In hierarchical organizations, authenticating data from multiple users can be complex and resource-intensive. Hierarchical Identity-Based Signature with Designated Aggregator (HIBS-DA) provides an efficient solution by allowing users at different levels to generate signatures that can be combined into a single, compact signature. We first introduce the HIBS-DA framework and present the {\em{first}} lattice-based construction of HIBS-DA. Our scheme allows users at different hierarchical levels to generate individual signatures that can be aggregated into a single, compact signature, reducing communication and verification costs. The proposed construction is secure, correct, and resistant to forgery, making it suitable for large-scale environments such as universities, corporations, and government agencies.