The asymptotic behaviour of the Dickey–Fuller tests under the crash hypothesis

Limiting behaviour of Dickey–Fuller F‐tests under the crash model alternative

“…We also present finite sample simulation evidence to assess the performance of all unit root statistics, namely, T (ρ −1), τ τ , 2 , and 3 . The simulation evidence for τ τ and T (ρ −1) complements the findings in Montañés and Reyes (1999). Our simulations indicate that the power of τ τ , T (ρ − 1), and 3 can be low in finite samples when the intercept break magnitude is large, especially if the break does not occur very early or late in the sample, or in the middle of the sample.…”

“…(0,σ u 2 ). Montañés and Reyes (1999) have shown that and , where A * =λ(1 − 4λ+ 6λ 2 − 3λ 3 ). Theorem describes the limiting behaviour of the unit root statistics Φ 2 and Φ 3 under the crash model alternative given in (4).…”

“…Otherwise their power can be quite low. Montañés and Reyes (1999) noted that the power of the normalized estimator is larger than the power of the pseudo t‐ratio. However, our results reveal that the power of the pseudo t‐ratio is larger than the power of the normalized estimator when the break occurs early in the sample, see the results for τ τ and when λ= 0.1.…”

“…For example, Perron (1989) used the changing growth model for the postwar quarterly real GNP, the mixed model to describe real wages and common stock prices, and the crash model for all other Nelson–Plosser series 2 . The limiting behaviour of the τ τ and statistics under the crash model was derived by Montañés and Reyes (1999) under the assumption that the error term is i.i.d. (0,σ u 2 ).…”

Unit root tests allowing for a break in the trend function at an unknown time under both the null and alternative hypotheses

“…Perron and Vogelsang (1992a) and Lanne et al (2003) exploit this asymptotic negligibility in deriving the limiting distribution of their test statistics. Since the boundary values 0 and 1 are also permitted, even a standard unit root test without break is asymptotically robust to a shift in the intercept (see Montañés and Reyes (1999), for details). However, it is important to use a consistent estimate of the break fraction because the test statistics will have better power when the correct break date is used.…”