Abstract

"The new approach of economic forecasting horizon"

author: - Sirchenko E.
Leader of work: associate professor of department of AMI Smirnov Alexander V

Time series forecasting   by his current and past values is the important applied task. One of the most widespread methods of prediction consists in extrapolation, I.e. in an extension in the future of tendency, observed in the past. Extrapolation is based on the followings assumptions:

1. development the phenomenon it can be with sufficient foundation described a fusible  trajectory - trend;
2. general conditions, determining a progress trend in the past, will not suffer substantial changes in the future.

1. review of methods of statistical forecasting and analysis of time series, types of processes of time series;
2. design offered approach for time series with different shaken, other casual descriptions of time series;
3. analysis of conduct of  autocorrelation functions of time series;
4. analysis of conduct of autocorrelation functions of time series;
5. research of dependence of exactness of the forecasting values ot the expected depth of forecast.

Presently in practical activity economists for the estimation of depth of economic forecast are use the following dependence:

Picture 1 -  The estimation of depth of economic forecast

Nevertheless, exactly it is used specialists in area of financial and economic forecasting. In [1] the known specialist in area of the strategic planning and forecasting - Fatkhutdinov
Table 1 is Estimation of depth of forecast for the different methods of forecasting

Method	Область применения
1. Normative	Term forestalling to 10-15 years.
2. Experimental	Term forestalling to 10-15 years.
3. Parametricheskiy	Term forecastings to 10 years.
4. Extrapolation	Term forecastings to 5 years
5. Index	Term forecastings to 5 years.
6. Expert	Term forecastings unreserved.
7. Estimation of technical strategies	Term forecastings unreserved.
8. Functional	Term forecastings unreserved.
9. Combined	Term forecastings unreserved.

Complete absence of clear mathematical ground of depth of forecast western school resists for native researchers. In works [7,8] Jurik M offers approach to determination of optimum depth of forecast through an analysis chaotic components of time series.

Offered approach is simple enough and at the same time to the very important instruments for the increase of exactness of economic forecast and allows to overcome the followings lacks of approaches used presently: not account of degree of shaken of levels of series round trend; not account of presence/absence of connection between the levels of series in a base period; Absence of clear border outside which an economic forecast does not make sense Offered new approach to the estimation of depth of economic forecast synthesizes quantitative and high-quality descriptions of initial values of dynamic series and allows grounded from the mathematical point of view to set the period of forestalling for the extrapolated time series. Essence offered approach is in the following. For determination of connection between the values of initial series the selective function of autocorrelation is used. For the construction of this function the method of theory of chances is used for the case of two selections. Temporal lag is characterized by the change of values of initial time series. In practice size to to limited to the small number of the first values of selective autocorrelation function of . So, the k-member of  autocorrelation function is determined as follows:

It is further necessary to find the area of plane, being under to the crooked function. This area  characterizes the optimum depth of forecast taking into account the closeness conditions of correlation connection between basic data. That the depth of forecast must not exceed the scopes of meaningful connection of levels of dynamic series.

Extrapolation of trend and intervals of confidences of forecast.
If there are reasons to adopt two base assumptions extrapolations about which we talked higher at the analysis of development of object forecast, the process of extrapolation consists in the substitution of the proper size of period of forestalling in a formula, describing trend. . Extrapolation, generally speaking, gives a point forecast estimation. Insufficiency of such estimation and necessity of receipt of interval estimation is intuitional felt with tem, that a forecast, engulfing some interval of forecasting variable values, would be more reliable. As it is already said higher, an exact coincidence of sheets of facts and forecast estimations of points, got by extrapolation of curves, characterizing a tendency, is the phenomena improbable. The proper error has the followings sources:

1. choice of form a curve, characterizing trend, contains the element of subjectivism. In any event often there is not hard basis in order to assert that the chosen form of curve is uniquely possible or the more so the best for extrapolation in these concrete terms;
2. evaluation of parameters of curves (otherwise speaking, evaluation of trend) is made on the basis of the limited aggregate of supervisions, each of which contains casual a component. By virtue of it to the parameters of curve, and consequently, and about space some vagueness is incident to its position;
3. trend is characterized by some middle level of series, on every moment of time. Separate supervisions, as a rule, deviated from him in the past. It is natural to expect that similar sort of rejection will take place and in the future.

An error, related to its second and third source, can be reflected as a confidence interval of forecast at adopting some assumptions about property of series. By such interval a point extrapolation forecast will be transformed in an interval. Cases are fully possible, when the form of curve, that describe tendency, is chosen wrong or when a progress trend in the future can substantially change and not follow the those type of curve, which was accepted at smoothing. In last case basic assumption of extrapolation falls short of actual position. The found curve only aligns a dynamic series and characterizes a tendency only within the limits of period, overcame a supervision. Extrapolation of such trend inevitably will result in an erroneous result, thus an error such can not be estimated beforehand. In this connection it is possible only to mark to, that, presumably, it is necessary to expect growth of such error (or probabilities of its origin) at multiplying the period of forestalling of forecast. One of basic tasks, arising up at extrapolation of trend, consists in determination of intervals of confidences of forecast. Intuitional clearly, that in basis of calculation of confidence interval of forecast a measuring device must be fixed to shaken of series of the looked after values of sign. What higher this shaken, tem position of trend is less certain in space a "level is time" and the wider owe to be interval for the variants of forecast at the same degree of trust. Consequently, question about the confidence interval of forecast it is necessary to begin with consideration of measuring device to shaken. Usually such measuring device is determined as middle quadratic deviations (standard deviation) of actual supervisions from calculations, got at aligning a dynamic series. In a general view mean quadratic deviation from trend it is possible to express as:

In a general view a confidence interval for trend is determined as:

If t = I + L, to equalization will be defined by the value of confiding interval for trend, prolonged on L of time units. Confidence interval for a forecast, obviously must take into account a vagueness, related to position of trend not only, but possibility of deviation from this trend. There are cases in practice, when more or less grounded for extrapolation can apply a few types of curves. Thus seasonings are sometimes taken to the following. As each of curves characterizes one of alternative trend, obviously, that space between extrapolated trend it is some natural confiding area for the forecasting size. It is impossible to consent with such assertion. Foremost because each on possible lines of trend answers to some beforehand accepted hypothesis of development. Space between it is unconnected trend with none of them - it is possible to conduct through him unlimited number of trend. It is necessary also to add that a confidence interval is related to some level of probability going beyond his scopes. Space between trends unconnected with no level of probability, and depends on the choice of types of curves. Besides at the prolonged enough period of forestalling it space, as a rule, becomes so substation, that a similar confidence interval is lost by every sense.

For comparison of quality of decision of tasks of forecasting for traditional and offered approach the intervals of confidences of forecast are used for linear trend. As an example analysis of influencing of high-quality descriptions of time series on the depth of forecast three time series were taken by the dimension of n equal 30 with different variance round trend. In the total calculations of values of area of areas of the crooked selective autocorrelation functions the followings estimations turned out for the optimum depth of forecast: for the bit variance series are 9 levels, for middle variance are 3 levels, for strong variance is a 1 level (Picture 2).

Picture 2  The  result depths of forecast

The analysis of results shows that even at middle shaken of values of series round trend a confidence interval appears very wide (at confiding probability 90%) for the period of forestalling, exceeding calculation by the offered method. Already for forestalling on 4 levels a confidence interval was almost 25% calculation level. Pretty quickly extrapolation results in indefinite in statistical sense results. It proves possibility of application offered approach.
As so as higher a calculation was conducted based on the estimations of sizes, is possible to build dependence of estimation of depth of economic forecast on the values of his base, setting the values of temporal lag to to and the proper by him values of depth of economic forecast.
So by appearance, offered new approach to to synthesizes the estimation depths of economic forecast quantitative and high-quality descriptions of initial values of dynamic series and allows grounded from the mathematical point of view to set the period of forestalling for the extrapolated time series.

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