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Does Inflation Uncertainty Really Increase With Inflation?
Manchester School of Management, UMIST, PO Box 88, Manchester M60 1QD, UK
This paper proposes a two-factor model for analysing inflation uncertainty. Inflation pressure is relatively more persistent than policy intervention, which characterises the pattern of inflation uncertainty. Inflation uncertainty is found to increase with inflation in the medium to high range of inflation. Beyond that range, inflation uncertainty does not increase with inflation.
Key words: stochastic dummy, time-varying probability, uncertainty
JEL No: C51, E3
It is claimed that uncertainty about future inflation is higher the higher the inflation rate. This may be true in a certain range of inflation but not true beyond that range. Empirical studies have not yet paid attention to this prospect, yielding unsettled results and findings. Various research on this issue includes, prominently, Fischer (1986), Engle (1982), Ball (1992), Golob (1993, 1994), and Brunner and Hess (1993).
In this paper, we first propose an economic analytical framework in section 2, illustrating why inflation uncertainty would be time-varying and related to the level of inflation; then we translate this analytical framework into a testable econometric specification. Section 3 presents an empirical case to support our propositions. Finally, section 4 is a brief summary.
2. Modelling Inflation Uncertainty
2.1. The economic model and analytical framework
There are several conceivable descriptions about a possible link between the level of inflation and inflation uncertainty. In general, when inflation is regarded as high, the possibility of policy intervention, differing from the prevailing policies, becomes larger. However, the intervention is only a possibility and takes place with a probability which may increase with the level of or the shock in inflation. Therefore, future inflation may be more uncertain when the current inflation is above or around a certain level to possibly trigger a policy intervention and, consequently change the future evolutionary path in inflation. The other source of uncertainty is inflation "pressure" which is naturally inherited in the economic system, in its production and distribution processes. Unlike policy intervention, inflation pressure remains in the same state when inflation is very high as well as very low. Total uncertainty is the combined effect of these two sets of variables.
Let pt denote the inflation rate in time period t; ~ pt+1 the expected inflation rate in t+l formed in time t. The future inflation is specified as:
Where dzt is the stochastic dummy for inflation pressure, dgt is the stochastic dummy for policy intervention, z(pt) and g(pt) are the effects of inflation pressure and policy intervention on the level of inflation respectively, conditioning on the information set It. pz(pt) and pg(pt) (written as pz,t and pg,t thereafter) are time-varying probability when the respective dummy taking a value of one. In general, It is a vector of state variables for inflation, but is simplified and represented by the current inflation in this study. ~ pt+1, the expected inflation in the next period, is the sum of the current inflation, and the estimates/expectations of the effects of inflation pressure and policy intervention.
The model can be called the two-factor model. The expected level of inflation would be:
the variance (assuming these two variables are orthogonal for simplicity) in pt+1, would be:
and the variance of dzt being pz,t(1-pz,t) and the variance of dg,t being pg,t. It is clear not only the expected level of inflation, but also the variance in inflation, is time-varying if the probabilities are time-varying.
It is reasonable to assume that pg,t and pz,t, the probabilities of a policy intervention and high inflation pressure, are increasing functions of inflation. The critical point is to what extent they would increase with inflation. Notice pz,t(1-pz,t) reaches the maximum value when Pz,t is equal to 0.5. Therefore, if pz,t is constrained to [0, 0.5], then the variance pz,t(1-pz,t) is a strict increasing function of inflation. When pz,t is not constrained to the limit of 0.5, the variance pz,t(1-pz,t) would decrease as inflation is above a certain level and further increases. The same analysis applies to pg,t(1-pg,t).
To analyse the pattern of inflation uncertainty, we take the different characteristics of policy intervention and inflation pressure into consideration. Pgg,t, the probability of a policy intervention, would increase with inflation and may not (well) exceed 0.5 even if inflation is very high, as a policy intervention is never sure under any situations. In this regard, inflation uncertainty is almost an increasing function of inflation. However, pz,t would increase with inflation and would be close to one when inflation is very high, as inflation pressure is inherited in production and distribution processes and cannot be mitigated easily; therefore, inflation uncertainty would decrease when inflation is above a certain level. The joint effects of policy intervention and inflation pressure, as captured equation (3), are displayed in Fig 1, with two possible curves for the evolutionary path of inflation uncertainty.
Fig 1: Combined effects of inflation pressure and policy intervention
The two-factor model is well posed to explain the mixed results in empirical studies. As Friedman (1977) puts, "A burst of inflation produces strong pressure to counter it. Policy goes from one direction to the other, encouraging wide variation in the actual and anticipated rate of inflation". Obviously, inflation uncertainty will be higher if inflation is higher and, if the behaviour and effect of policy intervention alone is considered. However, the inherited pressure in production and distribution processes will remain high and continue to influence the future path of inflation in a less uncertain way, no matter which kind of policy is adopted to counter high inflation. The model helps explain why inflation uncertainty may or may not increase with inflation levels.
2.2. Econometric specifications
We use an autoregression for the mean process, similar to the mean equations in Stockton and Glassman (1987) and Golob (1994). But, the assumptions underlined by equation (1) are different. The mean process in our model is stochastic in state transitions. Hamilton (1989) shows that such a process would appear as an autoregression, but unlike the conventional autoregression, it would have different variances in different states.
Let us have two alternative hypotheses, H0: inflation uncertainty increases with inflation; and H1: inflation uncertainty increases with inflation in the range of (pL,pH), it does not increase when pt-1> pH or pt-1< pL. Expressing them in testable statistical forms:
where w1 is the dummy taking value of unity when pt-1< pL and zero otherwise, w2 is the dummy taking value of unity when pL£ pt-1< pH. and zero otherwise, w3 is the dummy taking value of unity when pt-1³ pH and zero otherwise. The coefficients b1, b2 and b3 represent three slopes in the three ranges of inflation. It is obvious that the specification of equation (5) encompasses that of equation (4). There are two advantages in applying equation (5). First, it is able to detect a more general inflation-inflation uncertainty pattern and the turning point. Second, if b1, b2 and b3 have different values, b in equation (4) is likely to be less statistically significant or statistically indifferent from zero, incorrectly ruling out a time-varying variance and any links between inflation uncertainty and inflation levels. In the following section, the two hypotheses will be empirically tested.
3. Results and findings
The US consumer price index (CPI) is used in this study. The data set runs from January 1960 to March 1995. The index is of monthly frequency but inflation rate is calculated on the year-on-year basis as reported in the press. This would result in serial correlation of up to 12 lags and is carefully taken into consideration in the mean equation. The results are reported in table 1.
Table 1: Tests for inflation uncertainty and inflation relations
Panel A is for hypothesis H0 but we look at Panel B, the hypothesis H1, first. It is found that inflation uncertainty increases with inflation in a wide range of inflation from about 4 percent to 10 percent, confirmed by a highly significant b1, at 1 percent level. Inflation uncertainty seems to decrease when the inflation rate is higher than 10 percent; however the coefficent b3 is not significant at all, implying inflation uncertainty would not further increase nor decrease beyond the 10 percent level. Rather suprisingly, instead of increasing with inflation, inflation uncertainty is a decreasing function of inflation at the lower end of inflation, as suggested by a negative and significant b1, for the inflation rate lower than 4 percent. This prompted us to use a nonlinear function with a local minimum and a local maximum for the time-varying variance named H1', and the statistics are presented in Panel C. With a local minimum of 3.6 percent and a local maximum of 11.7 percent, it confirms a similar pattern as revealed by Panel B with the slightly different turning points: Inflation uncertainty increases with inflation in the range of 3.6 percent to 11.7 percent of inflation rates, it decreases when inflation is lower than 3.6 percent. As b3 is close to be marginally significant (0.12% level), inflation uncertainty would probably decrease when inflation is higher than 11.7 percent.
The residual in all three specifications displays no serial correlation by the criterion of the Q statistic, though H1 is the best and H0 the worst. Both H1 and H1' are superior to H0 by the criterion of likelihood ratio statistic, while H1 is slightly better than H1'. This suggests that H0 is mis-specified. Indeed b in equation (4) is only significant at a modest level of 5 percent as it has to accommodate for the whole range of inflation and compromise for both positive (major influence) and negative/no (minor influence) links between inflation uncertainty and inflation. In addition, the coefficient b in Panel A is larger than b2 in Panel B while the intercept in Panel A is larger than that in Panel B, suggesting that the variation in the time-varying variance is exaggerated with H0. Fig 2 presents the empirical curve for inflation uncertainty and inflation as against its theoretical counterpart in Fig 1.
Fig 2: Empirical inflation -uncertainty - inflation curve
This paper proposes a two-factor model for analysing inflation uncertainty. The model states that there are two influential factors underlying and inducing inflation uncertainty and the two factors have rather different property. Inflation pressure is relatively more persistent than policy intervention, which characterises the pattern of inflation uncertainty. To empirically
verify the model, the econometric specification of differential slopes in different ranges of inflation is applied. There is one major finding in our study. Inflation uncertainty is found to increase with inflation in the medium to high range of inflation. Beyond that range, inflation uncertainty does not increase with inflation. In addition, our results have two implications.
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The simplest variance equation to have one local minimum and one local maximum is ht= a+b1pt-1+ b2(pt-1)2+ b3(pt-1)3. With this specification, not only the variance is the continuous function of inflation, but also it first derivative. The local minimum and maximum are obtained by letting the first derivative equal zero, and the former (0.0036) with a positive second derivative value (0.00749) and the latter (0.117) with a negative second derivative value (-0.00749).