 Analyzing the provided declarations and their statuses to determine the root causes of performance issues in this process involves examining trends in frequency and performance values associated with each type of approval or rejection. From a purely data-specific perspective:

1. **Declaration Approved by ADMINISTRATION (frequency = 13, average performance score decrease from initial to final approvals)** - A notable trend here is the consistent approval by administration despite varied outcomes post-approval, indicating potential bottlenecks or delays after initial approval stages. This could suggest inefficiencies within certain administrative segments dealing with subsequent steps of the process (e.g., Budget Owner rejections, Supervisor final approvals).

2. **Declaration FINAL_APPROVED by SUPERVISOR (frequency = 6, performance score variability)** - The varying performances and frequency suggest inconsistency in the supervisory process. Certain cases might be efficiently managed to completion without issues, while others could encounter obstacles leading to delayed approvals or rejections post-final approval attempt. This inconsistency points towards a need for clearer guidelines or additional training/support at this stage of review.

3. **Declaration REJECTED by Various Entities (frequency ranging from 4 to 8)** - The high frequency and low performance scores in these scenarios highlight significant issues with the process at multiple stages leading up to rejection. Notably, repeated instances where declarations are rejected after receiving preliminary approvals suggest systemic or procedural flaws that allow for such reversals without just cause. This might indicate a need for more rigorous checks before moving forward in the approval chain, ensuring only thoroughly vetted requests proceed to later stages.

4. **Declaration Rejected by PRE_APPROVER (frequency = 3)** - The specific focus on pre-approval rejections points towards potential issues in earlier stages of screening or decision-making before formal approvals are sought. This might imply a need to evaluate the criteria used for initial assessments, ensuring they accurately reflect necessary requirements and lead to fewer post-preliminary steps rejections.

From these observations, specific considerations include:
- **Streamlining Process Post-Approval** - Identifying and addressing inefficiencies after the declaration is approved by administrative bodies, ensuring smoother transitions through subsequent stages.
  
- **Enhancing Supervisory Review Guidelines and Support** - Given the variability in handling final supervisor approvals, implementing clearer guidelines or offering additional support might reduce performance issues and delays at this critical juncture.

- **Strengthening Initial Checks Before Preliminary Approvals** - Focusing on early rejection patterns indicates a need for more stringent initial assessments to prevent unnecessary reversals in the process, aiming to improve both efficiency and performance outcomes from the start of the process.

These considerations are based strictly on observed frequencies and performance scores linked directly to each approval/rejection status within this dataset, highlighting specific areas where interventions might significantly impact overall process performance.