LECTURELECTURE 22.. GENERALIZED GENERALIZED LINEAR LINEAR ECONOMETRIC ECONOMETRIC MODEL AND MODEL AND METHODS OF ITS METHODS OF ITS CONSTRUCTIONCONSTRUCTION

PlanPlan2.1 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.

2.1 2.1 The Simple Linear ModelThe Simple Linear Model ,.,,..,,...,,| 2121 mm XXXaXXX

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Theoretical linear multiple regression

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

Matrix form of an algebraic linear equation system

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In general terms, the empirical model is written as:

2.2 2.2 Empirical model of Empirical model of multiple linear regression.multiple linear regression.

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The empirical model, which is a prototype of a theoretical model.

where e – random component of the regression equation.

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Pair linear regression:

where

2.3 Ordinary Least Squares. 2.3 Ordinary Least Squares.

statistic

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Example : relationship between the volume of bank loans and the cost of advertising

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Figure 2.1 - The relationship between the volume of bank loans and the cost of advertising

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The deviation of the theoretical values from the actual

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Lets solve a system of linear algebraic equations using the Kronecker-Capelli theorem.

We obtain a system of linear algebraic equations:

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where

The relation for the parameter α1 estimation:

To simplify the expression for α1 lets multiply numerator and denominator of this expression by 1 divided n. We obtain:

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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

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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:

Pair linear regressionPair linear regression

Dependent variable Independent variable1

the volume of bank reserves

the composition of the loan portfolio

2The volume of the bank costs

The volume of deposits

3change of rating of the bank

time factor

EXAMPLE of a regression equation illustration

Table 1 - Research on effectiveness of advertising costs

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Received linear equation will look like: 103 xyth

To calculate the unknown parameters α0,α1 we consistently have to make the following calculations:

2.2.4 OLS estimation operator4 OLS estimation operator

2.5 Preconditions of using OLS – 2.5 Preconditions of using OLS – Gaus-Markov conditionsGaus-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.

2.6 Nonlinear Model Construction on 2.6 Nonlinear Model Construction on the Basis of Linear Models.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;

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.

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For example, a power function

after logarithmation takes the form

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Exponential function

after logarithmation takes the linear form

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where lny – assessment of y; lna0 =α0– assessment of a0;

and after replacing ln хi = αi , i=1,2, …, m is linear relatively to parameters αi.

Hyperbolic function

and Quadratic function

change of variables

leads to a linear form

or

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

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

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Thank you Thank you for your for your

attention!attention!