Development of an Explicit Rule of Monetary Policy for the Economy of Ukraine

The paper explains the expediency of developing an explicit rule of monetary policy for the economy of Ukraine. It studies the stages of its development, proving the expediency of formation of monetary rules for money aggregates, evaluates equilibrium values of the rule's parameters based on the use of the modified Hodrick-Prescott filter, and determines the possible parameters of the monetary rule and their estimated coefficients by developing multivariate regression models.


Introduction
The most well-known rule of the monetary policy is the Taylor rule. This rule was first formulated by the Stanford University Professor John Taylor in 1993 [22]. In his subsequent works John Taylor continued his research of methodological approaches regarding the development and implementation of the monetary rules [23]. In the second half of the 90s of the 20th century and the beginning of the 21st century other scientists also studied the ways of development and improvement of monetary rules [2,15,16]. The Taylor rule and its main modifications are presented in Table 1.
Within this paper the terms "monetary policy rule", "monetary rule", "policy rule" and "rule" will be used as synonyms.  [22] , 2 ) 2 ( 5 , 0 5 , 0 p y p r where r is the federal funds rate; p is the level of inflation (GDP deflator) for the previous four quarters; y is the percentage deviation of real GDP from the target level (GDP gap).
The first monetary rule for interest rate proposed by John Taylor in 1993.
Generalized rule developed by Taylor for different economies [23] , ) ( * f r h gy r where r is a short-term interest rate; is the retrospective level of inflation; y is the percentage deviation of real GDP (Y) from the trend; g, h, r f * are constants.
The form of monetary rule generalized by John Taylor.
Modification of Richard H. Clarida [3,4] , ) ( * * f t t r h gy r where r is a short-term interest rate; * is the long-term equilibrium level of inflation; y is the deviation of real GDP from the trend, r f is the long-term equilibrium real interest rate; g, h are constants.
Monetary rule, which uses equilibrium inflation levels and a clear division of time horizons in the calculation of the rule's components.

Modification of Peters
Amos [1] ), log( where R t is a short-term interest rate; y t is the GDP; t is the inflation; t is the money supply growth; s t is the real exchange rate.
Modified Taylor rule used in the structural dynamic model of small open economies of the developing countries, which tested the data of the South Africa, Mexico, Indonesia and Thailand.
Modification of Laurence Ball [2] ), ( ) 1 ( 1 e b ay e w wr where e is a mean value of logarithms of the real exchange rate (higher values indicating appreciation); r is the average real interest rate; y is the logarithm of real GDP; is the inflation; is the coefficient of influence of exchange rates on inflation; w, a, b -coefficients.
It proves the efficiency of the use of exchange rate in the Taylor rule. It is developed taking into account the specifics of developed countries with small open economies such as Canada, New Zealand and Sweden.
Modification of the National Bank of Canada [5] ), where i t is a short-term target interest rate; * t i is the equilibrium value of short-term interest rates; t is the actual annual rate of core inflation; * t is the target annual rate of core inflation; y t is the logarithm of real GDP; * t y is the logarithm of real potential GDP.
Monetary rule that was applied for the quarterly forecasting model of the National Bank of Canada.
Modification of the Central Bank of Hungary [7] , ) ) ( ( ) 1 (  John Taylor defined [23] the monetary policy rule as a description (graphically formalized by a formula or numeric values) of a method for the use of monetary policy instruments, such as monetary base or federal funds rate, depending on the dynamics of economic parameters. In his other paper [24] he provides a similar definition of the monetary policy rule: a plan that determines, as clearly as possible, the conditions when a central bank should change its monetary policy instruments. He also clarifies that under this definition, the targeting of inflation or nominal GDP by a central bank is not considered a policy rule. These definitions explain the essence of explicit monetary rule used by the central bank.
In our opinion, there are two approaches to determine the parameters and estimation of monetary rules coefficients. The first approach involves the formation of a system of equations that describe the transmission mechanism of the monetary policy or the functioning of the economy as a whole. Monetary rule is used as an integral part of the model. Therefore, its coefficients and parameters will be formed in the process of development and implementation of the model in general. In this case, we can say that the monetary rule is used by the central bank implicitly (not explicitly) as one of the model's equations. Another way of the first approach realization is the use of the model to determine the coefficients of the monetary rule, which is used explicitly. That means the formation of a simple monetary rule directly used by the central bank to assess the expediency of changing the use of monetary policy instruments.
The second approach is based on developing a multivariate regression, which will connect the key parameter of the monetary policy (a basic rate or a monetary aggregate) with certain macroeconomic indicators (inflation, GDP gap, exchange rate, money turnover rate). In this case, the estimation of coefficients and parameters of the rule is based on the method of ordinary least squares [15]. This approach also requires the formation of an explicit monetary rule. Table 1 presents monetary rules used both explicitly and implicitly. The latter are characterized by more complex forms and methods of calculation. Experts of the National Bank of Ukraine (NBU) have developed a quarterly forecasting model of the transmission mechanism of monetary policy in Ukraine [12, pp. 55-93, 17]. This model implicitly uses the monetary policy rule for the key interest rate.
The aim of this paper is to substantiate the expediency of developing an explicit monetary rule for the Ukrainian economy. We also examine the main stages of its development: (1) determination of the rule's basic form; (2) estimation of equilibrium values of the rule's parameters; and (3) determination of the parameters that should be included in the rule. The paper's material is presented in the context of the above mentioned objectives of the study.

Substantiation of expediency of developing an explicit monetary rule for Ukraine
John Taylor identifies five major problems to be solved in order to apply monetary policy rules in countries with emerging economies [24]: introduction of inflation targeting; the use of exchange rate values in the formation of the monetary rule, the choice of instruments for the monetary rule (interest rate or monetary aggregates); the level of the rule's formalization; taking into account the development of the long-term securities market. In our opinion, the first two issues define the methodology of the monetary rules' development, while the latter three are mostly related to methodological aspects of development of the monetary policy rules. Therefore, the implementation of inflation targeting regime is a key prerequisite for the use of monetary rules. After the introduction of changes of July 9, 2010 to the Law of Ukraine "On the National Bank of Ukraine", the priority objective of the National Bank is to achieve and maintain price stability in the country. The introduction of these changes in the regulatory acts and the signing of the Memorandum with the IMF [26] demonstrated the strategic orientation of Ukraine to reform its monetary policy towards the implementation of inflation targeting regime. At this stage, the monetary system of Ukraine can be characterized as a "transitional one with a focus on price stability". That means a shift from exchange rate targeting to inflation targeting.
In our opinion, the introduction of any monetary policy regimes (with the exception of a strict exchange rate targeting) creates conditions for the limited use of monetary rules if one of the monetary policy priorities is the achievement of price stability. The US experience is important in this regard. The purpose of the monetary policy of the Federal Reserve System is formulated quite broadly: the increase of production, maximum employment, stable prices, and the formation of moderate longterm interest rates. On the other hand, the most wellknown monetary rule was developed by the American Professor John Taylor on the basis of a retrospective analysis of the U.S. macroeconomic data. Despite the fact that price stability is only one of the monetary policy goals in the United States, this country has extensive experience in the development and implementation of monetary rules. Thus, the development of an explicit monetary rule has a practical importance for Ukraine and is characterized by significant urgency level.
During the modeling of the monetary policy the National Bank of Ukraine for a long time has been implicitly using a modified Taylor rule (Table 1). However, we emphasize the need to develop a simplified explicit policy rule. The introduction of an explicit rule would make it possible to benefit from its direct use, not just using it as one of the forecasting models equations. The advantages of a partial transition from the purely discretionary monetary policy to a policy that uses explicit monetary rules have been studied in the following works [5; 22].
An attempt to introduce the explicit monetary rule for determining the discount rates was made by the National Bank of Ukraine in 2001: it approved a temporary provision "Definition of the interest rates on the National Bank of Ukraine operations" [19]. This provision was abrogated in August 2004 [20]. As shown in our paper [21, pp. 230-234], this monetary rule had a low efficiency and was seldom used by the National Bank of Ukraine.

Determination of the rule's basic form
After proving the expediency of the monetary rule we will proceed to its immediate formation. In our opinion, the choice of the monetary rule's basic form should be based on the relevant international experience. During the selection of the rule's basic form for the developing countries it is necessary to consider the recommendations given in the works of John Taylor [24]: substantiation for the choice of monetary rule instruments (interest rate or monetary aggregates); definition of an acceptable level of the rule's formalization; consideration of the development level of the long-term securities market. It should be noted that in selecting the monetary rule's basic form it is important to take into account some national peculiarities of the functioning of the money market, the financial system and the real economy.
The main problem in determining the rule's basic form is the choice of the key instrument of the monetary policy. The results of the study of the international experience lead to the conclusion regarding the prevalent use of monetary rules for determining the basic interest rates by central banks (Table 1). However, researchers note that under inefficient performance of the interest rate channel of transmission mechanism, a monetary rule can be developed to determine the equilibrium (optimal) monetary aggregates [2; 23; 24].
In their studies representatives of the National Bank of Ukraine [12; 13; 14] stress the lack of effectiveness of the interest rate channel of transmission mechanism in Ukraine. This inefficiency manifests itself in the lack of response (or reaction with considerable lags that makes impossible an effective application of interest rate policies) of rates of the wholesale and retail segments of the money market to the changes of basic rates by the National Bank of Ukraine. In particular, the analysis of functioning of the interest rate channel of transmission mechanism during the financial crisis demonstrates that the official rates of the National Bank of Ukraine (the discount rate and the average refinancing rate) do not have a significant influence on the interbank market rates. Researchers come to the conclusion about the gap in the interest rate channel of monetary policy transmission in Ukraine already during the first phase of the monetary impulse transmission. The impact of the discount rate growth on the increase of interest rates on loans and deposits with a lag exceeding 11 months has also been found. In our opinion, this lag eliminates even the weak influence of the basic rate on the dynamics of the money market retail rates. During a year (lag size) the dynamics of rates is influenced by a number of different factors (macroeconomic, political, social, bank internal, etc.). As a consequence, it is difficult to find a practical use for the impact of changes in the basic rate of the National Bank of Ukraine.
The results of the study of functioning of the interest rate channel in the pre-crisis period 2000-2007 are presented in the monograph of V. Stelmakh. In particular, this paper notes that the impact of the NBU discount rate as well as other interest rates on the rates of interbank credit market is statistically insignificant. Thus, the researchers come to the conclusion about the minimal impact of the National Bank of Ukraine on inflation through the instruments of interest rate policy.
At the current stage of development of the monetary policy in Ukraine it seems necessary to develop a monetary rule for money aggregates. It should be noted that paragraph 14 of the Memorandum of economic and financial policies signed by Ukraine and the IMF [26] states that the use of interest rates will gradually become the main instrument of the monetary policy. Therefore, it is likely that in the medium term some prerequisites for the development of monetary policy rule for the basic interest rate will be formed.
We will study the preconditions for the development of the policy rule for Ukraine that will determine the equilibrium dynamics of money aggregates. First of all, we should note that paragraph 15 of the Memorandum [26]  It is necessary to emphasize that the monetary rules for the money supply were developed earlier than the monetary rules for interest rates. The constant growth rate rule was formulated by M. Friedman in the 50s of the 20th century.
Milton Friedman substantiated the expediency of using monetary policy rules based on the arguments about the complexity of realization of discretionary monetary policy and taking into account the monetary history of the United States. In his opinion, the ineffectiveness of discretionary monetary policy is caused mainly by the presence of lags during its implementation. Milton Friedman conducted a study of the monetary history of the United States focusing on the relationship of changes in the money supply with the dynamics of prices and real GDP. Having concluded that the dynamics of the money supply has a significant impact on key macroeconomic indicators, he proposed to maintain a fixed nominal growth rate of money supply in the economy (according to the author, for the U.S. economy the money supply growth rate had to be 2 percent a year). The Friedman monetary rule is given as follows: , q m (1) where m is a growth rate of money supply; q is the growth rate of real GDP; v is the growth rate of money turnover; is the growth rate of inflation levels.
In order to compare them, all indicators of equation (1) are given in growth rates. Milton Friedman believed that on the basis of equation (1) it is possible to determine the constant growth rate of money supply by using the following indicators: target growth rate of inflation, growth rate of potential (equilibrium) GDP, and selected trend of changes in money turnover. It should be emphasized that this rule is based on the equation of exchange proposed by Irving Fisher: where, is the money supply, V is the money turnover rate, P is the level of prices, Y is the real GDP.
Based on the comparative analysis of the Taylor and Friedman rules Edward Nelson made a conclusion about the similar views of these authors on the mechanism of the economy's functioning [15].
The following monetary policy rule, which is based on the use of monetary aggregates, was proposed in 1988 by Bennett McCallum [10,11]. This rule is the result of dialectical development of Friedman's constant growth rate rule. The McCallum rule is given as follows: where m is the money supply growth rate; x * = = * + q * is the growth rate of equilibrium nominal GDP; v is the money turnover growth rate; x is the growth rate of nominal GDP; x is constant.
By applying this rule Bennett McCallum proved that the efficiency of the monetary policy in the US could be much higher. This is especially true for the 30s and 70s of the 20th century, when the Federal Reserve System committed the biggest mistakes in monetary regulation.
Despite the fundamental character of the relationship between the money supply and price levels (equation (2) there are many factors that determine the strength, direction and form of this relationship. These factors include: the level of monetization, the rate of money turnover, the structure of channels for the issue of money supply, money demand, the share of cash in the money supply, etc. We will study the dynamics of some indicators that have an impact on the parameters of the relationship between money supply and price dynamics in Ukraine (Table 2). According to the results presented in Table 2 Table 2), leading to the gradual renewal of a close relationship between the money base growth and price dynamics in Ukraine. Secondly, we can make the conclusion about the similarity of official reserves dynamics, money supply and money base growth rates during the study period ( Figure 1). Thus, interventions in the foreign exchange market of Ukraine were the main channel of money supply emission in the period of 2003-2010. We should note that in comparison with the dynamics of the broad money supply the dynamics of the monetary base is close to the dynamics of official reserves. This fact confirms our conclusions, because in most periods the National Bank of Ukraine conducted the sterilization of excess money supply through interventions in the foreign exchange market. This structure of money issue had significant advantages during the actual exchange rate targeting and initial accumulation of official reserves. However, after the revision of the monetary policy priorities in 2010 in favor of maintaining price stability and practical restoration of pre-crisis levels of reserves, the National Bank of Ukraine should increase the diversification of money emission channels in order to bind its dynamics to domestic economic factors.
Thirdly, the analysis of indicators in Table 2 reveals a very complex type of connections between the dynamics of money supply and price dynamics. To a large extent this connection is determined by the efficiency of functioning of transmission mechanism individual channels in Ukraine.

Assessment of equilibrium values of indicators, which are included in the monetary rule
One of the most accurate and the most common methods for assessing the equilibrium values of macroeconomic indicators is Hodrick-Prescott filter. This filter is widely used by researchers to determine the equilibrium dynamics of indicators, which are included in the monetary rule [5; 16; 23].
A classical approach to Hodrick-Prescott filter has certain difficulties for practical calculations. If only input values of the time series are known, it would be difficult to find their components by conventional mathematical methods. Therefore, implementation of the classical approach of the filter's use involves the formation of additional optimization task to be solved in several stages. An example of the classical approach implementation with the use of Hodrick-Prescott Filter with additional restrictions and the use of VARmodels is presented in the work of specialists of the Department of Economic Analysis and Forecasting of the National Bank of Ukraine [18].
To solve these problems, different versions of the Hodrick-Prescott Filter were developed. To estimate equilibrium levels of the nominal and real GDP we apply the modified Hodrick-Prescott filter proposed by Hyeongwoo Kim [8]. This approach is also used by the World Bank [6, pp. 2-3]. In general, the modified model of the filter can be presented as follows: , ) ( where, y are the input (non-smooth) time series; g is the trend (non-smooth component) of time series; I T is the identity matrix with × dimensions (square matrix); F is the special matrix of deviations of trend indicators.
It should be noted that matrix F (equation 5) is symmetrical and is determined on the basis of statistical selection. It has a standard form for any input time series.   Taking into consideration the above-mentioned aspects, we will study the growth rates of real GDP in Ukraine in the period from the first quarter of 2001 till the first quarter of 2012. In our study we will use the statistical data from the official cite of the National Bank of Ukraine [25]. The results of the filter's use for the above-mentioned time series are shown in Figure 2.

Determination of parameters, which can be included in the monetary rule
We will study the equilibrium dynamics of money supply. Figure 3 shows the equilibrium growth of M3 for Ukraine's economy calculated on the basis of the Friedman monetary rule (equation (1)). Equilibrium dynamics of the money supply growth is calculated as the sum of GDP deflator (or Consumer price index) and real GDP. This figure gives the results of the indicator's calculation made in four different ways: based on the GDP deflator, based on the consumer price index (CPI), by using growth rates and by using growth rates logarithms. As a result, we have received four deviation indicators. Negative values of deviations indicate the excess of the actual dynamics of money supply (M3) over its equilibrium dynamics, whereas positive values indicate the opposite situation.
On the basis of the data analysis the following conclusions can be made. Firstly, the indicators calculated on the basis of GDP deflator are closer to the real dynamics of money supply compared to the indicators calculated by using the consumer price index. This conclusion is explained by the adequate methodology for the calculation of GDP deflator (a wide range of goods and services, the dynamics of which is taken into account in determining the GDP deflator) in the context of the Friedman monetary rule. Since the monetary rule includes GDP indicators, it is better to use the indicator of GDP deflator in retrospective construction of the money supply equilibrium dynamics.
Secondly, no significant differences were found in the calculations with the use of growth rates or natural logarithms of growth rates for the related indicators. As a result, we consider it necessary to use growth rates of the related indicators without taking their logarithms. This method of calculation is more conducive for adequate interpretation of the results' economic content.
Thirdly, the results confirm our hypothesis regarding the expediency of using the equations of money turnover and the Friedman monetary rule for determining the dynamics of the money supply, which will help achieve the money market and macroeconomic equilibrium.  actual real GDP growth rate smoothed growth rate of real GDP

Fig. 3. Equilibrium annual dynamics of money supply in Ukraine and its deviation from the actual dynamics, decimal fraction
Analyzing the figures in Table 3 we can conclude that in periods of significant deviations of the money supply actual dynamics from this indicator's equilibrium dynamics, significant deviations from macroeconomic equilibrium were observed: inflation dynamics significantly exceeded the dynamics of economic growth. On the other hand, during the periods of insignificant deviations between the actual and equilibrium dynamics of the money supply (highlighted in bold in Table 3) the inflation was maintained within acceptable limits and was consistent with economic growth: the ratio of GDP growth and consumer prices was close to 1.  Table 3  growth rates of equilibrium money supply (1),as the sum of logarithms of growth rates, calculated using the GDP deflator growth rates of equilibrium money supply (2), as the sum of logarithms of growth rates, calculated using the CPI deviation of growth rates of equilibrium money supply (1) from actual growth rates of money supply deviation of growth rates of equilibrium money supply (2) from actual growth rates of money supply growth rates of equilibrium money supply (3), as the sum of growth rates, calculated using the GDP deflator growth rates of equilibrium money supply (4), as the sum of growth rates, calculated using the CPI deviation of growth rates of equilibrium money supply (3) from actual growth rates of money supply deviation of growth rates of equilibrium money supply (4) from actual growth rates of money supply We have established an algorithm for the construction of the money supply equilibrium dynamics and tested it on the basis of the annual data for the period of 2003-2011. The results confirm the thesis that focusing on the equilibrium dynamics of money supply will help in achieving macroeconomic equilibrium and in balancing the money market.
Further we will build a multiple regression to explain the dependence of the actual dynamics of money supply on factor characteristics, which are the components of the Bennett McCallum rule for money supply: money turnover rate, equilibrium real GDP, gap in nominal GDP, level of inflation.
In accordance with the McCallum rule (equation (3) (7) where m is the money supply growth (M3 aggregate); v is the money turnover growth; x is the growth of nominal GDP; * x is the growth of equilibrium nominal GDP; q * is the growth of equilibrium real GDP; is the growth of inflation level (consumer price index).
In general, the regression dependence is significant ( = 0,05). All coefficients are also significant (excluding inflation ). A regression model with inflation lag in the first quarter improves regression statistics, but inflation coefficient remains insignificant. The use of other leads (inflation expectations) and lags (lag effects of inflation) at economically substantiated intervals does not improve the model.
In our opinion, insignificant influence of consumer price index on the dynamics of money supply can be explained by the fact that the analysis was conducted during the period of 2004-2011, when the main priority for the National Bank of Ukraine was to ensure the exchange rate stability while the main channel for increasing the money supply was currency interventions in the interbank market. Therefore, we will withdraw inflation from the regression model in order to improve its quality. For further research, which will not be conducted on the basis of retrospective data, the inclusion of consumer inflation in the monetary rule is recommended.
We have built the following regression model:  (8) where m is the money supply growth (M3 aggregate); v is the money turnover growth; x is the growth of nominal GDP; x * is the growth of equilibrium nominal GDP; q is the growth of equilibrium real GDP.
Regression statistics for this model is shown in Table 4. Analyzing the data, we can conclude that the model is statistically significant ( = 0,05). All coefficients of the model are also significant. The analysis of the correlation matrix leads to the conclusion about the lack of multicollinearity. Based on the calculation of the Durbin-Watson criterion we have made the conclusion about the lack of autocorrelation of the regression model residuals. The average approximation error is 4.3%, while an acceptable error in economic research is up to 10%. Therefore, we can conclude that our multivariate model has a sufficient quality and can be used to analyze the relationship between its indicators as well as in forecasting.
In accordance with theoretical provisions, the growth of money turnover should result in the reduction of the money supply growth. Our regression model (equation (8) confirms the presence of a strong inverse relationship between these indicators.
The growth of real GDP has the most significant direct impact on the dynamics of money supply. A direct impact twice its size on the resultant indicator causes a gap in the GDP ( x x * ). It is used to determine the current levels of the GDP deviation from its equilibrium (smooth) trajectory. Theoretically, an inverse relation should exist between this indicator and the money supply.  Thus, the central bank should conduct anti-cyclical monetary policy, rapidly increasing the money supply during the periods of low growth or decline in GDP, while moderately increasing or reducing the money supply during the peaks of GDP growth, reflecting the economy's "overheating". In the monetary rule based on retrospective data there is a direct dependence between the GDP gap and the dynamics of money supply, which is explained by the lack of countercyclical monetary policy during the study period. Modified multivariate regression dependences are summarized in Table 5. Calculations were carried out for the period from the 4th quarter of 2003 to the 1st quarter of 2012 (34 observations). Table 5. Modified multivariate regression dependences of the money supply and the monetary rule parameters for Ukraine

Conclusions
We have substantiated the importance of developing the money supply explicit monetary rule for the economy of Ukraine. We have also analyzed the methodological approaches to its formation. In particular, the procedure for calculating the value of the equilibrium GDP, which is used in the formation of the policy rule, was determined. A multivariate regression model, which confirms the presence of a statistically significant relationship between the main parameters of the monetary rule, was developed.
In our opinion, the conception of "monetary rules" as any other economic conception that involves the use of certain stable relationships between macroeconomic parameters for solving the current economic problems, has several deficiencies. These deficiencies are caused by the unstable nature of socio-economic relations and the complexity of forecasting of the economic agents' behavior at macroeconomic level. Based on the general content of research publications Van Lear William identifies five major deficiencies in the development and implementation of monetary rules [27].
However, we believe that two key theses in support of monetary rules negate most of the deficiencies cited in the scientific literature. Firstly, all developers of monetary rules, including John Taylor, warn against their mechanical application and emphasize the importance of their use along with other decision making instruments (expert judgment, modeling, etc.) of the monetary authorities. Secondly, at the moment there is no theoretically substantiated and empirically confirmed alternative to monetary rules. This thesis is particularly important during the use of inflation targeting regime.
The findings of this study can be used by the National Bank of Ukraine to develop a monetary rule. It should be noted that an explicit monetary policy rule will have a limited sphere of application until the full implementation of inflation targeting regime in Ukraine. It should be used only as an additional instrument for analyzing the effectiveness of the monetary policy. However, after the transition to inflation targeting and the renewal of stable relationships between the money supply and price dynamics in the economy of Ukraine (which will cause the inclusion of inflation indicator in the monetary rule for money supply) this rule can become one of the main instruments in the development and implementation of the monetary policy.