N.H. Rev. Stat. Ann. § 14:35

When the position of the legislative budget assistant becomes vacant the fiscal committee shall meet not less than one month after the vacancy occurs for the purpose of filling said vacancy. The deputy legislative budget assistant shall perform the duties of the legislative budget assistant during the period that the position of legislative budget assistant is vacant or for any period that the legislative budget assistant is unable for any reason to perform the duties of the position. When the position of the deputy legislative budget assistant becomes vacant the legislative budget assistant with the approval of the fiscal committee may appoint a deputy legislative budget assistant to fill the vacancy.