Beveridge-Nelson Decomposition

The Beveridge-Nelson (BN) Decomposition breaks down a time series, $y_t$, into a stochastic trend and stationary cycle. The trend, or permanent component, represents the long-run forecast of $y_t$ at time $t$, after all transitory shocks have died out. In fact, the BN decomposition is defined so that this is true.

Consider the time series $y_t=z_t+c_t$ where $z_t=\mu +z_{t-1}+({\sum }_{j=0}^{\infty }{a}_j){ϵ}_t$ $-c_t=({\sum }_{j=1}^{\infty }{a}_j){ϵ}_t+({\sum }_{j=2}^{\infty }{a}_j){ϵ}_{t-1}+({\sum }_{j=3}^{\infty }{a}_j){ϵ}_{t-2}+...$ Here $z_t$ is the trend and $c_t$ is the cycle.

$z_t$ essentially counts the full impact of of the current shock, even future effects that have not yet occurred. Since it incorporates its own lag, it also by definition includes the full impact of all past shocks as well. In a sense, the permanent component “books all the revenue upfront”.

$c_t$ cumulates all the temporary impacts from past shocks that have not yet transpired in the current level of the series. In other words, it only books the “future revenue”, dropping anything that’s been booked already.

The net of “all impacts past, present, and future” and “impacts in the future” is “impacts past and present”, i.e. the time series itself.

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References

@cochraneHowBigRandom1988

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