Determination of the production curves for the construction industry in the Czech Republic
The construction industry is the first to be affected by developments in the economy. It is the first to contract during a period of economic recession and the first to expand during a period of economic growth. This brings to mind whether it is possible to predict the development of the economy bas...
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Інститут телекомунікацій і глобального інформаційного простору НАН України
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Vochozka, M. 2018-03-29T18:29:45Z 2018-03-29T18:29:45Z 2015 Determination of the production curves for the construction industry in the Czech Republic / M. Vochozka // Математичне моделювання в економіці. — 2015. — № 1(2). — С. 5-21. — англ. 2409-8876 https://nasplib.isofts.kiev.ua/handle/123456789/131758 004.02 The construction industry is the first to be affected by developments in the economy. It is the first to contract during a period of economic recession and the first to expand during a period of economic growth. This brings to mind whether it is possible to predict the development of the economy based on predictions for the construction industry. This is a fairly complex problem which can easily be divide into several sub-problems. First and foremost it is necessary to find the means that will allow us to predict the development of the construction industry. It is possible to apply a production function, when an independent variable (production factory) is known and we look for a dependent variable i.e. the volume of production. The production functions for the coming years - if we know the volume of production factors - may indicate the future development of the industry. The subject of this paper is to find an answer to the first of these partial problems. The purpose of this is to be able to predict the development of the economy based on the development of the construction industry and to find the means by which these predictions can be made specifically based on production functions for the construction industry in the Czech Republic. Галузь, на яку в першу чергу негативно впливає розвиток економіки в період економічного спаду та кризи, – будівельна галузь. Але це також перша галузь, що сигналізує про розвиток. Розглядається питання прогнозування розвитку економіки, засноване на прогнозуванні будівельної галузі. Наведено діагностичний приклад з використанням виробничої функції (за обсягом виробництва). Предметом роботи є знаходження інструментів для прогнозування розвитку будівельної галузі, а саме виробничих функцій будівельної галузі, на прикладі Чеської Республіки. Строительная промышленность бывает первой, где проявляется неблагоприятное влияние снижения темпов развития экономики, кризиса и стагнации. И наоборот – именно она становится первым индикатором улучшения экономической коньюнктуры. Поднят вопрос: возможно ли прогнозировать общее развитие экономики на основе отдельного прогноза строительной промышленности? Рассматривается вопрос прогнозирования развития экономики, основанный на прогнозировании строительной отрасли. Приведен диагностический пример с использованием производственной функции (по объему производства). Предметом работы является нахождение инструментов для прогнозирования развития строительной отрасли, а именно производственных функций строительной отрасли, на примере Чешской Республики. en Інститут телекомунікацій і глобального інформаційного простору НАН України Математичне моделювання в економіці Інформаційні технології в економіці Determination of the production curves for the construction industry in the Czech Republic Визначення виробничих кривих для будівельної галузі в Чеській Республіці Определение производственных кривых для строительной отрасли Чешской Республики Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Determination of the production curves for the construction industry in the Czech Republic |
| spellingShingle |
Determination of the production curves for the construction industry in the Czech Republic Vochozka, M. Інформаційні технології в економіці |
| title_short |
Determination of the production curves for the construction industry in the Czech Republic |
| title_full |
Determination of the production curves for the construction industry in the Czech Republic |
| title_fullStr |
Determination of the production curves for the construction industry in the Czech Republic |
| title_full_unstemmed |
Determination of the production curves for the construction industry in the Czech Republic |
| title_sort |
determination of the production curves for the construction industry in the czech republic |
| author |
Vochozka, M. |
| author_facet |
Vochozka, M. |
| topic |
Інформаційні технології в економіці |
| topic_facet |
Інформаційні технології в економіці |
| publishDate |
2015 |
| language |
English |
| container_title |
Математичне моделювання в економіці |
| publisher |
Інститут телекомунікацій і глобального інформаційного простору НАН України |
| format |
Article |
| title_alt |
Визначення виробничих кривих для будівельної галузі в Чеській Республіці Определение производственных кривых для строительной отрасли Чешской Республики |
| description |
The construction industry is the first to be affected by developments in the economy. It is the first to contract during a period of economic recession and the first to expand during a period of economic growth. This brings to mind whether it is possible to predict the development of the economy based on predictions for the construction industry. This is a fairly complex problem which can easily be divide into several sub-problems. First and foremost it is necessary to find the means that will allow us to predict the development of the construction industry. It is possible to apply a production function, when an independent variable (production factory) is known and we look for a dependent variable i.e. the volume of production. The production functions for the coming years - if we know the volume of production factors - may indicate the future development of the industry. The subject of this paper is to find an answer to the first of these partial problems. The purpose of this is to be able to predict the development of the economy based on the development of the construction industry and to find the means by which these predictions can be made specifically based on production functions for the construction industry in the Czech Republic.
Галузь, на яку в першу чергу негативно впливає розвиток економіки в період економічного спаду та кризи, – будівельна галузь. Але це також перша галузь, що сигналізує про розвиток. Розглядається питання прогнозування розвитку економіки, засноване на прогнозуванні будівельної галузі. Наведено діагностичний приклад з використанням виробничої функції (за обсягом виробництва). Предметом роботи є знаходження інструментів для прогнозування розвитку будівельної галузі, а саме виробничих функцій будівельної галузі, на прикладі Чеської Республіки.
Строительная промышленность бывает первой, где проявляется неблагоприятное влияние снижения темпов развития экономики, кризиса и стагнации. И наоборот – именно она становится первым индикатором улучшения экономической коньюнктуры. Поднят вопрос: возможно ли прогнозировать общее развитие экономики на основе отдельного прогноза строительной промышленности? Рассматривается вопрос прогнозирования развития экономики, основанный на прогнозировании строительной отрасли. Приведен диагностический пример с использованием производственной функции (по объему производства). Предметом работы является нахождение инструментов для прогнозирования развития строительной отрасли, а именно производственных функций строительной отрасли, на примере Чешской Республики.
|
| issn |
2409-8876 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/131758 |
| citation_txt |
Determination of the production curves for the construction industry in the Czech Republic / M. Vochozka // Математичне моделювання в економіці. — 2015. — № 1(2). — С. 5-21. — англ. |
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~ 5 ~
Математичне моделювання в економіці, №1, 2015
УДК 004.02
M. VOCHOZKA
DETERMINATION OF THE PRODUCTION CURVES FOR THE
CONSTRUCTION INDUSTRY IN THE CZECH REPUBLIC
Abstract. The construction industry is the first to be affected by developments in
the economy. It is the first to contract during a period of economic recession and
the first to expand during a period of economic growth. This brings to mind
whether it is possible to predict the development of the economy based on
predictions for the construction industry. This is a fairly complex problem which
can easily be divide into several sub-problems. First and foremost it is necessary
to find the means that will allow us to predict the development of the construction
industry. It is possible to apply a production function, when an independent
variable (production factory) is known and we look for a dependent variable i.e.
the volume of production. The production functions for the coming years - if we
know the volume of production factors - may indicate the future development of the
industry. The subject of this paper is to find an answer to the first of these partial
problems. The purpose of this is to be able to predict the development of the
economy based on the development of the construction industry and to find the
means by which these predictions can be made specifically based on production
functions for the construction industry in the Czech Republic.
Key words: production factors, production funtions, construction industry,
prediction.
Introduction
Even in 2015 it can be said that the Czech economy continues to be in recession, a
recession that has persisted for many years. Low GDP growth has been temporary
and there are no signs of a long term trend towards improvement. Economists
estimate the development of the economy primarily on the basis of macroeconomic
variables such as GDP, balance of trade, inflation, unemployment, investment rate,
etc.
Business and financial managers alike carefully observe such economic
predictions. Their task is to financially manage specific enterprises in accordance
with the objectives of their shareholders (according to the logic of increasing
shareholder value). If we put forward the argument that every company tries to
increase shareholder value (that is if they have no existential problems), this would
mean that if the majority of enterprises in the economy are successful and are
ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ В ЕКОНОМІЦІ
Ó M. Vochozka, 2015
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Математичне моделювання в економіці, №1, 2015
meeting their basic objectives (i.e. they add value to the economy as a whole), it
can be stated that the whole economy is successful and growing.
On the basis of this argument it would be logical to question whether it is then
possible to predict the development of the economy according to the developments
in, and forecasts from, the business environment. In other words, as the sum of the
economic performances of business entities. The answer may be twofold. Firstly,
we can assume that the development of the business environment heralds the
development of the entire economy. On this basis we can predict the development
of the entire economy on the results of business entities. Secondly, we can assume
that higher entities behave with indifference to lower level entities.
In either case it is an extremely important moment for obtaining the relevant
data and determining the production curve.
The aim of this paper is to determine the production curves for the
construction industry in the Czech Republic and in its individual regions.
1. Material and methodology
The information given below about companies comes from the Albertina database.
The data covers all the construction companies which operated on the Czech
market between the years 2003-2011 and which fall into the classification CZ-
NACE under Section F. The following activities are involved – construction of
buildings, civil engineering and specialized construction activities.
The file contains a total of 67,492 rows of data in columns labelled:
- company registration no.;
- company name (15,189 companies);
- region
- list of annual financial statements between 2002-2011 (some companies
did not submit financial statements in some years, some did it twice, three
times or four times a year);
- sales (dependent variable y - production),
- sales costs + power consumption (explanatory variable = regressor
production factor);
- labour costs (explanatory variable production factor),
- amortization of intangible and tangible assets (explanatory variable
production factor);
- other operating expenses (explanatory variable production factor),
- interest (explanatory variable production factor).
The production value and production factors are stated in TCZK (thousands of
Czech Crowns). These values can be zero or negative too. The data is completed
with the following variables: dic (data identification number, 1 – 67492, jv vector
of ones),
(1
)
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Математичне моделювання в економіці, №1, 2015
Linear models were derived by using the method of least squares (for this
purpose the introduction of logarithmic transformations is required). The programs
MS Office Excel, R, STATGRAPHICS and GiveWin2 were used. The data was
loaded into an R program in the sentence: >LNPF <- read.table ("clipboard",
header=TRUE, sep="\t", na.strings="NA", + dec=",", strip.white=TRUE).
Results – production function.
Based on the imported underlying data, basic data on variables was obtained
in an R program (year_k is a categorical variable year). The linear dependence of
production on production factors is indicated by the correlation matrix.
2. Linear single-factor production function
The linear single-factor production function where y is dependent on x1 is subjected
to an investigation into the shape of the regression line
,
where b0 and b1 are the estimated regression parameters (the absolute term
– the intercept – a regression coefficient describes the
absolute flexibility of the production factor y on x1)
ei is a random (error component), i = 1, ..., n (n is the number
of measurements).
A regression line is determined on the basis of an estimation (offsetting) using
the method of the least squares (hereinafter referred to as “LSM”). This method
aims to minimize the sum of the squared residues i.e. the differences between the
measured values and equivalent values measured on a straight line, using the
software R.
The output is the following result:
The resulting values are shown in Table 1.
Table 1 – Results from the application of the LSM on production factor x1
Estimate Std. error t-value Pr(<|t|)
Intercept 3.19e + 03 1.44e + 02 22.05 <2e - 16 ***
x1 1.216e + 00 3.470e - 04 3504.84 < 2e - 16 ***
Source: Author
Residual standard error: 37,400 on 67,490 degrees of freedom
Multiple R-squared: 0.9945 Adjusted R-squared: 0.9945
F-statistic: 1.228e+07 on 1 and 67,490 DF, p-value: < 2.2e-16
The estimated production function can be expressed as:
(2)
where e is a vector of residues (differences between measured and balanced
values), which represent the estimates for random elements.
From small p-value (risks of an incorrect rejection test null hypothesis)
significance tests it is evident that both the estimated regression parameters are
statistically significant (production factor significantly acts linearly on
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Математичне моделювання в економіці, №1, 2015
production function y and at an average initial level of production of TCZK 3,195
when x_1 = 0 is significantly set). The slope can be interpreted as the absolute
flexibility i.e. with every increase in production factor x by a total of one thousand,
output y increases on average by TCZK 1,216. The strength of dependence is
expressed by the coefficient of determination R2 = 0.9945This can be interpreted to
mean that 99.45% of changes in production factor can be explained. This fact
confirms the very strong linear dependence of production function y on factor .
This is evident from the close proximity of the dots on the dot plot, which represent
the measured values, to the balanced line. Other factors relating to changes in
production function y can only explain a maximum of 0.55% of changes. This
single-factor production function is therefore crucial. In mathematical statistics it
has been shown that the estimates obtained through the LSM are impartial and
consistent (approaching in probability the estimated parameters). In order to
minimise selection errors so that other statistical induction (tests and confidence
intervals) could be performed, the normality of the distribution of random
components with zero value and a constant variance (i.e. so-called
homoscedasticity of variances measurement y) and the linear independence of the
individual measurement of output (i.e. between measurements is not a serial
dependence - the so-called autocorrelation) are required. The verification of these
conditions is usually done on residues graphically using residual analysis (or by
statistical tests applied to the residues). Each program therefore counts residues and
their standardized values (i.e. studentized residues, which in the case of normality
should randomly fluctuate around zero, maximally within the range from -8 to +8).
The dependence is usually shown by the studentized residues on the balanced
values and serial numbers of measurement. If the residues do not move within that
band (having the shape of open scissors), there is heteroscedasticity (opposite
homoscedasticity) in the model. This can be dealt with by subjecting the estimate
of the regression function to a so-called weighted LSM. If there is inertia in the
residues there is an autocorrelation. In such cases estimates of the generalized LSM
(e.g. in the model CORC- methods with offered econometric applications
GiveWin2) are used instead of the ordinary LSM. Normality can be verified
through a QQ-diagram (the dependency of empirical and theoretical quantiles of
residues and normal quantiles lying on the line). If the points – showing the
empirical quantiles – are close to that line, the residues are normally distributed.
Graphs of residues are given in Figure 1, 2.
Residual Plot
-4 1 6 11 16 21 26
(X 1,E6)predicted y
-70
-40
-10
20
50
80
St
ud
en
tiz
ed
re
si
du
al
Residual Plot
0 2 4 6 8
(X 10000)dic
-70
-40
-10
20
50
80
St
ud
en
tiz
ed
re
si
du
al
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Математичне моделювання в економіці, №1, 2015
Figure 1 – Graphs of residues
Figure 1 – Graphs of residues
These graphs of studentized residues show that the normality of data is
breached and in the model there is a certain level of heteroscedasticity in variance
of residues (according to the previous dot plot, the level is not relatively high).
These problems can be partially explained by the presence of relatively large
amounts of low and high, outlying and extreme values in the data. This can also
been seen in the last picture which is a box plot for the course of studentized
residues. It is necessary - with discretion - to look at the possible use of other
methods of statistical induction in the model and at the evaluation of the size of the
standard errors of the estimated regression parameters. The static quality of the
model could be improved by deleting these values, but the values of the estimated
regression parameters would consequently change.
Further specification of the production function derived by year or by region
is also of interest. If a qualitative (categorical) variable is to be included in the
regression model, a so-called dummy variable must be used to determine its levels.
These dummy variables have a value of 1 if the qualitative variable comes in at this
level and otherwise the value 0. In order to prevent multicollinearity in the
regression model, the number of used dummy variables must be equal to the
number of levels minus one. Any level at which a dummy variable has been
omitted is called a reference variable (this is usually atthe first level). The
estimated regression coefficients of dummy variables indicates how much the
average value of the explanatory variable has changed while the level of dummy
variables passed from the reference level to the intended level (e.g. the region has
changed from JCK under consideration to another region, or the year changed from
2002 to another intended year). The model can include quantitative interactions
with a qualitative variable (the estimated coefficient is interpreted as an
amendment to the directive corresponding to the transition from the reference level
to the intended level of the qualitative variable).
First, we assume the dependence on production factor x1 and the year. The
year is a categorical variable to levels in 2002, 2003, 2004, 2005, 2006, 2007,
2008, 2009, 2010 and 2011. Dummy variables u02, u03,…,u11 can be introduced for
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Математичне моделювання в економіці, №1, 2015
each year. The first dummy variable u02 is omitted. This variable for the year is
considered to be level reference one. Other dummy variables u03 to u11 are included
in the model and a production function predicted:
(3
)
The equation can be established again in the program R1 using the LSM. The
resulting parameters are listed in Table 2.
Table 2 – Resulting parameters of a production function for production factor x1
according to the individual years in the reference period
Estimate Std. error t-value Pr(<|t|)
(Intercept) -3.160e+03 6.406e+03 -0.493 0.622
x1 1.373e+00 3.008e-03 456.574 <2e-16 ***
year_k[T.2003] 5.966e+03 6.428e+03 0.928 0.353
year _k[T.2004] 5.782e+03 6.423e+03 0.900 0.368
year _k[T.2005] 6.099e+03 6.420e+03 0.950 0.342
year _k[T.2006] 6.003e+03 6.418e+03 0.935 0.350
year _k[T.2007] 6.003e+03 6.416e+03 1.075 0.282
year _k[T.2008] 7.668e+03 6.416e+03 1.195 0.232
year _k[T.2009] 6.442e+03 6.416e+03 1.004 0.315
year _k[T.2010] 5.872e+03 6.417e+03 0.915 0.360
year _k[T.2011] 7.045e+03 6.466e+03 1.090 0.276
x1: year _k[T.2003] -1.408e-01 3.332e-03 -42.256 <2e-16 ***
x1: year _k[T.2004] -1.530e-01 3.223e-03 -47.491 <2e-16 ***
x1: year _k[T.2005] -1.547e-01 3.151e-03 -49.086 <2e-16 ***
x1: year _k[T.2006] -1.635e-01 3.147e-03 -51.968 <2e-16 ***
x1: year _k[T.2007] -1.766e-01 3.153e-03 -56.006 <2e-16 ***
x1: year _k[T.2008] -1.900e-01 3.136e-03 -60.577 <2e-16 ***
x1: year_k[T.2009] -1.448e-01 3.133e-03 -46.225 <2e-16 ***
x1: year_k[T.2010] -1.387e-01 3.175e-03 -43.687 <2e-16 ***
x1: year_k[T.2011] -1.502e-01 3.251e-03 <2e-16 <2e-16 ***
Source: Author
Residual standard error: 36,000 on 67,472 degrees of freedom
Multiple R-squared: 0.9949, Adjusted R-squared: 0.9949
F-statistic: 6.983e+05 on 19 and 67,472 DF, p-value: < 2.2e-16
1 Using the application R has the advantage that the necessary dummy variables will be
introduced into the model automatically. It is not necessary to generate them.
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Математичне моделювання в економіці, №1, 2015
Here, for example, year_k[T.2003] indicates u03, and x1:year_k[T.2003]
indicates x1u03. The estimated production function is:
(4
)
The production function for 2002 (reference period) can be determined if we
substitute all the dummy variables for 0. The production function for 2002 is
therefore as follows:
(5)
The production function for 2003 can be obtained by substituting u03=1, and
by substituting the other dummy variables for 0. The production function for 2003
is therefore as follows:
.
(6
)
By continuing this process (i.e. by adding the coefficient of the respective
dummy variable to the intercept, and by adding the coefficient u x1 mutipled by the
dummy variable to the slope) the equations for the production functions for years
are as follows:
(7)
In summary it can be stated that the average levels of production (at )
are not statistically significant in the individual years (in 2002 there were only 33
highly variable figures). On the contrary, absolute flexibilities are statistically
significant (marked by asterisks) and vary only slightly.
The R-square (coefficient of determination) rose to 0.9949, i.e. 99.49 % of all
changes to production y in the individual years is explained by factor x1. The
conclusions, subject to the verification of conditions, are similar to the previous
model.
In the following part, linearsingle-factor production functions will be
compared for production factor x1 with the variable region.
Region JCK (South Bohemia) represents the basic level (alphabetically it is in
the first place). Apart from the variable x1, the explanatory dummy variables for the
other regions (uJCK, uJMK, uKVK, uKV, uKK, uLK, uMK, uOK, uPK, uPK, uSK, uUK, uZK, uPHA)
~ 12 ~
Математичне моделювання в економіці, №1, 2015
will be inserted into the model (designated in R as region[T.JMK],
region[T.KK],…, region[T.ZK]).
The results of the analysis are presented in Table 3.
Table 3 – Resulting parameters of production function for production factor x1 in
the individual regions
Estimate Std. error t-value Pr(>|t|)
(Intercept) 4.739e+03 5.778e+02 8.201 2.42e-16 ***
x1 1.169e+00 2.972e-03 393.282 < 2e-16 ***
region[T.JMK] -1.365e+03 6.971e+02 -1.958 0.05028
region[T.KK] -2.422e+03 8.621e+02 -2.809 0.00497 **
region[T.KV] -5.905e+03 1.004e+03 -5.879 4.14e-09 ***
region[T.KVK] -1.647e+03 1.191e+03 -1.383 0.16670
region[T.LK] -1.289e+03 9.089e+02 -1.418 0.15620
region[T.MK] -3.432e+03 7.382e+02 -4.650 3.33e-06 ***
region[T.OK] -5.484e+03 9.453e+02 -5.801 6.61e-09 ***
region[T.PHA] -1.266e+03 6.426e+02 -1.970 0.04881 *
region[T.PK] -4.222e+03 7.562e+02 -5.583 2.37e-08 ***
region[T.SK] 6.726e+02 7.633e+02 0.881 0.37823
region[T.UK] -9.768e+02 7.817e+02 -1.249 0.21150
region[T.ZK] -3.605e+03 8.547e+02 -4.218 2.47e-05 ***
x1:region[T.JMK] 1.026e-02 3.195e-03 3.211 0.00132 **
x1:region[T.KK] 2.587e-02 8.474e-03 3.053 0.00227 **
x1:region[T.KV] 2.132e-01 8.621e-03 24.730 < 2e-16 ***
x1:region[T.KVK] 4.689e-02 1.533e-02 3.059 0.00222 **
x1:region[T.LK] 5.164e-03 5.198e-03 0.993 0.32051
x1:region[T.MK] 1.146e-01 3.467e-03 33.062 < 2e-16 ***
x1:region[T.OK] 1.529e-01 6.404e-03 23.883 < 2e-16 ***
x1:region[T.PHA] 4.890e-02 2.995e-03 16.327 < 2e-16 ***
x1:region[T.PK] 1.550e-01 4.939e-03 31.386 < 2e-16 ***
x1:region[T.SK] -1.815e-02 3.653e-03 -4.968 6.78e-07 ***
x1:region[T.UK] 5.045e-02 4.249e-03 11.872 < 2e-16 ***
x1:region[T.ZK] 1.016e-01 5.441e-03 18.680 < 2e-16 ***
Source: Author
Residual standard error: 35,980 on 67,466 degrees of freedom.
Multiple R-squared: 0.9949 Adjusted R-squared: 0.9949
F-statistic: 5.31e+05 on 25 and 67466 DF, p-value: < 2.2e-16
The production function for region JCK is:
(8)
The coefficients next to the dummy variables indicate how much the initial
average production changes during the transition from region JCK to another
region (it mostly declines, with the most significant decrease being TCZK 5,905
~ 13 ~
Математичне моделювання в економіці, №1, 2015
when moving from region JCK to region KV). However, what is more important
are the changes to absolute flexibility that occur with regards to the product
variables (the last twelve lines). For all the regions, except for region SK, the
absolute flexibility is higher than in region JCK. Compared to region JCK, the
highest increase (by TCZK 213) is represented by region KV. An overview of the
resulting parameters in TCZK for the production function is given
in Table 4. From this overvew it is clear that most of the regression parameters are
statistically significant.
Table4 – Overview of resulting parameters in TCZK for theproduction function
region change b0 b0 change b1 b1
JCK 0 4739 0 1.169
JMK -1365 3374 0.010 1.179
KK -2422 2317 0.026 1.195
KV -5905 -1166 0.213 1.382
KVK -1647 3092 0.047 1.216
LK -1289 3450 0.005 1.174
MK -3432 1307 0.115 1.284
OK -5484 -745 0.152 1.321
PHA -1266 3473 0.049 1.218
PK -4222 517 0.155 1.324
SK 673 5412 -0.018 1.151
UK -977 3762 0.050 1.219
ZK -3605 1134 0.102 1.271
Source: Author
The results of the conditions verification for the use of this model are
analogous to those of the first model.
A brief description follows of the results of the linear single-factor model for
production y dependence on the other production factors. The comments on the
results are analogous to the previous models.
3. Linear single-factor model for production y dependence on production
factor x2
The result of the LSM application to production factor x2 is given in Table 5.
Table 5 – Results of the LSM application to production factor x2
Estimate Std. error t-value Pr(>|t|)
(Intercept) 12087.12 1097.00 11.02 <2e-16 ***
x2 11.55 0.03 384.90 <2e-1 ***
~ 14 ~
Математичне моделювання в економіці, №1, 2015
Source: Author
Residual standard error: 283,100 on 67,490 degrees of freedom
Multiple R-squared: 0.687 Adjusted R-squared: 0.687
F-statistic: 1.482e+05 on 1 and 67490 DF, p-value: < 2.2e-16
The single-factor model equation can be estimated again in the program R
using the LSM. The resulting parameters are presented in Table 6.
Table 6 – Resulting parameters of the production function for production factor x2
according to the individual years in the reference period
Estimate Std. error t-value Pr(<|t|)
(Intercept) 7.600e+05 5.354e+04 14.195 < 2e-16 ***
x2 -1.634e+01 3.162e+00 -5.166 2.39< 2e-07 ***
year_k[T.2003] -7.138e+05 5.367e+04 -13.300 < 2e-16 ***
year_k[T.2004] -7.534e+05 5.364e+04 -14.046 < 2e-16 ***
year_k[T.2005] -7.398e+05 5.362e+04 -13.797 < 2e-16 ***
year_k[T.2006] 7.581e+05 5.361e+04 -14.140 < 2e-16 ***
year_k[T.2007] -7.560e+05 5.360e+04 -14.104 < 2e-16 ***
year_k[T.2008] -7.576e+05 5.360e+04 -14.136 < 2e-16 ***
year_k[T.2009] -7.589e+05 5.360e+04 -14.160 < 2e-16 ***
year_k[T.2010] -7.586e+05 5.360e+04 -14.151 < 2e-16 ***
year_k[T.2011] -7.608e+05 5.389e+04 -14.117 < 2e-16 ***
x2: year_k[T.2003] 2.111e+01 3.162e+00 6.674 2 <2.51e-11 ***
x2: year_k[T.2004] 3.033e+01 3.164e+00 9.586 < 2e-16 ***
x2: year_k[T.2005] 3.001e+01 3.165e+00 9.481 < 2e-16 ***
x2: year_k[T.2006] 3.149e+01 3.163e+00 9.956 < 2e-16 ***
x2: year_k[T.2007] 3.054e+01 3.163e+00 9.656 < 2e-16 ***
x2: year_k[T.2008] 2.977e+01 3.163e+00 9.412 < 2e-16 ***
x2: year_k[T.2009] 2.847e+01 3.162e+00 9.002 < 2e-16 ***
x2: year_k[T.2010] 2.788e+01 3.162e+00 8.817 < 2e-16 ***
x2: year_k[T.2011] 2.903e+01 3.163e+00 9.179 < 2e-16 ***
Source: Author
Residual standard error: 254,400 on 67,472 degrees of freedom
Multiple R-squared: 0.7472 Adjusted R-squared: 0.7471
F-statistic: 1.05e+04 on 19 and 67472 DF, p-value: < 2.2e-16
The results of the linear single-factor model for production y dependence on
production factor x2 and the region are shown in Table 7.
Table7 – Resulting parameters of the production function for production factor x2
according to the individual regions
Estimate Std. error t-value Pr(>|t|)
(Intercept) 10028.5392 3969.9714 2.526 0.0115 *
~ 15 ~
Математичне моделювання в економіці, №1, 2015
x2 9.7158 0.2067 47.011 < 2e-16 ***
region[T.JMK] -8169.2802 4794.1515 -1.704 0.0884
region[T.KK] -5034.6940 5923.3758 -0.850 0.3953
region[T.KV] -318.2433 6797.4757 -0.047 0.9627
region[T.KVK] -163.2542 8185.7353 -0.020 0.9841
region[T.LK] -2629.4228 6252.4120 -0.421 0.6741
region[T.MK] 25197.9974 5052.9068 4.987 6.15e-07 ***
region[T.OK] 11621.8814 6383.6814 1.821 0.0687
region[T.PHA] 7313.7758 4412.2246 1.658 0.0974
region[T.PK] -4971.1854 5193.9058 -0.957 0.3385
region[T.SK] -6915.7810 5273.9969 -1.311 0.1898
region[T.UK] -3053.9825 5371.1556 -0.569 0.5696
region[T.ZK] 5887.2073 5872.6174 1.002 0.3161
x2:region[T.JMK] 5.8584 0.2441 24.001 < 2e-16 ***
x2:region[T.KK] 1.3965 0.6113 2.284 0.0224 *
x2:region[T.KV] -0.4954 0.4454 -1.112 0.2660
x2:region[T.KVK] -1.2031 0.9804 -1.227 0.2197
x2:region[T.LK] 3.7006 0.4658 7.945 1.97e-15 ***
x2:region[T.MK] -6.3060 0.2163 -29.150 < 2e-16 ***
x2:region[T.OK] -3.1054 0.3047 -10.191 < 2e-16 ***
x2:region[T.PHA] 3.6460 0.2090 17.446 < 2e-16 ***
x2:region[T.PK] 1.4353 0.3319 4.325 1.53e-05 ***
x2:region[T.SK] 4.3173 0.3231 13.362 < 2e-16 ***
x2:region[T.UK] 1.4987 0.3009 4.981 6.35e-07 ***
x2:region[T.ZK] -0.3238 0.4061 -0.797 0.4253
Source: Author
Residual standard error: 246,600 on 67,466 degrees of freedom
Multiple R-squared: 0.7626 Adjusted R-squared: 0.7625
F-statistic: 8670 on 25 and 67466 DF, p-value: < 2.2e-16
4. Linear single-factor model for production y dependence on production
factor x3
The results of the application of the LSM to production factor x3 are given in Table
8.
Table 8 – Results of the application of the LSM to production factor x3
Estimate Std. error t-value Pr(>|t|)
(Intercept) 15664.223 1185.551 13.21 <2e-16 ***
x3 46.855 0.137 342.01 <2e-16 ***
Source: Author
Residual standard error: 306,000 on 67,490 degrees of freedom
Multiple R-squared: 0.6341 Adjusted R-squared: 0.6341
F-statistic: 1.17e+05 on 1 and 67490 DF, p-value: < 2.2e-16
~ 16 ~
Математичне моделювання в економіці, №1, 2015
The single-factor model equation is estimated again in the program R using
the LSM. The resulting parameters are presented in Table 9.
Table 9 – Resulting parameters of the production function for production factor x3
according to the individual years in the reference period
Estimate Std. error t-value Pr(<|t|)
(Intercept) 2.690e+04 5.283e+04 0.509 0.610638
x3 4.758e+01 8.615e-01 55.228 < 2e-16 ***
year_k[T.2003] -1.634e+04 5.302e+04 -0.308 0.757926
year_k[T.2004] -1.795e+04 5.297e+04 -0.339 0.734779
year_k[T.2005] -1.971e+04 5.295e+04 -0.372 0.709761
year_k[T.2006] -1.795e+04 5.293e+04 -0.339 0.734528
year_k[T.2007] -4.710e+03 5.292e+04 -0.089 0.929079
year_k[T.2008] -2.688e+03 5.291e+04 -0.051 0.959493
year_k[T.2009] -1.208e+04 5.291e+04 -0.228 0.819355
year_k[T.2010] -1.282e+04 5.292e+04 -0.242 0.808608
year_k[T.2011] -2.047e+04 5.333e+04 -0.384 0.701012
x3: year_k[T.2003] 1.154e+01 1.083e+00 10.657 < 2e-16 ***
x3: year_k[T.2004] 1.142e+01 1.021e+00 11.189 < 2e-16 ***
x3: year_k[T.2005] 9.608e+00 1.021e+00 11.189 < 2e-16 ***
x3: year_k[T.2006] 1.242e+01 9.691e-01 12.818 < 2e-16 ***
x3: year_k[T.2007] -2.720e+00 9.462e-01 -2.875 0.004040 **
x3: year_k[T.2008] -7.875e+00 9.270e-01 -8.495 < 2e-16 ***
x3: year_k[T.2009] -7.197e+00 9.154e-01 -7.862 3.85e-15 ***
x3: year_k[T.2010] -9.389e+00 9.280e-01 -10.118 < 2e-16 ***
x3: year_k[T.2011] 3.438e+00 9.767e-01 3.520 0.000431 ***
Source: Author
Residual standard error: 297,500 on 67,472 degrees of freedom
Multiple R-squared: 0.6544 Adjusted R-squared: 0.6543
F-statistic: 6724 on 19 and 67472 DF, p-value: < 2.2e-16
The results of the linear single-factor model for production y dependence on
production factor x3 and the region are given in Table 10.
Table 10 – Resulting parameters of the production function for production factor x3
according to the individual regions
Estimate Std. error t-value Pr(>|t|)
(Intercept) 12497.7201 4595.1744 2.720 0.00653 **
x3 45.1346 1.2388 36.433 < 2e-16 ***
region[T.JMK] 24710.4797 5535.5871 4.464 8.06e-06 ***
region[T.KK] -2580.7511 6794.7052 -0.380 0.70408
region[T.KV] 12672.1787 7746.6942 1.636 0.10188
~ 17 ~
Математичне моделювання в економіці, №1, 2015
region[T.KVK] 6186.4484 9465.0203 0.654 0.51336
region[T.LK] 19577.5231 7174.8805 2.729 0.00636 **
region[T.MK] 12562.4482 5848.4599 2.148 0.03172 *
region[T.OK] -3255.9070 7528.1553 -0.432 0.66538
region[T.PHA] 7267.2783 5105.1992 1.424 0.15459
region[T.PK] 9521.9270 5959.9292 1.598 0.11012
region[T.SK] -3366.2787 6067.6280 -0.555 0.57904
region[T.UK] 13776.5131 6184.6876 2.228 0.02592 *
region[T.ZK] 15453.6051 6686.1185 2.311 0.02082 *
x3:region[T.JMK] -22.7433 1.3178 -17.258 < 2e-16 ***
x3:region[T.KK] 0.1404 3.0459 0.046 0.96325
x3:region[T.KV] -18.5642 2.0231 -9.176 < 2e-16 ***
x3:region[T.KVK] -12.8074 6.9697 -1.838 0.06613
x3:region[T.LK] -33.8729 1.6180 -20.936 < 2e-16 ***
x3:region[T.MK] -16.5264 1.3171 -12.548 < 2e-16 ***
x3:region[T.OK] 23.0879 3.3662 6.859 7.00e-12 ***
x3:region[T.PHA] 9.0754 1.2479 7.272 3.57e-13 ***
x3:region[T.PK] -15.7783 1.5823 -9.972 < 2e-16 ***
x3:region[T.SK] 11.5434 1.6022 7.205 5.87e-13 ***
x3:region[T.UK] -17.5910 1.5401 -11.422 < 2e-16 ***
x3:region[T.ZK] -23.7210 1.4542 -16.312 < 2e-16 ***
Source: Author
Residual standard error: 284,800 on 67,466 degrees of freedom
Multiple R-squared: 0.6832 Adjusted R-squared: 0.6831
F-statistic: 5820 on 25 and 67466 DF, p-value: < 2.2e-16
5. Linear single-factor model for production y dependence on production
factor x4
The results of the application of the LSM to production factor x4 are given in Table
11.
Table 11 – The results of the application of the LSM to production factor x4
Estimate Std. error t-value Pr(>|t|)
(Intercept) 4.121e+04 1.693e+03 24.35 <2e-16 ***
x4 1.654e+01 1.105e-01 149.67 <2e-16 ***
Source: Author
Residual standard error: 438,400 on 67,490 degrees of freedom
Multiple R-squared: 0.2492 Adjusted R-squared: 0.2492
F-statistic: 2.24e+04 on 1 and 67490 DF, p-value: < 2.2e-16
The single-factor model equation is estimated in the program R using the
LSM. The resulting parameters are given in Table 12.
~ 18 ~
Математичне моделювання в економіці, №1, 2015
Table 12 – Resulting parameters of the production function for production factor x4
according to the individual years in the reference period
Estimate Std. error t-value Pr(<|t|)
(Intercept) -466066.99 78477.25 -5.939 2.88e-09 ***
x4 424.43 11.78 36.023 < 2e-16 ***
year_k[T.2003] 523006.24 78718.49 6.644 3.08e-11 ***
year_k[T.2004] 499575.35 78664.24 6.351 2.16e-10 ***
year_k[T.2005] 501582.58 78631.73 6.379 1.80e-10 ***
year_k[T.2006] 492918.73 78612.39 6.270 3.63e-10 ***
year_k[T.2007] 508932.28 78612.39 6.476 9.51e-11 ***
year_k[T.2008] 512025.04 78587.12 6.515 7.30e-11 ***
year_k[T.2009] 507968.62 78583.01 6.464 1.03e-10 ***
year_k[T.2010] 501930.24 78597.36 6.386 1.71e-10 ***
year_k[T.2011] 478717.01 79133.67 6.049 1.46e-09 ***
x4: year_k[T.2003] -414.23 11.79 -35.140 < 2e-16 ***
x4: year_k[T.2004] -414.23 11.79 -33.779 < 2e-16 ***
x4: year_k[T.2005] -398.27 11.79 -34.467 < 2e-16 ***
x4: year_k[T.2006] -406.20 11.79 -32.850 < 2e-16 ***
x4: year_k[T.2007] -387.29 11.79 -34.605 < 2e-16 ***
x4: year_k[T.2008] -410.87 11.79 -34.862 < 2e-16 ***
x4: year_k[T.2009] -414.01 11.79 -35.133 < 2e-16 ***
x4: year_k[T.2010] -410.13 11.79 -34.799 < 2e-16 ***
x4: year_k[T.2011] -375.80 11.81 -31.826 < 2e-16 ***
Source: Author
Residual standard error: 417,200 on 67,472 degrees of freedom
Multiple R-squared: 0.3202 Adjusted R-squared: 0.32
F-statistic: 1672 on 19 and 67472 DF, p-value: < 2.2e-16
The results of the linear single-factor model for production y dependence on
production factor x4 and the regions are given in Table 13.
Table 13 – The resulting parameters of the production function for production
factor x4 according to the individual regions
Estimate Std. error t-value Pr(>|t|)
(Intercept) 30474.2785 6878.5812 4.430 9.42e-06 ***
x4 22.0101 1.1595 18.983 < 2e-16 ***
region[T.JMK] 12549.1052 8306.1418 1.511 0.13084
region[T.KK] -7824.0283 10143.4349 -0.771 0.44051
region[T.KV] 9197.5592 11534.7329 0.797 0.42523
region[T.KVK] -3201.0797 13666.7446 -0.234 0.81481
region[T.LK] 3680.5467 10782.3806 0.341 0.73284
~ 19 ~
Математичне моделювання в економіці, №1, 2015
region[T.MK] 1629.7708 8798.4523 0.185 0.85305
region[T.OK] 1224.2068 11135.6782 0.110 0.91246
region[T.PHA] 30473.9353 7660.2720 3.978 6.95e-05 ***
region[T.PK] 6020.2783 8923.0732 0.675 0.49988
region[T.SK] 8920.6082 9077.8329 0.983 0.32577
region[T.UK] 8550.1696 9281.7580 0.921 0.35696
region[T.ZK] 3099.7348 10063.4451 0.308 0.75807
x4:region[T.JMK] -10.8739 1.1924 -9.119 < 2e-16 ***
x4:region[T.KK] -0.9454 3.4983 -0.270 0.78696
x4:region[T.KV] -14.7233 1.4047 -10.482 < 2e-16 ***
x4:region[T.KVK] -16.6689 1.8883 -8.828 < 2e-16 ***
x4:region[T.LK] -15.1489 1.3772 -11.000 < 2e-16 ***
x4:region[T.MK] -0.7168 1.3206 -0.543 0.58726
x4:region[T.OK] 8.6624 3.0027 2.885 0.00392 **
x4:region[T.PHA] -2.4618 1.1677 -2.108 0.03501 *
x4:region[T.PK] -8.7781 1.5818 -5.549 2.88e-08 ***
x4:region[T.SK] -12.9655 1.2073 -10.739 < 2e-16 ***
x4:region[T.UK] -8.0794 1.6312 -4.953 7.33e-07 ***
x4:region[T.ZK] -5.3507 1.5395 -3.476 0.00051 ***
Source: Author
Residual standard error: 432,400 on 67,466 degrees of freedom
Multiple R-squared: 0.2699 Adjusted R-squared: 0.2697
F-statistic: 997.8 on 25 and 67466 DF, p-value: < 2.2e-16
6. Linear single-factor model for production y dependence on production
factor x5
The results of the application of the LSM to production factor x5 are given in Table
14.
Table 14 – The results of the application of the LSM to production factor x5
Estimate Std. error t-value Pr(>|t|)
(Intercept) 4.979e+04 1.901e+03 26.20 <2e-16 ***
x5 4.689e+01 7.396e- 63.40 <2e-16 ***
Source: Author
Residual standard error: 491,500 on 67,490 degrees of freedom
Multiple R-squared: 0.05621 Adjusted R-squared: 0.0562
F-statistic: 4020 on 1 and 67490 DF, p-value: < 2.2e-16
The single-factor model equation is estimated in the program R using the
LSM. The resulting parameters are given in Table 15.
~ 20 ~
Математичне моделювання в економіці, №1, 2015
Table 15 – The resulting parameters of the production function for production
factor x5 according to the individual years in the reference period
Estimate Std. error t-value Pr(<|t|)
(Intercept) 729390.6 99265.6 7.348 2.04e-13 ***
x5 -581.5 241.4 -2.409 0.015992 *
year_k[T.2003] -670336.4 99527.0 -6.735 1.65e-11 ***
year_k[T.2004] -672406.0 99466.8 -6.760 1.39e-11 ***
year_k[T.2005] -671426.2 99431.8 -6.753 1.46e-11 ***
year_k[T.2006] -674466.9 99410.9 -6.785 1.17e-11 ***
year_k[T.2007] -682290.2 99389.5 -6.865 6.72e-12 ***
year_k[T.2008] -689796.7 99384.5 -6.941 3.94e-12 ***
year_k[T.2009] -687866.6 99379.9 -6.922 4.51e-12 ***
year_k[T.2010] -682673.9 99394.8 -6.868 6.55e-12 ***
year_k[T.2011] -711993.9 99978.3 -7.121 1.08e-12 ***
x5: year_k[T.2003] 650.3 241.4 2.693 0.007075 **
x5: year_k[T.2004] 620.3 241.4 2.570 0.010185 *
x5: year_k[T.2005] 615.3 241.4 2.549 0.010807 *
x5: year_k[T.2006] 620.0 241.4 2.568 0.010217 *
x5: year_k[T.2007] 637.3 241.4 2.640 0.008293 **
x5: year_k[T.2008] 650.8 241.4 2.696 0.007016 **
x5: year_k[T.2009] 627.2 241.4 2.598 0.009370 **
x5: year_k[T.2010] 604.6 241.4 2.505 0.012265 *
x5: year_k[T.2011] 888.9 241.5 3.680 0.000233 ***
Source: Author
Residual standard error: 486,400 on 67,472 degrees of freedom
Multiple R-squared: 0.07596 Adjusted R-squared: 0.0757
F-statistic: 291.9 on 19 and 67472 DF, p-value: < 2.2e-16
The results of the linear single-factor model for production y dependence on
production factor x5 and the regions are given in Table 16.
Table 16 – The resulting parameters of the production function for production
factor x5 according to the individual regions
Estimate Std. error t-value Pr(>|t|)
(Intercept) 15315.416 7781.765 1.968 0.04906 *
x5 213.256 12.011 17.756 < 2e-16 ***
region[T.JMK] 10699.029 9355.929 1.144 0.25281
region[T.KK] 2565.560 11365.226 0.226 0.82141
region[T.KV] 9700.561 13138.837 0.738 0.46033
region[T.KVK] 8093.617 15707.174 0.515 0.60636
region[T.LK] 6384.239 12077.346 0.529 0.59708
~ 21 ~
Математичне моделювання в економіці, №1, 2015
region[T.MK] 14345.605 9900.675 1.449 0.14736
region[T.OK] 7443.801 12512.836 0.595 0.55192
region[T.PHA] 61624.631 8639.664 7.133 9.94e-13 ***
region[T.PK] 26911.970 9990.610 2.694 0.00707 **
region[T.SK] 10017.302 10203.453 0.982 0.32622
region[T.UK] 18136.686 10431.187 1.739 0.08209
region[T.ZK] 17687.625 11335.737 1.560 0.11868
x5:region[T.JMK] -25.694 12.694 -2.024 0.04297 *
x5:region[T.KK] -102.818 17.683 -5.815 6.10e-09 ***
x5:region[T.KV] -49.488 22.262 -2.223 0.02622 *
x5:region[T.KVK] -111.359 47.026 -2.368 0.01789 *
x5:region[T.LK] -132.696 12.864 -10.315 < 2e-16 ***
x5:region[T.MK] -42.089 14.162 -2.972 0.00296 **
x5:region[T.OK] 6.939 21.838 0.318 0.75066
x5:region[T.PHA] -162.319 12.051 -13.469 < 2e-16 ***
x5:region[T.PK] -204.096 12.072 -16.906 < 2e-16 ***
x5:region[T.SK] -74.997 12.766 -5.874 4.26e-09 ***
x5:region[T.UK] -82.646 15.458 -5.346 9.00e-08 ***
x5:region[T.ZK] -95.747 17.291 -5.537 3.08e-08 ***
Source: Author
Residual standard error: 479,500 on 67,466 degrees of freedom
Multiple R-squared: 0.1024 Adjusted R-squared: 0.102
F-statistic: 307.8 on 25 and 67466 DF, p-value: < 2.2e-16
Conclusion
The objective of the paper was to determine the production curves for the
construction industry in the Czech Republic and in its individual regions. The
objective was achieved. The result is a production function for the construction
industry at a regional and national level.
A tool is now at our disposal which allows us to predict the development of
the construction industry in the Czech Republic in the years to come. By
continuing to predict the values for the coming years we can subsequently compare
them with the actual results in the field. In so doing, we can determine the degree
of accuracy of the predictions for production and the development of the
construction industry over time, both at regional and national levels.
Стаття надійшла до редакції 26.02.2015
Introduction
Introduction
1. Material and methodology
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