Standard methods, such as sequential procedures based on Johansen's(pseudo-)likelihood ratio (PLR) test, for determining the co-integration rankof a vector autoregressive (VAR) system of variables integrated of order onecan be significantly affected, even asymptotically, by unconditionalheteroskedasticity (non-stationary volatility) in the data. Known solutions tothis problem include wild bootstrap implementations of the PLR test or the useof an information criterion, such as the BIC, to select the co-integrationrank. Although asymptotically valid in the presence of heteroskedasticity,these methods can display very low finite sample power under some patterns ofnon-stationary volatility. In particular, they do not exploit potentialefficiency gains that could be realised in the presence of non-stationaryvolatility by using adaptive inference methods. Under the assumption of a knownautoregressive lag length, Boswijk and Zu (2022) develop adaptive PLR testbased methods using a non-parameteric estimate of the covariance matrixprocess. It is well-known, however, that selecting an incorrect lag length cansignificantly impact on the efficacy of both information criteria and bootstrapPLR tests to determine co-integration rank in finite samples. We show thatadaptive information criteria-based approaches can be used to estimate theautoregressive lag order to use in connection with bootstrap adaptive PLRtests, or to jointly determine the co-integration rank and the VAR lag lengthand that in both cases they are weakly consistent for these parameters in thepresence of non-stationary volatility provided standard conditions hold on thepenalty term. Monte Carlo simulations are used to demonstrate the potentialgains from using adaptive methods and an empirical application to the U.S. termstructure is provided.

We present a kinematic analysis of the main-sequence galaxy HZ4 at $z=5.5$.Our study is based on deep, spatially resolved observations of the [CII] 158$\mu$m transition obtained with the Atacama Large Millimeter/SubmillimeterArray (ALMA). From the combined analysis of the disk morphology, thetwo-dimensional velocity structure, and forward-modeling of the one-dimensionalvelocity and velocity dispersion profiles, we conclude that HZ4 has a regularrotating disk in place. The intrinsic velocity dispersion in HZ4 is high($\sigma_{0}=65.8^{+2.9}_{-3.3}$ km s$^{-1}$), and the ratio between therotational velocity and the intrinsic velocity dispersion is $V_{\rmrot}/\sigma_{0}=2.2$. These values are consistent with the expectations fromthe trends of increasing $\sigma_{0}$ and decreasing $V_{\rm rot}/\sigma_{0}$as a function of redshift observed in main-sequence galaxies up to $z\approx4$.Galaxy evolution models suggest that the high level of turbulence observed inHZ4 can only be achieved if, in addition to stellar feedback, there is radialtransport of gas within the disk. Finally, we find that HZ4 is baryon dominatedon galactic scales ($\lesssim2\times R_{\rm e}$), with a dark matter fractionat one effective radius of $f_{\rm DM}(R_{\rm e})=0.41^{+0.25}_{-0.22}$. Thisvalue is comparable to the dark matter fractions found in lower redshiftgalaxies that could be the descendants of HZ4: massive($M_{\star}\approx10^{11}~M_{\odot}$), star-forming galaxies at $z\sim2$, andpassive, early type galaxies at $z\approx0$.