CC BY
34UDC 338.2(476)+316.42(476) https://doi.org/10.33619/2414-2948/40/30
JEL classification: H10, J58, P35, Z13
FORECASTING OF SOCIO-ECONOMIC SECURITY INDICATORS BY MEANS
OF EXPONENTIAL SMOOTHING
©Shvaiba D., ORCID: 0000-0001-6783-9765, Ph.D., Belarusian Trade Union of workers of chemical, mining and oil industries, Belarusian national technical University, Minsk, Belarus, [email protected]
ПРОГНОЗИРОВАНИЕ ПОКАЗАТЕЛЕЙ СОЦИАЛЬНО-ЭКОНОМИЧЕСКОЙ
БЕЗОПАСНОСТИ СПОСОБОМ ЭКСПОНЕНЦИАЛЬНОГО СГЛАЖИВАНИЯ
©Швайба Д. Н., ORCID: 0000-0001-6783-9765, канд. экон. наук, Белорусский профсоюз работников химической, горной и нефтяной отраслей промышленности, Белорусский национальный технический университет, г. Минск, Беларусь, [email protected]
Abstract. The method of exponential smoothing is widely used in the forecasting of financial and economic characteristics in different sectors of the economy, departments, etc. In the construction of a forecast model by exponential smoothing time series of characteristics of socio-economic security is smoothed with the support of a weighted moving average, in which the weights obey the exponential law. In this case, the following levels of the series are given significant values in comparison with the past, because they carry more important information to determine the predicted values at the level of socio-economic security characteristics.
Аннотация. Способ экспоненциального сглаживания обширно используется при прогнозировании финансово-экономических характеристик в разных секторах экономики хозяйства, ведомствах и т. д. При построении прогнозной модели способом экспоненциального сглаживания временной ряд характеристик социально-экономической безопасности сглаживается с поддержкой взвешенной скользящей средней, в которой веса подчиняются экспоненциальному закону. В данном случае следующим уровням ряда придаются существенные значения в сравнении с прошлыми, т. к. они несут более важную информацию для определения прогнозируемых значений на уровне характеристик социально-экономической безопасности.
Keywords: socio-economic security, government, society, enterprise, employee, threat, security, interests, economics, analysis, system.
Ключевые слова: социально-экономическая защищенность, государство, общество, предприятие, работник, угроза, защищенность, интересы, экономика, анализ, система.
The method of exponential smoothing is widely used in forecasting financial and economic characteristics in different sectors of the economy, departments, etc. [1-2].
When constructing a predictive model by exponential smoothing, the time series of socioeconomic security characteristics is smoothed with the support of a weighted moving average, in which the weights obey the exponential law. In this case, the following levels of the series are given significant values in comparison with the past, since they carry more important information to determine the predicted values at the level of socio-economic security characteristics [3, p. 81].
The predictive model constructed by the exponential smoothing method contains the following form:
Yt+l = p(t) + at+l (1)
where: p(t) — the deterministic part;
at+i — the margin of error;
t + I — forecast period.
When using the method of exponential smoothing, the deterministic segment (the most possible variant of the forecast) is specified by the predictive model, the forecast error characterizing the upper and lower limit of the predicted values is calculated by formulas.
Let there be a time series of characteristics of socio-economic security Yt(t = 1,2, ...,n), which is possible to represent a polynomial of p-th degree.
Yt = a0 + a1t + ^2t2 + - + +St (2)
2! p!
Forecast levels of socio-economic security in the time period t + 1(t = ri) can be computed by decomposition in Taylor series:
Y -y(0) + 1Y(1)+—Y(2)+ + — V(k) + — y('P) (3)
yt+i = yt +lrt + 2! h +' +k! Yt + p! Y1
where Y^ — k-th derivative at the time point t. Any K-th (k=0, 1, 2, ..., p) derivative is expressed by a combination of exponential means up to (p+1) order:
Y[k] = aS[k-1] + BS[k] (4)
where a — smoothing factor.
In the scientific literature to select values of a were proposed various approaches [4, p. 118; 5, p. 92; 6, s. 78]. Thus, the best value of a is proposed to be found by comparing the variance of deviations of the actual values of the indicators of socio-economic security characteristics from the predicted ones formed by models constructed by the method of exponential smoothing using different smoothing characteristics. In this regard, the number of characteristics of socio-economic security are symbolically divided into a retrospective stage and a stage of pre-emption. According to the data of the pre-forecast period, models are formed and the forecast for the entire length of the 2nd part of the time series of socio-economic security characteristics is made. Following this, the differences between the actual values of socio-economic security characteristics and the forecast and variance of these deviations are revealed. The smallest variance characterizes the suitable variant at which it is obtained and is used for subsequent calculations.
The main defects of the approach are labour costs and the need for a long time series of socioeconomic security statistics.
Бюллетень науки и практики /Bulletin of Science and Practice http ://www.bulletennauki.com Т. 5. №3. 2019
In practical calculations, as a rule, a method is used to show the length of the smoothing interval with values that are calculated by empirical formulas:
a = —, ß = 1 — a m+1 r (5)
where: m — smoothing time (shown with the periodization of forecasting, specific cyclic characteristics of socio-economic security, etc.).
Based on the recurrent formula 4. you can represent an exponential mean expression of any order for expression 3.
^[(У) = ^(У) + ^-l(y) , (6)
s [2] 5 [(] ,o5 [2] °t(y) a°t(y) + ^t-l(y) (7)
ç И _ ™ç [fc-1] , or M ^t(y) = ttJt(y) + P°t-1(y) (8)
çM _ „cfa-ll 1 о çM °t(y) = a°t(y) + ^t-i(y) (9)
Using exponential averages, it is possible to obtain a system of equations that provide an opportunity to qualify the characteristics of predictive models of a linear and quadratic form used in the practice of forecasting the characteristics of socio-economic security.
Let us form a predictive model of the linear form by exponential smoothing. The predictive model of the linear form by means of exponential smoothing is created according to the parameters of Table 1. for a smoothing period of 4 years (m=4).
Then:
2
a = — = 0.4, B = 1 - a = 0,6
4+1 ' r
Using the system of normal equations the trend parameters are revealed:
Y = 1360.0 + 124.855t u at = 148.641.
Then the initial conditions are calculated and S[2^)
Table 1.
PRIMARY DATA AND INTERMEDIATE CALCULATIONS TO CALCULATE THE PARAMETERS OF THE LINEAR MODEL BY THE METHOD OF EXPONENTIAL SMOOTHING
Year Y ç[i] Vy) ç[2] "Vy) ao a( ?
t-9 1637
t-8 1683 1358.430 1134.634 1582.227 149.198 1731.425
t-7 1750 1488.258 1276.083 1700.433 141.449 1841.882
t-6 1796 1592.955 1402.083 1783.078 126.748 2909.826
t-5 1876 1674.173 1511.368 1836.977 108.536 1945.513
t-4 1983 1754.904 1608.782 1901.025 97.414 1998.439
t-3 2089 1846.142 1703.726 1988.558 94.943 2083.501
t-2 2225 1943.285 1799.550 2087.020 95.824 2182.844
t-1 2549 2055.971 1902.118 2209.824 102.568 2312.392
t 2879 2253.183 2042.544 2463.821 140.425 2604.246
t+1 2503.509 2226.930 2780.089 184.386 2964.475
Source: elaboration of author.
m ß 0.6
S0Z) = "ow-^arn = 1360.000 —124.855 = 1172.717
2ß 2x0,6
S[2]y) = am--^-a1(t) = 1360.000 --^j—124.855 = 985.435
In that case ^^ and S^) respectively are equal:
s[(]} = aYt-1 + fiS^ = 0.4 x 1637 + 0.6 x 1172.717 = 1358.435 sj^ = aS^ + fiS^ = 0.4 x 1358.430 + 0.6 x 985.435 = 1134.633
On this basis
a0 = 2S[(^) - S[^ = 2 x 1358.430 - 1134.633 = 1582.277
a m m 0.4 a(=jx (S^ - sj^) ^ x (1358.430 - 1134.633) = 149.198
S^ = aYt(y) + = 0.4 x 1683.0 + 0.6 x 1358.430 = 1488.258
IS^ = asfjjj + = 0.4 x 1488.258 + 0.6 x 1134.633 = 1276.082
Intermediate data are included in the Table 1.
The predictive model of exponential smoothing of the linear form for t+1 year will have the
form:
Y+ = 2780.089 + 184.386t
where: t=1 for t+1
Forecasted value Yt+i are calculated by putting the indicator t of the year in the forecast model:
= 2780.089 + 184.386 x 1 = 2964.475
Yt+i = 2780.089 + 184.386 x 1 = 2964.475 The forecast error is calculated by the formula:
u
ay(t+i) = ±Ost I (2-a)2 [1 + 4(1 -a) + 5(1 - a)2 + 2a(4 - 3a)e + 2d2e7-']
where: aEt — the mean square error calculated for deviations from the linear trend formed by the least square's method.
ay(t+() = ±148.641 x
I 04
x I--[1 + 4(1 - 0.4) + 5(1 - 0.4)2 + 2 x 0.4(4 -3x 0.4) x1 + 2x 0.42 x 12]
= ±129.396
ay(t+5) = ±148.641 x
I 04
x I--[1 + 4(1 - 0.4) + 5(1 - 0.4)2 + 2 x 0.4(4 -3x 0.4) x1 + 2x 0.42 x —■]
V )
= ±229.448
Forecast data of socio-economic security indicators are summarized in Table 2.
Table 2.
FORECASTED VALUE ? and ? ± ayin t+1-t+5 years
Год ^t+e °>(t+i) Lower and upper boundaries
^t+e — °y(t+1) ^t+e + °y(t+1)
t+1 2964.475 129.396 2835.079 3093.871
t+2 3148.861 153.778 2995.083 3302.639
t+3 3333.247 178.698 3154.549 3511.945
t+4 3517.633 203.959 3313.674 3721.592
t+5 3702.019 229.448 3472.571 3931.467
Source: elaboration of author.
By changing the value of the smoothing period using the exponential smoothing method, it is possible to create a number of linear predictive models. Thus, linear models of exponential smoothing formed at the smoothing period of 3, 4, 5 and 6 years have the following form:
Y^ = 2822.830 + 219.1524t yjj) = 2780.089 + 184.3860t yjjj = 2743.712 + 163.3480t
У(6) = 2715.079 + 150.5091t
where tx for t+1 year is 1.
The choice of a specific forecast model can be made by comparing the deviations of the calculated levels of socio-economic security characteristics from the actual ones in the pre-forecast period. Yj > Yj — the minus sign, Yj < Yj — plus sign (Table 3).
Table 3.
COMPARATIVE DEVIATION VALUES Yj FROM YJ IN THE CASE OF LINEAR EXPONENTIAL SMOOTHING MODELS, %
Year The smoothing period
m=3 m=4 m=5 m=6
t-8 -4,63 -2,87 -1,67 -0,81
t-7 -5,48 -5,25 -4,81 -4,35
t-6 -5,41 -6,34 -6,72 -5,84
t-5 -2,09 -3,71 -4,77 -5,47
t-4 +0,61 -0,78 -1,95 -2,87
t-3 +1,00 +0,26 -0,61 -1,43
t-2 +2,15 +1,89 +1,35 +0,73
t-1 +9,05 +9,28 +9,17 +8,89
t +7,80 +8,61 +10,57 +11,16
Source: elaboration of author.
The minimum deviations of the calculated values of the socio-economic security characteristics from the actual ones in the last year of the pre-forecast period are obtained by a linear model, a constant method of exponential smoothing at the smoothing period of 3 years (Table 3).
Бюллетень науки и практики / Bulletin of Science and Practice http ://www.bulletennauki.com
Т. 5. №3. 2019
We construct a predictive model of the quadratic form by exponential smoothing. To form a predictive model of a quadratic form by means of exponential smoothing, we apply the data of Table 1 and smoothing period of 3 years (a = 0.5; fi = 0.5).
Primary conditions S^), S[2y) and S[(]) equal.
m B B(2 — a) 0.5
sSy) = ao(t) —^«m + „ «2(t) = 1734.495 — — x (—62.395) +
L/C L/C U ■ J
(0.5(2 x 0.5) \ + (-^--17.023) = 1847.959
r2l 2B 2B(3 — 2d) 0.5
% = am— — am+—-~a2(t) = 1734.495 — 2 —x (—62.395) +
^ ^ 0. 5
/3 — 2x0.5 \ + (-—-17.023) = 1995.468
r3l 3B 3B(4 — 3a) 0.5
sSy) = a0(t)—-;-am+^—2--<*2(t) = 1734.495 — 3 — x (—62.395) +
LC LC U ■ J
/3x0.5(4—3x0.5) \ + (-05*- 17.023) = 2177.023
where a0(t), a((t), a2(t) — indicators of trends of models of the quadratic form calculated by the solution of systemthe of the normal equations — for the simulated number of levels of characteristics of social and economic safety the model will have the form:
f = 1734.495 - 62.395t + 17.023t2(at t( for t +1 years = 11) Calculated S^), S^) and S^)
S^ = aYt-( + = 0.5 x 1637 + 0.4 x 1847.959 = 1742.479
S^) = aSf^ + ps[2}1(y) = 0.5 x 1742.479 + 0.5 x 1995.461868.973 S^ = aS[2l) + = 0.5 x 1868.973 + 0.5 x 2177.023 = 2022.998
Model parameter a0, a(, a2 equal:
a0 = 3(s[t^ - S^) + S^ = 3(1742.479 - 1868.973) + 2022.998 = 1643.516
a
a1
2(1-a)2
x[(6- 5x 0.5) x 1742.479 -2(5-4x 0.5) x 1868.973 + (4 3 x 0.5) x 2022.988] = 57659
n - a2 " r[1] ?r[2] , r[3]
a2 = (1- a)2 x *t(y) t(y) + ^t(y)
0.52
-2 x (1742.479 -2 x 1868.973 + 2022.998) = 27.530
2(1-0.5)2
In that case Yt-8 equal 1599.611; Yj - Y for t-8 years equal 83.389 (Yj — calculated for t-8/t years).
Then, the integration calculation is repeated for S^), St[2y) and St[3y); a0, a(, a2; Yj, Yj - Yj. Intermediate calculations are formed in Table 4.
Forecasted value Yt-e are calculated by the formula:
о a2
Yt-i = aa + au + —t
(10)
where: tx for t+1 years equal 1,
1
ft+1 = 2870.749 + 342.341 Xi^X 50.579 X 12 = 3238.379
2
1
rt+5 = 2870.749 + 342.341 x 5^X 50.579 X 52 = 3238.379
The forecast error is found by the formula:
°y(t+1) = +0£tv2a + 3a2 + 3a2t2
where aet — the standard deviation of the actual equations of characteristics of social and economic security from the calculated, calculated by the trend model of the quadratic form, and equal ±58.2526. Then:
at+1 = ±5.255672 x 0,5 + 3 x 0,52 + 3 x 0.53 x 1 = ±84.917
at+5 = ±5.255672 x 0.5 + 3 x 0.52 + 3 x 0.53 x 52 = ±194.297
Indicators ft+1, ft+1oy(t+1), oy(t+1) are summarized in Table 4. The models of the quadratic form formed by the method of exponential smoothing at the period of smoothing in 3, 4, 5 and 6 years will have the form:
f(3) = 2870.749 + 342.341t + 1/2 x 50.579f2 f(4) = 2857.399 + 316.279t + 1/2 x 41.703C2 f(5) = 2845.564 + 300.589t + 1/2 x 57.714f2 f(6) = 2836.365 + 219.699t + 1/2 x 35.906f2
Table 4.
PRIMARY DATA AND INTERMEDIATE CALCULATIONS FOR CALCULATING THE PARAMETERS OF THE PREDICTIVE MODEL OF EXPONENTIAL SMOOTHING
OF THE QUADRATIC FORM
Year Y çM Vv) ç[2] Vv) ç[3] Vv) aa Й2
t-9 1637
t-8 1750 1742.479 1868.973 2022.998 1643.515 -57.659 27.530
t-7 1750 1712.739 1790.856 1906.927 1672.576 16.767 37.954
t-6 1796 1731.369 1761.113 1834.020 1744.790 78.165 43.164
t-5 1876 1763.684 1762.399 1798.209 1802.067 94.027 37.096
t-4 1983 1819.842 1791.120 1794.665 1880.830 109.387 32.266
t-3 2089 1901.421 1846.271 1820.468 1985.918 128.518 29.347
t-2 2225 1995.210 1920.740 1870.604 2094.013 135.303 24.333
t-1 2549 2110.105 2015.423 1943.013 2227.060 150.365 22.273
t 2879 2329.552 2172.487 2057.750 2528.945 262.884 42.328
t +1 2604.276 2388.382 2223.066 2870.749 342.341 50.579
Source: elaboration of author.
2
Table 5.
FORECAST DATA Yt+h ay(t+l} and ± ay(t+l}
Year Yt+e ay(t+e) Lower and upper boundaries
Yt+e — °y(t+e) Yt+e + °y(t-e)
t+1 3238.379 84.917 3153.462 3323.296
t+2 3635.589 105.016 3551.573 3761.605
t+3 4125.377 131.875 3993.502 4257.252
t+4 4644.745 162.168 4482.577 4806.913
t+5 5214.688 194.297 5020.391 5408.985
Source: elaboration of author.
Deviations of the actual levels of socio-economic security characteristics from the calculated ones calculated by quadratic models of exponential smoothing are given in Table 6 ( Yj > Yj — the minus sign, Yj < Yj — plus sign).
Table 6.
COMPARATIVE INDICATORS OF DEVIATIONS ^^ FROM Yj WHEN USING LINEAR MODELS
OF EXPONENTIAL SMOOTHING, %
Year The smoothing period
m=3 m=4 m=5 m=6
t-8 +4.95 +4.03 +3.41 +2.96
t-7 +2.38 +3.18 +3.52 +3.67
t-6 -2.70 -1.45 -0.59 +0.01
t-5 -2.05 -2.00 -1.67 -1.31
t-4 -1.18 -1.73 -1.88 -1.85
t-3 -1.92 -2.63 -3.06 -3.30
t-2 -0.74 -1.75 -2.54 +3.12
t-1 +6.28 +5.49 +4.68 +3.97
t +2.29 +3.45 +3.92 +4.04
Source: elaboration of author.
Thus, the minimum deviation of the actual values of the socio-economic security characteristics from the calculated ones in the pre-forecast period was obtained by applying the quadratic model of exponential smoothing built at the smoothing period of 3 years. The revealed results make sense to apply in the analysis of socio-economic security at the level of the country, region and economic entity to develop certain measures to counter the detected threats.
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7. Shvaiba D. Dynamic regression models of forecasting indicators of social and economic security // Бюллетень науки и практики. 2019. Т. 5. №1. С. 249-257.
8. Shvaiba, D. Socio-economic security of the hierarchical system // Бюллетень науки и практики. 2018. Т. 4. №6. C. 248-254.
Работа поступила в редакцию 17.02.2019 г.
Принята к публикации 21.02.2019 г.
Cite as (APA):
Shvaiba, D. (2019). Forecasting of socio-economic security indicators by means of exponential smoothing. Bulletin of Science and Practice, 5(3), 241-249. https://doi.org/10.33619/2414-2948/40/30.
Ссылка для цитирования:
Shvaiba D. Forecasting of socio-economic security indicators by means of exponential smoothing // Бюллетень науки и практики. 2019. Т. 5. №3. С. 241-249. https://doi.org/10.33619/2414-2948/40/30.
