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The effect of future Arctic amplification (AA) on the extratropical atmospheric circulation remains unclear in modeling studies. Using a collection of coordinated atmospheric and coupled global climate model perturbation experiments, we find an emergent relationship between the high‐latitude 1,000–500 hPa thickness response and an enhancement of the Siberian High in winter. This wave number‐1‐like sea level pressure anomaly pattern is linked to an equatorward shift of the eddy‐driven jet and a dynamical cooling response in eastern Asia. Additional simulations, where AA is imposed directly into the model domain by nudging, demonstrate how the sea ice forcing is insufficient by itself to capture the vertical extent of the warming and by extension the amplitude of the response in the Siberian High. This study demonstrates the importance of the vertical extent of the tropospheric warming over the polar cap in revealing the "warm Arctic, cold Siberia" anomaly pattern in future projections. Plain Language Summary: Surface air temperature averaged within the Arctic Circle is rising more than twice as fast as the rate of the globally averaged temperature. This is known as "Arctic amplification" and is the result of numerous feedbacks in the Earth system. This warming is not restricted to the surface, as the temperature is also increasing at least to the middle troposphere in the Arctic. The effects of this warming on weather patterns in the midlatitudes remain uncertain in observations and model simulations. In this study, we conduct a series of climate model experiments to better understand how the vertical extent of warming in the Arctic atmosphere connects to a strengthening of the Siberian High in boreal winter. This high‐pressure response leads to colder air temperatures in eastern Asia through advection. We find that climate models prescribed with only Arctic sea ice loss do not fully capture the vertical extent of warming in the atmosphere, and thus underestimate the linkage between projected Arctic amplification and Eurasian climate. Key Points: Climate models forced only by sea ice anomalies do not capture the vertical extent of Arctic warming during winterIncrease in thickness of the winter 1,000–500 hPa layer is linked to a strengthening of the Siberian High and cold anomalies in eastern AsiaRole of the stratosphere is unclear due to large internal variability at future global warming of 2° C above preindustrial levels [ABSTRACT FROM AUTHOR]