We develop dimension-reduction-free tests for the slope function in functional linear regression when the functional regressor may be endogenous or measured with error. The tests are based on a functional moment condition induced by an auxiliary functional variable and do not require estimation of the slope function. This feature is particularly useful in infinite-dimensional settings, where the identification and regularization conditions needed for consistent estimation are often strong and difficult to verify. The proposed procedures remain asymptotically valid under weak or even failed relevance of the auxiliary variable, and they are consistent against fixed alternatives that are detectable through the moment operator. We establish the asymptotic null distribution, consistency against detectable alternatives, and local power under drifting alternatives. We also derive the locally optimal test within a class of weighted test statistics. Feasible critical values for implementation of the tests are obtained from data. Simulations show reliable size control and competitive power, including under weak relevance. We illustrate the method using a functional regression analysis of residential electricity demand and temperature distributions in South Korea.

翻译：我们针对函数线性回归中斜率函数提出了免降维检验方法，该检验适用于函数回归变量存在内生性或测量误差的情形。该方法基于辅助函数变量诱导的函数矩条件，且无需估计斜率函数。这一特性在无限维设定下尤为实用，因为此时一致估计所需的识别条件和正则化条件通常较强且难以验证。所提程序在辅助变量相关性较弱甚至失效时仍保持渐近有效性，并且对可通过矩算子检测的固定备择假设具有一致性。我们建立了渐近零分布、可检测备择假设下的一致性以及漂移备择假设下的局部功效。同时，在一类加权检验统计量中推导了局部最优检验。检验实施所需的可行临界值可通过数据获得。模拟结果表明该方法具有可靠的尺寸控制和有竞争力的功效，包括在弱相关性情形下。我们通过韩国住宅电力需求与温度分布的函数回归分析对方法进行了实证说明。