Can someone explain how the Beveridge-Nelson Decomposition works? So far all I know is it estimates trend cycles in non stationary time series data.
I looked at multiple journal articles and I am still confused on how it works http://research.economics.unsw.edu.au/jmorley/bn.pdf
Beveridge-Nelson decomposition is a decomposition of $ARIMA(p,1,q)$ process. Such process has a unit root:
$$y_t=y_{t-1}+u_{t},$$
but $u_t$ is not a white noise process, it is an $ARMA(p,q)$ process. What Beveridge and Nelson in their original article observed is that it is possible to decompose this process into two parts:
$$y_t=\tau_t+\xi_t,$$
where $\tau_t$ is now "pure" random walk, i.e. $\tau_t=\tau_{t-1}+\varepsilon_t$, where $\varepsilon_t$ is a white noise proces. The term $\xi_t$ is another stationary process. This decomposition is algebraic identity (the details below), but it can lead to interesting interpretations.
The precise statement. Let $u_t=\sum_{j=0}^\infty \psi_{j}\varepsilon_{t-j}$, where $\varepsilon_t$ is a white noise process and $\sum j|\psi_j|<\infty$. Then
$$u_1+...+u_t=\psi(1)(\varepsilon_1+...+\varepsilon_t)+\eta_t-\eta_0,$$
where
$$\psi(1)=\sum_{j=0}^\infty\psi_j,\quad \eta_t=\sum_{j=0}^\infty\alpha_j\varepsilon_{t-j},\quad \alpha_j=-(\psi_{j+1}+\psi_{j+2}+...), \quad \sum|\alpha_j|<\infty.$$
This decomposition has nice application, for example
$$\frac{1}{\sqrt{T}}\sum_{t=1}^Tu_{t}=\frac{1}{\sqrt{T}}\psi(1)\sum_{t=1}^T\varepsilon_t+\frac{1}{\sqrt{T}}(\eta_t-\eta_0)\to N(0,[\psi(1)\sigma]^2),$$
where we apply the central limit theorem for the first term and observe that the second term goes to zero, due to stationarity (mean is zero and variance of term goes to zero, due to T in the denominator).
So we get that limiting behaviour of ARIMA(p,1,q) process is simply the same as for a ARIMA(0,1,0) process. This fact is used a lot in the time series literature. For example Phillips and Perron unit root test is based on it.
