One of the most basic and useful of the time series models is the order 1 (1 lag) autoregressive model, denoted AR(1) and given by Y t − μ = ρ(Y t − 1 − μ) + e t where Y t is the observation at time t, μ is the long run mean of the time series and e t is an independent sequence of random variables. We use this venerable model to illustrate the Dickey–Fuller test then mention that the results extend to a broader collection of models.
When written as Y t = μ(1 − ρ) + ρ Y t − 1 + e t , or more convincingly as Y t = λ + ρ Y t − 1 + e t , with e independent and identically distributed as N(0, σ 2), the AR(1) model looks like a regression with errors satisfying the usual assumptions. Indeed the least squares estimators of the coefficients are asymptotically unbiased and normally distributed under one key condition, namely that the true ρ satisfies | ρ | < 1. It appears that this assumption is quite often violated. Many prominent time series appear to have ρ = 1, in which case Y ...
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Dickey, D.G. (2011). Dickey-Fuller Tests. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_210
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DOI: https://doi.org/10.1007/978-3-642-04898-2_210
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