Neural architecture search (NAS) is a promising approach for automatically designing neural network architectures. However, the architecture estimation of NAS is computationally expensive and time-consuming because of training multiple architectures from scratch. Although existing zero-shot NAS methods use training-free proxies to accelerate the architecture estimation, their effectiveness, stability, and generality are still lacking. We present a novel training-free estimation proxy called weighted response correlation (WRCor). WRCor utilizes correlation coefficient matrices of responses across different input samples to calculate the proxy scores of estimated architectures, which can measure their expressivity and generalizability. Experimental results on proxy evaluation demonstrate that WRCor and its voting proxies are more efficient estimation strategies than existing proxies. We also apply them with different search strategies in architecture search. Experimental results on architecture search show that our zero-shot NAS algorithm outperforms most existing NAS algorithms in different search spaces. Our NAS algorithm can discover an architecture with a 22.1% test error on the ImageNet-1k dataset within 4 GPU hours. All codes are publicly available at https://github.com/kunjing96/ZSNAS-WRCor.git.