The practical significance of the multiple regression equation is assessed using the multiple correlation indicator and its square - the coefficient of determination.

The coefficient of determination shows the proportion of the variation of the resulting trait, which is under the influence of factor traits, i.e. determines what proportion of the trait variation at taken into account in the model and due to the influence of factors included in the model:

The multiple correlation coefficient can be found as the square root of the determination coefficient. The closer the correlation coefficient is to one, the closer the relationship between the result and all factors, and the regression equation better describes the actual data. If the multiple correlation coefficient is close to zero, then the regression equation does not describe the actual data well, and the factors have little effect on the result. This coefficient, unlike the pairwise correlation coefficient, cannot be used to interpret the direction of a relationship.

The value of the multiple correlation coefficient is greater than or equal to the value of the maximum pair correlation coefficient:

For linear multiple regression, the multiple correlation coefficient can be calculated using the following formula:

Accordingly, the multiple coefficient of determination:

There is another formula for calculating the multiple correlation coefficient for linear regression:

where is the determinant of the full matrix of linear paired correlation coefficients (i.e., including paired linear correlation coefficients of factors with the result and among themselves):

The determinant of the matrix of linear pair correlation coefficients of factors among themselves:

The adjusted coefficient of determination is also calculated:

where n is the number of observations;

m- the number of parameters of the regression equation without taking into account the free term (for linear regression, for example, this number is equal to the number of factors included in the model).

The corrected coefficient of determination is used to solve two problems: assessing the real tightness of the relationship between the result and factors and comparing models with a different number of parameters. In the first case, attention is paid to the proximity of the adjusted and uncorrected coefficients of determination. If these indicators are large and differ slightly, the model is considered good.

When comparing different models, other things being equal, preference is given to the one that has a larger adjusted coefficient of determination.

It should be noted that the scope of the adjusted coefficient of determination is limited only to these tasks. It cannot be used in formulas where the usual coefficient of determination is applied. The adjusted coefficient of determination cannot be interpreted as the fraction of the variance in the outcome explained by the variance in the factors included in the regression model.

To check the significance of the multiple correlation coefficient, use F- Fisher's criterion, which is determined by the formula:

where R2– multiple coefficient of determination;

m- the number of parameters with factors x in the multiple regression equation (in paired regression m=1).

The obtained value of the F-criterion is compared with the tabular one when certain level significance and m and n-m-1 degrees of freedom. If the calculated value F- the criterion is greater than the tabular one, the multiple regression equation is recognized as significant.

The overall quality of a multiple regression equation is assessed using the multiple correlation coefficient and its square, the multiple determination coefficient.

Similar to pairwise regression multiple determination coefficient can be defined as the proportion of the variance of the result, explained by the variation of the factors included in the model, in its total variance:

The values of the coefficient of multiple determination vary from zero to one (0≤ R 2 y x 1 x 2… x p≤1). The closer this coefficient is to one, the more the regression equation explains the behavior of the result.

characterizes the closeness of the connection of the considered set of factors with the studied feature or, in other words, evaluates the closeness of the connection between the joint influence of factors on the result.

Multiple correlation coefficient can be found as the square root of the multiple determination coefficient:

The values of the multiple correlation coefficients vary from zero to one (0≤ R yx 1 x 2… x p≤1). The closer the coefficient is to one, the closer the relationship between the result and all factors in the aggregate, and the regression equation better describes the actual data. If the multiple correlation coefficient R yx 1 x 2… x p is close to zero, then the regression equation poorly describes the actual data, and the factors have little effect on the result.

The value of the multiple correlation coefficient is greater than or equal to the value of the maximum pair correlation coefficient:

R y x1x2…x p ≥І r y x i (max) I , where i= 1,R.

If the regression equation takes into account any factor that has the strongest impact on the resulting attribute, then the partial correlation coefficient will be close enough to the multiple correlation coefficient, but in no case more than it.

Sometimes another formula is used to calculate the multiple correlation coefficient (it is applicable only for linear multiple regression):

where DetІ R+І, DetІ RІ - determinants of matrices, respectively, of paired correlation coefficients and interfactorial correlation.

These determinants will have the following form for a linear multiple regression equation with R number of factors:

1 r yx1 r yx2…r yx p

r yx1 1 rx1x2 … r x1x p

Det І R+І = r yx2 r x1x2 1 … r x2x p ,

… … … … …

r yx p r x1x p r x2x p … 1

those. the matrix includes all pairwise correlation coefficients for the regression equation;

1 rx1x2 … r x1x p

DetІ RІ = rx1x2 1 … r x 2 x p

… … … … ,

r x 1 x p r x 2 x p … 1

those. this matrix is obtained from the previous matrix by eliminating the pairwise correlation coefficients of the factors with the result (the first row and the first column are crossed out).

In order to prevent the possible exaggeration of the closeness of the connection, it is usually used adjusted multiple correlation coefficient. It contains a correction for the number of degrees of freedom. The residual sum of squared deviations is divided by the number of degrees of freedom of the residual variation (P- T- 1), and the total sum of squared deviations - by the number of degrees of freedom in the whole population (P- one). Formula with corrected multiple correlation coefficient has the following form:

where T- number of parameters for variables X(v linear dependence it will be equal to the number of factors included in the model = p); P- number of observations.

To determine the degree of dependence between several indicators, multiple correlation coefficients are used. They are then summarized in a separate table, which is called the correlation matrix. The names of the rows and columns of such a matrix are the names of the parameters whose dependence on each other is established. Corresponding correlation coefficients are located at the intersection of rows and columns. Let's find out how you can make a similar calculation using Excel tools.

It is customary to determine the level of relationship between various indicators as follows, depending on the correlation coefficient:

0 - 0.3 - no connection;
0.3 - 0.5 - weak connection;
0.5 - 0.7 - average connection;
0.7 - 0.9 - high;
0.9 - 1 - very strong.

If the correlation coefficient is negative, then this means that the relationship of the parameters is inverse.

In order to compile a correlation matrix in Excel, one tool is used, included in the package "Data analysis". That's what it's called - "Correlation". Let's see how it can be used to calculate multiple correlation scores.

Step 1: Activate Analysis Pack

It must be said right away that the default package "Data analysis" disabled. Therefore, before proceeding with the procedure for directly calculating the correlation coefficients, you need to activate it. Unfortunately, not every user knows how to do this. Therefore, we will focus on this issue.

After the specified action, the tool package "Data analysis" will be activated.

Stage 2: coefficient calculation

Now you can proceed directly to the calculation of the multiple correlation coefficient. Let's calculate the multiple correlation coefficient of these factors using the example of the table of indicators of labor productivity, capital-labor ratio and power-to-weight ratio at various enterprises.

Stage 3: analysis of the result

Now let's figure out how to understand the result that we got in the process of data processing by the tool "Correlation" in the Excel program.

As we can see from the table, the correlation coefficient of capital-labor ratio (Column 2) and power-to-weight ratio ( Column 1) is 0.92, which corresponds to a very strong relationship. Between labor productivity ( Column 3) and power-to-weight ratio ( Column 1) this indicator is equal to 0.72, which is a high degree of dependence. Correlation coefficient between labor productivity ( Column 3) and capital-labor ratio ( Column 2) is equal to 0.88, which also corresponds to high degree dependencies. Thus, we can say that the relationship between all the studied factors can be traced quite strong.

As you can see, the package "Data analysis" in Excel is a very convenient and fairly easy-to-use tool for determining the multiple correlation coefficient. It can also be used to calculate the usual correlation between two factors.

Multiple correlation coefficient (R) characterizes the tightness of the relationship between the performance indicator and a set of factor indicators:

where σ 2 - the total dispersion of the empirical series, which characterizes the general variation of the result indicator (y) due to factors

σ ost 2 - residual variance in the series y, reflecting the influence of all factors except x;

at- the average value of the effective indicator, calculated according to the initial observations;

s- the average value of the effective indicator, calculated by the regression equation.

The multiple correlation coefficient takes only positive values ranging from 0 to 1. The closer the value of the coefficient is to 1, the greater the closeness of the relationship. Conversely, the closer to 0, the less dependence. With R value< 0,3 говорят о малой зависимости между величинами. При значении 0,3 < R< 0.6 indicates the average tightness of the connection. At R > 0.6, one speaks of the presence of a significant relationship.

The square of the multiple correlation coefficient is called determination coefficient (D):D = R 2 . The coefficient of determination shows what proportion of the variation of the effective indicator is associated with the variation of factor indicators. The calculation of the determination coefficient and the multiple correlation coefficient is based on the rule for adding variances, according to which the total variance (σ 2) is equal to the sum of the intergroup variance (δ 2) and the average of the group variances σ i 2):

σ2 = δ 2 + σ i 2 .

The intergroup dispersion characterizes the fluctuation of the effective indicator due to the studied factor, and the average of the group dispersions reflects the fluctuation of the effective indicator due to all other factors except the studied one.

Mathematical models correlation analysis in the form of coefficients have limited analytical capabilities. Knowing only the direction of the covariance of the indicators and the closeness of the relationship, it is impossible to determine the patterns of formation of the level of the effective indicator under the influence of the studied factors, to assess the intensity of their influence, to classify the factors into primary and secondary. For these purposes are used regression analysis models. The linear model (equation) of regression analysis can be represented as

y= bo + b 1 x 1 + b 2 x 2 +... + b n x n ,

where at- performance indicator;

x 1 , x 2 , ..., x n- factor models;

b 0 , b 1 , b 2 , ..., b n- regression coefficients.

See also:

autocorrelation is the correlation dependence of the levels of the series on the previous values.

The additive time series model has the form: Y=T+S+E

Autocorrelation occurs when each subsequent value of the residuals

Additive Time Series Model is a model in which the time series is presented as the sum of the listed components.

An additive time series model is built: the amplitude of seasonal fluctuations increases and decreases

The analytical notation of the problem of finding the values of the arguments for which the values of the two given functions are equal is known as the .. equation.

Attribute variable 1 can be used when: the independent variable is qualitative;

Within what limits does the coefficient of the determinant change: 0 to 1.

The value of the confidence interval allows you to establish the assumption that: the interval contains an estimate of the parameter of the unknown.

Internally non-linear regression is a truly non-linear regression that cannot be reduced to a linear regression by transforming the variables and introducing new variables.

time series- this is a sequence of values of a sign (resulting variable) taken during successive points in time or periods.

sample value Rxy not > 1, |R|< 1

When is the model considered adequate? Fcalc>Ftable

As a result of autocorrelation, we have inefficient parameter estimates

In a well-fitted model, the residuals should and follow the normal law

In econometric analysisXjconsidered as random variables

The value calculated by the formular=…is an estimate pair odds Correlations

Sample correlation coefficientrby absolute does not exceed unity.

In what case is the function y called the multivalued argument x if the same value of x corresponds to several values of y.

In econometric models, the endogenous variable is considered as both random and non-random.

In the equation of the system of economic equations, D = 1, the number of endogenous variables, D is the number of missing variables. This equation is: identifiable.

Selective coefficient. correlationsrin absolute value: does not exceed one.

In the economic-mathematical model, processes that depend on external conditions, but are independent of the internal structure of the phenomenon or process under study, are described through exogenous variables.

The sample mean is...evaluation of the average theoretical (mathematical expectation).

Sample set V=(1,0,3,2,4,3,1,3,2,3,3,4,4,0,5,2,4,3,4,3,3) determine the sample coefficient kurtosis... 2.714

Choose a model with lags:Уt= a+b0x1…….(the longest formula)

Any function of the form g (x) = E (Y | X = x), which describes the regression dependence for the two-dimensional distribution of a pair of random variables (Y, X), and the symbol E - denotes the operation of calculating the average value, is called a correlation function.

Within what limits does the multiple correlation coefficient R change answer2

R≤0(answer1) -1≤R≤+1 (answer2) R≥0 (answer3)

Is it true that one of the goals of regression analysis is to test static hypotheses about regression? Answer: yes

Heteroskedasticity- violation of the constancy of the variance for all observations.

Heteroskedasticity is present when:* when the variance of the residuals is different. The variance of random residuals is not constant; we are building the wrong version of the true model; two or more are independent. variables are highly correlated; the independent variable is estimated with an error.

Homoscedasticity- when the variance of the residuals is constant and the same for all ... observations. the constancy of the variance for all observations, or the same variance of each deviation (residual) for all values of factor variables.

The hypothesis of the absence of autocorrelation of residuals is proven, if Dtable2...

Dispersion- indicator of variation.

To determine the parameters of an unidentified model, apply: not one of the entities. methods cannot be applied.

To evaluate … changesyfromxis entered: elasticity coefficient.

Fisher's F-criterion is used to assess the quality of the model. What can be said about the regression model if its F-ratio is greater than the F-critical model is adequate to the original data.

To test the significance of individual regression parameters, we use: t-test.

Confidence probability is the probability that the true value of the effective indicator will fall within the calculated forecast interval.

To determine the parameters, the structural form of the model must be converted into reduced form of the model

To determine the parameters of a precisely identifiable model: indirect OLS is used (indirect OLS);

For paired regression ơ²bequals….(xi-x¯)²)

For regressiony= a+ bxfromnobservations confidence interval (1-a)% for the coefficient.bwill be b±t…….ơb

For regression fromnobservations andmindependent variables, there is such a relationship betweenR² andF..=[(n-m-1)/m](R²/(1- R²)]

Let's assume that 2 models are suitable for describing one economic process. Both are adequatefFisher's criterion. which one to provide an advantage, that one has a cat .: greater value of F criterion

Let us assume that the dependence of expenses on income is described by the functiony= a+ bxmean value y=2…equals 9

Let's say we're addicted. expenses from income is described by a + b / x. Average value y=3, average value x=2, coefficient of elastic expenses from income is: -0.5

Fisher's F-criterion is used to assess the quality of the model. What can be said about the regression model if its F-ratio is greater than the F-critical model is adequate to the original data

To evaluate a linear statistical relationship between one random variable and a linear combination of other random variables, use ... Multiple correlation coefficient R

To define parameters ABOVE the identified model: a two-step MNC is used.

Add missing values to ANOVA table, calculate multiple correlation coefficient R(determinationsR V 2) and check its significance. What conclusion can be drawn about the quality of the model?

a source	Number of degrees of freedom	Sum of squares	Middle squares	F-value
regression

Answer: R2=0.719, the model is adequate to the data

Using the values of the table (Fig. above) of the analysis of variance, determine the significance of the regression usingF- criterion. The critical value Fa,v1,v2 =4.3 at significance level a=0.05 and degrees of freedom v1=1 and v2=23. What conclusion can be drawn about the quality of the regression models used.

Answer: F=2.5, the model is inadequate to the data

IfRxyis positive, then as x increases, y increases.

If the qualitative factor has 3 gradations, then the required number of dummy variables 2

If the regression model has an exponential relationship, then the least squares method is applicable after reduction to a linear form.

If there is an insignificant variable in the regression equation, then it reveals itself by a low value T statistics.

If the correlation coefficient is positive, then in the linear model as x increases, y increases.

If we are interested in using attribute variables to show the effect of different months, we should use: 11 attribute methods.

If the regression coefficient is 2.4 with a variance of 0.8, then the value of Student's t-test will be:

Answer: first choice

Significance of the regression equation- the actual presence of the dependence under study, and not just a random coincidence of factors that mimics a dependence that does not actually exist.

The significance of the regression equation as a whole is estimated: -F-Fisher test

Significance of private and pair odds. correlation is verified. via:-t-student test

The relationship between the coefficient of multiple determination (D) and correlations (R) is described by the following method R=√D

Intercorrelation and related multicollinearity- this is a close relationship between factors approaching a complete linear relationship.

Correlation- stochastic dependence, which is a generalization of a strictly determined functional dependence by including a probabilistic (random) component.

Autocorrelation coefficient: characterizes the closeness of the linear connection of the current and upcoming levels of the series.

Determination coefficient- indicator of closeness of stochastic connection in the general case of non-linear regression

Determination coefficient: is the square of the multiple coefficient. correlations. the square of the pairwise correlation coefficient.

Determination coefficient is a value that characterizes the relationship between dependent and independent variables.

Determination coefficientRshows the proportion of variations in the dependent variable y, explained by the influence of the factors included in the model.

The coefficient of determination varies within: - 0 to 1

Confidence ratio- this is the coefficient that connects the limiting and average errors by a linear dependence, finds out the meaning of the limiting error characterizing the accuracy of the estimate, and is an argument of the distribution (most often, the probability integral). It is this probability that is the degree of reliability of the estimate.

Confidence coefficient (normalized deviation)- the result of dividing the deviation from the mean by the standard deviation, meaningfully characterizes the degree of reliability (confidence) of the obtained estimate.

Correlation coefficientRxyused to determine the completeness of the connection X and Y.

A correlation coefficient of 1 means that: -there is a functional dependency.

A correlation coefficient of 0 means that: - no linear connection .

Coef. correlation, equal to zero, means that between the variables the situation is not defined.

Coef. correlation equal to -1 means that between variables functional dependency.

The correlation coefficient is calculated for measuring the degree of linear relationship between two random variables.

The correlation coefficient varies within: from -1 to 1

The correlation coefficient is used for: determining the tightness of the relationship between random variables X and Y.

The correlation coefficient isI:

Answer: a value that characterizes the relationship between the independent and dependent I dependent I variables;

Linear correlation coefficient- an indicator of the tightness of the stochastic relationship between the factor and the result in the case of linear regression.

Regression coefficient- coefficient at the factor variable in the linear regression model.

Regression coefficientbshows: by how many units does y increase if x increases by 1.

Which of the regression equations yavl. power: y=a˳aͯ¹a

What method is used to estimate the parameters of the regression model method of least squares least squares.

What variables are used in the regression model one exogenous and one or more endogenous.

The classic method for estimating regression parameters is based on:- least square method.

Coefficient regression varies within: any value is applied; from 0 to 1; from -1 to 1;

The coefficient of elasticity is measured in: immeasurable quantity.

The Darwin-Chotson criterion is applied to: - selection of factors in the model; or - definitions of autocorrelation in residuals

Student's criterion- checking the significance of individual regression coefficients and the significance of the correlation coefficient.

Fisher's criterion shows: the statistical significance of the model as a whole, based on the combined validity of all its coefficients

Fisher's criterion- a method of statistical verification of the significance of the regression equation, in which the calculated (actual) value of the F-ratio is compared with its critical (theoretical) value.

What statistical characteristic is expressed by the formulaR²=… coefficient of determination

What statistical characteristic is expressed by the formula: r xy = Ca(x; y) divided by the rootVar(x)* Var(y): coefficient. Correlations

Which function is used when modeling models with constant growth power

Which points are excluded from the time series by the smoothing procedure both at the beginning and at the end.

Number of degrees of freedom fortstatistics when testing the significance of regression parameters from 35 observations and 3 independent variables 31;

Number of degrees of freedom of the denominatorF-statistics in regression of 50 observations and 4 independent variables: 45

Vector componentsEiand have a normal law.

Which variable corresponds to the concept of a function dependent variable .

Which model does not belong to the class of econometric models physical model .

What economic and mathematical models do not belong to econometric theoretical and economic models.

Which model does not belong to the class of econometric models? Answer: physical model.

What is the measured value of a variable called? Answer: options

What statistics are used to estimate the theoretical value of the general population, determined by the formula D(x)=m((x-m(x)) 2 } Sample coefficient of asymmetry

What is the name of the statistical study of the structure, relationships of phenomena, trends, patterns of economic phenomena and processes? Statistical forecast

Lag variables: are variables related to previous points in time; or -these values are dependent. change. for the previous period of time.

Linear Regression- this is a relationship (regression), which is represented by a straight line equation and expresses the simplest linear relationship.