Bootstrap Unit Root Tests for Time Series With Nonstationary Volatility

Abstract

In this paper we aim to assess linear relationships between the non constant variances of economic variables. The proposed methodology is based on a bootstrap cumulative sum (CUSUM) test. Simulations suggest a good behavior of the test for sample sizes commonly encountered in practice. The tool we provide is intended to highlight relations or draw common patterns between economic variables through their non constant variances.The outputs of this paper is illustrated considering U.S. regional data. -C. (2009) Threshold effect of the economic growth rate on the renewable energy development from a change in energy price: Evidence from OECD countries. Energy Policy 37, 5796-5802. Dahlhaus, R. (1997) Fitting time series models to nonstationary processes.

Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

“…Beare (2008) shows that (iii) suffices for this purpose when using a nonparametric kernel estimator as is required here since we have not assumed a specific parametric model for the volatility process. As the conditions in Assumption 1 are stronger than those in Cavaliere and Taylor (2008) and Smeekes and Taylor (2011), the large sample validity of the bootstrap unit root tests discussed in the next section is guaranteed. The reader is directed to Cavaliere and Taylor (2008) and Beare (2008) for further discussion of the conditions imposed by Assumption 1.…”

“…As the conditions in Assumption 1 are stronger than those in Cavaliere and Taylor (2008) and Smeekes and Taylor (2011), the large sample validity of the bootstrap unit root tests discussed in the next section is guaranteed. The reader is directed to Cavaliere and Taylor (2008) and Beare (2008) for further discussion of the conditions imposed by Assumption 1. Notice that Assumption 1 contains unconditional homoskedasticity as a special case.…”

“…Assumption 1 corresponds to the set of conditions imposed on the shocks in Cavaliere and Taylor (2008) and Smeekes and Taylor (2011), strengthened by the addition of condition (iii). This additional condition is required because the new heteroskedasticityrobust information criteria, which we propose in section 3 below, require a consistent estimate of the volatility process.…”

Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

“…This additional condition ensures that the new heteroskedasticity-robust information criteria, which we propose in section 3 below, are based on a consistent estimate of the volatility process; see Beare (2008) who shows that (iii) suffices for this purpose when using a nonparametric kernel estimator. As the conditions in Assumption 1 are stronger than those in Cavaliere and Taylor (2008) and Smeekes and Taylor (2012), the large sample validity of the bootstrap unit root tests discussed in the next section is guaranteed. The reader is directed to Cavaliere and Taylor (2008) and Beare (2008) for further discussion of the conditions imposed by Assumption 1.…”

“…Assumption 1 corresponds to the set of conditions imposed on the shocks in Cavaliere and Taylor (2008) and Smeekes and Taylor (2012), strengthened by the addition of condition (iii). This additional condition ensures that the new heteroskedasticity-robust information criteria, which we propose in section 3 below, are based on a consistent estimate of the volatility process; see Beare (2008) who shows that (iii) suffices for this purpose when using a nonparametric kernel estimator.…”