Introduction

The purpose of this essay is to draw attention to some recent work on the statistical analysis of nonstationary processes that are integrated - in particular, integrated of order 2. A nonstationary process is integrated of order 1 if the differences are stationary, and it is integrated of order 2 if the second differences are stationary. In the analysis of some macroeconomic data, in particular price series, it is found that they are better described as I(2) series than as I(1) series. As an example, take the log of a price index; if the inflation rate is not stationary, it can perhaps be described as I(1), in which case the price series itself will be I(2).

Consider, for illustrative purposes, measurements of the consumer price index in Australia and the United States and the exchange rate and the bond rate from each country. The variables are ptau, pftus, excht, itau, and itus if, where the first three are in logs. The data are quarterly, 1972:1 to 1991:1, and were kindly provided by Tony Hall. The data were analyzed by Johansen (1996), where indication of I(2) was found.

Figure 13.1 shows the data in levels and differences. Figure 13.2 shows, for U.S. prices, levels, differences, and second differences in the left-hand panel; for comparison, the right-hand panel shows a sequence of independent identically distributed (iid) Gaussian variables and their cumulated sum (which is a random walk) and twice-cumulated sum. The price variable shows a development similar to the twice-cumulated e over the period, which indicates that the process may be I(2).