Statistical Analysis of Some Nonstationary Time Series
Søren Johansen · Cambridge University Press eBooks · 2012
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 p t au , pf t us , exch t , i t au , and i t us 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).