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UDC 330.43/519.862/331.5
THE MODELLING OF REGISTERED UNEMPLOYMENT RATE NONLINEAR DYNAMICS IN UKRAINE BY MEANS OF THRESHOLD AUTOREGRESSION
® 2015 LUKIANENKO I. H., OLISKEVYCH M. 0.
UDC 330.43/519.862/331.5
Lukianenko I. H., Oliskevych M. 0. The Modelling of Registered Unemployment Rate Nonlinear Dynamics in Ukraine by Means of Threshold Autoregression
In conditions of unstable economic development of the national economy the analysis of labour market and forecast of employment and unemployment rate dynamics is of primary importance for social responsibility provision and social risks reduction in society The purpose of the research is the empirical analysis of nonlinear dynamics of registered unemployment rate in Ukraine in conditions of increased risks and asymmetry of information and the development of corresponding complex of nonlinear threshold autoregressive models. Methodology approach is based on methods of economic theory and economic and mathematical tools. Econometric methods of analysis of nonlinear time series with non-traditional functions of distribution have been used. In the result of the empirical investigation the application of threshold autoregressive models has been justified and the row of nonlinear econometric specifications that help to explain asymmetric dynamics of unemployment has been evaluated. The resulting threshold function and value of threshold parameter, estimated on the basis of real information, determine the branching of TAR model into two different modes of behaviour that in different ways characterise dynamics of unemployment growth and fall and allow forecasting the phases of employment fall and growth in the short term. Application of the developed nonlinear econometric models of registered unemployment rate dynamics adds to characteristics of labour sector in Ukraine and allows forecasting more precisely the effects of government measures in the sphere. The defined nonlinear character of dynamics of unemployment rate certifies non-Walrasian features of labour market in Ukraine and reveals its uneven and asymmetric cyclic behaviour. Key words: unemployment, employment, autoregressive model, TAR model, threshold parameter, labour market Pic.: 2. Tabl.: 3. Formulae: 5. Bibl.: 16.
Lukianenko Iryna H. - Doctor of Science (Economics), Professor, Head of the Department, Department of Finance, National University of «Kyiv-Mohyla Academy» (vul. Skovorody, 2,04655, Ukraine) Email: luk@kse.org.ua
Oliskevych Marianna O. - Candidate of Sciences (Physics and Mathematics), Associate Professor, Associate Professor, Department of Mathematical Economics and Econometrics, Ivan Franko National University of Lviv (vul. Universytetska, 1, Lviv, 79000, Ukraine) Email: olisk@ukr.net
УДК 330.43/519.862/331.5 Лукьяненко И. Г., Олискевич М. А. Моделирование нелинейной динамики зарегистрированного уровня безработицы в Украине с помощью пороговых авторегрессий
В условиях нестабильности экономического развития национальной экономики анализ рынка труда и прогнозирования динамики уровня занятости и безработицы является важной составляющей для обеспечения социальной ответственности и уменьшения социальных рисков в обществе. Основной целью исследования является эмпирический анализ нелинейной динамики зарегистрированного уровня безработицы в Украине в условиях повышенных рисков и асимметричности информации, а также разработка соответствующего комплекса нелинейных пороговых авторегрессионных моделей. Методология исследования базируется на методах экономической теории и экономико-математическом инструментарии, в частности, использованы эко-нометрические методы анализа нелинейных временных рядов с нетипичными функциями распределения. В результате эмпирического исследования обосновано применение пороговых авторегрессионых моделей и оценен ряд нелинейных эконометрических спецификаций, которые позволяют объяснить асимметричную динамику безработицы. Полученная в результате эконометрического анализа пороговая функция и оцененное на основе реальной информации значение порогового параметра определяют разветвления ТАН-модели на два разных режима поведения, которые по-разному характеризуют динамичный ход роста и снижения безработицы и позволяют в краткосрочном периоде предусматривать фазы спада и повышения занятости. Использование разработанных нелинейных эконометрических моделей динамики зарегистрированного уровня безработицы дополняет исследования характерных свойств, присущих сектору труда в Украине, и позволяет более точно прогнозировать последствия государственных мер в этой сфере. Установленный нелинейный характер динамики уровня безработицы свидетельствует о невальра-
УДК 330.43/519.862/331.5 Лук'яненко I. Г., Олскевич М. О. Моделювання нелiнiйноi динамки зареестрованого рiвня безробття в Укран за допомогою порогових авторегресй
В умовах нестаб'шьност'! економ'много розвитку нацонально! еконо-мiки анал'в ринку прац та прогнозування динамiки р'вня зайнятостi та безробiття е важливою складовою для забезпечення сощально! в'дпов'дальност'! та зменшення со^альних ризит у сустльстви Основною метою дотдження е емтричний анал'в нел'шшноI динам'ь ки зареестрованого р'вня безробiття в УкраЫ в умовах тдвищених ризи^в та асиметричностi нформацп, а також розробка вiдповiд-ного комплексу нел'шшних порогових авторегресшних моделей. Ме-тодолог'т дотдження базуеться на методах економiчноiтеорПта економко-математичному 'шструментари, зокрема використано економетричш методи анал'зу нел'шшних часових ряд'в з нетипо-вими функ^ями розпод'шу. В результат'! емтричного дотдження обфунтовано застосування порогових авторегресшних моделей i оцнено низку нелшшних економетричних специфiкацiй, як дають змогу пояснити асиметричну динамку безробття. Одержана вна-сл'док економетричного анал'зу порогова функ^я й оц'нене на основ'! реально!iнформацi! значення порогового параметра визначають роз-галуження ТАН.-модел'! на два рiзнi режими поведши, як по^зному характеризують динамiчний переб'г зростання та зниження без-робття i дають змогу в короткостроковому пер'юд'! передбачати фази спаду та пiдвищення зайнятост'!. Використання розроблених нелшшних економетричних моделей динамiки зареестрованого р'вня безробiття доповнюе дотдження характерних властивостей, як притамант сектору прац в УкраЫ, та дозволяе б'шьш точно про-гнозувати натдки державнихзаход'юу цш сфер'!. Встановлений нел'>-ншний характер динамiки р'вня безробiття засв'дчуе невальраавсьш властивост'! ринку прац в УкраЫ та виявляе його нерiвномiрну й асиметричну циклiчну поведiнку.
совских свойствах рынка труда в Украине и свидетельствует о его неравномерном и асимметричном циклическом поведении. Ключевые слова: безработица, занятость, авторегрессионная модель, TAR-модель, пороговый параметр, рынок труда Рис.: 2. Табл.: 3. Формул: 5. Библ.: 16.
Лукьяненко Ирина Григорьевна - доктор экономических наук, профессор, заведующий кафедрой финансов, Национальный университет «Киево-Могилянская академия» (ул. Сковороды, 2, Киев, 04655, Украи-
Email: luk@kse.org.ua
Олискевич Марианна Александровна - кандидат физико-математических наук, доцент, доцент кафедры математической экономики и эконометрики, Львовский национальный университет имени И. Франко (ул. Утверситетська, 1, Львов, 79000, Украина) Email: olisk@ukr.net
Ключов'1 слова: безробття, зайнятсть, авторегресйнамодель, TAR-модель, пороговий параметр, ринок прац Рис.: 2. Табл.: 3. Формул:5. Б'бл.: 16.
Лук'яненко 1рина Григорiвна- доктор економiчних наук, професор, зав'дувач кафедри ф'шанав, Нацональний ушверситет «Киево-Моги-лянська академiя» (вул. Сковороди, 2, Кив, 04655, Украна) Email: luk@kse.org.ua
Олтевич Марiанна Олександрiвна - кандидат ф'вико-математичних наук, доцент, доцент кафедри математичноi економки та економе-трн, Льжський нацональний ушверситет iм. i. Франка (вул. Утверситетська, 1, Льв'в, 79000, Укра'ша) Email: olisk@ukr.net
The processes of market transformation of the Ukrainian economy aimed at its socio-economic development, competitiveness and improvement of human capital level need deep theoretical and empiric research of labour market characteristics, as well as modelling and forecasting of its basic indicators tendencies, in particular unemployment and employment rate. The analysis and forecast of its tendencies change acquire special topicality in the modern conditions of economic development instability of the national economy, as it allows to forecast in advance the phases of employment and unemployment growth and fall and develop the corresponding measures of socio-economic policy aimed at social responsibility provision and social risks reduction in society. Nowadays the search of new approaches is very active, in particular those that are based on modern economic and mathematical tools that would give an opportunity to represent adequately difficult nonlinear processes and asymmetry of information that characterize labour market condition not only in Ukraine but also in other countries of the world. One of the perspective approaches in this direction is the approach that is based on application of econometric methods and models of analysis of nonlinear time series with the non-traditional functions of distribution, in particular models of threshold autoregressions. Consequently, the development and implementation of these models nowadays are of primary topicality for the research of labour market in Ukraine, since it will allow not only to forecast adequately nonlinear processes and change of their dynamics on the labour market but also to evaluate the values of threshold parameters, that determine different modes of their behaviour, in particular dynamics of unemployment and employment growth and fall that gives an opportunity in a short-term period to forecast the phases of their growth and fall and to react to them in time.
Fundamental principles of labour-market functioning and econometric modelling of its basic indicators both for the countries with market and transformation economy are highlighted in the works of the prominent western scientists G. Bardsen, O. Blanchard, C. Bean, J. Gali, B. Hansen, D. Peel, M. Peeters, A. Speight, etc. [1—5]. The results of their researches state that statistical data of labour-market indexes for different countries of the world have different properties that predetermine the necessity of search of non-traditional approaches to their empiric analysis and modelling. On the whole, nowa-
days the analysis and forecast of tendencies change of indicators of unemployment and employment on the different phases of economic cycle and evaluation of basic factors of unemployment in a long-term period remain one of the important directions of macroeconomic researches of western scientists. Thus, one of the actively investigated problems is a problem to what extent unemployment represents imperfection of market and what are the causes and effects of it [1; 2; 4].
Numerous works of domestic scientists are devoted to theoretical and empiric labour market research, in particular S. Babych, D. Bohynja, V. Fedorenko, O. Grishnova, L. Gury-anova, T. Kiryan, T. Klebanova, O. Kolot, E. Libanova, I. Lukya-nenko, L. Lysohor, K. Petrenko, T. Umanez', O. Yermolenko, etc [6—12]. Considerable contribution to the research of this range of problems is made by the scientists of Research Institute of Labour and Employment of Population of Ministry of Labour and Social Policy of Ukraine and The National Academy of Sciences of Ukraine [12]. Important direction of modern researches deals with the recurrence of short-term behaviour of labour-market. Some scientific theoretical and empiric results certify that a real salary is not cyclical as a result of non-Walra-sian character of labour-market that determines its short-term behaviour [8, 11]. However, these researches haven't found a sufficient reflection in the works of the Ukrainian scientists. In addition, the problem of development and application of modern econometric methods and models for the analysis and forecast of nonlinear and asymmetric dynamics of employment and unemployment, in particular threshold autoregressions, inherent to the labour market of Ukraine are insufficiently investigated in the works of Ukrainian scientists [9]. Deepening of scientific researches in this direction will complement the analysis of characteristics, inherent to the labour market in Ukraine on the current stage of economic development and will allow more exactly and immediately to form government measures to reduce social tension in society.
In spite of sufficient number of researches of both western and Ukrainian scientists dedicated to the labour market and its basic indicators dynamics modelling, topical is the problem of improvement and development of different approaches to the construction of nonlinear econometric models of unemployment, the analysis of which would give an opportunity to control and forecast the possible tendencies changes on do-
mestic labour market and form the corresponding measures of socio-economic policy.
The aim of the research is an empiric analysis of nonlinear dynamics of unemployment and employment rate in Ukraine in the conditions of increased risks and asymmetry of information, and also the development of corresponding complex of nonlinear threshold autoregressive models, justification of choosing threshold function-indicator, that will determine the change of the cyclic behaviour mode of the investigated indicator, quantitative evaluation of threshold parameter and forecast of tendencies changes of basic labour market indicators for forming appropriate socio-economic policy and program of social responsibility provision in Ukrainian society.
Theoretical dynamic models that determine common behaviour of employment of economic agent are based on maximization of the expected present value of his income
XP Vt
t=0
where yt - an income of economic agent (person) in a period t,
P - a discount factor (0 < P < 1).
In case a person is unemployed and receives unemployment benefit c, his income is yt = c; in case a person has accepted a job offer with wage wt, that is a random variable with
some distribution F(w) = P {W < w}, then yt = wt.
Theoretical models of unemployment, in particular, the models of search and matching are based on the assumption, that an important characteristics of labour market is inconsistency between unemployed persons and vacancies, that they may apply for, and that is why the process of employment provision doesn't take place due to the market mechanisms, but is a result of difficult and lengthy process of matching search [13]. In such models a random parameter 0 is introduced. It characterizes the level of consistency of parties and corresponds to marginal labour product. The process of matching search is divided into a few stages. During the first stage, when an unemployed person decides whether to accept a job offer, the value of 0 is not observational, and a firm and a person are oriented only on value y = 0 + u, where u - a random noise that does not correlate with 0 . It is assumed that 0 ~ N[|i,o, u ~ N[0,oU]. If a person has accepted a job offer and begins to work, then during the next stage in the process of collaboration a firm and worker gradually examine each other, set a corresponding wage level that is determined by the formula m0 = E[0 /(0 + u)] = |+o U/(o U +o U)-(y-|).
During the third stage parties decide whether to contract and work further on these terms or an employee resigns from the job and looks for a new one responding or not to other job offers. Generally the implementation of such model is conducted by Bellman principle and results in the following equation during the second stage
V (m0) = max{m0 + pj J(0 ')dF(0' | m0, ©U), PQ},
where J(0) - a solution of Bellman equation J(0) =
= max{0 + PJ(0), PQ}, obtained during the third stage ©U = © U /(© U +o U)o U,
Q - expected present value of wage of a person, who is unemployed and behaves optimally.
Theoretical conclusions of research of search and matching models while studying unemployment confirm statistical observations that characterize labour market and show that the wage grows with the increase of duration of worker and firm collaboration, quits usually happens at the initial stages of such collaboration, and also, that the sequence of quits negatively correlates with the current growth rate of wage. Thus, the theoretical models of unemployment certify non-linearity and asymmetry of fluctuations on the labour market that causes different character of dynamics of unemployment rate in periods of its increase and gradual decline.
On the basis of theoretical implications of non-linearity of labour market indexes behaviour, we will model the dynamics of unemployment rate for the Ukrainian economy by means of analysis of nonlinear time series. We will consider a time series that describes a registered unemployment rate among working population in Ukraine. Dynamics of its monthly values during January 2000 - June 2015 is presented on Fig.1«. The data have been obtained on the basis of the statistical reports on the web-site of State Statistics Service of Ukraine and periodicals of State Employment Centre of Ukraine [14]. Visual analysis of the graph of this time series allows making a preliminary conclusion about its non-stationarity. Statistical testing of this time series for a unit root on the basis of the augmented Dickey-Fuller test (ADF = -0.8649) confirms, that the time series ut, that we are analysing, is integrated of first order, and that is why we will conduct modelling for the row of the first differences, that correspondingly is stationary. The graph of values of the first-order differences of the row is presented on Fig. 1b.
Investigating statistical properties and graphic presentation of values of unemployment rate in Ukraine, it is possible to assume that its behaviour is modelled by different processes during different time intervals. To model such type of non-linearity threshold autoregressive models are applied [5; 15; 16]. They give an opportunity to represent the change of behaviour of time series depending on the value of some function that is determined by the economic structure. One of the most widespread types of this class of models is a two-regime threshold autoregressive model (TAR) of unemployment of a standard form:
Aut = (a0 +a-|Aut-1 +a2Aut-2 +...+a pAut - p) - I(qt-1 <y) + +(P0 + PiAut-i + P2 Aut-2 + . + P PAu - p) - I(qt-1 > Y) + £t,
(1)
where I(-) denotes a function - indicator, qt-i = q(ut-1,..., ut_p) is a known data function. The value of parameter p > 1 determines the order of autoregression. A parameter Y is called a threshold parameter. Parameters aj are the autoregressive slope coefficients for qt- < y , and parameters P j are autoregressive coefficients for lagged variables for qt-1 > y . It is assumed that error et are equally distributed independent random values (et is idd [0,o2]).
Having defined yt = (1 Aut-1 ... Aut_p)' and
yt (Y) = (y 't - I(qt-1 < Y) y' t - I(qt-1 > Y))' •
■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i ■ ■ ■ i 00 01 02 03 04 05 06 07 08 09 10 11 12 13
i1111111111111111111111111111111111111111111111111 00 01 02 03 04 05 06 07 08 09 10 11 12 13
a) b)
Fig. 1: a) Dynamics of unemployment rate in Ukraine during 2000-2015; b) Dynamics of changes in unemployment rate in Ukraine during 2000-2015
Source, data of the State Statistics Service of Ukraine, elaborations of authors
The model (1) can be written in an alternative form [14]: Aut = y\ a-I(qt-i <y) + y\P- >y) + £f
or
Aut = y't (Y) S + £t, (2)
where S = (a' P')' .
The unknown parameters of model (2) that need to be estimated, are S and y , and since it is nonlinear in relation to parameters, one of adequate methods of evaluation is a maximum likelihood method (ML). By assumption that £t is idd N[0,o2], ML is equivalent to the least square method (LS). Taking into account that regression equation (2) is discontinuous, it is appropriate to use sequential conditional LS values for the evaluation of model parameters.
For a given value of y we obtain LS estimate of S as
[16]:
fT vY T \
S(Y ) =
S Yt (Y ) Y 't (Y )
f=i
S Yt (Y Ut
f=i
On the basis of residuals of estimated model et(y) = aut - y't(y )S(y )
we calculate the estimate of the variation of the unexplained part of the regression
^ (3)
I
cî2(Y ) = - E (êt(Y ))2. 1 t=i
Then LS estimate of threshold parameter y is a value that minimizes variance (3)
Y = argminYerâ 2(y )
where qt-1 er = [Y 1, Y 2] •
It should be noted that variance of residuals à 2(y ) can acquire maximum T of different values that depend on values that a parameter y is varied. These values correspond to à2(qt-i), t = \...,T • Therefore, we will apply OLS to regres-
sions of type (2), accepting that y = qt-i for each qt-1 e r to estimate model parameters. For every regression we calculate residual variance and choose a value y that corresponds to the lowest value of variance, i.e.
Y = argmiriqer 6j(qt _ 1),
Then we find estimate S as S =S(f). Corresponding
residuals we calculate as et =Aut - (yt (Y)) 'J5 with sampling
„2 „ j „ variance Oj = o 7-(Y).
For application of threshold model in the research of the behaviour of unemployment rate as a function - indicator we will consider different lags of changes of unemployment Aut-d for some d < 12 ,and also different lags of unemployment rates. Tables 1 and 2 present results of models estimation that take into account 24 different variants of threshold functions. Numeral calculations have been conducted on the basis of the program created by us in the software environment of GAUSS by means of matrix programming language, that is widely used by scientists, statisticians, econometrists and by financial analysts and is universal enough for various numeral tasks. The first column of tables presents the type of threshold function. The second - a minimum sum of squares of residuals for a corresponding threshold function. The best result in case of the use of delays of unemployment change (see Table 1) we have obtained for qt_ 1 = Auf_6 , and in case of the use of unemployment lags (see Table 2) - for qt_| = ut_2 .
It is important to check whether the developed nonlinear TAR models statistically significantly differ from the linear autoregressive models of certain order that is determined in the process of analysis of graphs of autocorrelation and partly autocorrelation function, and also more difficult procedures. A corresponding null hypothesis is Hg: a = P .If errors are independent and equally distributed, then on condition of the known value of threshold, testing of null hypothesis with a test power that is close to optimal, is possible to carry out, using F-statistics [16]
Table 1
Search and testing results of overall significance of TAR models of unemployment for a threshold function that characterizes lags
of changes in an unemployment rate
Type of threshold function The lowest value of variance J 2*1000 Estimation of threshold parameter y Testing of significance of TAR model Ft (Y) Probability of inadequacy of TAR model Prob
i i A 3.8187 0.065 80.24 0.000
i i A t 3.8493 0.005 78.36 0.000
i i A t 3.9610 -0.035 71.75 0.001
i i A t 4 3.8030 0.005 81.22 0.000
11 a t 4.2664 -0.073 55.44 0.001
qt-i = Au-e 3.2135 -0.105 124.73 0.000
11 a t 3.6485 -0.125 91.26 0.000
q-i = Auts 3.7097 -0.085 87.18 0.000
11 a t 9 3.8647 -0.098 77.42 0.000
q-i = Aut-io 3.6432 0.006 91.27 0.000
11 a t 3.6134 0.045 93.62 0.000
11 a t 3.3641 0.105 112.15 0.000
Source: estimations of authors
Table 2
Search and testing results of overall significance of TAR models of unemployment for a threshold function that characterizes lags
of unemployment rate
Type of threshold function The lowest value of variance cj 2*1000 Estimation of threshold parameter f Testing of significance of TAR model Fy (y ) Probability of inadequacy of TAR model Prob
11 A t 3.3183 2.296 115.86 0.000
11 A t 3.2383 2.050 131.82 0.000
11 A t 3.5030 1.835 101.52 0.000
11 A t 4 4.1370 2.175 62.06 0.000
i i A t 4.3347 2.390 52.11 0.006
q-i = Aut-e 4.4971 2.493 44.60 0.019
11 a t 4.5074 2.645 44.13 0.029
q-i = Auts 4.5414 3.150 42.64 0.040
tq 11 A t 9 4.6584 3.154 37.65 0.099
q-i = Aut-io 4.2658 2.427 39.01 0.081
tq 11 A t 4.0201 2.370 68.40 0.000
tq i i A t 3.9001 2.335 75.31 0.000
Source: estimations of authors
where
Ft (Y ) = T ((a a )2 -â 2(Y ))/(â 2(Y )),
(a a )2= 1/T £ ( Aut - Y \a)2 t=1
(4) and
( t \
( t A
S YtY' t S Yt Aut
Vf=i J Vt=l
is OLS evaluation a by assumption that a = P.Since Fy(y) is monotonous of a 2 function, then
matemathhhï metoflm ta mo^eni b ekohom^i
Fj = SUPY£r Fj(Y) is point F - statistics against the alternative H1: a ^ P if y is known.
However, for TAR model such testing is not correct, since a threshold y is not defined for Hg and asymptotic distribution Fj in this case is not chi-square distribution. Methodology of testing of hypothesis Hg for TAR models has been proposed by Hansen [16]. He showed that asymptotic distribution can be approximated by means of such bootstrap procedure.
Let e* is idd N[0,1] (t = 1,...,j) random draws. We will estimate regression ef on yt to obtain residual variance (o a)2, and on yt (y ) to obtain residual variance (O*)2(Y) .Then Fj *(y ) = 7((o _ (O*)2(Y ))/((O*)2(Y)) and Fj = supYGp Fj (y). Distribution F- converges weakly in probability to distributionFj* , thus the repetition of bootstrap procedure for Fj* can be used for approximation of asymptotic distribution Fj . Bootstrap approximation of test ^-value is formed on the basis of calculation of sample percent, for which Fj * exceeds Fj .
We will conduct the procedure of adequacy testing of TAR model (2) for different variants of threshold functions for which the point estimations of parameters and corresponding residual variances have already been calculated. We will also calculate p - values, that determine probability that AR model
has advantage over the corresponding TAR model. Values of F-statistics and corresponding probabilities (Prob) for 24 threshold functions are presented in the last two columns of Tables 1 and 2. Testing results certify that in most cases we haven't obtained statistically meaningful residuals that would show that unemployment rate modelling by means of linear AR-model is more adequate, than nonlinear TAR modelling.
The results of calculations show that unemployment has a nonlinear structure and while modelling its behaviour it is necessary to apply threshold models that give an opportunity to represent its different dynamics depending on the previous state of economic environment. In particular, it has been revealed that the behaviour change of unemployment takes place when its value two periods ago exceeded 2.05, or when unemployment change half a year ago was more than -0.105. In this respect the influence on the mode of behaviour of past values of unemployment changes is more significant than the influence of past values of unemployment rate.
The general specification of the evaluated nonlinear threshold autoregressive model of unemployment is:
Aut = (a0 + a-|Aut-1 +a2Aut-2 + . . .+apAut-12 )I(Aut-6 < -0,105) + +(Po + PM_1 + P2Aut_2 + - ■ - + PpAut_u )I(Aut_6 >-0,105) + et,
(5)
The values of evaluations of parameters of model (5) and their corresponding i-statistics are presented in Table 3. The values of Student's statistics state that most parameters are statistically significant.
Table 3
Parameter estimations and significance testing results of nonlinear threshold autoregressive model of unemployment
Lag order j Estimates of parameters aj Student's statistics t(a j ) Estimates of parameters ßj Student's statistics t( )
I 0.1304 2.5146* -0.0138 -2.2971*
2 1.4076 9.3539** 0.3652 5.1405**
3 -1.2280 -5.0466** -1.0824 -4.0247**
4 1.4460 5.2121** -0.0312 -0.4108
5 0.2589 0.7058 -0.1720 -2.3482*
6 -0.8418 -2.1706* 0.0087 0.1172
7 1.1212 2.7584** -0.1055 -1.3039
8 -0.2693 -1.0601 -0.0554 -0.7043
9 -0.5451 -2.1977* -0.0134 -0.1633
10 0.9135 4.5120** -0.1956 -2.1120*
11 -0.2866 -1.8516 0.1088 0.9256
12 0.2769 2.2000* 0.0715 0.6464
Source, estimations of authors
Threshold function defined on the basis of modelling and estimation of threshold parameter show that a model will change its regime when the unemployment rate change, that happened half a year ago, will exceed a value of - 0,105. Using point estimates of slope parameters and threshold parameter, we will find estimated values of unemployment rate in Ukraine. Fig. 2 presents the estimated values of unemployment rate in
Ukraine by means of constructed TAR model. The observations that are described by the different branches of threshold autoregression are presented by different symbols.
Modelling results state that during the investigated period 49 observations in the different months of different years are governed by the first regime and characterize the periods of unemployment growth, while 142 observations belong to the
4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0
/V
• А* •• А
V" V Ч i •
* V: А
*: д
V -
А • Д
а д : v -А J
V, £
2000
2002
2012
2014
2004 2006 2008 2010
• режим 1 д режим 2
Fig. 2. Branching unemployment rates into two different regimes of behaviour depending on the change of past values
Source: evaluations of authors
second regime of behaviour that describes the dynamics of employment growth respectively.
Conclusions. As a result of empiric research the row of nonlinear econometric specifications has been estimated and application of threshold autoregressive model for description of nonlinear and asymmetric behaviour of unemployment rate in Ukraine has been justified. Modelling states that the dynamics of unemployment rate in the national economy is characterized by different regimes of behaviour that change depending on the value acquired by the threshold function that is defined by econometric analysis. The estimated value of threshold parameter reveals branching of TAR model behaviour of unemployment rate in two directions, and transition between them is determined by the fact whether unemployment rate change, that was observed 6 months ago was higher than -0,105, or not. The first branch of developed nonlinear autoregression describes and gives an opportunity in a short-term period to forecast the dynamics of unemployment rate growth, and consequently and employment reduction, while the second set of estimations allows to characterise dynamics of unemployment fall and to forecast employment growth phases. The revealed nonlinear character of unemployment rate dynamics confirms non-Walrasian properties of labour market in Ukraine and states its uneven and asymmetric cyclic behaviour.
The research showed that development and application of threshold autoregressive models is a perspective direction of nonlinear character and tendencies changes of basic labour market indicators forecast, since it allows to obtain not only qualitative forecasts, but also to evaluate quantitatively the values of threshold parameters, that determine the different modes of their behaviour that gives an opportunity in advance to forecast the phases of their fall and growth and to react to them in time. Further directions of researches are development and estimation of the real information of more difficult modifications of threshold autoregressions with more than one threshold parameter.
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