Self‐normalization inference for linear trends in cointegrating regressions

In this article, statistical tests concerning the trend coefficient in cointegrating regressions are addressed for the case when the stochastic regressors have deterministic linear trends. The self‐normalization (SN) approach is adopted for developing inferential methods in the integrated and modified ordinary least squares (IMOLS) estimation framework. Two different self‐normalizers are used to construct the SN test statistics: a functional of the recursive IMOLS estimators and a functional of the IMOLS residuals. These two self‐normalizers produce two SN tests, denoted by TSNϵ$$ {T}^{\mathrm{SN}}\left(\epsilon \right) $$ and τδ1η^T⊥$$ {\tau}_{\delta_1}\left({\hat{\eta}}_T^{\perp}\right) $$ respectively. Neither test requires studentization with a heteroskedasticity and autocorrelation consistent (HAC) estimator. A trimming parameter ϵ$$ \epsilon $$ must be chosen to implement the TSNϵ$$ {T}^{\mathrm{SN}}\left(\epsilon \right) $$ test, whereas the τδ1η^T⊥$$ {\tau}_{\delta_1}\left({\hat{\eta}}_T^{\perp}\right) $$ test does not require any tuning parameter. In the simulation, the QSNϵ≡TSNϵ2$$ {Q}^{\mathrm{SN}}\left(\epsilon \right)\equiv {\left({T}^{\mathrm{SN}}\left(\epsilon \right)\right)}^2 $$ test exhibits the smallest size distortion among the inferential methods examined in this article. However, this may come with some loss of power, particularly in small samples.