Time series, multivariate statistical methods and methods of catastrophe theory. Goals, stages and methods of time series analysis Time series graphs
Introduction
This chapter examines the problem of describing ordered data obtained sequentially (over time). Generally speaking, ordering can occur not only in time, but also in space, for example, the diameter of a thread as a function of its length (one-dimensional case), the value of air temperature as a function of spatial coordinates (three-dimensional case).
Unlike regression analysis, where the order of rows in the observation matrix can be arbitrary, ordering is important in time series, and therefore, the relationship between values at different points in time is of interest.
If the values of a series are known at individual points in time, then such a series is called discrete, Unlike continuous, the values of which are known at any time. Let's call the interval between two consecutive moments of time tact (step). Here we will consider mainly discrete time series with a fixed clock cycle length, taken as a counting unit. Note that time series of economic indicators are, as a rule, discrete.
The series values can be directly measurable(price, profitability, temperature), or aggregated (cumulative), for example, output volume; distance traveled by cargo carriers during a time step.
If the values of a series are determined by a deterministic mathematical function, then the series is called deterministic. If these values can only be described using probabilistic models, then the time series is called random .
A phenomenon that occurs over time is called process, therefore we can talk about deterministic or random processes. In the latter case, the term is often used “stochastic process”. The analyzed segment of the time series can be considered as a particular implementation (sample) of the stochastic process being studied, generated by a hidden probabilistic mechanism.
Time series arise in many subject areas and have different natures. Various methods have been proposed for their study, which makes the theory of time series a very extensive discipline. Thus, depending on the type of time series, the following sections of the theory of time series analysis can be distinguished:
– stationary random processes that describe sequences of random variables whose probabilistic properties do not change over time. Similar processes are widespread in radio engineering, meteorology, seismology, etc.
– diffusion processes that take place during the interpenetration of liquids and gases.
– point processes that describe sequences of events, such as the receipt of requests for service, natural and man-made disasters. Similar processes are studied in queuing theory.
We will limit ourselves to considering the applied aspects of time series analysis, which are useful in solving practical problems in economics and finance. The main emphasis will be on selection methods mathematical model to describe a time series and predict its behavior.
1.Goals, methods and stages of time series analysis
The practical study of a time series involves identifying the properties of the series and drawing conclusions about the probabilistic mechanism that generates this series. The main goals in studying time series are as follows:
– description of the characteristic features of the series in a condensed form;
– construction of a time series model;
– prediction of future values based on past observations;
– control of the process that generates the time series by sampling signals warning of impending adverse events.
Achieving the set goals is not always possible, both due to the lack of initial data (insufficient duration of observation) and due to the variability of the statistical structure of the series over time.
The listed goals dictate, to a large extent, the sequence of stages of time series analysis:
1) graphical representation and description of the behavior of the series;
2) identification and exclusion of regular, non-random components of the series that depend on time;
3) study of the random component of the time series remaining after removing the regular component;
4) construction (selection) of a mathematical model to describe the random component and checking its adequacy;
5) forecasting future values of the series.
When analyzing time series, various methods are used, the most common of which are:
1) correlation analysis used to identify the characteristic features of a series (periodicities, trends, etc.);
2) spectral analysis, which makes it possible to find periodic components of a time series;
3) smoothing and filtering methods designed to transform time series to remove high-frequency and seasonal fluctuations;
5) forecasting methods.
2. Structural components of the time series
As already noted, in a time series model it is customary to distinguish two main components: deterministic and random (Fig.). The deterministic component of a time series is understood as a numerical sequence, the elements of which are calculated using a certain rule as a function of time t. By excluding the deterministic component from the data, we obtain a series oscillating around zero, which can, in one extreme case, represent purely random jumps, and in another, a smooth oscillatory motion. In most cases there will be something in between: some irregularity and some systematic effect due to the dependence of successive terms of the series.
In turn, the deterministic component may contain the following structural components:
1) trend g, which is a smooth change in the process over time and is caused by the action of long-term factors. As an example of such factors in economics, we can name: a) changes in the demographic characteristics of the population (numbers, age structure); b) technological and economic development; c) growth in consumption.
2) seasonal effect s , associated with the presence of factors that act cyclically with a predetermined frequency. The series in this case has a hierarchical time scale (for example, within a year there are seasons associated with the seasons, quarters, months) and similar effects take place at the same points in the series.
Rice. Structural components of a time series.
Typical examples of the seasonal effect: changes in highway congestion during the day, by day of the week, by time of year, peak sales of goods for schoolchildren in late August - early September. The seasonal component may change over time or be of a floating nature. So, on the graph of the volume of traffic by airliners (see figure) it can be seen that local peaks occurring during the Easter holiday “float” due to the variability of its timing.
Cyclic component c, describing long periods of relative rise and fall and consisting of cycles of variable duration and amplitude. A similar component is very typical for a number of macroeconomic indicators. Cyclical changes are caused here by the interaction of supply and demand, as well as the imposition of factors such as resource depletion, weather conditions, changes in tax policy etc. Note that the cyclic component is extremely difficult to identify by formal methods, based only on the data of the series being studied.
"Explosive" component i, otherwise intervention, which is understood as a significant short-term impact on the time series. An example of intervention is the events of “Black Tuesday” in 1994, when the dollar exchange rate rose by several tens of percent per day.
The random component of a series reflects the influence of numerous factors of a random nature and can have a varied structure, ranging from the simplest in the form of “white noise” to very complex ones, described by autoregressive-moving average models (more details below).
After selection structural components it is necessary to specify the form of their occurrence in the time series. At the top level of representation, highlighting only deterministic and random components, additive or multiplicative models are usually used.
The additive model has the form
multiplicative –
where is the value of the series at the moment t ;
The value of the deterministic component;
The value of the random component.
In turn, the deterministic component can be represented as an additive combination of deterministic components:
as a multiplicative combination:
,
or as a mixed combination, for example,
3.Models of components of the deterministic component of a time series
3.1.Trend models
The trend reflects the effect of constant long-term factors and is smooth in nature, so polynomial models, linear in parameters, are widely used to describe the trend
where are the exponent values k the polynomial rarely exceeds 5.
Along with polynomial models, economic data describing growth processes are often approximated by the following models:
– exponential
This model describes a process with a constant growth rate, that is
– logistics
For a process described by a logistic curve, the growth rate of the characteristic under study decreases linearly with increasing y, that is
– Gompertz
.
This model describes a process in which the growth rate of the characteristic under study is proportional to its logarithm
.
The last two models set the trend curves S-shaped, representing processes with an increasing growth rate in the initial stage with a gradual slowdown at the end.
When selecting a suitable functional relationship, or trend specification, a graphical representation of the time series is very useful.
Let us also note that the trend, reflecting the action of long-term factors, is decisive when constructing long-term forecasts.
3.2 Seasonal component models
The seasonal effect in a time series appears against the “background” of the trend and its identification becomes possible after a preliminary assessment of the trend. (Methods of spectral analysis, which make it possible to isolate the contribution of the seasonal component to the spectrum without calculating other components of the series, are not considered here.) Indeed, a linearly growing series of monthly data will have similar effects at the same points - smallest value in January and greatest in December; however, it is hardly appropriate to talk about a seasonal effect here: by eliminating the linear trend, we will get a series in which seasonality is completely absent. At the same time, the series describing the monthly sales volumes of New Year's cards, although it will have the same feature (minimum sales in January and maximum in December), will most likely have an oscillatory nature relative to the trend, which allows these fluctuations to be specified as a seasonal effect.
In the simplest case, the seasonal effect can manifest itself in the form of a strictly periodic dependence.
For anyone t, Where t- seasonal period.
In general, values separated by t may be related by functional dependence, that is
For example, the seasonal effect itself may contain a trend component, reflecting a change in the amplitude of fluctuations.
If the seasonal effect is additive in the series, then 
The seasonal effect model can be written as
where are Boolean, otherwise indicator, variables, one for each clock cycle within the period t seasonality. So, for a series of monthly data =0 for all t , except for January of each year, for which =1 and so on. The coefficient at shows the deviation of January values from the trend, - the deviation of February values, and so on up to . To remove ambiguity in the values of seasonality coefficients, an additional restriction is introduced, the so-called reparameterization condition, usually
In the case where the seasonal effect is multiplicative in nature, that is
a series model using indicator variables can be written as
The coefficients in this model are usually called seasonal indices.
For a fully multiplicative series
the linearization procedure is usually carried out using a logarithm operation
Let us agree to call the presented models of the seasonal effect “indicative”. If the seasonal effect is quite “smooth” - close to a harmonic - use the “harmonic” representation
,
Where d- amplitude, w- frequency conditions (in radians per unit time), a- wave phase. Because the phase is usually unknown in advance. The last expression is written as
Options A And IN can be estimated using usually regression. Angular frequency w considered famous. If the quality of the fit is unsatisfactory, along with the harmonic w fundamental wave, the model also includes the first harmonic (with double the fundamental frequency 2 w), if necessary, the second and so on harmonics. In principle, from two representations: indicator and harmonious, you should choose the one that requires fewer parameters.
3.3 Intervention model
An intervention that represents an impact significantly exceeding the fluctuations of a series can be of the nature of an “impulse” or a “step”.
The impulse effect is short-lived: once it begins, it ends almost immediately. The stepwise effect is long-lasting and sustainable. The generalized intervention model has the form
where is the value of the deterministic component of the series, described as intervention;
Moving average type coefficients;
An exogenous variable of one of two types;
(“step”), or (“impulse”)
where is a fixed point in time, called the moment of intervention.
4.Trend identification methods
The series specifications given in paragraph 3.1 are parametric functions of time. Parameters can be estimated using the method least squares the same as in regression analysis. Although the statistical prerequisites for regression analysis (see point) in time series are often not met (especially point 5 - uncorrelated disturbances), nevertheless, trend estimates turn out to be acceptable if the model is specified correctly and there are no large outliers among the observations. Violation of the assumptions of regression analysis affects not so much the coefficient estimates as their statistical properties; in particular, estimates of the variance of the random component and confidence intervals for the model coefficients are distorted.
The literature describes estimation methods under conditions of correlated disturbances, but their application requires additional information about the correlation of observations.
The main problem when identifying a trend is that it is often impossible to select a single specification for everything temporary, since the conditions of the process change. Accounting for this variability is especially important if the trend is being calculated for forecasting purposes. This is where the peculiarity of time series comes into play: data relating to the “distant past” will be irrelevant, useless or even “harmful” for estimating the parameters of the model of the current period. This is why data weighting procedures are widely used in time series analysis.
To take into account the variability of conditions, the series model is often endowed with the property of adaptability, at least at the level of parameter estimates. Adaptability is understood in the sense that parameter estimates are easily recalculated as new observations become available. Of course, the ordinary least squares method can also be given adaptive features by recalculating the estimates each time, involving old data plus fresh observations in the calculation process. However, each new recalculation leads to a change in past estimates, while adaptive algorithms free from this deficiency.
4.1 Moving averages
The moving average method is one of the oldest and most widely known methods for identifying the deterministic component of a time series. The essence of the method is to average the original series over a time interval, the length of which is selected in advance. In this case, the selected interval itself slides along the row, shifting one measure to the right each time (hence the name of the method). Due to averaging, it is possible to significantly reduce the dispersion of the random component.
The series of new values becomes smoother, which is why this procedure is called time series smoothing.
We will first consider the smoothing procedure for a series containing only a trend component, on which a random component is additively superimposed.
As is known, a smooth function can be locally represented as a polynomial with quite high degree accuracy. Let us postpone from the beginning of the time series a time interval of length (2 m+1) points and construct a polynomial of degree m for the selected values and use this polynomial to determine the trend value in ( m +1 )-th, middle, point of the group.
For definiteness, let us construct a 3rd order polynomial for an interval of seven observations. For the convenience of further transformations, we number the moments of time within the selected interval so that its middle has null value, i.e. t= -3, -2, -1, 0, 1, 2, 3. Let’s write down the required polynomial:
We find the constants using the least squares method:
We differentiate by coefficients:
;
The sums of odd orders t from -3 to +3 are equal to 0, and the equations are reduced to the form:
Using the first and third of the equations, we obtain at t=0:
Therefore, the trend value at the point t= 0 equals weighted average seven points with this point as the central one and weights
, which, due to symmetry, can be written shorter:
.
In order to calculate the trend value at the next (m+2)th point of the original series (in our case, the fifth), you should use formula (1), where the observation values are taken from the interval shifted one tick to the right, etc. to the point N - m .
number of points formula
9 
.
Properties of moving averages:
1) the sum of the weights is equal to one (since smoothing a series, all terms of which are equal to the same constant, should lead to the same constant);
2) the weights are symmetrical about the middle value;
3) formulas do not allow calculating trend values for the first and last m values of the series;
4) it is possible to derive formulas for constructing trends on an even number of points, but this would result in trend values in the middle of time steps. The trend value at observation points can be determined in this case as the half-sum of two adjacent trend values.
It should be noted that if the number 2 is even m cycles in the averaging interval (twenty-four hours a day, four weeks a month, twelve months a year), simple averaging with weights is widely practiced. Let there be, for example, observations on the last day of each month from January to December. Simply averaging the 12 weighted points gives the trend value in mid-July. To get the trend value at the end of July, you need to take the average trend value in mid-July and mid-August. It turns out that this is equivalent to averaging 13 months of data, but the values at the edges of the interval are taken with weights. So, if the smoothing interval contains the even number 2 m points, not 2 are involved in averaging m, and 2 m+1 row values:
Moving averages, smoothing the original series, leave trend and cyclical components in it. The choice of the smoothing interval value should be made based on meaningful considerations. If the series contains a seasonal component, then the value of the smoothing interval is chosen equal to or a multiple of the seasonality period. In the absence of seasonality, the smoothing interval is usually taken in the range of three to seven
Slutsky-Yul effect
Let us consider how the smoothing process affects the random component of the series, relative to which we will assume that it is centered and neighboring terms of the series are uncorrelated.
Moving average of a random series x There is:
.
Due to the centrality x and the absence of correlations between the members of the original series, we have:
AND 
.
From the obtained relations it is clear that averaging leads to a decrease in the dispersion of oscillations. In addition, the terms of the series obtained as a result of averaging are no longer independent. The derived, smoothed series has non-zero autocorrelations (correlations between series members separated by k-1 observations) up to order 2m. Thus, the derived series will be smoother than the original random series, and may exhibit systematic fluctuations. This effect is called the Slutsky-Yul effect.
4.2 Determining the order of a polynomial by the method of successive differences
If there is a series containing a polynomial (or locally represented by a polynomial) with a random element superimposed on it, then it would be natural to investigate whether the polynomial part could not be eliminated by computing successive differences of the series. Indeed, the differences of a polynomial of order k represent a polynomial of order k-1. Further, if the series contains a polynomial of order p, then the transition to differences, repeated (p+1) times, eliminates it and leaves the elements associated with the random component of the original series.
Consider, for example, the transition to differences in a series containing a third-order polynomial.
0 1 8 27 64 125
6 12 18 24
6 6 6
0 0
Taking the differences transforms the random component of the series.
In the general case we get:
;
.
From the last relation we get
Therefore, the method of successive differences of a variable consists of calculating the first, second, third, etc. differences, determining sums of squares, dividing by, etc. and detecting the moment when this relationship becomes constant. In this way we obtain estimates of the order of the polynomial contained in the original series and the variance of the random component.
4.3.Methods of exponential smoothing
Methods for constructing functions to describe observations have so far been based on the least squares criterion, according to which all observations have equal weight. However, it can be assumed that recent points should be given in some sense more weight, and observations dating back to the distant past should have less value in comparison. To some extent, we took this into account in moving averages with a finite length of the averaging segment, where the values of the weights assigned to a group of 2m+1 values do not depend on previous values. Now let's turn to another method of identifying more “recent” observations.
Let us consider a series of weights proportional to the factor b, namely, etc. Since the sum of the weights must be equal to one, i.e. , will actually be scales, etc. (assuming 0 4.3.1 Simple exponential smoothing Let's consider the simplest series equal to the sum of the constant (level) and random components: In the expression given, the discrepancies between the observed values of the series and the level estimate are taken with exponentially decreasing weights depending on the age of the data. The rating received at the time t let's denote ( t). Smoothed value at time t can be expressed through the smoothed value at the previous moment t-1 and new observation: The resulting ratio Let's rewrite it a little differently, introducing the so-called smoothing constant (0 £ a£1). From the resulting relationship it is clear that the new smoothed value is obtained from the previous one by correcting the latter for the share of error, mismatch, between the new and predicted values of the series. There is a kind of adaptation of the series level to the new data. 4.3.2 High-order exponential smoothing Let us generalize the method of exponential smoothing to the case when the process model is determined by a linear function. As before, for a given b we minimize: (Here, for ease of presentation, the signs ~ and Ù are omitted). Considering that we get Let's write down: This operation can be considered as 1st order smoothing. By analogy, we will construct 2nd order smoothing: The procedure discussed above can be generalized to the case of polynomial trends of higher order n, in which case the algebraic expressions will be more complex. For example, if the model is described by a parabola, then the triple exponential smoothing method is used. Seasonal components can be of independent interest or act as an interfering factor. In the first case, it is necessary to be able to isolate them from a series and estimate the parameters of the corresponding model. As for removing the seasonal component from the series, several methods are possible. Let us first consider the procedure for estimating seasonal effects. Let the original series be completely additive, that is It is necessary to evaluate based on observations. In other words, it is necessary to obtain estimates of the coefficients of the indicator model. As already noted, the seasonal effect manifests itself against the background of the trend, so first it is necessary to evaluate the trend component using one of the methods considered. Then, for each season, all the differences related to it are calculated where, as usual, is the observed value of the series, and is the estimated trend value. Each of these differences gives a joint estimate of the seasonal effect and the random component, which, however, differs from the original one due to the taking of the differences. By averaging the resulting differences, estimates of the effects are obtained. Assuming the original series contains an integer k seasonality periods and limiting ourselves to a simple average, we have Taking into account the reparameterization condition, which requires that the sum of seasonal effects be equal to zero, we obtain adjusted estimates In the case of a multiplicative seasonal effect, when the series model has the form They no longer calculate differences, but ratios The seasonal index is assessed by the average In practice, it is believed that in order to assess seasonal effects, a time series must contain at least five to six seasonality periods. Let's now move on to ways to remove the seasonal effect from a series. There are two such ways. Let’s call the first one “post-trend”. It is a logical consequence of the assessment procedure discussed above. For an additive model, removing the seasonal component is reduced to subtracting the estimated seasonal component from the original series. For the multiplicative model, the series values are divided by the corresponding seasonal indices. The second method does not require preliminary assessment of either the trend or seasonal components, but is based on the use of difference operators. Difference operators. When studying time series, it is often possible to represent deterministic functions of time by simple recurrence equations. For example, a linear trend can be written as The last relation is obtained from (1) by comparing two values of the series for adjacent moments t-1 and t. Considering that relation (2) is also valid for moments t-2 and t
-
1, so Model (3) does not explicitly contain parameters describing the trend. The described transformations can be described more compactly using the difference operators Models (2) and (3) can be written as It turns out that the second-order difference completely excludes the linear trend from the original series. It is easy to see that the order difference d excludes the polynomial trend of order from the series d-1. Let now the series contain a seasonal effect with a period t, So The procedure for moving from the series ( t
= 1,2,...,T) to a series is called taking the first seasonal difference, and the operator is a seasonal difference operator with a period t. From (4) it follows that It turns out that taking the seasonal difference eliminates any deterministic seasonal component from the time series. Sometimes higher order seasonal operators are useful. Thus, a second-order seasonal operator with period t There is If a series contains both a trend and a seasonal component, they can be eliminated by sequentially applying the and operators. It is easy to show that the order in which these operators are applied is not significant: We also note that a deterministic trend, consisting of a trend and a seasonal component, after applying the operators and completely degenerates, that is. However, writing the last equation in recurrent form, we get From the last relation it is clear how the series can be continued indefinitely, having at least t+1 consecutive values. linear series time system For convenience of presentation, we agree to denote random variables here as is customary in mathematical statistics - by lowercase letters. By a random process X
(
t
)
on the set T is a function whose values are random for each tÎT. If the elements of T are countable (discrete time), then random process often called a random sequence. A complete mathematical description of a random process involves specifying a system of distribution functions: - for each tОT, (1) – for each pair of elements and in general for any finite number of elements Functions (1), (2), (3) are called finite-dimensional distributions of a random process. It is almost impossible to construct such a system of functions for an arbitrary random process. Typically, random processes are specified using a priori assumptions about its properties, such as the independence of increments, the Markovian nature of trajectories, etc. A process for which all finite-dimensional distributions are normal is called normal (Gaussian). It turns out that for a complete description of such a process, knowledge of one- and two-dimensional distributions (1), (2) is sufficient, which is important from a practical point of view, since it allows us to limit ourselves to the study of the mathematical expectation and the correlation function of the process. In the theory of time series, a number of random component models are used, ranging from the simplest - “white noise”, to very complex types of autoregression - moving average and others, which are built on the basis of white noise. Before defining the white noise process, consider a sequence of independent random variables for which the distribution function is From the last relation it follows that all finite-dimensional distributions of a sequence are determined using one-dimensional distributions. If, moreover, in such a sequence the constituent random variables X
(t) have zero expected value and are distributed equally for all tÎT, then this is “white noise”. In the case of normal distribution X
(t) talk about Gaussian white noise. So, Gaussian white noise is a sequence of independent normally distributed random variables with zero mathematical expectation and the same (total) variance. More complex models, widely used in the theory and practice of time series analysis, are linear models: moving average processes, autoregression and mixed ones. Moving Average Process q represents a weighted sum of random disturbances: Where – numerical coefficients. It is easy to see from the definition that the moving average process has order q(abbreviated CC( q)) statistically dependent are ( q+1) consecutive quantities X
(t), X
(t
-1),..., X
(t
-
q). Members of the series spaced from each other by more than ( q+1) clock, are statistically independent, since different terms participate in their formation. where is a random disturbance acting at the current moment t
; – numerical coefficients. Expressing consistently in accordance with relation (5) X(t-1) through X(t-2), . . . , X(t-p-1), then X(t-2) through X(t-3), . . . , X(t-p-2), etc. we get that X(t) is an infinite sum of past disturbances. It follows from this that the terms of the autoregressive process X(t) and X(t-k) are statistically dependent for any k
. The process AP(1) is often called the Markov process, and AP(2) is the Yule process. In general, a Markov process is called a process whose future is determined only by its state in the present and the influences on the process that will be exerted in the future, while its state up to the present moment is unimportant. AP process(1) is Markovian, since its state at any moment is determined through the values of the process if the value at the moment is known. Formally, an autoregressive process of arbitrary order can also be considered Markovian if its state at the moment t count a set (X(t), X(t-1), . . . , X(t-p-1)) . The SS, AR models, as well as their composition: autoregressive - moving average models are discussed further (section 10.1.5). We only note that they all seem to be special cases of the general linear model where are the weighting coefficients, the number of which is, generally speaking, infinite. Among the models of the random component, we will highlight an important class - stationary processes, those whose properties do not change over time. A random process Y(t) is called stationary if for any n, the distributions of random variables and are identical. In other words, the functions of finite-dimensional distributions do not change with a time shift: The random variables that form the stationary sequence are equally distributed, so the white noise process defined above is stationary. When analyzing time series, numerical characteristics similar to those of random variables are used: – mathematical expectation (average value of the process) – autocovariance function – dispersion - standard deviation – autocorrelation function – partial autocorrelation function Note that in the function operator averaging occurs at a constant t, that is, there is a mathematical expectation over a set of realizations (generally speaking, potential ones since “you cannot enter the river of time twice”). Let us consider the introduced numerical characteristics for stationary processes. From the definition of stationarity it follows that for any s
,
t And putting = - t, we get It turns out that for a stationary process the mathematical expectation and variance are the same for any t, and the autocovariance and autocorrelation functions do not depend on the moment of time s or t, but only on their difference (lag). Note that the fulfillment of properties (1) does not yet imply stationarity in the sense of the definition from clause 6. Nevertheless, the constancy of the first two moments, as well as the dependence of the autocorrelation function only on the lag, definitely reflects some invariance of the process over time. If conditions (1) are met, then the process is said to be stationary in the broad sense, while conditions () are met means stationary in the narrow (strict) sense. The definition of white noise given above should be interpreted in a narrow sense. In practice, they are often limited to white noise in the broad sense, which is understood as a time series (random process) for which =0 and Note that a Gaussian process, stationary in the narrow sense, is also stationary in the broad sense. It is much easier to judge stationarity in the broad sense. For this purpose, various statistical criteria are used, based on one realization of a random process. Estimating the numerical characteristics of a random time series at each moment in time requires a set of realizations (trajectories) of the corresponding random process. Although time is not reproducible, the conditions of the process can sometimes be considered repeatable. This is especially typical for technical applications, for example, voltage fluctuations in the electrical network during the day. Time series observed on different days can be considered independent implementations of one random process. The situation is different when studying processes of a socio-economic nature. As a rule, there is only one implementation of the process available here, which cannot be repeated. Consequently, it is impossible to obtain estimates of the mean, variance, and covariance. However, for stationary processes such estimates are still possible. Let the observed values of the time series be at moments respectively. The traditional estimate of the mean can serve as an estimate of the mathematical expectation of a stationary (in the broad sense) random process. It is clear that such an estimate for a stationary series will be unbiased. The consistency of this estimate is established by Slutsky’s theorem, which, as a necessary and sufficient condition, requires that where is the autocorrelation function of the process. The accuracy of estimating the average depends on the length N row. It is believed that the length N must always be no less than the so-called correlation time, which is understood as the value Magnitude T gives an idea of the order of magnitude of the time period during which a noticeable correlation between two values of the series remains. Let us now consider obtaining estimates of the values of the autocorrelation function. As before, these are the observed values of the time series. Let's form ( N-1) par. These pairs can be thought of as a sample of two random variables for which an estimate of the standard correlation coefficient can be determined. Then we will compose ( N-2) pairs and determine the rating, etc. Since the sample size changes during the next calculation, the value of the mean and standard deviation for the corresponding set of values changes. To simplify, it is customary to measure all variables relative to the average value of the entire series and replace the dispersion terms in the denominator with the dispersion of the series as a whole, that is where is the average, equal to . At large N The differences in estimates are insignificant. On practice k they charge no more N
/4. If a series is considered as a general population of infinite length, then we talk about autocorrelations (theoretical) and denote them. An array of coefficients or their corresponding sample coefficients contain very valuable information about the internal structure of the series. A set of correlation coefficients plotted on a graph with coordinates k(lag) along the x-axis and either along the ordinate axis is called a correlogram (theoretical or sample, respectively). Accuracy estimation characteristics were obtained for Gaussian processes. In particular, for Gaussian white noise, for which all correlations are zero, . The mathematical expectation for Gaussian white noise turns out to be not equal to zero, namely, , that is, the estimate turns out to be biased. The magnitude of the bias decreases with increasing sample size and is not so significant in applied analysis. The estimate is asymptotically normal at , which provides a basis for constructing an approximate confidence interval. A widely used 95% interval is . The boundaries of the confidence interval plotted on the graph are called a confidence tube. If the correlogram of some random process does not go beyond the confidence tube, then this process is close to white noise. True, this condition can only be considered sufficient. Often, a sample correlogram of Gaussian white noise contains one or even two outliers among the first 20 estimates, which naturally complicates the interpretation of such a correlogram. Along with the autocorrelation function, when analyzing the structure of a random time series, a partial autocorrelation function is used, the values of which are partial correlation coefficients. The simplest hypothesis that can be put forward regarding a fluctuating series that does not have a clearly defined trend is the assumption that the fluctuations are random. In random series, according to the hypothesis, the observations are independent and can appear in any order. To test for randomness, it is desirable to use a criterion that does not require any restrictions on the type of distribution of the population from which the observed values are assumed to be drawn. 1. Turning point criterion consists of counting peaks (values that are greater than two neighboring ones) and troughs (values that are less than two neighboring ones). Consider the series y 1 ,...,y N . peak trough y t-1< y t
>y t+1 y t-1 > y t< y t+1 y t-1 y t y t+1 y t-1 y t y t+1 Rice. Turning points. Three consecutive values are required to determine the turning point. The start and end values cannot be turning points, since y 0 and y N+1 are unknown. If the series is random, then these three values can occur in any of the six possible orders with equal probability. Only four of them will have a turning point, namely when the largest or smallest of the three values is in the middle. Therefore, the probability of finding a turning point in any group of three values is 2/3. Rice. Options relative position three points. For a group of N quantities, we define a countable variable X. М 1, if y t-1< y t
>y t+1 or y t-1 > y t< y t+1 î 0, otherwise. Then the number of turning points p in the series is simply , and their mathematical expectation is M[p]=2/3(N-2). The variance of the number of turning points is calculated using the formula D[p]=(16N-29)/90, and the distribution itself is close to normal. 2. Criterion based on determining phase length The interval between two turning points is called phase. In order to establish the presence of a phase of length d (for example, ascending), it is necessary to detect d+3 terms containing a decrease from the first term to the second, then a sequential rise to the (d+2)th term and a decrease to (d+3) -his dick. 1 2 3 4 d+1 d+2 d+3 N rice. 3. Phase length d. Consider a group of d+3 numbers arranged in ascending order. If, without touching the two extreme terms, we extract a pair of numbers from the remaining d+1 and put one of them at the beginning and the other at the end, we obtain a phase of length d. There are ways to choose a pair of numbers in this way and each member of the pair can be placed at any end, therefore the number of ascending phases is equal to d(d+1). In addition, turning points will occur if the first term of the sequence is placed at the end, and any of the remaining ones, with the exception of the second, are placed at the beginning. The number of such sequences will be (
d
+1)
. The same number of sequences will be obtained if the last term in the original, increasing sequence is placed at the beginning, and any other, except the last, at the end. To avoid double counting, the case where the first term is placed in last place and the last in first place should be excluded. Thus, in sequence from (
d
+3)
numbers with phase length d the number of cases of increase will be d
(d
+1)+2(d
+1)-1 =+3d
+1 . Number of possible sequences from (
d
+3)
numbers equals the number of permutations (
d
+3)
!, so the probability of either an ascending or descending phase is In a series of length N, one can successively identify N-2-d groups of d+3 members. That. mathematical expectation of the number of phases of length d It can be shown that the mathematical expectation of the total number of phases of length from 1 to N-3 3
.Criterion based on signs of differences This criterion consists of counting the number of positive first-order differences in a series, in other words, the number of increasing points of the series. For a series of N terms we get N-1 differences. Let us define the counting variable as If we now denote by With the number of increasing points of a random series, then The distribution tends to normal quite quickly with variance Basically, this criterion is recommended to check the presence of a linear trend. On the other hand, a criterion based on turning points is poorly suited for trend detection because superimposing noticeable random fluctuations on a moderate trend results in approximately the same number of turning points as in the absence of a trend. A more advanced, but more complex test for detecting a linear trend is to regress y on t and test the significance of the regression coefficient. 4.Criterion based on rank comparisons The idea of comparing neighboring values in a series can be developed to comparing all values. For a given series, we count the number of cases when the next member of the series exceeds all subsequent ones. In total there are N(N-1) pairs for comparison. Let n total number of cases exceeded. Calculate the Kendal rank correlation coefficient If this coefficient is significant and positive, then the series is increasing; if it is negative, then it is decreasing. A general linear model of a stochastic process is considered where is white noise – weighting coefficients. Recall that =0, , Let's introduce the shift operator one step back IN
: Multiple (to be specific j-multiple) application of the operator IN, denoted as , gives Taking into account the introduced notation, the general linear model can be written as where is a linear operator. Let's find the mathematical expectation, variance and autocovariance function for process (1): For the model to make sense, the variance must be finite, that is, the series is assumed to converge. In addition, it is assumed that the so-called reversibility condition holds: where instead of IN appear complex numbers. This condition implies the existence of the inverse operator where, that is, such that Expanding the product in the last expression, grouping homogeneous terms and equating them to zero, we obtain expressions for determining the coefficients. So, Multiplying () by on the left, we obtain that the reversible process can be written as Entry (2) corresponds to an autoregressive scheme of infinite order. This same ratio can be interpreted as a linear predictor for all past values of the time series, and the term can be interpreted as a random error of this predictor. If all past values of the series are known, then using form (2) it is possible to predict the future value of the series. Let us consider in more detail the random component models, which are special cases of the general linear model, namely the autoregressive, moving average and mixed models, which are widely used in practice. The AR(1) model has the form The model will take the form Considered as the sum of an infinitely decreasing geometric progression with a denominator A
IN we get that Thus, the Markov process is special case general linear model, the coefficients of which change according to the law of geometric progression, that is. Expression (2) can also be obtained from (1) directly, expressing through , through, etc. The variance in accordance with () is It turns out that white noise with dispersion generates a random process in the Markov scheme with increased dispersion equal to . To find the autocovariance function of a Markov process, you can use the general expression (). However, it is more clear next way. Let us multiply equation (1) of the Markov process by and take the mathematical expectation Since the second term on the right side is equal to zero due to the uncorrelated nature of the disturbance at the current moment with the past values of the series, we obtain ( From the last relation we have that is A coincides with the autocorrelation coefficient of the average terms of the series. Let us now multiply (1) by and take the mathematical expectation: Replacing A by and dividing by , we get Giving k values 2,3,... we get So, in a Markov process, all autocorrelations can be expressed in terms of the first autocorrelation. Since , the autocorrelation function of the Markov process decreases exponentially with growth k
. Let us now consider the partial autocorrelation function of the Markov process. We found that the correlation between two terms of the series separated by two clock cycles, that is, between and is expressed by the value . But it depends on , and on . The question arises whether the dependence between and will remain if the dependence on the median term is eliminated. The corresponding partial correlation coefficient is Because the Concluding our consideration of the AR(1) model, we note that it is very often used in economic and mathematical research to describe the residuals of a linear regression connecting economic indicators. Using the shift operator IN the model will be written as The properties of the model depend on the roots and the polynomial which can also be written in the form (1-IN)(1-IN)=0. For process (1) to be stationary, it is necessary that the roots and lie inside unit circle(the case of complex roots), or were less than unity (the case of real roots), which is ensured when Let them both be valid and different. Let's break it down into simple fractions Where Considering the individual terms in (3) as sums of infinite geometric progressions, we get It turns out that AR(2) is a special case of the general linear model () with coefficients Let us now consider the autocorrelation function of the Yule process. Let's multiply (1) in turn by and , take the mathematical expectations and divide by . As a result we get These equations are sufficient to determine through the first two autocorrelations and, conversely, using the known ones one can find . Now multiplying (1) by we obtain the recurrent equation from which high order autocorrelations can be found through the first autocorrelations. Thus, the correlogram of the Yule process is completely determined. Let us examine the form of the correlogram of the AR(2) process. Expression (4) can be considered as a second-order difference equation with respect to r with constant coefficients. The general solution to such an equation has the form where are the roots of the characteristic equation It is easy to see that equations (2) and (5) are equivalent up to replacement IN on z and dividing both sides by so that the roots of these equations coincide, that is The general solution to difference equation (4) is where are the coefficients A And IN found from the boundary conditions at j=0 and j
=1. Thus, in the case of real roots, the correlogram AP(2), as can be seen from (6), is a mixture of two damped exponentials. In the case of completeness of roots, the correlogram of the process AR(2) turns out to be a damped harmonic. Let us now consider how the partial autocorrelation function of the Yule process behaves. Only the coefficient equal to is different from zero. Partial correlations of higher orders are equal to zero (this process is discussed in more detail later). Thus, a partial correlogram of the process is cut off immediately after a lag equal to one. In conclusion, we note that the AR(2) models turned out to be acceptable in describing the behavior of a cyclic nature, the prototype of which is a pendulum, which is affected by small random impulses. The amplitude and phase of such an oscillatory process will change all the time. The solution to the difference expression (1) or () with respect to y consists of two parts: the general solution containing R arbitrary constants, and a particular solution. There is a general solution where there is constant odds, (j
=1,2,...,R) are the roots of the characteristic equation. Stationarity of series (2) occurs if the roots of equation (3) have a modulus less than one. In other words, the roots must lie inside the unit circle. Assuming that the series has a sufficiently long history, the general solution (2) can be neglected due to attenuation. A frequent solution, as can be seen from (), is The last relation is a form of representing the autoregressive process in the form of a general linear model. We successively multiply equation (1) by , take the mathematical expectation and divide by . We obtain a system of equations for correlation coefficients: , k
=1, 2, ..., p
(4) Considering that , and introducing matrix notation we write (4) in the form Pa
=
r
(5) The system of equations (5) is called the Yule-Walker system. From it we find that a
=
r
(6) Thus, knowing the first p autocorrelations of a time series, one can find higher-order autocorrelations from (3), that is, completely restore the autocorrelation function (which was already noted when analyzing the processes AR(1) and AR(2)). The behavior of the autocorrelation function depends on the roots of the characteristic polynomial. Usually the correlogram of the AR process( R) consists of a set of damped sinusoids. If the process AP(2) has a partial autocorrelation of series members separated by 2 or a large number members is equal to zero, then the process AP( R) autocorrelations of order p and higher are equal to zero. It turns out that the partial correlogram of the AR process( R) must be equal to zero starting from a certain moment. However, it should be noted that this fact holds for an infinite series. For finite implementations, it is often difficult to indicate the break point of the correlogram. So, for the process AP( R) the partial autocorrelation function terminates at the lag R, while the autocorrelation function smoothly decreases. 10.1.4 Moving average processes The generalized linear model for moving average processes contains only a finite number of terms, that is, in (): =0 k
>
q
. The model takes the form (IN(1) coefficients are redesignated by.) Relation (1) defines the moving average order process q, or abbreviated SS( q). Reversibility condition () for the process SS( q) is satisfied if the roots of the polynomial b
(IN) lie outside the unit circle. Let us find the variance of the process SS( q): All mixed works type are equal to zero due to the uncorrelated nature of disturbances at different times. To find the autocorrelation function of the CC process( q) sequentially multiply (1) by and take the mathematical expectation On the right side of expression (2) only those terms that correspond to the same time steps will remain (see figure) Therefore, expression (2) is dividing (3) by , we get The fact that the autocorrelation function of the process CC(q) has a finite extent ( q clock cycles) is a characteristic feature of such a process. If known, then (4) can in principle be resolved with respect to the parameters . Equations (4) are nonlinear and in the general case have several solutions, but the reversibility condition always selects a single solution. As already noted, reversible SS processes can be considered as infinite AP processes -AP(¥). Consequently, the partial autocorrelation function of the process CC( R) has infinite extent. So, the process CC( q) the autocorrelation function terminates at the lag q, while the partial autocorrelation function smoothly decreases. Although AR models ( R) and SS( q) make it possible to describe many real processes; the number of estimated parameters can be significant. To achieve greater flexibility and cost-effectiveness of description when selecting models for observed time series, mixed models containing both autoregression and moving average have proven to be very useful. These models were proposed by Box and Jenkins and were called the autoregressive moving average model (abbreviated ARMC( R,
q)): Using the shift operator IN model (1) can be presented more compactly: b
(IN)-moving average order operator q
. Model() can also be written like this: Let's consider the simplest mixed process ARSS(1,1) According to From relation (2) it is clear that the ARCC(1,1) model is a special case of the general linear model () with coefficients From (2) it is easy to obtain an expression for the variance: To obtain the correlation function, we will use the same technique as when analyzing autoregressive models. Let's multiply both parts of the model representation of the process ARSS(1,1) on and take the mathematical expectation: or (taking into account that the second term on the right side of the equality is equal to zero) Dividing covariance by variance we obtain expressions for autocorrelation the resulting relations show that it decreases exponentially from the initial value, depending on and, and if >, then the attenuation is monotonic; at< – затухание колебательное. Similarly, the autocorrelation function can be constructed for general model ARSS( R,
q). Let's multiply all terms (1) by . Let's take the mathematical expectation and as a result we get the following difference equation. Where It follows that for the values q+1 autocovariance and autocorrelation satisfy the same relations as in the AR model( R): As a result, it turns out that when q
<р
the entire autocorrelation function will be expressed by a set of damped exponentials and/or damped sine waves, and when q
>
p will q
-
p values falling out of this scheme. The ARSS model can be generalized to the case when the random process is nonstationary. A striking example such a process are “random walks”: Using the shift operator, model (1) takes the form From (2) it is clear that process (1) is divergent, since In the general case, it is assumed that the nonstationary autoregressive operator in the ARCC model has one or more roots equal to one. In other words, is a non-stationary autoregressive operator of order p
+
d
;
d the roots of the equation =0 are equal to one, and the rest R roots lie outside the unit circle. Then we can write that Where a
(B) – stationary autoregressive operator of order R(with roots outside the unit circle). Let us introduce a difference operator such that =(1- B), then the non-stationary process ARSS will be written as Where b
(B) is an invertible moving average operator (its roots lie outside the unit circle). For order difference d, that is, the model describes the already stationary reversible process ARSS( R,
q). In order to return from a series of differences to the original series, the operator is required s, reverse: This operator is called the summation operator because If the initial difference is the order d, then to restore the original series you will need d- multiple iteration of the operator s
,
otherwise d- multiple summation (integration). Therefore, process (3) is usually called the ARISS process, adding the term integrated to ARISS. Briefly, model (3) is written as ARISS( R,
d
,
q), Where R– order of autoregression, d– order of difference, q– order of the moving average. It is clear that when d=0 the ARISS model goes into the ARSS model. On practice d usually does not exceed two, that is d
. The ARISS model allows a representation similar to the general linear model, as well as in the form of a “pure” autoregressive process (of infinite order). Consider, for example, the process ARISS (1, 1, 1): From (4) it follows that In expression (5), the coefficients, starting from the third, are calculated using the formula. Representation (5) is interesting because the weights, starting from the third, decrease exponentially. Therefore, although formally it depends on all past values, several “recent” values of the series will make a real contribution to the current value. Therefore, equation (5) is most suitable for forecasting. As already noted, ARISS processes can be represented in the form of a generalized linear model, that is It is natural to look for the future (forecast) value of the series at the moment in the form The expected value, which we will denote as = The first sum on the right side of the last relation contains only future disturbances (the forecast is made at the moment t, when the past values of both the series and disturbances are known) and for them the mathematical expectation is equal to 0 by definition. As for the second term, the disturbances have already taken place here, so Thus The forecast error, which represents the discrepancy between the forecast value and its expectation, is The error variance from here is Forecasting using relation (1) is possible in principle, but it is difficult because it requires knowledge of all past disturbances. In addition, for stationary series the decay rate is often insufficient, not to mention non-stationary processes for which the series diverge. Since the ARISS model also allows other representations, we will consider the possibilities of using them for forecasting. Let the model be given directly by the difference equation Based on known values of the series (observational results) When predicting for two cycles, you should again use recurrence relation(3), where as the observed value of the series at the moment t+1 should take the value predicted by (4), that is, and so on. Finally, forecasting is possible based on the representation of the ARISS process in the form of autoregression (). As already noted, despite the fact that the order of autoregression is infinite, the weighting coefficients in the series representation decrease quite quickly, so a moderate number of past values of the series is sufficient to calculate the forecast. The variance of the forecast error for steps ahead is and according to expression (2) is given by the expression Assuming that random disturbances are Gaussian white noise, that is, we can consider the confidence interval for the predicted value of the series in a standard way. The theoretical schemes described above were built on the assumption that the time series has an infinite prehistory, whereas in reality a limited amount of observations is available to the researcher. The model has to be selected experimentally, fitting it to the available data. Therefore, from the standpoint theoretical application theory of time series analysis, the issues of correct specification of the ARISS model are of decisive importance ( p
,
d
,
q) (its identification) and subsequent assessment of its parameters. At the identification stage, observed data are used to determine a suitable class of models and preliminary estimates of its parameters are made, that is, a trial model is built. The trial model is then fitted to the data more carefully; wherein initial estimates, obtained at the identification stage act as initial values in iterative parameter estimation algorithms. And finally, at the third stage, the resulting model is subjected to diagnostic testing to identify possible inadequacy of the model and develop suitable changes in it. Let us consider the listed stages in more detail. Model identification The purpose of identification is to gain some idea of the quantities p
,
d
, q and about the model parameters. Model identification breaks down into two stages 1. Determination of the order of the difference d original series. 2. Identification of the ARSS model for a number of differences. The main tool used at both stages is the autocorrelation and partial autocorrelation functions. In the theoretical part we saw that stationary models autocorrelations decrease with growth k very quickly (according to the correlation law). If the autocorrelation function decays slowly and almost linearly, then this indicates that the process is nonstationary, however, perhaps its first difference is stationary. Having constructed a correlogram for a number of differences, the analysis is repeated again, and so on. It is believed that the order of the difference d, ensuring stationarity, is achieved when the autocorrelation function of the process decreases quite quickly. In practice, it is enough to look at about 15-20 first autocorrelation values of the original series, its first and second differences. After a stationary series of differences of order d is obtained, the general form of the autocorrelation and partial autocorrelation functions of these differences is studied. Based on the theoretical properties of these functions, you can choose the values p And q for AP and CC operators. Next, with the selected p And q initial estimates of autoregression parameters are constructed For mixed APCC processes, the assessment procedure becomes more complicated. So for the process ARSS(1,1) considered in paragraph 1, the parameters and, more precisely, their estimates, are obtained from () with replacement and their sample estimates. In the general case, the calculation of initial estimates of the ARCC process ( p
,
q) represents a multi-step procedure and is not discussed here. We only note that for practice, the AR and SS processes of the 1st and 2nd orders and the simplest mixed process ARCC(1,1) are of particular interest. In conclusion, we note that estimates of autocorrelations, on the basis of which identification procedures are based, can have large variances (especially in conditions of insufficient sample size - several dozen observations) and be highly correlated. Therefore, there is no need to talk about strict correspondence between the theoretical and empirical autocorrelation functions. This leads to difficulties when choosing p
,
d
,
q
,
therefore, multiple models may be selected for further study. linear series time series system Posted at http://www.
.
; 
; 
.
.
,
, 
,
.
; 
.
5. Estimation and elimination of the seasonal component
.
.
,
.
.
, model (1) can also be written in the form
6. Models of the random component of a time series
– independent identically distributed random variables (white noise);
7.Numerical characteristics of the random component
;
(1)
8.Evaluation of numerical characteristics of a time series
,
,
9. Criteria for checking a series for randomness, free from the distribution law
.
.
.
.
.
10. Theoretical analysis of a stationary random component of linear form
;
,
and so on.
10.1\. Autoregressive models
due to stationarity)
,
.
, numerator equal to zero. Similarly, it can be shown that the partial correlation coefficients for members of the series, separated by 3, 4, and so on cycles, are also equal to zero. Thus, autocorrelation exists only due to the correlation of neighboring terms, which, however, follows from the mathematical model of the Markov process.
,
.
, (3)
.
,
(5)
(6)
,
(1)
(3)
(4)
, ()
(2)
(j
>0)
- mutual covariance function between y And . Since the disturbances at the moment t and the values of the series in past moments (cm(2)) do not correlate, 0 for k>0.
(2)
. The characteristic equation of this process has a root equal to unity, that is, there is a boundary case when the root of the characteristic equation is on the boundary of the unit circle. At the same time, if we go to first differences, the process will turn out to be stationary.
,
, (3)
11.Forecasting using the ARISS model
=
and estimated values of disturbances , based on the recurrent formula (3), one can estimate the expected value of the series at the moment t
+1:
12.Technology for constructing ARISS models
and moving average b=(). For autoregressive processes, the Yule-Walker equations are used, where theoretical autocorrelations are replaced by their sample estimates. For moving average order processes q only the first ones q autocorrelations are different from zero and can be expressed through parameters (see). Replacing them with sample estimates and solving the resulting equations for , we obtain the estimate . These preliminary estimates can be used as seeds to obtain more efficient estimates in the next steps.
1 Types and methods of time series analysis
A time series is a series of observations of the values of a certain indicator (attribute), ordered in chronological order, i.e. in ascending order of the t-time parameter variable. Individual observations in a time series are called levels of that series.
1.1 Types of time series
Time series are divided into moment and interval. In momentary time series, levels characterize the values of an indicator as of certain points in time. For example, time series of prices for certain types goods, time series of stock prices, the levels of which are fixed for specific numbers. Examples of moment time series can also be series of population or value of fixed assets, since the values of the levels of these series are determined annually on the same date.
In interval series, levels characterize the value of an indicator for certain intervals (periods) of time. Examples of series of this type are time series of product production in physical or value terms for a month, quarter, year, etc.
Sometimes series levels are not directly observed values, but derived values: average or relative. Such series are called derivatives. The levels of such time series are obtained through some calculations based on directly observed indicators. Examples of such series are series of average daily production of the main types of industrial products or series of price indices.
Series levels can take deterministic or random values. An example of a series with deterministic level values is a series of sequential data on the number of days in months. Naturally, series with random level values are subject to analysis, and subsequently to forecasting. In such series, each level can be considered as a realization random variable- discrete or continuous.
1.2 Time series analysis methods
Time series analysis methods. There are a large number of different methods to solve these problems. Of these, the most common are the following:
1. Correlation analysis, which makes it possible to identify significant periodic dependencies and their lags (delays) within one process (autocorrelation) or between several processes (cross-correlation);
2. Spectral analysis, which allows you to find periodic and quasiperiodic components of a time series;
3. Smoothing and filtering, designed to transform time series in order to remove high-frequency or seasonal fluctuations from them;
5. Forecasting, which allows, based on a selected model of the behavior of a temporary rad, to predict its values in the future.
2 Basics of forecasting the development of processing industries and trade organizations
2.1 Forecasting the development of processing enterprises
Agricultural products are produced at enterprises of various organizational forms. Here it can be stored, sorted and prepared for processing; at the same time, there may be specialized storage facilities. Then the products are transported to processing plants, where they are unloaded, stored, sorted, processed, and packaged; From here transportation to commercial enterprises takes place. At the trading enterprises themselves, after-sales packaging and delivery are carried out.
All types of technological and organizational operations listed must be predicted and planned. In this case, various techniques and methods are used.
But it should be noted that food processing enterprises have some planning specifics.
The food processing industry occupies an important place in the agro-industrial complex. Agricultural production provides this industry with raw materials, that is, in essence, there is a strict technological connection between spheres 2 and 3 of the agro-industrial complex.
Depending on the type of raw materials used and the characteristics of the sale of the final product, three groups of food and processing industries have emerged: primary and secondary processing of agricultural resources and mining Food Industry. The first group includes industries that process poorly transportable agricultural products (starch, canned fruits and vegetables, alcohol, etc.), the second group includes industries that use agricultural raw materials that have undergone primary processing (baking, confectionery, food concentrates, refined sugar production, etc.). The third group includes salting and fishing industries.
Enterprises of the first group are located closer to areas of agricultural production; here production is seasonal. Enterprises of the second group, as a rule, gravitate towards areas where these products are consumed; they work rhythmically throughout the year.
Along with common features enterprises of all three groups have their own internal ones, determined by the range of products, the technical means, technologies used, the organization of labor and production, etc.
An important starting point for forecasting these industries is taking into account the external and internal features and specifics of each industry.
The food and processing industries of the agro-industrial complex include grain processing, baking and pasta, sugar, low-fat, confectionery, fruits and vegetables, food concentrates, etc.
2.2 Forecasting the development of trade organizations
In trade, forecasting uses the same methods as in other sectors of the national economy. The creation of market structures in the form of a network of wholesale food markets, improvement of branded trade, and the creation of a wide information network are promising. Wholesale trade allows you to reduce the number of intermediaries when bringing products from the producer to the consumer, create alternative sales channels, and more accurately predict consumer demand and supply.
In most cases, the plan for the economic and social development of a trading enterprise consists mainly of five sections: retail and wholesale trade turnover and commodity supply; financial plan; development of material and technical base; social development of teams; labor plan.
Plans can be developed in the form of long-term - up to 10 years, medium-term - from three to five years, current - up to one month.
Planning is based on trade turnover for each assortment group of goods.
Wholesale and retail trade turnover can be forecast in the following sequence:
1. evaluate the expected implementation of the plan for the current year;
2. calculate the average annual rate of trade turnover for two to three years preceding the forecast period;
3. based on the analysis of the first two positions, using the expert method, the growth (decrease) rate of sales of individual goods (product groups for the forecast period) is established as a percentage.
By multiplying the volume of expected turnover for the current year by the projected sales growth rate, the possible turnover in the forecast period is calculated.
The necessary commodity resources consist of the expected turnover and inventory. Inventories can be measured in physical and monetary terms or in days of turnover. Inventory planning is typically based on extrapolation of fourth-quarter data over a number of years.
Commodity supply is determined by comparing the need for necessary commodity resources and their sources. The necessary commodity resources are calculated as the sum of trade turnover, the probable increase in inventory minus the natural loss of goods and their markdown.
The financial plan of a trading enterprise includes a cash plan, a credit plan and estimates of income and expenses. I draw up a cash plan quarterly, the credit plan determines the need for various types of credit, and the estimate of income and expenses - by items of income and cash receipts, expenses and deductions.
The objects of planning the material and technical base are the retail network, technical equipment, and storage facilities, that is, the general need for retail space, retail enterprises, their location and specialization, the need for mechanisms and equipment, and the necessary storage capacity are planned.
Indicators of social development of the team include the development of plans for advanced training, improvement of working conditions and health protection of workers, housing and cultural conditions, development of social activity.
A rather complex section is the labor plan. It must be emphasized that in trade the result of labor is not a product, but a service; here the costs of living labor predominate due to the difficulty of mechanizing most labor-intensive processes.
Labor productivity in trade is measured by the average turnover per employee over a certain period of time, that is, the amount of turnover is divided by the average number of employees. Due to the fact that the labor intensity of the sale of various goods is not the same, when planning, changes in trade turnover, price indices, and the assortment of goods should be taken into account.
The development of trade turnover requires an increase in the number of trade and public catering enterprises. When calculating the quantity for the planning period based on the standards for the provision of the population with trading enterprises for urban and rural areas.
As an example, we give the content of the plan for the economic and social development of a fruit and vegetable trading enterprise. It includes the following sections: initial data; main economic indicators of the enterprise; technical and organizational development of the enterprise; plan for storing products for long-term storage; product sales plan; retail turnover plan; distribution of costs for import, storage and wholesale sales by groups of goods; distribution costs of retail sales of products; costs of production, processing and sales; number of employees and payroll plans; profit from wholesale sales of products; profit plan from all types of activities; income distribution; profit distribution; social development team; financial plan. The methodology for drawing up this plan is the same as in other sectors of the agro-industrial complex.
3 Calculation of the economic time series forecast
There are data on the export of reinforced concrete products (to countries outside the CIS), billion US dollars.
Table 1
Export of goods for 2002, 2003, 2004, 2005 (billion US dollars)
Before proceeding with the analysis, let us turn to graphic image initial data (Fig. 1).
Rice. 1. Export of goods
As can be seen from the plotted graph, there is a clear trend toward an increase in import volumes. After analyzing the resulting graph, we can conclude that the process is nonlinear, assuming exponential or parabolic development.
Now let's do a graphical analysis of quarterly data for four years:
table 2
Export of goods for the quarters of 2002,2003, 2004 and 2005
Rice. 2. Export of goods
As can be seen from the graph, seasonality of fluctuations is clearly expressed. The amplitude of the oscillation is rather unfixed, which indicates the presence of a multiplicative model.
In the source data we are presented with an interval series with equally spaced levels in time. Therefore, to determine the average level of the series, we use the following formula:
Billion dollars
To quantify the dynamics of phenomena, the following main analytical indicators are used:
· absolute growth;
· rates of growth;
· growth rate.
Let's calculate each of these indicators for an interval series with equally spaced levels in time.
Let us present the statistical indicators of dynamics in the form of Table 3.
Table 3
Statistical indicators of dynamics
| t | y t | Absolute growth, billion US dollars | Growth rate, % | Growth rate, % | |||
| Chain | Basic | Chain | Basic | Chain | Basic | ||
| 1 | 48,8 | - | - | - | - | - | - |
| 2 | 61,0 | 12,2 | 12,2 | 125 | 125 | 25 | 25 |
| 3 | 77,5 | 16,5 | 28,7 | 127,05 | 158,81 | 27,05 | 58,81 |
| 4 | 103,5 | 26 | 54,7 | 133,55 | 212,09 | 33,55 | 112,09 |
The growth rates were approximately the same. This suggests that the average growth rate can be used to determine the forecast value:
Let's check the hypothesis about the presence of a trend using Foster-Stewart test. To do this, fill out auxiliary table 4:
Table 4
Auxiliary table
| t | yt | mt | lt | d | t | yt | mt | lt | d |
| 1 | 9,8 | - | - | - | 9 | 16,0 | 0 | 0 | 0 |
| 2 | 11,8 | 1 | 0 | 1 | 10 | 18,0 | 1 | 0 | 1 |
| 3 | 12,6 | 1 | 0 | 1 | 11 | 19,8 | 1 | 0 | 1 |
| 4 | 14,6 | 1 | 0 | 1 | 12 | 23,7 | 1 | 0 | 1 |
| 5 | 12,9 | 0 | 0 | 0 | 13 | 21,0 | 0 | 0 | 0 |
| 6 | 14,7 | 1 | 0 | 1 | 14 | 23,9 | 1 | 0 | 1 |
| 7 | 15,5 | 1 | 0 | 1 | 15 | 26,9 | 1 | 0 | 1 |
| 8 | 17,8 | 1 | 0 | 1 | 16 | 31,7 | 1 | 0 | 1 |
Let's apply the Student's test:
We get, that is 
, hence the hypothesis N 0 is rejected, there is a trend.
Let's analyze the structure of the time series using the autocorrelation coefficient.
Let us find the autocorrelation coefficients sequentially:
–
first-order autocorrelation coefficient, since the time shift is equal to one (-lag).
We similarly find the remaining coefficients.
– second order autocorrelation coefficient.
– third-order autocorrelation coefficient.
– fourth-order autocorrelation coefficient.
Thus, we see that the highest is the fourth-order autocorrelation coefficient. This suggests that the time series contains seasonal variations with a periodicity of four quarters.
Let's check the significance of the autocorrelation coefficient. To do this, we introduce two hypotheses: N 0: , N 1: .
It is found from the table of critical values separately for >0 and<0. Причем, если ||>||, then the hypothesis is accepted N 1, that is, the coefficient is significant. If ||<||, то принимается гипотеза N 0 and the autocorrelation coefficient is insignificant. In our case, the autocorrelation coefficient is quite large, and it is not necessary to check its significance.
It is required to smooth the time series and restore lost levels.
Let's smooth the time series using a simple moving average. We present the calculation results in the form of the following table 13.
Table 5
Smoothing the original series using a moving average
| Year No. | Quarter number | t | Import of goods, billion US dollars, yt | moving average, | |
| 1 | I | 1 | 9,8 | - | - |
| II | 2 | 11,8 | - | - | |
| III | 3 | 12,6 | 12 , 59 | 1,001 | |
| IV | 4 | 14,6 | 13,34 | 1,094 | |
| 2 | I | 5 | 12,9 | 14,06 | 0,917 |
| II | 6 | 14,7 | 14,83 | 0,991 | |
| III | 7 | 15,5 | 15,61 | 0,993 | |
| IV | 8 | 17,8 | 16,41 | 1,085 | |
| 3 | I | 9 | 16 | 17,36 | 0,922 |
| II | 10 | 18 | 18,64 | 0,966 | |
| III | 11 | 19,8 | 20,0 | 0,990 | |
| IV | 12 | 23,7 | 21,36 | 1,110 | |
| 4 | I | 13 | 21 | 22,99 | 0,913 |
| II | 14 | 23,9 | 24,88 | 0,961 | |
| III | 15 | 26,9 | - | - | |
| IV | 16 | 31,7 | - | - |
Now let's calculate the ratio of actual values to the levels of the smoothed series. As a result, we obtain a time series whose levels reflect the influence of random factors and seasonality.
We obtain preliminary estimates of the seasonal component by averaging the levels of the time series for the same quarters:
For the first quarter:
For the second quarter:
For the second quarter:
For the fourth quarter: 
The mutual cancellation of seasonal impacts in multiplicative form is expressed in the fact that the sum of the values of the seasonal component for all quarters must be equal to the number of phases in the cycle. In our case, the number of phases is four. Summing up the average values by quarter, we get:
Since the sum turned out to be unequal to four, it is necessary to adjust the values of the seasonal component. Let's find an amendment to change the preliminary estimates of seasonality:
We determine the adjusted seasonal values; we summarize the results in Table 6.
Table 6
Estimation of the seasonal component in a multiplicative model .
| Quarter number | i | Preliminary assessment of the seasonal component, | Adjusted value of the seasonal component, |
| I | 1 | 0,917 | 0,921 |
| II | 2 | 0,973 | 0,978 |
| III | 3 | 0,995 | 1,000 |
| IV | 4 | 1,096 | 1,101 |
| 3,981 | 4 |
We carry out a seasonal adjustment of the source data, that is, we remove the seasonal component.
Table 7
Construction of a multiplicative trend seasonal model.
| t | Import of goods, billion US dollars | Seasonal component, | Deseasonalized import of goods, | Estimated value | Estimated value of imports of goods,  |
| 1 | 9,8 | 0,921 | 10,6406 | 11,48 | 10,57308 |
| 2 | 11,8 | 0,978 | 12,0654 | 11,85 | 11,5893 |
| 3 | 12,6 | 1 | 12,6 | 12,32 | 12,32 |
| 4 | 14,6 | 1,101 | 13,2607 | 12,89 | 14,19189 |
| 5 | 12,9 | 0,921 | 14,0065 | 13,56 | 12,48876 |
| 6 | 14,7 | 0,978 | 15,0307 | 14,33 | 14,01474 |
| 7 | 15,5 | 1 | 15,5 | 15,2 | 15,2 |
| 8 | 17,8 | 1,101 | 16,1671 | 16,17 | 17,80317 |
| 9 | 16 | 0,921 | 17,3724 | 17,24 | 15,87804 |
| 10 | 18 | 0,978 | 18,4049 | 18,41 | 18,00498 |
| 11 | 19,8 | 1 | 19,8 | 19,68 | 19,68 |
| 12 | 23,7 | 1,101 | 21,5259 | 21,05 | 23,17605 |
| 13 | 21 | 0,921 | 22,8013 | 22,52 | 20,74092 |
| 14 | 23,9 | 0,978 | 24,4376 | 24,09 | 23,56002 |
| 15 | 26,9 | 1 | 26,9 | 25,76 | 25,76 |
| 16 | 31,7 | 1,101 | 28,792 | 27,53 | 30,31053 |
Using OLS we obtain the following trend equation:3
Let's graphically depict a series of residues:
Rice. 3. Residual graph
After analyzing the resulting graph, we can conclude that the fluctuations of this series are random.
The quality of the model can also be checked using indicators of asymmetry and kurtosis of the residuals. In our case we get:
,
then the hypothesis about the normal distribution of residuals is rejected.
Since one of the inequalities is satisfied, it is appropriate to conclude that the hypothesis about the normal nature of the distribution of residuals is rejected.
The final step in applying growth curves is to calculate forecasts based on the chosen equation.
To forecast the import of goods next year, let’s estimate the trend values at t =17, t =18, t =19 and t =20:
4. Lichko N.M. Planning at agribusiness enterprises. – M., 1996.
5. Finam. Events and markets, – http://www.finam.ru/
Types and methods of time series analysis
A time series is a collection of sequential measurements of a variable taken at equal time intervals. Time series analysis allows you to solve the following problems:
- explore the structure of a time series, which, as a rule, includes a trend - regular changes in the average level, as well as random periodic fluctuations;
- explore cause-and-effect relationships between processes that determine changes in series, which manifest themselves in correlations between time series;
- build a mathematical model of the process represented by a time series;
- transform the time series using smoothing and filtering tools;
- predict the future development of the process.
A significant part of the known methods are intended for the analysis of stationary processes, the statistical properties of which, characterized by a normal distribution by the mean value and variance, are constant and do not change over time.
But the series often have a non-stationary character. Non-stationarity can be eliminated as follows:
- subtract the trend, i.e. changes in the average value, represented by some deterministic function that can be selected by regression analysis;
- perform filtering with a special non-stationary filter.
To standardize time series for uniformity of methods
analysis, it is advisable to carry out their general or seasonal centering by dividing by the average value, as well as normalization by dividing by the standard deviation.
Centering a series removes a non-zero mean that can make the results difficult to interpret, for example in spectral analysis. The purpose of normalization is to avoid operations with large numbers in calculations, which can lead to a decrease in the accuracy of calculations.
After these preliminary transformations of the time series, its mathematical model can be built, according to which forecasting is carried out, i.e. Some continuation of the time series was obtained.
In order for the forecast result to be compared with the original data, transformations that are inverse to those performed must be made on it.
In practice, modeling and forecasting methods are most often used, and correlation and spectral analysis are considered as auxiliary methods. It's a delusion. Methods for forecasting the development of average trends make it possible to obtain estimates with significant errors, which makes it very difficult to predict the future values of a variable represented by a time series.
Methods of correlation and spectral analysis make it possible to identify various, including inertial, properties of the system in which the processes under study are developing. The use of these methods makes it possible to determine with sufficient confidence from the current dynamics of processes how and with what delay the known dynamics will affect the future development of processes. For long-term forecasting, these types of analyzes provide valuable results.
Trend analysis and forecasting
Trend analysis is intended to study changes in the average value of a time series with the construction of a mathematical model of the trend and, on this basis, forecasting future values of the series. Trend analysis is performed by constructing simple linear or nonlinear regression models.
The initial data used are two variables, one of which is the values of the time parameter, and the other is the actual values of the time series. During the analysis process you can:
- test several mathematical trend models and choose the one that more accurately describes the dynamics of the series;
- build a forecast of the future behavior of the time series based on the selected trend model with a certain confidence probability;
- remove the trend from the time series in order to ensure its stationarity, necessary for correlation and spectral analysis; for this, after calculating the regression model, it is necessary to save the residuals to perform the analysis.
Various functions and combinations are used as trend models, as well as power series, sometimes called polynomial models. The greatest accuracy is provided by models in the form of Fourier series, but not many statistical packages allow the use of such models.
Let us illustrate the derivation of a series trend model. We use a series of data on US gross national product for the period 1929-1978. at current prices. Let's build a polynomial regression model. The accuracy of the model increased until the degree of the polynomial reached the fifth:
Y = 145.6 - 35.67* + 4.59* 2 - 0.189* 3 + 0.00353x 4 + 0.000024* 5,
(14,9) (5,73) (0,68) (0,033) (0,00072) (0,0000056)
Where U - GNP, billion dollars;
* - years counted from the first year 1929;
Below the coefficients are their standard errors.
The standard errors of the model coefficients are small, not reaching values equal to half the values of the model coefficients. This indicates the good quality of the model.
The coefficient of determination of the model, equal to the square of the reduced multiple correlation coefficient, was 99%. This means that the model explains 99% of the data. The standard error of the model turned out to be 14.7 billion, and the significance level of the null hypothesis - the hypothesis of no connection - was less than 0.1%.
Using the resulting model, it is possible to give a forecast, which, in comparison with actual data, is given in Table. PZ. 1.
Forecast and actual size of US GNP, billion dollars.
Table PZ.1
The forecast obtained using the polynomial model is not very accurate, as evidenced by the data presented in the table.
Correlation analysis
Correlation analysis is necessary to identify correlations and their lags - delays in their periodicity. Communication in one process is called autocorrelation, and the connection between two processes characterized by series - cross-correlations. A high level of correlation can serve as an indicator of cause-and-effect relationships, interactions within one process, between two processes, and the lag value indicates a time delay in the transmission of interaction.
Typically, in the process of calculating the values of the correlation function on To The th step calculates the correlation between the variables along the length of the segment / = 1,..., (p - k) first row X and the segment / = To,..., P second row K The length of the segments thus changes.
The result is a value that is difficult for practical interpretation, reminiscent of the parametric correlation coefficient, but not identical to it. Therefore, the possibilities of correlation analysis, the methodology of which is used in many statistical packages, are limited to a narrow range of classes of time series, which are not typical for most economic processes.
Economists in correlation analysis are interested in studying lags in the transfer of influence from one process to another or the influence of an initial disturbance on the subsequent development of the same process. To solve such problems, a modification of the known method was proposed, called interval correlation".
Kulaichev A.P. Methods and tools for data analysis in the Windows environment. - M.: Informatics and computers, 2003.
The interval correlation function is a sequence of correlation coefficients calculated between a fixed segment of the first row of a given size and position and equal-sized segments of the second row, selected with successive shifts from the beginning of the series.
Two new parameters are added to the definition: the length of the shifted fragment of the series and its initial position, and the definition of the Pearson correlation coefficient accepted in mathematical statistics is also used. This makes the calculated values comparable and easy to interpret.
Typically, to perform an analysis, it is necessary to select one or two variables for autocorrelation or cross-correlation analysis, and also set the following parameters:
Dimension of the time step of the analyzed series for matching
results with a real timeline;
The length of the shifted fragment of the first row, in the form of the number included in
of the elements of the series;
The shift of this fragment relative to the beginning of the row.
Of course, it is necessary to choose the option of interval correlation or another correlation function.
If one variable is selected for analysis, then the values of the autocorrelation function are calculated for successively increasing lags. The autocorrelation function allows us to determine to what extent the dynamics of changes in a given fragment are reproduced in its own segments shifted in time.
If two variables are selected for analysis, then the values of the cross-correlation function are calculated for successively increasing lags - shifts of the second of the selected variables relative to the first. The cross-correlation function allows us to determine to what extent changes in the fragment of the first row are reproduced in fragments of the second row shifted in time.
The results of the analysis should include estimates of the critical value of the correlation coefficient g 0 for a hypothesis "r 0= 0" at a certain significance level. This allows you to ignore statistically insignificant correlation coefficients. It is necessary to obtain the values of the correlation function indicating the lags. Graphs of auto- or cross-correlation functions are very useful and visual.
Let us illustrate the use of cross-correlation analysis with an example. Let us evaluate the relationship between the growth rates of GNP of the USA and the USSR over the 60 years from 1930 to 1979. To obtain characteristics of long-term trends, the shifted fragment of the series was chosen to be 25 years long. As a result, correlation coefficients were obtained for different lags.
The only lag at which the correlation turns out to be significant is 28 years. The correlation coefficient at this lag is 0.67, while the threshold, minimum value is 0.36. It turns out that the cyclicality of the long-term development of the USSR economy with a lag of 28 years was closely related to the cyclicality of the long-term development of the US economy.
Spectral analysis
A common way to analyze the structure of stationary time series is to use the discrete Fourier transform to estimate the spectral density or spectrum of the series. This method can be used:
- to obtain descriptive statistics of one time series or descriptive statistics of dependencies between two time series;
- to identify periodic and quasiperiodic properties of series;
- to check the adequacy of models built by other methods;
- for compressed data presentation;
- to interpolate the dynamics of time series.
The accuracy of spectral analysis estimates can be increased through the use of special methods - the use of smoothing windows and averaging methods.
For analysis, you must select one or two variables, and the following parameters must be specified:
- the dimension of the time step of the analyzed series, necessary to coordinate the results with the real time and frequency scales;
- length To the analyzed segment of the time series, in the form of the number of data included in it;
- shift of the next segment of the row to 0 relative to the previous one;
- type of smoothing time window to suppress the so-called power leakage effect;
- a type of averaging of frequency characteristics calculated over successive segments of a time series.
The results of the analysis include spectrograms - values of amplitude-frequency spectrum characteristics and values of phase-frequency characteristics. In the case of cross-spectral analysis, the results are also the values of the transfer function and the spectrum coherence function. The results of the analysis may also include periodogram data.
The amplitude-frequency characteristic of the cross-spectrum, also called cross-spectral density, represents the dependence of the amplitude of the mutual spectrum of two interconnected processes on frequency. This characteristic clearly shows at what frequencies synchronous and corresponding in magnitude changes in power are observed in the two analyzed time series or where the areas of their maximum coincidences and maximum discrepancies are located.
Let us illustrate the use of spectral analysis with an example. Let us analyze the waves of economic conditions in Europe during the period of the beginning of industrial development. For the analysis, we use an unsmoothed time series of wheat price indices averaged by Beveridge based on data from 40 European markets over 370 years from 1500 to 1869. We obtain the spectra
series and its individual segments lasting 100 years every 25 years.
Spectral analysis allows you to estimate the power of each harmonic in the spectrum. The most powerful are the waves with a 50-year period, which, as is known, were discovered by N. Kondratiev 1 and received his name. The analysis allows us to establish that they were not formed at the end of the 17th - beginning of the 19th centuries, as many economists believe. They were formed from 1725 to 1775.
Construction of autoregressive and integrated moving average models ( ARIMA) are considered useful for describing and forecasting stationary time series and nonstationary series that exhibit uniform fluctuations around a changing mean.
Models ARIMA are combinations of two models: autoregression (AR) and moving average (moving average - MA).
Moving Average Models (MA) represent a stationary process as a linear combination of successive values of the so-called “white noise”. Such models turn out to be useful both as independent descriptions of stationary processes and as an addition to autoregressive models for a more detailed description of the noise component.
Algorithms for calculating model parameters MA are very sensitive to the incorrect choice of the number of parameters for a specific time series, especially in the direction of their increase, which may result in a lack of convergence of calculations. It is recommended not to select a moving average model with a large number of parameters at the initial stages of analysis.
Preliminary assessment - the first stage of analysis using the model ARIMA. The preliminary assessment process is terminated upon acceptance of the hypothesis about the adequacy of the model to the time series or upon exhaustion of the permissible number of parameters. As a result, the results of the analysis include:
- values of parameters of the autoregressive model and the moving average model;
- for each forecast step, the average forecast value, the standard error of the forecast, the confidence interval of the forecast for a certain level of significance are indicated;
- statistics for assessing the significance level of the hypothesis of uncorrelated residuals;
- time series plots indicating the standard error of the forecast.
- A significant part of the materials in the PZ section is based on the provisions of the books: Basovsky L.E. Forecasting and planning in market conditions. - M.: INFRA-M, 2008. Gilmore R. Applied theory of disasters: In 2 books. Book 1/ Per. from English M.: Mir, 1984.
- Jean Baptiste Joseph Fourier (Jean Baptiste Joseph Fourier; 1768-1830) - French mathematician and physicist.
- Nikolai Dmitrievich Kondratiev (1892-1938) - Russian and Soviet economist.
TIME SERIES ANALYSIS
INTRODUCTION
CHAPTER 1. TIME SERIES ANALYSIS
1.1 TIME SERIES AND ITS BASIC ELEMENTS
1.2 AUTOCORRELATION OF TIME SERIES LEVELS AND IDENTIFICATION OF ITS STRUCTURE
1.3 TIME SERIES TREND MODELING
1.4 Least SQUARE METHOD
1.5 REDUCING THE TREND EQUATION TO A LINEAR FORM
1.6 ESTIMATION OF REGRESSION EQUATION PARAMETERS
1.7 ADDITIVE AND MULTIPLICATE TIME SERIES MODELS
1.8 STATIONARY TIME SERIES
1.9 APPLYING THE FAST FOURIER TRANSFORM TO A STATIONARY TIME SERIES
1.10 AUTOCORRELATION OF RESIDUALS. DURBIN-WATSON CRITERION
Introduction
In almost every field there are phenomena that are interesting and important to study in their development and change over time. In everyday life, for example, meteorological conditions, prices for a particular product, certain characteristics of an individual’s health status, etc. may be of interest. All of them change over time. Over time, business activity, the mode of a particular production process, the depth of a person’s sleep, and the perception of a television program change. The totality of measurements of any one characteristic of this kind over a certain period of time represents time series.
The set of existing methods for analyzing such series of observations is called time series analysis.
The main feature that distinguishes time series analysis from other types of statistical analysis is the importance of the order in which observations are made. If in many problems observations are statistically independent, then in time series they are, as a rule, dependent, and the nature of this dependence can be determined by the position of observations in the sequence. The nature of the series and the structure of the process generating the series can predetermine the order in which the sequence is formed.
Target The work consists in obtaining a model for a discrete time series in the time domain, which has maximum simplicity and a minimum number of parameters and at the same time adequately describes the observations.
Obtaining such a model is important for the following reasons:
1) it can help to understand the nature of the system generating time series;
2) control the process that generates the series;
3) it can be used to optimally predict future values of time series;
Time series are best described non-stationary models, in which trends and other pseudo-stable characteristics, possibly changing over time, are considered statistical rather than deterministic phenomena. In addition, time series associated with the economy often have noticeable seasonal, or periodic, components; these components may vary over time and must be described by cyclic statistical (possibly non-stationary) models.
Let the observed time series be y 1 , y 2 , . . ., y n . We will understand this entry as follows. There are T numbers representing the observation of some variable at T equidistant moments in time. For convenience, these moments are numbered with integers 1, 2, . . .,T. A fairly general mathematical (statistical or probabilistic) model is a model of the form:
y t = f(t) + u t , t = 1, 2, . . ., T.
In this model, the observed series is considered as the sum of some completely deterministic sequence (f(t)), which can be called a mathematical component, and a random sequence (u t ), which obeys some probabilistic law. (And sometimes the terms signal and noise are used for these two components, respectively). These components of the observed series are unobservable; they are theoretical quantities. The exact meaning of this decomposition depends not only on the data themselves, but partly on what is meant by the repetition of the experiment from which these data are the result. The so-called “frequency” interpretation is used here. It is believed that, at least in principle, it is possible to repeat the entire situation, obtaining new sets of observations. Random components, among other things, may include observational errors.
This paper considers a time series model in which a random component is superimposed on the trend, forming a random stationary process. In such a model it is assumed that the passage of time does not affect the random component in any way. More precisely, it is assumed that the mathematical expectation (that is, the average value) of the random component is identically equal to zero, the variance is equal to some constant, and that the values of u t at different times are uncorrelated. Thus, any time dependence is included in the systematic component f(t). The sequence f(t) may depend on some unknown coefficients and on known quantities that change over time. In this case, it is called the “regression function”. Statistical inference methods for regression function coefficients prove useful in many areas of statistics. The uniqueness of methods related specifically to time series is that they study those models in which the above-mentioned quantities that change over time are known functions of t.
Chapter 1. Time series analysis
1.1 Time series and its main elements
A time series is a collection of values of any indicator for several consecutive moments or periods of time. Each level of a time series is formed under the influence of a large number of factors, which can be divided into three groups:
· factors shaping the trend of the series;
· factors that form cyclical fluctuations in the series;
· random factors.
With different combinations of these factors in the process or phenomenon under study, the dependence of the levels of the series on time can take different forms. Firstly, most time series of economic indicators have a trend that characterizes the long-term cumulative impact of many factors on the dynamics of the indicator being studied. It is obvious that these factors, taken separately, can have a multidirectional impact on the indicator under study. However, together they form an increasing or decreasing trend.
Secondly, the indicator being studied may be subject to cyclical fluctuations. These fluctuations may be seasonal, since the activities of a number of economic and agricultural sectors depend on the time of year. If large amounts of data are available over long periods of time, it is possible to identify cyclical fluctuations associated with the overall dynamics of the time series.
Some time series do not contain a trend or a cyclical component, and each subsequent level is formed as the sum of the average level of the series and some (positive or negative) random component.
In most cases, the actual level of a time series can be represented as the sum or product of trend, cyclical and random components. A model in which a time series is presented as the sum of the listed components is called additive model time series. A model in which a time series is presented as a product of the listed components is called multiplicative model time series. The main task of a statistical study of an individual time series is to identify and quantify each of the components listed above in order to use the information obtained to predict future values of the series.
1.2 Autocorrelation of time series levels and identification of its structure
If there is a trend and cyclical fluctuations in a time series, the values of each subsequent level of the series depend on the previous ones. The correlation dependence between successive levels of a time series is called autocorrelation of series levels.
It can be measured quantitatively using a linear correlation coefficient between the levels of the original time series and the levels of this series shifted by several steps in time.
One of the working formulas for calculating the autocorrelation coefficient is:
(1.2.1)As a variable x, we will consider the series y 2, y 3, ..., y n; as a variable y – the series y 1, y 2, . . . ,y n – 1 . Then the above formula will take the form:
(1.2.2)Similarly, autocorrelation coefficients of the second and higher orders can be determined. Thus, the second-order autocorrelation coefficient characterizes the closeness of the connection between the levels y t and y t – 1 and is determined by the formula
(1.2.3)The number of periods for which the autocorrelation coefficient is calculated is called lagom. As the lag increases, the number of pairs of values from which the autocorrelation coefficient is calculated decreases. Some authors consider it advisable to use the rule to ensure statistical reliability of autocorrelation coefficients - the maximum lag should be no more than (n/4).
Goals of time series analysis. In the practical study of time series based on economic data over a certain period of time, the econometrician must draw conclusions about the properties of this series and the probabilistic mechanism that generates this series. Most often, when studying time series, the following goals are set:
1. Brief (compressed) description of the characteristic features of the series.
2. Selection of a statistical model that describes the time series.
3. Predicting future values based on past observations.
4. Control of the process that generates the time series.
In practice, these and similar goals are far from always and far from being fully achievable. This is often hampered by insufficient observations due to limited observation time. Even more often, the statistical structure of a time series changes over time.
Stages of time series analysis. Typically, in practical analysis of time series, the following stages are sequentially followed:
1. Graphic representation and description of the behavior of a temporary rad.
2. Identification and removal of regular components of a time rad, depending on time: trend, seasonal and cyclical components.
3. Isolation and removal of low- or high-frequency components of the process (filtration).
4. Study of the random component of the time series remaining after removing the components listed above.
5. Construction (selection) of a mathematical model to describe the random component and checking its adequacy.
6. Forecasting the future development of the process represented by a time series.
7. Study of interactions between different time bands.
Time series analysis methods. There are a large number of different methods to solve these problems. Of these, the most common are the following:
1. Correlation analysis, which makes it possible to identify significant periodic dependencies and their lags (delays) within one process (autocorrelation) or between several processes (cross-correlation).
2. Spectral analysis, which makes it possible to find periodic and quasiperiodic components of a time series.
3. Smoothing and filtering, designed to transform time series to remove high-frequency or seasonal fluctuations from them.
5. Forecasting, which allows, based on a selected model of the behavior of a temporary rad, to predict its values in the future.
Trend models and methods for extracting them from time series
The simplest trend models. Here are the trend models most often used in the analysis of economic time series, as well as in many other areas. First, it's a simple linear model
Where a 0, a 1– trend model coefficients;
t – time.
The unit of time can be an hour, a day(s), a week, a month, a quarter or a year. Model 3.1. Despite its simplicity, it turns out to be useful in many real-life problems. If the non-linear nature of the trend is obvious, then one of the following models may be suitable:
1. Polynomial :
(3.2)
where is the degree of the polynomial P in practical problems it rarely exceeds 5;
2. Logarithmic:
This model is most often used for data that tends to maintain a constant growth rate;
3. Logistics :
(3.4)
Gompertz
(3.5)
The last two models produce S-shaped trend curves. They correspond to processes with gradually increasing growth rates in the initial stage and gradually decaying growth rates at the end. The need for such models is due to the impossibility of many economic processes to develop for a long time at constant growth rates or according to polynomial models, due to their rather rapid growth (or decrease).
When forecasting, the trend is used primarily for long-term forecasts. The accuracy of short-term forecasts based only on a fitted trend curve is usually insufficient.
The least squares method is most often used to estimate and remove trends from time series. This method was discussed in some detail in the second section of the manual in problems of linear regression analysis. The time series values are treated as a response (dependent variable), and time t– as a factor influencing the response (independent variable).
Time series are characterized by mutual dependence its members (at least not far apart in time) and this is a significant difference from ordinary regression analysis, for which all observations are assumed to be independent. However, trend estimates under these conditions are usually reasonable if an adequate trend model is chosen and if there are no large outliers among the observations. The above-mentioned violations of the restrictions of regression analysis affect not so much the values of the estimates as their statistical properties. Thus, in the presence of a noticeable dependence between the terms of the time series, variance estimates based on the residual sum of squares (2.3) give incorrect results. The confidence intervals for the model coefficients, etc., also turn out to be incorrect. At best, they can be considered very approximate.
This situation can be partially corrected by applying modified least squares algorithms, such as weighted least squares. However, these methods require additional information about how the variance of observations or their correlation changes. If such information is not available, researchers must use the classical least squares method, despite these disadvantages.
