LECTURELECTURE 55 ECONOMETRIC ECONOMETRIC MODELS OF MODELS OF DYNAMICSDYNAMICS

PlanPlan5.1 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.

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.

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) 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.

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.

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.

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.

Concerning the above we explained that the statistical analysis of time series is faced with research.

f(t)trend

seasonal component

cyclical component

random component

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.

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.

5.1 The notion of time 5.1 The notion of time seriesseriesDynamic 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.

nyyyy 21,

nttt 21 ,

ty nt ,...,2,1

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:

Together with the time series sometimes are considered the variational series that 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 series is called a number of levels n, which form a time series.

Momental time series 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 Interval time series If the time series level is formed by aggregating over a certain period (interval) of time

Date of the loan 01.10 05.10 12.10 23.10 03.11 07.11

The size of the loans, thousand UAH 3747 3710 3839 3783 3747 3710

Month January February March April May

GDP, mln UAH

6578 7016 7353 7353 7941

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

MonthJanuary February March April May

The average salary in General, UAH/month.

152,2 153,7 165,8 161,6 163,71

The main characteristics of the dynamics of the time series

Characteristics

1. Absolute increase

2. Growth factor

3. Increment factor

4. Growth rate

5. Rate of increase

Calculation formulas

or

6. Arithmetical average

7. Average value

8. Average absolute increase

9. Average growth rate

10. Average rate of increment

5.1.3 Systematic and random components of the time series

ttttt EVSUfY ,,,

ttttt EVSUY

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:

ttt ESY

tttt ESUY

ttttt EVSUY

ttt EUY

Decompositions of time series occurs in the following version of the model

The trend model

The seasonality model

The trend-seasonal model

Multiplicative model

Additive models

The main components of the time series

y ty t

y t

t

tt 0

0

0а б

в

Seasonal component

The random component

A trend that is growing

An example of filtering components of some time series

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

ttttt EVSUfY ,,,

Systematic

components of the

time series

trend, seasonal and cyclical

components

Random

component

(errors)Et.

a part of the time series that

remains after removing from

it the systematic component

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.

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.

Testing hypotheses about the existence of a trend

1 1 22

12 2

1 1 22 2

,

n

t tt

n n

t tt t

y y y yr

y y y y

1 2 12 2

1 1, .

1 1

n n

t tt t

y y y yn n

The formula for calculation the autocorrelation coefficient has the form

where

3 2 43

22 2

3 2 43 3

,

n

t tt

n n

t tt t

y y y yr

y y y y

3 4 23 3

1 1, .

2 2

n n

t tt t

y y y yn n

Second order autocorrelation coefficient is determined by the formula:

where

The properties of the The properties of the autocorrelation coefficientautocorrelation 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.

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.

Autocorre la tion F unctionVAR1

(Standard e rro rs a re wh ite -no ise estimates)

Conf. L imit-1 .0 -0 .5 0 .0 0 .5 1 .00

15 +.588 .0624

14 +.623 .0625

13 +.655 .0626

12 +.687 .0628

11 +.715 .0629

10 +.740 .0631

9 +.764 .0632

8 +.788 .0633

7 +.816 .0635

6 +.842 .0636

5 +.868 .0637

4 +.893 .0639

3 +.921 .0640

2 +.949 .0641

1 +.976 .0643

Lag Corr. S.E.

0

2364. 0.000

2275. 0.000

2176. 0.000

2067. 0.000

1947. 0.000

1818. 0.000

1680. 0.000

1534. 0.000

1379. 0.000

1213. 0.000

1038. 0.000

852.5 0.000

656.8 0.000

449.6 0.000

230.5 0.000

Q p

Autocorre la tion F unctionNEW VAR1

(Standard e rro rs a re wh ite -no ise estimates)

Conf. L imit-1 .0 -0 .5 0 .0 0 .5 1 .00

15 -.025 .0625

14 +.066 .0626

13 -.007 .0628

12 +.068 .0629

11 +.079 .0630

10 -.023 .0632

9 +.007 .0633

8 -.014 .0635

7 +.017 .0636

6 -.048 .0637

5 +.045 .0639

4 -.068 .0640

3 +.057 .0641

2 +.039 .0643

1 +.088 .0644

Lag Corr. S.E.

0

9.48 .8509

9.32 .8098

8.22 .8288

8.21 .7685

7.06 .7944

5.48 .8567

5.35 .8029

5.34 .7211

5.29 .6248

5.22 .5163

4.65 .4600

4.15 .3867

3.00 .3916

2.21 .3313

1.85 .1738

Q p

Autocorre la tion F unctionNEW VAR3

(Standard e rro rs a re wh ite -no ise estimates)

Conf. L imit-1 .0 -0 .5 0 .0 0 .5 1 .00

15 -.113 .0744

14 -.077 .0746

13 +.184 .0749

12 +.296 .0751

11 +.029 .0754

10 -.170 .0756

9 -.064 .0759

8 -.096 .0761

7 +.029 .0764

6 +.156 .0766

5 +.031 .0769

4 -.020 .0771

3 -.089 .0774

2 -.216 .0776

1 +.086 .0778

Lag Corr. S.E.

0

47.32 .0000

45.02 .0000

43.95 .0000

37.90 .0002

22.36 .0218

22.21 .0141

17.16 .0464

16.44 .0365

14.84 .0382

14.69 .0228

10.55 .0612

10.38 .0345

10.31 .0161

8.99 .0112

1.21 .2713

Q p

The main rules for The main rules for identifying the trend and identifying the trend and seasonalityseasonality1. 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

The main rules for The main rules for identifying the trend and identifying the trend and seasonalityseasonality 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

Problems of seasonality Problems of seasonality analysis (and / or cycling)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) determine the 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.

Filtration of seasonal components Filtration of seasonal components with use of seasonal indexwith use of seasonal index

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

Filtration of seasonal components Filtration of seasonal components with use of seasonal indexwith use of seasonal index

tmt ss

mkn

ijjijij Iuy

The seasonal component st has a period m

In addition, it is known that m multiple of n, namely

Consider the following model

k

II

k

iij

j

1 %1001

k

II

k

iij

j

i

ij

ij y

yI

m

yy

m

jij

i

1

Approximate evaluation indexes are calculated as

where

ij

ij

ij

ijij

ij

u

y

u

suI

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

iju

ijs

ijI

The method of time series decompositionThe method of time series decomposition

ttij yys ~

k

ss

k

iij

j

1

2. Calculation of the difference between the input and centre medium, i.e. deviations, which characterize the seasonal factor:

1. Time series is smoothed by the moving average method.

The sequence of construction phases additive or multiplicative trend-seasonal model:

3. Calculation of the assessments the seasonal component

js

To do this it is found average values for each period j:js

, j = 1, 2, …, m

and average seasonal value: s

m

jjss

1

01

m

jjs

msm

jj

1

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.

jj ss ms

jj ss sm

Corrected estimate of seasonal components for the additive model is measured in absolute terms and equal to

,

For the multiplicative model, this value is

,

tt su

tt su

The sequence of construction phases additive or The sequence of construction phases additive or multiplicative trend-seasonal model: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

tu

Graphical analysis of changes in Graphical analysis of changes in lending by the additive modellending by the additive model

Methods of Social and Economic Forecasting

Methods of forecasting

Quantitative methods

Qualitative methods

Causal methods Time series analysis

Multivariate regression model

Econometrics models

Computer simulation

Extrapolation based on the average level

yy n

)(

nty n

11)(

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:

Extrapolation of the average absolute growth

yyy nn

)(

It can be done in the case where the general trend of development is considered to be linear. Predictive estimates obtained by the formula:

Extrapolation of the average growth rate

gr)( Tyy nn

Extrapolation can be done in the case when there is reason to believe that the general trend of the dynamic series is characterized by an exponential curve. Forecast calculated by the formula:

The method of moving average

m

yy

k

kiit

t

p

i

iit taay

10

To determine the smoothed values used formula:

More precise results obtained by the use of smoothing weighted moving average:

)31217123(35

1214120 ttttt yyyyya

The values of weighting coefficients w depending on the length of segments averaging k and order of approximating polynomials p

The method of exponential smoothing

...),)1()1(()1( 22

1

nnnn yyyy 10

ttt yyy

)1()1(

)()1( tttt yyyy

Exponential smoothing method makes it possible to describe the progress of a process where the most important position provides the latest observation, and the weight of the remaining observations decreases geometrically

Practical exponential average calculation is carried out by the recurrence formula

or

Forecasting trends in time Forecasting trends in time series for analytical methodsseries for analytical methods Regression analysisRegression analysis

,...,,2,1 ntvy ttt

Estimation of parameters of the growth curves is carried out on the basis of building a regression model in which the explanatory variable is the time

Adaptive forecasting Adaptive forecasting methodsmethods

6. Using the obtained model for future prediction

Yes

No

4. Calculation of forecast error 111

ttt yye

3. The prediction of one step 1

ty

2. Modification of the model on the basis of forecast error

1. Calculation of initial coefficients model

5. Chek: complete the process of the model

adaptation

Braun model Braun model

tt aat

y ,2,1ˆ

,1 21,21,1,1 tttt eaaa

ttt eaa 2

1,2,2 1

1ˆ ttt yye

If there is a time series of observations yt, t=1,…,n the forecast at time t on τ steps forward can be made according to the formula:

In Brown's model modification (adaptation) coefficients of the linear model carried out as follows:

where β - the discount rate data

et - Error of prediction ( ).

)(1)( Cy et

)125,1(1)( C

The point forecast is calculated after substituting the value τ into valued model. The limits of reliability prediction interval can be defined as follows:

where the value

Holt modelHolt model

tt aat

y ,2,1ˆ

tttt eaaa 11,21,1,1

ttt eaa 21,2,2

The model coefficients of the linear model

are modified as follows

Thank you for Thank you for your attention!your attention!