Theoretical Foundations of Macroeconomics with Impulse and Jump Characteristics

1. Introduction

There are a lot of problems in this area. But two main questions are the following: “Why are some  countries rich and others poor?”, and “Why does the economy in some countries  grow fast and in others slowly?” [1]. To answer these questions, we need to analyze data  relating to the economic performance of various countries.

Currently, there  are different models of economic growth. Let's look at the most famous of them.

The foundations of  modern growth theory were laid by Nobel Prize winner R. Solow, having developed  the neoclassical theory of economic growth [2], in which the  main role was played by the accumulation of physical capital. Solow also showed  for the first time that the key factor of economic growth is technical  progress, which he set exogenously [3]. The main focus of the Solow model, along with  traditional issues of capital accumulation, is on the relationship between the  two main factors of production - labor and capital, as well as their  relationship with the exogenous source of changes in productivity - technical  progress. Solow's growth theory, like all classical growth theories,  is based on the law of diminishing returns of factors, which is the main  condition for achieving an equilibrium state and sustainable development.  According to this law, under constant technical conditions, a consistent  increase in any of the production factors by an additional unit, with the  others remaining unchanged, leads to a decreasing increase in production. This  occurs under conditions of perfect competition. The production function in the  economic growth model is the Cobb-Douglas function, which includes a power  function, the basis of which is physical capital, and the exponent varies from  zero to one. The mathematical basis of Solow's economic growth model is the  first-order Bernoulli differential equation.

From the point of view of this article, the  disadvantage of the Solow model is that it is based on continuous functions and  does not take into account impulse, shock, step, spasmodic and other types of  rapidly changing processes characteristic of modern macroeconomics. In  addition, this model includes a multiplier that expresses a measure of  productivity. To this day, economists have been unable to quantify this  multiplier with a high enough degree of realism. Solow himself called this  measure “a measure of our ignorance.” [4].  This drawback sharply reduces the possibility of practical use of the model for  solving real problems and predicting economic events.

In the Mankiw-Romer-Weil model [5], human capital is introduced into the basic  neoclassical Solow model of economic growth, which improves the theoretical  provisions of the model and is more consistent with empirical data than the  similar result of the Solow model without human capital. However, the main  results of the basic neoclassical model remain unchanged; sustainable economic  growth depends on external technological progress, while the savings rate,  institutional and behavioral parameters do not affect it. Therefore,  sustainable economic growth remains exogenous in nature.

The shortcomings of this model are similar to those  of the Solow macroeconomic model. Just like the Solow model, the  Mankiw-Romer-Weil model is based on continuous power functions, which does not  allow it to be used to describe abrupt macroeconomic processes.

Recursive macroeconomic models of  Stokey-Lucas-Prescott [6] are based on dynamic  programming methods developed by R. Bellman [7].  Dynamic programming is associated with the ability to represent the management  process in the form of a chain of sequential actions or steps, unfolded over  time and leading to a goal. The control process can be divided into parts and  represented as a dynamic sequence and interpreted as a step-by-step program.  The dynamic programming problem is formulated as follows: it is required to  determine a control that transfers the system from the initial state to the  final state, in which the objective function takes an extreme value.

The goals and methods of the proposed macroeconomic  theory are completely different. It is not about a dynamic management process  with a specifically defined goal and an assigned target function. This article  is about formulating the foundations of macroeconomic theory and its methods,  which, with some degree of accuracy, could give a forecast about the impending  possibility of abrupt changes (impulse, shock, stepwise and others) in the  macroeconomic situation, both in a negative and positive direction. What  management decisions will be made to correct the situation based on the  forecasts made remains outside the scope of the proposed approach to the  analysis of fast-moving processes in macroeconomics.

The economies of  many countries are strongly influenced by rapidly changing processes and  phenomena, crisis events, epidemics, military operations, and so on. This is  especially evident in recent years [8,91011,12,13]. Examples include the global pandemic, events in  Ukraine, Russia, Israel, Palestine and Europe, fuel and food crises, and  others. The new paradigm requires the development of new economic theories,  economic and mathematical models that take into account impulse, step and  spasmodic changes in economic indicators. The article describes new  mathematical methods that make it possible to analyze macroeconomic processes  with impulse, generalized and piecewise linear functions. Methods for  approximating these functions by means of analytical expressions are considered.  The possibilities of forecasting macroeconomic processes with impulse and jump  characteristics are presented.

Let us note some  publications in which shock situations are studied in various aspects of  macroeconomic theory [14,15,16,1718,19,20,21,22,23,24,25]. Note that these works represent only a small part of  the publications on the selected topic. This once again indicates the relevance  of developing a macroeconomic theory with rapidly changing characteristics.

2. Main Types of Impulse and Step Characteristics

In the scientific  literature, there are discrepancies in the definitions of impulse and step  characteristics depending on the areas of their use (signal transmission,  electrical engineering, automatic control theory, dynamic systems, catastrophe  theory, and others) [26,27,28,29]. For definiteness,  in this paper, by jump characteristics we mean functions with a sharp change in  values with subsequent stabilization of values at a new level (shifts). By  impulse responses we mean functions with a sharp short-term change in values  (shocks). The words “sharp”, “short-term” have a subjective meaning, however,  the definitions can be clarified using the terms “outliers”, “standard  deviation” and other statistical concepts.

Next, we present idealized models of these  characteristics and their applications in the field of economic analysis.  Naturally, in practice, fluctuations caused by many random causes are  superimposed on trend lines.

• Single jump function (single step function, Heaviside function, “step”).

One way to write this function is:

$$f\left(x\right)=\left\{\begin{array}{c}0, x<0;\\ 1, x\ge 0.\end{array}\right.$$

The graph of the function is shown in Figure 1a).

In practice, a sharp change in the values of the  function does not occur instantly, but with a certain transitional time period  x* (Figure 1b). For example, an abrupt  change in the exchange rate of the national currency tenge in Kazakhstan in  February 2014 against the US dollar and the euro took place over several days (Table 1). The official website of the National  Bank of the Republic of Kazakhstan is www.nationalbank.kz.

The unit jump function is the antiderivative function for the delta function (Dirac function) [28]. The delta function graph is shown in Figure 1c).

Examples of the manifestation of the unit jump function in the economy are shown in Figure 2.

In September 2020, the incidence of COVID-19 began to increase in most countries. In June 2020, the National Bureau of Economic Research (USA) announced a recession or recession in the economy since February. In absolute terms, the number of unemployed increased from 5787 thousand people. (February) to 23078 thousand people. (April). In August it amounted to 13550 thousand people. The employment rate is shown in Figure 2b).

In order to highlight the trend in more detail, it is possible to use a statistical procedure for smoothing the average value (Figure 3).

A unit pulse (rectangular function, normalized rectangular window) is given by the following expression:

$$f\left(x\right)=\left\{\begin{array}{c}0, \left|x\right|>\mathrm{0,5}; \\ \mathrm{0,5},\left|x\right|=\mathrm{0,5}; \\ 1,\left|x\right|<\mathrm{0,5}. \end{array}\right.$$

The graph of the function is shown in Figure 4a).

Triangular impulse (triangular function) is a piecewise linear function given by:

$$f\left(x\right)=\left\{\begin{array}{c}1-\left|x\right|, \left|x\right|<1;\\ 0, \left|x\right|\ge 1.\end{array}\right.$$

The graph of the function is shown in Figure 4b).

Examples of impulse functions in the economy are shown in Figure 5.

The graph in Figure 5a largely corresponds to the delta function.

On Figure 5b shows the macroeconomic shock caused by the September 11, 2001 terrorist attacks and reflected in the Dow Jones Industrial Average. After the initial panic, the DJIA rose rapidly, falling only marginally from its pre-attack levels.

To describe rapidly changing macroeconomic processes, various combinations of impulse and jump models are possible. For example, Figure 6 shows the stepwise transformation of the Russian ruble exchange rate from one stable state to another through an impulse transition process. This shows the effect of the interaction of the step and impulse functions. The trend line is highlighted in red. Figure 7 shows the dynamics of the world cycle: industrial production, 2019-2022, seasonally adjusted quarterly growth rates. In the figure we see a combination of two oppositely directed impulses. It is also possible to display various variants of piecewise linear functions.

3. Approximations of Step, Impulse and Generalized Functions

3.1. Description of the Methods

Approximations of step, impulse and generalized functions makes it possible to more accurately identify trends in rapidly changing processes and describe existing processes in macroeconomics. These functional forms may become useful here as predictive tools.

Currently, there are many methods for approximating piecewise linear functions using analytic expressions. This article describes only the methods proposed by the author and having a number of advantages over known approximations. These methods, like Fourier series, are based on the use of trigonometric expressions. The difference lies in the fact that trigonometric expressions in the developed methods have the form of nested, recursive functions [30,31,32,33]. The methods were further developed in [34,35] and many other scientific papers.

For example, let's take a step function, which will explain the main idea of the approximation:

$$f_0(x)=\mathrm{sign}(\sin \mathrm{x})$$

(1)

The step function is often used to describe Fourier series. Therefore, the choice of a step function is convenient for comparing and identifying the advantages and disadvantages of expansion into Fourier series and the methods developed by the author.

It is known that the Fourier expansion of the proposed function (1) is characterized by certain disadvantages [36,37]. The Gibbs effect can be noted. In the developed methods, the initial step function is approximated by a recursive sequence of periodic trigonometric functions, which eliminates the shortcomings of the Fourier series expansion:

$$\left\{f_n(x)\hspace{0.33em}\left|f_n(x)=\sin \left((\pi /2)\cdot f_{n-1}(x)\right),\hspace{0.33em}{f}_1(x)=\sin x;\hspace{0.33em}n-1\in N\right.\right\}\subset C^{\infty }[-\pi ,\pi ]$$

(2)

In Figure 8, the thickened line marks the initial step function. The thin lines show the graphs of five successive approximations by the proposed method.

Figure 8 shows that even the first approximations of the proposed iterative sequence quickly converge to the step function (1). At the same time, the shortcomings of the expansion into Fourier series, in particular, the Gibbs effect [36], are completely leveled.

The developed methods of trigonometric approximation using an iterative recursive procedure differ in some of their properties. Let's note some of them.

Periodic functions $f_n(x)$ and $f_0(x)$ with period $2\pi$ are odd. The periodic functions $f_n(x+\pi /2)$ and $f{}_0(x+\pi /2)$ are even. It follows from this fact that one can only study the approximating sequence of functions (2) on the interval on the interval $\left[0,\hspace{0.33em}\pi /2\right]$. It would be enough.

Let $\left\{f_n(x)\right\}\subset L_2[0,\pi /2]$ and $f_0(x)\in L_2[0,\pi /2]$. Because $\underset{n\in N}{\sup }\underset{x\in [0,\hspace{0.33em}\pi /2]}{\sup }\left|f_n(x)\right|=1<\infty$ (because the function is limited) $f_n(x)$) and $\underset{n\in N}{\sup }\underset{0}{\overset{\mathsf{\pi }/2}{\mathrm{Var}}}{f}_n=1<\infty$ (since the function $f_n(x)$ is monotonic on the segment $[0,\pi /2]$), then by Helly's theorem we can extract a subsequence from the given sequence $\left\{f_n(x)\right\}$. This subsequence converges to some function $f$ at each point from $[0,\hspace{0.33em}\pi /2]$, and, additionally, $\underset{0}{\overset{\pi /2}{\mathrm{Var}}}f\le \underset{n\to \infty }{\overset{\_\_\_}{\lim }}\underset{0}{\overset{\pi /2}{\mathrm{Var}}}{f}_n$. The original function may be such a function $f_0(x)$. Let's prove this statement.

Theorem 1.

The sequence$f_n(x)$converges pointwise to the function$f_0(x)$, but not uniformly.

Proof. 

In $x=0$ and $x=\pi /2$ there is $f_n(x)-f_0(x)=0,\hspace{0.33em}\forall n\in N$. Therefore $f_n(x)\underset{n\to \infty }{\to }{f}_0(x)$ since $\forall \epsilon >0\hspace{0.33em}\exists n^{\ast }\in N\hspace{0.33em}\forall n:n>n^{\ast }\Rightarrow \hspace{0.33em}\left|f_n(x)-f_0(x)\right|=0<\epsilon$. For instance, $n^{\ast }=1$.

$f_n(x)=\sin \left((\pi /2)\cdot f_{n-1}(x)\right)>f_{n-1}(x)>\dots >f_1(x)>0$ for any $x\in (0, \pi /2)$, since $\sin x>(2/\pi )\cdot x,\hspace{0.33em}\forall x\in (0,\hspace{0.33em}\pi /2)$. $f_n(x),\hspace{0.33em}\forall x\in (0,\hspace{0.33em}\pi /2)$ has a finite limit $\underset{n\to \infty }{\lim }{f}_n(x)=A\in R$. $A=\underset{n\to \infty }{\lim }\sin ((\pi /2)\cdot f_{n-1}(x))=\sin ((\pi /2)\cdot \underset{n\to \infty }{\lim }{f}_{n-1}(x))=\sin ((\pi /2)\cdot A)$, $A=0$ or $A=1$. $A=1=f_0(x)$, since $f_n(x)$ is increasing and positive. For $x\in (0, \pi /2)$, $f_n(x)\underset{n\to \infty }{\to }{f}_0(x)$. $f_n(x)$ converges at $x=0$, $x=\pi /2$. Therefore, $f_n(x)\underset{n\to \infty }{\to }{f}_0(x),\hspace{0.33em}\forall x\in [0,\hspace{0.33em}\pi /2]$. $f_0(x)$ is not continuous on $\left[0,\hspace{0.33em}\pi /2\right]$. Therefore, the convergence is pointwise, but not uniform. $∎$

Theorem 2.

$f_n(x)$ converges in norm to $f_0(x)$ in the spaces $L_1[0,\pi /2]$ and $L_2[0,\pi /2]$.

Proof. 

We take a sequence of minorant functions for $f_n(x)$:

$$\left\{{\mathit{\eta }}_n(x)\hspace{0.33em}\left|\hspace{0.33em}{\mathit{\eta }}_n(x)=(2/\pi )\cdot \mathrm{arctg}(n\pi );\hspace{0.33em}n\in N\right.\right\}\subset C^{\infty }[0,\pi /2]$$

$f_n(x)\ge {\mathit{\eta }}_n(x), \forall n\in N,\hspace{0.33em}\forall x\in [0,\hspace{0.33em}\pi /2]$. The measure for the set of discontinuity points of $f_0(x)$ is zero. $f_n(x)$ and ${\mathit{\eta }}_n(x)$ are non-negative and limited on the segment $\left[0,\hspace{0.33em}\pi /2\right]$ in $L_1[0,\hspace{0.33em}\pi /2]$:

$\left|\left|f_0(x)-f_n(x)\right|\right|={\displaystyle \underset{0}{\overset{\pi /2}{\int }}(1-f_n(x))\mathrm{dx}}\le {\displaystyle \underset{0}{\overset{\pi /2}{\int }}(1-{\mathit{\eta }}_n(x))\mathrm{dx}}=\frac{\pi }{2}-\mathrm{arctg}\frac{\pi n}{2}+\frac{1}{\pi n}\cdot \ln \left(1+{(\pi n)}^2/4\right)$$\underset{n\to \infty }{\lim }\left(\frac{\pi }{2}-\mathrm{arctg}\frac{\pi n}{2}+\frac{1}{\pi n}\cdot \ln \left(1+{(\pi n)}^2/4\right)\right)=0$, therefore, $\left|\left|f_0(x)-f_n(x)\right|\right|\underset{n\to \infty }{\to }0$.

The same is for $f_n(x)$ in $L_2[0,\pi /2]$. ■

$f_n(x)$ is fundamental in $L_1[-\pi ,\pi ]$ and $L_2[-\pi ,\pi ]$, but not in $C[-\pi ,\pi ]$.

The angular function $f_1(x)$ may not necessarily be the sine, but another one, including non-periodic ones. For $\left|f_1(x)\right|<2$, $\underset{n\to \infty }{\lim }{f}_n(x)=\mathrm{sign}(f_1(x))$. With (2) it is possible to approximate an arbitrary step function. For example, consider

$$f(x)=\left\{\begin{array}{l}h,\hspace{0.33em}x\in (x_1,x_2),\\ 0,\hspace{0.33em}x\notin (x_1,x_2)\end{array}\right.$$

(3)

and take the function $f_1(x)=\exp (1-{(ax+b)}^2)-1$ as the angular one. $f_1(x_1)=f_1(x_2)=0$, $a=2/(x_1-x_2);\hspace{0.33em}\hspace{0.33em}b=(x_1+x_2)/(x_2-x_1)$, so

$\left\{f_n(x)\hspace{0.33em}\left|\hspace{0.33em}{f}_n(x)=(h/2)\cdot (1+\sin {\varphi }_n(x)),\hspace{0.33em}{\varphi }_n(x)=(\pi /2)\cdot \sin {\varphi }_{n-1},\hspace{0.33em}{\varphi }_1(x)=(\pi /2)\cdot f_1(x),\hspace{0.33em}n-1\in N\hspace{0.33em}\right.\right\}$ converges to $f(x)$. It is shown in Figure 9.

Any step function with $h_i$ on $(x_{1i},x_{2i})$ can be approximated by the sum ${\displaystyle \sum _{i=1}^k{\left\{f_n(x)\right\}}_i}$.

$f\left(x\right)=\sum _{i=1}^k{h}_i·f_{0i}\left(x\right),h_i\in R,$ where $f_{0i}\left(x\right)=\mathrm{sign}\left(\sin \left(l_{i}x-x_i\right)\right),l_i,x_i\in R$, can approximate any periodic step function. The convergence of $f_n\left(x\right)=\sum _{i=1}^k{h}_i·f_{ni}\left(x\right)$ to $f\left(x\right)$ in the norm is ensured by the proved theorem, since $‖f_{0i}\left(x\right)-f_{ni}\left(x\right)‖\underset{n\to \infty }{\to }0$ in $L_1\left[-\pi ,\pi \right]$ and $L_2\left[-\pi ,\pi \right]$, and $‖f\left(x\right)-f_n\left(x\right)‖=‖\sum _{i=1}^k{h}_i·f_{0i}\left(x\right)-\sum _{i=1}^k{h}_i·f_{ni}\left(x\right)‖\le \sum _{i=1}^k\left|h_i\right|·‖f_{0i}\left(x\right)-f_{ni}\left(x\right)‖\underset{n\to \infty }{\to }0.$

3.2. Generalized Functions. Approximation

Generalized functions [38] began to be used in the 20th century to solve problems in the field of quantum physics. The solution of such tasks required supplementing the generally recognized mathematical concept of a function. This concept suggested that by a function we understand a certain statement, according to which for each value of an independent variable belonging to a certain set, one value of the dependent variable is indicated. To solve the problems of quantum physics, it was necessary to introduce functions that go beyond the usual definitions. At present, generalized functions are used not only for solving problems of quantum physics, but also for solving other problems in various fields of science and engineering.

Let us consider the concept of a generalized function in more detail.

Let's take a linear space $Y$. Functions in the traditional mathematical sense are points in this space. A functional is given on $Y$, if for each point $y\in Y$ from the space under consideration we can specify a certain number. The defined functional will be written $I:Y\to R$, or $I\left(y\right)$).

The functional is linear under the assumption $I\left(\alpha y_1+\beta y_2\right)=\alpha I\left(y_1\right)+\beta I\left(y_2\right),\forall y_1,y_2\in Y,\forall \alpha ,\beta ϵR,$ and continuous when $y_n\Rightarrow y^{*}$ follows $I\left(y_n\right)\to I\left(y^{*}\right),\forall y_n,y^{*}\in Y$. We consider our functions on the set of real numbers R.

A function $\phi \left(x\right)$ is finite, if it is equal to zero outside $\left[a,b\right]$, and the boundaries of this segment are determined by $\phi \left(x\right)$. Any continuous function with compact support is basic. $C_0$ is a set of basic functions.

Next, consider a continuous function $f\left(x\right)$, which can have a finite number of discontinuity points. The function is bounded on any finite interval.

A functional is defined by the integral $I\left(\phi \right)={\int }_{-\infty }^{+\infty }f\left(x\right)\phi \left(x\right)\mathrm{dx}$. Any basic function $\phi \left(x\right)$ will be finite. In this case we have a regular functional.

Definition [32]. Any linear continuous functional $I\left(\phi \right)$ on $C_0$ is called a generalized function, if

• $I$(α${\phi }_1$+ β${\phi }_2$)$=\alpha I\left({\phi }_1\right)+\beta I\left({\phi }_2\right),\forall {\phi }_1,{\phi }_2\in C_0,\forall \alpha ,\beta ϵR;$
• $I\left({\phi }_n\right)\to I\left(\phi \right),$ if ${\phi }_n\Rightarrow \phi$ in $C_0$.

A generalized function that cannot be represented by the integral $I\left(\phi \right)={\int }_{-\infty }^{+\infty }f\left(x\right)\phi \left(x\right)\mathrm{dx}$, is called singular. For example, the δ-function or the Dirac function is a singular generalized function $I\left(\phi \right)=\phi \left(0\right)$.

The basic idea of singular generalized functions can be visualized if we consider a generalized function from the point of view of the limit to which a sequence of approximating functions with traditional definitions tends. In this case, generalized functions are perceived through the prism of their approximating procedures. For example, suppose

$${\delta }_n(x)=\left\{\begin{array}{c}n/2,\hspace{0.33em}\forall x\in [-1/n,\hspace{0.33em}1/n],\\ 0,\hspace{0.33em}\forall x\notin [-1/n,\hspace{0.33em}1/n].\end{array}\right.$$

The functions of this sequence have graphs corresponding to the graph of the step function shown in Figure 10.

It is easy to see that for any area of the figure under the graph of such a step function is equal to one.

The limit of the sequence of the functions, the graph of one of them is schematically shown in Figure 10 is the delta function. However, step functions have discontinuity points of the 1st kind. At these points the step functions are not differentiable. Therefore, this approach prevents the analytical representation of the approximation dependences for the derivatives of the delta function. Note that the derivatives of the delta function also belong to the class of generalized functions. To enable the analytical representation of derivatives, the author proposed a procedure using an approximating sequence that allows one to find derivatives of delta functions of any order. Let us write an expression that implements this approach

$f\left(x\right)=\cos \left(A\left(A\left(\dots A\left(x\right)\right)\right)\right)$, where $A\left(x\right)=\frac{\pi }{2}\sin \left(x\right)$.

For example, the graph of the function

$f\left(x\right)=310\cos \left(A\right(A\left(A\right(A\left(A\right(A\left(A\right(A\left(A\right(A\left(A\right(A\left(A\right(A\left(A\right(A\left(A\right(A\left(A\right(A\left(x\right)\left)\right)\left)\right)\left)\right)\left)\right)\left)\right)\left)\right)\left)\right)\left)\right)\left)\right)\left)\right)$ from the proposed sequence is shown in Figure 11.

The approximation error of the δ-function by the proposed methods is much smaller compared to the Fourier series. Moreover, the approximation error can be arbitrarily small with an increase in the number of nested functions in the proposed expression. Using the integral condition written in the definition of the δ-function, one can find the amplitude of the approximation dependence.

Let us show how to find the height of the approximation peak. We know that the δ-function is the derivative of the unit jump function

$$H\left(x\right)=\left\{\begin{array}{c}1, \forall x>0;\\ 0, \forall x<0.\end{array}\right.$$

We can approximate the unit jump function (Heaviside function) by a sequence of analytic functions $H_n\left(x\right)=\mathrm{0,5}\left(1+f_n\left(x\right)\right)$. The sequence $f_n\left(x\right)$ is defined by (2) on $[-\pi /2,\pi /2$].

For instance, we have (Figure 12)

$$H_9(x)=0,5(1+\sin (A(A(A(A(A(A(A(A(x))))))))))$$

$$H_{10}(x)=0,5(1+\sin (A(A(A(A(A(A(A(A(A(x))))))))))$$

$$H_{11}(x)=0,5(1+\sin (A(A(A(A(A(A(A(A(A(A(x)))))))))))$$

where $A(x)=\frac{\pi }{2}\sin x$.

The larger the approximation number, the greater the graph thickness (Figure 12).

Graphs of the first derivatives of successive functions $H_9\left(x\right), H_{10}\left(x\right), H_{11}\left(x\right)$ are shown in Figure 13. They characterize successive approximations of the delta function.

For the sequence $H_n(x)=0,5(1+f_n(x))$, we get

$\frac{d{H}_n(x)}{dx}=\frac{{\pi }^{n-1}}{2^n}{\displaystyle \prod _{k=1}^{n-1}\cos \hspace{0.33em}\left(\frac{\pi }{2}{f}_k(x)\right)}\cdot \cos \hspace{0.33em}x$.

Therefore, we have

$A_n=\frac{{\pi }^{n-1}}{2^n}$.

The performed analytical approximations of step, impulse and generalized functions make it possible to use the methods of mathematical analysis in the analysis and forecasting of rapid changes in the macroeconomic situation.

The developed methods are universal and have found application in various fields of scientific research and their practical applications [39,40,41,42,43,44].

4. Choice of Macroeconomic Indicators for the Analysis of Impulse and Spasmodic Processes

The totality of macroeconomic parameters is controversial. Nevertheless, it is possible to single out macroeconomic parameters with the importance of which the overwhelming majority of macroeconomists agree. We will divide the parameters into two groups (Table 2) [8]. Note that macroeconomic indicators are largely interconnected.

As a rule, macroeconomic indicators have annual values, in some cases, quarterly and monthly. This is the difficulty of applying macroeconomic indicators for the analysis of fast-moving (impulsive and spasmodic) economic processes. As practice shows, in some cases the macroeconomic situation can change dramatically in a matter of days. Examples of macroeconomic indicators with daily values are the exchange rate, energy prices (and even by hours and minutes) on stock exchanges, and others. Table 3, for example, shows prices for Brent crude oil.

The daily change in the situation on some macroeconomic indicators can be estimated indirectly. For example, the daily unemployment rate can be estimated by the number of welfare claims filed in a country during the day. Modern computer technology makes it quite easy to do this.

To improve the accuracy of forecasting crisis situations, economic analysis should be carried out on a set of macroeconomic indicators, which provides a more holistic view of pre-crisis realities. First of all, macroeconomic indicators with greater inertia to changes are of interest. Note that macroeconomic shifts can also occur in a positive direction, which can be determined, for example, by investments and a high level of innovative development.

5. Approaches to Methods for Forecasting Macroeconomic Events

5.1. Control Rules

In quality control systems [45,46,47], control rules have been developed (Table 4), which should be paid attention to if violations of specified standards are suspected in the near future. The probability of manifestation of each of these rules within the framework of a random process is small, therefore, each of these manifestations indicates a probable violation of the course of a normal process. Therefore, with some probability, a loss of quality should be expected.

The problem of applying similar rules in the analysis of macroeconomic processes lies in the fact that, in practice, changes in macroeconomic processes can occur not only for random reasons, but also as a result of purposeful managerial actions. Therefore, for application in macroeconomic theory with impulsive and abrupt changes, these rules must be formulated with some “margin”, for example, not one value above +3 sigma, but two, not one value above + 3 sigma, but one value above + 4 sigma, etc. At the same time, the probability of an error of the first kind decreases (false prediction of an impulse or abrupt development of an event series), but the probability of an error of the second kind increases (ignoring such a development). For example, Figure 14 shows graphs of the same data set in the ∓3σ range (Figure 14a) and in the ∓4σ range (Figure 14b). The range boundaries are marked with dotted lines. Some points that are critical in the range ∓3σ are not critical in the range ∓4σ. Therefore, the standard quality control rules for predicting possible crisis phenomena in the framework of the new macroeconomic theory need to be reformulated.

Some of the rules can be left unchanged, for example, 6 points in a row in ascending (descending) order. This rule indicates the existence of a trend, which can lead to abrupt changes in the macroeconomic process.

The creation of a new system of rules for predicting rapid changes in macroeconomic indicators is possible on the basis of statistical processing of information and heuristic methods. Pay attention to weak signals. Note that predictions made on the basis of a system of attention-grabbing criteria can only be realized with a certain probability, which is common when using statistical methods.

In the context of the frequent lack of daily data, we will give examples of forecasts for a rapid change in macroeconomic indicators by months.

Figure 15a shows a graph of the Brent oil price ($US per barrel) for several consecutive months in 2007-08. Source of information: 1. https://investfunds.ru, 2. https://ru.investing.com. It follows from the graph that 3 points break the rules. Of particular interest is point number 3, since at this point 3 rules are violated at once (Table 5).

Figure 15b shows the graph of inflation in Russia by months as a percentage of the previous period. Source of information: Rosstat data, https://rosstat.gov.ru/folder/10705. Of critical points 1 and 2, point 2 is of particular interest, since in the subsequent period (in March 2022), the inflation rate jumped sharply to a value of 7.61.

Let's take another example. Table 6 contains the values of the average monthly nominal accrued wages in rubles of employees in the whole economy of the Russian Federation. Source: Rosstat, https://rosstat.gov.ru. The December 2019 salary figure is omitted as the annual bonus is usually paid in the month of December, which greatly distorts the overall picture.

The graph of the average monthly salary with critical points 1,2 and 3 is shown in Figure 16.

Violation of control rules at critical points is explained in Table 7.

It should be noted that in the considered examples (Figure 15, 16) the control rules of the theory of quality were applied. Within the framework of the new macroeconomic paradigm, it is necessary to revise and adjust the system of control rules based on the accumulated statistics and heuristic considerations.

We note the main issues that require further development within the framework of the new macroeconomic theory:

Creation of a system of macroeconomic indicators with fixing daily values.

Detailing the set of statistical rules for predicting crisis, fast-moving, impulsive and spasmodic changes in macroeconomic processes.

5.1. Theoretical Foundations of the Technique

For each macroeconomic indicator $X$ we define:

$n$ − number of consecutive days with daily observations $X_1, X_2, \dots , X_n$;

$\overline{X}$ − sample mean, $\overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_1;$$\sigma$ − sample standard (mean square) deviation, $\sigma =\sqrt{\frac{\sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}{n-1}}$. We construct the central line Mean corresponding to the sample mean $\overline{X}$, and the lines ±σ; ±2σ; ±3σ; ±4σ;

UCL − upper control limit;

LCL − lower control limit.

The upper UCL and lower LCL reference lines can correspond to values of ±3σ or ±4σ depending on our choice, as explained in the previous Section 5.1. Next, to forecast possible abrupt changes in the macroeconomic situation, we use the control rules from Section 5.1.

As noted earlier, in order to improve the accuracy of forecasts based on the provisions of the new macroeconomic theory, macroeconomic indicators should be considered not separately, but in their totality. In this case, it is required to apply methods of multivariate analysis, such as: multivariate scaling, factor analysis, cluster analysis, and others.

The new macroeconomic theory requires the accumulation of statistical data on the daily values of macroeconomic indicators due to their possible rapid change. This work is only at the beginning of the journey and requires additional efforts. However, the development of computer technology does not raise doubts about the possibility of successfully solving this problem. At present, however, the overwhelming majority of macroeconomic indicators are measured by years, and only at best by quarters and months. Therefore, it is still difficult to give specific examples of forecasts based on the daily values of a set of macroeconomic indicators. It is only possible to outline some analogies of such approaches based on the annual values of macroeconomic indicators. As such analogies, let us consider the dynamics of a certain set of macroeconomic indicators of the Russian Federation for the period 2012-22 (Table 8). Source: compiled by the author based on data from the Federal State Statistics Service: https://rosstat.gov.ru/folder/10705.

Using the ALSCAL multidimensional scaling procedure, you can get a visual representation of multidimensional tabular information (Figure 17a). The figure shows that 2020, 2021 and 2022 stand out sharply in terms of the totality of their indicators. Further analysis shows the economic crisis in 2020 and a sharp improvement in the situation in 2021 and 2022.

The conclusions made are confirmed by cluster analysis. On Figure 17b is a dendrogram showing the possibility of splitting the variable “Year” into 5 clusters (section along the dotted line).

Clustering is presented in Table 9. Average values of variables by clusters are presented in Table 10.

Multidimensional scaling methods have a great advantage, they allow you to process multidimensional information and present the results of processing visually. This allows the use of macroeconomic variables in their totality to identify possible pre-crisis and forecast other rapidly changing macroeconomic situations. In addition to the previous examples, let's consider a more complete set of Russia's macroeconomic indicators in the dynamics of their changes over the period 2004-2011 (Table 11).

A visual representation of the results of applying multidimensional scaling is given in Figure 18.

It can be seen from the figure that the years 2005 and 2005, 2007 and 2008, 2010 and 2011 are pairwise close in terms of the totality of initial macroeconomic indicators. The years 2006 and 2009 occupy rather isolated positions and, therefore, differ from the rest and from each other.

A similar processing of information for predicting rapid macroeconomic changes is assumed in the framework of the new macroeconomic theory in the presence of a set of variables with daily values and daily monitoring.

Currently, work is also underway to improve the mechanism of foresight controlling [48,49,50].

6. Conclusions

• This article describes only the main approaches to the creation of a macroeconomic theory with rapidly changing characteristics of an impulse and jump type, and formulates its main provisions. These provisions need to be further developed and improved on the basis of statistical analysis.
• The current macroeconomic paradigm is characterized by rapid impulsive and spasmodic changes, in some cases within days, so a new macroeconomic theory should be based on the analysis of macroeconomic characteristics with daily values.
• For daily monitoring of the macroeconomic situation within the framework of the new macroeconomic theory, it is required to form a system for the daily collection and processing of macroeconomic information, which can be done on the basis of automated systems with the widespread use of computer technology.
• Daily values of macroeconomic indicators are best expressed in relative terms, in the form of the dynamics of their changes for comparative analysis and identification of critical points.
• It is necessary to clarify the system of criteria for predicting possible rapid changes in the macroeconomics and identifying critical points. This may require extended ranges of acceptable changes compared to conventional statistical methods and control rules, for example, a point going outside the ∓4σ range, not one but several points going beyond the ∓3σ range. The system of criteria may include such rules as several (6 or 8) points in a row in ascending (or descending) order to identify an emerging trend, and other control rules. Refinement of the system should be carried out using the accumulated statistics on rapidly changing macroeconomic processes and using heuristic methods and rules.
• To improve the accuracy of forecasts using the methods of the new macroeconomic theory, macroeconomic indicators should be considered not separately, but in their totality. At the same time, to identify pre-crisis conditions and possible rapid positive changes, multidimensional information processing methods, for example, multidimensional scaling, cluster analysis, factor analysis, and others, may be useful.
• The paper describes the methods developed by the author for analytical approximation of stepwise and generalized functions, which makes it possible to describe impulsive and jumpy functions by conventional methods of mathematical analysis.