Structural Exchange Rate Modeling: The Case of a Small Open Economy

Abstract

A new mathematical structural model of the exchange rate and a new nonlinear multifactorial dependence of the exchange rate dynamics on a wide system of fundamental economic factors are the main results of the work. The model includes a wide system of economic factors based on the balance of payments. The proposed mathematical modeling methodology (International Flows Equilibrium Exchange Rate; IFEER) makes it possible to include additional fundamental factors in the model. This is the main theoretical achievement of this study, which also has a clear practical application. The constructed model made it possible to reveal the mechanism and structure of dynamic pricing of the exchange rate from updated and improved positions. The period of the last completed major financial and economic crisis in recent Russian history was chosen for conducting empirical research. During the period under review, the proposed model (applied to the dynamics of the Russian ruble against the US dollar exchange rate) showed its sufficiently high practical applicability, which is confirmed by the calculated quality indicators.

1. Introduction and Critical Literature Review

Over the past decades, the economic scientific literature has maintained a steady interest in mathematical structural models of exchange rates based on a system of fundamental factors.

One of the first models, which is the basis of modern macroeconomic theory, is the Mundell–Fleming model (Mundell 1963, 1968; Fleming 1962). The mathematical methods used, and the system of fundamental factors, largely predetermined the development of economic theory in this direction.

The next important step in the development of the theory of exchange rate determination can be considered the monetary approach in structural and analytical exchange rate modeling (monetary approach). It is important that, in the mathematical models that have already become classics (Hooper and Morton 1980; Mussa 1974, 1976; Frenkel 1976; Frenkel and Johnson 1978), structural modeling of the exchange rates was based on balance modeling. In parallel, a model with tight prices was being developed by R. Dornbusch (Dornbusch 1976). It has the mixed features of the Keynesian Mundell–Fleming model and the monetary model.

The monetarist approach is aimed at studying the monetary value, which is the exchange rate itself, which makes it very fruitful in scientific use.

Other studies (Stockman 1980; Clark and MacDonald 2000) use mathematical modeling methods to study the structural fundamental equilibrium dynamics of exchange rates.

The development of structural models includes the portfolio balance models (Branson 1984; Taylor 2004, etc.). According to them, the equilibrium exchange rate in an open economy is determined primarily by the equilibrium of supply and demand in the market of financial instruments with free movement of capital.

The next step, which determined the development of the theory of exchange rate determination for decades, is Dynamic Stochastic General Equilibrium Models (DSGE-models). Very significant in this class is the Obstfeld–Rogoff model of monopolistic competition between two equal open economies (Obstfeld and Rogoff 1995), aimed at integrating the intertemporal approach of many predecessor models.

The high applicability of models of fundamental determinants in practice is shown in the studies of (MacDonald and Taylor 1994; Chen and Mark 1995; Cheung et al. 2004; Engel et al. 2007, etc.). Studies were conducted on various, both active and defunct, dual currency pairs in different periods of history.

On the other hand, it should be noted that econometric studies (Rebitzky 2010; Heiden et al. 2013; Iwatsubo and Marsh 2014) emphasize the impact of the emergence of fundamental data on the dynamics of exchange rates. Similarly, Li et al. (2021) used a portfolio approach to assess the effect of Chinese yuan exchange rate expectation, with significant attention to capital flows. Adekoya et al. (2022) used LSTM-Networks (Long Short-Term Memory Networks) to predict exchange rates Ghanaian cedis to US dollars, Euro, and British pounds, primarily based on historical fundamental macroeconomic data and, importantly, commodity prices.

When modeling the exchange rate, different methods and approaches are used. Several models use the principles of the theory of purchasing power parity (PPP). However, the studies of (Engel et al. 2007; Cheung et al. 2004, etc.) do not confirm the practical applicability of the PPP approach for predicting the dynamics of the considered dual currency pairs.

A number of studies have highlighted the issues of exchange rate dynamics in the context of monetary policy (Brahmi and Zouari 2015; Mishkin 2000). First of all, this concerns the inflation targeting regime, which essentially requires a transition to a free floating exchange rate. This is important in the context of the fact that the Bank of Russia is implementing an inflation targeting regime.

Considerable attention has been paid to modeling the ruble exchange rate in both nominal and real terms from the perspective of fundamental justification of dynamics (Bozhechkova and Trunin 2015), in order to predict dynamics on various horizons (Ageev et al. 2022).

Mathematical methods of stochastic analysis (Gorskaya et al. 2022), neural networks (Polbin and Kropocheva 2022), and econometric methods (Kuzmin 2022) are widely used.

Mathematical modeling results, in the context of the formation of a significant system of ruble exchange rate drivers, are widely used in intersectoral and macroeconomic analysis (Kuranov et al. 2021; Kuzmin 2021), in studies of the transformation of monetary policy in Russia and international monetary relations (Pechalova 2023), and national competitiveness in modern conditions (Ishkhanov and Linkevich 2022). However, all of the models reviewed are not structural–analytical models.

Each model is based on its own system of factors. At the same time, the basic factors among them are fundamental macroeconomic factors related to real output, financial price, and percentage indicators; many models also use balance of payments indicators.

In most well-known models, the analysis and modeling of the exchange rate are structurally limited by the parameters of the current account balance, without considering the impact of capital flows. Attempts have been made in the scientific literature to formalize this problem. In the model of R. McDonald and M. Taylor (MacDonald and Taylor 1994), an approach to estimating the equilibrium exchange rate determined by capital flows (capital enhanced equilibrium exchange rate; CHEER) is conceptually formulated. However, in the CHEER model and in the portfolio balance models, the analytical flow factor functions of capital movements are not explicitly specified.

2. Goals and Methods

In this paper, the main goal is to develop a new mathematical structural dynamic model of the nominal exchange rate of a small open economy. The study was conducted according to the methodology “Small open economy—the rest of the world”. This approach distinguishes the developments and results of this article from the exchange rate model of two equal interconnected economies (Kuzmin 2022).

The study further develops the concept of modeling exchange rates, defined by the author as International Flows Equilibrium Exchange Rate modeling (IFEER). The presented model of the nominal exchange rate of a small open economy explicitly includes balance of payments flows, the level of real aggregate output, an indicator of international competitive advantages, price and percentage financial indicators, etc. Note that the model includes a very broad system of economic factors. It is important to note that the proposed mathematical modeling methodology makes it possible to include additional fundamental factors such as capital movements and interest rate differential in the model. This is the main theoretical achievement of this study, which also has a clearly practical application.

Structurally, the introduction presents a critical overview of the subject area and literature under study. In Section 2, the goals and methods are described. Section 3 sets out an integral version of the conceptual approach. In Section 4, a mathematical modeling of the ruble exchange rate is conducted. Section 5 is devoted to empirical verification and analysis of modeling results. In Section 6, we discuss the limitations of the results obtained and the future directions of research. In Section 7, an overview of the main results of the work is presented.

3. IFEER System Approach: An Integral Variant

A discrete and integral approach to the formation of the exchange rate at the conceptual level is systematically described in sufficient detail in the studies of Kuzmin (2021, 2022). Next, we propose an integral version, which can be considered a conceptual generalization.

Under consideration is a situation when, in a certain period in the foreign exchange market, participants conducted N market transactions at the appropriate exchange rates $e_i,i\in \left(1,N\right).$

The following designations $e_i,D_i,R_i$ are used in the i-th operation: the nominal exchange rate (direct quote), the volume of the transaction in the selected foreign currency, and the volume of the transaction in the national currency, respectively.

The variables in the i-th transaction are related by the following relations: $e_i{D}_i=R_i$.

As a consequence,

$$e_i=R_i/D_i$$

(1)

Under this approach, R(t) and D(t) are related flows in national and foreign currencies.

In the differential form, for the exchange rate, the equivalent of Equation (1) at period t is equal to:

$${\displaystyle \frac{\raisebox{1ex}{$\partial R(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}{\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}}=e(t).$$

(2)

Next, the weight function w(t) is introduced. Here w(t) defines the share of cash flow in foreign currency for the period T:

$$w(t)=\left[{\displaystyle \frac{\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}}\right].$$

(3)

Considering these assumptions, we define the average exchange rate in the period T as an integral value:

$$e(T)={\displaystyle \underset{T}{\int }w(t)}e(t)dt.$$

(4)

Substituting the abovementioned weight function of Equation (3) into Equation (4), we obtain the following:

$$\begin{gathered} e(T)={\displaystyle \underset{T}{\int }w(t)}e(t)dt={\displaystyle \underset{T}{\int }\left[\frac{\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}\right]}e(t)dt= \\ ={\displaystyle \underset{T}{\int }\left[\frac{\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.e(t)}{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}\right]}dt. \end{gathered}$$

After this, the denominator is taken outside the integral:

$$\begin{array}{l}e(T)={\displaystyle \underset{T}{\int }\left[\frac{\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.e(t)}{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}\right]}dt={\displaystyle \underset{T}{\int }\left[\frac{\raisebox{1ex}{$\partial R(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}\right]}dt=\\ ={\displaystyle \frac{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial R(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}}.\end{array}$$

As a result, we can disaggregate the flow determinants of the balance of payments:

$$e(T)={\displaystyle \frac{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial R(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}}={\displaystyle \frac{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial R^{CA}(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt+{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial R^K(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}{{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D^{CA}(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt+{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D^K(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt}}.$$

(5)

In Formula (5), the upper indices CA and K signal that the flows belong to the current balance and the capital flow balance, respectively.

The components of Formula (5) are defined as follows:

$$\begin{array}{l}{\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial R^{CA}(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt=I_T,\\ {\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial R^K(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt={K_T}^{-},\\ {\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D^{CA}(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt=E_T,\\ {\displaystyle \underset{T}{\int }\raisebox{1ex}{$\partial D^K(t)$}\!\left/ \!\raisebox{-1ex}{$\partial t$}\right.}dt={K_T}^{+}.\end{array}$$

At this conceptual level, in period T, the time path of the exchange rate has a dynamic form:

$$e(T)=e_T={\displaystyle \raisebox{1ex}{$(I_T+{K_T}^{-})$}\!\left/ \!\raisebox{-1ex}{$(E_T+{K_T}^{+})$}\right.}.$$

(6)

4. Structural Exchange Rate Modeling: Small Open Economy

Next, in the two-period model, we will use the accepted scientific designation t instead of T dynamically in Formulas (3)–(6).

The production programs of exporters are the basis for the formation of export foreign exchange earnings in the period t:

$$E_t=k_E{P}_t^{\ast }{({Q_t}^{{\displaystyle \frac{1}{x+1}}}{Q_{t-1}}^{{\displaystyle \frac{x}{x+1}}})}^{1+\delta }{\left(\left(\Delta {y_t}^{{\displaystyle \frac{1}{x+1}}}\right)\left(\Delta {y_{t-1}}^{{\displaystyle \frac{x}{x+1}}}\right)\right)}^{\beta }{(e_{t-1}^R)}^z$$

(7)

In Equation (7) the basic system of fundamental factors used is presented:

• $Q_t$—the real gross national product,
• $P_t^{\ast }$—the aggregate foreign price level,
• $\Delta y_t=f(y_t-y_t^{\ast })$—a factor related to the function of the difference in interest rates. The proposed methodology for using this factor for modeling purposes is described in Section 4.

The content of dependence (7) is explained in detail: in the financial market, export foreign exchange earnings are part $k_E$ of total production in foreign prices. At the same time, it is a constant and $x\ne -1$.

Here, the total output $Q_t$ is dynamically averaged with the appropriate weights:

$${\displaystyle \frac{1}{x+1}}+{\displaystyle \frac{x}{x+1}}=1$$

When deciding, an important fundamental factor in dependence (7) in the t − 1 period is the indicator of international competitive advantages. In the model, it is represented by a nominal exchange rate adjusted for the ratio of foreign and domestic price aggregates. Note that it is structurally identical to the real exchange rate:

$${e^R}_{t-1}=e_{t-1}{\displaystyle \raisebox{1ex}{$P_{t-1}^{*}$}\!\left/ \!\raisebox{-1ex}{$P_{t-1}$}\right.}.$$

(8)

Here and from now on, the parameters δ, θ, α, β, x, y and z in exponents show the response of export volumes and other introduced macroeconomic aggregates to the dynamics of changes in the main relevant modeling factors.

Within the framework of this model, we will accept the hypothesis that domestic import demand in period t is part of the domestic product at domestic prices in the context of established international advantages:

$$I_t=k_I{P}_t({Q_t}^{{\displaystyle \frac{1}{x+1}}}{Q_{t-1}}^{{\displaystyle \frac{x}{x+1}}}){(e_{t-1}^R)}^y.$$

(9)

In Equations (7) and (9), k_I is a constant and the parameters are related: z − y = x.

An important fact is that the author’s IFEER concept explicitly incorporates cross-country capital flows into mathematical modeling, which is a determining factor in the dynamics of exchange rates.

Within the framework of the model, we will also accept the hypothesis that capital outflow has an domestic investment character. It makes up a part of the total income in domestic prices:

$$K_t^{-}=k_{K^{-}}{P}_t({Q_t}^{{\displaystyle \frac{1}{x+1}}}{Q_{t-1}}^{{\displaystyle \frac{x}{x+1}}}){(e_{t-1}^R)}^y$$

(10)

Capital inflows are also of an investment nature. We assume that the inflow is determined by the decisions of foreign investors who want to purchase a share k_K+ of total national output in their prices $P_t^{\ast }$. Another fundamental factor in this dependence is the difference in interest rates:

$$K_t^{+}=k_{K^{+}}{P}_t^{\ast }{({Q_t}^{{\displaystyle \frac{1}{x+1}}}{Q_{t-1}}^{{\displaystyle \frac{x}{x+1}}})}^{1+\theta }{\left(\left(\Delta {y_t}^{{\displaystyle \frac{1}{x+1}}}\right)\left(\Delta {y_{t-1}}^{{\displaystyle \frac{x}{x+1}}}\right)\right)}^{\alpha }{(e_{t-1}^R)}^z.$$

(11)

An increase in the interest rate differential will lead to an increase in capital inflows, which is strongly supported by both economic theory and empirical evidence.

The indicator of international competitive advantages will also have a significant impact on dependence (11), which is because foreign investors positively perceive the improvement of the country’s competitive advantages not only in terms of production, but also in terms of investment.

From an economic point of view, in functional dependencies (9) and (10), the indicator of international competitive advantages should have a significantly stronger effect on the dynamics of aggregates than in dependencies (7) and (11).

Then, substituting functional dependencies (7)–(11) in (6), we obtain:

$$\begin{array}{c}{e}_t\end{array}={\displaystyle \frac{k_I{P}_t({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}}){(e_{t-1}^R)}^y+k_{K^{-}}{P}_t({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}}){(e_{t-1}^R)}^y}{k_E{P}_t^{\ast }{({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}})}^{1+\delta }{\left(\left(\Delta {y_t}^{\frac{1}{x+1}}\right)\left(\Delta {y_{t-1}}^{\frac{x}{x+1}}\right)\right)}^{\beta }{(e_{t-1}^R)}^z+k_{K^{+}}{P}_t^{\ast }{({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}})}^{1+\theta }{\left(\left(\Delta {y_t}^{\frac{1}{x+1}}\right)\left(\Delta {y_{t-1}}^{\frac{x}{x+1}}\right)\right)}^{\alpha }{(e_{t-1}^R)}^z}}=$$

$$={\displaystyle \frac{P_t({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}}){(e_{t-1}^R)}^y(k_I+k_{K^{-}})}{P_t^{\ast }{({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}})}^{1+\theta }{\left(\left(\Delta {y_t}^{\frac{1}{x+1}}\right)\left(\Delta {y_{t-1}}^{\frac{x}{x+1}}\right)\right)}^{\alpha }{(e_{t-1}^R)}^z\left[k_E{({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}})}^{\delta -\theta }{\left(\left(\Delta {y_t}^{\frac{1}{x+1}}\right)\left(\Delta {y_{t-1}}^{\frac{x}{x+1}}\right)\right)}^{\beta -\alpha }+k_{K^{+}}\right]}}.$$

A term is allocated in the numerator and denominator:

$${\displaystyle \frac{(k_I+k_{K^{-}})}{\left[k_E{({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}})}^{\delta -\theta }{\left(\left(\Delta {y_t}^{\frac{1}{x+1}}\right)\left(\Delta {y_{t-1}}^{\frac{x}{x+1}}\right)\right)}^{\beta -\alpha }+k_{K^{+}}\right]}}.$$

Let us now evaluate several of its properties. Firstly, there is a greater stability of the average term ${({Q_t}^{{\displaystyle \frac{1}{x+1}}}{Q_{t-1}}^{{\displaystyle \frac{x}{x+1}}})}^{\delta -\theta }$ compared to the volatility in the medium term of internal and external price aggregates. Also, from the logic of the construction: δ ≈ θ.

Secondly, from an economic point of view, when conducting our modeling, the situation is important when capital flows significantly prevail over trade balance flows. In this case, mathematically k_K+ >> k_E. These assumptions allow us to use the mathematical term (12) as a constant in the medium term:

$${\displaystyle \frac{(k_I+k_{K^{-}})}{\left[k_E{({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}})}^{\delta -\theta }{\left(\left(\Delta {y_t}^{\frac{1}{x+1}}\right)\left(\Delta {y_{t-1}}^{\frac{x}{x+1}}\right)\right)}^{\beta -\alpha }+k_{K^{+}}\right]}}=(k^{x+1})\approx const.$$

(12)

It is possible to rewrite the dynamic dependence of the exchange rate in the form:

$$\begin{array}{l}{e}_t={\displaystyle \frac{P_t({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}})(k^{x+1})}{P_t^{\ast }{({Q_t}^{\frac{1}{x+1}}{Q_{t-1}}^{\frac{x}{x+1}})}^{1+\theta }{\left(\left(\Delta {y_t}^{\frac{1}{x+1}}\right)\left(\Delta {y_{t-1}}^{\frac{x}{x+1}}\right)\right)}^{\alpha }{(e_{t-1}\frac{P_{t-1}^{\ast }}{P_{t-1}})}^x}}=\\ ={\displaystyle \frac{P_t(k^{x+1})}{P_t^{\ast }({Q_t}^{\frac{\theta }{x+1}}{Q_{t-1}}^{\frac{x\theta }{x+1}})\left(\left(\Delta {y_t}^{\frac{\alpha }{x+1}}\right)\left(\Delta {y_{t-1}}^{\frac{x\alpha }{x+1}}\right)\right){(e_{t-1}\frac{P_{t-1}^{\ast }}{P_{t-1}})}^x}}.\end{array}$$

After transferring the advantages’ indicator ${(e_{t-1})}^x$ to the left side and rearranging the variables:

$$\begin{array}{l}{e}_t{(e_{t-1})}^x={\displaystyle \frac{P_t{(P_{t-1})}^x(k^{x+1})}{P_t^{\ast }{(P_{t-1}^{\ast })}^x\left({Q_t}^{\frac{\theta }{x+1}}\Delta {y_t}^{\frac{\alpha }{x+1}}\right)\left({Q_{t-1}}^{\frac{x\theta }{x+1}}\Delta {y_{t-1}}^{\frac{x\alpha }{x+1}}\right)}}=\\ ={\displaystyle \frac{P_t{(P_{t-1})}^x(k^{x+1})}{P_t^{\ast }{(P_{t-1}^{\ast })}^x\left({Q_t}^{\frac{\theta }{x+1}}\Delta {y_t}^{\frac{\alpha }{x+1}}\right){\left({Q_{t-1}}^{\frac{\theta }{x+1}}\Delta {y_{t-1}}^{\frac{\alpha }{x+1}}\right)}^x}}=\\ =\left(k{\displaystyle \frac{P_t}{P_t^{\ast }\left({Q_t}^{\frac{\theta }{x+1}}\Delta {y_t}^{\frac{\alpha }{x+1}}\right)}}\right){\left(k{\displaystyle \frac{P_{t-1}}{P_{t-1}^{\ast }\left({Q_{t-1}}^{\frac{\theta }{x+1}}\Delta {y_{t-1}}^{\frac{\alpha }{x+1}}\right)}}\right)}^x.\end{array}$$

We will carry out a temporary period-by-period separation of the variables involved in the modeling:

$$\begin{array}{c}{e}_t=\end{array}k{\displaystyle \frac{P_t}{P_t^{\ast }\left({Q_t}^{\frac{\theta }{x+1}}\Delta {y_t}^{\frac{\alpha }{x+1}}\right)}}.$$

As a result, after performing parameter reassignments ${\alpha }^{\prime }={\displaystyle \frac{\alpha }{x+1}},$ ${\theta }^{\prime }={\displaystyle \frac{\theta }{x+1}}$ for convenience, we obtain the formula:

$$\begin{array}{c}e(t,Q(t),P(t),P^{\ast }(t))=e_t=\end{array}k{\displaystyle \frac{P(t)}{P^{\ast }(t)}}Q{(t)}^{-{\theta }^{\prime }}\Delta y^{-{\alpha }^{\prime }}.$$

(13)

As a result, a dynamic dependence of the exchange rate on the selected system of the main macroeconomic internal and external fundamental factors is obtained, which, according to its construction, has a medium-term character.

After the logarithmic operation, the resulting dependence (13):

$$\begin{array}{c}\ln e_t=\end{array}\ln k+\ln P(t)-\ln P^{\ast }(t)-{\theta }^{\prime }\ln Q(t)-{\alpha }^{\prime }\ln \Delta y.$$

(14)

From the standpoint of the conducted modeling, we would like to note that this form of dynamic dependence of the exchange rate (14) is structurally quite close to the canonical monetary model (Mussa 1974, 1976; Frenkel 1976).

5. Dynamics of the Russian Ruble Exchange Rate: A Bit of Empiricism

The financial and economic crisis of 2014–2015 has been investigated earlier by the author in the context of the ruble exchange rate modeling (Kuzmin 2021). Therefore, it is quite important to obtain an assessment of the results from the perspective of the newly developed model.

For the purposes of the simulation, we note that the floating exchange rate regime was applied in the Russian Federation during the selected period. In accordance with the official position of the Bank of Russia, the exchange rate of foreign currency against the ruble is determined by the balance of supply and demand of foreign currency in the foreign exchange market (the Bank of Russia: “The main directions of the unified state monetary policy for 2023 and the periods 2024 and 2025”1).

It is important to note that the USD/RUB dual currency pair was the core pair in Russia, to which the rest of the pairs (EUR/RUB, CNY/RUB and others) are tied.

The determinants P and Q in the model are represented by the consumer price index and the index of aggregate real output in the Russian Federation (data from the Federal State Statistics Service2).

In accordance with the fact that a multifactorial nonlinear model is being studied, it is necessary to evaluate the parameters θ and α of Formula (13).

At the initial point of research, taken as equilibrium (December 2013), the coefficient k in (13) is identical to the observed exchange rate of the Russian ruble against the US dollar with a condition imposed on the initial values of the main determinants P (started) = 1, Q (started) = 1, P* (started) = 1.

In accordance with the set goals, it is necessary to solve the following two-parameter optimization problem:

$${\displaystyle \frac{1}{N}}{{\displaystyle \sum _{t=1}^N\left(ND{(e)}_t\right)}}^2\underset{{\theta }^{\prime },{\alpha }^{\prime }}{\to }\min$$

(15)

where $ND{(e)}_t={\displaystyle \frac{e_t({\theta }^{\prime },{\alpha }^{\prime })-e_t}{e_t}}$ are the normalized deviations,

• $e_t({\theta }^{\prime },{\alpha }^{\prime })$—the theoretical ruble exchange rate according to the Formula (13),
• $e_t$—the nominal ruble exchange rate.

As before, during the selected research period, the nominal exchange rate of the US dollar calculated by the Bank of Russia at the end of the period in monthly terms of Russia was adopted as the nominal exchange rate3.

At the same time, the pricing mechanism for Russian export products is based on the prices of world commodity exchanges in US dollars. So, the price index of the Brent oil mixture (Intercontinental Exchange (ICE), data from the terminal of the Bloomberg News agency) was P*. This is determined by the significant predominance of petroleum products and natural gas in exports at that time and the close correlation of other Russian export products with oil prices. The significance of the influence of oil prices on the exchange rate of the Russian ruble is confirmed by the results of several studies (Kuzmin 2021; Polbin and Kropocheva 2022; Gorskaya et al. 2022, etc.).

The difference $\Delta y_t=e^{\left(y_t-{y_t}^{\ast }\right)}$ between the key rates of the Bank of Russia4 and the US Federal Reserve System5 in the form of exponential functional dependence was used as a determinant.

As a result of solving the two-parameter optimization problem (15), the values of the main modeling parameters θ’ = 0.376 and α’ = 0.0021 were estimated (Figure 1). The calculations were conducted by the author using the Microsoft Excel Solver tool.

We suggest using the average of normalized deviations (AND) and the average of absolute normalized deviations (AAND) as the model quality indicators because they are directly related to the parameter estimation procedure (15):

$$\begin{array}{cc}AND={\displaystyle \frac{1}{N}}{\displaystyle {\displaystyle \sum }_{t=1}^N}\left(ND{(e)}_t\right),& AAND={\displaystyle \frac{1}{N}}{\displaystyle {\displaystyle \sum }_{t=1}^N}\left(ND{(e)}_t\right)\end{array}.$$

As a result of the numerical simulation, the quality indicators were respectively: AAND (average of absolute normalized deviations) = 0.0014 and AND (average of normalized deviations) = 0.0254. These values confirm the high quality of the dynamic model (13). In the context of the conducted modeling, it should be emphasized that, in comparison with the truncated previously used model (Kuzmin 2021) (dependence variant (13) without the factor $\Delta y_t$; Figure 1), this model slightly improved the quality indicators that previously amounted to: AAND = 0.0017 and AND = 0.03. In addition, the non-zero value of the parameter α’ = 0.0021 for the factor $\Delta y_t=e^{\left(y_t-{y_t}^{\ast }\right)}$ shows a certain effect of the difference in interest rates on the result, although incomparably less than other fundamental factors of the model. This involves a virtually twofold drop in export prices of Russian oil and other energy resources on international markets.

6. Discussion, Limitations and Future Research

We assume that this model works well primarily on the economic data of the crisis period, which is confirmed by the results shown in Section 5.

Unfortunately, currently, it is not possible to directly apply the modeling results to the data of the crisis of 2022–2024 in Russia. This is due to tectonic changes in the Russian foreign exchange market. In the period up to February 2022, the USD/RUB exchange rate could be considered as the main exchange rate, as described in Section 5. The introduction of a significant number of sanctions since that time (including those related to the financial market and financial payments) has led to a change in the vector of Russia’s monetary policy towards the Chinese yuan. As a result, the CNY/RUB exchange rate has become the key object of the monetary policy since March 2022.

Thus, the development of a new methodology that would make it possible to apply mathematical modeling results in this situation of a dramatic change in the fundamentals of the foreign exchange market functioning is an obvious area of future research.

A second area of development of this work is the application of modeling results to countries with similar economic development features to Russia.

Another important scientific area of research is the expansion of the determinant system of exchange rate formation. The mathematical incorporation of new factors into the model will make it possible to clarify the dependence of the exchange rate and increase its practical applicability.

7. Conclusions and Results

Structural modeling of the exchange rate dynamics of a small open country is performed from the standpoint of “Small open economy—the rest of the world” within the framework of the author’s approach to modeling based on international flows (IFEER). An integral version of this approach is described in the paper.

As a result, we have obtained a new nonlinear multifactorial dependence of the exchange rate dynamics on an expanded system of fundamental economic factors.

At the same time, the proposed mathematical modeling methodology makes it possible to include additional fundamental factors in the model. This is the main theoretical achievement of this study, which also has a clear practical application. A new mathematical model from updated and improved positions explained the dynamic mechanism and structure of the exchange rate formation.

We chose the period of the last completed major financial crisis in Russia to conduct empirical research and verify the model. During the period under review, the proposed model showed its sufficiently high practical applicability, which is confirmed by the calculated quality indicators.

It is important to note that the model includes a broad system of economic factors. At the same time, the proposed mathematical modeling methodology makes it possible to include additional fundamental factors in the model. This is the main theoretical achievement of this study.

This system includes macroeconomic aggregates of export and import operations, capital movements, real gross domestic product, actual export prices, consumer prices, an indicator of international competitive advantages, interest rate differences, etc. The mathematical incorporation of the latter factor into the system of determinants and the structural dynamic dependence of the exchange rate of a small open country are the main achievements of this work, which also has a clearly practical application.

At the same time, this model is subject to development, both theoretically and practically, as discussed above.