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Distribution of the Estimators for Autoregressive Time Series With a Unit Root

Опубликовано на портале: 06-04-2004
Journal of the American Statistical Association. 1979.  Vol. 74. No. 366. P. 427-431. 
Тематический раздел:
Abstract Let $n$ observations $Y_1, Y_2, \ldots, Y_n$ be generated by the model $Y_t = \rho Y_{t - 1} + e_t$, where $Y_0$ is a fixed constant and $\{e_t\}_{t = 1}^n$ is a sequence of independent normal random variables with mean 0 and variance $\sigma^2$. Properties of the regression estimator of $\rho$ are obtained under the assumption that $\rho = \pm 1$. Representations for the limit distributions of the estimator of $\rho$ and of the regression $t$ test are derived. The estimator of $\rho$ and the regression $t$ test furnish methods of testing the hypothesis that $\rho = 1$.

David A. Dickey - Testing for Unit Roots Definition of a Unit Root Process and How to Test for Unit Roots

  • AR(1) models
    • Model: Yt - m = r ( Yt-1 - m ) + et
    • Yt = observation at time t
    • et = error or "shock" at time t (assumed iid normal)
    • m = series mean (assumed constant over time)
    • r = Autoregressive coefficient

  • Test of Ho: r =1
    • If r =1 then mean m drops out of model.
    • If r =1 forecast does NOT revert to mean.
    • Stock price example: if r =1 then can't make money by buying low and selling high.

  • Test construction
    • subtract ( Yt-1 - m ) from both sides of model
    • Reparameterized Model: Yt - Yt-1 = (r -1)( Yt-1 - m ) + et
    • Compute **First Difference** Dt = Yt-Yt-1
    • Regress Dt on 1, Yt-1
    • Test coefficient of Yt-1
      • Distribution is nonstandard. n( r _hat - 1) is Op(1).
      • Tables in Fuller, Introduction to Statistical Time Series
      • Distribution (of coefficient)does NOT hold in higher order models (more lags)
    • t-test on coefficient of Yt-1
      • Distribution is nonstandard. t test is Op(1).
      • Tables in Fuller, Introduction to Statistical Time Series
      • Distribution of t DOES hold (is same asymptotically) in higher order models.

  • Detrending

  • Higher Order Models
  • Yt - m = a 1 [Yt-1 - m ] + a 2 [Yt-2 - m ] +, ..., + a p[Yt-p -m ]+et

    • Characteristic polynomial is:
    • mp- a 1 mp-1 - a 2 mp-2 - ... - a p
      • If m=1 is a root then 1 - a 1 - a 2 - ... - a p = 0 (definition of root)
      • If m=1 then m [1 - a 1 - a 2 - ... - a p] is 0 regardless of m
      • If m=1 then no mean m in model (not identifiable)
      • No mean reversion
      • Unit root is generalization of r =1 to higher order models.
    • Compute first difference Dt = Yt -Yt-1 and its lagged values.
    • Regress Dt= Yt -Yt-1 on 1, Yt-1, Dt-1 ,...,Dt-p-1
      • True coefficient on Yt-1 is -[1 - a 1 - a 2 - ... - a p] (do the algebra!)
      • Test coefficient of Yt-1 against t (t ) tables in Fuller.
      • Test available in SAS (and other packages)
      • Test discussed in Box and Jenkins (newest edition), Fuller, and many other texts.

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