In the empirical analysis time series data was used, with monthly frequency from January 1995 to December 2009. This time range was chosen because of the availability of the necessary data as well as the fact that there is no such research on this subject during this particular period. All data have been obtained from the databases of the International Monetary Fund's International Financial Statistics, the Organisation for Economic Co-operation and Development (OECD), the Vienna Institute for International Economic Studies (WIIW) and the Bank of Russia (CBR).
For the purpose of the empirical estimation the data on refinancing rates (short-term interest rates), consumer price index for the calculation of inflation, industrial production index as a proxy for output, monetary aggregates M1, and real effective exchange rate were used.
In order to construct series for target values of output the traditional Hodrick-Prescott (HP) filter was employed. Moreover, instead of levels the growth rates of consumer price index, real effective exchange rate, monetary aggregate M1 and industrial production index were used in the models. The rates of growth were calculated by the differences of the logarithms:
Before running any regressions, it is important to check whether the time-series data is stationary or not. For this purpose both informal (graphical presentations) and formal unit root tests (Augmented Dickey-Fuller tests) were performed. From the graphs it was decided which type of the Augmented Dickey-Fuller test had to be used (with intercept, with intercept and trend, or without intercept and trend). The null hypothesis in this test is that there is unit root and the series are not stationary. And the alternative hypothesis is that there is no unit root and the series are stationary.
Based on the low values of the probability (p-values for all variables, except that of monetary aggregate growth, are less than 0.05) and large negative values of the t-statistic (t-statistic for all variables, except that of monetary aggregate growth, is less than the critical value at 5% level of significance), we reject the null hypothesis that each of the variables individually is non-stationary and accept the alternative hypothesis that it is stationary at 5% significance level. The p-value for monetary aggregate growth is 0.0862; and the t-statistic of approximately -2.6444 is lower than the t-critical for 10% level of significance, indicating that the variable is stationary at 10% significance level. Since all the variables are stationary we can proceed to the estimation of the equations. Table 1 illustrates the results of the unit root test for all variables and the results from each individual test are included in Appendix A-Unit root tests.
Empirical Results
Results for the Taylor rule for open economy
The results of the estimation of the Taylor rule version for open economy indicate that only the coefficient of inflation is significant and shows the expected sign. The p-value of 0.061 means that the coefficient of inflation is significant at 6.1% significance level, or higher. The coefficient of the output gap shows the expected sign but the p-value of 0.8219 shows that it is highly insignificant. This could possibly be caused by the fact that for the first periods of the transition stage the data on the output gap was overestimated or by the fact that the output gap was not one of the objectives of the Central Bank of Russia. Finally, the coefficient of the real effective exchange rate does not show the expected sign and its p-value of 0.2118 indicates that it is also highly insignificant.
The interpretation of the coefficient of the lagged value of the short-term interest rate of approximately 0.91 is that the interest rate in the current period is about 91% of the interest rate in the previous period plus the effect of the other explanatory variables. The high R-squared of approximately 0.938 means that the variation in the refinancing rate can be explained 93.8% by the variation in the inflation, output gap and exchange rate. However, the fact that most of the explanatory variables are insignificant and some of them show wrong signs suggests that the Taylor rule for open economy does not explain the behaviour of the Central Bank of Russia well. The results from the estimations are shown in table 2.
Results for the McCallum rule
The results of the estimation of the original version of the McCallum rule demonstrate that although the coefficient of the target nominal output growth shows the expected sign it is insignificant since the p-value of 0.574 is higher than 0.05. However, both the coefficients of money velocity growth and that of the change between the target nominal output growth and the nominal output growth in the previous period are showing the expected signs. The p-values of these coefficients of 0.0000 and 0.0002 respectively indicate that they are significant at 5% significant level. The results from the estimation are shown in Appendix B. The interpretation of the R-squared of approximately 0.599 is that the variation in the growth of the monetary aggregate M1 is 59.9% explained by the variation in the explanatory variables. On the other hand, the insignificance of the coefficient of the target nominal output growth may be caused by a model misspecification or an inappropriate use of some of the variables.
Moreover, with the purpose of stabilizing the monetary aggregate data from any seasonal fluctuations M1 variable was deflated with the consumer price index.
This adjustment for inflation generates a new series M1_d, which was then used to calculate the real growth of the monetary aggregate dM1_d.
A new model was estimated with this deflated monetary aggregate growth as a dependent variable, leaving all the independent variables the same. The results show a better performance of the model. All the coefficients are showing expected signs. Not only the coefficients of the money velocity growth and that of the change between the target nominal output growth and the nominal output growth in the previous period are significant, but also the coefficient of the target nominal output growth is statistically significant. The p-values of all coefficients are very small numbers close to zero, which indicate that all explanatory variables are highly significant. The results from the estimations are shown in the table 3.
The high R-squared of approximately 0.742 shows that about 74.2% of the variation in the monetary aggregate is explained by the model. In addition, comparing to the previous estimations the information criterion now is lower, indicating that this model is preferred. Akaike info criterion in the McCallum rule with deflated monetary aggregate is approximately 5.028 comparing to 5.28 in the McCallum rule with the original value of monetary aggregate. Schwarz and Hannan-Quinn criterion are 5.1 and 5.058 respectively in the model with deflated value of M1 and 5.353 and 5.31 in the original version. All this illustrates that the modified version of the McCallum rule is a better model and it explains the behaviour of the Bank of Russia quite well. However, in order to detect if these results are reliable few tests have to be carried out.
Testing for serial correlation
Breusch-Godfrey Serial Correlation LM Test
This test is performed in order to see if there is serial correlation in the residuals. The null hypothesis in this test is that there is no serial correlation. The alternative is that there is serial correlation on the residuals. The number of lags is chosen based on the frequency of the data used. In this estimation monthly data was use, so the number of lags should be 12. The LM test statistic of 44.76869, which in E-views is shown as Obs*R-squared, is clearly higher than the ?2(12) critical value at 95% confidence interval of 21.026, so we can reject the null hypothesis of no serial correlation up to lag 12 and conclude that serial correlation on the residuals in present. The p-value of ?2(12) of 0.0000 represents the probability with which it would be incorrect to reject the null hypothesis of no serial correlation up to lag 12 at the 95% confidence interval. The full output from the test is shown in Appendix C.
Correlogram - Q-statistics
The null and the alternative hypothesis are the same as in the previous LM test for serial correlation. The p-values of the Q-statistic for all 12 lags are approximately equal to zero, which leads to a rejection of the null hypothesis of no autocorrelation and to conclusion that autocorrelation is present.
The result obtained from Q-statistics test is the same as the one from the Breusch-Godfrey LM-test, so there is serial correlation on the residuals. This unsatisfactory result often is present in time-series data, and if serial correlation is nor corrected the estimations may not be reliable. The consequences that may occur if the presence of autocorrelation is ignored are incorrect standard errors, which lead to misleading confidence intervals and hypothesis tests, and least squares estimators which do not have the lowest variance.
Accounting for serial correlation
A way to solve these problems is to compute heteroskedasticity and autocorrelation consistent (HAC) or Newey-West standard errors. Furthermore, lags can be added according to the significance of the coefficient and information criterion. These modifications were taken into account and a new model was estimated, where Newey-West standard errors were computed and lagged values of the growth rates of monetary aggregate and the money velocity were included. This version of the McCallum rule with deflated monetary aggregate that also accounts for autocorrelation is called full model.
The results from the estimation demonstrate that all the coefficients are showing the expected signs. In addition, since their p-values are very small numbers close to zero all coefficients are statistically significant at 1% level of significance. The high R-squared of approximately 0.834 indicates that the variation in the growth of monetary aggregate is 83.4% explained by the model. Comparing to the previous estimation of the McCallum rule with deflated monetary aggregate, where the effects of autocorrelation were not taken into consideration, the information criterions are lower in the full model. Awake info criterion is now equal to 4.609, Schwarz criterion is 4.717 and Hannan-Quinn criterion is 4.653, comparing to those of 5.028, 5.1 and 5.058 respectively. The results are shown in the table 4.
The results from Breusch-Godfrey serial correlation LM test show that the LM test statistic of 17.79556 is lower than the ?2(12) critical value at 95% confidence interval of 21.026, so the null hypothesis of no serial correlation up to lag 12 cannot be reject. In addition, the p-value of ?2(12) of 0.122 indicate that the null hypothesis can be accepted at 5% level of significance. This point to the fact that in the full model autocorrelation is not present. The full output from the test is shown in Appendix C.
Testing for heteroskedasticity
Breusch-Pagan-Godfrey Heteroskedasticity Test
The full model also has to be tested for the presence of heteroskedasticity. Breusch-Pagan-Godfrey test is used to check whether there is serial correlation in the squared residuals, which is equivalent to checking if heteroskedasticity is present. The null hypothesis is that there is no serial correlation on the squared residuals, meaning no heteroskedasticity. The alternative hypothesis is that there is serial correlation on the residuals, meaning that heteroskedasticity is present. Since the LM test statistic of 3.729826 is lower than the ?2(12) critical value at 95% confidence interval of 21.026, the null hypothesis of no heteroskedasticity cannot be rejected, leading to conclude that heteroskedasticity is not present. The p-value of ?2(12) of 0.5889 indicate that the null hypothesis on no heteroskedasticity can be accepted at 5% level of significance. The full output from the test is shown in Appendix D.
Correlogram - Squared Residuals
The null and the alternative hypothesis are the same as in the previous heteroskedasticity test. The p-values show that for all 12 lags the null hypothesis should be accepted at 1% significance level, and so heteroskedasticity does not exist. This result is the same as the one obtained from the Breusch-Pagan-Godfrey test, so we can conclude that heteroskedasticity is not present and no further actions are required.
Testing for normality
Jarque-Bera test and histogram
The last test to be carried out in order to check the reliability of the results obtained in the full model is a normality test. Jarque-Bera test is used for this purpose. The null hypothesis in this model is that the residuals are normally distributed. The alternative is that the errors do not follow normal distribution. The large value of Jarque-Bera statistic of 535.8341 is much higher than the 5% ?2 (2) critical value of 5.99, which leads to a rejection of the null hypothesis of normally distributed residuals. Identical conclusion can be made by observing the p-value of zero, which also indicates that the residuals are not normally distributes. The full output from the test is shown in Appendix E. As a consequence, these residuals should not be employed in hypothesis tests as it may lead to a misleading conclusion. The possible cause of not normally distributed residuals could be omitting an important variable, using a wrong functional form of some variable or a general misspecification of the model.
Testing for seasonal patterns
Since monthly data was used in the regression analysis, it is highly likely that there are some seasonal patterns in the data. To correct this we include seasonal dummies for every January and December and estimate a new regression. According to Dabrowski (2002) this seasonal patterns could possibly be caused due to control prices increase at the start of each year. Furthermore, he proposes that money supply decrease in the beginning of each year can be explained by accounting measures.
The results are illustrated in table 5. The results demonstrate that all coefficients have signs according to prior expectations. The p-values of all coefficients are lower that the critical value at 5% level of significance of 0.05, so we can reject the individual null hypothesis that each partial regression coefficient is zero and conclude that all independent variables individually have impact on the growth of monetary aggregate. The probability of F-statistic is close to zero, which leads to a rejection of the null hypothesis that all coefficients simultaneously are insignificant. Therefore, the coefficients together also influence the dependent variable.
Comparing to the previous full model, where the seasonality of the data was not taken into account, this model is showing a better performance. Firstly, the adjusted R-squared has increased from 82.89% to 85.79%. Secondly, values of the information criterions have decreased from 4.609, 4.717 and 5.653 to 4.435, 4.579 and 4.493 for Akaike, Schwarz and Hannan-Quinn information criterion respectively. Finally, the standard errors have dropped from 2.384 to 2.174. All this indicated a better performance of the full model with seasonal dummies.
