The multivariate Beveridge-Nelson decomposition with I(1) and I(2) series

- We consider joint estimation of the natural rates and gaps of macroeconomic variables.
- By the consumption Euler equation, if the real interest rate is I(1), then log output is I(2).
- We develop the multivariate Beveridge-Nelson decomposition when some series are I(1) and others are I(2).
- We apply the method to Japanese data during 1980Q1-2013Q3.

The consumption Euler equation implies that the output growth rate and the real interest rate are of the same order of integration; i.e., if the real interest rate is I(1), then so is the output growth rate and hence log output is I(2). To estimate the natural rates and gaps of macroeconomic variables jointly, this paper develops the multivariate Beveridge-Nelson decomposition when some series are I(1) and others are I(2). The paper applies the method to Japanese data during 1980Q1-2013Q3 to estimate the natural rates and gaps of output, inflation, interest, and unemployment jointly.