=Paper=
{{Paper
|id=Vol-1726/paper-06
|storemode=property
|title=Econometric Modeling of the Dynamics of the Integration Activity of Russian Companies
|pdfUrl=https://ceur-ws.org/Vol-1726/paper-06.pdf
|volume=Vol-1726
|authors=Vladimir S. Mkhitarian,Mariia G. Karelina
}}
==Econometric Modeling of the Dynamics of the Integration Activity of Russian Companies==
https://ceur-ws.org/Vol-1726/paper-06.pdf
Econometric Modeling of the Dynamics of the
Integration Activity of Russian Companies ?
Vladimir S. Mkhitarian1 and Mariia G. Karelina2
1
National Research University Higher School of Economics, Russia
http://www.hse.ru
2
Nosov Magnitogorsk State Technical University, Russia
http://www.magtu.ru
Abstract. This paper presents an empirical analysis of the Russian
market of mergers and acquisitions in 2003-2012. This analysis allowed
for the conclusion that, to assess and forecast the integration activity
of Russian companies, the most precise and appropriate models are sea-
sonal autoregressive integrated moving average models built on weighted
observations to eliminate the effect of the structural changes which are
characteristic of developing economies. Forecasting the values of develop-
ment of the market for corporate control may serve as “input” informa-
tion to form a prompt regulation system for the mergers and acquisitions
of holding companies, which meets current needs.
Keywords: harmonic analysis, forecasting, autoregressive integrated
moving average (ARIMA) models, mergers and acquisitions (M&A),
structural change
1 Introduction
Few problems of economic theory and practice give rise to more heated discus-
sions than problems of integration. However, global practice evidences that it
is major corporate entities which generate an aggregate demand and supply,
and define the most important prerequisites to beat the competition. To achieve
competitive advantage, a firm needs advanced technologies and a high rate of
capital turnover. This is practical only when holding companies have the most
favorable conditions for the generation, use and renewal of resources.
The Latin word “integratio” means “joining, consolidation of separate parts
in one”. There are many criteria for classifying M&A deals (the American ab-
breviation of ‘merger and acquisition deals’) and it should be noted that they
are not uniform. Analysis reveals the following main types of integration deals,
classified according to their direction (see Table 1).
However, the literature in economics as well as on practitioners focuses on
forecasting the integration activity of Russian holdings. In recent years, thanks
?
The work is performed under the grant of the President of the Russian Federation
for the state support of young Russian scientists – PhDs (MK-5339.2016.6)
Table 1. Main types of M&A deal, according to the direction of integration
Integration deal Description
types
Horizontal Merger (acquisition) between companies in the same sector man-
ufacturing the same products or performing the same production
stages
Vertical Merger (acquisition) between companies in different economic sec-
tors connected with the manufacturing process of certain finished
products
Conglomerate Merger (acquisition) between companies in different sectors having
no common manufacturing processes (merger between a company
from one sector and a company from another which is not a supplier,
a customer, nor a competitor)
to accumulated statistical information about integration deals on the part of
holding companies, methodological approaches to forecasting the rate of inte-
gration in Russia are starting to develop. Therefore, it is interesting to study
the course of research into corporate integration processes in countries with a
developed institutional environment.
2 Research on trends in M&A
Melicher in his works suggests forecasting integration activity by means of time
series models and the log transformation of series [11]. Weston in his early works
empirically came to support the hypothesis that the M&A of corporate entities
had a wave-like formation [15].
Tollison and Shughart [14] reject the wave-like pattern of integration activ-
ities to simulate M&A processes by an autoregressive model of the 1st order.
However, the fact that M&A may be described by a random-walk process gives
no reason to reject the hypothesis of wave-like behavior by M&A.
In recent times, now that the hypothesis of a wave pattern of M&A is no
longer provoking lively discussion in the Western research environment, but has
rather became a point to support the simulation of a time pattern of M&A,
new approaches have been developed. However, recent developments focus on
explaining the integration activity using models with a switching parameter,
where waves of M&A are caused by “switching” a discrete parameter. Markov
regime switching models define the specific time intervals of M&A waves more
accurately and set a unified structural process which may show waves of inte-
gration activity in industrial and cross-border companies.
Barkoulas [3] simulates the integration activity of holding companies using
fractional integration tests. As he states, ARFIMA processes have hyperbolically
decaying autocorrelations. Therefore, such processes have a long forecasting hori-
zon. Fluctuations in M&A are influenced not only by the latest historical values,
but also by M&A values far back in the past. Such behavior by integration pro-
cesses indicates the influence of fundamental factors on trends in the integration
activity of holding companies.
As regards the Russian M&A market, some rare attempts have been made
to forecast integration activity, notably, studies by Musatova and Ignatishin.
Musatova focuses on quarterly trends in the number of completed national M&A
deals for the period from early 1995 to late Q2 2010. Due to the incomplete data
available, the total value of M&A deals was not forecast [12].
The analysis of papers suggests that this focus on identifying the time behav-
ior of the integration activity among Russian holdings and simulating integration
processes in Russia is a new field of study.
3 Forecasting the trend in the number of M&A deals
It is a general practice to use finite samples in econometrics, and it is conse-
quently important to analyze the properties of various methods of estimation
and forecasting when samples are limited. The widely used approach to a com-
parison of methods and models is a cross-validation method [8]. In general the
cross-validation method may be described as the implementation of the two
following stages:
1. All available basic statistical data X are randomly divided into two subsets:
Xtr and Xval . Xtr is used to select (“adjust”) a model.
2. Each of the model options calculated at a previous stage is validated on the
validation sample (Xval ).
The available series of the trend in the number of completed M&A deals
is represented in absolute values for the period from January 2003 to March
2013. The series falls into interval series because it characterizes the value of
the indicator each month. Following a general principle in the cross-validation
method, a monthly trend in the number of completed M&A deals for the period
from January 2003 to December 2012 was used as a training sample Xtr (see
Figure 1).
The analysis revealed that, starting from the period of time t∗ = 69 (Septem-
ber 2008) the trend in the indicator under study had changed its structure. Such
a period of time is characterized by the beginning of a financial and economic
crisis in Russia. Buyers’ and sellers’ reasons for M&A during periods of economic
growth differ from possible reasons for integration during a period of economic
crisis, which cannot but influence the integration activity of holding companies.
Foreign research has endorsed this rationale3 . In the context of a financial and
economic crisis, the reasons for integration are influenced by a significant im-
balance between demand and supply in the M&A market: a sharp decrease in
buyers and a great number of assets offered for sale.
3
For example, Yagil J. Mergers and macro-economic factors // Review of financial
economics. 1996. Vol. 5, No. 2. pp. 181–190
Fig. 1. Trends in the number of completed M&A deals, 2003-2012
In the case of a stable and growing economy, the main reasons for consoli-
dation (operations and strategy) are related to potential synergetic effects, the
rates of growth of business, entrance to new markets, cost reductions due to
the scale of business, and others, but during an unstable period the dominating
reasons would become financial and investment ones. A decline in the risk of
bankruptcy and reduction of accounts payable, in particular for sellers of assets,
are brought into prominence, while for buyers, the important features are di-
versification, tax benefits and the opportunity to pay a lower market price for
assets.
Econometrics has developed several formal tests allowing us to determine
whether there is a structural change in the data and at what observation point
it has occurred: the Chow test, a sliding window method, the Zivot-Andrews
test and others [2,6]. In the present study the Chow test was performed to test
the hypothesis on the feasibility of breaking down the initial sample by subsets
[5]. As Fobserv = 43.48 > Fcr (0.05; 3; 116) = 2.68, the results show that the
structural change occurred in September 2008. Therefore, it is feasible to divide
the initial set into two subsets to improve the quality of the model with respect
to t∗ = 69.
Testing the hypothesis on the trend anticipates the determination and high-
lighting of the trend. The main approaches to this task are based on statistical
hypothesis testing. To test the hypothesis on the randomness of a series, vari-
ous approaches can be used; they differ from each other in terms of the power
and complexity of their mathematical tools; for example, the square successive
difference test (the Abbe test); a series test based on the median of a set; an
“ascending and descending” series test; a testing method for means; the Foster-
Stewart method and others.
According to the Foster-Stewart method, the hypothesis H0 on a constant
mathematical expectation for a period from January 2003 to September 2008
(t = 1, 69) may be rejected. Consequently, the series of completed M&A deals
has a trend component:
x̂t = 12.21 + 0.4t.
t-statistics4 (7.20) (9.58)
The coefficient of determination R2 = 57.18% and the regression equation
are significant, since Fobserv = 91.78 > Fcr = 3.13.
Following the Foster-Stewart method for the period from October 2008 to
December 2012 (t = 70, 120), the hypothesis H0 on a constant mathematical
expectation with an evaluation of 21.16 was adopted. Consequently, in the series
of completed M&A deals the trend component is:
(
12.21 + 0.4t, t ≤ t∗
x̂t = , where t∗ = 69.
21.16, t > t∗
At a preliminary stage of time series modeling a seasonal component is to be
studied. The spectral analysis and spectral density distribution by periods (the
density being estimated according to the Parzen window method) revealed that
a maximum value of the spectral density was within a 6-month period (Figure
2).
We next review statistical models to forecast the number of M&A deals
completed by Russian holding companies.
The model of the number of completed M&A deals using a harmonic
analysis
The harmonic analysis is the expansion of the periodic function f (x) (f (x+2l) =
f (x)) into a Fourier series. If the function f (x) is set analytically, its harmonic
analysis will be fully performed by applying well-known Euler-Fourier formulas
to calculate the coefficients of the Fourier series.
A main objective of the harmonic analysis is the representation of f (x) as
the series:
∞
a0 X
f (x) = + (an cos(nx) + bn sin(nx)) (1)
2 n=1
4
Regression coefficients are statistically significant at a 5% level, as tcr (0.05; 67) =
1.68
Fig. 2. Spectral density for a time series of the number of completed M&A deals
or
a0 X
f (x) = + cn sin(nx + ϕn ), (2)
2
p
where cn = a2n + b2n – harmonic amplitude, ϕn – harmonic phase.
As compared to other research methods for the seasonality, harmonic analysis
has some considerable advantages. It allows us to determine both a period of
fluctuations (frequency) and their rate (amplitude). The seasonal component is
a sum of a mean value and a series of sinusoids and cosinusoids:
n
X n
X
st = s̄ + ai cos(ωi t) + bi sin(ωi t), (3)
i=1 i=1
where a and b represent parameters of harmonic representation; ωi represents
angular frequency measured in radians per unit of time and equal to ω = 2πf
(0 ≤ ω ≤ 2π); n = N/2 (N – the length of the time series).
Given that the series of the number of completed M&A deals revealed the
trend, the Fourier series was used to describe a series generated after finding a
trend component in the original series. To select the best harmonic represen-
tation, coefficients of determination for equations with a different number of
harmonics were calculated (see Table 2).
The model with 12 harmonics describes quite a good seasonal component of
the trend in the number of M&A deals completed by holding companies in the
Table 2. Harmonic functions for the model of the number of completed M&A deals
involving Russian holdings
Harmonic Number of Cumulative coefficient of
Harmonic functions
number harmonics determination (R2 ), %
1 0.37 cos(t − 2.99) 1 4.56
2 0.87 cos(2t − 1.30) 2 19.73
3 1.54 cos(3t − 2.05) 3 24.86
4 0.62 cos(4t − 2.33) 4 41.25
5 1.50 cos(5t − 1.85) 5 43.89
6 1.32 cos(6t − 0.93) 6 54.29
7 1.33 cos(7t − 2.07) 7 62.56
8 2.04 cos(8t − 0.57) 8 68.45
9 1.3 cos(9t − 1.23) 9 71.87
10 1.43 cos(10t − 1.66) 10 74.26
11 1.28 cos(11t − 2.62) 11 79.23
12 0.68 cos(12t − 1.68) 12 85.92
Russian Federation, explaining an 85.92% variation of levels. Then:
12.21 + 0.4t + 0.37 cos(t − 2.99) + 0.87 cos(2t − 1.30) +
+ 1.54 cos(3t − 2.05) + 0.62 cos(4t − 2.33) +
+ 1.50 cos(5t − 1.85) + 1.32 cos(6t − 0.93) +
1.33 cos(7t − 2.07) + 2.04 cos(8t − 0.57) +
+
+ 1.3 cos(9t − 1.23) + 1.43 cos(10t − 1.66) +
+ 1.28 cos(11t − 2.62) + 0.68 cos(12t − 1.68), t ≤ t∗
ŷt =
21.16 + 0.37 cos(t − 2.99) + 0.87 cos(2t − 1.30) +
+ 1.54 cos(3t − 2.05) + 0.62 cos(4t − 2.33) +
+ 1.50 cos(5t − 1.85) + 1.32 cos(6t − 0.93) +
+ 1.33 cos(7t − 2.07) + 2.04 cos(8t − 0.57) +
+ 1.3 cos(9t − 1.23) + 1.43 cos(10t − 1.66) +
+ 1.28 cos(11t − 2.62) + 0.68 cos(12t − 1.68), t > t∗ ,
where t∗ = 69.
A diagram of the original series of the trend in the number of M&A deals
and the series generated by the model is given (see Figure 3).
To test the normality of distribution for this model, the Pearson test was
applied. Since χ2observ = 2.46 < χ2cr (0.05; 3) = 7.8, there are no grounds to reject
the hypothesis on a normal distribution of residuals of the model. To test for
autocorrelation in residuals, the asymptotic Breusch-Godfrey serial correlation
test was used; it is based on an idea that, if neighboring observations are corre-
lated, it will be obvious to expect that in the equation the coefficient ρ will be
Fig. 3. The original series of the number of completed M&A deals and data from the
harmonic analysis, units
significantly different from zero:
et = ρ0 + ρ1 et−1 , t = 2, n, (4)
where et represents residuals of the model.
The available data show that:
et = −0.91 − 0.017et−1 , (5)
i. e. the coefficient ρ = −0.017 is not significantly different from 0; consequently,
there is no first-order autocorrelation in the residuals.
Econometrics has developed formal tests to determine the existence of het-
eroscedasticity in residuals: the Goldfeld-Quandt test, White test, Glaser test
and others [9]. In this paper we performed the Goldfeld-Quandt test; according
to its results Fobserv = 1.54 < Fcr (0.05; 43; 43) = 1.66, there are no grounds to
reject the hypothesis on the homoscedasticity of the residuals of the model.
To test for conditional heteroscedasticity, i. e. ARCH effects in errors of the
model, an asymptotic test was used; it is based on regression e2t by e2t−1
e2t = λ0 + λ1 e2t−1 , t = 2, n.
The available data show that:
e2t = 11.67 − 0.0008e2t−1 ,
i. e. the coefficient λ = −0.0008 is not significantly different from 0, evidencing
that there are no ARCH effects in errors of the model.
Thus, the test results show that the model generated by performing the
harmonic analysis is relevant to the integration activity of the Russian holding
companies under study.
A seasonal model of the autoregressive integrated moving average
(the Box-Jenkins model) of the number of completed M&A deals
When the series under analysis includes non-random polynomial or harmonic
components, their formal approximation may require too many parameters, i. e.
the resulting parameterization of the model may be inefficient. In this case the
Box–Jenkins method may be used
As shown above, the Russian economy is undergoing a difficult period of eco-
nomic transformation accompanied by structural economic changes occurring,
inter alia, in the integration activity of holdings. In the past 20 years the objec-
tive of determining the structural changes is actively pursued by, for example,
D. Andrews, J. Bai, P. Perron you should give the date of each of these papers
and others. The objective of adapting forecasts to the structural changes made
in the data was not covered in detail.
However, Pesaren and Timmermann show in their work that this subject is
important, since a failure to take into account structural changes entails wrong
forecasts: data from different structural modes may cause a major shift in the
forecast. The paper “Market Timing and Return Prediction under Model Insta-
bility” by Pesaren and Timmermann proposes a two-stage approach to forecast-
ing, which is stable towards structural changes. The first stage determines when
the most recent break has occurred, and the second stage uses the most relevant
data from the most recent structural mode to make a forecast. If the number of
observations in the mode is not large, the coefficients of the ARIMA model will
determine inaccurately, with high dispersion, causing wrong results.
To eliminate this disadvantage, the paper “Selection of estimation window
in the presence of breaks” by Pesaren and Timmermann proposes a method of
making an optimal choice of the length of the observation window applied to
make a forecast. The method uses a number of observations to estimate the
coefficients of the model with low dispersion, but not so much as to give rise
to any high shift of the forecast due to an increased factor of observations from
other structural modes.
In the Lomonosov Moscow State University, Kitov proposed a new method
of stable forecasting subject to structural changes in the regression model, which
gives more accurate forecasts than the method of choosing the window length.
This makes the forecasts more accurate due to its more flexible approach to data,
applying continuous optimization and all the information available in the sample.
This method allows for the generalization of a random number of structural
changes in data [1].
An idea of the proposed approach to structural changes in forecasting the
integration activity is that an optimum ratio between dispersion and a shift in
forecasts is adjusted not by discrete optimization, depending not on the choice
of an optimal number of observations, but on continuous optimization, by the
selection of optimal weighing coefficients, which are considered in observations
from every structural mode. It is assumed that:
– at the time point t+1 there is no structural change; otherwise, the probability
of changes and distribution of model coefficients after changes would have
to be taken into account,
– in asymptotics a share of the observations attributed to each of the structural
modes in a total volume of the sample is constant [10].
Analytically, using Annex of the paper “The Forecasting Method with
Weighted Account for Observations” by Kitov, we determined the optimal weigh-
ing coefficient α1 = 0.98 for one structural change for a series of M&A deals.
Thus, when building the model, the data for the period from January 2003 to
September 2008 were taken into account with a weighing coefficient of 0.98;
consequently, the data from October 2008 to December 2012 – with a weighing
coefficient of 1.
Constructing ARIMA models for the time series under study includes the fol-
lowing key stages: identification of a test model; assessment of model parameters
and a diagnostic test of the adequacy of the model; use of the model to make a
forecast [4]. Initially, following the Box-Jenkins methodology, analyzing the au-
tocorrelation function (ACF) and the partial autocorrelation function (PACF)
of the original time series is recommended (see Figure 4).
Fig. 4. Autocorrelation (ACF) and partial autocorrelation functions (PACF) of the
series of the number of closed M&A deals
Figure 4 shows that the series of the number of closed M&A deals is not
stationary. The Dickey-Fuller test applied for the series characterizing the num-
ber of completed integration deals proves its non-stationarity (as tobserv =
−1.79 > tcr (0.05; 119) = −1.95 for the parameter δ = −0.603 from the equation
∆yt = −0.603yt−1 , evidencing that the null hypothesis H0 on a unit root can-
not be rejected). Therefore, the series under study was adjusted by passing to
a first-order difference of events and generating the stationary series, where the
observation unit is ∆yt = yt − yt−1 [7].
The Box-Jenkins seasonal model was selected and fitted to the available data
using a three-stage iterative procedure, consisting of identification, assessment
and a diagnostic test of the model. To make an original adjustment of model
parameters, let us consider ACF and PACF of the differentiated series (see Fig-
ure 5).
Fig. 5. Autocorrelation (ACF) and partial autocorrelation functions (PACF) of the
differentiated series of the number of closed M&A deals
Figure 5 shows that ACF dies out, and PACF has peaks at lags 1, 2, and 3,
while ACF has also a peak at lag 6, and PACF has peaks at lags 6 and 12. This
confirms the inference made on the analysis of spectral density (see Figure 2),
of a seasonal component with a six-month period. An optimal model was finally
determined by searching through model parameters to minimize the statistic
R2 , the mean square error and the statistic χ2 , characterizing a near-normal
distribution of residuals [13]. Calculations were made in packages of application
software Statistica 10.0 and SPSS 20.0.
This resulted in the following models:
1. ARIMA(3;1;0)(1;0;0);
2. ARIMA(3;1;0)(2;1;0);
3. ARIMA(0;1;3).
Because several tested models turned to be appropriate for the basic data,
two requirements were followed in the final selection:
1. growth of precision (the quality of fit of the model),
2. reduction of the number of model parameters.
Regarding ARIMA(3;1;0)(1;0;0), we used the Akaike criterion AIC=6.25 and
the Bayes criterion SIK=6.49; regarding ARIMA(3;1;0)(2;1;0), AIC=7.05,
SIK=6.97; regarding ARIMA(0;1;3), AIC=8.93, SIK=7.89. Consequently, the
ARIMA(3;1;0)(1;0;0) model was selected.
In general terms, the seasonal process of the autoregressive integrated moving
average ARIMA(p;d;q)(Ps ;Ds ;Qs ) may be written with the lag operator B as
follows:
(1 − α1 B − . . . − αp B p − β1 B S − . . . − βPs B S+Ps −1 )∆d ∆DS yt =
= (1 − Q1 B − . . . − Qq B q − W1 B S − . . . − WQS B S+Qs −1 )εt
Then ARIMA(3;1;0)(1;0;0) is:
(1 − α1 B − α2 B 2 − α3 B 3 − β1 B 6 )(1 − B)yt = εt . (6)
To make an assessment of M&A and a forecast by ARIMA(3;1;0)(1;0;0), a
transition to the ARIMA model was made as follows:
(1 + 0.553B + 0.389B 2 + 0.314B 3 − 0.344B 6 )(1 − B)yt = εt
or
ŷt = 0.447yt−1 + 0.164yt−2 + 0.076yt−3 + 0.314yt−4 + 0.344yt−6 − 0.344yt−7 +εt .
(4.325) (3.123) (3.785) (3.967) (5.121) (4.442)
Since in this expression tobserv > tcr (0.95; 113) = 1.981, all the coefficients in
the model provided are significant.
A diagram of the original series and the series generated by
ARIMA(3;1;0)(1;0;0) is given in Figure 6.
To test the adequacy of the model, a histogram of residuals (see Figure 7)
was drawn; following the diagram the distribution of residuals complies with the
normal law. This conclusion is endorsed by the Pearson test (χ2observ = 2.28 <
χ2cr (0.05; 3) = 7.8). The available data show that et = 0.05 − 0.008et−1 , i. e.
ρ = −0.008 is not significantly different from 0; consequently, according to the
Breusch-Godfrey test there is no autocorrelation in the residuals.
According to the Goldfeld-Quandt test, Fobserv = 1.41 < Fcr (0.05; 43; 43) =
1.66; consequently, there are no grounds to reject the hypothesis of the ho-
moscedasticity of residuals of the model. The asymptotic test for conditional
heteroscedasticity showed no ARCH effects in model errors, since e2t = 2.03 −
0.006e2t−1 .
Thus, all the studied characteristics of the model evince its adequacy for the
M&A process under study.
Precision of models of completed M&A deals
The most important characteristics of the model are the indicators of its pre-
cision. Precision may be assessed by the value of an error (imprecision) in the
forecast characterizing the differences between the actual and the expected num-
bers of completed integration deals involving Russian holding entities. Table 3
Fig. 6. The original series of the number of completed M&A deals and data generated
by ARIMA(3;1;0)(1;0;0), units
Fig. 7. Histogram of the residuals generated by ARIMA(3;1;0)(1;0;0)
characterizes the precision of the models of completed M&A deals using data of
the training sample Xtr (January 2003 – December 2012).
According to a general procedure of the cross-validation method, all model
options calculated at a previous stage are validated using data from the valida-
tion sample Xval (January-March 2013). Table 4 characterizes the precision of
5
The literature often specifies that MAPE ≤ 10% evidences the high precision of the
model; if this parameter is within a range of 10%–20%, precision may be deemed
good, and when 20% < MAPE ≤ 50%, it is satisfactory.
Table 3. Precision of models of completed M&A deals using data from the training
sample (January 2003 – December 2012)
Precision characteristics
Error Sum Mean absolute Mean
Model
mean squared percentage error square
square error (MAPE)5 error
1) Model of the trend in the 11.7 1404 12.71% 3.42
number of completed M&A
deals, generated by performing
the harmonic analysis
2) ARIMA(3;1;0)(1;0;0) 6.39 761 9.28% 2.53
Table 4. Precision of models of completed M&A deals using data from the validation
sample (January 2013 – March 2013)
Precision characteristics
Error Sum Mean absolute Mean
Model
mean squared percentage error square
square error (MAPE) error
1) Model of the trend in the 15.7 47 18.93% 5.39
number of completed M&A
deals, generated by performing
the harmonic analysis
2) ARIMA(3;1;0)(1;0;0) 7.67 23 14.23% 4.05
models of the completed M&A deals using data from the validation sample Xval
(January 2003 – December 2012).
As models are adjusted to the data of the training sample, values of the
quality criterion adjusted to the data of the validation sample Xval are less
optimistic than those adjusted to the data of Xtr .
In view of the data generated, it may be concluded that a trend in the number
of completed M&A deals is better described with an autoregressive integrated
moving average model. Figure 8 gives the autocorrelation (ACF) and partial
autocorrelation functions (PACF) of the residuals of the ARIMA(3;1;0)(1;0;0)
model.
An important aspect of modeling any time series is forecasting.
ARIMA(3;1;0)(1;0;0) may be used to forecast the number of M&A deals of Rus-
sian holdings for several periods ahead. Forecasting of the integration activity of
the holding entities in future was performed for Q2 2013 (see Table 5).
It should be noted that the generated confidence interval bounds of the fore-
cast for the number of closed M&A deals for Q2 2013 are fairly wide. A value of
the confidence interval at the set confidence probability γ = 95% and dispersion
of the forecasting error σe calculated by ARIMA(3;1;0)(1;0;0) significantly de-
Fig. 8. Autocorrelation (ACF) and partial autocorrelation functions (PACF) of resid-
uals of the ARIMA(3;1;0)(1;0;0) model of the series of the number of closed M&A
deals
Table 5. Forecasting by ARIMA(3;1;0)(1;0;0) for Q2 2013, units
ARIMA Confidence intervals
Actual Forecasting
No. Period forecast
value, units error Left border Right border
value, units
1 April 2013 17 20 3 10.69 29.31
2 May 2013 21 22 1 11.42 32.58
3 June 2013 18 15 3 3.64 26.36
pends on the scope of sample n. A lessening in the confidence interval by m times
is ensured by an increase in the number of measurements n by m2 times. Thus,
whereas the statistical data on the Russian market of mergers and acquisitions
are accumulated, all other conditions being equal, the confidence interval will be
lessened.
In this case, in a period from April 2013 to June 2013 this model forecast
the number of closed M&A deals with a MAPE of 13.03%, which confirmed that
the model was acceptable for forecasting the completion of integration deals by
holdings in the Russian M&A market.
4 Conclusions
This paper proposes a solution to the optimal forecasting of the trend in M&A
deals of holding entities, subject to structural shifts in the national economy.
Having analyzed the above results, we concluded that to analyze and forecast
the integration activity of Russian holdings, the most precise and appropriate
models were seasonal autoregressive integrated moving average ones. To evaluate
the trend in M&A deals subject to structural shifts, we applied the forecasting
method with a weighted account for observations, since a uniform accounting
of all available observations resulted in a shift in the forecast, while accounting
for the most relevant observations after the structural shift might entail a high
variance in the forecast.
In particular, to forecast the trend in the number of closed M&A deals, we
proposed seasonal ARIMA(3;1;0)(1;0;0) based on the weighted account for ob-
servations to eliminate the effect of the structural shift. Using ARIMA seasonal
models, and factoring into the available observations with some optimally se-
lected weights, we made the forecast for Q2 2013 on the number of closed M&A
deals. Having compared the forecast to the actual data, we proved that models
were acceptable for forecasting the development of the Russian M&A market to
increase the efficiency and competitiveness of the Russian economy.
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