In this paper, we propose a streaming model to distinguish voice queries intended for a smart-home device from background speech. The proposed model consists of multiple CNN layers with residual connections, followed by a stacked LSTM architecture. The streaming capability is achieved by using unidirectional LSTM layers and a causal mean aggregation layer to form the final utterance-level prediction up to the current frame. In order to avoid redundant computation during online streaming inference, we use a caching mechanism for every convolution operation. Experimental results on a device-directed vs. non device-directed task show that the proposed model yields an equal error rate reduction of 41% compared to our previous best model on this task. Furthermore, we show that the proposed model is able to accurately predict earlier in time compared to the attention-based models.