Введение

In recent years, the number of public contracts for the purchases of laptops has increased. Due to the increasing number of disruptions in the public procurements system, a large number of scientific researches are being conducted into the background causes and for avoiding negative effects of an inefficient use of budgetary funds. Single and clear mechanism of monitoring activities during the state tenders has not yet been developed in Russia. So Rybnikova G.I. states that it is important «to study those stages of public procurement at which the supervisory authorities have difficulties» [Rybnikova G.I., Tevosyan K.M., 2016].

One of the key stages of any public procurement is planning in which the major difficult lies in analysis the price calculation and justification of proposed procurements. Therefore, the purpose of this article is to analyze the factors affecting the initial (maximum) price of laptops at the planning stage of state and municipal procurements with subsequent econometric model specification.

• часть

The core of any econometric analysis is based on the correct determining of model parameters which are more likely accounted for estimated value.

Modern laptops are delicate pieces of technology that consist of many elements. The significant factors affecting the laptop’s price are listed in Table 1.

Table 1

Data, their designations and units of measurement

Таблица 1

Данные, их обозначения и единицы измерения

Fig. 1. Multiple Linear Regression (MLR) model with all variables

Рис. 1. Модель множественной линейной регрессии (МЛР)
со всеми объясняющими переменными

It is necessary to assess the quality of the resulting model for further analysis.

1. The significance of the coefficients of the explanatory variables. It is considered in the Gretl program that coefficient is significant at a significance level of 10%», if the achieved level of significance (p-value) of the coefficient is less than 0.1 [Econometrics. Regression analysis using the package Gretl: Laboratory Workshop, 2014]. This requirement is satisfied by the variables const, memory, HDD, pixel, videocard_performance, OS, keyboard, material, Nvidia, usb2 and usb3 satisfy. For the other variables, the p-value of the coefficients has turned out to be greater than 0.1, they were insignificant and therefore were excluded from further analysis.

2. t-statistics. Since the sample was 102 observations, so t-critical by t-Student is equal to 1.659 at a significance level of 10%. Comparing the obtained values, it appeared that the coefficients of const, memory, HDD, pixel, videocard_performance, OS, keyboard, material, Nvidia, usb2 and usb3 have t-statistics modulo more than t-critical, which indicates their statistical significance. And the remaining coefficients of CPU, core, SSD, screen_size, DVDRW, battery and AMD have t-statistics less than t-critical, it means they are not significant.

3. The significance of the regression equation in general according to Fisher’s F-test (P-value (F)). «If the P-value (F) is less than 0.01, the equitation is significant at a significance level of 1% (at an assurance level of 99%)» [Econometrics. Regression analysis using the package Gretl: Laboratory Workshop, 2014]. Because of the P-value(F)=2.33e-26<0.01, the regression equitation is significant and it can be used in further analysis.

4. Standard error of the estimate is 9006.38 RUB with an average laptop’s price at 50998.5 RUB (or 17.66%), which indicates the satisfactory accuracy of the model.

5. Goodness of fit to selected data by the adjustable coefficient of determination (adj. R-squared). Using the coefficient of determination, one can be defined «the matching rate of the found equation to the actual data. Adj. R-squares in this model was 0.9484, so the factor of laptop’s price change is explained sum of squares by 94.84%. Thus, the quality of the fit equation is very accurate.

6. Goodness of fit to selected data by the mean absolute percentage error (MAPE). In this regression equation was 16.4%. «If the model is fitted with high accuracy, so MAPE<10%, good – 10% <MAPE<20%, satisfactory – 20%<MAPE<50%, unsatisfactory – MAPE>50%». That was, the goodness of fit is good [Absolute approximation error, 2018].

The MLR model after eliminating insignificant explanatory variables is presented in Figure 2.

Fig. 2. MLR model after eliminating insignificant explanatory variables

Рис. 2. Модель МЛР после исключения незначимых объясняющих переменных

Adj. R-squared improved from 0.9484 to 0.9493. Thus, the eliminating insignificant explanatory variables has turned out to be true. All coefficients were significant (p-value less than 0.1).

At the next stage, «the existence of the strong correlation between explanatory variables» was determined [Econometrics. Regression analysis using the package Gretl: Laboratory Workshop, 2014], so multicollinearity test was performed (Fig. 3).

All values of variance inflation factors (VIF) of explanatory variables were less than 10; it has shown the absence of multicollinearity between these variables.

Further, the data were checked for the unequal spread (heteroscedasticity), to the White test for heteroscedasticity was carried out (Fig. 4).

Fig. 3. Multicollinearity test

Рис. 3. Тест на мультиколлинеарность

Fig. 4. White test for heteroscedasticity

Рис. 4. Тест Вайта на гетероскедастичность

Since the p-value was equal to 0.0337 and it was less than 0.05; this indicated the presence of heteroscedasticity. Therefore, calculations of robust errors were conducted, which has corrected the values of standard errors of the coefficient estimates. The MLR model adjusted on heteroscedasticity is shown in Figure 5.

Thus, from an economic point of view the interpretation of all coefficients of the explanatory variables was correct. All coefficients were significant. The standard error of the model was 8923.05 with a mean of 50998.5 (or 17.5%). Adjusted R-square has increased to 0.9484 (more than 0.9 is considered to be highly accurate) [Afanasev V.N., Semenychev E.V., 2014].

At the final stage, it was important to understand whether the specification of the model was correct or whether it was switching from a linear to a non-linear model. To this end, Regression Equation Specification Error Test (RESET-test Ramsey) was carried out on the correctness of the linear specification. RESET-test is shown in Figure 6.

Fig. 5. The model adjusted on heteroscedasticity

Рис. 5. Модель МЛР с поправкой на гетероскедастичность

Fig. 6. RESET-test

Рис. 6. RESET тест

Since p-value was equal to 0.0175 and less than 0.05; the MLR model in Figure 5 was presented in the wrong functional form (with a 95% assurance level). Therefore, non-linear terms were added to this regression equation, so a polynomial was considered (Fig. 7).

Polynomial model. Adding squared variables haven not helped to improve the goodness of the model: all variables were insignificant. As for the multiplication of explanatory variables, there were several significant coefficients (in this case, quantitative variables were multiplied by dummy):

1.  Multiplication of battery (quantitative) and Intel (dummy): battery_OS;

2.  Multiplication of SSD (quantitative) and Intel (dummy): SSD_Intel;

3.  Multiplication of CPU (quantitative) and OS (fictitious): CPU_OS;

4.  Multiplication of screen_size (quantitative) and Intel (dummy): screen_size_Intel.

When non-linear terms were introduced, it was also necessary to introduce the desired variables into the model, so as not to disturb the economic interpretation. The model with new non-linear variables (polynomial model) is presented in Figure 7.

Fig. 7. Polynomialmodel

Рис. 7. Полиномиальная модель

From an economic point of view the interpretations of all coefficients of the explanatory variables have turned out to be correct. Compared to the model in Figure 5, all coefficients were significant; the standard error of the estimate decreased from 8923.05 RUB to 8131.81 RUB (or 15.94%); adj. R-squared increased to 0.9579, which means the inclusion of new variables has turned out to be true. MAPE was evaluated to be 14.44% that was a little better than 16.4% in Figure 5. That was, the goodness of fit to selected data was good [Absolute approximation error, 2018].

Because of the result of RESET-test, the hypothesis of a well-chosen model specification has declined, models in nonlinear forms were considered: exponential, logarithmic, power-law, semi-logarithmic.

1. Exponential model: the dependent variable was represented by the logarithm, and the independent variables were in the original form. Verification of all problems: multicollinearity, heteroscedasticity, equality of coefficients. After their recovering the following model was obtained (Fig. 8).

Fig. 8. Exponentialmodel

Рис. 8. Экспоненциальная модель

All coefficient were significant. The standard error of estimate was low: 0.2106 with a mean of 10.6261 (or 2%). Adj. R-squared was equal to 0.8918 (about 0.9 is considered to be highly accurate) [Afanasev V.N., Semenychev E.V., 2014]. MAPE was 1.55%, that was significantly lower than 10%, therefore it indicated a high accuracy fit of the model to the sample data.

2. The logarithmic model: the dependent variable was presented in its original form, and the independent variables were represented by the logarithm (Fig. 9).

Fig. 9. Logarithmicmodel

Рис. 9. Логарифмическая модель

All coefficient were significant. The standard error of estimate was high: 19043.20 RUB with a mean of 50998.50 RUB (or 37.34%). Adj. R-squared was equal to 0.7693 (about 0.75 is considered to be the lower acceptable value) [Afanasev V.N., Semenychev E.V., 2014]. MAPE was 34.33%, so the model fit was satisfactory.

3. The power-law model: dependent and independent variables were presented in logarithm (Fig. 10).

Fig 10. The power-law model

Рис. 10. Степенная модель

All coefficient were significant. The standard error of estimate was low: 0.2471 with a mean of 10.6261 (or 2.32%). Adj. R-squared was equal to 0.8511 (less than 0.9 indicates a lack of high accuracy). MAPE was 1.87%, that was significantly lower than 10%, therefore it indicated a high accuracy fit of the model to the sample data.

4. The semi-logarithmic model is shown in Figure 11.

Fig. 11. The semi-logarithmic model

Рис.11. Полулогарифмическая модель

All coefficient were significant. The standard error of estimate was low: 0.1717 with a mean of 10.6261 (or 1.62%). Adj. R-squared was equal to 0.9281(more than 0.9 is considered to be highly accurate). MAPE was 1.21%, that was significantly lower than 10%,
therefore it indicated a high accuracy fit of the
model to the sample data.

The next stage, the comparison of the resulting models was carried out. Comparison is possible only if the models are presented in the same type of dependent variables, so in the same units of measurement. The comparative Table 2 is below.

Table 2

Comparative table on models

Таблица 2

Сравнительная таблица моделей

Thus, according to all indicators the polynomial model was better in the first case and the semi-logarithmic – in the second case. The choice of the best model was carried out to the non-nested models test (PE-test). Figure 12 shows the models with predictions of a competing model.

For the PE-test, the coefficient of the variable lin (the difference between the logarithm of the forecast of the polynomial model and the forecast of the semi-logarithmic model) and log (the difference between the forecast of the polynomial model and the exposed forecast of the semi-logarithmic) were calculated [Econometrics. Regression analysis using the package Gretl: Laboratory Workshop, 2014]. For the polynomial model, the coefficient for the variable lin has turned out to be significant (the p-value was 0.0041), so the model can be improved.

For the semi-log model, the coefficient of the variable log has also turned out to be significant (the p-value was 0.0009), so the model can be improved too. Since the coefficients in the both models are significant, it was not possible to make a definite conclusion. Therefore the models were compared by the value of significance. Since the p-value (0.0041) was larger in the polynomial model than in the semi-logarithmic model (0.0005), then the coefficient in the polynomial was less significant.

Fig. 12. PE-test results

Рис. 12. Результаты PE-теста

Thus, the polynomial model was chosen or further econometric analysis (Fig. 7). Let us give an economic interpretation of the coefficients of the explanatory variables in the total influence of all factors:

1. If the amount of RAM is increased by 1GB, the laptop’s price will increase on average by 2,462.94 RUB, other things being equal.
2. If the number of pixels is increased by 1000000, the laptop’s price will increase on average by 7168.90 RUB, other things being equal
3. If the performance of the video card is increased by 1% the laptop’s price will increase on average by 429.53 RUB, other things being equal.
4. If the number of usb-ports 2.0 is increased by 1 unit, the laptop’s price will increase on average by 5233.82 RUB, other things being equal.
5. If the number of usb-ports 3.x is increased by 1 unit, the laptop’s price will increase on average by 9,261.76 RUB, other things being equal.
6. If the battery life is increased by 1 hour, the laptop’s price will increase on average by 3330.49 RUB.
7. If the CPU’s frequency is increased by 1 GHz, the laptop’s price will increase on average by 11206.7 RUB.
8. The laptops with keyboard lightning will cost more on average by 5636.04 RUB than laptops with non-lightning.
9. The laptops with a metal case will cost more on average by 6,510.67 RUB than laptops with a plastic case.
10. The price of the laptops with the Windows OS is on average by 34994.60 RUB higher than other operating systems, provided that:

• if the battery life is increased by 1 hour, the price of the laptops with Windows OS will grow less on average by 3,258.75 RUB than laptops with another OS;
• if the CPU’s frequency is increased by 1 GHz, the price of the laptops with Windows OS will grow less on average by 8591.80 RUB than laptops with another OS;

1. If the volume of solid-state drive (SSD) is increased be 1 GB, the laptop’s price will increase on average by 46.99 RUB;
2. If the diagonal of the screen is increased by 1 inch, the laptop’s price will increase on average by 3094.64 RUB;
3. The price of the laptops with Intel graphics card will cost more on average by 59380.60 RUB higher than laptops with AMD and Nvidia under two conditions:

• if the value of SSD is increased by 1 GB, the price of the laptops with Intel graphics cards will grow less on average by 45.82 RUB less than laptops with AMD and Nvidia videocards;
• if the diagonal of the screen is increased by 1 inch, the price of the laptops with Intel graphics cards  will grow less on average by 3397.57 less than laptops with AMD and Nvidia videocards.

So during the research the model was built based on econometric analysis which allowed to draw the conclusion on the optimal laptop’s price under the influence of various factors. These factors were explained the laptop’s price by 95,79%.

The model has an error of 15,95%. The error can be reduced by increasing the number of observations and the number of factors. If an econometric model is built for using in practice, we can add factors such as laptop weight, processor generation, battery capacity, RAM frequency, display matrix type, memory card support, the presence of Kensington lock slot, etc.

Use the resulting econometric model on a specific example. Take from official website of the State Procurement the Purchase №31807033837 to deliver the computers and one laptop, posted on 10.23.2018 [5]. Under the contract, one laptop is purchased with a stated initial contract price of 134750.49 RUB. These terms of purchase are suitable for the resulting econometric model as the delivery is carried out at retail. According to the information that is specified in the technical project, it is expected to purchase a laptop model Dell XPS15 15.6". In this laptop model uses the following parameters, listed in Table 3.

Table 3

Dell XPS15 15.6" laptop settings shown in the information card

Таблица 3

Параметры ноутбука DellXPS15 15.6", указанные в информационной карте