N.H. Rev. Stat. Ann. § 14:33

The legislative budget assistant may be dismissed at any time by vote of the fiscal committee for good cause shown in a complaint brought by the said committee. The legislative budget assistant shall be afforded due notice of any such complaint and a fair hearing before said committee before dismissal. Should the fiscal committee vote to dismiss the legislative budget assistant, the legislative budget assistant may appeal the dismissal to a special committee consisting of the house finance and senate finance committees which shall have the power to uphold or reverse the decision of the fiscal committee.