Multiple regression equation in natural and standardized form. Standardized regression coefficients
The coefficients of the regression equation, like any absolute indicators, cannot be used in comparative analysis if the units of measurement of the corresponding variables are different. For example, if y – family expenses for food, X 1 – family size, and X 2 is the total family income, and we define a relationship like = a + b 1 x 1 + b 2 x 2 and b 2 > b 1 , then this does not mean that x 2 has a stronger effect on y , how X 1 , because b 2 is the change in family expenses when income changes by 1 ruble, and b 1 – change in expenses when family size changes by 1 person.
Comparability of regression equation coefficients is achieved by considering a standardized regression equation:
y 0 = 1 x 1 0 + 2 x 2 0 + … + m x m 0 + e,
where y 0 and x 0 k – standardized variable values y And x k :
S y and S – standard deviations of variables y And x k ,
k (k=) – -coefficients of the regression equation (but not parameters of the regression equation, in contrast to the previous notations). -coefficients show by what part of its standard deviation (S y) the dependent variable will change y , if the independent variable x k will change by the value of its standard deviation (S). Estimates of the parameters of the regression equation in absolute terms (b k) and β-coefficients are related by the relation:
The coefficients of a regression equation on a standardized scale provide a realistic representation of the impact of independent variables on the modeled indicator. If the value of the -coefficient for any variable exceeds the value of the corresponding -coefficient for another variable, then the influence of the first variable on the change in the performance indicator should be considered more significant. It should be borne in mind that the standardized regression equation, due to the centering of variables, does not have a free term by construction.
For simple regression, the -coefficient coincides with the pair correlation coefficient, which makes it possible to give the pair correlation coefficient a meaningful meaning.
When analyzing the impact of indicators included in the regression equation on the modeled characteristic, along with -coefficients, elasticity coefficients are also used. For example, the average elasticity indicator is calculated by the formula
and shows by what percentage on average the dependent variable will change if the average value of the corresponding independent variable changes by one percent (all other things being equal).
2.2.9. Discrete Variables in Regression Analysis
Typically, variables in regression models have continuous ranges of variation. However, the theory does not impose any restrictions on the nature of such variables. Quite often there is a need to take into account in regression analysis the influence of qualitative characteristics and their dependence on various factors. In this case, it becomes necessary to introduce discrete variables into the regression model. Discrete variables can be either independent or dependent. Let's consider these cases separately. Let us first consider the case of discrete independent variables.
Dummy Variables in Regression Analysis
To include qualitative features in regression as independent variables, they must be digitized. One method for quantifying them is to use dummy variables. The name is not entirely apt - they are not fictitious, but for these purposes it is more convenient to use variables that take only two values - zero or one. So they were called fictitious. Typically, a qualitative variable can take on several levels of values. For example, gender – male, female; qualification – high, medium, low; seasonality - I, II, III and IV quarters, etc. There is a rule according to which, to digitize such variables, you need to enter the number of dummy variables, one less in number than the number of levels of the modeled indicator. This is necessary so that such variables do not turn out to be linearly dependent.
In our examples: gender is one variable, equal to 1 for men and 0 for women. Qualification has three levels, which means two dummy variables are needed: for example, z 1 = 1 for a high level, 0 for others; z 2 = 1 for the average level, 0 for others. A third similar variable cannot be introduced, because in this case they would turn out to be linearly dependent (z 1 + z 2 + z 3 = 1), the determinant of the matrix (X T X) would turn to zero and it would not be possible to find the inverse matrix (X T X) -1 it would be possible. As is known, estimates of the parameters of the regression equation are determined from the relation: T X) -1 X T Y).
The coefficients on dummy variables show how much the value of the dependent variable differs at the analyzed level compared to the missing level. For example, if the salary level was modeled depending on several characteristics and skill level, then the coefficient for z 1 would show how the salary of specialists with a high level of qualification differs from the salary of a specialist with a low level of qualification, all other things being equal, and the coefficient for z 2 – a similar meaning for specialists with an average level of qualifications. In the case of seasonality, three dummy variables would have to be entered (if quarterly data are considered) and the coefficients on them would show how the value of the dependent variable differs for the corresponding quarter from the level of the dependent variable for the quarter that was not entered when digitizing them.
Dummy variables are also introduced to model structural changes in the dynamics of the studied indicators when analyzing time series.
Example 4. Standardized Regression Equation and Dummy Variables
Let's consider an example of using standardized coefficients and dummy variables using the example of analyzing the market for two-room apartments based on a multiple regression equation with the following set of variables:
PRICE – price;
TOTSP – total area;
LIVSP – living space;
KITSP – kitchen area;
DIST – distance to the city center;
WALK – equal to 1 if you can walk to the metro station and equal to 0 if you need to use public transport;
BRICK – equal to 1 if the house is brick and equal to 0 if it is panel;
FLOOR – equal to 1 if the apartment is not on the first or last floor and equal to 0 otherwise;
TEL – equal to 1 if there is a telephone in the apartment and equal to 1 if not;
BAL – equals 1 if there is a balcony and equals 0 if there is no balcony.
Calculations were carried out using the STATISTICA software (Figure 2.23). The presence of -coefficients allows you to order variables according to the degree of their influence on the dependent variable. Let us carry out a brief analysis of the calculation results.
Based on Fisher statistics, we conclude about the significance of the regression equation (p-level< 0,05). Обработана информация о 6 286 квартирах (n–m–1 = 6 276, а m = 9). Все коэффициенты уравнения регрессии (кроме при переменной BAL) значимы (р-величины для них < 0,05), а наличие или отсутствие балкона в этом случае существенно не сказывается на цене квартиры.
Figure 2.24 – Apartment market report based on STATISTICA PPP
The coefficient of multiple determination is 52%, therefore, the variables included in the regression determine the price change by 52%, and the remaining 48% of the change in the price of the apartment depends on unaccounted factors. Including from random price fluctuations.
Each of the coefficients for a variable shows how much the price of an apartment will change (all other things being equal) if this variable changes by one. So, for example, when the total area changes by 1 sq. m, the price of an apartment will change on average by 0.791 USD, and if the apartment moves 1 km from the city center, the price of an apartment will decrease by 0.596 USD on average. etc. Dummy variables (last 5) show how much the average price of an apartment will change if you move from one level of this variable to another. So, for example, if the house is brick, then the apartment in it costs an average of 3,104 USD. That is, more expensive than the same in a panel house, and the presence of a telephone in the apartment raises its price by an average of 1,493 USD. e., etc.
Based on the -coefficients, the following conclusions can be drawn. The largest -coefficient, equal to 0.514, is the coefficient for the “total area” variable, therefore, first of all, the price of an apartment is formed under the influence of its total area. The next factor in terms of influence on the change in the price of an apartment is the distance to the city center, then the material from which the house is built, then the kitchen area, etc.
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The standardized regression coefficients show how many sigmas the result will change on average if the corresponding factor x changes by one sigma, while the average level of other factors remains unchanged. Due to the fact that all variables are set as centered and normalized, the standardized refession coefficients D are comparable to each other. Comparing them with each other, you can rank the factors according to the strength of their impact on the result. This is the main advantage of the standardized recourse coefficients, in contrast to the pure recourse coefficients, which are incomparable among themselves.
The consistency of partial correlation and standardized regression coefficients is most clearly seen from a comparison of their formulas in a two-factor analysis.
The consistency of partial correlation and standardized regression coefficients is most clearly seen from a comparison of their formulas in a two-way analysis.
To determine the values of estimates at of standardized regression coefficients a (the following methods for solving a system of normal equations are most often used: the method of determinants, the square root method and the matrix method. Recently, the matrix method has been widely used to solve problems of regression analysis. Here we will consider solving a system of normal equations equations by the method of determinants.
In other words, in two-factor analysis, partial correlation coefficients are standardized regression coefficients multiplied by the square root of the ratio of the shares of residual variances of the fixed factor to the factor and to the result.
There is another possibility of assessing the role of grouping characteristics and their significance for classification: on the basis of standardized regression coefficients or coefficients of separate determination (see Chap.
As can be seen from table. 18, the components of the studied composition were distributed according to the absolute value of the regression coefficients (b5) with their square error (5br) in a series from carbon monoxide and organic acids to aldehydes and oil vapors. When calculating the standardized regression coefficients (p), it turned out that, taking into account the range of concentration fluctuations, ketones and carbon monoxide generally come to the fore in the formation of the toxicity of the mixture, while organic acids remain in third place.
The conditional pure regression coefficients bf are Named Numbers expressed in different units of measurement and are therefore not comparable to each other. To convert them into comparable relative indicators, the same transformation is applied as for obtaining the pair correlation coefficient. The resulting value is called the standardized regression coefficient or coefficient.
Conditional pure regression coefficients A; are named numbers expressed in different units of measurement and are therefore incomparable with each other. To convert them into comparable relative indicators, the same transformation is applied as for obtaining the pair correlation coefficient. The resulting value is called the standardized regression coefficient or coefficient.
In the process of developing headcount standards, initial data on the headcount of managerial personnel and the values of factors for selected basic enterprises are collected. Next, significant factors are selected for each function based on correlation analysis, based on the value of the correlation coefficients. Factors with the highest value of the pair correlation coefficient with the function and the standardized regression coefficient are selected.
The results of the above calculations make it possible to arrange in decreasing order the regression coefficients corresponding to the mixture under study, and thereby quantify the degree of their danger. However, the regression coefficient obtained in this way does not take into account the range of possible fluctuations of each component in the mixture. As a result, degradation products with high regression coefficients, but fluctuating in a small range of concentrations, may have a lesser effect on the total toxic effect than ingredients with relatively small b, the content of which in the mixture varies over a wider range. Therefore, it seems appropriate to perform an additional operation - the calculation of the so-called standardized regression coefficients p (J.
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Exercise.
- For a given data set, build a linear multiple regression model. Evaluate the accuracy and adequacy of the constructed regression equation.
- Give an economic interpretation of the model parameters.
- Calculate the standardized coefficients of the model and write the regression equation in standardized form. Is it true that the price of a good has a greater impact on the volume of supply of the good than the wages of employees?
- For the resulting model (in natural form), check whether the residuals are homoscedastic by applying the Goldfeld-Quandt test.
- Test the resulting model for autocorrelation of residuals using the Durbin-Watson test.
- Check whether the assumption of homogeneity of the original data in the regression sense is adequate. Is it possible to combine two samples (for the first 8 and the remaining 8 observations) into one and consider a single regression model of Y on X?
1. Estimation of the regression equation. Let's determine the vector of regression coefficient estimates using the Multiple Regression Equation service. According to the least squares method, the vector s obtained from the expression: s = (X T X) -1 X T Y
Matrix X
| 1 | 182.94 | 1018 |
| 1 | 193.45 | 920 |
| 1 | 160.09 | 686 |
| 1 | 157.99 | 405 |
| 1 | 123.83 | 683 |
| 1 | 152.02 | 530 |
| 1 | 130.53 | 525 |
| 1 | 137.38 | 418 |
| 1 | 137.58 | 425 |
| 1 | 118.78 | 161 |
| 1 | 142.9 | 242 |
| 1 | 99.49 | 226 |
| 1 | 116.17 | 162 |
| 1 | 185.66 | 70 |
Matrix Y
| 4.07 |
| 4 |
| 2.98 |
| 2.2 |
| 2.83 |
| 3 |
| 2.35 |
| 2.04 |
| 1.97 |
| 1.02 |
| 1.44 |
| 1.22 |
| 1.11 |
| 0.82 |
Matrix X T
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 182.94 | 193.45 | 160.09 | 157.99 | 123.83 | 152.02 | 130.53 | 137.38 | 137.58 | 118.78 | 142.9 | 99.49 | 116.17 | 185.66 |
| 1018 | 920 | 686 | 405 | 683 | 530 | 525 | 418 | 425 | 161 | 242 | 226 | 162 | 70 |
Multiply matrices, (X T X)
Find the inverse matrix (X T X) -1
| 2.25 | -0.0161 | 0.00037 |
| -0.0161 | 0.000132 | -7.0E-6 |
| 0.00037 | -7.0E-6 | 1.0E-6 |
The vector of regression coefficient estimates is equal to
| Y(X) = |
| * |
| = |
|
Regression equation (estimation of regression equation)
Y = 0.18 + 0.00297X 1 + 0.00347X 2
2. Matrix of paired correlation coefficients R. Number of observations n = 14. The number of independent variables in the model is 2, and the number of regressors taking into account the unit vector is equal to the number of unknown coefficients. Taking into account the sign Y, the dimension of the matrix becomes equal to 4. The matrix of independent variables X has a dimension (14 x 4).
Matrix composed of Y and X
| 1 | 4.07 | 182.94 | 1018 |
| 1 | 4 | 193.45 | 920 |
| 1 | 2.98 | 160.09 | 686 |
| 1 | 2.2 | 157.99 | 405 |
| 1 | 2.83 | 123.83 | 683 |
| 1 | 3 | 152.02 | 530 |
| 1 | 2.35 | 130.53 | 525 |
| 1 | 2.04 | 137.38 | 418 |
| 1 | 1.97 | 137.58 | 425 |
| 1 | 1.02 | 118.78 | 161 |
| 1 | 1.44 | 142.9 | 242 |
| 1 | 1.22 | 99.49 | 226 |
| 1 | 1.11 | 116.17 | 162 |
| 1 | 0.82 | 185.66 | 70 |
Transposed matrix.
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 4.07 | 4 | 2.98 | 2.2 | 2.83 | 3 | 2.35 | 2.04 | 1.97 | 1.02 | 1.44 | 1.22 | 1.11 | 0.82 |
| 182.94 | 193.45 | 160.09 | 157.99 | 123.83 | 152.02 | 130.53 | 137.38 | 137.58 | 118.78 | 142.9 | 99.49 | 116.17 | 185.66 |
| 1018 | 920 | 686 | 405 | 683 | 530 | 525 | 418 | 425 | 161 | 242 | 226 | 162 | 70 |
Matrix A T A.
| 14 | 31.05 | 2038.81 | 6471 |
| 31.05 | 83.37 | 4737.04 | 18230.79 |
| 2038.81 | 4737.04 | 307155.61 | 995591.55 |
| 6471 | 18230.79 | 995591.55 | 4062413 |
The resulting matrix has the following correspondence:
| ∑n | ∑y | ∑x 1 | ∑x 2 |
| ∑y | ∑y 2 | ∑x 1 y | ∑x 2 y |
| ∑x 1 | ∑yx 1 | ∑x 1 2 | ∑x 2 x 1 |
| ∑x 2 | ∑yx 2 | ∑x 1 x 2 | ∑x 2 2 |
Let's find pair correlation coefficients.
| Features x and y | ∑(xi) | ∑(yi) | ∑(x i y i ) | |||
| For y and x 1 | 2038.81 | 145.629 | 31.05 | 2.218 | 4737.044 | 338.36 |
| For y and x 2 | 6471 | 462.214 | 31.05 | 2.218 | 18230.79 | 1302.199 |
| For x 1 and x 2 | 6471 | 462.214 | 2038.81 | 145.629 | 995591.55 | 71113.682 |
| Features x and y | ||||
| For y and x 1 | 731.797 | 1.036 | 27.052 | 1.018 |
| For y and x 2 | 76530.311 | 1.036 | 276.641 | 1.018 |
| For x 1 and x 2 | 76530.311 | 731.797 | 276.641 | 27.052 |
Matrix of pair correlation coefficients R:
| - | y | x 1 | x 2 |
| y | 1 | 0.558 | 0.984 |
| x 1 | 0.558 | 1 | 0.508 |
| x 2 | 0.984 | 0.508 | 1 |
To select the most significant factors x i, the following conditions are taken into account:
- the connection between the resultant characteristic and the factor one must be higher than the interfactor connection;
- the relationship between factors should be no more than 0.7. If the matrix has an interfactor correlation coefficient r xjxi > 0.7, then there is multicollinearity in this multiple regression model.;
- with a high interfactor connection of a characteristic, factors with a lower correlation coefficient between them are selected.
In our case, all pairwise correlation coefficients |r| Regression model on a standard scale A regression model on a standard scale assumes that all values of the characteristics under study are converted into standards (standardized values) using the formulas:
where x ji is the value of the variable x ji in the i-th observation.
Thus, the origin of each standardized variable is combined with its mean value, and its standard deviation is taken as the unit of change S.
If the relationship between variables on a natural scale is linear, then changing the origin and unit of measurement will not violate this property, so the standardized variables will also be related by a linear relationship:
t y = ∑β j t xj
To estimate β-coefficients, we use OLS. In this case, the system of normal equations will have the form:
r x1y =β 1 +r x1x2 β 2 + ... + r x1xm β m
r x2y =r x2x1 β 1 + β 2 + ... + r x2xm β m
...
r xmy =r xmx1 β 1 + r xmx2 β 2 + ... + β m
For our data (we take it from the matrix of pair correlation coefficients):
0.558 = β 1 + 0.508β 2
0.984 = 0.508β 1 + β 2
We solve this system of linear equations using the Gaussian method: β 1 = 0.0789; β 2 = 0.944;
The standardized form of the regression equation is:
y 0 = 0.0789x 1 + 0.944x 2
The β-coefficients found from this system make it possible to determine the values of the coefficients in regression on a natural scale using the formulas:
Standardized partial regression coefficients. Standardized partial regression coefficients - β-coefficients (β j) show by what part of its standard deviation S(y) the result will change y with a change in the corresponding factor x j by the value of its standard deviation (S xj) with the constant influence of other factors (included in the equation).
By the maximum β j one can judge which factor has a stronger influence on the result Y.
The elasticity coefficients and β-coefficients can lead to opposite conclusions. The reasons for this are: a) the variation of one factor is very large; b) multidirectional influence of factors on the result.
The coefficient β j can also be interpreted as an indicator of direct (immediate) influence j-th factor (x j) on the result (y). In multiple regression j The th factor has not only a direct, but also an indirect (indirect) effect on the result (i.e., influence through other factors of the model).
Indirect influence is measured by the value: ∑β i r xj,xi , where m is the number of factors in the model. Full Impact jth factor on the result equal to the sum of direct and indirect influences measures the linear pair correlation coefficient of this factor and the result - r xj,y.
So for our example, the direct influence of the factor x 1 on the result Y in the regression equation is measured by β j and amounts to 0.0789; the indirect (mediated) influence of this factor on the result is defined as:
r x1x2 β 2 = 0.508 * 0.944 = 0.4796
In econometrics, a different approach is often used to determine the parameters of multiple regression (2.13) with the excluded coefficient:
Let's divide both sides of the equation by the standard deviation of the explained variable S Y and present it in the form:
Let's divide and multiply each term by the standard deviation of the corresponding factor variable to get to standardized (centered and normalized) variables:
where the new variables are denoted as
.
All standardized variables have a mean of zero and the same variance of one.
The regression equation in standardized form is:
Where 
- standardized regression coefficients.
Standardized regression coefficients 
differ from the coefficients 
ordinary, natural form in that their value does not depend on the scale of measurement of the explained and explanatory variables of the model. In addition, there is a simple relationship between them:
, (3.2)
which gives another way to calculate the coefficients 
by known values 
, more convenient in the case of, for example, a two-factor regression model.
5.2. Normal system of least squares equations in standardized
variables
It turns out that to calculate standardized regression coefficients, you only need to know the pairwise linear correlation coefficients. To show how this is done, let us exclude the unknown from the normal system of least squares equations 
using the first equation. Multiplying the first equation by ( 
) and adding it term by term with the second equation, we get:
Replacing the expressions in parentheses with the notations for variance and covariance
Let's rewrite the second equation in a form convenient for further simplification:
Let's divide both sides of this equation by the standard deviation of the variables S Y And ` S X 1 , and divide each term and multiply by the standard deviation of the variable corresponding to the number of the term:
Introducing the characteristics of a linear statistical relationship:
and standardized regression coefficients
,
we get:
After similar transformations of all other equations, the normal system of linear equations of least squares (2.12) takes the following, simpler form:
(3.3)
5.3. Standardized Regression Options
Standardized regression coefficients in the special case of a model with two factors are determined from the following system of equations:
(3.4)
Solving this system of equations, we find:
, (3.5)
. (3.6)
Substituting the found values of the pair correlation coefficients into equations (3.4) and (3.5), we obtain 
And 
. Then, using formulas (3.2), it is easy to calculate estimates of the coefficients 
And 
, and then, if necessary, calculate the estimate 
according to the formula 
6. Possibilities of economic analysis based on a multifactor model
6.1. Standardized Regression Coefficients
Standardized regression coefficients show how many standard deviations 
the average explained variable will change Y, if the corresponding explanatory variable X i
will change by the amount 
one of its standard deviations while maintaining unchanged the average level of all other factors.
Due to the fact that in standardized regression all variables are specified as centered and normalized random variables, the coefficients 
comparable to each other. By comparing them with each other, you can rank the factors corresponding to them X i by the strength of impact on the explained variable Y. This is the main advantage of standardized regression coefficients from coefficients 
regressions in natural form, which are incomparable.
This feature of standardized regression coefficients makes it possible to use when eliminating the least significant factors X i with values of their sample estimates close to zero 
. The decision to exclude them from the linear regression model equation is made after testing the statistical hypotheses that its average value is equal to zero.
A beta coefficient equal to 0.074 (Table 3.2.1) shows that if real wages change by the value of their standard deviation (σх1), then the coefficient of natural population growth will change on average by 0.074 σу. A beta coefficient of 0.02 shows that if the crude marriage rate changes by the value of its standard deviation (by σх2), then the rate of natural population growth will change on average by 0.02 σу. Similarly, a change in the number of crimes per 1000 people by the value of its standard deviation (by σх3) will lead to a change in the resulting characteristic by an average of 0.366 σу, and a change in the input of square meters of residential premises per person per year by the value of its standard deviation (by σх4) leads to a change in the effective characteristic by an average of 1.32σу.
The elasticity coefficient shows by what percentage on average y changes with a change in the factor attribute by 1%. From the analysis of time series it is known that the value of a 1% increase in an effective characteristic is negative, since in all units of the population there is a natural population decline. Therefore, growth actually means a reduction in loss. This means that negative elasticity coefficients in this case reflect the fact that with an increase in each of the factor characteristics by 1%, the natural loss coefficient will decrease by the corresponding number of percent. With an increase in real wages by 1%, the natural decline rate will decrease by 0.219%; with an increase in the overall marriage rate by 1%, it will decrease by 0.156%. An increase in the number of crimes per 1000 population by 1% is characterized by a reduction in natural population decline by 0.564. Of course, this does not mean that increasing crime can improve the demographic situation. The results obtained indicate that the more people remain per 1000 population, the correspondingly more crimes occur per thousand. Increasing input sq.m. housing per person per year by 1% leads to a reduction in natural loss by 0.482%
Analysis of elasticity coefficients and beta coefficients shows that the greatest influence on the rate of natural population growth is exerted by the factor of commissioning sq.m of housing per capita, since it corresponds to the highest value of the beta coefficient (1.32). However, this does not mean that the greatest opportunities for changing the rate of natural population growth are associated with changes in this of the factors considered. The result obtained reflects the fact that demand in the housing market corresponds to supply, that is, the greater the natural population growth, the greater the need of this population for housing and the more it is built.
The second largest beta coefficient (0.366) corresponds to the number of crimes per 1000 people. Of course, this does not mean that by increasing crime the demographic situation can be improved. The results obtained indicate that the more people remain per 1000 population, the correspondingly more crimes occur per thousand.
The largest of the remaining indicators, beta coefficient (0.074), corresponds to the indicator of real wages. The greatest opportunities for changing the rate of natural population growth are associated with changes in this of the factors considered. The indicator of the overall marriage rate is inferior in this regard to real wages due to the fact that the natural population decline in Russia is due, first of all, to the high mortality rate of the population, the growth rate of which can be reduced rather by material security than by an increase in the number of marriages.
3.3 Combined grouping of regions by real wages and overall marriage rate
Combined or multidimensional grouping is a grouping based on two or more characteristics. The value of this grouping lies in the fact that it shows not only the influence of each factor on the result, but also the influence of their combination.
Let us determine the influence of the value of real wages and the general marriage rate on the birth rate per 1000 people.
Let us identify typical groups according to the intended characteristics. To do this, we will construct and analyze ranked and interval series according to the factor attribute (salary value), determine the number of groups and the size of the interval; then, within each group, we will construct a ranked and interval series based on the second criterion (marriage rate) and also set the number of groups and the interval. The procedure for carrying out this work is presented in Chapter 2, therefore, omitting the calculations, we present the results. For the value of real wages, 3 typical groups have been identified, for the overall marriage rate - 2 groups.
We will draw up a layout of a combination table in which we will provide for the division of the population into groups and subgroups, as well as columns for recording the number of regions and the birth rate per 1000 people of the population. For the selected groups and subgroups, we calculate birth rates (Table 3.3.1)
Table 3.3.1
The influence of real wages and the overall marriage rate on the birth rate.
Let us analyze the obtained data on the dependence of the birth rate on real wages and the marriage rate. Since one characteristic is being studied - the fertility rate, we will write the data about it in a chess combination table of the following form (Table 3.3.2)
The combined grouping allows us to assess the degree of influence on the birth rate of each factor separately and their interaction.
Table 3.3.2
Dependence of the birth rate on real wages and marriage rates
Let us first study the effect on the birth rate of the value of real wages at a fixed value of another grouping characteristic - the marriage rate. Thus, with a marriage rate from 13.2 to 25.625, the average birth rate increases as wages increase from 9.04 in the 1st group to 9.16 in the 2nd group and 9.56 in the 3rd group; the increase in the birth rate from wages in the 3rd group compared to the 1st is: 9.56-9.04 = 0.52 people per 1000 population. With a marriage rate of 25.625-38.05, the increase from the same wage is equal to: 10.27-9.49 = 0.78 people per 1000 population. The increase from the interaction of factors is equal to: 0.78-0.52 = 0.26 people per 1000 population. A completely natural conclusion follows from this: an increase in well-being motivates, or rather allows, with confidence in the future, a person’s desire to get married and start a family with children to be realized. This shows the interaction of factors.
In the same way, we will estimate the impact on the fertility rate of the marriage rate at a fixed wage level. To do this, let’s compare the birth rate for groups “a” and “b” within each group according to the value of real wages. The increase in the birth rate with an increase in the marriage rate to 25.625-38.05 per 1000 population compared to group “a” is: in the 1st group with a salary of 5707.9 - 6808.7 rubles. per month - 9.49-9.04 = 0.45 people per 1000 population, in the 2nd group - 10.01-9.16 = 0.85 people per 1000 population and in the 3rd - 10.27- 9.56=0.71 people per 1000 population. As you can see, the decision to have a child depends on marital status, i.e. there is an interaction of factors, giving an increase of 0.26 people per 1000 population.
With a joint increase in both factors, the birth rate increases from 9.04 in subgroup 1 “a” to 10.27 people per 1000 population in subgroup 3 “b”.
Representatives of the UN Economic Commission for Europe recently announced that the age of first marriage in European countries has increased by five years. Guys and girls prefer to get married after 30. Russians do not dare to tie the knot before 24-26 years old. Also common to Europe and Russia is a trend towards a reduction in the number of marriages. Young people increasingly prefer careers and personal freedom. Domestic experts see in these processes signs of a deep crisis in the traditional family. In their opinion, she is literally living her last days. Sociologists say that private life is now going through a period of restructuring. The family in the usual sense of the word, living according to the “mom-father-children” scheme, is gradually becoming a thing of the past. In private life, Russians are increasingly experimenting, inventing more and more new forms of family that would meet the needs of the time. “Now a person more often changes his job, profession, interests, place of residence,” Anatoly Vishnevsky, director of the Center for Demography and Human Ecology, told Novye Izvestia. “He also often changes spouses, which was considered unacceptable 20 years ago.”
Sociologists note that one of the reasons for the increase in divorces in Russia is the low standard of living of the population. “According to statistics, there are approximately 10–15% more divorces in Russia than in Europe,” Mr. Gontmakher (scientific director of the Center for Social Research and Innovation) told NI. – But the reasons for divorce are different for us and for them. Our primacy is dictated mainly by the fact that economic problems are increasingly affecting the lives of Russians. Spouses quarrel more often if they have cramped living conditions. Young people do not always manage to live independently. In addition, in the regions, many men drink, do not work and cannot provide for their families. This is also a reason for divorce.”
Conclusion
In this work, a statistical and economic analysis of the influence of the standard of living of the population on the processes of natural growth was carried out.
An analysis of the dynamics showed that over the past 10 years there has been an increase in real wages and the cost of living. In general, over these 10 years, the effective attribute - the coefficient of natural increase - is stationary. The stability of the emerging processes of change in the selected characteristics is such that making a forecast is possible only for the value of real wages and the mortality rate. According to the built-up parabolic trend, by 2010 the forecast value of the average real wage will be 17,473.5 rubles, and the mortality rate will decrease to 12.75 people per 1000.
The analytical grouping showed a direct relationship between the indicators: with an increase in wages, the indicators of natural growth improve.
However, a family of two workers with an average salary can provide a minimum level of consumption for 2 children - in the lowest typical group, 3 children - in the middle and highest typical groups. Considering that two children “replace” the lives of their parents in the future, a slight increase in population is possible only in the middle and highest typical groups, and then under the condition of a low mortality rate compared to the birth rate. The fertility potential, which comes with wages in Russia, is low for improving the demographic situation in the country. This precisely reveals the need for the introduced national demographic project in Russia. An increase in wages has a more favorable effect on the mortality rate than on the birth rate.
The construction of a correlation-regression model revealed that the simultaneous influence of factor characteristics (wages, marriage rates, crime rates and housing commissioning) on the productive (natural increase) is observed with an average strength of connection. The variation in the rate of natural population growth by 44.9% is characterized by the influence of selected factors, and 55.1% by other unaccounted and random reasons. The greatest opportunities for changing the rate of natural population growth are associated with changes in the value of real wages.
The combined group confirmed that an increase in well-being motivates, or rather allows, with confidence in the future, a person’s desire to get married and start a family with children to be realized.
And finally, we need to assess the effectiveness of solving the demographic problem in our country. In general, the positive and effective influence of material incentives on the process of natural population movement has been proven. Another thing is that there is a complex of socio-psychological problems (alcoholism, violence, suicide) that are inexorably reducing our population. Their main reason is a person’s attitude towards himself and others. But these problems cannot be solved by the state alone; civil society must come to its own aid in the problem of extinction, forming moral values focused on creating a prosperous family.
And the state can and should do everything to improve the level and quality of life in the country. It cannot be said that our state neglects these responsibilities. It is doing everything possible, looking for and trying different ways out of the demographic crisis.
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SYSTEM OF INDICATORS (KEY TO CHIPS)
1-average monthly nominal wage in 2006 (in rubles)
2-consumer price indices for all types of goods and paid services in 2006 as a percentage compared to December last year
3 - average monthly real wage in 2006 (in rubles)
4 – population at the beginning of 2006
5 – population at the end of 2006
6 – average annual population in 2006
7 – number of births in 2006, people
8 – number of deaths in 2006, people
9 – birth rate in 2006 per 1000 population
10 – mortality rate in 2006 per 1000 population
11 – rate of natural increase in 2006 per 1000 population
12 – the cost of living for 2006 (in rubles)
13 – number of crimes committed per 1000 people
14 – commissioning of sq.m of housing per person per year
15 – overall marriage rate per 1000 population
Annex 1
Table
Real wages, rub.
Appendix 2
The cost of living, rub.
Appendix 3
