MODELLING INFLATION UNCERTAINTY IN TRANSITION … · MODELLING INFLATION UNCERTAINTY IN TRANSITION...

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* Faculty of Economics and Administrative Sciences, Hacettepe University, Turkey** Faculty of Economics and Administrative Sciences, Hacettepe University, Turkey

JEL CLASSIFICATION: C 32, E 31

ABSTRACT: This study investigates the linkage between inflation and inflation-uncertainty in seven transitional economies (Armenia, Azerbaijan, Georgia, Kazakhstan, the Kyrgyz Republic, the Russian Federation and the Ukraine) which experienced hyper-inflation until the mid-1990s. This linkage is investigated in the ARCH modelling framework by using both conventional Granger non-causality testing and the Holmes-Hutton approach, which has significant small- and large-sample power advantages over the

former. The results support the Friedman-Ball hypothesis in Azerbaijan, the Russian Federation and the Ukraine. The Cukierman-Meltzer hypothesis is favoured in the Kyrgyz Republic and in the Russian Federation using a different model. In Azerbaijan, greater inflation uncertainty preceded lower rates of inflation, indicative of the strong monetary stabilization policies pursued in this economy.KEY WORDS: Inflation, inflation-uncertainty, Granger-causality, conditional variance

Serkan Erkam*, Tarkan Cavusoglu** DOI:10.2298/EKA0879044E

MODELLING INFLATION UNCERTAINTY IN TRANSITION ECONOMIES: THE CASE OF RUSSIA AND THE FORMER SOVIET REPUBLICS

SCIENTIFIC PAPERS

INFLATION UNCERTAINTY IN TRANSITIONAL ECONOMIES

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1. Introduction

Inflation uncertainty, as well as the rate of inflation, has been of interest to economists since Okun’s (1971) emphasis on the costs of a “highly variable and uncertain state of price movements” to the income and wealth of individuals. In his “stop-go world”, Okun (1971: 493) argued that countries with high rates of inflation would also experience highly variable inflation rates, empirical evidence being provided for seventeen industrial OECD countries. However, Gordon (1971: 505) claimed this evidence to be “far from universal” due to its bias resulting from the choice of the sample period, 1951-1968. Logue and Willet (1976) and Foster (1978) confirmed Okun’s findings, providing further empirical evidence on the positive relationship between average rates of inflation and the variability of inflation, with the exception of a weak relationship reported in the former study for countries with moderate inflation. This ongoing debate gained significance with Milton Friedman’s Nobel lecture on inflation and unemployment. According to Friedman (1977), high variability in inflation rates which is preceded by high rates of inflation is an outcome of the political cohesiveness in counteracting inflation. Adjustments to “a normal price level” in institutional arrangements and long-term contracts cause wide variations in the rates of actual and anticipated inflation, which lower economic efficiency through rigidities that reduce the effectiveness of markets and through distortions that weaken the signalling function of relative prices (Friedman, 1977: 466-7). However, the argument that higher rates of inflation lead to more inflation uncertainty was first formalized by Ball (1992) within a game-theoretic model framework, and therefore called the Friedman-Ball hypothesis in the literature. In Ball’s model, monetary authority is characterized by two different types of policymakers - conservative and liberal - who alternate in charge stochastically. When the rate of inflation is low, the public is certain that both types of policymaker will try to keep the inflation low. However, when the rate of inflation is high, uncertainty about future inflation arises, because the public does not know which type of policymaker will be in charge: If the conservative is in charge he will prefer to disinflate, whereas the liberal will be reluctant to disinflate for fear of causing a recession.

The opposite of the Friedman-Ball hypothesis was described by Cukierman and Meltzer (1986) through a model in which policy credibility and ambiguity are related to the quality of monetary control. In this setting, the low credibility and high ambiguity that characterize the policymakers with unstable objectives increase the average rate of inflation. Using a different approach, Devereux (1989) introduced a theoretical framework that justified the causal link from the variance of inflation to the average rate of inflation, by exploiting a model of discretionary

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Serkan Erkam, Tarkan Cavusoglu

monetary policy which incorporated real disturbances and endogenous wage indexation. It was argued that a rise in the variance of real disturbances in the economy lowers the optimal degree of wage indexation in the labour market and encourages monetary authorities to cause surprise inflation, and thereby increases the mean rate of inflation in a discretionary equilibrium (Devereux, 1989: 106). Explanations of a causal link between inflation uncertainty and the rate of inflation are broadly known as the Cukierman-Meltzer hypothesis. Alternatively, Holland (1995) presented empirical evidence that greater inflation uncertainty precedes lower inflation, the possible explanation being the stabilization motive of policymakers, who view inflation uncertainty as a welfare cost.

Some early empirical studies (Engle (1982), Engle (1983) and Cosimano and Jansen (1988)) did not find any evidence of a link between inflation and inflation- uncertainty. However, Ball and Cecchetti (1990) and Evans (1991) provided supporting evidence on the Friedman-Ball hypothesis, particularly for long-term uncertainty while Brunner and Hess (1993) suggested using an asymmetric model to produce results consistent with the Friedman-Ball view. Ungar and Zilberfarb (1993), Arnold and den Hertog (1995) and Davis and Kanago (2000) confirmed the Friedman-Ball hypothesis, but only for countries experiencing inflation rates above a certain threshold level.

Recent studies examining the link between inflation and inflation-uncertainty fall into two groups on the basis of their econometric methodology. The first group tests the Friedman-Ball hypothesis through including the level or the one-period-lag of the inflation rate into the conditional-variance equation of a GARCH process (Baillie et al., 1996; Caporale and McKiernan, 1997; Fountas, 2001; Kontonikas, 2004; Thornton, 2006; Thornton, 2008). The second group rely on Granger causality tests involving the rate of inflation and the conditional-variance estimates (Grier and Perry, 1998; Nas and Perry, 2000; Telatar and Telatar, 2003; Apergis, 2004; Fountas et al., 2004; Daal et al., 2005; Berument and Dincer, 2005; Conrad and Karanasos, 2005; Thornton, 2007). The conclusions of these studies are summarized in Table 1 and Table 2. The Friedman-Ball hypothesis is supported only for six of the eleven countries in the first group of studies (see Table 1). However, in the second group, Granger causality tests computed with a varying lag-structure (from 4 to 24 lags) provide strong evidence in favour of the Friedman-Ball hypothesis at all lags for almost all countries (see Table 2). The findings listed in Table 2 clearly demonstrate that the Granger causality between inflation and inflation-uncertainty is not uni-directional in most of the cases, i.e. in a considerable number of cases, the Friedman-Ball hypothesis is supported together with the Cukierman-Meltzer hypothesis, indicating a feedback


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Table 1. First-group studies (based on the conditional variance equation)Study Country Lags

Baillie et al. (1996)Argentina, Brazil, Israel, UK

1Canada, France, Germany, Italy, Japan, US

Caporale and McKiernan (1997) US 1

Fountas (2001) UK (Sample-1) 0UK (Sample-2) 0

Kontonikas (2004) UK (Quarterly, Monthly) 1Thornton (2006) S. Africa 0Thornton (2008) Argentina 0

between the rate of inflation and inflation-uncertainty. Moreover, when the lag-length and the sign of the Cukierman-Meltzer-type relationship are considered in Table 2, it appears that the negative relationship suggested by Holland (1995) is valid, especially at relatively longer horizons.

The very high rates of price inflation experienced in transitional economies before the mid-1990s created an environment of uncertainty, due to the peculiarities of the transition from a command-driven economy to a market-oriented economy. Distorted relative prices due to high inflation generate uncertainty about key macroeconomic indicators, discouraging investment and encouraging unproductive activities. The objective of this paper is to investigate the relationship between inflation and inflation-uncertainty in seven transitional economies. To our knowledge, there is no similar empirical study in the literature concerning the inflation and inflation-uncertainty relationship in transitional economies, with the exception of the conference paper by Mladenovic (2007), concerning inflation in Serbia. Within this framework, the relevant hypotheses concerning this relationship are tested empirically for Armenia, Azerbaijan, Kazakhstan, the Kyrgyz Republic, the Russian Federation and the Ukraine. Our results based on conditional-variance estimates provide evidence that inflation itself is a source of inflation uncertainty in Azerbaijan, the Russian Federation and the Ukraine, while at the same time inflation uncertainty preceded higher inflation in the Russian Federation and in the Kyrgyz Republic. Moreover, causality running from inflation uncertainty towards lower rates of inflation is found in Azerbaijan, consistent with Holland’s (1995) proposition.

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The plan of the study is as follows. Part 2 provides an overview about inflation and monetary policy in the transitional economies of interest. Part 3 comprises a description of the data and a brief explanation of the methodology. The results are presented in part 4, followed by conclusions.

Table 2. Second-group studies ( based on Granger-causality tests)

Study Country Hypothesis / LagsF-B C-M Holland

Grier and Perry (1998)

US All - 8,12Germany All - 4,8Japan 8,12 All -UK All - 12Canada All - -Italy All 8 4France All All -

Nas and Perry (2000)

Turkey All - All Turkey (Subsample-1) All - 4,12Turkey (Subsample-2) All 4,8 12,16,24Turkey (Subsample-3) All 4,8,12 16

Telatar and Telatar (2003) Turkey All - -

Apergis (2004) Panel of G7 Countries All All -

Fountas et al. (2004)

UK All 2,6 4,8France All 2 -Germany - - 4,6,8Italy All 4,6,8 -Netherlands All - AllSpain All 6,8 -

Daal et al. (2005)

Korea, Pakistan, Sri Lanka, Thailand, Canada, France, Japan, US, Mexico, Morocco, Turkey

8 - -

Indonesia, Italy, UK, Bahrain, Egypt 8 8 -

India, Argentina, Colombia, Venezuela 8 - 8

Germany - 8 -Peru - - -


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Berument and Dincer (2005)

Canada, UK, US All - 8,12France 8,12 - All Germany 12 - -Italy 8,12 - -Japan All 4,8 -

Conrad and Karanasos (2005)

US All - -UK All 8 12Japan All All -

Thornton (2007)

S. Africa, Thailand, Jordan, India, All - -

Mexico, Turkey All - All Colombia All - 12Korea, Indonesia, Hungary All 4,8 -Israel - 8 4,12Malaysia - - -

2. An Overview of Economies in the Sample

One immediate outcome of the transition of the former Soviet Socialist Republics to market-oriented economies was the very high rates of inflation at the beginning of the 1990s. The liberalization of the price and exchange rate systems and a

Figure 1. Consumer price inflation in seven transitional economies

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lack of monetary and fiscal controls trigger hyper-inflation in transitional economies (Dąbrowski, 1999). However, price liberalization and stabilization are necessary components of reform for achieving budgetary balance and for controlling monetary expansion (Åslund et al., 1996), especially in economies with monetary overhang.

The consumer price indices reported for the countries in this study show that rates of inflation in all of these economies were reduced to two-digit rates only after the mid-1990s. As shown in Figure 1, inflation was stabilized at single-digit rates after 2000 in all countries except the Russian Federation. Average rates of inflation within the 1996-2000 period indicate that Armenia, Azerbaijan and, to a lesser extent, Georgia and Kazakhstan successfully managed to insulate themselves from the contagious effects of the Russian financial crisis in August 1998. In particular, Azerbaijan experienced negative inflation rates in 1998 and 1999 (-0.8 % and -8.6 % respectively) resulting from policies to avoid the depreciation of the Manat in the wake of the Russian crisis. In fact, unlike the other economies in transition, Azerbaijan managed to reduce inflation sharply in 1996 by conducting an excessively tight monetary policy with an IMF-supported stabilization plan (see Figure 1). According to Singh and Laurila (1999), the reduction in the Azeri inflation rate was faster than planned because banks were inefficient in maintaining their intermediation function, which constituted a restriction on the credit creation process, evidenced by the very low money-multiplier. In an analysis of the monetary policy during transition, de Melo and Denizer (1997) classify transitional economies as of end-1994 by the extent of market-orientation in the use of monetary policy instruments, namely high, substantial, moderate and low. According to this research, the countries included in this study have moderate or low market orientation. Azerbaijan, Georgia, Kazakhstan and the Russian Federation are in the moderate class while Armenia, the Kyrgyz Republic and the Ukraine are low. Moreover, all these countries except for the Kyrgyz Republic are argued to be slowly stabilizing economies compared to the Eastern European economies in transition.

Monetary policies in the disinflation era of the transitional economies in the post-2000 period seem to have been commonly based on either IMF-supported monetary programs or strategies directed towards monitoring various monetary indicators as before. In achieving price stability, monetary aggregates were used as intermediate targets by the monetary authorities in most cases. The supply of money was coordinated by setting controls on net domestic and foreign assets of the central banks. According to the de facto classification of the IMF, managed floating was the most commonly used exchange rate regime in these


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countries, except for the Ukraine, which implemented a fixed peg arrangement, and Azerbaijan, which moved between managed-floating and pegging regimes. However, in the last few years rates of inflation have begun to climb towards 10 % in five of the seven economies in question, with the exception of Armenia whose Central Bank has started moving to a fully-fledged inflation targeting strategy since 2006, and the Kyrgyz Republic which implicitly set the target inflation rate for 2007 to a maximum of 5-6 %, well above the 2 % rate of inflation achieved in 2002. These increased rates of inflation can be explained through looser policies, designed to slow down the nominal appreciation of the local currencies, from 2003 onwards.

As implied in the overview above, economies in transition are frequently exposed to policy trade-offs which cause variations in inflation rate and result in uncertainty about monetary policy.

In this context, in part 3 the relationship between inflation and inflation-uncertainty in the seven transitional economies of interest is investigated empirically.

3. Data and the Estimation Methodology

The rate-of-inflation data used in this study are computed from the monthly CPI indices obtained from the IMF-IFS database. Inflation rates are the first-

Table 3. Summary statistics for monthly rate of inflation

Armenia Azerbaijan Georgia Kazakhstan Kyrgyz Rep.

Russian Fed. Ukraine

Mean 0.0036 0.0010 0.0056 0.0066 0.0078 0.0159 0.0094Maximum 0.0527 0.0310 0.1147 0.0465 0.0766 0.3248 0.0620Minimum -0.0566 -0.0590 -0.0236 -0.0095 -0.0286 -0.0037 -0.0180Std. Dev. 0.0201 0.0110 0.0165 0.0074 0.0158 0.0313 0.0123Skewness -0.1631 -1.5116 2.7164 2.3684 1.4147 8.1154 0.9407Kurtosis 3.4007 10.619 19.276 13.939 7.9972 78.219 5.6964

JB 1.5017 294.01 a 1349.4 a 740.17 a 170.38 a 30594 a 56.301 a

Seasonality 28.403 b 5.366 b 9.247 b 4.319 b 13.286 b 1.426 12.058 b

Moving Seasonality 3.303 b 3.283 b 1.108 5.485 b 5.621 b 4.198 b 2.227 c

LM(4) 5.915 b 0.144 0.187 12.918 b 11.673 b 0.118 1.784LM(12) 2.562 b 0.061 0.068 5.520 b 5.619 b 0.037 0.667

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LB(4) 23.269 b 0.602 0.810 31.058 b 41.479 b 0.531 7.166LB(12) 30.398 b 0.959 1.138 33.862 b 57.395 b 0.574 9.813

BDS(2) 0.0219 b -0.0002 0.0337 a 0.0787 a 0.0465 a 0.1339 a 0.0535 a

BDS(3) 0.0445 a -0.0004 0.0485 a 0.1405 a 0.0814 a 0.2225 a 0.0819 a

BDS(4) 0.0602 a -0.0007 0.0446 a 0.1762 a 0.1029 a 0.2779 a 0.1031 a

BDS(5) 0.0577 a -0.0013 0.0351 b 0.1958 a 0.1139 a 0.3051 a 0.1116 a

BDS(6) 0.0553 a -0.0020 0.0319 c 0.2006 a 0.1183 a 0.3167 a 0.1071 a

Observations 135 105 110 125 124 124 125Superscripts a, b and c denote 1, 5 and 10 % levels of statistical significance, respectively. JB is the Jarque-Bera statistic to test whether the series is normally distributed. LM is the Lagrange multiplier statistic for testing the null of ‘no ARCH effects’ in the residuals. LB denotes the Ljung-Box Q-statistics to test the null of ‘no serial correlation’ in the squared residuals. BDS is a portmanteau test to check whether or not the residuals are independent and identically distributed.

differences of the CPI series in natural logarithms and denoted by πt henceforth. The countries and the time-span of the data in the sample are as follows: Armenia, 1996:1-2007:3; Azerbaijan, 1996:1-2004:9; Georgia, 1997:1-2006:2; Kazakhstan, 1997:1-2007:5; the Kyrgyz Republic, 1997:1-2007:4; the Russian Federation, 1997:1-2007:4; the Ukraine, 1997:1-2007:5. The starting date of each series corresponds to the beginning of the period with relatively moderate rates of inflation while the end-date is determined by data availability.

Summary statistics for the monthly rate of inflation in the sample are given in Table 3. The first three rows of the table show that monthly average rates of inflation are lower than 1 % in all countries except for the Russian Federation, which has both extremes, with a minimum of -0.37% and a maximum of 32.48%. The distributions of the inflation series are all leptokurtic, exhibiting leftward skewness in Armenia and Azerbaijan and rightward skewness in the rest. Due to outliers resulting from the 1998 Russian crisis, Jarque-Bera normality is violated for the series of all countries except Armenia. Two types of variance-ratio tests were used to test seasonality within the framework of the X-12-ARIMA seasonal adjustment approach, where the latter is for moving seasonality, and both have F-distribution. According to the test statistics, all inflation series need to be adjusted seasonally because the null hypothesis of ‘no seasonality’ is rejected in one or both of the two tests, as seen in Table 3. Plots of the seasonally adjusted inflation series are presented in Figure 2. When the seasonally adjusted series are tested by the Lagrange multiplier (LM) test and the Ljung-Box (LB) Q-statistic by using four and twelve lags, the ARCH effects and serial correlations are observed only for Armenia, Kazakhstan and the Kyrgyz Republic. However, Bollerslev et


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al. (1994: 2975) assert that the ARCH tests carried out above “… are tests for volatility clustering rather than general conditional heteroskedasticity, or general non-linear dependence” and hence recommend a more powerful test, i.e. the BDS test introduced by Brock et al. (1987). Thus, the BDS test was utilized to show whether or not there is a form of dependence in the residuals of the simplest univariate specification of each of the series, which may not be detected with the ARCH-LM test. In Table 3, the BDS tests computed in each case for dimensions 2 to 6 exhibit evidence for departures from independent and identically distributed observations for all countries, except for Azerbaijan.1 When the outlier fall observed in the inflation rate at the end of 1998 in Azerbaijan (see in Figure 2) is eliminated through an impulse dummy variable, Azerbaijan no longer appears as an exception in the test results.2

As the first step of the estimation methodology used in the study, other univariate tests were computed in order to identify the integration characteristics of the seven inflation-rate (πt) series. Four different types of tests were used to determine the integration order of the series, i.e. Dickey and Fuller (1979) ADF, Elliot et al. (1996) DF-GLS, Phillips and Perron (1998) PP and Kwiatkowski et al. (1992) KPSS tests. It must be noted that it is the ‘unit root’ or the ‘non-stationarity’ null hypothesis that is being tested in the ADF, DF-GLS and PP tests while it is the ‘stationarity’ hypothesis in the KPSS test. Thus, the rejection of the null hypothesis implies stationarity in the ADF, DF-GLS and PP but non-stationarity in the KPSS test.

1 Note that the critical values used in the BDS tests are bootstrapped values in order to compensate for the divergence from the asymptotic normal distribution, which may occur in small samples or in series that have unusual distributions.

2 The LM, LB and BDS test statistics obtained for Azerbaijan after the elimination of the outlier are as follows: ARCH-LM(4)=8.9108 and ARCH-LM(12)=3.9856; Q2-LB(4)=20.468

and Q2-LB(12)=36.652; BDS(2)=0.0529, BDS(3)=0.1036, BDS(4)=0.1424, BDS(5)=0.1745 and

BDS(6)=0.1918. They are all statistically significant at 1 % level of significance.

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Figure 2. The rate of CPI inflation


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The next step of the estimation methodology is to fit an appropriate autoregressive (AR) representation for the monthly inflation rates. In this respect, a lag length is determined for the AR(s) model through testing the statistical significance of the highest lag s in a sequential reduction process, which is started from an AR(12) model down to an AR(1) model. Accordingly, the residuals of the chosen AR(s) model are initially tested for the presence of ARCH effects, serial correlation and linear or non-linear dependence through LM, LB and BDS tests, respectively.

The AR(s) models, in which the residuals are found relevant for ARCH-type modelling, are checked to determine whether or not they can be modelled through exploiting the alternative types in ARCH class models. The basic model to start with is the ARCH(1) model introduced by Engle (1982):

(1)

(2)

In the mean equation denoted by (1), vt is a white-noise process such that . Both the conditional and unconditional means of residuals εt are zero, while the conditional variance of εt is equal to ht. For a positive conditional variance, the parameters in equation (2) must satisfy the conditions α0 >0 i α1 ≥ 0.

Bollerslev (1986) proposed a generalized version of this ARCH(1) model, denoted by GARCH(1,1), in which the conditional variance equation is re-specified as

(3)

In the above equation, α1+α2<1 and α0 >0, α1, α2 ≥ 0 are the necessary and sufficient conditions for a covariance stationary model and for a non-negative conditional variance, respectively.

Subsequently, Nelson (1991) discussed a conditional variance that can capture the possible “leverage effect” through consideration of the sign of lagged residuals. This model is known as the exponential GARCH (EGARCH) model. The conditional variance of an EGARCH(1, 1) model is represented as

(4)

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A significant and negative α12 implies the presence of the “leverage effect”. Stationarity requires α2<1 to hold. However, the logarithmic formulation of ht guarantees a non-negative conditional variance, regardless of the signs of the parameters given in equation (4).

Finally, Ding et al. (1993) introduced the power ARCH (PARCH) model, in which a conditional variance with asymmetry is specified, such as

(5)

The simplest form of this PARCH(1,1) representation is the one obtained through the assumption of δ=1, as suggested by Taylor (1986) and Schwert (1989). Note that when δ=2, the PARCH(1,1) model is reduced simply to the GARCH(1,1) model.

The final step of the estimation methodology used in this study is to investigate the relationship between the rate of inflation and inflation-uncertainty through a Granger non-causality analysis. Accordingly, the estimate of the conditional variance series , which may be obtained from either of the equations (2)-(5), is used as a measure for inflation uncertainty, and subsequently exploited within the p-th order vector autoregressive [VAR(p)] framework for the Granger non-causality analysis:

(6)

(7)

In the single equations of the above VAR(p) model, which in fact are estimated by ordinary least squares separately, Granger non-causality hypotheses are tested with Wald statistics at lags p = 4, 8, 12 to account for short- and long-horizon differences, as done in the relevant literature. Tests are based on the null hypotheses of : β1=β2=…..=βp=0 in equation (6) and : φ1=φ2=…..=φp=0 in equation (7). The rejection of the former (latter) restriction means the rejection of the null hypothesis that πt ( ) does not Granger-cause (πt), which is accepted as evidence for the Friedman-Ball (Cukierman-Meltzer) hypothesis in the literature. However, if the Cukierman-Meltzer type of Granger causality exhibits an inverse relationship between πt and ht, such a finding is interpreted in terms of Holland’s (1995) proposition. The sign-consistency of the findings is checked with respect to the sign corresponding to the sum of the estimated βi or φi coefficients.


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Since the distributional assumptions in time series models are often violated, leading to inconsistent parameter estimates, a robust alternative estimation approach was used in this study while carrying out the Granger non-causality tests.

This approach, developed by Holmes and Hutton (1988), is argued to separate the causality from the functional form of the variables and from the assumptions of the homoscedasticity and normality of the errors.3 The corresponding test is known as the multiple rank F test. In this testing procedure, the Granger non-causality hypotheses are tested in equations (6) and (7) as usual, after replacing the data of and πt with their ranks, R( ) and R(πt):

(8)

(9)

While computing the ranks for equations (8) and (9), current and lagged values of and πt are treated as separate variables. Holmes and Hutton (1988) argued that the result of the multiple rank F test is invariant to strictly monotonic transformations and independent of the error distribution, and hence, a reliable substitute for Granger parametric tests for causal relationships. Moreover, Holmes and Hutton (1990b) found that in linear models with lagged dependent variables the test has significant power advantages both in smaller and larger samples when error distributions are non-normal.

4. Estimation Results

Computations of the ADF, DF-GLS, PP and KPSS test statistics, given in Table 4, show that all seven πt series are stationary in the given sample periods. However, the stationarity of the inflation series of the Kyrgyz Republic is not confirmed by the KPSS figure, despite the fact that the other three tests indicate stationarity for that series.

3 See Holmes and Hutton (1990a and 1992) for the use of this approach in testing the expenditure-income and money-income causalities, respectively.

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Table 4. Unit root and stationarity tests ADF DF-GLS PP KPSS

Armenia -8.25 b (0) -8.17 b (0) -8.23 b (3) 0.17 (9)Azerbaijan -7.91 b (0) -3.81 b (1) -8.09 b (8) 0.16 (9)Georgia -5.50 b (1) -5.18 b (1) -9.63 b (4) 0.07 (4)Kazakhstan -4.91 b (1) -3.17 b (1) -8.27 b (5) 0.09 (7)Kyrgyz Rep. -5.47 b (0) -5.12 b (0) -5.37 b (6) 0.48 b (12)Russian Fed. -8.23 b (0) -8.12 b (0) -8.91 b (13) 0.32 (12)Ukraine -5.61 b (0) -5.63 b (0) -5.76 b (10) 0.26 (12)Superscripts a, b and c denote 1, 5 and 10 % levels of statistical significance, respectively. Figures in parentheses show selected lags in ADF and DF tests and the bandwidth in PP and KPSS tests using Parzen Kernel.

The estimation results of the first part of our empirical investigation on the linkage between the rate of inflation and inflation-uncertainty are presented in Table 5. In order to achieve a proxy for inflation uncertainty for use in the next part, alternative ARCH-type specifications are estimated for each of the countries of interest except Georgia, where estimations do not lend support for any statistically valid conditional variance representation for the inflation rate.4 In fact, LM, LB and BDS test statistics computed for the AR(s) models (not reported here for brevity) already provide relevant evidence on the presence or absence of the ARCH effects and non-linear dependencies in these AR(s) models, consistent with the results given in Table 5. That is, for all countries except for the Kyrgyz Republic and Georgia, computed LM, LB and BDS statistics result in rejection of the null hypotheses of no ARCH effects, no serial correlation and no non-linear dependence, respectively. However, results are mixed for the Kyrgyz Republic. The residuals of the corresponding AR(12) model are found serially correlated and exhibit no ARCH effects, with a statistic which is slightly below the 10 % critical value, whereas no non-linear dependence is justified by the BDS test results. Consequently it was decided to include the Kyrgyz Republic in modelling the conditional variance of inflation.

In models, the outlier effects of the Russian crisis of August 1998 and other relatively extreme episodes of inflation are taken into account through impulse dummy variables, where relevant. The outlier inflation rates due to depreciations stemming from the Russian crisis are observed in the following countries and

4 All computations are carried out with the Eviews 5.1 software. Estimations are carried out with the maximum-likelihood approach based on the Marquardt optimization algorithm.


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months: Georgia, 1998:12; the Kyrgyz Republic, 1998:11; the Russian Federation, 1998:8, 1998:9, 1998:12; Ukraine, 1998:10. However, inflation sharply decreased in Azerbaijan in 1998:12 owing to the extremely tight monetary policy and increased in Kazakhstan in 1999:4 and 1999:6 due to depreciations of the Tenge following the transition from the crawling-peg to the freely floating exchange rate system. Additionally, the slowdown of the retail trade in the Kyrgyz Republic and the tense political situation in Georgia, together with the rise in corn prices in the global markets, led to sharp increases in inflation in Kyrgyzstan in 1999:6 and in Georgia in 2003:11, respectively. All these outlier effects are modelled with impulse dummy variables in relevant regressions, the estimates of which are given in Table 5.

In the attempts to match models from the ARCH class to each of the inflation series, none of the conditional variances is suitable to be modelled by an EGARCH process. However, the PARCH process is suitable for the series of the Kyrgyz Republic and Ukraine, in addition to the ARCH and GARCH processes for the latter. ARCH and GARCH processes seem to fit well to the inflation series of Armenia, Azerbaijan and the Russian Federation, whereas only the GARCH process fits the series of Kazakhstan. However, according to the residual-diagnostic tests given in the lower part of Table 5 (continued), although the LM and LB tests do not indicate any ARCH effects in the residuals, there seems to be a nonlinear dependence left unmodelled in the GARCH specification of the Russian Federation. This can be observed both in the BDS test applied to the standardized-residual series and in the plot of the estimated GARCH variance series of the Russian Federation (GARCH_RUS) given in Figure 3.5 When the LM and LB tests results are evaluated together with those previously obtained from the residuals of the pure AR(s) models of inflation, there seem to be no ARCH effects left unmodelled after ARCH, GARCH or PARCH structures are fitted to the residuals. Another failure observed in the diagnostic tests is the violation of the normality assumption in all models, except for the GARCH model estimates of the Russian Federation. However, the insignificant LM, LB and BDS test statistics reported in Table 5 (continued) imply that the detected non-normality is not associated with any remaining ARCH effects or non-linearity but may be

5 We are aware of the fact that the asymptotic distribution of the BDS statistic obtained from the standardized residuals of the ARCH-type models should not be used for inference (Hsieh, 1991:1870). However, Monte Carlo simulations necessary to obtain the appropriate critical values are computationally demanding, and therefore, will be beyond the scope of this study. For this reason, BDS statistics given in Table 5 were used only as a naïve indicator of misspecification due to unmodelled non-linear dependence.

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0.00

26 a

(0.0

008)

0.00

21 a

(0.0

007)

0.00

40 a

(0.0

003)

0.00

42 a

(0.0

005)

0.00

37 a

(0.0

009)

0.00

31 a

(0.0

009)

0.00

33 a

(0.0

009)

π t-1

0.46

84 a

(0.0

995)

0.28

52 a

(0.0

870)

0.21

14 a

(0.0

299)

0.19

68 a

(0.0

619)

0.13

65 b

(0.0

649)

0.32

01 a

(0.0

426)

0.40

79 a

(0.0

741)

0.07

99 a

(0.0

053)

0.11

03 a

(0.0

335)

0.50

02 a

(0.0

853)

0.53

34 a

(0.0

957)

0.49

19 a

(0.0

839)

π t-2

-0.1

126

(0.1

201)

-0.0

409

(0.0

941)

0.10

95 a

(0.0

289)

0.10

28 b

(0.0

403)

0.23

04 a

(0.0

676)

0.04

86(0

.083

4)-0

.058

7(0

.104

2)0.

1032

a

(0.0

062)

0.12

17 a

(0.0

249)

-0.1

616 c

(0.0

954)

-0.0

863

(0.0

975)

-0.0

799

(0.0

961)

π t-3

0.07

39(0

.094

9)0.

0179

(0.0

917)

-0.0

130

(0.0

454)

0.03

59(0

.039

2)0.

0463

(0.0

377)

-0.0

314

(0.0

612)

0.07

45(0

.102

4)0.

1931

a

(0.0

416)

0.14

77 a

(0.0

398)

0.18

52 b

(0.0

887)

0.15

04(0

.113

9)0.

1426

(0.1

099)

π t-4

-0.0

882

(0.0

869)

-0.0

579

(0.1

003)

0.06

47(0

.042

2)0.

0238

(0.0

271)

-0.0

523

(0.0

363)

-0.0

441

(0.0

727)

-0.0

220

(0.0

996)

0.12

77 a

(0.0

227)

0.13

26 a

(0.0

142)

π t-5

-0.0

236

(0.0

913)

-0.0

908

(0.1

139)

-0.0

628

(0.0

469)

-0.0

922 a

(0.0

249)

-0.1

347 a

(0.0

374)

0.11

74 b

(0.0

537)

0.17

58 c

(0.1

016)

π t-6

0.07

60(0

.073

4)-0

.021

9(0

.085

9)0.

0522

(0.0

419)

0.02

51(0

.024

8)-0

.046

5(0

.049

9)0.

0317

(0.0

768)

-0.0

094

(0.0

982)

π t-7

-0.0

415

(0.0

659)

-0.0

261

(0.0

725)

0.00

11(0

.036

5)0.

0254

(0.0

197)

-0.0

974

(0.0

756)

0.09

71(0

.064

0)0.

0458

(0.0

805)

π t-8

0.00

33(0

.058

0)-0

.044

7(0

.069

3)-0

.046

6(0

.046

5)0.

0020

(0.0

216)

0.08

49(0

.060

3)-0

.061

0(0

.080

0)0.

0539

(0.0

715)

π t-9

-0.0

554

(0.0

551)

-0.0

262

(0.0

576)

-0.0

508

(0.0

417)

-0.0

429

b

(0.0

172)

0.06

91(0

.045

8)0.

1045

(0.0

773)

0.03

39(0

.075

3)

π t-10

0.06

05(0

.057

1)0.

0697

(0.0

590)

0.04

49(0

.037

8)0.

0199

(0.0

214)

-0.0

009

(0.0

482)

-0.0

367

(0.0

529)

-0.0

266

(0.0

799)

π t-11

-0.1

361 b

(0.0

585)

-0.0

627

(0.0

639)

0.05

44(0

.033

4)0.

0404

c

(0.0

218)

-0.0

124

(0.0

409)

0.03

69(0

.062

6)

π t-12

-0.1

288 c

(0.0

698)

-0.2

357 a

(0.0

773)

-0.0

451

(0.0

354)

-0.0

359

(0.0

223)

-0.1

634 b

(0.0

724)

-0.1

533 a

(0.0

578)

D1

-0.0

579 a

(0.0

005)

-0.0

578 a

(0.0

006)

0.09

63 a

(0.0

019)

0.04

18 a

(0.0

013)

0.04

22 a

(0.0

033)

0.02

81 a

(0.0

004)

0.02

84 a

(0.0

006)

0.04

02 a

(0.0

017)

0.03

94 a

(0.0

020)

0.03

79 a

(0.0

039)

D2

0.04

12 a

(0.0

010)

0.03

75 a

(0.0

041)

0.04

31 a

(0.0

077)

0.30

79 a

(0.0

003)

0.30

69 a

(0.0

011)

D3

0.02

27 a

(0.0

135)

0.03

17 a

(0.0

120)


61

Tabl

e 5.

(con

tinu

ed)

Arm

enia

Aze

rbai

jan

Geo

rgia

Kaz

akhs

tan

Kyrg

yz R

ep.

Russ

ian

Fede

ratio

nU

krai

ne

ARC

HG

ARC

HA

RCH

GA

RCH

AR

GA

RCH

PARC

HA

RCH

GA

RCH

ARC

HG

ARC

HPA

RCH

c5.

4×10

-5 a

(1.6×1

0-5)

2.4×

10-5

(1.9×1

0-5)

6.3×

10-6

a

(1.4×1

0-6)

1.8×

10-6

(1.1×1

0-6)

1.0×

10-6

(8.3×1

0-7)

0.00

21(0

.001

7)9.

4×10

-6 a

(2.3×1

0-6)

2.8×

10-7

c

(1.6×1

0-7)

3.5×

10-5

a

(7.3×1

0-6)

1.2×

10-5

(1.3×1

0-5)

0.00

14(0

.001

4)

ε2t-1

0.69

28 b

(0.2

751)

0.45

60 b

(0.1

909)

0.66

31 b

(0.3

031)

0.57

79 c

(0.3

126)

0.24

24 c

(0.1

324)

0.77

15 a

(0.1

969)

0.11

81 b

(0.0

624)

0.34

67 c

(0.1

898)

0.13

86 c

(0.0

784)

h t-1

0.38

44 b

(0.1

891)

0.41

13 b

(0.1

820)

0.64

57 a

(0.1

399)

0.84

36 a

(0.0

532)

0.62

57 b

(0.2

804)

|εt-1

|0.

2574

c

(0.1

315)

0.17

12 b

(0.0

718)

0.51

79 b

(0.2

632)

0.67

45 a

(0.2

126)

AIC

-6.1

482

-6.1

794

-7.9

877

-7.9

644

-7.0

579

-8.6

887

-6.8

359

-7.6

470

-7.7

687

-7.0

214

-7.0

048

-7.0

241

log-

L39

3.12

396.

0338

7.43

387.

3436

0.84

515.

5940

0.81

468.

8247

7.12

435.

3043

5.29

436.

47

LM(4

)1.

9300

0.66

980.

4824

1.25

881.

2323

0.63

481.

6152

0.43

451.

1226

1.31

901.

1779

0.94

72LM

(12)

0.90

690.

4228

1.56

251.

1544

1.16

210.

6019

0.77

891.

3648

0.65

090.

6993

0.66

760.

6255

LB(4

)8.

3940

c3.

1085

3.12

906.

9406

4.98

442.

7964

6.91

251.

7675

5.40

765.

5340

5.24

064.

3829

LB(1

2)12

.432

6.44

5416

.583

13.8

2716

.789

5.94

5010

.341

19.6

72 c

10.8

847.

7475

7.61

637.1

647

BDS(

2)-0

.005

8-0

.007

90.

0062

0.01

000.

0135

0.00

29-0

.000

9 0

.032

4 b 0

.001

6 a0.

0002

0.00

310.

0004

BDS(

3)-0

.000

1-0

.005

3-0

.001

7-0

.001

90.

0116

0.00

09-0

.001

4 0

.117

6 0

.021

6 b-0

.001

40.

0059

-0.0

035

BDS(

4)0.

0161

0.00

330.

0050

3.4×

10-5

-0.0

034

0.00

090.

0006

0.1

400

0.0

748 c

-0.0

005

0.00

48-0

.008

4BD

S(5)

0.02

180.

0036

0.01

010.

0012

-0.0

091

0.00

58-1

.9×1

0-5 0

.218

8 0

.322

40.

0105

0.01

42-0

.000

6BD

S(6)

0.02

330.

0037

0.01

620.

0024

-0.0

140

0.00

050.

0037

0.1

228

0.2

632

0.01

400.

0176

0.00

30

JB10

.293

a18

.589

a3.

7460

5.90

37 c

3.35

528.

7895

b26

.058

a10

.682

a0.

3139

8.65

31 b

13.1

75 a

12.1

42 a

Supe

rscr

ipts

a , b and

c den

ote

1, 5

and

10

% le

vels

of s

tatis

tical

sign

ifica

nce,

resp

ectiv

ely.

Fig

ures

in p

aren

thes

es a

re B

olle

rsle

v-W

oold

ridg

e he

tero

sked

astic

ity-c

onsi

sten

t sta

ndar

d er

rors

. AIC

: Aka

ike

Info

rmat

ion

Cri

teri

on; l

og-L

: log

-lik

elih

ood;

LM

: Lag

rang

e m

ultip

lier t

est f

or A

RCH

effe

cts i

n re

sidu

als;

LB: L

jung

-Box

Q-s

tatis

tic fo

r tes

ting

seri

al

corr

elat

ion

in sq

uare

d re

sidu

als;

BDS:

A p

ortm

ante

au te

st to

che

ck w

heth

er o

r not

the

resi

dual

s are

inde

pend

ent a

nd id

entic

ally

dis

trib

uted

(boo

tstr

appe

d cr

itica

l val

ues a

re u

sed)

; JB

: Jar

que-

Bera

stat

istic

for t

esti

ng th

e re

sidu

al n

orm

ality

.

62


associated with outliers that are not captured by the impulse dummy variables already used in the estimations.

Given the estimates of the inflation models in Table 5, hypotheses concerning the relationship between inflation and inflation-uncertainty are tested through Granger-causality tests. These tests are carried out between the rate of inflation

Figure 3. Estimated conditional variance series (ht)


63

(πt) and the conditional-variance ( ) series, with and without rank transformation to enable the comparison of the causality implications between the conventional and Holmes-Hutton approaches. In this respect, bivariate VAR estimations at four, eight and twelve lags are exploited to test the following two null hypotheses of the Granger non-causality:

Table 6. Granger-causality testslags ARCH GARCH PARCHp

Armenia4 13.71 a (+) 7.04 24.14 a (+) 5.188 13.29 9.70 21.43 a (+) 10.0512 16.94 16.94 19.25 c (+) 21.23 b (+)

Azerbaijan4 1.86 67.37 a (-) 2.62 78.08 a (-)8 5.34 64.02 a (-) 7.30 75.49 a (-)12 6.70 57.02 a (-) 9.07 74.71 a (-)

Kazakhstan4 4.32 2.238 5.01 31.08 a (+)12 27.07 a (+) 94.53 a (+)

Kyrgyz Rep.4 7.97 c (+) 3.048 23.68 a (+) 13.38 c (-)12 28.07 a (+) 19.67 c (-)

Russian Fed.

4 0.55 17.72 a (+) 9.06 c (+) 32.55 a (+)8 23.37 a (+) 32.93 a (+) 20.16 a (-) 50.48 a (+)

12 26.99 a (+) 122.25 a (+) 25.56 b (+) 128.14 a (+)

Ukraine4 11.20 b (+) 2.58 10.89 b (+) 2.73 10.77 b (+) 1.108 14.92 c (+) 3.88 14.71 c (+) 4.07 15.26 c (+) 1.7912 17.24 4.04 15.36 4.22 16.06 2.05

Superscripts a, b and c denote 1, 5 and 10 % levels of statistical significance respectively. Granger-causality test statistics have the χ2-distribution with the degrees of freedom equal to p. They are computed with keeping the sample periods fixed for p=4, 8 and 12 in each case. (+) and (-) indicate the signs of the summed coefficients on lagged variables corresponding with each hypothesis.

(1) : Inflation does not Granger-cause inflation uncertainty.

(2) : Inflation uncertainty does not Granger-cause inflation.

64


Computed statistics for testing the Granger non-causality hypotheses and the signs corresponding to the sum of the coefficients on lagged variables are presented in Table 6. However, as seen in Table A in the Appendix, the causality regressions with which the tests are performed cannot pass the residual diagnostics. Normality of the errors is violated for all cases while serial correlation exists for some of the causality regressions. The empirical literature concerning the inflation and inflation-uncertainty relationship usually ignores the residual diagnostics of the causality regressions. In contrast, this study follows a robust approach over alternative distributions of errors for non-causality testing. The corresponding test results are given in Table 7.

When the figures in Table 6 and Table 7 are compared, it is seen that most of the outcomes change after the rank-transformed data are used instead of the original. The rejection of the hypothesis together with a positive sign

Table 7. Multiple-rank F test for causalitylags ARCH GARCH PARCHp

Armenia4 7.68 2.09 3.75 1.458 9.28 7.69 6.82 8.2712 13.87 10.89 8.49 15.66

Azerbaijan4 3.88 3.63 1.82 10.78 b (-)8 10.09 6.58 15.87 b (+) 15.04 c (-)12 23.73 b (+) 10.99 16.19 16.71

Kazakhstan4 1.54 0.988 3.58 5.3012 3.93 7.36

Kyrgyz Rep.4 4.87 10.52 b (+)8 9.43 12.3312 10.49 21.25 b (+)

Russian Fed.

4 3.79 1.09 1.73 8.20 c (+)8 6.54 5.35 4.75 15.48 c (+)12 28.72 a (+) 13.82 13.09 19.44 c (+)

Ukraine4 8.21 c (+) 0.76 4.72 3.58 5.59 1.908 13.85 c (+) 3.42 10.46 3.85 11.23 2.7212 13.98 8.40 14.25 5.07 16.27 6.07

Superscripts a, b and c denote 1, 5 and 10 % levels of statistical significance respectively. χ2(p) version of the multiple rank F test is used for comparability with results given in Table 6. The test statistics are computed with keeping the sample periods fixed for p=4, 8 and 12 in each case. (+) and (-) indicate the signs of the summed coefficients on lagged variables corresponding with each hypothesis.


65

now lends support for the Friedman-Ball hypothesis only for Azerbaijan, the Russian Federation and the Ukraine with respect to the multiple rank F test, whereas it lends support for all countries except for Azerbaijan with respect to the conventional test of non-causality. On the other hand, when causality that runs from uncertainty towards inflation is considered, the hypothesis is rejected for Azerbaijan, the Kyrgyz Republic and the Russian Federation according to the multiple rank F test. For Armenia and Kazakhstan, conclusions based on the conventional approach, which were in favour of the rejection of the hypothesis, are not supported by the alternative approach. Moreover, according to the multiple rank F test results, there appears to be no other statistically significant relationship between inflation and inflation uncertainty for Armenia and Kazakhstan.

Rejection of the hypothesis for Azerbaijan, the Kyrgyz Republic and the Russian Federation lends support for the Cukierman-Meltzer hypothesis only for the last two. For Azerbaijan the rejections favour Holland’s (1995) proposition that greater inflation uncertainty may lead to lower inflation, as indicated by the negative signs found for the summed coefficients. This is one of the conclusions unchanged with respect to both the conventional test and the multiple rank F test. This outcome implies the presence of a strong motive for stabilization policy of the monetary authorities in Azerbaijan. In fact, this finding appears to be consistent with the overview given in the second part of the study. That is, of the seven economies in question (including Georgia), Azerbaijan is the only country that managed to keep inflation at moderate levels during the transition to a market economy, and that reduced it sharply during the subsequent Russian crisis.

Finally, two other striking differences appear in the results of the Holmes-Hutton approach. Firstly, the multiple rank F test provides additional evidence in favour of the Friedman-Ball hypothesis in the case of Azerbaijan. Secondly, the statistical evidence in favour of the Holland proposition in the case of the Kyrgyz Republic is transformed into a Cukierman-Meltzer type of causality with the multiple rank test results.

To summarize, when the power advantages of the multiple rank F test demonstrated by Holmes and Hutton (1990b) are considered, the results presented in Table 7 appear to be more reliable than those in Table 6.

66


5. Concluding Remarks

This paper examines the possible relationship between inflation and inflation-uncertainty for seven transitional economies in their post-hyperinflation periods. The two principal published models (the Friedman-Ball and the Cukierman-Meltzer hypotheses) were tested using a methodology that exploits the conditional-variance estimation and Granger non-causality procedures. Non-causality tests were performed not only with the conventional tools but also with the multiple rank F test, which is argued to be robust to different functional forms of the variables, and to be more powerful when heteroskedasticity and non-normality exist. According to this approach, the Friedman-Ball hypothesis (inflation precedes inflation uncertainty) is supported in Azerbaijan, the Russian Federation and the Ukraine. Support is provided for the Cukierman-Meltzer hypothesis in the Kyrgyz Republic and also in the Russian Federation using a different model. Uniquely in Azerbaijan, greater inflation uncertainty preceded lower inflation rates, which is attributed to the strong monetary stabilization policies pursued in the Azeri economy between 1996 and 2003.

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Appendix

Table A. Residual diagnostics for causality regressionslags(p) test

ARCH GARCH PARCH

Armenia

4LBLMJB

0.18110.17663294.1a

4.33513.54666.9176 b

0.12940.15472083.7 a

4.63893.51798.9727 b

8LBLMJB

0.40810.36042558.2 a

10.1568.16087.8485 b

0.37650.34751053.6 a

9.33677.58789.1420 b

12LBLMJB

0.69980.59282456.5 a

8.13297.649618.747 a

1.01260.8082927.39 a

8.49337.366218.118 a

Azerbaijan

4LBLMJB

0.75080.67262127.7 a

5.70275.8954176.84 a

0.33000.28542510.6 a

5.94476.1133144.04 a

8LBLMJB

1.20071.00961616.0 a

6.56846.0472219.25 a

0.72890.63661745.4 a

7.39696.8967183.33 a

12LBLMJB

0.75590.38131353.4 a

6.56442.8834168.96 a

0.44260.42031314.5 a

6.52193.541984.037 a

Kazakhstan

4LBLMJB

0.36760.29251014.8 a

16.027 a

12.844 b 1416.5 a

8LBLMJB

4.59149.0417270.79 a

46.521 a

44.885 a

254.25 a

12LBLMJB

3.70935.0394304.68 a

28.757 a 9.19886.7229 b

Kyrgyz Rep.

4LBLMJB

29.424 a

28.561 a

422.05 a

27.567 a

21.919 a

55.798 a

8LBLMJB

18.156 b 15.706 b

114.94 a

12.4205.43966.6169 b

12LBLMJB

12.85417.84893.408 a

18.942 c

12.2891.9932

Thornton, J. (2007), “The Relationship between Inflation and Inflation Uncertainty in Emerging Market Economies”, Southern Economic Journal, Volume 73, Issue 4, pp. 858-870.

Thornton, J. (2008), “Inflation and Inflation Uncertainty in Argentina, 1810-2005”, Economics Letters, Volume 99, Issue 3, pp. 247-252.

Ungar, M. and B. Zilberfarb (1993), “Inflation and its Unpredictability-Theory and Empirical Evidence”, Journal of Money, Credit, and Banking, Volume 25, Issue 4, pp. 709–720.


71

Table A. (continued)

Russian Fed.

4LBLMJB

12.136 b

14.214 a

415.89 a

0.04420.040121150 a

12.210 b

13.611 a

514.92 a

0.09990.089213715 a

8LBLMJB

45.754 a

50.621 a

314.37 a

0.04402.881811927 a

34.590 a

31.154 a

641.89 a

0.30433.93077515.5 a

12LBLMJB

46.705 a

42.358 a

288.00 a

0.471422.079 b

1513.9 a

45.550 a

41.939 a

319.26 a

0.562624.402 b

1075.3 a

Ukraine

4LBLMJB

2.05971.8427453.69 a

6.74547.8754 c

85.204 a

2.25712.0324435.03 a

6.83728.1024 c

83.013 a

6.10515.594484.566 a

7.35428.6776 c

87.692 a

8LBLMJB

4.00404.5895309.21 a

6.81359.316486.328 a

2.11432.5984339.18 a

6.57468.962789.936 a

4.83592.795955.971 a

7.375612.38789.908 a

12LBLMJB

5.04886.8396224.65 a

7.73417.944198.049 a

3.24143.7334262.08 a

7.20127.918896.469 a

12.71412.78931.164 a

7.97039.255892.802 a

Superscripts a, b and c denote 1, 5 and 10 % levels of statistical significance respectively. LB denotes the Ljung-Box Q-statistics to test the null of ‘no serial correlation’ in the squared residuals. LM is the Lagrange multiplier statistic for testing the null of ‘no ARCH effects’ in the residuals. JB is the Jarque-Bera statistic to test whether the series is normally distributed.

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