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

1. LECTURE2. 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 wherey – variable to be explained (dependent variable) or rehresant; х1, x2,...,хm – independent explaning variables or regressors; a1, a2,..., am – model parameters (theoretic, nonstatistic data);

4. Matrix form of an algebraic linear equation system

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

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

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

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

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. The relation for the parameter α1 estimation: To simplify the expression for α1lets multiply numerator and denominator of this expression by 1 divided n. We obtain: where

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 a0 and a1. 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

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

17. To calculate the unknown parameters α0,α1 we consistently have to make the following calculations: Received linear equation will look like:

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: wherey –variable to be explained or rehresant; х1, x2,...,хm – independent explanatory variables or regressors; α1, α2,..., αm – 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; lna0 =α0– assessment of a0; and after replacing ln хi = αi , i=1,2, …, mis linear relatively to parameters αi.

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

24. Table 2.1 - Reduction of nonlinear econometric models to the linear form

25. Table 2.1 - Reduction of nonlinear econometric models to the linear form

26. Thank you for your attention!