TEST

discipline "Planning and forecasting

in market conditions"

on the topic: Confidence intervals of the forecast

Assessment of the adequacy and accuracy of models

Chapter 1. Theoretical part

Confidence intervals of the forecast. Assessment of the adequacy and accuracy of models

1.1 Forecast confidence intervals

final stage The application of growth curves is to extrapolate the trend based on the chosen equation. The predicted values ​​of the indicator under study are calculated by substituting time values ​​into the equation of the curve t corresponding to the lead time. The forecast obtained in this way is called a point forecast, since only one value of the predicted indicator is determined for each point in time.

In practice, in addition to a point forecast, it is desirable to determine the boundaries of a possible change in the predicted indicator, to set a "fork" of possible values ​​of the predicted indicator, i.e. calculate interval forecast.

The discrepancy between the actual data and the point forecast obtained by extrapolating the trend from growth curves can be caused by:

1. subjective fallacy of choosing the type of curve;

2. error in estimating the parameters of the curves;

3. the error associated with the deviation of individual observations from the trend that characterizes some middle level series for each moment of time.

The error associated with the second and third sources can be reflected in the form of a confidence interval of the forecast. The confidence interval, which takes into account the uncertainty associated with the position of the trend, and the possibility of deviation from this trend, is defined as:

where n is the length of the time series;

L - lead time;

y n + L -point forecast at the moment n+L;

t a - the value of Student's t-statistics;

S p - root mean square error of the forecast.

Let's assume that the trend is characterized by a straight line:

Since the parameter estimates are determined by sampling frame, represented by a time series, they contain an error. The error of the parameter a o leads to a vertical shift of the straight line, the error of the parameter a 1 - to a change in the angle of inclination of the straight line relative to the x-axis. Taking into account the scatter of specific implementations relative to the trend lines, the variance can be represented as:

(1.2.),

where is the variance of deviations of actual observations from calculated ones;

t 1 - lead time for which extrapolation is made;

t 1 = n + L ;

t- serial number of levels of the series, t = 1,2,..., n;

Serial number level in the middle of the row

Then the confidence interval can be represented as:

(1.3.),

Let us denote the root in the expression (1.3.) through K. The value of K depends only on n and L, i.e. on the length of the row and the lead time. Therefore, you can make tables of values ​​K or K * \u003d t a K. Then the interval estimate will look like:

(1.4.),

An expression similar to (1.3.) can be obtained for a second-order polynomial:

(1.5.),

(1.6.),

The dispersion of deviations of actual observations from calculated ones is determined by the expression:

(1.7.),

where y t- actual values ​​of the series levels,

Estimated values ​​of the levels of the series,

n- the length of the time series,

k- number of estimated parameters of the leveling curve.

Thus, the width of the confidence interval depends on the level of significance, the lead period, the standard deviation from the trend, and the degree of the polynomial.

The higher the degree of the polynomial, the wider the confidence interval for the same value Sy, since the variance of the trend equation is calculated as the weighted sum of the variances of the corresponding parameters of the equation

Figure 1.1. Forecast confidence intervals for a linear trend

Confidence intervals for predictions obtained using the exponential equation are determined in a similar way. The difference is that both when calculating the parameters of the curve and when calculating the mean square error, not the values ​​of the time series levels themselves are used, but their logarithms.

The same scheme can be used to determine confidence intervals for a number of curves with asymptotes, if the value of the asymptote is known (for example, for a modified exponential).

Table 1.1. values ​​are given TO* depending on the length of the time series n and lead time L for straight lines and parabolas. Obviously, as the length of the series ( n) values TO* decrease, with an increase in the lead time L values TO* increase. At the same time, the influence of the lead period is not the same for different meanings n: the longer the row length, the less influence the lead period has L .

Table 1.1.

K* values ​​for evaluation confidence intervals forecast based on a linear trend and a parabolic trend when confidence level 0,9 (7).

Linear trend	parabolic trend
Length row (n)	Lead time (L)	row length (p)	lead time (L)
7	2,6380 2,8748 3,1399	7	3,948 5,755 8,152
8	2,4631 2,6391 2,8361	8	3,459 4,754 6,461
9	2,3422 2,4786 2,6310	9	3,144 4,124 5,408
10	2,2524 2,3614 2,4827	10	2,926 3,695 4,698
11	2,1827 2,2718 2,3706	11	2,763 3,384 4,189
12	2,1274 2,2017 2,2836	12	2,636 3,148 3,808
13	2,0837 2,1463 2,2155	13	2,536 2,965 3,516
14	2,0462 2,1000 2,1590	14	2,455 2,830 3,286
15	2,0153 2,0621 2,1131	15	2,386 2,701 3,100
16	1,9883 2,0292 2,0735	16	2,330 2,604 2,950
17	1,9654 2,0015 2,0406	17	2,280 2,521 2,823
18	1,9455 1,9776 2,0124	18	2,238 2,451 2,717
19	1,9280 1,9568 1,9877	19	2,201 2,391 2,627
20	1,9117 1,9375 1,9654	20	2,169 2,339 2,549
21	1,8975 1,9210 1,9461	21	2,139 2,293 2,481
22	1,8854 1,9066 1,9294	22	2,113 2,252 2,422
23	1,8738 1,8932 1,9140	23	2,090 2,217 2,371
24	1,8631 1,8808 1,8998	24	2,069 2,185 2,325
25	1,8538 1,8701 1,8876	25	2,049 2,156 2,284

Chapter 2 Practical part

Task 1.5. Using adaptive methods in economic forecasting

1. Calculate the exponential average for the time series of the stock price of the company UM. As initial value exponential average take the average of the first 5 levels of the series. The value of the adaptation parameter a is taken equal to 0.1.

Table 1.2.

IBM stock price

t	y t	t	y t	t	y t
1	510	11	494	21	523
2	497	12	499	22	527
3	504	13	502	23	523
4	510	14	509	24	528
5	509	15	525	25	529
6	503	16	512	26	538
7	500	17	510	27	539
8	500	18	506	28	541
9	500	19	515	29	543
10	495	20	522	30	541

2. According to task No. 1, calculate the exponential average with the value of the adaptation parameter a equal to 0.5. Compare graphically the original time series and the series of exponential averages obtained with a=0.1 and a=0.5. Indicate which row is smoother.

3. Forecasting the price of IBM shares was carried out on the basis of an adaptive polynomial model of the second order

,

where is the lead time.

At the last step, the following coefficient estimates are obtained:

1 day ahead (=1);

2 days ahead (=2).

Task 1.5 solution

1. Let's define

Let us find the values ​​of the exponential average at a =0,1.

. a=0.1 - according to the condition;

; S 1 \u003d 0.1 x 510 + 0.9 x 506 \u003d 506.4;

; S 2 \u003d 0.1 x 497 + 0.9 x 506.4 \u003d 505.46;

; S 3 \u003d 0.1 x 504 + 0.9 x 505.46 \u003d 505.31, etc.

a=0.5 - according to the condition.

; S 1 \u003d 0.5 x 510 + 0.5 x 506 \u003d 508;

; S 2 \u003d 0.5 x 497 + 0.5 x 508 \u003d 502.5, etc.

The calculation results are presented in Table 1.3.

Table 1.3.

Exponential Averages

t	Exponential Average		t	Exponential Average
	a =0,1	a =0,5		a =0,1	a =0,5
1	506,4	508	16	505,7	513,3
2	505,5	502,5	17	506,1	511,7
3	505,3	503,2	18	506,1	508,8
4	505,8	506,6	19	507,0	511,9
5	506,1	507,8	20	508,5	517
6	505,8	505,4	21	509,9	520
7	505,2	502,7	22	511,6	523,5
8	504,7	501,4	23	512,8	523,2
9	504,2	500,7	24	514,3	525,6
10	503,4	497,8	25	515,8	527,3
11	502,4	495,9	26	518,0	532,7
12	502,0	497,5	27	520,1	525,8
13	502,0	499,7	28	522,2	538,4
14	502,7	504,4	29	524,3	540,7
15	505,0	514,7	30	525,9	540,9

Figure 1.2. Exponential Smoothing time series of the stock price: A - actual data; B - exponential average at alpha = 0.1; C - exponential average at alpha = 0.5

At a=0.1 exponential average has a smoother character, because in this case, random fluctuations of the time series are absorbed to the greatest extent.

3. The forecast for the adaptive polynomial model of the second order is formed at the last step by substituting into the model equation latest values coefficients and values ​​- lead time.

Forecast 1 day ahead (= 1):

Forecast 2 days ahead (= 2):

Bibliography

1. Dubrova T.A. Statistical Methods forecasting in the economy: Tutorial/ Moscow State University economics, statistics and informatics. - M.: MESI, 2003. - 52p.

2. Afanasiev V.N., Yuzbashev M.M. Time series analysis and forecasting M.: Finance and statistics, 2001.

3. Lukashin Yu.P. Regression and adaptive forecasting methods. Tutorial. – M.: MESI, 1997.

If, when analyzing the development of the forecast object, there are reasons to accept the two basic extrapolation assumptions that we discussed above, then the extrapolation process consists in substituting the corresponding value of the lead period into the formula describing the trend.

Extrapolation, generally speaking, gives a point predictive estimate. Intuitively, there is an insufficiency of such an assessment and the need to obtain interval estimation so that the forecast, covering a certain range of values ​​of the predicted variable, would be more reliable. As mentioned above, the exact match between actual data and forecast point estimates obtained by extrapolating trend curves is an unlikely occurrence. The corresponding error has the following sources:

1) the choice of the shape of the curve characterizing the trend contains an element of subjectivity. In any case, there is often no firm basis for asserting that the chosen form of the curve is the only possible one, or even the best for extrapolation under given specific conditions;

2) estimation of curve parameters (in other words, trend estimation) is based on a limited set of observations, each of which contains a random component. Because of this, the parameters of the curve, and consequently, its position in space, are characterized by some uncertainty;

3) the trend characterizes some average level of the series for each moment of time. Individual observations tended to deviate from it in the past. It is natural to expect that such deviations will occur in the future.

The error associated with its second and third sources can be reflected in the form of a confidence interval of the forecast when making certain assumptions about the property of the series. With the help of such an interval, a point extrapolation forecast is converted into an interval one.

It is quite possible that the shape of the curve describing the trend is chosen incorrectly or when the development trend in the future may change significantly and not follow the type of curve that was adopted during the alignment. AT last case the basic extrapolation assumption does not correspond to the actual state of affairs. The found curve only equalizes the dynamic series and characterizes the trend only within the period covered by the observation. Extrapolation of such a trend will inevitably lead to an erroneous result, and an error of this kind cannot be estimated in advance. In this regard, we can only note that, apparently, one should expect an increase in such an error (or the probability of its occurrence) with an increase in the forecast lead period.

One of the main tasks that arise when extrapolating a trend is to determine the confidence intervals of the forecast. It is intuitively clear that the calculation of the confidence interval of the forecast should be based on the meter of fluctuation of a number of observed values ​​of the feature. The higher this fluctuation, the less certain is the position of the trend in the “level - time” space and the wider should be the interval for forecast options with the same degree of confidence. Therefore, when constructing the confidence interval of the forecast, one should take into account the assessment of the fluctuation or variation in the levels of the series. Typically, this estimate is the average standard deviation(standard deviation) of actual observations from the calculated ones obtained during the alignment dynamic series.

Before proceeding to the determination of the confidence interval of the forecast, it is necessary to make a reservation about some conventionality of the calculation considered below. What follows is, to some extent, an arbitrary extension of the results found for the regression of sample measures to time series analysis. The point is that the assumption regression analysis about the normality of the distribution of deviations around the regression line cannot, in essence, be unconditionally asserted in the analysis of time series.

The parameters obtained in the course of statistical estimation are not free from the error associated with the fact that the amount of information on the basis of which the estimation was made is limited, and in a sense this information can be considered as a sample. In any case, shifting the observation period by only one step, or adding or eliminating members of the series due to the fact that each member of the series contains a random component, leads to a change in the numerical estimates of the parameters. Hence, the calculated values ​​bear the burden of uncertainty associated with errors in the value of the parameters.

AT general view the confidence interval for the trend is defined as

where ¾ standard error of the trend;

¾ design value yt;

¾ meaning t-Student statistics.

If a t = i+ L then the equation will determine the value of the confidence interval for the trend extended by L units of time.

The confidence interval for the forecast, obviously, should take into account not only the uncertainty associated with the position of the trend, but the possibility of deviation from this trend. In practice, there are cases when several types of curves can be applied more or less reasonably for extrapolation. In this case, the reasoning sometimes comes down to the following. Since each of the curves characterizes one of the alternative trends, it is obvious that the space between the extrapolated trends represents a certain “natural trust region” for the predicted value. One cannot agree with such a statement. First of all, because each of the possible trend lines corresponds to some previously accepted development hypothesis. The space between the trends is not associated with any of them - an unlimited number of trends can be drawn through it. It should also be added that the confidence interval is associated with a certain level of probability of going beyond its boundaries. The space between trends is not related to any level of probability, but depends on the choice of curve types. Moreover, with a sufficiently long lead time, this space, as a rule, becomes so significant that such a “confidence interval” loses all meaning.

Provided that the standard errors of the estimates of the parameters of the trend equation are taken into account (which, by definition, are selective, and therefore may not be estimates of unknown general parameters due to the manifestation random error representativeness), and without considering the sequence of transformations, we obtain general formula confidence interval of the forecast.

where - the value of the forecast calculated by the trend equation for the period t+L

¾ standard error of the trend;

K - coefficient taking into account the errors of the coefficients of the trend equation

¾ meaning t-Student statistics.

Coefficient To calculated as follows

n ¾ the number of observations (the length of the series of dynamics);

L is the number of predictions

The value of K depends only on n and L, i.e., the duration of observation and the forecasting period.

An example of calculating the forecast and constructing the confidence interval of the forecast.

The optimal trend is a linear trend . It is necessary to calculate the forecasts of import volumes in Germany for 1996 and 1997. To do this, it is necessary to determine the values ​​of the trend levels for the values ​​of the time factor 14 and 15.

Import volume in 1996:

Import volume in 1997:

standard error trend Sy = 30.727. The coefficient of confidence of Student's distribution at a significance level of 0.05 and the number of degrees of freedom is 2.16. The K coefficient is 1.428:

Thus, the lower limit of the first confidence interval is 378.62: 473.452-30.727*2.16*1.428.

The upper limit is 568.28: 473.452+30.727*2.16*1.428.

The results of the calculations must be presented in the form of a table and graphically.

	The actual value of the volume of imports in Germany for 1996	Forecast value of import volume in Germany for 1996
Lower bound of the 95% confidence interval	The actual value of the volume of imports in Germany for 1997	Forecast value of import volume in Germany for 1997	Upper bound of the 95% confidence interval

This graph is drawn as follows:

1) it is necessary to make a copy of the already existing graph of smoothing the dynamic series with a linear trend

2) complete the missing values ​​(actual levels of the series for 1996 and 1997, forecasts for 1996 and 1997, as well as the boundaries of confidence intervals).

The schedule is to some extent conditional, since accurate scale unlikely to be exposed. You can draw both by hand and using Excel drawing tools.

TEST

discipline "Planning and forecasting

in market conditions"

on the topic: Confidence intervals of the forecast

Assessment of the adequacy and accuracy of models

Chapter 1. Theoretical part

Confidence intervals of the forecast. Assessment of the adequacy and accuracy of models

1.1 Forecast confidence intervals

The final step in applying growth curves is to extrapolate the trend based on the chosen equation. The predicted values ​​of the indicator under study are calculated by substituting time values ​​into the equation of the curve t corresponding to the lead time. The forecast obtained in this way is called a point forecast, since only one value of the predicted indicator is determined for each point in time.

In practice, in addition to a point forecast, it is desirable to determine the boundaries of a possible change in the predicted indicator, to set a "fork" of possible values ​​of the predicted indicator, i.e. calculate interval forecast.

The discrepancy between the actual data and the point forecast obtained by extrapolating the trend from growth curves can be caused by:

1. subjective fallacy of choosing the type of curve;

2. error in estimating the parameters of the curves;

3. the error associated with the deviation of individual observations from the trend characterizing a certain average level of the series at each moment of time.

The error associated with the second and third sources can be reflected in the form of a confidence interval of the forecast. The confidence interval, which takes into account the uncertainty associated with the position of the trend, and the possibility of deviation from this trend, is defined as:

(1.1.),

where n is the length of the time series;

L - lead time;

y n + L -point forecast at the moment n+L;

t a - the value of Student's t-statistics;

S p - root mean square error of the forecast.

Let's assume that the trend is characterized by a straight line:

Since the parameter estimates are determined by the sample population represented by the time series, they contain an error. The error of the parameter a o leads to a vertical shift of the straight line, the error of the parameter a 1 - to a change in the angle of inclination of the straight line relative to the x-axis. Taking into account the scatter of specific implementations relative to trend lines, the variance

can be represented as: (1.2.), - dispersion of deviations of actual observations from calculated ones;

t 1 - lead time for which extrapolation is made;

t 1 = n + L ;

t- serial number of levels of the series, t = 1,2,..., n;

- the serial number of the level in the middle of the row,

Then the confidence interval can be represented as:

(1.3.),

Let us denote the root in the expression (1.3.) through K. The value of K depends only on n and L, i.e. on the length of the row and the lead time. Therefore, you can make tables of values ​​K or K * \u003d t a K. Then the interval estimate will look like:

(1.4.),

An expression similar to (1.3.) can be obtained for a second-order polynomial:

(1.5.), (1.6.),

The dispersion of deviations of actual observations from calculated ones is determined by the expression:

(1.7.),

where y t- actual values ​​of the series levels,

- calculated values ​​of the levels of the series,

n- the length of the time series,

k- number of estimated parameters of the leveling curve.

Thus, the width of the confidence interval depends on the level of significance, the lead period, the standard deviation from the trend, and the degree of the polynomial.

The higher the degree of the polynomial, the wider the confidence interval for the same value Sy, since the variance of the trend equation is calculated as the weighted sum of the variances of the corresponding parameters of the equation

Figure 1.1. Forecast confidence intervals for a linear trend

Confidence intervals for predictions obtained using the exponential equation are determined in a similar way. The difference is that both when calculating the parameters of the curve and when calculating the mean square error, not the values ​​of the time series levels themselves are used, but their logarithms.

The same scheme can be used to determine confidence intervals for a number of curves with asymptotes, if the value of the asymptote is known (for example, for a modified exponential).

Table 1.1. values ​​are given TO* depending on the length of the time series n and lead time L for straight lines and parabolas. Obviously, as the length of the series ( n) values TO* decrease, with an increase in the lead time L values TO* increase. At the same time, the influence of the lead period is not the same for different values n: the longer the row length, the less influence the lead period has L .

Table 1.1.

K* values ​​for estimating forecast confidence intervals based on a linear trend and a parabolic trend with a confidence level of 0.9 (7).

Linear trend	parabolic trend
Length row (n)	Lead time (L) 1 2 3	row length (p)	lead time (L) 1 2 3
7	2,6380 2,8748 3,1399	7	3,948 5,755 8,152
8	2,4631 2,6391 2,8361	8	3,459 4,754 6,461
9	2,3422 2,4786 2,6310	9	3,144 4,124 5,408
10	2,2524 2,3614 2,4827	10	2,926 3,695 4,698
11	2,1827 2,2718 2,3706	11	2,763 3,384 4,189
12	2,1274 2,2017 2,2836	12	2,636 3,148 3,808
13	2,0837 2,1463 2,2155	13	2,536 2,965 3,516
14	2,0462 2,1000 2,1590	14	2,455 2,830 3,286
15	2,0153 2,0621 2,1131	15	2,386 2,701 3,100
16	1,9883 2,0292 2,0735	16	2,330 2,604 2,950
17	1,9654 2,0015 2,0406	17	2,280 2,521 2,823
18	1,9455 1,9776 2,0124	18	2,238 2,451 2,717
19	1,9280 1,9568 1,9877	19	2,201 2,391 2,627
20	1,9117 1,9375 1,9654	20	2,169 2,339 2,549
21	1,8975 1,9210 1,9461	21	2,139 2,293 2,481
22	1,8854 1,9066 1,9294	22	2,113 2,252 2,422
23	1,8738 1,8932 1,9140	23	2,090 2,217 2,371
24	1,8631 1,8808 1,8998	24	2,069 2,185 2,325
25	1,8538 1,8701 1,8876	25	2,049 2,156 2,284

Chapter 2. Practical part

Task 1.5. Using adaptive methods in economic forecasting

1. Calculate the exponential average for the time series of the stock price of the company UM. As the initial value of the exponential average, take the average value of the first 5 levels of the series. The value of the adaptation parameter a is taken equal to 0.1.

Table 1.2.

IBM stock price

t	y t	t	y t	t	y t
1	510	11	494	21	523
2	497	12	499	22	527
3	504	13	502	23	523
4	510	14	509	24	528
5	509	15	525	25	529
6	503	16	512	26	538
7	500	17	510	27	539
8	500	18	506	28	541
9	500	19	515	29	543
10	495	20	522	30	541

2. According to task No. 1, calculate the exponential average with the value of the adaptation parameter a equal to 0.5. Compare graphically the original time series and the series of exponential averages obtained with a=0.1 and a=0.5. Indicate which row is smoother.

TEST

discipline "Planning and forecasting

in market conditions"

on the topic: Confidence intervals of the forecast

Assessment of the adequacy and accuracy of models

Chapter 1. Theoretical part. 3

Chapter 2. Practical part. nine

List of used literature.. 13

Chapter 1. Theoretical part

Confidence intervals of the forecast. Assessment of the adequacy and accuracy of models

1.1 Forecast confidence intervals

The final step in applying growth curves is to extrapolate the trend based on the chosen equation. The predicted values ​​of the indicator under study are calculated by substituting the values ​​of time t corresponding to the lead period into the equation of the curve. The forecast obtained in this way is called a point forecast, since only one value of the predicted indicator is determined for each point in time.

In practice, in addition to a point forecast, it is desirable to determine the boundaries of a possible change in the predicted indicator, to set a "fork" of possible values ​​of the predicted indicator, i.e. calculate interval forecast.

The discrepancy between the actual data and the point forecast obtained by extrapolating the trend from growth curves can be caused by:

1. subjective fallacy of choosing the type of curve;

2. error in estimating the parameters of the curves;

3. the error associated with the deviation of individual observations from the trend characterizing a certain average level of the series at each moment of time.

The error associated with the second and third sources can be reflected in the form of a confidence interval of the forecast. The confidence interval, which takes into account the uncertainty associated with the position of the trend, and the possibility of deviation from this trend, is defined as:

where n is the length of the time series;

L - lead time;

y n + L -point forecast at the moment n+L;

t a - the value of Student's t-statistics;

S p - root mean square error of the forecast.

Let's assume that the trend is characterized by a straight line:

Since the parameter estimates are determined by the sample population represented by the time series, they contain an error. The error of the parameter a o leads to a vertical shift of the straight line, the error of the parameter a 1 - to a change in the angle of inclination of the straight line relative to the x-axis. Taking into account the scatter of specific implementations relative to the trend lines, the variance can be represented as:

(1.2.),

where is the variance of deviations of actual observations from calculated ones;

t 1 is the lead time for which extrapolation is made;

t - serial number of levels of the series, t = 1,2,..., n;

The serial number of the level in the middle of the row,

Then the confidence interval can be represented as:

(1.3.),

Let us denote the root in the expression (1.3.) through K. The value of K depends only on n and L, i.e. on the length of the row and the lead time. Therefore, you can make tables of values ​​K or K * \u003d t a K. Then the interval estimate will look like:

(1.4.),

An expression similar to (1.3.) can be obtained for a second-order polynomial:

(1.5.),

(1.6.),

The dispersion of deviations of actual observations from calculated ones is determined by the expression:

(1.7.),

where y t are the actual values ​​of the levels of the series,

Estimated values ​​of the levels of the series,

n is the length of the time series,

k is the number of estimated parameters of the leveling curve.

Thus, the width of the confidence interval depends on the level of significance, the lead period, the standard deviation from the trend, and the degree of the polynomial.

The higher the degree of the polynomial, the wider the confidence interval for the same value of S y , since the variance of the trend equation is calculated as a weighted sum of the variances of the corresponding parameters of the equation

Figure 1.1. Forecast confidence intervals for a linear trend

Confidence intervals for predictions obtained using the exponential equation are determined in a similar way. The difference is that both when calculating the parameters of the curve and when calculating the mean square error, not the values ​​of the time series levels themselves are used, but their logarithms.

The same scheme can be used to determine confidence intervals for a number of curves with asymptotes, if the value of the asymptote is known (for example, for a modified exponential).

Table 1.1. the values ​​of K* are given depending on the length of the time series n and the lead period L for a straight line and a parabola. Obviously, with an increase in the length of the rows (n), the values ​​of K* decrease, with an increase in the lead period L, the values ​​of K* increase. At the same time, the effect of the lead period is not the same for different values ​​of n: the longer the row length, the less influence the lead period L has.

Table 1.1.

K* values ​​for estimating forecast confidence intervals based on a linear trend and a parabolic trend with a confidence level of 0.9 (7).

	Linear trend		parabolic trend
Row length (p)	Lead time (L)	row length (p)	lead time (L)
7	2,6380 2,8748 3,1399	7	3,948 5,755 8,152
8	2,4631 2,6391 2,8361	8	3,459 4,754 6,461
9	2,3422 2,4786 2,6310	9	3,144 4,124 5,408
10	2,2524 2,3614 2,4827	10	2,926 3,695 4,698
11	2,1827 2,2718 2,3706	11	2,763 3,384 4,189
12	2,1274 2,2017 2,2836	12	2,636 3,148 3,808
13	2,0837 2,1463 2,2155	13	2,536 2,965 3,516
14	2,0462 2,1000 2,1590	14	2,455 2,830 3,286
15	2,0153 2,0621 2,1131	15	2,386 2,701 3,100
16	1,9883 2,0292 2,0735	16	2,330 2,604 2,950
17	1,9654 2,0015 2,0406	17	2,280 2,521 2,823
18	1,9455 1,9776 2,0124	18	2,238 2,451 2,717
19	1,9280 1,9568 1,9877	19	2,201 2,391 2,627
20	1,9117 1,9375 1,9654	20	2,169 2,339 2,549
21	1,8975 1,9210 1,9461	21	2,139 2,293 2,481
22	1,8854 1,9066 1,9294	22	2,113 2,252 2,422
23	1,8738 1,8932 1,9140	23	2,090 2,217 2,371
24	1,8631 1,8808 1,8998	24	2,069 2,185 2,325
25	1,8538 1,8701 1,8876	25	2,049 2,156 2,284

Chapter 2. Practical part

Task 1.5. Using adaptive methods in economic forecasting

1. Calculate the exponential average for the time series of the stock price of the company UM. As the initial value of the exponential average, take the average value of the first 5 levels of the series. The value of the adaptation parameter a is taken equal to 0.1.

Table 1.2.

IBM stock price

1	510	11	494	21	523
2	497	12	499	22	527
3	504	13	502	23	523
4	510	14	509	24	528
5	509	15	525	25	529
6	503	16	512	26	538
7	500	17	510	27	539
8	500	18	506	28	541
9	500	19	515	29	543
10	495	20	522	30	541

2. According to task No. 1, calculate the exponential average with the value of the adaptation parameter a equal to 0.5. Compare graphically the original time series and the series of exponential averages obtained at a=0.1 and a=0.5. Indicate which row is smoother.

If, when analyzing the development of the forecast object, there are reasons to accept two basic extrapolation assumptions, then the extrapolation process consists in substituting the corresponding value of the lead period into the formula describing the trend. Moreover, if, for some reason, during extrapolation, it is more convenient to set the start of the countdown to a moment that differs from initial moment accepted when estimating the parameters of the equation, then for this it is sufficient to change the constant term in the corresponding polynomial. So in the equation of a straight line, when the time reference is shifted for t years ahead, the constant term will be equal to a + bm, for a parabola of the second degree it will be a + bt + st2.

Extrapolation, generally speaking, gives a point predictive estimate. Intuitively, there is an insufficiency of such an estimate and the need to obtain an interval estimate so that the forecast, covering a certain range of values ​​of the predicted variable, would be more reliable. As mentioned above, an exact match between the actual data and predictive point estimates obtained by extrapolating trend curves is unlikely. The corresponding error has the following sources: the choice of the shape of the curve characterizing the trend contains an element of subjectivity. In any case, there is often no firm basis for asserting that the chosen form of the curve is the only possible one, or even the best one for extrapolation under given specific conditions;

• 1. Estimation of curve parameters (in other words, trend estimation) is based on a limited set of observations, each of which contains a random component. Because of this, the parameters of the curve, and, consequently, its position in space, are characterized by some uncertainty;
• 2. The trend characterizes some average level of the series for each moment of time. Individual observations tended to deviate from it in the past. It is natural to expect that such deviations will occur in the future.

The error associated with its second and third sources can be reflected in the form of a confidence interval of the forecast when making certain assumptions about the property of the series. With the help of such an interval, a point extrapolation forecast is converted into an interval one. There are quite possible cases when the shape of the curve describing the trend is chosen incorrectly or when the development trend in the future may change significantly and not follow the type of curve that was adopted during the alignment. In the latter case, the basic extrapolation assumption does not correspond to the actual state of affairs. The found curve only equalizes the dynamic series and characterizes the trend only within the period covered by the observation. Extrapolation of such a trend will inevitably lead to an erroneous result, and an error of this kind cannot be estimated in advance. In this regard, we can only note that, apparently, one should expect an increase in such an error (or the probability of its occurrence) with an increase in the forecast lead period. One of the main tasks that arise when extrapolating a trend is to determine the confidence intervals of the forecast. It is intuitively clear that the calculation of the confidence interval of the forecast should be based on the meter of fluctuation of a number of observed values ​​of the feature. The higher this fluctuation, the less certain is the position of the trend in the space "level - time" and the wider should be the interval for forecast options with the same degree of confidence. Therefore, the question of the confidence interval of the forecast should begin with a consideration of the variability meter. Typically, such a meter is defined as the standard deviation ( standard deviation) actual observations from the calculated ones obtained by equalizing the time series. In general, the standard deviation from the trend can be expressed as:

In general, the confidence interval for a trend is defined as:

If t = i + L, then the equation will determine the value of the confidence interval for the trend extended by L units of time. The confidence interval for the forecast, obviously, should take into account not only the uncertainty associated with the position of the trend, but the possibility of deviation from this trend. In practice, there are cases when several types of curves can be applied more or less reasonably for extrapolation. In this case, the reasoning sometimes comes down to the following. Since each of the curves characterizes one of the alternative trends, it is obvious that the space between the extrapolated trends is some natural confidence region for the predicted value. One cannot agree with such a statement.

First of all, because each of the possible trend lines corresponds to some previously accepted development hypothesis. The space between the trends is not associated with any of them - an unlimited number of trends can be drawn through it. It should also be added that the confidence interval is associated with a certain level of probability of going beyond its boundaries. The space between trends is not related to any level of probability, but depends on the choice of curve types. Moreover, with a sufficiently long lead time, this space, as a rule, becomes so significant that such a confidence interval loses all meaning.

Figure 2 - Finding the maximum correlation interval

Animation: Frames: 20, Number of repetitions: 7, Volume: 55.9 Kb

To compare the quality of solving forecasting problems in the traditional and proposed approaches, forecast confidence intervals for a linear trend are used. As an example of the analysis of the influence of qualitative characteristics of time series on the depth of the forecast, three time series with a dimension of n equal to 30 with different fluctuations around the trend were taken. As a result of calculating the values ​​of the area of ​​the sections of the curves of the sample autocorrelation functions the following estimates were obtained for the optimal forecast depth: for a weakly oscillating series - 9 levels, for a medium oscillating series - 3 levels, for a highly oscillating series - 1 level (Figure

Figure 3 - Obtained results of forecast depth estimation

An analysis of the results shows that even with an average fluctuation of the values ​​of the series around the trend, the confidence interval turns out to be very wide (with a confidence probability of 90%) for the lead period exceeding the calculated one by the proposed method. Already for the lead by 4 levels, the confidence interval was almost 25% of the calculated level. Quite quickly, extrapolation leads to statistically uncertain results. This proves the possibility of applying the proposed approach.

Since the calculation above was carried out based on estimates of values, it seems possible to plot the dependence of the estimate of the depth of the economic forecast on the values ​​of its base by setting the values ​​of the time lag k and the corresponding values ​​of the depth of the economic forecast.

Thus, the proposed new approach to assess the depth of the economic forecast synthesizes quantitative and quality characteristics initial values ​​of the dynamic series and allows reasonable mathematical point view to set the lead period for the extrapolated time series.

forecast extrapolation strategic planning