LECTURE 5 ECONOMETRIC MODELS OF DYNAMICS

1. LECTURE5ECONOMETRIC MODELS OF DYNAMICS

2. Plan 5.1 Basic concepts and preliminary analysis of the time series. 5.1.1 The notion of time series. 5.1.2 Main characteristics of the dynamics of the time series (self-directed learning). 5.1.3 Systematic and random components of the time series. 5.2 Testing hypotheses about the existence of a trend. 5.3 Methods of filtering the seasonal components. 5.3.1 Problems of seasonality analysis (and / or cycling). 5.3.2 Filtering seasonal components with use of the seasonality index. 5.3.3 The method of time series decomposition. 5.4 Methods for time series prediction (self-directed learning). 5.4.1 Methods of social and economic forecasting. 5.4.2 Forecasting trends in time series for average characteristics. 5.4.3 Forecasting trends in time series by mechanical methods. 5.4.4 Forecasting trends in time series by analytical methods.

3. Todays lecture is devoted to the analysis of time series. • There are classic tasks that involve the usage of the dynamics econometric models. • For example: 1) the sale of tickets for buses.

4. 2) formation of tourist flows (the number of people who cross the border of one or another country). • 3) a similar example with rail and road transport between countries. • 4) it is well described with use of time series demographic questions (fertility and mortality; the number of pupils in schools, which will be in the country in 10 years).

5. 5) it is well described with use of time series the prediction of diseases development. • For example, in Italy, the number of overweight children is increased on 30%. • Italian doctors worry and say that about the number of cardiovascular diseases among indigenous populations in Italy will increase the same percentage in the next 15-20 years.

6. We gave examples of economic indicators forecasting related to the change in time. • But it is interesting not only analysis and forecasting on the basis of statistical information, but the following questions: • 1) The process is slowly faded or activated or stood still.

7. In other words, do we have the presence of trend component, or can we approximate the process by a straight line. • 2) the Second question for analysis - is there a seasonal component, that is the influence of the season (spring, autumn, winter, summer), in order to make predictions separately for each astronomical period.

8. It is conducted a study on the existence of cyclical components: what is it. • For example, the flooding of a catastrophic nature in Sumy region, when the water rises above the critical value, is passed once in a decade. It is suggested that if we are going to build a house on the territory, where 8 years, the water did not rise, this does not mean that the house is in a safe place against the flood.

9. Concerning the above we explained that the statistical analysis of time series is faced with research. trend f(t) seasonal component cyclical component random component

10. Separately, we analyze the behavior of the random component, which includes all unaccounted factors in the model. • We have to understand that the random component in real economic processes can explain any share variations and 30%, and 40% and 50%. • Therefore, on the base of given values of variation we can immediately say whether we can use the model or not.

11. So, in other words, if the share of random component exceeds permissible for the actual process value, this suggests that it is impossible to model the process. • Random component includes all unaccounted factors: for example, the politics, the weather, the mood, the mentality of partners and others. • So it is a bunch of factors that not directly but indirectly can affect the monitoring process. • And now turn to mathematical formalization.

12. 5.1 The notion of time series Dynamic series is a set of one indicator observations, ordered by the values of another indicator, which is consistently rise or fall. Time series is a dynamic series, ordered by time, or a set of economic values observations at different time moments.

13. It is typical for the time series that the order in the sequence is essential for the analysis, that is the time acts as one of the determining factors. This distinguishes the time series from random sampling, where the indexes are usre only for ease of identification. Time series can be written in a compressed form:

14. Together with the time series sometimes are considered the variational seriesthat is derived from input series through a streamlining values by the series levels. So, in the variational series on the first place is not the first time of observation, but the first value, that is, the last will be the minimum value. • The length of time series, is a time from the first to the last moment of observation. Often The length of time seriesis called a number of levels n, which form a time series.

15. Momental time series The time series formed by the indicators of the economic phenomenon at a particular time moments. For example, information about bank loans Interval time series If the time series level is formed by aggregating over a certain period (interval) of time

16. The time series can be generated as the absolute values of economic indicators and avarage or relative values - is a derivative series For example, time series which is formed from the average values of indicator

17. The main characteristics of the dynamics of the time series Characteristics Calculation formulas 1. Absolute increase 2. Growth factor 3. Increment factor 4. Growth rate or 5. Rate of increase

18. 6. Arithmetical average 7. Average value 8. Average absolute increase 9. Average growth rate 10. Average rate of increment

19. 5.1.3 Systematic and random components of the time series Typical time series can be represented as a decomposition of the four structural elements: trend (Ut), seasonal component (St), cyclical component (Vt), the random component (Et) Obviously, the actual data is not completely correspond to only one of the following functions, so that the time series yt, t=1,2,…n can be represented in the form of decomposition:

20. Decompositions of time series occurs in the following version of the model Additive models The trend model The seasonality model The trend-seasonal model Multiplicative model

21. The main components of the time series Seasonal component A trend that is growing The random component

22. An example of filtering components of some time series

23. Let we know filtered components of time series that are graphically represented in the figures а) trend component U(t)=3t1,3–5t+12 b) seasonal component c) cyclical component d) random component

26. Evolutionary factors determine the direction of economic index, a leading trend. • The trend is non-random component of the time series, which change slowly, and is described by a certain function, which is called the function of a trend or just a trend. • The trend reflects the impact on the economic indicator some constant factors, the effect of which is accumulated over time. • In the broadest sense, the trend is any orderly process that is different from the random. Sometimes the trend is understood as time shift of mathematical expectation.

27. Among the factors, which determine regular fluctuations of the time series, distinguish the following: • Seasonal component, corresponding to fluctuations that have periodic or close-to-it during one year. Seasonal factors may cover causes associated with human activities (holidays, religious traditions, etc.). • Cyclic variations are similar to seasonal fluctuations, but are exist on long time intervals. Cyclical fluctuations are explained by the effect of long-term cycles of economic, demographic, or astrophysical nature.

28. Testing hypotheses about the existence of a trend The formula for calculation the autocorrelation coefficient has the form where

29. Second order autocorrelation coefficient is determined by the formula: where

30. The properties of the autocorrelation coefficient • 1.It is based on the analogy with the linear correlation coefficient and thus characterizes the closeness of the only linear relation of the current and previous levels of series. Therefore, the autocorrelation coefficient can indicate the presence of a linear (or close to linear) trend. • 2.The sign of the autocorrelation coefficient cannot indicate the increasing or decreasing trends in the levels of the series.

31. The number of periods for calculation of the autocorrelation coefficient, is called lag. • The sequence of the first order autocorrelation coefficients, second order autocorrelation coefficients, etc. is called the autocorrelation function of the time series. • The graph of relationship of its values from the value of the lag (of the order autocorrelation coefficient) is called correlogram.

35. The main rules for identifying the trend and seasonality • 1. Time series hasn’t a trend, when the autocorrelation coefficients between the levels of time series does not depend on the time lag (statistically insignificant) • 2. Time series has a linear additive trend in the case when autocorrelation analysis indicates the linear dependence of autocorrelation coefficients change from a time lag, and the transition to first differences eliminates this dependence

36. The main rules for identifying the trend and seasonality • 3. Time series contains a seasonal component, if there isn’t a linear relationship of autocorrelation coefficients changes from a time lag, but correlogram contains a large number of significant maximum and minimum values of the autocorrelation coefficients, indicating the significant dependence between observations shifted the same time interval • 4. Time series has a linear trend and seasonal component, if its correlogram indicates the linear dependence of autocorrelation coefficients change from a lag and contains a large number of significant maximum and minimum values of the autocorrelation coefficients, but the transition to first differences excludes linear trend, but the statistical significance of certain autocorrelation coefficients remains

37. Problems of seasonality analysis (and / or cycling) • The problem of analysis of the seasonality or cyclicality is to study the seasonal fluctuations and the external cyclical mechanism. For the study of purely seasonal fluctuations we should • 1) determinethe trend and the degree of smoothness; • 2) detect the seasonal fluctuations presence in the time series; • 3) filter seasonal components in case of seasonal process confirmation; • 4) analize the dynamics (evolution) seasonal wave; • 5) research the factors that determine seasonal variations; • 6) develop the forecast trend seasonal process.

38. Filtration of seasonal components with use of seasonal indexThe easiest way, which characterizes the volatility of the research parameters level, is the calculation of each level share in the General annual volume, or the index of seasonality.Seasonality index Іj characterizes the deviation degree of the seasonal time series level relatively the average-(trend) value or, in other words, the degree of changes relatively 100 %.

39. Filtration of seasonal components with use of seasonal index The seasonal component st has a period m In addition, it is known that m multiple of n, namely Consider the following model

40. Approximate evaluation indexes are calculated as where

41. If you know the estimates of trend and seasonal components in the additive model, can estimate more accurately

42. The method of time series decomposition The sequence of construction phases additive or multiplicative trend-seasonal model: 1. Time series is smoothed by the moving average method. 2. Calculation of the difference between the input and centre medium, i.e. deviations, which characterize the seasonal factor: 3. Calculation of the assessments the seasonal component To do this it is found average values ​​for each period j: , j = 1, 2, …, m and average seasonal value:

43. To addition, it is suggested that the seasonal influence over the entire annual cycle cancel out each other, that is, for an additive model and for the multiplicative model. If these conditions are not valid, the average assessment of the seasonal component will correct.

44. Corrected estimate of seasonal components for the additive model is measured in absolute terms and equal to , For the multiplicative model, this value is ,

45. The sequence of construction phases additive or multiplicative trend-seasonal model: • 4. Withdrawal of the seasonal component from the original time series is a deseasonal series. • 5. Analytic smoothing of the deseasonal series, and obtaining estimates of the trend • 6. Calculation of the non-random component in the additive model or multiplicative model • 7. Calculation of absolute or relative errors and validation of the model. • 8. Calculation of the predictions

48. Graphical analysis of changes in lending by the additive model

49. Methods of Social and Economic Forecasting Methods of forecasting Quantitative methods Qualitative methods Time series analysis Causal methods Multivariate regression model Econometrics models Computer simulation

50. Extrapolation based on the average level During the extrapolation of the socio-economic processes based on the average number of predicted value taking as the average arithmetic value previous levels of a number which is calculated by the formula: The confidence interval for the projected series estimates is: