1. LECTURE 7 CONSTRUCTION OF ECONOMETRIC MODELS WITH AUTOCORRELATED RESIDUES

2. Plan 7.1. The nature and consequences of autocorrelation. 7.2. Methods for determining autocorrelation. Durbin- Watson criterion. Von Neumann criterion. 7.3. Autocorrelation coefficients and their applications. 7.4. Models with autocorrelated residues. 7.5. The method of for Aitken parameter estimation. 7.6. Kochren-Orkatt method. 7.7. The method of converting the output information. Durbin method (self-directed learning).

3. After proposed for the algebraic dependence of the investigated process are conducted factor analysis; investigated for relevant and notrelevant factors; researched, are there factors that affect the constant of the free member variance; the value of the determination coefficient is received more than 0,6 - 0,7; conducted a study of multicolinearity factors, the next step of the research model is verification of the third Gauss-Markov precondition about absence of correlation between a free member in the i-th observation and in the j-th observation.

4. Why the question arises: 1) If residues, or in other words the set of residues in n step are not correlated among themselves, in other words, the correlation coefficient between the residues ε 1, ε 2, …, ε n ε 1+k, ε 2+k, …, ε n-k are taken a value of less than 0.5 or more than -0.5, then it is suggested that we could use the algebraic view of the model for forecasting y.

5. 2) If the value of the correlation coefficient between two series, we will provide statistically significant, this will talk about the presence of the k-th order autocorrelation. Examples: There are economic processes, in the study of which we assume that the autocorrelation can take place.

6. This is very clearly we can observe in forecasting sales of ice-cream, beer sales, sales of drinking water, i.e. in these processes, we can assume that selling is affected by ambient temperature Temperature increases - sales of beer, ice cream increases, the temperature is decreasing - sales goes down In the sale of some drugs, for example, anti-inflammatory action, when the temperature drops, the sales of such products is growing

7. Winter Summer 100 packes This is an example of positive first- order autocorrelation We gave an example, when we could assume But there are studied processes, where it is very difficult to do it

8. So if the quality of the model does not suit us, we have to test for autocorrelation of residuals without any assumptions Example, flowers sales: is depend the growth of flowers sales of on Friday from sales on Thursday. In Sumy, we obtained results on the example of statistical information in the context of flower shops

9. We conclude that the increase in sales of flowers on Friday does not depend on the previous days of weeks, and is explained by the fact that in Sumy as practically in all cities of Ukraine, on Friday are marriage registrations And this is an example of factor analysis, and not autocorrelation research.

10. 7.1. The nature and consequences of autocorrelation (7.1) (7.2) (7.3) Let us consider the classical linear multifactor model: or in matrix form Y – a column vector of the dependent variable of dimension (n  1); X – a matrix of independent variables of dimension (n  (m + 1)); a – a column vector of unknown parameters of dimension ((m + 1)  1); ε – a column vector of random errors of dimension (n  1);

11. Gauss-Marcov Assumption 3. Absence of systematic relation between the values ​​ of the random errors in any two observations 4. Random errors must be distributed independently of explanatory variables 7.1. The nature and consequences of autocorrelation

12. Autocorrelation of residues – a phenomenon, which occurs in case of violation of the assumption for the classical regression analysis on the independence of random variables (although the variance of residuals is constant there is the homoscedasticity of residues).

13. Causes of autocorrelation 1. Autocorrelation of residues occurs when the econometric model is based on time series. 2. Autocorrelation occurs in the context of the inertia and the cyclical nature of many economic processes. 3. Autocorrelation occurs due to specification of functional dependence in regression models incorrectly.

14. (7.4) (7.5) (7.6) Assume that the model has autocorrelated residues, that the random variables ε i dependent among themselves So, as in the case of heteroscedasticity, dispersion of residues equals to: Note that the presence of residual autocorrelation, as in the presence of heteroscedasticity, dispersion residues has the form

15. First order Autoregressive model (7.7) For example, if you note first order autoregressive model ρ characterizes the strength of residues connection in t period from residues values in t-1 period

16. Table 7.1 - Comparative analysis of Gauss-Markov assumptions violations in the case of heteroskedasticity and autocorrelation of residues Gauss-Markov Condition heteroskedastic ity autocorrelation Variance of residuals changeconst Covariance of residues absencepresence

17. The consequences of ignoring the matrix Ω when determining residual variances by estimating the parameters of the model by OLS The estimates of the model parameters can be unbiased, but inefficient, that is the sample variance estimation vector can be unnecessarily large Statistical criteria for t - and F-statistics obtained from the classical linear model cannot be used for analysis of variance, because their calculation does not consider the presence of residues covariance. The inefficiency of the estimation of the econometric models parameters, as a rule, leads to inefficient forecasts, so the expected value will have greate sample variance

18. 7.2. Durbin-Watson criterion Step 1. Calculation the d-statistics value (7.8)

19. Step 2. We set the significance level . With use of Durbin-Watson table for a given significance level , the number of factors m and n number of observations we have to find two values DW1 і DW2: Positive Autocorrelation is absent Negative Zone of uncertainty 0 DW 1 DW 2 2 4- DW 2 4- DW 1 4

21. 7.2. Von Neumann criterion (7.9) In von Neumann criterion we have to calculate the actual value of the criterion Hence Consequently, In case ifthere is a positive autocorrelation

22. Table of critical values ​​ for the ratio of von Neumann

23. 7.3 Autocorrelation coefficients and their applications Let's calculate residues, i.e. the deviation of actual and theoretical values for each of the i-th observation, which are located in the following sequence Let’s shift values of random deviations by one item and receive the following sequence

24. On the basis of application of random residues sequences let’s calculate the correlation coefficient between their values, which is called the first order autocorrelation coefficient, because it determines the relationship between the values of random deviations and values of the same variance, but shifted by one element

25. Let’s shift values of random deviations by two elements, we obtain the following sequence On the basis of application of random residues sequences let’s calculate the correlation coefficient, which is called the second order autocorrelation coefficient, because it determines the relationship between the values of random deviations and values of the same variance, but shifted by two elements:

26. k-th order autocorrelation coefficient

27. Noncyclical autocorrelation coefficient It reflects the degree of correlation of the series, and is calculated by the formula

28. Cyclical autocorrelation coefficient Since it is difficult to establish the probability distribution of r*, in practice is calculated the cyclical autocorrelation coefficient r 0

29. Cyclical autocorrelation coefficient If the last member of a series equals to the first one, that is, noncyclical autocorrelation coefficient equal to cyclic autocorrelation coefficient

30. 7.4 Models with the autocorelated residues 1) Aitken (Generalized OLS) method; 2) converting the original information; 3) Kochren-Orkatt method; 4) Durbin method.

31. 7.4 Models with the autocorelated residues The first two methods appropriate to apply when the residues describes by the first order autoregressive model Iterative Kochrane-Orcutt and Darbin method can be applied to estimate the parameters of econometric models, when the residues describes by autoregressive model of the highest order:

32. 7.5 The method of Aitken parameter estimation the matrix inverse to dispersion- covariance matrix of residuals Ω the matrix inverse to the matrix

34. 7.6 Kochren-Orkatt method Steps of Kochren-Orcutt method realization 1 step. Choice arbitrarily the parameter ρ value, for example, ρ =r 1. Putting it in we obtain

35. Step 2 Put and, substituting them into equation of the previous step, we can calculate Step 3 Putting into equation of the first step the value, we can calculate and

36. Step 4 We can use and to minimize the sum of squared residuals in the equation of the first step in the context of unknown parameter. Repeat the procedure, until the following parameter values and do not differ by less than a specified amount.

37. Advantages of Kochren- Orcutt method 1.Give an opportunity to find a global optimum; 2.Have relatively good convergence.

38. 7.7. The method of converting the output information Alternative approach to model parameters estimates Step 1. Transformation of the input information with use of parameter ρ (covariance of each residues value with previous one). Step 2. Application OLS for parameters estimation on the basis of the conversed data.

40. Durbin method

41. Thank you for your attention!