Abstract
We propose a test to discern between an ordinary autoregressive model, and a random coefficient one. To this end, we develop a full-fledged estimation theory for the variances of the idiosyncratic innovation and of the random coefficient, based on a two-stage WLS approach. Our results hold irrespective of whether the series is stationary or nonstationary, and, as an immediate result, they afford the construction of a test for ”relevant” randomness. Further, building on these results, we develop a randomised test statistic for the null that the coefficient is non-random, as opposed to the alternative of a standard RCA(1) model. Monte Carlo evidence shows that the test has the correct size and very good power for all cases considered.
Highlights
In this paper we study the Random Coefficient Autoregressive (RCA) modelXt = (φ + bt) Xt−1 + et, 1 ≤ t < ∞, where X0 is an initial value. (1.1)Model (1.1) has been paid considerable attention by the literature, mainly due to its flexibility and analytical tractability
This set-up is arguably of great importance: when τ 2 = 0, equation (1.1) is the standard unit root process; the case in which φ = 1 and τ 2 > 0 - known in the literature as a Stochastic Unit Root (STUR) process - represents as a series which has periods of explosive and stationary dynamics
We state and discuss the main assumptions, and derive the weighted least squares (WLS) estimator in Section 2, where we study the asymptotics of the estimator of τ 2
Summary
The econometric literature has investigated the use of RCA models as a flexible alternative to a unit root specification; in particular, tests have been developed for H0 : τ 2 = 0 when φ = 1 This set-up is arguably of great importance: when τ 2 = 0, equation (1.1) is the standard unit root process; the case in which φ = 1 and τ 2 > 0 - known in the literature as a Stochastic Unit Root (STUR) process - represents as a series which has periods of explosive and stationary dynamics (see the seminal contribution by Granger and Swanson, 1997). The proofs of the main results, further discussion and numerical evidence are reported in the online supplement
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