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Stochastic trend analysis

Berveridge-Nelson Decomposition

The Beveridge-Nelson decomposition[1] divides a time series into two stochastic components, given that the time series is non-stationary (has a unit root) or in other terms that the series can be modeled as an ARIMA(p, 1, q) process with drift.

Unlike in other models the decomposition of time series $y$ in terms of trend $\tau$ and cycle $c$ components is defined as:

\[ y_t = \tau_t - c_t\]

This package offers two ways of computing the decomposition, one by Miller[2] for autoregressive models and one by Newbold[3] for ARMA models. For the latter it is also possible to calculate the decomposition with already estimated parameters of an ARIMA(p, 1, q) model, otherwise the function armaNR (see ARMA(p, q)) is used.

TrendDecomposition.beveridgeNelson — Function
beveridgeNelson(X :: Vector, p :: Int; method::Symbol=:Newbold, estimator::Symbol=:OLS)

Computes the Beverdige-Nelson decomposition of a non-stationary time series X using an autoregressive model for estimation with p-lags. As estimators :OLS, :burg, :yuleWalker can be used.

Either uses Newbold's (1990) or Miller's (1987) method of computation.

Returns a (Nx1) vector of the trend component.

source
beveridgeNelson(X :: Vector, p :: Int, q :: Int)

Computes the Beverdige-Nelson decomposition of a non-stationary time series X using an ARMA(p, q) model.

Uses Newbold's (1990) method of computation.

Returns a (Nx1) vector of the trend component.

source
beveridgeNelson(X :: Vector, ϕ :: Vector, θ :: Vector)

Computes the Beverdige-Nelson decomposition of a non-stationary time series X using an ARMA(p, q) model of ΔX given the AR parameter ϕ and MA parameter θ.

Uses Newbold's (1990) method of computation.

Returns a (Nx1) vector of the trend component.

source
  • 1

    Beveridge, S., & Nelson, C. R. (1981). A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’. Journal of Monetary economics, 7(2), 151-174.

  • 2

    Miller, S. M. (1988). The Beveridge-Nelson decomposition of economic time series: Another economical computational method. Journal of Monetary Economics, 21(1), 141-142.

  • 3

    Newbold, P. (1990). Precise and efficient computation of the Beveridge-Nelson decomposition of economic time series. Journal of Monetary Economics, 26(3), 453-457.