3.1 Specification of a large- and medium-sized enterprise’s econometric model

Study of company’s production process, in large part, is falls within a realm of econometrics applicability. Productive process is a structure created by a person equipped with means of production. In market economy, it adapts the production profile to services and customer’s expectations. Requirement of rationality in all decision-making and actions means that this person must be aware of cause and effect relationships occurring between various relevant economic variables in a given enterprise.

In company-management practice, strong multilateral relationships between various economic processes exist. Such links occur only within a company. They are subject to influence of various external factors. A mechanism of economic links that occur in large- and medium-sized enterprise is presented in Figure 3.1.¹

Production is the final process described in a mechanism of inter-relations in a large (medium)² enterprise. This type of a company has a complex structure containing specialized services, which deal with individual elements occurring in Figure 3.1. The concept of production here will be understood as the sum of net⁴ sales income. Production of finished goods is subordinate to a generated demand for company’s goods and services, as observed by marketing services. Execution of liabilities for sold goods (services) is dealt with by debt recovery specialists. Therefore, a time interval between manufacture of goods (production of a service), its invoicing, and receiving a payment from a client is relatively short. It can be assumed that it is a period negligible from the perspective of a large enterprise’s operation time.

Production volume results from the impact of human labor, presented in the model by employment magnitude and efficiency of live-labor inputs, as well as from activity of services concentrated within the field specified as marketing. Marketing can be regarded as an instrument created within an enterprise, where it can be applied as a management tool used, among other things, for production volume formation.

Employment is a variable inside a company, which influences its final result. It is also a subject to influence of factors, some of which are located within the enterprise and other outside it. The second of important production factors – efficiency of labor input (labor productivity) – is similar in character. Efficiency depends on conditions inside the company as well as outside it. What draws attention is feedback between efficiency of live-labor input and competitiveness of working-conditions⁵ of an enterprise. This signifies simultaneous reciprocal influence of the pair of variables mentioned here. The level of technology and production organization, located within the term technological-organizational progress, is an important agent in efficacy of a labor factor.

Widely interpreted competitiveness of working conditions of an enterprise is shaped by many factors. Two particularly important ones are shown here: staff qualifications as well as the autonomous process of living-conditions improvement ongoing outside the enterprise,⁶ independently of its inside processes.

Employment volume is shaped, above all, by the available fixed assets. Complementarity and substitutability relations occur between employment and fixed assets. This means that increment of company’s fixed assets can cause employment increase (complementarity) or decrease (substitution). Generally, in an enterprise, complementarity and substitution occur simultaneously. Changes in employment volume are the outcome of these processes.

Enterprise’s fixed assets undergo physical and moral (economic) consumption; thus, it must be regenerated. It follows that assets consumption causes quantitative and qualitative changes in the mass of those assets. On the other hand, enterprise’s development requires investment outlays, which increase and modify the company’s existent fixed assets.

Investment outlays result from a general atmosphere in the economy and within the country, as those can cause inclinations to invest or to refrain from such projects. Economic growth, being a most synthetic conceptualization of given area’s economic conditions in a given time period, is an expression of that economic and national climate. The size and the structure of investment outlays are determined by needs and opportunities arising from a progressive consumption of fixed assets as well as by defined within company goals, which are later on translated into the language of company’s investment objectives.

3.2 Structural form of an econometric model of a large- and medium-sized enterprise

Econometric model’s endogenous variables describing a large-sized (medium-sized) enterprise can be as follows⁷:

• Y₁ – net sales income of an enterprise (in millions PLN)⁸ for a t (t = 1, …, n) time-period
• Y₂ – the average annual⁹ number of employees, in full-time equivalent,
• Y₃ – labor productivity calculated as the ratio¹⁰ of ,
• Y₄ – average monthly wage, in PLN monthly¹¹ for one employee,
• Y₅ – initial value of active fixed assets, in millions PLN,
• Y₆ – technical labor devices measured by the initial value of the fixed assets, for one employee (in thousands PLN/1 employee), adjusted by a shift coefficient,
• Y₇ – value of investment outlays in a t period, that is, in thousands PLN.

The following can be exogenous variables of a model for a large (medium) enterprise:

• X₁ – cost of marketing activity,¹²
• X₂ – production volume in a natural measure unit,¹³ that is, in thousands of tons,
• X₃ – number of manufactured product range; production entropy H_t in a t (t = 1, …, n) period can be an alternative and is calculated using the following formula:
  (3.1)
  where p_ti represents the share of ith product range¹⁴ in production value, whereas m_t is the number of manufactured assortments in a t period.
• X₄ – value of special-order production¹⁵ in millions PLN,
• X₅ – number of employees with higher education,¹⁶
• X₆ – annual depreciation of fixed assets¹⁷ in millions PLN,
• X₇ – GDP growth rate,¹⁸
• X₈ – number of the unemployed having qualifications necessary in the company, who are on the market on which the company functions,¹⁹
• X₉ – value of sales on new markets,²⁰ in millions PLN,
• X₁₀ – time variable²¹ t (t = 1, …, n).

Using Figure 3.1, hypothetical equations of structural-form econometric model for a large-sized enterprise can be written as follows²²:

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

In the system of theoretical structural-form equations,²³ Equations 3.2–3.8, symbols η₁, η₂, …, η₇ represent random components of each equation.

It can be noticed that the equation describing variable Y₇ has characteristics of a detached equation, since only explanatory variables having characteristics of predetermined ones can occur in that equation. A hypothetical system of reduced-form equations of the above model can be written as follows²⁴:

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

It is noticeable that in the system of Equations 3.9–3.15, each equation is characterized by an identical set of explanatory variables.²⁵

3.3 Empirical econometric model of a medium-sized enterprise

3.3.2 Equation of the sales income

A hypothetical equation describing a mechanism of the net sales income (SNET) took into account autoregression up to and including the fourth order. None of the autoregressive variants proved to be statistically significant. Labor productivity (EFEMP) in the form of an unbiased index of dynamics as well as the impact of an EFEMP variable delayed by 1, 2, 3, and 4 quarters was entered into the hypothetical equation. The delayed variables proved to be statistically insignificant. Similarly, variables representing employment (EMP) delayed by 1–4 quarters were insignificant. Equation of sales income also contained three variants of variables describing the company’s marketing activity. These were advertisement expenses, representative expenses, and the sum of advertisement and representative expenses. Simultaneously, those variables’ values as well as delays from 1 to 4 quarters were considered. Finally, the variable MARK(−1) representing advertisement expenses delayed by 1 quarter proved to be statistically significant. The results of parameters’ estimation of an empirical equation of net sales income are shown in Table 3.2. Graphically, estimation results are shown in Figure 3.2.

Dependent variable: SNET
Method: least squares
Date: 04/01/2014, time: 15:09
Sample (adjusted): 2008:2 2012:4
Included observations: 19 after adjusting endpoints
Variable	Coefficient	Standard error	t-Statistic	Probability
C	−17114.14	1621.608	−10.55381	0.0000
EFEMP	71.74060	4.018011	17.85476	0.0000
EMP	123.2627	7.381952	16.69784	0.0000
MARK(−1)	−9.898086	3.333035	−2.969692	0.0095
R-squared	0.962390	Mean dependent var		14445.00
Adjusted R-squared	0.954868	S.D. dependent var		3134.381
S.E. of regression	665.8806	Akaike info criterion		16.02476
Sum squared resid	665.0955	Schwarz criterion		16.22359
Log likelihood	−148.2352	F-statistic		127.9419
Durbin–Watson statistic	1.821667	Prob (F-statistic)		0.000000

As expected, labor productivity and employment volume have turned out to be simulators of the amount of net sales income. In contrast, advertisement expenses delayed by 1 quarter had negative influence on sales income. This probably resulted from a high intensity of advertisement expenses in the beginning of the study period, before 2008. Since 2010, advertisement expenses decreased significantly. A decrease in those expenses was accompanied by an increase in sales income, which probably resulted from previously established business relations.

3.3.3 Equation of employment

The company was characterized by high labor costs before 2010. Savings achieved by reduction in employment level were necessary. As a consequence of a downward trend in the number of employees – along with an established customer range – there was no impact of the fixed assets as well as of the demographic situation on the company’s area. Employment was characterized only by autoregressive processes of the first, second, and fourth order. Figure 3.3 presented actual quarterly employment, theoretical employment, and residuals calculated based on the equation from Table 3.3.

Dependent variable: EMP
Method: least squares
Date: 04/01/2014, time: 17:34
Sample(adjusted): 2009:1 2012:4
Included observations: 16 after adjusting endpoints
Variable	Coefficient	Standard error	t-Statistic	Probability
C	−19.07222	9.272138	−2.056938	0.0621
EMP(−1)	1.994066	0.179455	11.11179	0.0000
EMP(−2)	−1.372568	0.258369	−5.312427	0.0002
EMP(−4)	0.466590	0.159079	2.933073	0.0125
R-squared	0.993331	Mean dependent var		139.7500
Adjusted R-squared	0.991664	S.D. dependent var		35.43915
S.E. of regression	3.235600	Akaike info criterion		5.398624
Sum squared resid	125.6293	Schwarz criterion		5.591771
Log likelihood	−39.18899	F-statistic		595.8284
Durbin–Watson statistic	2.153871	Prob (F-statistic)		0.000000

Employment volatility in any enterprise is specified by inertia, which is manifested by autoregressive dependencies of various orders. Most frequently, there is a positive autoregression of the first order, which can be seen in the case of the analyzed company ENERGY. Negative autoregression of the second order primarily results from the mode of employment contracts’ termination, which is regulated by Labor Law. Employment equation was dropped from the chain that composes a recursive mechanism of the model. It became the so-called detached equation, meaning a specific type of an equation for the simple model.

3.3.4 Equation of labor productivity

Team-labor productivity in the company ENERGY is affected by a significant number of explanatory variables. There are autoregressive mechanisms of the first, second, and fourth order. A significant level of autoregressive dependency of the fourth order draws attention. There was no feedback between the average wage and labor productivity.²⁷ Only delayed impact of the average wage on productivity occurs. At the same time, the impact of average wage, delayed by 1 and 3 quarters, is negative. In contrast, impact of the average wage delayed by 2 quarters on labor productivity is positive. What draws attention is the bigger amount of negative coefficients than that of the positive one. This may signify defectiveness of a motivational mechanism dominating in the company. Lack of a simultaneous impact of the average wage on employees’ efficiency also confirms such thesis.

Negative impact of the technical equipment delayed by 4 quarters on current labor productivity is significant. This may result from the aimed at reduction of business operating costs reduction of fixed assets and employment. A positive trend in labor productivity is a favorable development. It turns out, that labor productivity has increased, on average, by 3000 PLN per one employee in 1 quarter.

An equation of labor productivity does not co-create the system of interdependent equations with an average wage equation. It remains, however, in an interdependency relationship of chain character, typical for a recursive model. Empirical results of modeling are provided in Table 3.4 and in Figure 3.4.

Dependent variable: EFEMP
Method: least squares
Date: 04/04/2014, time: 15:57
Sample (adjusted): 2009:1 2012:4
Included observations: 16 after adjusting endpoints
Variable	Coefficient	Standard error	t-Statistic	Probability
C	140.8303	54.57531	2.580477	0.0364
EFEMP(−1)	0.622322	0.133667	4.655762	0.0023
EFEMP(−2)	−0.481733	0.114267	−4.215850	0.0040
EFEMP(−4)	1.041329	0.120529	8.639625	0.0001
APAY(−1)	−5.644315	1.601560	−3.524261	0.0097
APAY(−2)	11.88797	2.266970	5.243989	0.0012
APAY(−3)	−8.988088	1.827948	−4.917036	0.0017
TAL(−4)	−2.241678	0.683094	−3.281654	0.0135
TIME	2.795396	1.125612	2.483446	0.0420
R-squared	0.977291	Mean dependent var		106.9183
Adjusted R-squared	0.951337	S.D. dependent var		31.68854
S.E. of regression	6.990365	Akaike info criterion		7.025264
Sum squared resid	342.0564	Schwarz criterion		7.459845
Log likelihood	−47.20211	F-statistic		37.65562
Durbin–Watson statistic	2.099497	Prob (F-statistic)		0.000045

3.3.5 Equation of the average wage

In an equation describing a mechanism of the average wage formation, autoregression of the first, second, and fourth orders as well as impact of labor productivity have occurred. Autoregression of the first order is positive, while autoregressions of the second and fourth order are negative in character. A tendency to lower the wages in the company by elimination of inertia in wages is seen. Appearance of a positive impact of labor productivity informs about untaken actions aimed at activation of “healthy” motivational role of wages, according to the principle higher wages for more productive work.

Accuracy of wage mechanism’s description in the equation is significantly lower , compared to the earlier presented empirical equations, in which R² significantly exceeded the level of 0.95. This may suggest a significant impact of other factors on the average wage level in the analyzed company.

Empirical results of the average wage modeling are presented in Table 3.5 and in Figure 3.5. Seasonal wage fluctuations in each quarter, with a tendency to increase, can be noticed.

Dependent variable: APAY
Method: least squares
Date: 04/01/2014, time: 17:46
Sample (adjusted): 2009:1 2012:4
Included observations: 16 after adjusting endpoints
Variable	Coefficient	Standard error	t-statistic	Probability
C	11.23787	2.765680	4.063329	0.0019
APAY(−1)	0.700296	0.163949	4.271424	0.0013
APAY(−2)	−0.477972	0.166112	−2.877415	0.0150
APAY(−4)	−0.678725	0.190013	−3.571994	0.0044
EFEMP	0.059336	0.011705	5.069207	0.0004
R-squared	0.768637	Mean dependent var		12.07144
Adjusted R-squared	0.684505	S.D. dependent var		1.988078
S.E. of regression	1.116682	Akaike info criterion		3.308907
Sum squared resid	13.71677	Schwarz criterion		3.550341
Log likelihood	−21.47126	F-statistic		9.136088
Durbin–Watson statistic	2.732821	Prob (F-statistic)		0.001667

3.3.6 Equation of the fixed assets

The company ENERGY was equipped with the fixed assets (FIXAS) through the deed of foundation of a holding to which it belongs. In years 2008–2012, a systematic and significant decrease in the value of that variable is observed. Equation describing the variable FIXAS has characteristics of a detached one, although, the model hypothesis assumed recursiveness of that endogenous variable’s contribution.

Table 3.6 and Figure 3.6 present a mechanism of the volatility of company’s fixed assets. In the equation, negative autoregression of the fourth order occurred, which resulted in systematic decrease in the variable FIXAS’s value. A negative tendency informing about an average quarterly decrease of fixed assets, by around 320 thousands PLN, occurs as well. At the same time, the value of fixed assets’ depreciation has a positive influence on the growth of tangible assets. This means that attention to replacement of fully exploited fixed assets occurs.

Dependent variable: FIXAS
Method: least squares
Date: 04/04/2014, time: 15:56
Sample (adjusted): 2009:1 2012:4
Included observations: 16 after adjusting endpoints
Variable	Coefficient	Standard error	t-Statistic	Probability
C	15.81238	1.011460	15.63324	0.0000
FIXAS(−4)	−0.249517	0.103405	−2.413016	0.0327
DEPR(−4)	0.003374	0.001431	2.358760	0.0361
TIME	−0.320004	0.014186	−22.55784	0.0000
R-squared	0.989627	Mean dependent var		10.05000
Adjusted R-squared	0.987034	S.D. dependent var		1.441296
S.E. of regression	0.164117	Akaike info criterion		−0.564158
Sum squared resid	0.323212	Schwarz criterion		−0.371011
Log likelihood	8.513264	F-statistic		381.6291
Durbin–Watson statistic	2.141813	Prob (F-statistic)		0.000000

3.3.7 Equation of the technical labor equipment

Technical labor equipment belongs to characteristics of technical development in an enterprise. Empirical equation describing the volatility mechanism of labor technical equipment (TAL) is presented in Table 3.7 and in Figure 3.7. From Figure 3.7 results a dynamic growth of a variable TAL after 2010.

Dependent variable: TAL
Method: Least squares
Date: 04/01/14 time: 18:10
Sample (adjusted): 2009:1 2012:4
Included observations: 16 after adjusting endpoints
Variable	Coefficient	Standard error	t-Statistic	Probability
C	−38.19772	19.68906	−1.940049	0.0935
TAL(−1)	3.219578	0.368494	8.737126	0.0001
TAL(−2)	−2.731791	0.407198	−6.708751	0.0003
TAL(−4)	0.876021	0.186679	4.692666	0.0022
FIXAS	11.97294	1.892871	6.325280	0.0004
FIXAS(−1)	−20.33832	2.661017	−7.643062	0.0001
FIXAS(−2)	14.41811	2.286484	6.305803	0.0004
FIXAS (−4)	−7.373084	1.407030	−5.240176	0.0012
DEPREMP(−3)	0.018297	0.003847	4.756120	0.0021
R-squared	0.996096	Mean dependent var		74.56903
Adjusted R-squared	0.991634	S.D. dependent var		11.88248
S.E. of regression	1.086854	Akaike info criterion		3.302773
Sum squared resid	8.268763	Schwarz criterion		3.737354
Log likelihood	−17.42219	F-statistic		223.2409
Durbin–Watson statistic	1.991177	Probability (F-statistic)		0.000000

Table 3.7 indicates that the variable TAL is characterized by autoregression of the first, second, and fourth order. Autoregression of the first and fourth order is positive, while autoregression of the second order is negative. The sum of coefficients for the variables delayed by 1 and 4 quarters significantly exceeds the negative value of a coefficient for the variable TAL delayed by 2 quarters. This results in an increase in the level of technical labor equipment.

The value of tangible assets’ depreciation, delayed by 3 quarters exerted a positive impact on the company’s labor technical equipment. Consumption of fixed assets forced positive adjustments of tangible assets. It resulted in an increase in the value of labor technical equipment.

The equation in Table 3.7 is characterized by very good stochastic properties. The value of coefficient R² is very high and the value of Durbin–Watson’s statistic is close to 2. Such values of stochastic characteristics often occur in case of equations of econometric micromodels.

The above presented empirical multiple-equation micromodel of an average enterprise can be a useful tool in estimation of forecasts of economic variables that are important for the company. It can be used for current decision-making. Decisions prepared in such a way will bear a relatively low risk of inaccuracies.

Collection of statistical information about the variables presented in Section 3.2 will increase security of empirical model’s application. Information enrichment of the company’s model should bring significant benefits for the managing business entity.²⁸

3.4 Application of the company’s model during a decision-making process

A multiple-equation model of an enterprise can be applied multidirectionally in that company’s decision-making processes. Most commonly, the model or its equations are useful in forecast estimation of economic variables. It is also possible to conduct a simulation of the results of various decisions, both in a given equation as well as in the enterprise’s entire system described in Figure 3.1.

Cases when an empirical equation is autoregressive in character with a trend are relatively uncomplicated. It signifies inertia of the endogenous variable. In such a case, it is easy to estimate a forecast for the period following the last observation or for few successive periods. Let us consider the empirical equation of employment presented in Table 3.3.

Predictor will have the following form:

(3.16)

Forecast of employment volumes for the first quarter of 2013 will be calculated as follows:

The average prediction error, calculated using formula 1.55, here is equal to persons. The relative prediction error, calculated using formula 1.58, is equal to . Typical value of the relative limiting prediction error is often equal to . Assuming such a limiting value, employment forecast for the first quarter of 2013 can be deemed as admissible, because the following inequality occurs:

The results of employment forecast estimation for the first and the second quarter of 2013 in the company ENERGY are presented in Figure 3.8.

Employment forecast for the second quarter of 2013 will be as follows:

The average prediction error, calculated using formula 1.55, here is equal to persons. The relative prediction error, calculated using formula 1.58, is equal to . Forecast for the second quarter of 2013 is characterized by relatively small precision. The following inequality occurs:

This signifies an inadmissible forecast. However, further declines in employment volumes can be expected. After the first quarter of 2013, evaluation of the prognosis’ relevance will be necessary. It may turn out that respecification of the employment equation, done through reconsideration of the hypothetical mechanism from Figure 3.1, will also is required. In case such an attempt fails, it is worth to consider changing the analytical form of the employment equation. Application of one of the nonlinear forms of the employment equation can prove to be an effective way of establishing its good predictive qualities.

The empirical model of a medium-sized enterprise presented in this chapter is recursive in character. Hypothetical feedback was not sustained; it was “broken.” The following chains of links between the total interdependent variables occur in this model:

(3.17)

Consequently, it is necessary to use chain prediction, interspersed with sequential prediction resulting from the delays of the endogenous variables. Possibilities of estimating the forecasts are dependent on the following necessities:

1. SNET_Tp is having forecasts of EMP_Tp and EFEMP_Tp,
2. APAY_Tp requires having the forecast of EFEMP_Tp,
3. TAL_Tp can be obtained while knowing the forecast of FIXAS_Tp.

The entire predictive proceeding requires a lot of attention to take into account both the delays of the system’s endogenous variables as well as the existing chains of links forming a recursive mechanism.

Let us take a look at the prediction process in the following chain of links:

We have at our disposal previously estimated forecasts of EMP_2013.1p and EMP_2013.1p, with an indication that only the forecast of EMP_2013.1p is admissible. Possibility of estimating the forecasts of SNET_Tp will only appear when forecasts of EFEMP_Tp are estimated in before. As such, it is necessary to estimate the forecasts of EFEMP_Tp using the predictor from the equation in Table 3.4:

(3.18)

The values of the variable APAY delayed by 1, 2, and 3 quarters are provided for the first forecasted period. Successive delays can be estimated from a predictor equation for the equation in Table 3.5, while having forecasts of EFEMP_Tp. Figure 3.9 presented forecasts of labor efficiency (EFEMP_Tp) in the company ENERGY obtained for three quarters of 2013.

Relative accuracy of the forecasts of EFEMP_Tp can be noticed. As a result, only forecast estimation for the first forecasted period, that is for the first quarter of 2013, will be relatively safe. The predictor of SNET_Tp will be formed on the basis of the equation in Table 3.2:

(3.19)

Figure 3.10 illustrates a forecast compared with the actual values of the sales income in the past.

Forecast of the sales income is characterized by a moderately high prediction error, which is equal to

The estimated forecast is characterized by higher relative prediction error than the previously established limiting error. However, it is worth to consider large volatility amplitude of the variables of a streaming character in each of the company’s models that are based on quarterly or monthly data. Therefore, the relative limiting prediction error in such cases is often established at the level of . With such an assumption, forecast thousands PLN becomes admissible,²⁹ because . In such a case, management of the company ENERGY obtains information that with such formulated assumptions, a decline in the company’s net sales income in the first quarter of 2013, even down to the level of 8 million PLN, should be expected. Having this information allows the company to adequately prepare for anticipated characteristics of the forecasted variables. These forecasts can be regarded as a warning, if they are contrary to the company’s strategy. Some actions can, therefore, be undertaken for the forecasted variables’ realizations to form at more optimistic levels. This challenge can only be met while having an empirical econometric model of the company.