Econometrics assignment. Building a correlation field that characterizes the dependence of the price of an apartment on living space

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St. Petersburg State University economy and finance

Correspondence Faculty, Department of Statistics and Econometrics

Test

Econometrics

Student group №351

Hop Valentin Alexandrovich

Option 3

1. Task 1

2. Task 2

3. Task 3

4. Task 4

5. Task 5

Literature

1. Task 1

We study the relationship between the price of an apartment (y - thousand dollars) and the size of its living area (x - sq.m.) according to the following data:

	The price of the apartment, thousand dollars	Living area, sq.m

Exercise

1. Build a correlation field that characterizes the dependence of the price of an apartment on living space.

2. Determine the parameters of the steam room equation linear regression. Give an interpretation of the regression coefficient and the sign of the free term of the equation.

3. Calculate the linear correlation coefficient and explain its meaning. Determine the coefficient of determination and give its interpretation.

4.Find average error approximations.

5.Calculate standard error regression.

6. With a probability of 0.95, evaluate the statistical significance of the regression equation as a whole, as well as its parameters. Draw your own conclusions.

7. With a probability of 0.95 build confidence interval the expected value of the price of the apartment, assuming that the living area of the apartment will increase by 5% of its average value. Draw your own conclusions.

Decision

1. Building a correlation field that characterizes the dependence of the price of an apartment on living space

We build the correlation field by plotting the observational data on the coordinate plane:

When examining two factors, this constructed graph already shows whether there is a dependence or not, the nature of this dependence. In particular, the above graph already shows that with the growth of the factor x, the value of the factor y also increases. True, this dependence is fuzzy, blurry, or, correctly speaking, statistical.

2. Determination of the parameters of the equation of paired linear regression

Let us define the equation of paired linear regression by the method least squares.

The essence of the least squares method is to find the model parameters a 0 , a 1 , at which the sum of the squared deviations of the empirical (actual) values of the resulting feature from the theoretical ones, obtained by sampling equation regressions:

For linear model

A function of two variables S(a 0 , a 1) can reach an extremum when its partial derivatives are equal to zero. Calculating these partial derivatives, we obtain a system of equations for finding the parameters a 0 , a 1 linear equation regression.

In the case when the perturbing variable e has normal distribution, the coefficients a 0 , a 1 , obtained by the least squares method for linear regression, are unbiased effective estimates of the parameters b 0 , b 1 of the original equation.

We build a table of intermediate calculations, given that n=10:

We get a system of equations:

We decide this system with respect to the variables a 0 and a 1 by the Cramer method.

By Cramer's formulas we find:

;

We substitute the obtained values into the equation and get the equation:

Interpretation of the regression coefficient and sign at the free term of the equation.

The parameter a 1 =0.702 shows the average change in the result y with a change in factor x by one. Parameter a 0 =11.39=y when x=0. Since a 0 >0, the relative change in the result is slower than the change in the factor, that is, the variation in the result is less than the variation in the factor.

3. Calculate the linear correlation coefficient

The correlation coefficient of x and y (r xy) - indicates the presence or absence of a linear relationship between the variables:

If: r xy = -1, then there is a strict negative relationship; r xy = 1, then there is a strict positive relationship; r xy = 0, then linear connection is absent.

We find the necessary values:

Determine the coefficient of determination

The coefficient of determination is the square of the correlation coefficient:

The higher the determination index, the better model describes the source data. Therefore, the quality of the description of the initial data in this model is 69.8%

4. Find the average approximation error

The average approximation error is the average relative deviation of the calculated values from the actual ones:

Average approximation error:

5. Calculate the standard error of the regression

Regression standard error:

where n is the number of population units; m - number of parameters for variables. For linear regression, m = 1.

6. With a probability of 0.95, we evaluate the statistical significance of the regression equation as a whole, as well as its parameters

To assess the statistical significance of the linear regression coefficients and linear coefficient pair correlation r xy Student's t-test is applied and confidence intervals of each indicator are calculated.

According to the t-criterion, the hypothesis H 0 is put forward about the random nature of the indicators, that is, about their insignificant difference from zero. Next, the actual values of the criterion t fact are calculated for the estimated regression coefficients and the correlation coefficient r xy by comparing their values with the value of the standard error.

We make a table of intermediate calculations:

The residual sum of squares is: , and its standard deviation:

Find the standard error of the regression coefficient:

Find the standard error of parameter a 0:

We calculate the actual value of Student's criterion for the regression coefficient:

We find the tabular values of the Student's t-test at a significance level? = 0.05

The assessment of the significance of the entire regression equation as a whole is carried out using the Fisher F-test.

Fisher's F-test is to test the hypothesis H about the statistical insignificance of the regression equation. For this, a comparison of the actual F fact and the critical (tabular) F table of the values of the Fisher F-criterion is performed.

Finding the actual value of the F-criterion:

We find table value F-criterion, given k 1 = m=1, k 2 = n - m - 1=8:

Since F table< F факт, то Н 0 -гипотеза о случайной природе оцениваемых характеристик отклоняется и признается их statistical significance and reliability.

7. With a probability of 0.95, we build a confidence interval of the expected value of the apartment price, assuming that the living area of the apartment will increase by 5% of its average value

We build a table of intermediate calculations:

2. Task 2

For 79 regions of the country, the following data are known on retail trade turnover y (% of the previous year), real money incomes of the population x 1 (% of the previous year) and average nominal wages per month x 2 (thousand rubles):

; ; ; ; ;

; ; ; .

1.Build a linear multiple regression equation

2.Find the coefficient of multiple determination, including the corrected one. Draw your own conclusions.

3. Assess the significance of the regression equation through the Fisher F-test with a probability of 0.95. Draw your own conclusions.

4. Estimate the expediency of additional inclusion in the model of the factor x 2 in the presence of the factor x 1 using a private F-criterion.

1. Linear multiple regression equation

Multiple regression - a link equation with several independent variables: y=f(x 1 ,x 2 ,...,x p), where y is the dependent variable (resultant sign); х 1 ,х 2 ,…,х p - independent variables (factors).

In this problem, the multiple regression equation has the form:

Multiple regression is used in situations where it is impossible to single out one dominant factor from a variety of factors affecting the resultant trait and it is necessary to take into account the influence of several factors.

The calculation of multiple regression parameters is carried out by the least squares method, by solving a system of equations with parameters a, b 1 , b 2 .

We get a system of equations:

We solve the resulting system with respect to the variables a, b 1 , b 2 by the Cramer method

Expanded matrix of the system of equations:

We find the determinant of the matrix of coefficients:

We successively replace the columns of the matrix of coefficients with a column of free members and find the determinants of the resulting matrices:

According to Cramer's formulas, we find the values a, b 1, b 2:

.

We write the linear equation of multiple regression:

2. We find the coefficient of multiple determination, including the corrected one.

The coefficient of multiple determination is found by the formula:

Find the pair correlation coefficients: ; ; .

;

where

;

where

;

Got: ; ;

The adjusted multiple determination coefficient contains a correction for the number of degrees of freedom and is calculated as follows:

where n=79, m=2 is the number of factor features in the regression equation.

3. We check the significance of the regression equation through the Fisher F-test with a probability of 0.95

;

The tabular value of the Fisher criterion is equal to

Since F table< F факт, то Н 0 -гипотеза о случайной природе оцениваемых характеристик отклоняется и признается их статистическая значимость и надежность.

4. Evaluate the feasibility of additional inclusion of factor x 2 in the model in the presence of factor x 1 using a private F-criterion

In the previous paragraphs, the coefficient of multiple correlation was obtained, while the coefficients of pair correlation were; ; the pair regression equation y \u003d f (x) covered 27.0639% of the fluctuations of the effective trait under the influence of the factor x 1, and the additional inclusion of the factor x 2 in the analysis reduced the share of the explained variation to 15.4921%

5. Determine the partial correlation coefficients and draw conclusions.

Partial correlation coefficients are determined by the f-le:

The multiple correlation coefficient is determined by the formula:

6. Determine private and average coefficients of elasticity and draw conclusions.

Calculate the average coefficients of elasticity according to the formula:

; ;

Confidence intervals determine the limits within which the exact values of the determined indicators lie with a given degree of confidence corresponding to a given level of significance b..

To calculate a point forecast, we substitute the given value of the factor attribute x i into the regression equation. The confidence interval of the forecast is determined with the probability (1 - ??), as, where is the standard error of the point forecast.

where x k is the predicted value of x. According to the condition, the living area of the apartment (x i) should increase by 5%. Then

;

Then the confidence interval is

or

With a reliability of 0.95, the average predicted living space of apartments is contained in a confidence interval of 21.1479
3. Task 3

The model of demand and supply of goods "A" is considered:

q d - demand for goods;

q s - offer of goods;

P - the price of the goods;

Y - per capita income;

W - the price of goods in the previous period.

The reduced form of the model was:

2. Specify the method for estimating the parameters of the structural model

1.Identify the model using the necessary and sufficient condition for identification.

This model is a system of simultaneous equations, as it contains interdependent variables.

Let us check the fulfillment of the necessary identification condition for each equation of the model.

In this model, there are two endogenous variables located on the left side. These are q d and q s . The remaining variables - P, Y, W - are exogenous variables. Thus, the total number of predefined variables is 3.

For the first equation, H=1, it includes the endogenous variable q d and D=1 (the equation does not include the predefined variable W).

D+1=1+1=2>1

Therefore, the first equation is overidentifiable.

For the second equation H=1 (q s); D=2(P; Y).

D+1=1+1=2>1

The second equation is also over-identifiable

The third equation is an identity, so it is not identified.

To check for a sufficient condition, we fill in the following table of coefficients with coefficients missing in the first equation:

Matrix determinant:

The rank of the matrix is 2, that is, not less than the number of endogenous variables in the system without one. Therefore, the sufficient condition is satisfied.

2. Specify a method for estimating the parameters of the structural model

Since the system under study is precisely identifiable and can be solved by the indirect method of least squares.

3.Find the structural coefficients of the model.

The given form of the model looks like:

Here 3; - 2; 5; 1 - reduced coefficients of the model; u 1 ; u 2 - random errors.

Calculation of the structural coefficients of the model:

1) From the second equation of the reduced form, we express W (since it is not in the first equation of the structural form)

This expression contains the variables P and Y, which are included in the right side of the first equation of the structural form of the model (SFM). We substitute the resulting expression W into the first equation of the reduced form of the model (RFM)

From where we get the first SFM equation in the form:

2) There is no variable Y in the second SFM equation. From the first equation of the reduced form, we express Y

Let us substitute the resulting expression W into the second equation of the reduced form of the model (RFM):

From where we get the second SFM equation in the form:

Thus, the SFM will take the form

4. Task 4

The dynamics of the passenger turnover of transport enterprises in the region is characterized by the following data:

Billion passenger-km.

Exercise

3. Using the Durbin-Watson test, draw conclusions about the autocorrelation in the residuals in the equation under consideration.

1. Determine the first order autocorrelation coefficient and give its interpretation.

First order autocorrelation coefficient:

,

;

We make a table of intermediate calculations:

Billion passenger-km. y t

Billion passenger-km. y t-1

; ; ,

2.Build a trend equation in the form of a second order parabola. Explain the interpretation of the parameters.

The second order parabola has the form: , values t =1, 2, 3…

The second order parabola has 3 parameters b 0 , b 1 , b 2 , which are determined from a system of three equations:

We make a table of intermediate calculations:

We solve the system of equations with respect to the variables b 0 , b 1 , b 2 by the Cramer method.

Expanded matrix of the system of equations:

We find the determinant of the matrix of coefficients:

We successively replace the columns in the matrix of coefficients with a column of free terms and find the determinants of the resulting matrices:

By Cramer's formulas we find:

;;.

The second-order parabola for this case has the form:

.

We build a table of values:

3. Using the Durbin-Watson test, draw conclusions about the autocorrelation in the residuals in the equation under consideration.

Autocorrelation in the residuals is found using the Durbin-Watson test and the calculation of the value:

The value of d is the ratio of the sum of squared differences of successive residual values to the residual sum of squares according to the regression model. In almost all statistical PPPs, the value of the Durbin-Watson test is indicated along with the coefficient of determination, the values of t- and F-criteria.

The autocorrelation coefficient of the first order residuals is defined as

Between the Durbin-Watson test and the autocorrelation coefficient of the first-order residuals, the following relationship takes place:

Thus, if there is a complete positive autocorrelation in the residuals and, then d=0. If there is a complete negative autocorrelation in the residuals, then and, therefore, d=4. If there is no autocorrelation of residuals, then d=2. Hence, .

The actual value of the Durbin-Watson criterion for this model is

Let's formulate hypotheses:

H 0 - there is no autocorrelation in the residuals;

H 1 - there is a positive autocorrelation in the residuals;

H 1 * - there is a negative autocorrelation in the residuals.

We compare the actual value with the table: d L and d U , for a given number of observations n, the number of independent variables k and significance level??

We get: d L \u003d 0.66; d U ,=1.60, i.e.

4. Give an interval forecast of the expected level of passenger traffic for 2005.

We calculate the forecast error:

where S is the standard error of the second degree parabola.

We get:

5. Task 5

We study the dependence of the retail trade turnover in the region (y i - billion rubles) on the real cash expenditures of the population (x i - % compared to December of the previous year) according to the following data:

Retail trade turnover, billion rubles, y t

Real cash income of the population, % compared to December of the previous year, x t

September

Exercise

1. Determine the correlation coefficient between time series using:

a) directly the initial levels,

Correlation coefficient of x t and y t (r xy):

We find the necessary values, given that n=12. We make a table of intermediate calculations:

September

The resulting value of the correlation coefficient is close to 1, therefore, there is a fairly close relationship between X and Y.

b) the first differences in the levels of the series.

We pass from the initial data to the first level differences

September

2. Justify the difference between the results obtained and draw a conclusion about the tightness of the relationship between the time series.

These values diverge due to the intervention of the time factor. The interference of the time factor can lead to a false correlation. In order to eliminate it, there are methods, one of which was applied here.

3.Build a regression equation, including the time factor. Give an interpretation of the parameters of the equation. Make an assumption about the statistical significance of the regression coefficient at the x factor.

September

We solve the system of equations with respect to the variables a, b, c by the Cramer method.

Expanded matrix of the system of equations:

We find the determinant of the matrix of coefficients:

We successively replace the columns in the matrix of coefficients with a column of free terms and find the determinants of the resulting matrices:

By Cramer's formulas we find:

The model including the time factor has the form:

Literature

correlation regression determination trend

1. Econometrics (guidelines for the study of the discipline and the implementation of the test), Moscow INFRA-M 2002 - 88 p.;

2. Eliseeva I.I. Econometrics Moscow “Finance and statistics” 2002.-344 p.;

3. Eliseeva I.I. Workshop on econometrics Moscow “Finance and statistics” 2003.-192 p.;

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Here are free examples of conditions for solved problems in econometrics:

Solving problems in econometrics. Task number 1. Single Variable Paired Linear Regression Equation Example

The task:

For seven territories of the Ural region, the values of two signs for 201_ are known:

Posted on www.site

1. To characterize the dependence of y on x, calculate the parameters of the paired linear regression equation;
2. Calculate the linear coefficient of pair correlation and give its interpretation;
3. Calculate the coefficient of determination and give its interpretation;
4. Evaluate the quality of the resulting linear regression model through the average approximation error and Fisher's F-test.

An example of solving a problem in econometrics with explanations and an answer. An example of constructing a paired linear regression equation:

To construct a paired linear regression equation, we will compile a table of auxiliary calculations, where the necessary intermediate calculations will be made:

district number Average daily wage per worker, rub., x yx
1 66.3 41.5 2751.45
2 59.9 57.7 3456.23
3 57.3 55.8 3197.34
4 53.1 59.4 3154.14
5 51.7 56.7 2931.39
6 50.7 44.6 2261.22
7 48 52.7 2529.6
Total 387 368.4 20281.37
Mean 55.29 52.63 2897.34
σ 5.84 6.4 -
σ2 34.06 40.93 -

The coefficient b is calculated by the formula:

An example of calculating the coefficient b of the paired linear regression equation: b = (2897.34-55.2952.63)/40.93 = -0.31

Coefficient a calculate according to the formula:

Coefficient calculation example a paired linear regression equations: a = 55.29 - -0.3152.63 = 71.61

We obtain the following paired linear regression equation:

Y = 71.61-0.31x

The linear pair correlation coefficient is calculated by the formula:

An example of calculating the linear coefficient of pair correlation:

r yx = -0.316.4 / 5.84 = -0.3397

The interpretation of the value of the linear coefficient of pair correlation is carried out on the basis of the Chaddock scale. According to the Chaddock scale, there is a moderate inverse relationship between the expenditure on the purchase of food products in total expenditure and the average daily wage per worker.

r 2 yx = -0.3397-0.3397 = 0.1154 or 11.54%

Interpretation of the value of the coefficient of determination: according to the obtained value of the coefficient of determination, the variation in expenses for the purchase of food products in total expenditures is only 11.54% determined by the variation in the average daily wage of one worker, which is a low indicator.

An example of calculating the value of the average approximation error:

district number Expenses for the purchase of food products in total expenditures, %, y Y y-y A i
1 66,3 58,7 7,6 11,5
2 59,9 53,7 6,2 10,4
3 57,3 54,3 3 5,2
4 53,1 53,2 -0,1 0,2
5 51,7 54 -2,3 4,4
6 50,7 57,8 -7,1 14
7 48 55,3 -7,3 15,2
Total - - - 60,9
Mean - - - 8,7
Interpretation of the value of the average approximation error: the obtained value of the average approximation error of less than 10% indicates that the constructed paired linear regression equation has a high (good) quality.

An example of calculating the Fisher F-test: F = 0.1154 / 0.8846 * 5 = 0.65.

Interpretation of the value of Fisher's F-test. Since the obtained value of Fisher's F-criterion is less than the tabular criterion, the resulting paired linear regression equation is statistically insignificant and not suitable for describing the dependence of the share of expenditures on the purchase of food products in total expenditures only on the average daily wage of one worker. The indicator of closeness of connection is also recognized as statistically insignificant.

Consider an example of solving the previous econometrics problem in Excel. There are several ways in Excel to define the parameters of a pairwise linear regression equation. Consider an example of one of the ways to determine the parameters of a paired linear regression equation in Excel. To do this, we use the LINEST function. The solution procedure is as follows:

1. We enter the initial data into the Excel sheet

Initial data in an Excel sheet for building a linear regression model
2. Select the area of empty cells on the Excel worksheet with a range of 5 rows by 2 columns:

Building a linear regression equation in MS Excel
3. We execute the command "Formulas" - "Insert function" and in the window that opens, select the LINEST function:

4. Fill in the function arguments:

Known_values_y - a range with food spending data y

Known_values_y - range with data on average daily wages x

Const = 1, because the free term must be present in the regression equation;

Statistics = 1 because the required information should be displayed.

5. Press the "OK" button

6. To view the results of calculating the parameters of the paired linear regression equation in Excel, without removing the selection from the area, press F2 and then simultaneously CTRL + SHIFT + ENTER. We get the following results:

According to the results of calculations in Excel, the linear regression equation will look like: Y = 71.06-0.2998x. Fisher's F-test will be 0.605, determination coefficient - 0.108. Those. the parameters of the regression equation calculated using Excel slightly differ from those obtained by the analytical solution. This is due to the lack of rounding when performing intermediate calculations in Excel.

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Solving problems in econometrics. Task number 2. An example of a hyperbolic regression equation (equilateral hyperbola equation)

The task:

We study the dependence of the material consumption of products on the size of the enterprise for 10 homogeneous plants:

Factory No. Consumed materials per unit of production, kg. Output, thousand units
1 9,9 113
2 7,8 220
3 6,8 316
4 5,8 413
5 4,5 515
6 5,5 614
7 4,3 717
8 6,9 138
9 8,8 138
10 5,3 262
Based on initial data:
1. Determine the parameters of the hyperbolic regression equation (the equation of an equilateral hyperbola);
2. Calculate the value of the correlation index;
3. Determine the elasticity coefficient for the hyperbolic regression equation (equilateral hyperbola equation);
4. Assess the significance of the hyperbolic regression equation (equilateral hyperbola equation).

Free example of solving the problem in econometrics No. 2 with explanations and conclusions:

To construct a hyperbolic regression equation (the equation of an equilateral hyperbola), it is necessary to linearize the variable x. Let's make a table of auxiliary calculations:

Factory No. Consumed materials per unit of production, kg., y Output, thousand units, z yz
1 9,9 0,00885 0,087615
2 7,8 0,004545 0,035451
3 6,8 0,003165 0,021522
4 5,8 0,002421 0,014042
5 4,5 0,001942 0,008739
6 5,5 0,001629 0,00896
7 4,3 0,001395 0,005999
8 6,9 0,007246 0,049997
9 8,8 0,007246 0,063765
10 5,3 0,003817 0,02023
Total 65,6 0,042256 0,31632
Mean 6,56 0,004226 0,031632
σ 1,75 0,002535 -
σ2 3,05 0,000006 -
The parameter b of the hyperbolic regression equation is calculated by the formula:

An example of calculating the parameter b of the equation of an equilateral hyperbola:

b = (0.031632-6.560.004226)/0.000006 = 651.57

Parameter a hyperbolic regression equations are calculated by the formula:

Parameter calculation example a equations of an equilateral hyperbola:

a = 6.56-651.570.004226 = 3.81

We get the following hyperbolic regression equation:

Y = 3.81+651.57 / x

The value of the correlation index for the equation of an equilateral hyperbola is calculated by the formula:

To calculate the correlation index, we will build a table of auxiliary calculations:

Factory No. y Y (y-Y) 2 (y-y avg) 2
1 9,9 9,6 0,09 11,16
2 7,8 6,8 1 1,54
3 6,8 5,9 0,81 0,06
4 5,8 5,4 0,16 0,58
5 4,5 5,1 0,36 4,24
6 5,5 4,9 0,36 1,12
7 4,3 4,7 0,16 5,11
8 6,9 8,5 2,56 0,12
9 8,8 8,5 0,09 5,02
10 5,3 6,3 1 1,59
Total 65,6 65,7 6,59 30,54
An example of calculating the correlation index:

ρxy = √(1-6.59 / 30.54) = 0.8856

The interpretation of the correlation index is based on the Chaddock scale. According to the Chaddock scale, there is a very close relationship between output and material consumption.

The elasticity coefficient for the equation of an equilateral hyperbola (hyperbolic regression) is determined by the formula:
The formula for the elasticity coefficient for the equation of an equilateral hyperbola (hyperbolic regression)
An example of calculating the elasticity coefficient for hyperbolic regression:

E yx = -(651.57 / (3.81344.6+651.57)) = -0.33%.

Interpretation of the elasticity coefficient: The calculated elasticity coefficient for hyperbolic regression shows that with an increase in output by 1% from its average value, the consumption of materials per unit of production decreases by 0.33%% from its average value.

We will evaluate the significance of the hyperbolic regression equation (the equation of an equilateral hyperbola) using the Fisher F-test for non-linear regression. Fisher's F-test for non-linear regression is determined by the formula:

An example of calculating Fisher's F-test for non-linear regression. Fact = 0.7843 / (1-0.7843) 8 = 29.09. Since the actual value of Fisher's F-test is greater than the tabular one, the resulting hyperbolic regression equation and the indicators of closeness of connection are statistically significant.

Solving problems in econometrics. Task number 3. An example of assessing the statistical significance of regression and correlation parameters

The task:

For the territories of the region, data are given for 199x y (see table for an option):

Required:
1. Build a linear pair regression equation at from X
2. Calculate the linear pair correlation coefficient and the average approximation error
3. Assess the statistical significance of the regression and correlation parameters.
4. Run salary forecast at with the predicted value of the average per capita subsistence minimum X, which is 107% of the average level.
5. Assess the accuracy of the forecast by calculating the forecast error and its confidence interval.

To build a linear pair regression equation y from x, we will compile a table of auxiliary calculations:

region number X at yx Y dY A i
1 72 117 8424 135,63 -18,63 13,74
2 73 137 10001 136,94 0,06 0,04
3 78 125 9750 143,49 -18,49 12,89
4 73 138 10074 136,94 1,06 0,77
5 75 153 11475 139,56 13,44 9,63
6 93 175 16275 163,14 11,86 7,27
7 55 124 6820 113,36 10,64 9,39
Total 519 969 72819 969,06 -0,06 53,73
Mean 74,14 138,43 10402,71 - - 7,68
σ 10,32 18,52 - - - -
σ2 106,41 342,82 - - - -
Let's calculate the parameter b of the equation of pair regression according to the given value specified in the solution of problem 1 in econometrics:

b = (10402.71-138.4374.14)/106.41 = 1.31

Let us determine the parameter a of the pair regression equation for the given :

a = 138.43-1.3174.14 = 41.31

We get the following pair regression equation:

Y = 41.31+1.31x

Calculate the linear coefficient of pair correlation according to the data specified in the solution of problem 1 in econometrics

An example of calculating the value of the correlation coefficient:

r yx = 1.3110.32 / 18.52 = 0.73

The interpretation of the value of the linear coefficient of pair correlation is carried out on the basis of the Chaddock scale. According to the Chaddock scale, there is a direct close relationship between the per capita subsistence minimum per day of one able-bodied person and the average daily wage.

An example of calculating the value of the coefficient of determination:

r 2 yx = 0.730.73 = 0.5329 or 53.29%

Interpretation of the value of the coefficient of determination: according to the obtained value of the coefficient of determination, the variation in the average daily wage by 53.29% is determined by the variation in the average per capita subsistence minimum per day of one able-bodied person.

A = 53.73 / 7 = 7.68%.

Interpretation of the value of the average approximation error: the obtained value of the average approximation error of less than 10% indicates that the constructed pair regression equation has a high (good) quality.

We will evaluate the statistical significance of the regression and correlation parameters based on the t-test. To do this, we determine the random errors of the parameters of the linear pair regression equation.

Random parameter error a define by the formula:

An example of calculating the random error of a parameter of a paired regression equation:

m a = √(1124.58 / 5)(39225 / 5214.02) = 41.13

The random error of the coefficient b is determined by the formula:

An example of calculating the random error of the coefficient b of the paired regression equation:
m b = √((1124.58 / 5)/744.86) = 0.55
The random error of the correlation coefficient r is determined by the formula:
An example of calculating the random error of the correlation coefficient:
ta = 41.31 / 41.13 = 1.0044. Since t a a of the linear pair regression equation is statistically insignificant.
t b = 1.31 / 0.55 = 2.3818. Since t b b of the linear pair regression equation is statistically insignificant.
tr = 0.73 / 0.3056 = 2.3887. Since t r
Thus, the resulting equation is not statistically significant.
Define the marginal error for the regression parameter a: Δ a = 2.570641.13 = 105.73
The marginal error for the regression coefficient b will be: Δ b = 2.57060.55 = 1.41
ϒ amin = 41.31 - 105.73 = -64.42
ϒ amax = 41.31+105.73 = 147.04
a a.
ϒ bmin = 1.31 - 1.41 = -0.1
ϒ bmax = 1.31+1.41 = 2.72
Confidence Interpretation: Analysis of the Obtained Regression Parameter Interval b indicates that the received parameter contains a null value, i.e. confirms the conclusion about the statistical insignificance of the regression parameter b.
If the forecast value of the per capita subsistence minimum x is 107% of the average level, then the forecast value of wages will be Yп = 41.31+1.3179.33 = 145.23 rubles.
We calculate the standard error of the forecast by the formula:
Forecast error calculation example:
m yp \u003d 16.77 * 1.0858 \u003d 18.21 rubles.
The marginal forecast error will be: Δ yp = 18.212.5706 = 46.81 rubles.
ϒ pmin \u003d 145.23 - 46.81 \u003d 98.42 rubles.
ϒ pmax = 145.23+46.81 = 192.04 rubles
The range of the upper and lower boundaries of the forecast confidence interval:
D = 192.04 / 98.42 = 1.95 times.
Thus, the calculated forecast of the average daily wage turned out to be statistical, which shows the characteristics of the parameters of the regression equation, and inaccurate, which shows the high value of the range of the upper and lower boundaries of the forecast confidence interval.

Solving problems in econometrics. Task #4

For 20 territories of Russia, the following data are studied (table): dependence of the average annual per capita income at(thousand rubles) of the share of those employed in heavy physical labor in the total number of employed x 1 (%) and of the share of the economically active population in the total population x 2 (%).

Mean

Standard deviation

Tightness characteristic

Relationship equation

R yx 1 x 2* = 0,773

At x 1 x 2= -130.49 + 6.14 * x 1 + 4.13 * x 2

At x1\u003d 74.4 + 7.1 * x 1,

r yx2 = 0.507
r x1 x2 = 0.432

Y x2\u003d -355.3 + 9.2 * x 2

Required:
1. Compile an analysis of variance table to test at a significance level a= 0.05 of the statistical significance of the multiple regression equation and its indicator of closeness of connection.
2. With the help of private F- Fisher's criteria to evaluate whether it is expedient to include the factor x 1 in the multiple regression equation after the factor x 2 and how expedient it is to include x 2 after x 1.
3. Rate with t- Student's test statistical significance of the coefficients for the variables x 1 and x 2 of the multiple regression equation.

Solving problems in econometrics. Task #5

The dependence of the demand for pork x 1 on the price of it x 2 and on the price of beef x 3 is represented by the equation:
lg x 1 \u003d 0.1274 - 0.2143 * lg x 2 + 2.8254 * Igx 3
Required:
1. Present this equation in natural form (not in logarithms).
2. Assess the significance of the parameters of this equation, if it is known that the criterion for the parameter b 2 at x 2 . amounted to 0.827, and for the parameter b 3 at x 3 - 1.015

An example of solving problem No. 5 in econometrics with explanations and conclusions (formulas are not given):

The presented power equation of multiple regression is reduced to a natural form by potentiating both parts of the equation: x 1 \u003d 1.3409 * (1/ x 2 0.2143) * x 3 2.8254. The values of the regression coefficients b 1 and b 2 in the power function are equal to the elasticity coefficients of the results x 1 from x 2 and x 3: Ex 1 x 2 = - 0.2143%; Eh 1 x 3 = - 2.8254%. Demand for pork x 1 is more strongly associated with the price of beef - it increases by an average of 2.83% with a price increase of 1%. The demand for pork is inversely related to the price of pork: with a price increase of 1%, consumption decreases by an average of 0.21%. The tabular value of the t-test for a = 0.05 usually lies in the range of 2 - 3 depending on the degrees of freedom. In this example, t b2 = 0.827, t b3 = 1.015. These are very small values of the t-criterion, which indicate the random nature of the relationship, the statistical unreliability of the entire equation, so it is not recommended to use the resulting equation for forecasting.

Solving problems in econometrics. Task #6

For 20 enterprises of the region (see table), we study the dependence of output per worker y (thousand rubles) on the commissioning of new fixed assets x 1 (% of the value of funds at the end of the year) and on the proportion of highly skilled workers in the total number of workers x 2 (%).

Company number

Company number

Required:
1. Evaluate the variation indicators of each trait and draw a conclusion about the possibilities of using the least squares method to study them.
2. Analyze the linear coefficients of pair and partial correlation.
3. Write a multiple regression equation, evaluate the significance of its parameters, explain their economic meaning.
4. Using F-Fisher's test to evaluate the statistical reliability of the regression equation and R 2 yx1x2 . Compare the values of the adjusted and unadjusted linear multiple determination coefficients.
5. Using private F- Fisher's criteria to evaluate the feasibility of including the factor x 1 after x 2 and the factor x 2 after x 1 into the multiple regression equation.
6. Calculate the average partial elasticity coefficients and, on their basis, give a comparative assessment of the strength of the influence of factors on the result.

Solving problems in econometrics. Task #7

The following model is considered:
C t \u003d a 1 + b 11 Y t + b 12 * C t-1 + U 1(consumption function);
I t \u003d a 2 + b 21 * r t + b 22 * I t-1 + U 2(investment function);
r t \u003d a 3 + b 31 * Y t + b 32 * M t + U 3(money market function);
Y t = C t + I t + G t(identity of income),
where:
C t* t;
Y t- total income in the period t;
I t- investments in the period t;
r t- interest rate in the period t;
M t- money supply in the period t;
G t- government spending during the period t,
C t-1- consumption expenditure during the period t - 1;
I t-1- investments in the period t - 1;
U 1 , U 2 , U 3- random errors.
Required:
1. Assuming that there are time series of data for all variables of the model, suggest a way to estimate its parameters.
2. How will your answer to question 1 change if income identity is excluded from the model?

Solving problems in econometrics. Task #8

Based on the data for 18 months, the regression equation for the dependence of the profit of the enterprise at(million rubles) from raw material prices x 1(thousand rubles per 1 ton) and labor productivity x 2(unit of production per 1 employee):
y \u003d 200 - 1.5 * x 1 + 4.0 * x 2.
When analyzing the residual values, the values given in the table were used:

SUM E 2 t = 10500, SUM (E t - E t-1) 2 = 40000
Required:
1. For three positions, calculate y, E t, E t-1, E 2 t, (E t - E t-1) 2.
2. Calculate the Durbin-Watson criterion.
3. Evaluate the result obtained at a 5% significance level.
4. Indicate if the equation is suitable for the prediction.

Solving problems in econometrics. Task #9

The following data are available on the amount of income per family member and expenditure on goods BUT:

Indicator

Product costs BUT, rub.

Income per family member, % by 1985

Required:
1. Determine the annual absolute increase in income and expenses and draw conclusions about the development trend of each series.
2. List the main ways to eliminate the trend to build a demand model for the product BUT depending on income.
3. Build a linear demand model using the first differences in the levels of the original dynamic series.
4. Explain the economic meaning of the regression coefficient.
5. Build a linear model of product demand BUT, including the time factor. Interpret received parameters.

Solving problems in econometrics. Task #10

According to machine-building enterprises, use the methods of correlation analysis to investigate the relationship between the following indicators: X 1 - profitability (%); X 2 - bonuses and remuneration per employee (million rubles); X 3 - return on assets

2. Calculate the vectors of mean and standard deviations, the matrix of paired correlation coefficients
3. Calculate partial correlation coefficients r 12/3 and r 13/2
4. Using the correlation matrix R, calculate the estimate of the multiple correlation coefficient r 1/23
5. If a=0.05, check the significance of all paired correlation coefficients.
6. If a=0.05, check the significance of the partial correlation coefficients r 12/3 and r 13/2
7. If a=0.05, check the significance of the multiple correlation coefficient.

Solving problems in econometrics. Task #11

According to the agricultural areas of the region, it is required to build a regression model of yield based on the following indicators:
Y is the yield of grain crops (c/ha);
X 1 - the number of wheeled tractors per 100 ha;
X 2 - the number of combine harvesters per 100 ha;
X 3 - the number of tools for surface tillage per 100 ha;
X 4 - the amount of fertilizer used per hectare (t/ha);
X 5 - the amount of chemical plant protection products consumed per hectare (c / ha)

1. From the proposed data, cross out the line with the number corresponding to the last digit of the record book number.
2. Carry out a correlation analysis: analyze the relationships between the resulting variable and factor characteristics using the correlation matrix, identify multicollinearity.
3. Build regression equations with significant coefficients using a stepwise regression analysis algorithm.
4. Choose the best of the obtained regression models, based on the analysis of the values of the coefficients of determination, residual variances, taking into account the results of the economic interpretation of the models.

Solving problems in econometrics. Task #12

For the period from 1998 to 2006 for the Russian Federation, information is also given on the number of the economically active population - W t , million people, (materials of a sample survey of the State Statistics Committee).

Exercise:
1. Plot the actual levels of the time series - W t
2. Calculate the parameters of the second order parabola W t =a 0 +a 1 t+a 2 t 2
3. Evaluate the results:
- with the help of indicators of closeness of communication
- the significance of the trend model through the F-criterion;
- quality of the model through the corrected average approximation error, as well as through the autocorrelation coefficient of deviations from the trend
4. Run the forecast up to 2008.
5. Analyze the results.

Solving problems in econometrics. Task #13

It is proposed to study the interdependence of the socio-economic indicators of the region.
Y1 - expenditures of the population of the region for personal consumption, billion rubles.
Y2 - the cost of products and services of the current year, billion rubles.
Y3 - wage fund employed in the economy of the region, billion rubles.
X1 - share of employed in the economy among the total population of the region, %
X2 is the average annual cost of fixed production assets in the regional economy, billion rubles.
X3 - investments of the current year in the economy of the region, billion rubles.
At the same time, the following initial working hypotheses were formulated:
Y1=f(Y3,X1)
Y2=f(Y3,X1,X2,X3)
Y3=f(Y1,Y2,X1,X3)
Exercise:
1. On the basis of working hypotheses, construct a system of structural equations and identify them;
2. Indicate under what conditions the solution of each of the equations and the system as a whole can be found. Give a rationale for possible options for such decisions and justify the choice of the optimal variant of working hypotheses;
3. Describe the methods by which the solution of the equations will be found (indirect least squares, two-step least squares).

Solving problems in econometrics. Task #14

To test the working hypotheses (No. 1 and No. 2) on the relationship of socio-economic indicators in the region, statistical information for 2000 on the territories of the Central Federal District is used:
Y1 - average annual cost of fixed assets in the economy, billion rubles;
Y2 - the value of the gross regional product, billion rubles;
X1 - investments in fixed capital in 2000, billion rubles;
X2 is the average annual number of people employed in the economy, million people;
X3 - average monthly accrued wages of the 1st employed in the economy, thousand rubles.
Y1=f(X1;X2) - №1
Y2=f(Y1,X3) - #2
A preliminary analysis of the initial data on 18 territories revealed the presence of three territories (Moscow, Moscow region, Voronezh region) with anomalous values of features. These units should be excluded from further analysis. The values of the given indicators were calculated without taking into account the indicated anomalous units.
When processing the initial data, the following values of the linear pair correlation coefficients, average and standard deviations were obtained:
N=15.

To test the working hypothesis No. 1. To test the working hypothesis No. 2.

Exercise:
1. Make a system of equations in accordance with the put forward working hypotheses.

3. Based on the values of the matrices of pair correlation coefficients, mean and standard deviations given in the condition:
- determine the beta coefficients and build multiple regression equations on a standardized scale;
- give a comparative assessment of the strength of the influence of factors on the result;
- calculate parameters a1, a2 and a0 of multiple regression equations in natural form; - using the pair correlation coefficients and beta coefficients, calculate for each equation the linear coefficient of multiple correlation (R) and determination (R 2);
- Evaluate the statistical reliability of the identified relationships using Fisher's F-test.
4. Conclusions draw up a brief analytical note.

Solving problems in econometrics. Task #15

An analysis is made of the values of socio-economic indicators for the territories of the North-Western Federal District of the Russian Federation for 2000:
Y - investments in 2000 in fixed capital, billion rubles;
X1 is the average annual number of people employed in the economy, million people;
X2 is the average annual value of fixed assets in the economy, billion rubles;
X3 - investments in 1999 in fixed capital, billion rubles.
It is required to study the influence of these factors on the value of the gross regional product.
A preliminary analysis of the initial data on 10 territories revealed one territory (St. Petersburg) with anomalous values of features. This unit should be excluded from further analysis. The values of the given indicators are calculated without taking into account the indicated anomalous unit.
When processing the initial data, the following values were obtained:
A) - linear pair correlation coefficients, mean and standard deviations: N=9.

B) - partial correlation coefficients

Exercise
1. Based on the values of the linear pair and partial correlation coefficients, select non-collinear factors and calculate the partial correlation coefficients for them. Perform a final selection of informative factors in a multiple regression model.
2. Calculate the beta coefficients and use them to construct a multiple regression equation on a standardized scale. Analyze the strength of the relationship of each factor with the result using beta coefficients and identify strong and weak factors.
3. Use the values of the beta coefficients to calculate the parameters of the natural form equation (a1, a2 and a0). Analyze their meanings. Give a comparative assessment of the strength of the relationship of factors using general (average) elasticity coefficients
2. Determine the type of equations and system.
4. Assess the tightness of the multiple relationship using R and R 2 , and the statistical significance of the equation and the closeness of the identified relationship - through Fisher's F-test (for significance level a=0.05).
Let there be the following regression model characterizing the dependence of y on x: y = 3+2x. It is also known that rxy = 0.8; n = 20. Calculate the 99 percent confidence interval for the regression parameter b.

Solving problems in econometrics. Task #18

The macroeconomic production function model is described by the following equation: lnY = -3.52+1.53lnK+0.47lnL+e. R2 = 0.875, F = 237.4. (2.43), (0.55), (0.09). The values of standard errors for the regression coefficients are given in parentheses.
Task: 1. Evaluate the significance of the coefficients of the model using the Student's t-test and draw a conclusion about the appropriateness of including factors in the model.
2. Write the equation in power form and give an interpretation of the parameters.
3. Is it possible to say that the increase in GNP is more related to the increase in capital costs than with the increase in labor costs?

Solving problems in econometrics. Task #19

The structural form of the model looks like:
Ct = a1+b11Yt+b12Tt+e1
It = a2+b2Yt-1+e2
Tt=a3+b31Yt+e
Yt=Ct+It+Gt
where: Ct - total consumption in period t, Yt - total income in period t, It - investment in period t, Тt - taxes in period t, Gt - government spending in period t, Yt-1 - total income in period t- one.
Task: 1. Check each equation of the model for identifiability by applying the necessary and sufficient conditions for identifiability.
2. Write down the reduced form of the model.
3. Determine the method for estimating the structural parameters of each equation.

Solving problems in econometrics. Task #20

Rate on placed in the table. 6.5 statistical data from the Russian economy (%) covariance and correlation coefficient between changes in unemployment in the country in the current period x t and the growth rate of real GDP in the current period y t . What does the sign and value of the correlation coefficient r xy indicate?
Table 6.5.

Unemployment rate, U t 2) evaluate each model through the average relative approximation error and Fisher's F-test;
3) choose the best regression equation and give its justification (also take into account the linear model).

Solving problems in econometrics. Task #23

Determine the type of dependency (if any) among the data presented in the table. Choose the most adequate model for its description.
When answering a task, adhere to the following algorithm:
1) Build the correlation field of the result and the factor and formulate a hypothesis about the form of the relationship.
2) Determine the parameters of the paired linear regression equations and give an interpretation of the regression coefficient b. Calculate the linear correlation coefficient and explain its meaning. Determine the coefficient of determination and give its interpretation.
3) With a probability of 0.95, evaluate the statistical significance of the regression coefficient b and regression equations in general.
4) With a probability of 0.95, build a confidence interval of the expected value of the resultant feature if the factor feature increases by 5% of its average value.
5) Based on the table data, the correlation fields, choose an adequate regression equation;
6) Find the parameters of the regression equation using the least squares method, evaluate the significance of the relationship. Estimate the tightness of the correlation dependence, evaluate the significance of the correlation coefficient using the Fisher criterion. Draw a conclusion about the results obtained, determine the elasticity of the model and make a prediction of y t with an increase in the mean X by 5%, 10%, with a decrease in the average value X by 5%.
Make brief conclusions about the obtained values and about the model as a whole.
Budget survey data from 10 randomly selected families.

Family number

Real family income (thousand rubles)

Real household expenditure on food products (thousand rubles)

Solving problems in econometrics. Task #24

The researchers, having analyzed the activities of 10 firms, obtained the following data on the dependence of the volume of output (y) on the number of workers (x1) and the cost of fixed assets (thousand rubles) (x2)

Required:
1. Determine paired correlation coefficients. Make a conclusion.
2. Build a multiple regression equation in a standardized scale and natural form. Draw an economic conclusion.
3. Determine the multiple correlation coefficient. Make a conclusion.
4. Find the multiple coefficient of determination. Make a conclusion.
5. Determine the statistical significance of the equation using the F-test. Make a conclusion.
6. Find the predicted value of the volume of production, provided that the number of workers is 10 people, and the cost of fixed assets is 30 thousand rubles. The forecast error is 3.78. Conduct point and interval forecast. Make a conclusion.

Solving problems in econometrics. Task #25

There is a hypothetical model of the economy:
C t = a 1 + b 11 Y t + b 12 Y t + ε 1 ,
J t \u003d a 2 +b 21 Y t-1 + ε 2,
T t = a 3 + b 31 Y t + ε 3 ,
G t = C t + Y t ,
where: C t - total consumption in period t;
Y t - total income in period t;
J t - investment in period t;
T t - taxes in period t;
G t - government revenues in period t.
1. Using the necessary and sufficient identification condition, determine whether each equation of the model is identified.
2. Define the model type.
3. Determine the method for estimating model parameters.
4. Describe the sequence of actions when using the specified method.
5. Write the results in the form of an explanatory note.

Solving problems in econometrics. Task #26

The sample contains data on the price (x, c.u.) and quantity (y, c.u.) of this good purchased by households during the year:

1) Find the linear correlation coefficient. Make a conclusion.
2) Find the coefficient of determination. Make a conclusion.
3) Find the least squares estimates for the parameters of the paired linear regression equation of the form y = β 0 + β 1 x + ε. Explain the economic meaning of the results obtained.
4) Check the significance of the coefficient of determination at a significance level of 0.05. Make a conclusion.
5) Check the significance of estimates of the parameters of the regression equation at a significance level of 0.05. Make a conclusion.
6) Find a prediction for x = 30 with a confidence level of 0.95 and determine the remainder e 5 . Make a conclusion.
7) Find the confidence intervals for the conditional mean M and the individual value of the dependent variable y * x for x = 9.0. Make a conclusion.

Solving problems in econometrics. Problem #27

In table. the results of observations for x 1 , x 2 and y are presented:

1) Find the least squares estimates for the parameters of the multiple linear regression equation of the form y = β 0 + β 1 x 1 + β 2 x 2 + ε. Explain the meaning of the obtained results.
2) Check the significance of estimates of the parameters of the regression equation at a significance level of 0.05. Draw conclusions.
3) Find confidence intervals for the parameters of the regression equation with a confidence level of 0.95. Explain the meaning of the obtained results.
4) Find the coefficient of determination. Make a conclusion.
5) Check the significance of the regression equation (coefficient of determination) at a significance level of 0.05. Make a conclusion.
6) Check for the presence of homoscedasticity at a significance level of 0.05 (using Spearman's rank correlation test). Make a conclusion.
7) Check for autocorrelation at a significance level of 0.05 (using the Durbin-Watson test). Make a conclusion.

Solving problems in econometrics. Task #28

The enterprise has data for 3 years on a quarterly basis on the level of labor productivity (y, in thousand dollars per employee) and the share of the active part of fixed assets (x, in%):

Build a regression model with the inclusion of the time factor t as a separate independent variable. Explain the meaning of regression coefficients. Evaluate autocorrelation in the residuals. Give a forecast for the first quarter of the fourth year.
Gladilin A.V. Econometrics: textbook. - M.: KNORUS.
Prikhodko A.I. Workshop on econometrics. Regression analysis using Excel. - ed. Phoenix
Prosvetov G.I. Econometrics. Tasks and solutions: Educational-methodical manual. - M.: RDL.
Tikhomirov N.P., Dorokhina E.Yu. Economics: Textbook. - M.: Exam.
Polyansky Yu.N. etc. Econometrics. Problem solving using Microsoft Excel spreadsheets. Workshop. - M.: AEB MIA of Russia
Other tutorials and workshops for solving problems in econometrics.
The use of the materials given in the section without the permission of the site administration is prohibited.

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Econometrics is a science that gives a quantitative expression of the interconnections of economic phenomena and processes. Solutions to the following econometrics problems are currently available online:
Correlation-regression method of analysis

Non-parametric indicators of communication

Heteroscedasticity of the random component

autocorrelation

Autocorrelation of time series levels. Checking for autocorrelation with the construction of a correlogram;

Econometric methods for conducting expert research

Using the method of dispersion analysis, check the null hypothesis about the influence of the factor on the quality of the object.

The resulting solution is drawn up in Word format. Immediately after the solution, there is a link to download the template in Excel, which makes it possible to check all the received indicators. If the task requires a solution in Excel, then you can use the statistical functions in Excel.
Time series components

The Analytical Leveling service can be used for analytical smoothing of a time series (in a straight line) and for finding the parameters of the trend equation. To do this, you must specify the amount of initial data. If there is a lot of data, they can be inserted from Excel.

Calculation of Trend Equation Parameters.
When choosing the type of trend function, you can use the finite difference method. If the general trend is expressed by a second-order parabola, then we obtain constant second-order finite differences. If the growth rates are approximately constant, then an exponential function is used to equalize.
When choosing the form of the equation, one should proceed from the amount of information available. The more parameters the equation contains, the more observations should be for the same degree of estimation reliability.

Smoothing by the moving average method. Using

This section contains free econometrics tasks with solutions on various topics. Solutions to problems can be viewed for free; for this, screenshots of the solution (pictures) are posted. You can get the solution of the problem in Word format by paying the specified cost of the .doc file.

Here you can order an econometrics test without prepayment

Econometrics problem with Ek-8 solution

Task number: Ek-8

Solution: Free

Topic: coefficient of determination, confidence interval, forecasting

According to the condition of the previous problem for the regression equation:

Calculate deviations between actual and predicted values:

Calculate the forecast of gross production with the value of the average annual number of employees, which is 115% of the average level.

Assess the accuracy of the forecast by calculating the forecast error and its confidence interval.

Average annual number of employees (persons) The cost of gross output, (thousand rubles)
96 4603
58 4053
135 9665
153 5146
108 4850
105 7132
76 6257
119 7435
118 7560
149 4110
99 2988
128 4443
95 2198
283 15503
71 2258

For those specialties, in universities with a more in-depth study of the course of econometrics, which provides for the implementation coursework in econometrics- contact us through the order form or in any way convenient for you, and our specialists will assist in its implementation. Application programs specified by your instructor may be used.

The cost of solving problems in econometrics is from 300 rubles, depending on the complexity. Online assistance - from 1500 rubles per ticket.

For those who could not prepare for the exam, we offer:

Examples of completed work on econometrics:
When solving problems in econometrics, it is often necessary to use applied econometric software packages. We note the most common:
- data analysis package in Microsoft Excel;
- program Gretl;
- econometric package Eviews;
- Statistica package.
Let us briefly highlight the advantages and disadvantages of the listed software tools:
-Data analysis in Excel. Advantage: available and easy to use. Disadvantage: does not contain the simplest econometric tests for autocorrelation and heteroscedasticity, we do not mention other more complex econometric tests - they are not there.
-Gretl(download). Advantages: a free version is freely available, simple and easy to use, Russian interface. Disadvantage: does not contain a number of cointegration econometric tests.
-Eviews(download). Advantages: contains a lot of tests, ease of their implementation. Disadvantages: English interface, only the old version of Eviews 3 is freely available, all newer versions are paid.
-Static. Little used it, did not find advantages. Disadvantages - English interface, and the absence of many tests in econometrics.
Below are freely available examples of solving problems in econometrics in these software tools, which will contain a report on the solution of the problem and a file for implementing the problem in the econometric package. Also on this page are free versions of programs.

	Retail trade turnover, billion rubles, y t	Real cash income of the population, % compared to December of the previous year, x t








September

district number		Average daily wage per worker, rub., x	yx
1	66.3	41.5	2751.45
2	59.9	57.7	3456.23
3	57.3	55.8	3197.34
4	53.1	59.4	3154.14
5	51.7	56.7	2931.39
6	50.7	44.6	2261.22
7	48	52.7	2529.6
Total	387	368.4	20281.37
Mean	55.29	52.63	2897.34
σ	5.84	6.4	-
σ2	34.06	40.93	-

district number	Expenses for the purchase of food products in total expenditures, %, y	Y	y-y	A i
1	66,3	58,7	7,6	11,5
2	59,9	53,7	6,2	10,4
3	57,3	54,3	3	5,2
4	53,1	53,2	-0,1	0,2
5	51,7	54	-2,3	4,4
6	50,7	57,8	-7,1	14
7	48	55,3	-7,3	15,2
Total	-	-	-	60,9
Mean	-	-	-	8,7

Factory No.	Consumed materials per unit of production, kg.	Output, thousand units
1	9,9	113
2	7,8	220
3	6,8	316
4	5,8	413
5	4,5	515
6	5,5	614
7	4,3	717
8	6,9	138
9	8,8	138
10	5,3	262

Factory No.	Consumed materials per unit of production, kg., y	Output, thousand units, z	yz
1	9,9	0,00885	0,087615
2	7,8	0,004545	0,035451
3	6,8	0,003165	0,021522
4	5,8	0,002421	0,014042
5	4,5	0,001942	0,008739
6	5,5	0,001629	0,00896
7	4,3	0,001395	0,005999
8	6,9	0,007246	0,049997
9	8,8	0,007246	0,063765
10	5,3	0,003817	0,02023
Total	65,6	0,042256	0,31632
Mean	6,56	0,004226	0,031632
σ	1,75	0,002535	-
σ2	3,05	0,000006	-

Factory No.	y	Y	(y-Y) 2	(y-y avg) 2
1	9,9	9,6	0,09	11,16
2	7,8	6,8	1	1,54
3	6,8	5,9	0,81	0,06
4	5,8	5,4	0,16	0,58
5	4,5	5,1	0,36	4,24
6	5,5	4,9	0,36	1,12
7	4,3	4,7	0,16	5,11
8	6,9	8,5	2,56	0,12
9	8,8	8,5	0,09	5,02
10	5,3	6,3	1	1,59
Total	65,6	65,7	6,59	30,54

region number	X	at	yx	Y	dY	A i
1	72	117	8424	135,63	-18,63	13,74
2	73	137	10001	136,94	0,06	0,04
3	78	125	9750	143,49	-18,49	12,89
4	73	138	10074	136,94	1,06	0,77
5	75	153	11475	139,56	13,44	9,63
6	93	175	16275	163,14	11,86	7,27
7	55	124	6820	113,36	10,64	9,39
Total	519	969	72819	969,06	-0,06	53,73
Mean	74,14	138,43	10402,71	-	-	7,68
σ	10,32	18,52	-	-	-	-
σ2	106,41	342,82	-	-	-	-

Average annual number of employees (persons)	The cost of gross output, (thousand rubles)
96	4603
58	4053
135	9665
153	5146
108	4850
105	7132
76	6257
119	7435
118	7560
149	4110
99	2988
128	4443
95	2198
283	15503
71	2258

Factory No.	Consumed materials per unit of production, kg.	Output, thousand units
1	9,9	113
2	7,8	220
3	6,8	316
4	5,8	413
5	4,5	515
6	5,5	614
7	4,3	717
8	6,9	138
9	8,8	138
10	5,3	262

Econometrics assignment. Building a correlation field that characterizes the dependence of the price of an apartment on living space

Send your good work in the knowledge base is simple. Use the form below

1. Task 1

We study the relationship between the price of an apartment (y - thousand dollars) and the size of its living area (x - sq.m.) according to the following data:

Exercise

4.Find average error approximations.

6. With a probability of 0.95, evaluate the statistical significance of the regression equation as a whole, as well as its parameters. Draw your own conclusions.

Decision

1. Building a correlation field that characterizes the dependence of the price of an apartment on living space

We build the correlation field by plotting the observational data on the coordinate plane:

2. Determination of the parameters of the equation of paired linear regression

The essence of the least squares method is to find the model parameters a 0 , a 1 , at which the sum of the squared deviations of the empirical (actual) values ​​of the resulting feature from the theoretical ones, obtained by sampling equation regressions:

A function of two variables S(a 0 , a 1) can reach an extremum when its partial derivatives are equal to zero. Calculating these partial derivatives, we obtain a system of equations for finding the parameters a 0 , a 1 linear equation regression.

In the case when the perturbing variable e has normal distribution, the coefficients a 0 , a 1 , obtained by the least squares method for linear regression, are unbiased effective estimates of the parameters b 0 , b 1 of the original equation.

We build a table of intermediate calculations, given that n=10:

We get a system of equations:

We decide this system with respect to the variables a 0 and a 1 by the Cramer method.

By Cramer's formulas we find:

;

We substitute the obtained values ​​into the equation and get the equation:

Interpretation of the regression coefficient and sign at the free term of the equation.

2. Task 2

For 79 regions of the country, the following data are known on retail trade turnover y (% of the previous year), real money incomes of the population x 1 (% of the previous year) and average nominal wages per month x 2 (thousand rubles):

; ; ; ; ;

; ; ; .

3. Assess the significance of the regression equation through the Fisher F-test with a probability of 0.95. Draw your own conclusions.

4. Estimate the expediency of additional inclusion in the model of the factor x 2 in the presence of the factor x 1 using a private F-criterion.

Multiple regression - a link equation with several independent variables: y=f(x 1 ,x 2 ,...,x p), where y is the dependent variable (resultant sign); х 1 ,х 2 ,…,х p - independent variables (factors).

The calculation of multiple regression parameters is carried out by the least squares method, by solving a system of equations with parameters a, b 1 , b 2 .

We solve the resulting system with respect to the variables a, b 1 , b 2 by the Cramer method

Expanded matrix of the system of equations:

According to Cramer's formulas, we find the values ​​a, b 1, b 2:

.

We write the linear equation of multiple regression:

2. We find the coefficient of multiple determination, including the corrected one.

Find the pair correlation coefficients: ; ; .

;

;

;

where

;

;

;

where

;

;

;

Got: ; ;

where n=79, m=2 is the number of factor features in the regression equation.

3. We check the significance of the regression equation through the Fisher F-test with a probability of 0.95

;

The tabular value of the Fisher criterion is equal to

Since F table< F факт, то Н 0 -гипотеза о случайной природе оцениваемых характеристик отклоняется и признается их статистическая значимость и надежность.

4. Evaluate the feasibility of additional inclusion of factor x 2 in the model in the presence of factor x 1 using a private F-criterion

The multiple correlation coefficient is determined by the formula:

Calculate the average coefficients of elasticity according to the formula:

; ;

Confidence intervals determine the limits within which the exact values ​​of the determined indicators lie with a given degree of confidence corresponding to a given level of significance b..

To calculate a point forecast, we substitute the given value of the factor attribute x i into the regression equation. The confidence interval of the forecast is determined with the probability (1 - ??), as, where is the standard error of the point forecast.

where x k is the predicted value of x. According to the condition, the living area of ​​the apartment (x i) should increase by 5%. Then

;

or

3. Task 3

The model of demand and supply of goods "A" is considered:

q d - demand for goods;

q s - offer of goods;

P - the price of the goods;

Y - per capita income;

W - the price of goods in the previous period.

1.Identify the model using the necessary and sufficient condition for identification.

In this model, there are two endogenous variables located on the left side. These are q d and q s . The remaining variables - P, Y, W - are exogenous variables. Thus, the total number of predefined variables is 3.

For the first equation, H=1, it includes the endogenous variable q d and D=1 (the equation does not include the predefined variable W).

D+1=1+1=2>1

Therefore, the first equation is overidentifiable.

For the second equation H=1 (q s); D=2(P; Y).

D+1=1+1=2>1

To check for a sufficient condition, we fill in the following table of coefficients with coefficients missing in the first equation:

The rank of the matrix is ​​2, that is, not less than the number of endogenous variables in the system without one. Therefore, the sufficient condition is satisfied.

2. Specify a method for estimating the parameters of the structural model

Since the system under study is precisely identifiable and can be solved by the indirect method of least squares.

The essence of the least squares method is to find the model parameters a 0 , a 1 , at which the sum of the squared deviations of the empirical (actual) values of the resulting feature from the theoretical ones, obtained by sampling equation regressions:

We substitute the obtained values into the equation and get the equation:

According to Cramer's formulas, we find the values a, b 1, b 2:

Confidence intervals determine the limits within which the exact values of the determined indicators lie with a given degree of confidence corresponding to a given level of significance b..

where x k is the predicted value of x. According to the condition, the living area of the apartment (x i) should increase by 5%. Then

The rank of the matrix is 2, that is, not less than the number of endogenous variables in the system without one. Therefore, the sufficient condition is satisfied.

The second order parabola has the form: , values t =1, 2, 3…

Factory No.	Consumed materials per unit of production, kg.	Output, thousand units
1	9,9	113
2	7,8	220
3	6,8	316
4	5,8	413
5	4,5	515
6	5,5	614
7	4,3	717
8	6,9	138
9	8,8	138
10	5,3	262