned for the use of Hybrid methodology since there was no single method that could
effectively select both the correlation matrix and set of covariates. In line with their
recommendation, the study proposes a Two-Step Hybrid methodology and applies it in
modelling the drivers of firm value for firms listed in the Nairobi Securities Exchange
(NSE), Kenya. Further, the performance of the proposed method is compared to the
QIC-only benchmark method.
The rest of the paper is organized as follows: Section 2 on Materials and Methods
provides data description, a review of the GEE method in the context of the drivers
of shareholder value creation, GEE model selection in which the Hybrid methodology
is presented in Algorithms 1 and 2 and finally the efficiency measures used. Section 3
presents the results and discussion while section 4 provides the conclusions.
2 Materials and Methods
2.1 Data Description
Data for the study was collected for a sample of 53 firms out of 61 listed in the NSE
that were in operation as at January 2012. The representative sample was obtained
using proportionate stratified Sampling method. The corresponding number of firms per
cluster were: 8 agricultural; 3 auto-mobile and accessories; 10 banking; 8 commercial
services; 4 construction and allied ; 3 energy and petroleum; 5 insurance; 3 investment;
8 manufacturing and 1 telecommunications. Data collected covered a period of 6 years
(2012-2017)which resulted to 318 binary outcomes for Mv
Bv and all the covariates captured
in equation (5).
Electronic Journal of Applied Statistical Analysis 157
2.2 The GEE Model
For each firm i, let Yit ∈(0,1) be the value creation history, i= 1 · · · ,53 and t= 1,· · · ,6
such that:
Yit =(1 if MV
BV >1
0 if M V
BV ≤1
Therefore, Yi={yi1,· · · , yim }Tis the m×1 random vector of value creation history
for the ith firm, i= 1,· · · , n. Further, let Xjit = (X1it, X2it ,· · · , Xkit) be the vector
of explanatory variables wherek= 1,2,· · · ,10, t= 1,2,· · · , m,X1=economic profitabil-
ity, X2=growth in earnings, X3=interaction between growth in earnings and economic
profitability, X4=logarithm of total assets, X5=leverage (debt-equity ratio), Return on
Assets(ROA), X6=Dividend payout Ratio, X7=Level of fincial health calculated us-
ing the Altman’s Z score given by Z= 6.56Wc
Ta+ 3.26RE
Ta+ 6.72EBI T
Ta+ 1.02Mv
TLwhere
WC=Working Capital; RE=Retained Earnings; EBIT=earnings before interest and tax;
TL=Total value of Liabilities and Ta=Total assets, X8= Number of Board members, X9=
ROA, X10=Liquidity ratio representing the working capital policy.
For the set of data {Yit,Xj it}, let E(Yit|Xj it) = µit relate to Xjit through an appro-
priate link function so that
g(µit) = XT
jit β(6)
V ar(Yit|Xjit ) = ϕV (µit) (7)
where µit =P r(Yit = 1|Xjit) represents the chances of a firm i creating value for its
stockholders at time t hence (1 −µit) is the probability that it fails to create value for
its shareholders at time t and
V ar(Yi) =
C(1,1) C(1,2) . . . C(1,m)
C(2,1) C(2,2) . . . C(2,m)
.
.
..
.
.....
.
.
C(m,1) C(m,2) . . . C(m,m)
(8)
Where C(t,t),t= 1,· · · ,6 are variances and the off-diagonal elements are covariances.
β={β0, β1,· · · , β10 }Tis a p×1 vector of regression coefficients such that;
µit =g−1(β0+
k
X
j=1
βjXjit ) (9)
and considering that Yit is a binary response, the logit link function can be applied hence
equation (9) can be re-written as:
log(µit
1−µit
) = β0+
k
X
j=1
βjXjit (10)
µit =Exp(β0+Pk
j=1 βjXjit )
1 + Exp(β0+Pk
j=1 βjXjit )(11)
158 Nyabwanga
Liang and Zeger (1986), established that by solving the system of generalized estimating
equations in Equation (12) iteratively, the vector of parameters βare obtained.
U(β) =
n
X
i=1
∂µi
∂βTV−1
i(yi−µi) = 0 (12)
where
Vi=A1
2Ri(ρ)A1
2(13)
is the model based variance-covariance matrix for Yiand R(ρ) is the working correlation
matrix for cluster i whose order is mi×mifully specified by ρ.
Ai=
σ2
(Yi1)0. . . 0
0σ2
(Yi2). . . 0
.
.
..
.
.....
.
.
0 0 . . . σ2
(Yim)
(14)
Remark 2.1. Since GEE is not a likelihood based method of estimation, a solution to
equation (12) may be established by using the Iterative Weighted Least Squares in which
the estimating equations are solved by linearizing µiaround an initial estimate say β0
and also evaluating Viat the same β0. Let h(µi) = ηi=XT
jit βjsuch that;
∂µi
∂β=∂µi
∂ηi
×∂ηi
∂β×∂µi
∂ηi
Xi
then, by first-order Taylor approximation of µiin the neighborhood of µ(0)
i
µi=µ(0)
i+∂µi
∂β(β−β0)
=µ(0)
i+∂µi
∂ηi
XT
i{β−β0}(15a)
and
yi−µi=yi−µ0
i−∂µi
∂ηi
XT
i(β−β0) (15b)
Plugging (15a) and (15b) into (12) results into:
n
X
i=1
∂µi
∂η i
V−1
iXi(yi−µ0
i−∂µi
∂ηi
XT
i(β−β0)) = 0 (16)
This solves for βin the next iterate hence the updating formula is given as;
β(1) = (XTWX−1)XTWκ(17)
Where Xis a matrix of covariates, W={∂ ηi
∂µ i}2Viand κis the adjusted dependent
variable such that κ1
(1) =h(µ(0)
i+∂ηi
∂µ i(yi−µ(0)
i)). This procedure will produce a sequence
of estimates β(1),β(21) · · · ,β(t)and the iterations are stopped when || ˆ
β(t+1) −β(t)|| =ε,
where εis the set threshold value.
Electronic Journal of Applied Statistical Analysis 159
Lemma 2.2. For a given βfrom equation (12), let λi=Pm
t=1 yit be the total number of
times a firm creates value in the ’m’ financial years, then λi∼Binomial(m, Pi(ˆ
β)) and
the probability that a firm i creates value for the shareholders at least once given Xjis;
P r(Yit = 1|λi≥1) = 1 − {1−Pi(ˆ
β)}m(18)
Such that,
µit =E(Yit = 1|λi≥1) = Pit(ˆ
β)
1−Qm
t=1(1 −Pit (ˆ
β)) (19a)
V ar(Yit|λi≥1) = Pit(ˆ
β){1−Pit(ˆ
β)−Qm
t=1(1 −Pit (ˆ
β))}
[1 −Qm
t=1(1 −Pit (ˆ
β))]2(19b)
Theorem 2.3. If log(µit
1−µit ) = β0such that µit =g−1(β0) = P0, the estimating equation
for β0is:
n
X
i=1
{λi−mP0
1−(1 −P0)m}(20)
Proof. Assuming the independence correlation matrix, then, Di=∂ µi
∂βT=AiXi. From
equation (12) we have;
U(β) =
n
X
i=1
(AiXi)TV−1
i(yi−µi)=0
=
n
X
i=1
(Xi)TAiA−1
i(yi−µi) =
n
X
i=1
(Xi)T(yi−µi)=0
=
n
X
i=1
{1,· · · ,1} ×
E(Yi1)
E(Yi2)
.
.
.
E(Yim)
−
µi1
µi2
.
.
.
µim
=
n
X
i=1
λi−
m
X
t=1
µit = 0 (21)
However, based on the results in Equation (19a) and since Pit =g(β0) = P0, then
µit =P0
1−(1−P0)m. Hence the estimating equations for β0can be expressed as:
n
X
i=1
{λi−mP0
1−(1 −P0)m}
which ends the proof.
160 Nyabwanga
2.3 Shareholder Value creation GEE Model Selection
In GEE, model selection centers on selecting a working correlation structure R(ρ) and a
suitable set of covariates. The study proposes the use of empirical likelihood based AIC
(EAIC) proposed by Chen and Lazar (2012) in choosing the correct correlation matrix
and QIC in selecting the covariates.
2.3.1 Two-Step EAIC-QIC Hybrid Methodology
The proposed hybrid methodology is implemented using Algorithms 1 and 2 presented
below in which its proposed that the best correlation matrix be first selected and then
followed by the choice of the best subsets of covariates with some regularization that
will ensure that only the informative set of regressors is selected.
STEP I:Selection of Intra-Class Correlation Matrix
Algorithm 1 Application of EAIC for Intra-Class Correlation Matrix Selection
if ℜfrepresent the Empirical Likelihood Ratio(ELR) defined based on the full model then;
Formulate the full model ELR in terms of ρand βwhere ρ={ρ1,· · · , ρm−1}whose maxi-
mization is taken with respect to the probabilities p1,· · · , pn.
ℜf(β) = Sup{
n
Y
i=1
npi|pi≥0,
n
X
i−1
pi= 1,
n
X
i−1
pig(XT
iβ)=0}
g(XT
iβ) = ∂µi
∂βTV−1
i(µ)(yi−µi)
V−1
i(µ) = A−0.5
iR−1(ρ)A−0.5
i(22)
if there are ’s’ correlation structures considered, say R1(ρ),· · · , RS(ρ)then
[i] define the ELR of the GEE estimator based on each Rs, s = 1 · · · Si.e. ℜf
R1,· · · ,ℜf
RS=
ℜf{βR1,ˆρR1},· · · ,ℜf{βRS,ˆρRS}.
[ii] For each Rs, obtain the MELE(ˆ
βs). This is the same as the GEE estimator (βs
G) in
Equation (22).
for each of the s models do;
[i] Compute the EAIC values of based on the formula by Chen and Lazar (2012) i.e.
EAI Cs=−2logℜf(ˆ
θs
GEE )+2dim(θs)
where the s candidate models based on the s intra-class correlations are parameterized by
θs(s= 1,· · · , S ) and (ˆ
θs
GEE ) = ˆ
βs
GEE
ˆρs
GEE is the GEE estimator based on the correlation matrix
Rs(ρ). ˆρs
GEE is the method of moment estimator of ρgiven ˆ
βs
GEE and Rs
[ii] Select the best correlation matrix for the data. i.e. Rs
best(ρ) = argmin(EAICs)
end for
end if
end if
Electronic Journal of Applied Statistical Analysis 161
Remark 2.4. Use of the full model to select the best correlation matrix (Rs
best(ρ)) is
plausible as it contains most information to predict the outcome variable.
STEP II: Selection of Covariates
Algorithm 2 Application of QIC for Covariates Selection
for ’k’ regressor variables and Rs
best(ρ) from Algorithm 1 do
[i] Generate using the ”Dredge” procedure in Multi-Model Inference (MuMIn) R package,
all the 2k, p =k+ 1 GEE models (this excludes the constant only model).
[ii] ∀,Mi, i ∈Z, i = 1,2,· · · 2kmodels, compute the QIC values based on the formula;
QICRs
best (ρ) = −2Q{ˆ
βRs
best (ρ)|(Yi, Xij )}+ 2tr(ˆ
ΩIˆ
VRs
best (ρ)) (23)
Where, ˆ
ΩI=1
nPn
i=1(∂ µi
∂βT)TV−1
i
∂µi
∂βTis the model based variance-covariance matrix and
ˆ
VRsbest (ρ)=ˆ
Ω−1
I{(∂µi
∂βT)TV−1
i(yi−µi)(yi−µi)TV−1
i
∂µi
∂βT}ˆ
Ω−1
Iis the robust variance estimator
under the best working correlation matrix (Rs
best)
for Each of Mi, i = 1,· · · ,2kdo
[i] Rank the models based on the criteria suggested by Burnham and Anderson, (2002)
[ii] extract models whose ∆i≤2 where ∆i=QICMi−QICmin. In this case the best
model is {Mi: ∆i= 0}.
for The the best model, {Mi: ∆i= 0}do
Re-fit using Penalized GEE to obtain pnalized estimators (βj) and the corresponding
standard errors. PGEE employs the SCAD penalization to further remove uninformative
regressors.
end for
end for
end for
Remark 2.5. Fitting the final using PGEE is informed by findings by Nyabwanga et
al. (2019b) that QIC has high over-fitting probabilities hence part (4) in Algorithm 1
is meant to ensure that non-informative regressors are removed from he final selected
model so as to reduce model variability while keeping the model bias at minimum.
Definition 2.6. Let U(ˆ
β) denote the original GEE estimates and S(ˆ
β) the PGEE
estimates. Then;
S(ˆ
β) =
n
X
i=1
∂µi
∂βTV−1
i(yi−µi)−q′
λ(|β|)◦sign(β) (24)
where qλ(|β|) = {qλ(|β1|),· · · , qλ(|βp|)}Tis a p-dimensional vector of penalty functions
and sign(β) = {sign(β1),· · · , sig n(βp)}Twith sign(t) = I(t > 0) −I(t < 0).q′
λ(|β|)◦
sign(β) denotes the Hadamard product of the two vectors.
The smoothly clipped absolute deviation (SCAD) penalty by Fan and Li, (2001),
is employed owing to its ability to simultaneously achieve unbiasedness, sparsity and
162 Nyabwanga
continuity. The SCAD penalty function is zero for lager coefficients and is relatively
large if βjis close to or equal to zero hence will guard against under-fitting. For more
details on SCAD penalization, see Fan and Li, (2001). To obtain the SCAD regularized
estimates, the study employed the Minorization-maximization algorithm together with
the Newton-Raphson algorithm. See Wang et al. (2012).
2.4 Relative Efficiency of the Proposed Hybrid method
To establish the relative efficiency (RE) of the new procedure compared to the standard
QIC only procedure, the study employed V-fold Cross-Validation to obtain the MSE of
each procedure in which
MSEv=1
vX
i∈v
(yi−ˆyi)2
and the V-fold cross-validation estimate is the average of the V estimates of the test
errors MSEi,· · · , M SEvwhere
CVv=1
V
V
X
i
MSEi
and
RE =M SE(ˆ
βQIC
GEE )
MSE(ˆ
βEAI C−QIC
GEE )
If RE > 1, GEE estimates under the Hybrid method have higher efficiency than its
counterpart procedure. The GEE estimates’ efficiency for the new hybrid procedure will
be lesser compared to the exclusive use of QIC If RE < 1 and the methods will yield
the same results if RE=1.
3 Results and Discussion
3.1 Choosing the Correct Correlation Matrix for Firm Value Data
Based on the procedure outlined in Algorithm 1, the full GEE model is fit using five
correlation matrices: Independence (IN), Exchangeable (EX), Order one Autoregressive
(AR(1)), Toeplitz (TOEP) and Unstructured (UN). For each model, the EAIC and QIC
values are obtained. The results are shown in Figures 1. and 2.
Figure 1 show that the model formulated with the AR(1) correlation matrix had a
minimum EAIC value (967.7) hence was considerd the best for the data. The Unstruc-
tured matrix was the second least preferred by EAIC with the least preferred being the
independence matrix. The second best preferred correlation matrix was Exchangeable.
Figure 2 show that, for the same data, QIC preferred the unstructured correla-
tion matrix with a minimum value of 799.7 thus indicating QIC’s preference for over-
parameterized matrices. The Unstructured matrix preferred by QIC was the second
Electronic Journal of Applied Statistical Analysis 163
Figure 1: EAIC Values for Various Correlation Matrices
Figure 2: QIC Values for Various Correlation Matrices
164 Nyabwanga
least preferred by EAIC. AR(1) structure was ranked second by QIC with a value of
844.3. Based on Algorithm 1, it is concluded Rs
best =AR(1). The results corroborate
those of Chen and Lazar (2012) which suggested that EAIC had the tendency to choose
a parsimonious structure more often and was more effective than the QIC in selecting
the true correlation structure. It is therefore concluded that the AR(1) structure was a
better fit for the data.
3.2 Selection of Regresssors Based on the AR(1) Matrix
Using Rs
best =AR(1) and algorithm 2, the top ranked models based on their minimum
QIC values and whose ∆i≤2 are in Table 1.
Table 1: Model Selection Table for QIC under AR(1) Matrix
MKInt X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 QIC ∆k
479 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 394.00 0.00
351 ✓ ✓ ✓ ✓ ✓ ✓ ✓ 394.71 0.71
349 ✓ ✓ ✓ ✓ ✓ ✓ 395.26. 1.26
477 ✓ ✓ ✓ ✓ ✓ ✓ ✓ 395.38 1.38
223 ✓ ✓ ✓ ✓ ✓ ✓ ✓ 395.44 1.44
95 ✓ ✓ ✓ ✓ ✓ ✓ 395.97 1.97
Table 1 shows that QIC selected the model with the regressors X1,X2,X3,X4,X5,X6
and X7as the best model for the shareholder value creation data. This implies that X8,
X9and X10 which represented Board size, accounting profitability and working capital
policy respectively were dropped as drivers of value creation for firms in the NSE. To
compliment the results in Table 1, variable relative importance (VRIMP) values were
determined and are shown in Table 2.
Table 2: Variable Relarive importance measures for QIC under AR(1) Matrix
Covariate X2 X1 X4 X7 X3 X6 X5 X9 X8 X10
VIMP 1.00 1.00 0.88 0.81 0.62 0.62 0.56 0.27 0.08 0.01
The results shows that X1and X2which represent growth rate of earnings and eco-
nomic profitability respectively were the key drivers of value creation for firms in the
NSE. Other important drivers in order of importance are X4,X7,X3,X6and X5all with
VRIMP values of greater than 0.5. Just like the results in Table 1, Board size, Account-
ing profitability and working capital policy had low VRIMP values hence considered not
key drivers to value creation. However, it is worth noting that working capital ratio is a
key component of the Altman’s Z value that measures the financial health of the firms
hence can be said to have an indirect effect on firm value. To ascertain the importance
of selecting the true within subject correlation matrix in GEE modelling, the procedure
Electronic Journal of Applied Statistical Analysis 165
in algorithm 2 was repeated using the unstructured correlation matrix that was chosen
by QIC. Models whose ∆i≤2 are given in Table 3:
Table 3: Model Selection Table for QIC under AR(1) Matrix
MKInt X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 QIC ∆k
479 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 401.00 0.00
351 ✓ ✓ ✓ ✓ ✓ ✓ ✓ 402.49 0.49
The results indicate that the QIC-only benchmark method leads to the model with
explanatory variables X1,X2,X4,X5,X6,X7and X8being ranked as the top model
for the data with a QIC value of 401 which was much greater than that of the models
whose ∆i≤2 when the AR(1) structure was used. Notably, the top model included X8
which was not in any of the top six models when the EAIC-QIC strategy was used, and
that X3was excluded from the top model. Accounting profitability and working capital
policy were both not included as drivers of value creation for firms in the NSE for the
top two ranked models under this strategy. Likewise, under the QIC-only strategy, only
two models had ∆i≤2.
The relative efficiency of estimators from the top-ranked models under the two ap-
proaches was compared using 5-fold cross-validation. The M SEEAI C−QIC ,M SEQICOnly
and relative efficiency values for 10 iterations are given in Table 4
Table 4: Model Selection Table for QIC under AR(1) Matrix
Iteration 1 2 3 4 5 6 7 8 9 10
MSEQI COnly 9.59 9.22 10.02 10.30 10.03 7.13 9.82 9.73 10.33 9.83
MSEE AIC −QI C 7.57 7.94 1.02 1.02 1.01 5.22 8.03 7.79 1.01 8.60
RE 1.27 1.16 9.88 10.14 9.93 1.28 1.22 1.25 10.18 1.14
Table 4 shows that all the RE values were greater than one. This implies that the
model selected under the proposed EAIC-QIC Hybrid methodology yielded more efficient
estimates compared to the model selected under the QIC-only strategy. It is therefore
inferred that the proposed Hybrid methodology greatly enhanced the GEE estimates’
efficiency. The high mean squared error for the model under the QIC-only benchmark
approach could be attributed to the use of the unstructured matrix which had more
nuisance parameters to estimate thus costing efficiency (Chen and Lazar , 2012). The
greater efficiency of the hybrid methodology can be attributed to the intracluster corre-
lation structure selected by EAIC which if accurately modeled, enhances the efficiency of
analysis using generalized estimation equations (Hin et al. , 2007). Based on the results
in the selection table for the best model using the Hybrid methodology and Equation
(16), the probability that a firm ’i’ creates value for its shareholders in the tth period
166 Nyabwanga
will be:
µit =Exp(β0+β1X1it +β2X2it +β4X4it +β5X5it +β6X6it +β7X7it )
1 + Exp(β0+β1X1it +β2X2it +β4X4it +β5X5it +β6X6it +β7X7it )(25)
3.3 Regularization of the best model Selected using the Hybrid
Methodology
The best model selected using the hybrid methodology is subjected to SCAD regular-
ization to weed out non-informative covariates. The regularization parameter λwas
obtained using 5-fold cross-validation with 30 iterations. Findings are illustrated in
Figure 3.
Figure 3: Optimum value of λfor SCAD Regularization
The results in Figure 3 show that λ= 0.23 was the optimum tuning parameter and was
therefore applied in fitting the PGEE model. With the SCAD penalization, the regressors
X4,X5and X6which represent firm size, leverage and dividend policy are dropped
from the model and only the regressors X1,X2,X3and X7which respectively represent
economic profitability, growth rate of earnings, the interaction between the two and level
of financial health were retained as the main predictors of firm value. The deletion of
financial leverage from the final model supports MM capital structure irrelevance policy
Electronic Journal of Applied Statistical Analysis 167
of 1963 by Modigliani and Miller which suggested that financing decision does not matter
in value creation. The deletion of firm size from the SCAD regularized model confirms
findings by Mule et al. (2015) who established that firm size had no significant effect
on firm value. Also, the deletion of dividend policy meaning that it is not an important
predictor of value creation corroborates findings by Hansda, et al. (2020) and Nguyen
and Faff, (2007) but is contrary to findings by Agung et al. (2021a) and Bataha et al.
(2023) who both established a positive significant effect. The deletion of dividend policy
can however be justified since dividend pay-out ratio has a direct effect on growth in
earnings (g) which also depend on Return on Equity (ROE) since g=ROE ×γ, where
γis the retention ratio such that (1 −γ) is the dividend pay-out ratio. The selected
covariates, their coefficients, naive and standard errors are shown in Table 5.
Table 5: PGEE Regression Coefficients and Standard Errors
Criteria Int X1 X2 X3 X7
Penalized GEE Estimate −1.076∗∗∗ −0.3723∗∗∗ −0.6044∗∗∗ 0.0183∗0.0000143∗
Naive SE 0.1752 0.0923 0.1459 0.0373 0.0001
Robust SE 0.1731 0.0948 0.1509 0.0078 0.00004
UnPenalized GEE Estimate −2.781∗∗ 0.0781∗∗ −7.406∗∗ 0.0054 0.0184
Naive SE 1.5040 0.0273 2.475 0.0048 0.0979
Robust SE 1.0352 0.0311 2.7347 0.00247 0.0604
Signif. codes ′∗∗∗′,0.001 ′∗∗′,0.01 ′∗′,0.0
The PGEE estimators indicate that the four regressors retained were all statistically
significant at 5% level of significance with economic profitability and growth rate of
earnings exhibiting a negative relationship with Mv
Bvratio. This meant that a firm that
seeks to increase its economic profitability and growth in earnings will compromise value
creation for their shareholders. On the other hand, financial distress had a positive
effect on value creation. This means that firms that had smaller financial distress values
meaning higher Altman’s Z score values will have higher likelihoods of creating value
for their shareholders compared to distressed firms. Based on the PGEE estimates, the
model for the chances of a firm i creating value at time t takes the form;
µit =Exp(β0+β1X1it +β2X2it +β3X3it +β7X7it )
1 + Exp(β0+β1X1it +β2X2it +β3X3it +β7X7it
=Exp(−1.076 −0.3723X1it −0.6044X2it + 0.0183X3it + 0.0000143X7it )
1 + Exp(−1.076 −0.3723X1it −0.6044X2it + 0.0183X3it + 0.0000143X7it
(26)
Based on the results in Theorem 2.3, assuming that the regressors X1,X2,X3and X7
are not at play, the probability of a firm ’i’ in the NSE creating value for its shareholders
is;
P0=Exp(−1.076)
1 + Exp(−1.076) = 0.2543 (27)
168 Nyabwanga
which implies that the number of times the said firm is expected to create value in m
years will be;
λi=mP0
1−(1 −P0)m
=0.2543m
1−(1 −0.2543)m
Example 3.1. If m=3, λi=0.2543×3
1−(0.7457)3≊1, if m=6, λi=0.2543×6
1−(0.7457)6≊2, If m=10,
λi=0.2543×10
1−(0.7457)10 ≊3 e.t.c
The corresponding cumulative distribution function curve is shown in Figure 4
Figure 4: Binomial distribution for various ’m’ values and P0=0.2543
Electronic Journal of Applied Statistical Analysis 169
According to the plots in Figure 4, the graphs are positively skewed, and the skewness
decreases as n (years considered) increases.
4 Conclusions
In this article, a Two-Step hybrid methodology that combines EAIC and QIC to select
the true correlation matrix and set of regressors respectively was proposed. The proposed
procedure was used to develop a GEE model for the drivers of firm value. Compared
to the QIC-only benchmark method, the proposed method performed well in selecting a
GEE model whose estimates were more efficient than those under the QIC-only approach.
Also, the use of penalized GEE, ensured sparsity of the final model.
a In relation GEE modelling, it can be concluded that;
i For efficiency improvement, it is pertinent that the correct correlation matrix
be selected first based on the full model and then applied in the selection
of the correct set of regressors. This will enhance efficiency of the resultant
model.
ii Hybridization of the model selection procedure in GEE modelling improves
the GEE estimators’ efficiency. This is true since the combination of EAIC
and QIC yielded a versatile tool that ably overcame the established weak-
nesses of QIC in selecting the true correlation matrix, but capitalized on its
strength of being impressive in variable selection.
iii Sparsity in GEE model selection can only be achieved if techniques that in-
corporate penalization such as PGEE are used in model fitting. This ensures
that redundant variables that only add noise to the final model are discarded.
b In relation to drivers of shareholder value creation, it can be concluded that;
i The Gordon-Constant growth model continues to be vital in evaluating value
creation by listed firms
ii Dividend policy has no direct effect on shareholder value creation but rather
an indirect one through the growth rate of earnings (g), which is a function
of the dividend pay-out ratio.
iii Since working capital is a key component in the computation of the Altman’s
Z score used o determine the financial health status of the firms, it is concluded
that working capital policy will have an indirect effect on shareholder value
creation. This follows from the results that financial health has a significant
effect on value creation.
Acknowledgement
I would like to thank the anonymous editor and the reviewers for their constructive
comments that helped improve this paper.
170 Nyabwanga
References
Agung, G. and Hasnawati, S. and Huzaimah, R. F. (2021a). The Effect of Investment
Decision, Financing Decision, Dividend Policy on Firm Value. Jurnal Bisnis dan
Manajemen (JBM), 17(1): 1-12.
Bataha, I. E., Ilat, V. and Kalalo, M. Y. B. (2023) The Effect of Dividend Policy and
Market to Book Value of Equity on Firm Value in Energy Sector Companies Listed on
Indonesia Stock Exchange. Jurnal Ekonomi dan Bisnis Digital (MINISTAL), 2(1):29–
42
Burnham, K. and Anderson, D. (2002). Model Selection and MultimodelInference: a
practical information-theoretic approach 2nd edition. Springer-Verlag, New York.
Carey, V. J. and Wang, Y. G. (2011). Working Covariance Model Selection for GEE.
Journal of Statistics and Medicine, 70(26): 3117-3124.
Chen, J. and Lazar, N. A. (2012). Selection of Working correlation structure in Gener-
alized Estimating Equations via Empirical Likelihood. Journal of computational and
graphical Statistics, 21(1):18–41.
Chen, L. J. and Chen S. Y. (2011). The influence of profitability on firm value with
capital structure as the mediator and firm size and industry as moderators. Investment
Management and Financial Innovations, (8, Iss. 1): 121–129.
Cui, J. and Qian, G. (2007). Selection of Working Correlation Structure and Best Model
in GEE Analyses of Longitudinal Data. Communications in Statistics—Simulation
and Computation, 36(5): 987–996.
Fan, J. and Li, R. (2001). Variable Selection via Nonconcave Penalized Likelihood and its
Oracle Properties. Journal of American Statistical Association, 96(456:) 1348–1360.
Gordon, M. J. (1959). Dividends, Earnings, and Stock Prices The Review of Economics
and Statistics, Vol. 41(2): 99-105.
Hansda, S., Sinha, A., and Bandopadhyay, K. (2020). Impact of Dividend Policy on
Firm Value with Special Reference to Financial Crisis SIT Journal of Management,
10(2): 158-175
Hin, L.-Y. and Wang, Y.-G. (2009). Working Correlation Structure identification in
generalized estimating equations. Statistics in Medicine, 28(4): 642-658.
Hin, L.-Y. Carey, V. J. and Wang, Y.-G. (2007). Criteria for Working Correlation
Structure Selection in GEE. Assessment Via Simulation. The American Statistician,
61(4): 360-364.
Honjo, Y. and Harada, N. (2006). SME Policy, financial structure and firm growth.
Evidence from Japan. Small Business Economies, 27(4): 298–300.
Liang, K.-Y and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear
models, Biometrika,73(1):13-22.
Mule, K. R., Mukras M. S., and Nzioka, O. M. (2015). Corporate Size, profitability
and market value: An econometric panel analysis of listed firms in Kenya. European
Scientific Journal, 11(13):376–396.
Electronic Journal of Applied Statistical Analysis 171
Nguyen, H. and Faff, R. (2007). Impact of Board size and oard diversity on firm value:
AUstralian Evidence. Corporate Ownership and Control, 4(2):24–32.
Nyabwanga, R. N., Onyango, F., and Otumba, E. O. (2019a). Consistyency Inference
Property of QIC in Selecting the True Working Correlation Structure for Generalized
Estimating Equations. American Journal of Theoretical and Applied Statistics, 8:74-
84.
Nyabwanga, R. N., Onyango F. and Otumba E. O.(2019b). Inference properies of QIC
in the selection of Covariates fo Generalized Estimating Equations. Internartional
Mathematical Forum, 14(4):149–157
Olaniran, O. R. and Abdullah, M. A. A. (2019). Bayesian Variable Selection for Multi-
class classification using Bootstrap Prior Technique. Australian Journal of Statistics,
48(2):63–72.
Pan, W. (2001). Akaike Information Criteria in generalized estimating equations. Bio-
metrics, 57(1): 120–125.
Pandey, I. M. (2005). What drives Shareholder Value. Indian Institute of Management,
Ahmedabab, India Working Paper 2005-09-04.
Wang, L., Zhou, J., and Qu, A. (2012). Penalized Generalized Estimating Equations for
high dimensional longitudinal data analysis. Biometrics, 68(2):353–360.
