1. LECTURE 2. GENERALIZED LINEAR ECONOMETRIC MODEL AND METHODS OF ITS CONSTRUCTION

2. Plan 2.1 The Simple Linear Model 2.2 The empirical model of multidimensional linear regression. 2.3 Ordinary Least Squares. 2.4 OLS estimation operator. 2.5 Preconditions of using OLS – Gaus- Markov conditions. 2.6Nonlinear Model Construction on the Basis of Linear Models.

3. 2.1 The Simple Linear Model Theoretical linear multiple regression where y – variable to be explained (dependent variable) or rehresant; х 1, x 2,...,х m – independent explaning variables or regressors; a 1, a 2,..., a m – model parameters (theoretic, nonstatistic data);

4. Matrix form of an algebraic linear equation system

5. In general terms, the empirical model is written as: 2.2 Empirical model of multiple linear regression.

6. The empirical model, which is a prototype of a theoretical model. where e – random component of the regression equation.

7. Pair linear regression: where 2.3 Ordinary Least Squares. statistic theoretic

8. Example : relationship between the volume of bank loans and the cost of advertising X Y 0 1 2 3 Figure 2.1 - The relationship between the volume of bank loans and the cost of advertising y st i

9. The deviation of the theoretical values from the actual

11. Lets solve a system of linear algebraic equations using the Kronecker-Capelli theorem. We obtain a system of linear algebraic equations:

12. where The relation for the parameter α 1 estimation: To simplify the expression for α 1 lets multiply numerator and denominator of this expression by 1 divided n. We obtain:

13. To determine the parameter alpha lets return to the previous formula. We have: The expression gives us, firstly, to confirm that the amount of error is zero. In fact, secondly, dividing it into n we have an expression for determining

14. So we found a formula to determine the unknown parameters a 0 and a 1. We can write in the explicit form the regression equation y from x in which the parameters are calculated by the Ordinary least squares method, sometimes called the Ordinary least squares regression y from x. So, we have:

15. Pair linear regression Dependent variableIndependent variable 1 the volume of bank reserves the composition of the loan portfolio 2 The volume of the bank costs The volume of deposits 3 change of rating of the bank time factor

16. EXAMPLE of a regression equation illustration Table 1 - Research on effectiveness of advertising costs

17. Received linear equation will look like: To calculate the unknown parameters α 0,α 1 we consistently have to make the following calculations:

18. 2.4 OLS estimation operator

19. 2.5 Preconditions of using OLS – Gaus-Markov conditions 1. The mathematical expectation of random deviations must be equal to zero. 2. The variance of the random deviations must be a constant. 3. Random deviations should be independent each from other. 4. Random vector deviations must be independent from repressors. 5. Components of a random vector should have a normal distribution law. 6. There is no linear (correlation) relationship between repressors of matrix X. 7. Econometric models are linear relative to its parameters.

20. 2.6 Nonlinear Model Construction on the Basis of Linear Models. The influence of many factors on the variable to be explained can be described by a linear model: where y –variable to be explained or rehresant; х 1, x 2,...,х m – independent explanatory variables or regressors; α 1, α 2,..., α m – model parameters, which waas counted using OLS (practice, statistic data); e – random component of the regression equation.

21. For example, a power function after logarithmation takes the form

22. Exponential function after logarithmation takes the linear form where lny – assessment of y; lna 0 =α 0 – assessment of a 0 ; and after replacing ln х i = α i, i=1,2, …, m is linear relatively to parameters α i.

23. Hyperbolic function and Quadratic function change of variables leads to a linear form or

24. Table 2.1 - Reduction of nonlinear econometric models to the linear form № Type of functionThe change of variable 1 − 2 3 4 5

25. Table 2.1 - Reduction of nonlinear econometric models to the linear form № Type of functionThe change of variable 6 7 8 9

26. Thank you for your attention!