# RDP 9611: A Markov-Switching Model of Inflation in Australia 2. Markov-Switching Models

There has been some debate in the literature about the correct characterisation of inflation dynamics. A framework emphasising the integrated nature of inflation has been popular for some time. An integrated process is one which is non-stationary: shocks to the level of the series are permanent rather than temporary. This paper makes use of an alternative time-series characterisation for inflation that allows for distinct and differing periods of inflationary behaviour, each characterised by its own time-series properties. This alternative approach has both intuitive and empirical support. It describes the inflation process as being governed by two different regimes where switches between them are based on a probabilistic process. This approach is intuitively appealing, as the behaviour of economic time series often seems to go through distinct phases. It is also consistent with the fact that inflation is often found to be integrated of order 1 (ie non-stationary) – breaks in the mean of a series could lead to that series appearing to be non-stationary.[2]

The methodology employed is a ‘Markov-switching model’. A Markov process is one where the probability of being in a particular state is only dependent upon what the state was in the previous period. Transitions between differing regimes are governed by fixed probabilities. Similar analysis in the literature has commonly been univariate – no independent variables have been included in modelling the series of interest. Initial work was done by Hamilton (1989,1990) with applications to business cycles. Recent work by Evans and Wachtel (1993) and Ricketts and Rose (1995) has applied the technique to inflation. This technique has several advantages, including endogenising structural breaks and encompassing ARCH models, each of which is discussed in more detail below. The technical details of the ‘Hamilton filter’ estimation are discussed in the Appendix, and the particular Markov-model specification is discussed in Section 3 below.

## 2.1 Structural Breaks

The Markov-switching model posits that two (or more) regimes could have prevailed over the course of history. However, it differs from models with imposed breaks in that the timing of breaks is entirely endogenous. Indeed, breaks are not explicitly imposed, but inferences are drawn on the basis of probabilistic estimates of the most likely state prevailing at each point in history.

Estimates of parameters for the two most likely regimes are generated using maximum likelihood techniques. With the parameters identified, it is then possible to estimate the probability that the variable of interest (in this case inflation) is following one of the alternative regimes. This involves identifying where in the probability distribution of each regime the observation falls at each point in time. That is, the likelihood is calculated for each possible state. The probability that a particular state is prevailing is obtained by dividing the likelihood of that state by the total likelihood for both states. Thus, the sum of all the probabilities will equal one. With this estimate of the probabilities it is common to infer that a state is prevailing when the probability estimate for that state is greater than 50 per cent. In the models considered in this paper, values close to zero or one tend to occur, making identification of the prevailing state relatively easy.

## 2.2 Inflation Uncertainty

In practice, univariate models of inflation have commonly been characterised as having non-constant variance (commonly modelled as autoregressive heteroskedastic errors (ARCH)).[3] It is possible that these findings are related to the common observation that inflation is more variable during periods of high inflation. Ball (1992) posits a model where high inflation is associated with regime uncertainty whereas low inflation is not. Consequently, he suggests that the observation that high inflation has higher variance than low inflation reflects regime uncertainty. Another angle on the correlation between inflation and inflation variability is presented by Taylor (1981). His paper suggests that countries which place a high weight on output and employment stabilisation will have correspondingly higher and more variable inflation and vice versa. Thus, high inflation may be associated with high variability because it is correlated with choices by policy makers to focus on output and employment rather than inflation.

Whichever is the true explanation, regime switches, posited as the cause of higher volatility in high-inflation periods, could result in the identification of ARCH errors in single regime models. That is, ARCH processes suggest that volatility in one period is related to volatility in previous periods and that this volatility changes over time. If we allow for two possible states, one with a higher volatility than the other, then single regime models (ie. traditional models) might incorrectly identify the changes in volatility as symptomatic of ARCH. Instead, by correctly modelling the regime shifts, one possible reason for the finding of ARCH errors is eliminated. The estimation procedure thus allows us to investigate separately two questions with respect to inflation uncertainty or variability. First, are high-inflation states associated with high variability of inflation? And second, are ARCH effects present within regimes, implying that volatility within regimes tends to persist following a shock? These issues are discussed in the light of the results obtained in Section 4.

## Footnotes

The point has been made by Perron (1989) and reiterated frequently within this literature. Clearly, given enough breaks, any I(1) series could be indistinguishable from an I(0) series around a broken trend. However, by allowing a finite number of processes, the Markov methodology is not directly open to this sort of criticism. Also, it is still modelling the time-series properties within each regime and obtaining estimates which are statistically distinguishable from one I(1) series. If a series were truly I(1), it should be found to be I(1) within each sub-period; it is only statistical error that would prevent definitive findings. [2]

For example, Mishkin and Simon (1995) find significant ARCH in their investigation of inflation in Australia. [3]