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 seasonal autoregressive integrated moving average models built on weighted observations to eliminate the e ect of the structural changes which are characteristic of developing economies. Forecasting the values of development of the market for corporate control may serve as \input" information to form a prompt regulation system for the mergers and acquisitions of holding companies, which meets current needs.
Few problems of economic theory and practice give rise to more heated discussions than problems of integration. However, global practice evidences that it is major corporate entities which generate an aggregate demand and supply, and de ne the most important prerequisites to beat the competition. To achieve competitive advantage, a rm 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 abbreviation of `merger and acquisition deals') and it should be noted that they are not uniform. Analysis reveals the following main types of integration deals, classi ed 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) Integration deal
types Horizontal
Vertical Conglomerate
Description Merger (acquisition) between companies in the same sector manufacturing the same products or performing the same production stages Merger (acquisition) between companies in di erent economic sectors connected with the manufacturing process of certain nished products Merger (acquisition) between companies in di erent 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 integration 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
Melicher in his works suggests forecasting integration activity by means of time
series models and the log transformation of series [
Tollison and Shughart [
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 de ne the speci c time intervals of M&A waves more accurately and set a uni ed structural process which may show waves of integration activity in industrial and cross-border companies.
Barkoulas [
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 [
The analysis of papers suggests that this focus on identifying the time behavior of the integration activity among Russian holdings and simulating integration processes in Russia is a new eld of study. 3
Forecasting the trend in the number of M&A deals
It is a general practice to use nite samples in econometrics, and it is
consequently important to analyze the properties of various methods of estimation
and forecasting when samples are limited. The widely used approach to a
comparison of methods and models is a cross-validation method [
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 (September 2008) the trend in the indicator under study had changed its structure. Such a period of time is characterized by the beginning of a nancial and economic crisis in Russia. Buyers' and sellers' reasons for M&A during periods of economic growth di er from possible reasons for integration during a period of economic crisis, which cannot but in uence the integration activity of holding companies. Foreign research has endorsed this rationale3. In the context of a nancial and economic crisis, the reasons for integration are in uenced by a signi cant imbalance between demand and supply in the M&A market: a sharp decrease in buyers and a great number of assets o ered for sale. 3 For example, Yagil J. Mergers and macro-economic factors // Review of nancial economics. 1996. Vol. 5, No. 2. pp. 181{190
In the case of a stable and growing economy, the main reasons for consolidation (operations and strategy) are related to potential synergetic e ects, 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 nancial 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 diversi cation, tax bene ts 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 [
Testing the hypothesis on the trend anticipates the determination and highlighting 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, various approaches can be used; they di er from each other in terms of the power and complexity of their mathematical tools; for example, the square successive di erence 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 FosterStewart 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: t-statistics4 x^t = 12:21 + 0:4t:
(7:20) (9:58)
The coe cient of determination R2 = 57:18% and the regression equation are signi cant, 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: x^t = (12:21 + 0:4t; t
t 21:16; t > t ; where t = 69:
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 coe cients of the Fourier series.
A main objective of the harmonic analysis is the representation of f (x) as the series: 4 Regression coe cients are statistically signi cant at a 5% level, as tcr (0:05; 67) = 1:68 or where cn = pa2n + b2n { harmonic amplitude, 'n { 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 uctuations (frequency) and their rate (amplitude). The seasonal component is a sum of a mean value and a series of sinusoids and cosinusoids: st = s + n X ai cos(!it) + i=1 n X bi sin(!it); i=1 where a and b represent parameters of harmonic representation; !irepresents angular frequency measured in radians per unit of time and equal to ! = 2 f (0 ! 2 ); 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 nding a trend component in the original series. To select the best harmonic representation, coe cients of determination for equations with a di erent 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 (2) (3) 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 o2bserv = 2:46 < c2r(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 correlated, it will be obvious to expect that in the equation the coe cient will be signi cantly di erent from zero: where et represents residuals of the model.
The available data show that: et = 0 + 1et 1; t = 2; n; et = 0:91 0:017et 1; (4) (5) i. e. the coe cient = 0:017 is not signi cantly di erent from 0; consequently, there is no rst-order autocorrelation in the residuals.
Econometrics has developed formal tests to determine the existence of
heteroscedasticity in residuals: the Goldfeld-Quandt test, White test, Glaser test
and others [
To test for conditional heteroscedasticity, i. e. ARCH e ects in errors of the model, an asymptotic test was used; it is based on regression et2 by et 1 2 et2 =
0 + 1et2 1; t = 2; n: The available data show that: et2 = 11:67 0:0008et2 1; i. e. the coe cient = 0:0008 is not signi cantly di erent from 0, evidencing that there are no ARCH e ects 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 ine cient. In this case the Box{Jenkins method may be used
As shown above, the Russian economy is undergoing a di cult period of economic transformation accompanied by structural economic changes occurring, inter alia, in the integration activity of holdings. In the past 20 years the objective 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 di erent structural modes may cause a major shift in the forecast. The paper \Market Timing and Return Prediction under Model Instability" by Pesaren and Timmermann proposes a two-stage approach to forecasting, which is stable towards structural changes. The rst 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 coe cients 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 coe cients 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 exible 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 [
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 coe cients, 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 coe cients 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 [
Analytically, using Annex of the paper \The Forecasting Method with Weighted Account for Observations" by Kitov, we determined the optimal weighing coe cient 1 = 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 coe cient of 0.98; consequently, the data from October 2008 to December 2012 { with a weighing coe cient of 1.
Constructing ARIMA models for the time series under study includes the
following key stages: identi cation 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 [
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
number of completed integration deals proves its non-stationarity (as tobserv =
1:79 > tcr(0:05; 119) = 1:95 for the parameter = 0:603 from the equation
yt = 0:603yt 1, evidencing that the null hypothesis H0 on a unit root
cannot be rejected). Therefore, the series under study was adjusted by passing to
a rst-order di erence of events and generating the stationary series, where the
observation unit is yt = yt yt 1 [
The Box-Jenkins seasonal model was selected and tted to the available data using a three-stage iterative procedure, consisting of identi cation, assessment and a diagnostic test of the model. To make an original adjustment of model parameters, let us consider ACF and PACF of the di erentiated series (see Figure 5).
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
con rms 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 nally
determined by searching through model parameters to minimize the statistic
R2, the mean square error and the statistic 2, characterizing a near-normal
distribution of residuals [
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 nal selection: 1. growth of precision (the quality of t 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 1B : : : pBp 1BS : : :
Ps BS+Ps 1) d DS yt = The most important characteristics of the model are the indicators of its precision. Precision may be assessed by the value of an error (imprecision) in the forecast characterizing the di erences between the actual and the expected numbers of completed integration deals involving Russian holding entities. Table 3 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 validation sample Xval (January-March 2013). Table 4 characterizes the precision of 5 The literature often speci es 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. 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 Russian 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 con dence interval bounds of the forecast for the number of closed M&A deals for Q2 2013 are fairly wide. A value of the con dence interval at the set con dence probability = 95% and dispersion of the forecasting error e calculated by ARIMA(3;1;0)(1;0;0) signi cantly deFig. 8. Autocorrelation (ACF) and partial autocorrelation functions (PACF) of residuals of the ARIMA(3;1;0)(1;0;0) model of the series of the number of closed M&A deals pends on the scope of sample n. A lessening in the con dence 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 con dence 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 con rmed that the model was acceptable for forecasting the completion of integration deals by holdings in the Russian M&A market. 4
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 observations to eliminate the e ect of the structural shift. Using ARIMA seasonal models, and factoring into the available observations with some optimally selected 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 e ciency and competitiveness of the Russian economy.