Feng Jiang for The 2004 Moffatt Prize in Economics

Introduction

One of the most important issues in recent research in theoretical and empirical macroeconomics is to explore the relationship between the inflation rate and inflation uncertainty. A large number of previous and recent works have established the relationship between the inflation and inflation uncertainty based on the different background and different data set. Contradictory empirical evidence suggests that both positive and negative relationship may exist between inflation and inflation uncertainty. Moreover, there may be insignificant like between them. According to Friedman (1997), a positive influence exists between the inflation rate and inflation uncertainty, that is, higher inflation will cause the higher inflation uncertainty by the erratic policy responses. On the contrary, Pourgerami and Maskus (1990) have argued that there may be a negative influence between them. Engle (1983) and Cosimano and Jansen (1988) find little evidence of a link between them. The opposite direction of causality has also been analyzed in the theoretical literature. Cukierman and Meltzer (1986) shows that the increase in uncertainty about money growth and inflation will induce the higher average inflation rate because of the behavior of the policy-maker to stimulate the output growth. However, Holland (1995) claims that the increase of the inflation uncertainty will lead to the lower average inflation rate by the tight monetary policy response to minimize the real costs of inflation uncertainty. Given the above theoretical ambiguity, it is quite clear that the ambiguous statistical evidence plays a significant role in the determination of the relationship between the inflation and inflation uncertainty.

This paper mainly focuses on three main aspects. Firstly, it analyzes the inflation dynamics of eight countries belonging to the east and south Asia and of the Australia. One group of the countries consists of Australia, Hong Kong, Japan, Korea and Singapore that represent the developed countries and region in the East Asia Area. While the other group consisting of India, Indonesia, Malaysia and Thailand represents the developing countries in the south Asia. The first group except Australia belonging to the newly-developed countries in the east Asia is of distinct interest and the second group representing most other developing countries in the south Asia gives us a main description of the inflation and inflation uncertainty in the third world. We estimate the two main parameters measuring the degree of persistence in inflation and inflation uncertainty by using the ARFIMA-FIGARCH process. The model, developed in Baillie et al. (2002), provides a general and flexible framework with which to study a complicated process like inflation series. In other words, under the ARFIMA-FIGARCH framework, we can easily and flexibly handle the dual long memory behavior encountered in inflation series.

Secondly, it investigates the possible existence of heterogeneity in inflation dynamics across the nine countries in the east and south Asia area. Inflation differentials have important implications for the design of the optimal monetary policy. For example, as Benigno and Lòpez-Salido (2002) point out, an inflation targeting policy that assigns higher weight to countries with higher degrees of persistence benefits those countries since once the policy of the central bank is credible, it produces lower inflation rates for them simply because it cares more about those inflation rates.

Thirdly, it shed further light on the current issue of the relationship between inflation and inflation uncertainty. As has been mentioned by Grier and Perry (1998), the two-step method is employed in this paper. First, by estimating the ARFIMA-FIGARCH model, the conditional variance of inflation is generated and regarded as the proxy of inflation uncertainty. Second, with the known series of inflation uncertainty, we perform the Granger- causality tests to find the evidence on bi-directional causality relationship between inflation and inflation uncertainty.

Several interesting findings emerge from our empirical work. Our first finding is that seven out of the nine chosen countries’ inflation rates have the long memory properties in both their first and second conditional moments. This empirical evidence is consistent with the evidence provided by Baillie et al. (2002) for eight industrial countries.

The second result that comes out from this study is the existence of heterogeneity in inflation dynamics across the east and south Asia countries. These countries are falling into two groups in terms of the difference in the dynamics of the second conditional moment of their inflation rates. The first group of countries includes Australia, Hong Kong, Korea, Singapore, India, Indonesia and Malaysia that are characterized by the presence of long memory in the inflation uncertainty. The second group of countries includes Japan and Thailand characterized by an IGARCH behavior in the second conditional moment of inflation rates. This finding is of some significance since inflation differentials are not irrelevant for monetary policy.

Thirdly, we provide strong evidence that increased inflation raises inflation uncertainty for most of those developed countries and region, confirming the theoretical predictions made by Friedman. On the other hand, the evidence for the causal effect from inflation uncertainty to inflation reveals that for the developed countries, both Cukierman-Meltzer theory that greater uncertainty about money growth and inflation causes a higher average rate of inflation and the stabilization hypothesis of Holland that an increase in inflation uncertainty will invite a decrease in inflation rate are well supported by the different developed countries’ empirical evidence. The result brings out an important implication about the different policy-maker’s attitude and monetary policy in east and south Asia developed countries in addition to the difference in the economic sizes of the countries. However, for most of those developing countries in south Asia, the mixed impacts are found for the bi-directional causality relationship between inflation and inflation uncertainty. In other words, the increased inflation uncertainty significantly affects average inflation in developing countries but not all in the same manner. These differential responses to inflation uncertainty are not correlated with measures of central bank independence.

The rest of the paper is organized as follows. Section 2 briefly summarizes several empirical studies that investigate the short-term inflation dynamics, and briefly reviews some of the important advances in the theoretical modeling of inflation dynamics. In section 3, we introduce two different time series models for inflation and uncertainty about inflation and discuss its merits. Section 4 discusses the economic theory concerning the link between inflation rates and inflation uncertainty and the previous empirical testing of the inflation and inflation uncertainty hypothesis. The empirical results are reported in Section 5, and Section 6 points out several possible extensions of the time series model for inflation. Section 7 gives out the summary remarks and conclusions.

Theoretical Analysis

Long Memory Process

A lot of previous studies have concerned with the properties of the univariate time series representation of monthly inflation. In those studies, a popular topic is concerned with the degree of persistence of shocks and is related to its debate about the possible existence of a unit root in inflation rate. Some of them, for example, Rose (1988) and Grier and Perry (1998), claim that the inflation rate is I (0) stationary process, while most others, like Nelson and Schwert (1977), Ball and Cecchetti (1990), Kim (1993), Banerjee et al. (2001) and Banerjee and Russell (2001), find the evidence of a unit root in the inflation rate. Kirchgässner and Wolters (1993) obtain the mixed results for four European countries and the U.S. From the previous studies, it is obvious that most of them provide the evidence of a unit root in the inflation rate.

There are quite a lot of studies specialized in examining the characteristics of US inflation rate. Klein (1976) and Nelson and Schwert (1977) impose a unit root on the inflation process; while Ball and Cecchetti (1990) and Kim (1993) model inflation as a transitory and a permanent component, which is represented as a random walk. Barsky (1987) and Brunner and Hess (1993), using quarterly and monthly data respectively, argue that U.S. inflation rate contains a unit root subsequent to 1960, but is covariance stationary prior to this time. Evans and Watchel (1993) develop a time series model of inflation to describe the behavior of inflation uncertainty that take account of the factor of changing inflation regime. It switches the inflation series from the pure transitory shocks in the late 1960s to purely permanent shocks in 1970s, and back to transitory shocks in the late 1980s. According to Evans and Watchel (1993), Holland (1995) performs three different Granger Causality tests in that three periods between the two variables, each corresponding to one of the three different assumptions.

The concept of long memory and fractionary Brownian motion is originally developed by Hurst (1951) and Mandelbrot Van Ness (1968). However, the ideas were essentially operationalized for applications with discrete time representations since around 1980 by Granger (1980), Granger (1981), Granger and Joyeux (1980), and Hosking (1981).

In the recent studies, much of the empirical evidence supports that inflation series is fractionally integrated with a differencing parameter that is significantly different from zero and unity. For US monthly inflation data, Bachus and Zin (1993) find there is a fractional root in it. They conjecture that aggregation across agents with heterogeneous beliefs result in the long memory in inflation rate. Furthermore, they show that the fractional difference process describes the short-term interest rates well and suggest that the fractional unit root in the short rates is inherited from money growth and inflation. Hassler (1993) and Delgado and Robinson (1994) provide the strong evidence in favor of the presence of long memory properties in the Swiss and Spanish inflation rates respectively. Subsequently, Hassler and Wolters (1995), Baillie, Chung and Tieslau (1996), and Baum, Barkoulas and Caglayan (1999) have all found evidence that inflation is fractionally integrated and the autocorrelation and impulse response weights of inflation exhibit very slow hyperbolic decay.

All the findings in current studies suggest that the traditional ARMA and ARIMA specifications are incapable of imparting the persistence of inflation rate existing in the data. Instead of dealing it with I (0) process or I (1) process, we turn to I (d) process which has a better description of inflation series.

Although all the previous works have provided the consistent evidence across time periods and countries that inflation rate exhibits long memory property in its first conditional moment with the long memory parameter differing from zero and unity significantly, the property of time-dependent heteroskedasticity in the second conditional moment of the inflation rate is not explored until Baillie et al. (1996), in which they use the ARFIMA-GARCH model to describe the inflation dynamics for ten countries and find strong evidence of long memory with mean reverting behavior in all countries except Japan. With the same ARFIMA-GARCH type models, Hwang (2001) also find the strong evidence of long memory property in the inflation dynamic process for US monthly inflation data.

As is shown in those studies, the sum of the estimated GARCH (1,1) parameters is nearly close to one. On one hand, it implies the second conditional moment of inflation rate may possess integrated GARCH (IGARCH) behavior. For example, Baillie et al. (1996) shows that inflation series are completely persistent in its conditional variance for all ten countries. On the other hand, it may shade the underlying facts the existence of the long memory property. Baillie, Bollerslev and Mikkelsenthe (1996) apply the Monte Carlo simulations and find that it maybe easily mistaken for IGARCH behavior when the data are generated from a process exhibiting long memory behavior in its conditional variance. Furthermore, Baillie (2002) focus on the long memory specification and firstly explore the long memory property in the first and second conditional moments of inflation rates simultaneously. They employ the ARFIMA-FIGARCH process much better describing the inflation rates and find that the inflation series for many industrial countries exhibit the apparent long memory properties, where the integrated order is significantly differing from zero and unity and lies between them. Subsequently, Conrad and Karanasos (2002) use ARFIMA-FIGARCH model to estimate the monthly CPI inflation rates for the US and the UK and obtain the similar conclusion with Baillie (2002).

ARFIMA-FIGARCH PROCESS

In baillie (2002), he shows that the long memory property not only exists in the mean equation, but also exists in the second moment of inflation rate. In particular, the squared and absolute values of inflation residuals, from applying a fractional filter to the conditional mean, also possess long memory property. Hence, it is possible to model the conditional variance of inflation as a long memory autoregressive conditional heteroscedastic process. Together with the long memory conditional mean equation, it creates the joint model that incorporates long memory in both conditional mean and conditional variance of inflation, namely, ARFIMA-FIGARCH process. At the same time, he also finds that except the US inflation series from Jan, 1967 through Sept, 1998, all other countries’ inflation series exhibit the considerable seasonality. In order to avoid the higher order seasonal autoregressive moving average structure with more parameters, he introduces the seasonal ARFIMA model that produces a more parsimonious model.

Based on the above background, we describe the time series model for inflation and inflation uncertainty. Here, we denote the inflation rate by and define the conditional mean equation as:

,

Where  and have all their roots outside the unit circle with  This is the inflation rate following an ARFIMA (p, d_t, 24) specification. The fractional differencing operator, , has a binomial expansion conveniently expressed in terms of the hypergeometric function,

with  and F (a, b; c; z) is the hypergeometric function defined as:

F (a, b; c; z)

Where  denotes the Gamma function. Here we assume that the error term is serially uncorrelated, conditionally normal with mean zero and variance . That is | W_t-1 ~ N (0, h_t), where W_t-1 is the information set up to time t-1. A long memory conditional variance process can be set up from the same foundations as the ARCH model of Engle (1982). It is natural to define a discrete time, real valued stochastic process ,

where z_t is i.i.d. with E_t-1 (z_t) = 0 and Var_t-1 (z_t) = 1. The variable h_t is a time-varying, positive, and measurable function of the information set at time t-1, denoted by W_t-1 , and is known as an ARCH process. The FIGARCH process for conditional variance of inflation hence is defined as:

where ,  and .

If ht = w, a constant, the process reduces to the ARFIMA (p, d, 24) model. Then the inflation rate  will be covariance stationary and invertible for  and  is fractionally integrated of order d, or I(d). When , the inflation process does not have a finite variance and is a non-stationary process, but for , the impulse response weights are finite, which indicates the shocks to the level of the inflation series are mean reverting. In particular, when  the inflation series have a unit root and have a complete persistence. In this case, it would be more difficult for policy-maker to make a optimal decision. As mentioned by McCallum (1988), the non-stationary inflation process will complicate the derivation of optimal monetary policy rules.

The ARFIMA-FIGARCH model has a distinctive feature. It allows us to estimate the degree of persistence in both inflation and inflation uncertainty simultaneously. It also has the advantage of keeping the analytical elegance of the ARMA-GARCH model while enhancing its dynamics. In other words, ARFIMA-FIGARCH model has at least two important implications for our understanding of inflation and inflation uncertainty. First, it focus on the long memory aspect of the inflation rate and provides an empirical measure of inflation uncertainty that accounts for long memory in the second conditional moment of the inflation process. Second, it allows for a more systemic comparison of many possible models that can capture the features of the inflation series.

ARMA-APARCH PROCESS

Ding (1993) investigates a long memory property of the stock market returns series and finds not only there is substantially more correlation between absolute returns than returns themselves, but the power transformation of the absolute return r also has quite high autocorrelation for long lags. In order to capture the long memory property of the conditional variance process, Bollerslev (1986) introduced the GARCH (p, q) model, which defines the conditional variance equation as follows:

where

Taylor (1986) models the conditional standard deviation function instead of conditional variance. Schwert (1989), following the argument of Davidian and Carroll (1987), models the conditional standard deviation as a linear function of lagged absolute residuals. The Taylor / Schwert GARCH (p,q) model defines the conditional standard deviation equation as follows:

It is found from Monte-Carlo study that both Bollerslev’s GARCH and Taylor / Schwert’s model with appropriate parameters can produce the special pattern of autocorrelation existing in many stock market returns data. Under the confusion of the selection of these two different models, a more general class of ARCH model, Asymmetric Power ARCH denoted as APARCH, is developed by Ding (1993) which includes Bollerslev’s GARCH, Taylor / Schwert and five other models in the literature as special cases, including Higgins and Bera’s (1992) NARCH, Geweke’s (1986) and Pantula’s (1986) log – ARCH, and simple asymmetric and threshold GARCH models. The general structure is as follows:

,

where

if we assume the distribution of residuals are conditionally normal, then the condition for existence of  and  is

if this condition is satisfied, then when , we have  covariance stationary. But this is a sufficient condition for  to be covariance stationary.

Empirical studies by Nelson (1991), Glosen, Jaganathan and Runkle (1989) and Engle and Ng (1992) show it is crucial to include the asymmetric term in financial time series models. In order to explore whether the same property exists in the inflation rate and inflation uncertainty, we introduce the ARMA-APARCH time series models for inflation rate and inflation uncertainty. Here, we denote the inflation rate by   and define the conditional mean equation as:

Where   and have all their roots outside the unit circle

The main feature of the model is that it imposes a Box-Cox power transformation of the conditional standard deviation process and the asymmetric absolute residuals, with which we can linearize otherwise nonlinear models. Hence it couples the flexibility of a varying exponent with the asymmetric coefficient (to take the “ leverage effect ” into account). The two extra parameter, power parameter  and asymmetric parameter  , are important in reproducing a stylized fact in the data, that is, the first order autocorrelation function of power-transformed observation and may be used for drawing the conclusions about the existence of the integer moments based on the estimated models.

Inflation and Inflation Uncertainty

Theory

Economic theory provides the interpretation for the predicted relationship between inflation and inflation uncertainty from the economic view. Friedman (1977) in his NOBEL LECTURE argued that inflation uncertainty affects the trade-off between the inflation and unemployment because inflation volatility and uncertainty produce the negative influence on the real economic activity. Moreover, high inflation rates might result in more variable inflation and hence create more uncertainty about future inflation. He also suggested that uncertainty concerning the inflation regime may be the underlying source of the observed positive relation between inflation and inflation volatility. His NOBEL lecture stimulated much research on the link between inflation and inflation uncertainty.

Ball (1992) subsequently derived the Friedman’s results formally in an asymmetric repeated information game where the public faces uncertainty about the monetary authority. According to the different attitudes toward the economic costs of reducing inflation, we have two types of policy-maker. For tough type, it will apply the contradictory monetary policy when the inflation rate is low. Under the Ball’s assumption that the two types of policy-maker are in power in a stochastic condition, a higher inflation rate creates more uncertainty about the future inflation in terms of the uncertainty about the monetary policy.

Holland (1993a) finds that another potential cause of uncertainty about the parameters of the inflation process is the uncertainty about the timing of the effect of changes in the money supply on the price level. Furthermore, he suggests that the inflation rate and inflation uncertainty may be related with each other by the influence of uncertainty about the parameters of the inflation process. According to Holland (1993b), under the Evans and Wachtel (1993) framework, if the regime changes cause unpredictable changes in the persistence of inflation, then lagged inflation squared is positively related to inflation uncertainty. If the regime changes do not affect the persistence of inflation, no relationship between the rate of inflation and inflation uncertainty is discovered.

However, the opposite conclusion are raised by Pourgerami and Maskus (1990), they find there is a negative relation between the inflation rate and inflation uncertainty, that is, the higher inflation rate may lead to lower nominal uncertainty. Ungar and Zilberfarb (1993) setup a mechanism that may weaken, offset or even reverse the traditional view of the link between the inflation and inflation uncertainty.

In short, the models of Ball, Evans and Wachtel, and Holland imply that higher nominal uncertainty is part of the welfare costs of inflation because inflation causes inflation uncertainty.

The causal effect of inflation uncertainty on inflation rate has been analyzed in the theoretical macro literature. Cukierman and Meltzer (1986) and Cukierman (1992) using the Barro-Gordon model of Fed behavior show that greater uncertainty about money growth and inflation causes a higher average rate of inflation by increasing the incentive for the policy-maker to create inflation surprises to stimulate output growth. In addition, Devereux (1989) emphasizes the fact that higher variability of real shocks lowers the optimal degree of wage indexation and increases the incentives of the policy-maker to create surprise inflation. Therefore, if changes in the degree of wage indexation take time to occur then higher nominal uncertainty precedes greater inflation.

On the contrary, Holland (1995) claims that in the presence of a stabilization motive on the part of the policy-maker, an increase in inflation uncertainty will invite a tight monetary policy response and a lower average inflation rate in order to minimize the real costs of inflation uncertainty. This is more likely to happen under Central Bank independence and a commitment to long run stability. Hence, the stabilization hypothesis implies a negative causal effect of inflation uncertainty on inflation rejecting the Cukierman-Meltzer theory.

Empirical Evidence

Earlier studies

The empirical evidence for testing of the inflation rate and inflation uncertainty hypothesis might be totally different based on the different framework and inflation data set. However, the earlier work for testing of the inflation and its uncertainty hypothesis with the simple measures of inflation uncertainty does give us a good guide and comparison for our further research.

Pourgerami and Maskus (1987) find only two of seven Latin American countries have a significant positive influence of inflation on inflation uncertainty measured by absolute forecast error. On the following paper, Pourgerami and Maskus (1990) find a significant negative effect for Argentina and a considerably weaker negative effect for some other countries with the inflation uncertainty measured by the relative forecast error. Ungar and Zilberfarb (1993) show that a significant positive effect exists in Israel only in periods of high inflation but not in periods of relatively moderate inflation. Ball and Cecchetti (1990) find the similar results that a stronger relationship between inflation and inflation uncertainty exists in high inflation countries than that in moderate inflation countries.

From the earlier studies, Holland (1993b) collects all those works involving the empirical estimation for the United States of the relationship between the inflation rate and inflation uncertainty. Based on those surveys, he points out that almost all the studies have provided a strong evidence for the positive relationship between the inflation and inflation uncertainty with both simple measures of inflation uncertainty and time-varying parameters except that Katsimbris (1985) fails to find a significant effect. However, when most studies are restricted with the fixed parameters, it is quite interested to find there is insignificant relationship between inflation and inflation uncertainty. Only the earlier work of Holland (1984) finds the mixed results. The further studies of Holland (1995), together with those earlier surveys, simple measures of inflation variability and Granger causality tests, find strong evidence that high inflation causes greater inflation uncertainty and weak evidence that greater uncertainty precedes lower inflation for the post war US inflation series.

Current findings

Recent time series studies of inflation uncertainty have shed more lights on GARCH-type models which estimate the conditional variance as the proxy of inflation uncertainty. Under this framework, different methods are employed in the empirical work to find the relationship between the inflation and inflation uncertainty.

Some studies have estimated the GARCH type models with a function of lagged inflation rate in the conditional variance equation. Brunner and Hess (1993) apply the asymmetric effects of inflation shocks on inflation uncertainty and find a weak link between US inflation and its uncertainty. Subsequently, Caporale and McKierman (1997) find a positive relationship between US inflation and inflation uncertainty. Fountas (2001) use annual UK inflation data to examine the link between the inflation and inflation uncertainty and find a strong support for the Friedman-Ball hypothesis.

Some studies estimate the GARCH type models with a joint feedback between the conditional mean and conditional variance of inflation. Baillie et al. (1996) (for three high inflation countries and the UK) and Fountas, Karanasos, Karanassou (2001) (for the US) find a significant positive bi-directional relationship between the inflation and its uncertainty in accordance with the predictions of economic theory. On the other hand, Hwang (2001) points out that the US inflation affects its uncertainty weakly negatively whereas the uncertainty affects the inflation insignificantly. Hence his results support the Engle’s results that a high inflation rate does not necessarily imply a high inflation uncertainty.

Some studies use two-step method to find the link between the inflation and inflation uncertainty where the conditional variance is estimated as inflation uncertainty from a GARCH type model first and then Granger Causality Test is employed to test for bi-directional effects. Grier and Perry (1998) find that in all G7 countries’ inflation series have a significant positive relation with its uncertainty. At the same time, they also find that empirical evidence is in favor of the Cukierman-Meltzer hypothesis for some countries while in favor of the Holland hypothesis for other countries. Fountas et al. (2001) also find a significant positive relation between inflation and inflation uncertainty in five European countries by using quarterly data and employing the EGARCH model. On the other hand, their results regarding the inverse direction, the impact of the changes in inflation uncertainty on inflation rate, are generally consistent with the exist rankings of Central Bank independence. Conrad and Karanasos (2002) find that both Friedman and the Cukierman and Meltzer hypothesis are supported by strong evidence for three industrial countries.

Among those limited recent works, Conrad and Karanasos (2002) employ the ARFIMA-FIGARCH model to capture the dual long memory properties of inflation process. As is known, the long memory models imply different long run predictions and effects of shocks to conventional macroeconomic approaches. This study aims to fill the gaps arising from the lack of interests in the east and south Asia cases where the results would have interesting implications for the successful implementation of both developed and developing countries’ monetary policy and from the methodological shortcomings of the previous studies.

Empirical Analysis

Data

In this empirical work, we use monthly data of the Producer Price Index (PPI) for total nine countries. These nine countries except Australia belong to the east and south Asia. Also we divide these nine countries into two groups: developed countries and region (Australia, Hong Kong, Japan, Korea and Singapore) and developing countries (India, Indonesia, Malaysia and Thailand). The inflation series is measured by the monthly difference in the natural log of the PPI. The data for India, Korea and Thailand range from 1960:01 to 2003:04, for Australia range from 1960:01 to 1997:06, for Hong Kong range from 1976:01 to 2003:06, for Japan and Indonesia range from 1970:01 to 2003:03, for Malaysia range from 1986:01 to 2003:03 and for Singapore range from 1974:01 to 2003:05.

Figure 1 and Figure 2 allow us to examine the characteristics of Korea inflation graphically by presenting the autocorrelation function of inflation and changes in inflation. Among other things, the figures make clear the long memory property of inflation, that is the inflation series appear to be nonstationary, while the differenced series appears to follow an MA (1) process. Figure 3 and figure 4 plot the autocorrelations of the squared and absolute values of the residuals from an estimated ARFIMA (0, d_m, 24) model. Interestingly, these autocorrelations display extremely persistent autocorrelation, which is also suggestive of a form of long memory behavior. The plots for the other eight countries (Australia, Japan, Korea, Singapore, India, Indonesia, Malaysia and Thailand) being quite similar, they will not be reported here.

Next, we employ the PP and KPSS unit-root tests suggested by Phillips and Perron (1988) and Kwiatkowski et al. (1992) respectively. In the Phillip-Perron test, it provides a simple test for a unit root in univariate time series against stationary and trend alternatives. However, in the KPSS test, it tests the null hypothesis of stationarity, either around a level or around a linear trend against the alternative of a unit root, which yields the contradictory results. Therefore, rejecting the both null hypotheses imply that the fractional integrated models for long memory could be a possible choice. The two different testing results for all nine countries are presented below in Table 1. With all the inflation series, we find strong evidence against the unit root as well as against the stationary hypothesis for all the countries. It is conveniently for us to employ all kinds of the fractional integrated models. Of course, these unit root tests still have their own drawbacks. Lee and Amsler (1997) show that the KPSS statistic cannot distinguish consistently between nonstationary long memory and unit root.

Model Estimation

Estimates of the ARFIMA-FIGARCH model and ARFIMA-IGARCH model are shown in Table 2. The results are obtained by using quasi-maximum likelihood (QML) estimation provided by Lauren and Peters (2002) in Ox.

Table 2a provides the simultaneous estimation of ARFIMA-FIGARCH models for four developed countries and region, Australia, Hong Kong, Korea and Singapore. Except Hong Kong with the weak evidence, the evidence for all the other three countries significantly exhibits the long memory property. It is obvious from the results that in the mean equation, the long memory parameter d_m for these four developed countries and region lies between 0.228224 and 0.492346, which implies that all these four mean equations for ARFIMA model are stationary process. For the conditional variance estimated by FIGARCH model, we find that all the long memory parameters d_v for these four countries, Australia (0.625607), Hong Kong (0.752993), Korea (0.734246) and Singapore (0.791434) are significantly greater than 0.5, which reveals a nonstationary process.

Table 2b reports the estimation of ARFIMA-FIGARCH models for three developing countries, India, Indonesia and Malaysia. For Japan and Thailand, we also employ the ARFIMA-FIGARCH model and find the initial estimation of the long memory parameter d_v for conditional variance process nearly equal to and not significantly different from one implying an integrated process in its conditional variance. Hence we turn to apply the ARFIMA-IGARCH model to capture the long memory property in the mean equation with the complete persistence in its conditional variance process. The estimated results of d_m parameter in the mean equation for all those four countries are significantly lying between 0.087485 and 0.339610 smaller than 0.5 implying a stationary process. In particular, the d_m parameter for Japan (0.087485) and Thailand (0.139321) is quite small, however it is significantly different from zero and suggests a weaker long memory in its conditional mean process. The estimation of the long memory parameter d_v for India and Malaysia in the conditional variance process shows that it is smaller that 0.5. Therefore, the conditional variances of these two countries’ inflation series are characterized by the stationary FIGARCH behavior. While for Indonesia, the estimated long memory parameter d_v is larger than 0.5, which is described as a nonstationary FIGARCH process.

From the above reports of table 2, we can obtain several interesting findings. In all nine developed and developing countries and region, the estimates for long memory parameter d_m in mean equation are below 0.5, which indicates that all inflation series are stationary. The estimated long memory conditional mean parameter is in the range 0.087485 < d_m  <0.492434. The value of long memory parameter for Japan (0.087485) is obviously lower than the corresponding value for Australia (0.492434). However, although the value of d_m for Japan and Thailand is quite small, it is significantly away from one. Furthermore, except two extreme conditions: Australia (strong long memory property) and Japan (weak long memory property), the long memory conditional mean parameters for all those developed countries and region in east Asia are quite close to each other changing in the small range from 0.228224 to 0.277378. While in those developing countries, we divide them into two groups, in which the first group represented by India and Indonesia has the similar long memory conditional mean parameters and the second group represented by Malaysia and Thailand possesses the similar weak long memory property. In short, we find that the long memory effects are present in the inflation mean process for all of those countries and can be better described by the ARFIMA process.

The estimation of the FIGARCH and IGARCH models for developed countries and region reveals the nonstationary conditional variance process. In contrast, the long memory conditional variance parameters for two developing countries India and Malaysia imply the stationary conditional variance process. For the other two developing countries Indonesia and Thailand, we still get the nonstationary processes for their conditional variances. Based upon the Akaike and Schwarz information criteria (AIC, SIC respectively), the evidence in favor of the FIGARCH (0, d_v , 0)  model for Australia, Singapore, India, Indonesia and Malaysia, while for Hong Kong and Korea, the FIGARCH (1, d_v , 0) and FIGARCH(1, d_v ,1) models are the preferred specifications respectively. In addition, the estimated FIGARCH parameter (β) for Korea is significantly negative and satisfies the set of conditions sufficient to guarantee the nonnegative conditional variance. Only for Japan and Thailand, the estimated d_v parameter is close to one implying a complete persistence of the shocks and consequently the IGARCH (1,1) model can be conveniently applied for these two conditional variance processes of inflation rates. In a whole, the empirical evidence supports that the dual long memory properties does exist in inflation process for seven out of the total nine chosen countries, suggesting that dual long memory properties are important characteristics of inflation data. Finally, under the well supports of the hypothesis of uncorrelated standardized and squared standardized residuals for all those nine countries , we can avoid the statistically significant evidence of misspecification.

ARMA-APARCH Model

In this section, we estimate a different general class of model ARMA-APARCH to capture the ‘long memory’ property of the power transformation of the conditional standard deviation process. The long memory property is the strongest when the power δ is around one. The estimation results are shown in table 3, where table 3a gives the results for those developed countries and region in the east Asia area, while table 3b shows the results for those developing countries in south Asia. The results are also obtained by using quasi-maximum likelihood (QML) estimation. For those developed and developing countries, most of them have the power δ value for the conditional heteroskedasticity function that lies between one and two, only for Australia and India, the δ value is greater than 2.

In table 3a, we find that for Japan, Korea and Singapore, the δ values are 1.206652, 1.125932 and 1.897462 respectively which is significantly different from one (Taylor/Schwert model) or two (Bollerslev GARCH). For Hong Kong, it is found that the δ value is 0.921245 and nearly equal to one. It makes the Taylor/Schwert model a better choice. For Australia, the δ value is 2.290008 and is greater than 2 implying that the conditional variance process of inflation rate is a stationary process. The t statistics of the asymmetric term for Australia, Hong Kong, Korea and Singapore are 2.66103, 2.01615, 2.667305 and 2.075554 respectively which strongly supports that the leverage effect does exist in these four countries’ inflation uncertainty process. For Japan, it is reported that the t statistic of the asymmetric term is 1.313534 insignificantly accepting the null hypothesis, α=0. Hence there is no leverage effect in its conditional variance process.

In table 3b, the results report that for Thailand, the δ value (1.117843) lies significantly between one (Taylor/Schwert model) or two (Bollerslev GARCH). For Indonesia and Malaysia, the evidence weakly supports that the δ value (1.240237 and 1.705671 respectively) differs from one and two. Only for India, the δ value (2.213856) is significantly greater than two implying a stationary process of the power transformation of conditional standard deviation of inflation rate. As to the asymmetric term, it is clear that the t statistics of α parameter for Indonesia (2.801453), Malaysia (2.105657) and Thailand (2.976636) imply a significant leverage effect in these three developing countries. While for India, the t statistic is 0.781076 providing the strong evidence for the null hypothesis, α=0. Hence we are not expected to find the asymmetric effect in the inflation process for India.

After the comparison of the estimated parameters for those developed countries and region with developing countries in the east and south Asia area, we find several interesting results. Firstly, for eight of those nine countries, the estimated power parameter δ for the conditional heteroskedasticity function is significantly differing from one and two with the weak evidence for Indonesia and Malaysia. Secondly, the estimated β parameter for all those countries provides the strong evidence that the power transformation of conditional standard deviation has quite high autocorrelation except Korea and Indonesia with relative small β parameter (0.349844 and 0.376054 respectively). The small β parameter suggests that the power transformation of the conditional standard deviation do have a short memory and there is a portion of inflation rates that is predictable. Thirdly, the estimated α parameter provides the evidence that a significant leverage effect exists in seven of the total nine countries’ inflation processes. Only for Japan and India, there is no leverage effect in their inflation processes. Generally speaking, with the high autocorrelation, the power transformation of the conditional standard deviation can be characterized to be long memory. The general class of ARMA-APARCH model not only better describes the power effects and asymmetric effect but capture the important long memory properties existing in the inflation process as well. Finally, the well supports of the hypothesis of uncorrelated standardized and squared standardized residuals for all those nine countries help us avoid the statistically significant evidence of misspecificatioin.

Granger-causality tests

In this section, Granger causality tests are employed to provide the statistical evidence for the relationship between the inflation and inflation uncertainty. In table 4 we will report the F statistics of Granger-causality tests using four, eight and twelve lags, as well as the sign of the sums of the lagged coefficients in case of the statistical significance.

Panel A provides the results of Granger causality tests from inflation rates to inflation uncertainty. It exhibits a significant positive effect of inflation rates on inflation uncertainty for three of the five developed countries ( Japan, Korea and Singapore ) and one of the four developing countries (Indonesia ). The evidence is quite strong for Korea, Singapore and Indonesia and weaker for Japan, where F statistics are at the 15% significant level rejected the null hypothesis. The results for these four countries support the Friedman hypothesis, that is, inflation influences its uncertainty positively, predicted by Friedman (1977) and Ball (1992). For Australia, India, Malaysia and Thailand, we find the mixed results, where it reports the significant positive or negative relationship for different lags. Only for Hong Kong, we find that the evidence strongly supports the null hypothesis that inflation does not granger-cause inflation uncertainty. It follows that there is no necessary relation from inflation to inflation uncertainty. In a whole, from the results of panel A, we find for most developed countries, inflation rate affects the inflation uncertainty positively strongly supporting the Friedman hypothesis, while for most developing countries, we find the mixed results. For all of the nine countries, the evidence suggests a significant positive or negative effect of inflation rate on inflation uncertainty except Hong Kong insignificant.

Panel B reports the results of the causality tests where causality is run from inflation uncertainty to inflation rate. A significant negative effect of inflation uncertainty on inflation rate is found for three of the developed countries (Australia, Korea and Singapore) and three of the developing countries (Indonesia, Malaysia and Thailand). The results from these six countries support the Holland’s stabilization hypothesis that an increase in inflation uncertainty will induce a tight monetary policy response and a lower average inflation rate in order to minimize the real costs of inflation uncertainty. However a significant positive effect of inflation uncertainty on inflation rates is also reported for Japan and a weaker positive effect for Hong Kong where the null hypothesis is rejected at 5% significant level only for the first 8 lags and all the other F statistics for 4 lags and 12 lags are insignificant. It, on the contrary, supports the Cukierman-Meltzer hypothesis that increases in inflation uncertainty raise the optimal average inflation by increasing the incentive for the policy-maker to create the inflation surprises. For Indonesia and Thailand, we find the mixed results, where evidence suggests that for the first 4 lags, the inflation uncertainty significantly positively affects the inflation rate, while for 8 and 12 lags, the inflation uncertainty significantly negatively affects the inflation rate. Only for India, the evidence obviously supports the null hypothesis that inflation uncertainty does not Granger-cause inflation. Hence, the inflation uncertainty is independent of the inflation rate. The results of panel B clearly reveal that the increased inflation uncertainty significantly affects the future inflation rate in all chosen developed and developing countries except India, but differing in the different manners.

Possible Extensions

The main goal of this paper is to investigate the link between the rate of inflation and uncertainty about inflation, and estimate the two main parameters driving the degree of their persistence, for nine countries in the east and south Asia area. In that respect the paper achieves its goal. As Baum et al. (1999) point out, a likely explanation of the significant persistence in the inflation rate series is the aggregation argument put forth by Granger (1980), which states that persistence can arise from aggregation of constituent processes, each of which has short memory. Alternatively, Baum et al. (1999) conjecture that the long memory property of monetary aggregates will be transmitted to inflation, given the dependence of long-run inflation on the growth rate of money. However, one might also ask why it is necessary to allow for long memory in the conditional variance of inflation. To answer this we must enquire into the possible theoretical sources of heteroscedasticity in the inflation shocks. It will be very useful to provide a theoretical rationale for the dynamics of inflation. Here the choice of the ARFIMA-FIGARCH model is justified solely on empirical grounds.

There is substantial evidence that all the chosen inflation rates have long memory, a feature which can be captured by a fractional integrated I(d) model. On the other hand, as Hyung and Franses (2001) point out, inflation rates may perhaps show long memory because of the presence of neglected occasional breaks in the series rather than being really I(d). Caporale and Gil-Alana (2002) investigate the stochastic behavior of the inflation series in three hyperinflation countries. They test for fractional integration and find that when allowing for structural breaks the order of integration of the series decreases considerably.

However, Bos et al. (1999) find that the apparent long memory in monthly G7 inflation rates is quite resistant to mean shifts. Along these lines, Bos et al. (2002) using recursive estimation show that the order of integration in US postwar inflation series has remained quite stable around 0.3. They also point out that the long memory behavior found in time series analysis of inflation rates is consistent with the fact that many lags of inflation are statistically significant in reduced form equations of economic models. Recently, Hyung and Franses (2001) have compared time series models with structural breaks and models with long memory for 23 monthly US inflation rates in terms of out-of-sample forecasting for various horizons. They found that the two models perform equally well. Subsequently, Hyung and Franses (2002) put forward a joint model that incorporates both long memory and occasional level shifts. Overall, however, they find that the dominant feature in 23 US inflation rates is long memory and that the level shifts are less important. These results suggest several avenues for further research. One promising avenue would be to adapt the ARFIMA-FIGARCH model in a way that incorporates occasional level shifts in both the conditional mean and the conditional variance.

Moreover, Stock and Watson (1999) find that unemployment rate Phillips curve forecasts of inflation although more accurate than forecasts based on other macroeconomic variables, such as interest rates, money, and commodity prices, can be improved upon using a generalized Phillips curve based on other measures of aggregate activity. However, since it seems unlikely that any single measure of real economic activity completely captures all the economically relevant aspects of inflation, they point out that the best-performing forecast was the one that uses a new composite index of aggregate activity comprised of 168 individual activity measures. Subsequently, Bos et al. (2002) have emphasized that the introduction of two macroeconomic leading indicators namely, the unemployment rate and the short term interest rate, in the ARFIMA model lower the estimate of the fractional parameter and thus account partly for the persistence in inflation. More importantly, they argue that the multi-step forecast intervals of the ARFIMAX model are more realistic than of the ARIMAX model where ARFIMAX denotes an ARFIMA model with explanatory variables in the mean equation. In the context of our analysis, incorporating macroeconomic variables either in the ARFIMA or in the FIGARCH specification or in both could be at work. We look forward to sorting this out in future work.

Finally, Morana (2002) suggests that long memory in inflation is due to the output growth process. His model implies that inflation and output growth must share a common long memory component. Using a bivariate ARFIMA-FIGARCH model, which allows the measurement of uncertainty about inflation and output growth by the respective conditional variances, one can test for the empirical relevance of several theories that have been advanced on the relationship between inflation, output growth, real and nominal uncertainty.

Conclusion

In this paper, we estimate the various ARFIMA-FIGARCH-type models for total nine countries’ monthly inflation to discover the link between inflation rate and inflation uncertainty. The generated conditional variance of inflation that is the proxy of the inflation uncertainty has been conveniently applied in the Granger causality tests together with inflation rate to examine the bi-directional relationship between the two variables: inflation rate and inflation uncertainty. The empirical evidence revealed that in most developed countries in east Asia, the increased inflation raises inflation uncertainty supporting the theoretical prediction by Friedman. However for most of those developing countries in south Asia, the inflation has the mixed impacts on inflation uncertainty. As to the causal effect from inflation uncertainty to inflation, we find for the developed countries, both Cukierman-Meltzer theory that greater uncertainty about money growth and inflation causes a higher average rate of inflation and the stabilization hypothesis of Holland that an increase in inflation uncertainty will invite a decrease in inflation rate are well supported by the different developed countries’ empirical evidence. It may depend on the different policy-maker’s attitude toward the inflation rates and different monetary policy. While for those developing countries, we still get the mixed impacts. Hence the division of countries by how their inflation rates respond to inflation uncertainty does not appear to be closely related to existing rankings of central bank independence especially for those developing countries.

The aim in this paper focuses on the importance of modeling long memory not only in the conditional mean of inflation but in its conditional variance as well. We find that in most of the cases there is a need to consider the joint ARFIMA-FIGARCH model, because only in very few countries does one of its nested versions yield a better fit. Overall, these findings suggest that we need to shed more lights on the consequences of dual long memory when estimates of inflation uncertainty are used in applied research. In other words, as our results indicate, estimates of inflation uncertainty that ignore the effects of dual long memory may seriously underestimate both the degree of persistence of uncertainty and its consequences for the inflation and inflation uncertainty hypothesis.

Possible extensions of this paper could go in different directions. One could provide an enrichment of the dual long memory model by allowing the conditional variance to affect the inflation or by incorporating lagged values of inflation in the conditional variance equation. Finally, it is worth pointing to an important issue that we have not addressed. The dual long memory model used in this paper ignores the possibility of structural instability caused by changing regimes. One could develop a dual long memory Markov switching model that explains the changing time series behavior of inflation in the different sample period. This is undoubtedly a challenging yet worthwhile task.  

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