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Changing inflation dynamics and uncertainty in the United States.

1. Introduction

After the great inflation of the 1970s, the level of inflation declined in the 1980s and has remained moderate since. However, despite the relatively low level of price changes, there have been, in the current decade, a number of shocks with the potential to affect inflation. These include the bursting of the tech stock bubble, the rise and fall of the housing bubble, the 9/11 terrorist attacks, the Iraq War, and high energy prices.

While these events have not yet appeared to increase the mean of inflation substantially, they may have an impact on the persistence of inflation shocks. Even more importantly, they could raise uncertainty over the future path of price changes. Inflation uncertainty can decrease the information content of prices (Friedman 1977) and negatively affect investment (Cabellero 1991). Thus, while a moderate level of inflation may have no direct effect on Gross Domestic Product (GDP), inflation uncertainty can negatively impact growth. Grief et al. (2004) and Grier and Grief (2006) find a negative effect of inflation uncertainty on output in the United States and Mexico, respectively.

There have been previous papers on inflation uncertainty in the United States, but few have allowed for many structural changes to examine the level of uncertainty for different time periods. Moreover, no study has investigated whether uncertainty or any other aspect of inflation has changed in response to the aforementioned shocks of the current decade.

We will accordingly investigate changes in U.S. inflation, inflation persistence, and inflation uncertainty over time with two estimation strategies. We will first employ a Markovswitching model that allows for shifts in the mean, persistence, and variance. Results will indicate that uncertainty has indeed increased in the present decade.

Secondly, as a check on these results, we will employ a generalized autoregressive conditional heteroskedasticity (GARCH) (1) model. While other papers have employed the GARCH technique to investigate inflation and uncertainty, most assume parameter constancy. We will instead employ dummies and test for structural breaks. Results will confirm the Markov-switching finding that uncertainty has risen since the end of the 1990s. Moreover, by analyzing headline inflation as well as the CPI less energy costs with both models, our results suggest that more volatile energy prices are at the root of the higher uncertainty.

This paper proceeds as follows: Section 2 summarizes the previous literature on U.S. inflation, its persistence, and uncertainty. Section 3 describes in detail the methodology to be employed. Results are discussed in section 4, and section 5 concludes.

2. Previous Literature

There have been a number of attempts at modeling the dynamics of inflation, in particular its mean, persistence, and uncertainty. Some papers have examined only one of these three important properties, ignoring the others, often to the detriment of correct inference.

Whether the mean, or level, of inflation is subject to breaks because of policy has attracted interest from researchers. Perron (1989) examines the time series properties of the CPI by allowing for two changes in the mean. Clark (2003) and Levin and Piger (2003) both find that allowing for structural changes in inflation affects other properties, such as estimated persistence.

The persistence of inflation shocks will typically be high when the credibility of the central bank is low and vice-versa. If agents have confidence in the inflation-fighting intentions of the Federal Reserve (the Fed), a temporary spike in prices will not typically cause a prolonged rise in inflation. Along these lines, Alogoskoufis and Smith (1991) examine changes in inflation persistence resulting from departures from exchange rate pegs. The authors find that episodes such as the United States leaving the gold standard and the collapse of the Bretton Woods system did in fact lead to palpable increases in inflation persistence. Burdekin and Siklos (1999) conjecture that while the exchange rate regime has an effect on credibility, other events can also impact persistence. These authors find that events such as oil shocks have larger impacts than changes in currency pegs.

Persistence is often measured as the size of autoregressive moving average (ARMA) (2) coefficients in the time series model of inflation. By using a Markov-switching model, Kim (1993) finds that infrequent permanent shocks contribute most to the persistence in the inflation rate. Some recent papers, such as Levin and Piger (2003), find that persistence has declined in recent years in the United States. Cecchetti and Debelle (2006) note that, in examining inflation dynamics, it is vital to distinguish between changes in the level and changes in persistence. Failure to control for a change in one will lead to a bias in testing, in which there appears to have been a change in the other, even if no such change has actually occurred. These authors control for both mean and persistence change in their paper, and find that, while there has been a structural decline in persistence in the United States in recent years, it is of small magnitude.

The last, but perhaps most important, aspect of the inflation process to be investigated is uncertainty. As long as inflation is moderate, its level is likely neutral for output. However, Friedman (1977) argues that uncertainty concerning future inflation lowers the information content of prices, and thus hampers commerce and lowers income. Moreover, by making future price changes more difficult to forecast, inflation uncertainty can lower investment, and hence output (Cabellero 1991). Grier et al. (2004) find that, for the United States, an increase in inflation uncertainty indeed lowers GDP growth. Grier and Grier (2006) find this negative impact of inflation uncertainty on growth holds for Mexico as well.

Given the importance of inflation uncertainty, there have been many papers investigating it for the United States. Many question how much an increase in the level of inflation raises uncertainty. In a model in which shocks are decomposed into transitory and permanent components, Kim (1993) finds that high uncertainty about permanent shocks is associated with a positive shift in inflation. Caporale and McKiernan (1997) and Grier and Perry (1998) both find that an increase in the level of inflation raises inflation uncertainty in the United States, as Friedman (1977) claimed would be the case. In order to investigate inflation uncertainty, some empirical proxy must be obtained. Traditionally, the measure of uncertainty has been taken to be either the unconditional variance of inflation or the conditional variance of estimated inflation shocks (GARCH models; see Thornton 2007). GARCH models have become fairly standard in investigating inflation. However, Markov-switching models have become increasingly employed in macroeconomics and have been recommended as an alternative to GARCH (Kim and Nelson 1999, p. 3). (3) This is particularly the case since Kim and Nelson have developed a model that allows for changes in the mean, persistence, and variance of the series. As we detail below, we will employ the Markov-switching method first, and then use the GARCH model as a check.

While inflation and its properties in the United States have been widely analyzed, most assume parameter constancy in uncertainty or allow for at most a one-time break. Many changes have occurred in the last 8 to 10 years: the tech bubble, the end of Alan Greenspan's tenure, the rise and bursting of the housing bubble, and perhaps most importantly, the 9/11 terrorist attack, the Afghanistan and Iraq wars, and volatile energy prices. Given these changes, it is a delusion to presume constancy in uncertainty. Higher energy prices may well affect the uncertainty surrounding inflation, and no paper has focused on how the changes of the last decade may have had an impact. This paper aims to fill that gap.

3. Methodology

There are two ways to model changing variance (or uncertainty). The changing variance of a Markov-switching model results from regime changes in the variance structure. That differs from the varying variances of a GARCH model, which are dependent on the squares of previous innovations and previous conditional variances within a given structure (regime). A state-dependent Markov-switching model is useful in modeling structural change where the parameters vary as the regime changes. If the dates of the regime switching are known, the state variable is equivalent to a dummy variable. In that case, a dummy variable incorporated in a GARCH model to reflect changes in mean and variance of inflation should not be inferior to the Markov-switching model and both models should capture the structural change, if there is indeed a change. For example, to investigate structural change in inflation uncertainty during the Greenspan rule, the dates of Greenspan rule are known and a dummy variable can be incorporated to reflect changes in mean and variance of a GARCH model. However, in the case of unknown regime switching dates, a Markov-switching model is essential in capturing the changes in mean, persistence, and variance as the regime changes.

Markov-Switching Model

Given that much macroeconomic data is characterized by regime changes and nonlinearity, the Markov-switching model has become a popular modeling technique. Hamilton's (1989) canonical paper employs the model to analyze U.S. output. While most Markov-switching models allow for changes in the conditional mean of the variable, Kim (1993) used a Markov-switching model to incorporate changes in mean and variance of inflation simultaneously. For an AR(p) model,

[[pi].sub.1] = [[alpha].sub.0] + [[beta].sub.1][[pi].sub.t-1] + [[beta].sub.2][[pi].sub.t-2] + [[beta].sub.3][[pi].sub.t-3] + ... + [[beta].sub.p][[pi].sub.t-p] + [[mu].sub.t],

where [pi].sub.t] and [pi].sub.t-i] are the inflation rates at time t and t - i, we can transform it by arranging the terms and get the error correction format

[DELTA][[pi].sub.t] = [[alpha].sub.0] + ([rho] - 1)[[pi].sub.t-1] + [[gamma].sub.1][DELTA][[pi].sub.t-1] + [[gamma].sub.2][DELTA][[pi].sub.t-2] + ... + [[gamma].sub.p-1][DELTA][[pi].sub.t-p] + [u.sub.t],

where [rho] = [summation].sup.p.sub.i = 1] [beta].sub.i], and [rho] is a measure of persistence.

To examine the changes in the dynamics of the inflation process, we use a Markov-switching model that can simultaneously handle changes in the mean, persistence, and variance of inflation. The previous model can thus be expressed as


where [S.sub.t] is the state variable that denotes the state of the economy at time t, and [phi].sub.1] = [rho] - 1. A two-state Markov process has the following transition probabilities: Pr([S.sub.t] = 0 | [S.sub.t-1] = 0) = [??] and Pr([S.sub.t] = 1 | [S.sub.t-1] = [??]. In this specification, we have two different regimes: regime 1 (i.e., [S.sub.t] = 0) and regime 2 (i.e., [S.sub.t] = 1). The parameters [[alpha].sub.1], [[phi].sub.2], and [h.sub.1] capture the changes in the mean of inflation, the persistence of a shock to inflation, and the variance during regime 2 relative to regime 1. Since variance denotes uncertainty, a positive [h.sub.i] implies a shift from low inflation uncertainty of regime 1 to a high inflation uncertainty in regime 2. In this paper, we will use p = 3. (4) The parameters of the model are thus ([[alpha].sub.0], [[alpha].sub.1], [[phi].sub.1], [[phi].sub.2], [[gamma].sub.1], [[gamma].sub.2], [h.sub.0], [h.sub.j] [??], [??]).


While this Markov-switching model was developed as an alternative to the GARCH model (Kim and Nelson 1999), traditional GARCH models have been widely employed to investigate U.S. inflation, and we will employ GARCH models here, mainly as a check on the Markov-switching results. The GARCH results can also be compared to the results of previous authors, such as Grier and Perry (2000), and we will be able to test for a larger number of structural breaks with the GARCH than with the Markov-switching model.

A GARCH model begins with an ARMA portion, which captures the conditional mean of inflation:

[[pi].sub.1] = [p.summation over (i=1)][[alpha].sub.i][[pi].sub.t-i] + [q.summation over (i=0)][[beta].sub.i][[epsilon].sub.t-i]. (1)

Here, [[pi].sub.t] represents the inflation rate, and the larger the estimated [[alpha].sub.i] and [[beta].sub.i] parameters, the more persistent inflation is. The squared residuals from the ARMA model "correspond well to the notion of uncertainty in Cukierman and Meltzer (1986)" (Grier and Perry 2000, p. 47). Thus the ARCH portion of the model,

[[epsilon].sup.2.sub.t] = [[alpha].sub.0] + [q.summation over (i=1)][[alpha].sub.i][[epsilon].sup.2.sub.t-i] + [u.sub.t], (2)

is employed to yield an estimate of time-varying inflation uncertainty. Ever since Bollerslev (1986), the conditional variance is often modeled as an ARMA, rather than a pure AR process, as in Equation 2, yielding a Generalized ARCH, or GARCH model:

[h.sub.t] = [[alpha].sub.0] + [q.summation over (i=1)][[alpha].sub.i][[epsilon.sup.2.sub.t-i] + [p.summation over (i=1)][[beta].sub.i][h.sub.t-i]), (3)

in which [h.sub.t] represents the conditional variance.

In estimating the conditional variance of inflation, it is important to allow for the possibility that inflation shocks have an asymmetric effect on the conditional volatility. Specifically, a negative inflation shock may have a very different effect than a positive one. A negative shock to price changes may raise volatility by less than a positive innovation (indeed a negative shock may decrease rather than raise uncertainty). An extension to the basic GARCH model that allows for such asymmetry is known as the Threshold-GARCH or TARCH model (Glosten, Jagannathan, and Runkle 1993). In this model, the conditional variance is modeled as follows:


The term [gamma][epsilon].sup.2.sub.t-1][d.sub.t-1] represents the asymmetric portion of the conditional variance. Here, the dummy variable [d.sub.t-1] is equal to one if [[epsilon]t-1] < 0 and is equal to zero otherwise. If the estimated coefficient y is negative, negative shocks don't raise uncertainty as much as positive shocks. Grief and Perry (1998) test for asymmetric effects in U.S. inflation but find no significant asymmetry. Their sample ends in 1993, however, and since our sample is larger and includes more recent years, we will test for asymmetry and determine whether such an effect belongs in our GARCH model.

Since our focus is to examine whether and how uncertainty, as well as the level and persistence of inflation, have changed in the present decade, we will allow for structural breaks. Most previous studies

of inflation uncertainty assume parameter constancy over four to six decades or allow for at most a one-time structural break. Caporale and McKiernan employ a sample from November 1947 through August 1994 and include one dummy for the high inflation years of 1973-1980. This dummy is only included in the conditional mean, AR portion of the model, however, and not in the GARCH process. Grief and Perry (1998) do not allow for any structural change over their 1948-1993 sample.

Given that it is unrealistic to expect parameter constancy over six decades, and that the large shocks of the past decade may have had palpable effects on the properties of inflation, we will adjust the traditional GARCH estimation in the following ways. We will add dummies to the model for each break point. Unlike Caporale and McKiernan, we will add dummies not just for the level of inflation, to test for a break in the mean, but will also add dummies for changes in persistence and changes in uncertainty. For each break, three dummies will thus be included in the GARCH model.

First, a dummy will be added to the ARMA model to test for changes in the level of inflation. Next, there will be an interaction term added to the ARMA model with the dummy multiplied by the AR(1) term. This latter term will measure changes in persistence. As noted, it is important to allow for changes in both the mean and persistence of inflation. Cechetti and Debelle (2006) discuss a number of prior studies and note that many which purport to find large changes in persistence may do so spuriously, as they fail to also control for changes in the level of inflation. Both the level and/or persistence may change over time, and attempting to control for only the one gives a misleading picture of how much the other has changed. The authors find that persistence has only modestly declined in recent years once a level break is allowed for. We do note that interacting only the first AR coefficient is not the optimal way to test for changes in persistence, as, optimally, all AR coefficients would be interacted to see how persistence has changed. Kontonikas (2004) employs an AR(4) model and also uses the interaction with only the first AR coefficient to measure changes in persistence. We are constrained by degree of freedom considerations in measuring persistence and again take the Markov-switching model results as most reliable, with these GARCH results mostly as a check and a comparison with other GARCH inflation papers.

Lastly, the GARCH process itself is augmented with a dummy to see if there is a break in uncertainty over a certain time period. The conditional variance will thus be estimated as


In this way, we can examine whether a particular break affected the level, persistence, and uncertainty of inflation.

As the focus of the paper is on how the properties of inflation may have changed since the end of the 1990s, our first dummy variable will cover January 2000 through September 2007. Then, in order to see how the present decade compares to the 1990s, we will create a dummy for the 1990s, specifically June 1991-December 1999. We omit the first 18 months of the decade due to fears of high oil prices caused by the first Gulf War. The 1990s may have been a golden era for the macroeconomy, as unemployment fell below what had been considered the noninflation accelerating (NAIRU) rate without triggering price increases. This second dummy will allow us to test just how "golden" the properties of inflation were over this decade.

While the focus is of course on inflation in the current decade, it could certainly be that other events have had large effects on inflation properties. It would give a good perspective to compare the effects of the current decade with those of some prior, well-known inflation shocks. Accordingly, we include dummies for the 1973-2007 years and the high inflation period of 1973-1980. These dummies are also relevant as Caporale and McKiernan (1997) included a dummy in the ARMA portion of their model for the latter years, and Grier and Perry (1998) split their sample at 1973.

Finally, the Greenspan years of 1987-2005 may or may not have exhibited greater inflation properties, and if so, it may or may not have been due to this particular chairman's leadership, but we will include a dummy for this period.

4. Empirical Results

Due to the special role of energy prices in the inflation dynamics, two different kinds of consumer prices are used. One is the monthly seasonally adjusted Consumer Price Index for All Urban Consumers (all items) in the United States (i.e., CPIALL), while the other one is the same consumer price excluding energy prices (i.e., CPILEG). In both cases, the inflation rate is calculated as the percentage change of the consumer price index a year ago. Both data sets are from the FREDS website of the St. Louis Federal Reserve Bank. (5) CPIALL inflation rates run from January 1948 through September 2007, while CPILEG inflation rates are available from January 1958 to September 2007.

Markov-Switching Results

By using the Markov-switching model, we will examine the time series properties of U.S. inflation over the past 60 years with a focus on the post-1996 period. (6) To examine the change before and after 1997, three time periods are adopted: (7) January 1948-September 2007, January 1948-December 1996 and January 1997-September 2007, referred to as periods I, I.a, and I.b for CPIALL; and January 1958-September 2007, January 1958-December 1996 and January 1997-September 2007, referred to as periods II, II.a, and II.b for CPILEG.


We will first examine the inflation dynamics over the past 60 years and see if they are consistent with the results of other research. Figure 1 shows the actual inflation rate (the solid line) and the probability that [S.sub.t] = 1 (the dashed line) for the period of April 1948-September 2007. The periods of high probability of being in [S.sub.t] = 1 occur in most of the 1950s, 1973-1975, 1981-1983, and 2005-2007, with a few spikes in March 1986-May 1986 and April 2000-August 2000. The high inflation variance in the 1950s is consistent with the findings in the literature that the high variance during that period may be caused by the price controls during World War II and the Korean War (Barro 1989). As for the high inflation variances in 1973-1975 and 1981-1983, the evidence is consistent with the results of Balke and Fomby (1991), Kim (1993), and Byrne and Davis (2004). Balke and Fomby used GNP deflator and found evidence of large shocks in 1968 and 1983. By distinguishing between transitory and permanent shocks, Kim (1993) found major positive shifts in variance of permanent shocks around those time periods. Byrne and Davis (2004) used Kim's model and found that the high variance of permanent shocks occurred in the 1970s and early 1980s. A high variance of transitory shocks occurred in the 1960s through the 1970s and again in the late 1980s. Our model does not distinguish between permanent and temporary shocks. However, the spike in the late 1980s seems to coincide with the findings of Byrne and Davis' high variance of temporary shocks. In our model, for the post-1996 period, the positive shift in the inflation variance tends to cluster around the period of 2005-2007.

Table 1 provides the estimates of six different models: two different price indices in three different time periods, respectively. We will focus primarily on shifts in mean, persistence, and variance of inflation as the regime changes. First, the estimate of [[alpha].sub.1] signals a shift in the mean of inflation as [S.sub.t] varies from 0 to 1. In general, the shift in the mean of inflation rate in different regimes is not statistically significant. The only exception is the one in II.b, indicating that there is a significant decline in the mean inflation rate in regime 2 (i.e., [S.sub.t] = 1). Thus, for the post-1996 period, given that the inflation rate excluding energy prices has a significant negative shift, but the inflation rate including energy prices has an insignificant positive shift, energy prices seem to contribute to the insignificant shift in inflation mean as regime changes (i.e., [S.sub.t] varies from 0 to 1).

Second, as mentioned, [rho] is a measure of persistence, and [[phi].sub.1] = [rho] - 1 (or [[phi].sub.1] + [[phi].sub.2][S.sub.t] = [rho] - 1). The value of [[phi].sub.2] indicates a shift in persistence as the value of St shifts from 0 to 1. When energy prices are not included in the inflation rate, the persistence of shock is approximately the same in both high and low inflation uncertainty regimes because [[phi].sub.2] is not significantly different from zero in II, II.a, and II.b. However, when energy prices are included in the inflation rate, the estimate of [[phi].sub.2] being negative and significantly different from zero in the periods of I and I.a indicates that the persistence of a shock declines in the regime that inflation uncertainty is high (i.e., [rho] goes to a lower value as St varies from 0 to 1). Though the post-1996 period has a larger absolute value of [[phi].sub.2] (-0.293 compared to -0.026), indicating that the inflation shock is not as persistent as it used to be, the estimate (-0.293) is not significantly different from zero. However, the estimates of [[phi].sub.2] in I and I.a (i.e., -0.030 compared with -0.026), which are both significantly different from zero, suggest that the persistence of a shock may indeed decline very slightly for the post-1996 period, comparing with the period before January 1997.

Third, for the dynamics of inflation uncertainty, the estimate of [h.sub.1] indicates the significance and magnitude of the change in variance (or uncertainty) as the regime changes. Both including and excluding energy prices, there is an apparent change of inflation uncertainty from regime 1 to regime 2. This is obvious because all the estimates of [h.sub.1] are significantly different from zero at the 1% level, except for the estimate in I.b, where it is still significantly different from zero at the 10% level. In the corresponding high variance regimes, the uncertainty is larger when energy prices are included in the inflation rate than when they are excluded. This can be seen from all the larger [h.sub.1] estimates in I, I.a, and I.b, compared to those corresponding estimates in II, II.a, and II.b. Even though it is only significant at the 10% level, the estimate of [h.sub.1] in the post-1996 period is larger than that of the period before 1997 (0.384 compared to 0.318), indicating a larger variance in the high uncertainty regime in the post-1996 period for prices including energy prices. Compared to their counterparts in prices excluding energy prices, where the estimate of hi is higher in II.a than that in II.b, energy prices seem to be the contributor to the larger variance in the high uncertainty regime (i.e., [S.sub.t] = 1) of the post-1996 period.

GARCH Results

For the GARCH model, we employ twelve AR lags in the ARMA portion for both the CPIALL and the CPILEG. This will capture annual effects. Moreover, Grier and Perry (1998) employed this specification in their seminal paper. We then test the residuals for ARCH effects, and in each series we can reject the null hypothesis of no such effects at less than the 5% level (results available upon request). We fit a GARCH(1,1) model to each series. This is the most standard GARCH specification and again was employed in Grier and Perry (1998) and Caporale and McKiernan (1997). A priori, it may not be adequate, so we test for any remaining GARCH effects. We cannot reject the null of no such effects for either series (results available upon request), and thus the GARCH(1,1) specification adequately captures the conditional volatility of each series.

We next test for a threshold, or asymmetric effect, in each GARCH series, as explained in Equation 4. The T-GARCH term is negative and significant for both the CPIALL and the CPILEG, indicating that negative shocks to inflation increase uncertainty by less than positive shocks. Table 2 displays our baseline models for our two inflation measures.

Note that the estimates of "DUM," "AR1DUM," and "ARCHDUM" in Table 3 indicate the changes in the level, persistence, and uncertainty of inflation. In the first column of Table 3, the dummy for the current decade indicates no significant change in level or persistence of inflation, but a significant, positive effect on uncertainty. This confirms the Markov-switching model results. In Table 4, the CPILEG model fails to obtain convergence when the 2000s dummy is included.

The 1990s dummy has a slight, positive effect on the level, but a negative, significant impact on both the persistence and uncertainty of inflation. This is further evidence that the 1990s was a "golden era" for inflation, in which unemployment fell below the rate that was previously thought of as the natural rate but inflation remained low. Both the 2000s dummy and the 1990s dummy give formal confirmation of the picture presented in Figure 2, which is a plot of the baseline GARCH model for the CPI. As observed, there is a very flat period for the 1990s and then a pronounced and disturbing increase in the current decade. The GARCH model for CPILEG again fails to attain convergence when the 1990s dummy is included. This may be attributable to more stable energy prices over the 1990s and hence less volatility, conditional or otherwise.


For the post-1973 years, CPIALL displays a positive reaction in persistence but a negative display in the level, with no change in uncertainty. When the relatively high inflation years (1973-1980) are investigated, there is no significant effect on any inflation property. For the CPILEG, the post-1973 years are characterized by lower uncertainty but no difference in the mean or persistence. The 1973 1980 dummy has a positive impact on persistence, a negative effect on the mean, and a negative effect on uncertainty. Finally, the Greenspan years of 1987-2005 had no significant effect on inflation properties for CPIALL but a slightly positive impact on the mean and a negative impact on uncertainty for CPILEG.

5. Conclusion

Several recent working papers (Mishkin 2007; Razin and Binyamini 2007), as well as a report in The Economist (Curve Ball 2006), indicate that the Phillips Curve relationship between inflation and unemployment has flattened in the United States since 1998. This is reflected in the decline in unemployment below the level many economists believed would trigger price changes, without actually causing a significant increase in inflation. Unfortunately, according to the article in The Economist, the corollary is that if inflation indeed rises in the future it will be very costly to reduce in terms of increased unemployment. It is then crucial, according to the article, that the central bank be credible in such a situation. The Fed focuses on core prices and its personal consumption expenditure index, but headline prices may go higher because of commodity price inflation. The article concludes that if agents become concerned over high commodity prices, they may demand higher wages and make the Fed's job of maintaining price stability quite difficult.

Our findings that inflation uncertainty has risen in the present decade, and that higher energy prices are the likely culprit, are thus not good news. Moreover, policy implications are unclear. This is especially the case in light of the possibility that much of the credit for the improved inflation performance, both in the United States and throughout the world, is allocated to an improved environment, rather than just better policy (Rogoff 2007). If a calmer global environment was partially responsible for the lower uncertainty of the 1990s, the greater turbulence of recent years may be in whole or in part at the root of higher uncertainty of this decade. Indeed, Rogoff wonders about the effect of terrorist attacks and wars on current inflation.

Some would suggest an inflation target as a means to lower uncertainty. A clear, formal target may in principle anchor expectations more than the Fed's "just do it" approach, and decrease uncertainty. However, Ball and Sheridan (2005) find, contrary to some previous research, that inflation targets are not associated with lower inflation or expectations, so there may be no significant effect on uncertainty arising from a formal target. In any event, given the negative effect on investment, contracting, and output, the recent increase in inflation uncertainty complicates policy-making and is an unwelcome development.

We would like to thank the Barton School of Business at Wichita State University for support and two anonymous referees for helpful comments.

Received June 2007; accepted January 2008.


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William Miles * and Chu-Ping C. Vijverberg ([dagger])

* Department of Economics, Wichita State University, 1845 Fairmount, Wichita, KS 67260-0078, USA; E-mail; corresponding author.

([dagger]) Department of Economics, Wichita State University, 1845 Fairmount, Wichita, KS 67260-0078, USA; E-mail

(1) For further details on the specifics of this model, see Bollerslev 1986.

(2) For an explanation of ARMA models see Box and Jenkins 1976.

(3) We benefit from the Gauss programs provided in the book of Kim and Nelson (1999) regarding the Markov-switching Model. However, all the Markov-switching programs used in this paper are coded in S+ and R.

(4) Due to the programming complexity in dealing with various state variable statuses at different time points, it is not trivial to have a model with a long lag. As the lag length of the model increases, the time-dimension of the state variable increases substantially. For example, if the lag length is l (= p - 1 as shown in the equation above), the time- dimension of the state variable is a [2.sup.l+t] x l + l matrix. Thus, for a lag length of l = 2, the time-dimension of the state variable G is an 8 x 3 matrix, i.e., [G.sup.t] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the three columns of G represent the statuses of the state variable at t - 2, t - 1, and t, respectively. Then for l = 3, G is a 16 x 4 matrix; for l - 4, G is a 32 x 5 matrix.


(6) We actually prefer to use January 2000 as the starting point for a "recent" period. However, given the number of the parameters in the model and the number of observations after December 1999 (=93 observations), the computer program is not able to find a Hessian matrix to calculate the standard errors and t values for the estimates of the model. The additional 36 observations (i.e., starting at January 1997) are needed for our computer program to generate the Hessian matrix.

(7) Though the structural change is endogenous in a Markov-switching model, the structural change is characterized by the parameters of the model. The reason for estimating the model over different time periods is to observe if there are any significant changes in the parameters before and after 1996.
Table 1. Parameter Estimates of the Six Markov-Switching Models
                          I                     I.a

Parameter Estimates   CPIALL 1948:1-2007:9   CPIALL 1948:1-1996:12

[[alpha].sub.0]           0.000(0.992)           0.007(0.818)
[[alpha].sub.1]           0.078(0.238)           0.061(0.384)
[[phi].sub.1]             0.003(0.624)           0.002(0.826)
[[phi].sub.0]            -0.030(0.012)          -0.026(0.032)
[[gamma].sub.1]           0.338(0.000)           0.347(0.000)
[[gamma].sub.0]          -0.002(0.952)           0.062(0.139)
[h.sub.0]                 0.269(0.000)           0.268(0.000)
[h.sub.1]                 0.319(0.000)           0.318(0.000)
[??]                      0.976(0.159)           0.980(0.144)
[??]                      0.987(0.103)           0.990(0.126)
nob                     714                    585
Log-like                -271.77                -220.96

                         I.b                    II

Parameter Estimates   CPIALL 1997:1-2007:9   CPILEG 1958:1-2007:9

[[alpha].sub.0]           0.105(0.370)           0.033(0.226)
[[alpha].sub.1]           0.876(0.229)           0.018(0.674)
[[phi].sub.1]            -0.043(0.305)          -0.013(0.116)
[[phi].sub.0]            -0.293(0.225)           0.004(0.719)
[[gamma].sub.1]           0.278(0.005)           0.153(0.000)
[[gamma].sub.0]          -0.411(0.000)           0.098(0.017)
[h.sub.0]                 0.248(0.000)           0.132(0.000)
[h.sub.1]                 0.384(0.081)           0.208(0.000)
[??]                      0.385(0.121)           0.997(0.379)
[??]                      0.897                  0.998(0.280)
nob                     126                    594
Log-like                 -32.48                  34.35

                        II.a                   II.b

Parameter Estimates   CPILEG 1948:1-1996:12  CPILEG 1997:1-2007:9

[[alpha].sub.0]           0.065(0.3l2)           0.352(0.009)
[[alpha].sub.1]          -0.023(0.757)          -0.269(0.042)
[[phi].sub.1]            -0.019(0.246)          -0.105(0.042)
[[phi].sub.0]             0.011(0.535)           0.043(0.437)
[[gamma].sub.1]           0.158(0.00l)          -0.025(0.795)
[[gamma].sub.0]           0.118(0.0l4)           0.019(0.865)
[h.sub.0]                 0.157(0.000)          -0.082(0.000)
[h.sub.1]                 0.186(0.000)           0.163(0.000)
[??]                      0.998(0.395)           0.736(0.189)
[??]                      0.997(0.459)           0.454(0.013)
nob                     465                    126
Log-like                 -54.38                  96.68

Numbers in the parentheses are the p-values. nob = number of

Table 2. GARCH Baseline Models

             CPI           CPILEG

ARCH     0.09 (0.01)    0.05 (0.00)
GARCH   0.923 (0.000)   0.984 (0.00)
TARCH   -0.08 (0.04)    -0.1 (0.00)

p-values are in parentheses.

Table 3. GARCH Models of CPIALL Inflation

            2000-2007:9     1991:06-1999:12      87:9-2005

DUM       -0.088 (0.79)      0.500 (0.00)      0.059 (0.89)
ARIDUM     0.0005 (0.99)    -0.205 (0.006)    -0.079 (0.38)
ARCH       0.044 (0.03)      0.084 (0.01)      0.098 (0.01)
GARCH      0.968 (0.00)      0.914 (0.00)      0.915 (0.00)
TARCH     -0.07 (0.01)      -0.08 (0.04)      -0.086 (0.049)
ARCHDUM    0.0044 (0.00)    -0.003 (0.014)    -0.0012 (0.324)

                73               73-80

DUM        -1.64 (0.00)      0.23 (0.256)
ARIDUM     0.453 (0.00)      0.025 (0.491)
ARCH       0.103 (0.00)      0.092 (0.01)
GARCH      0.918 (0.00)      0.915 (0.00)
TARCH      -0.094 (0.01)     -0.074 (0.06)
ARCHDUM   -0.0005 (0.595)    0.003 (0.47)

p-values are in parentheses. The periods at the top are the dummy
variables. DUM refers to the dummy in the AR model, ARIDUM is the
interaction between the dummy and the AR(1) coefficient in the AR
model, and ARCHDUM is the dummy in the GARCH model.

Table 4. GARCH Models of CPILEG Inflation

           2000-2007:9     1991:06-1999:12       87-05

DUM                                           0.296 (0.07)
ARCDUM                                       -0.11 (0.108)
ARCH                                          0.05 (0.002)
GARCH                                         0.9577 (0.000)
TARCH                                        -0.08 (0.008)
ARCHDUM                                      -0.001 (0.06)

                73              73-80

DUM       -0.07 (0.833)    -3.11 (0.001)
ARCDUM    -0.033 (0.715)    0.858 (0.004)
ARCH       0.04 (0.000)     0.26 (0.022)
GARCH      1.000 (0.000)   -0.183 (0.000)
TARCH     -0.13 (0.000)    -0.431 (0.000)
ARCHDUM   -0.0014 (0.03)   -1.64 (0.000)

p-values are in parentheses. The periods at the top are the dummy
variables. DUM refers to the dummy in the AR model, AR 1 DUM is the
interaction between the dummy and the AR(1) coefficient in the AR
model, and ARCHDUM is the dummy in the GARCH model. There are no
results for the first two columns, as convergence was not obtained
in those cases.
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Comment:Changing inflation dynamics and uncertainty in the United States.
Author:Miles, William; Vijverberg, Chu-Ping C.
Publication:Southern Economic Journal
Geographic Code:1USA
Date:Jan 1, 2009
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