We propose three novel consistent specification tests for quantile regression models which generalize former tests in three ways. First, we allow the covariate effects to be quantile-dependent and nonlinear. Second, we allow parameterizing the conditional quantile functions by appropriate basis functions, rather than parametrically. We are thereby able to test for general functional forms, while retaining linear effects as special cases. In both cases, the induced class of conditional distribution functions is tested with a Cramér–von Mises type test statistic for which we derive the theoretical limit distribution and propose a bootstrap method. Third, a modified test statistic is derived to increase the power of the tests. We highlight the merits of our tests in a detailed MC study and two real data examples. Our first application to conditional income distributions in Germany indicates that there are not only still significant differences between East and West but also across the quantiles of the conditional income distributions, when conditioning on age and year. The second application to data from the Australian national electricity market reveals the importance of using interaction effects for modeling the highly skewed and heavy-tailed distributions of energy prices conditional on day, time of day and demand.

中文翻译：

---

分位数回归模型中的灵活规格测试

我们为分位数回归模型提出了三种新颖的一致规范测试，它们以三种方式概括了以前的测试。首先，我们允许协变量效应是分位数相关的和非线性的。其次，我们允许通过适当的基函数而不是参数化来参数化条件分位数函数。因此，我们能够测试一般函数形式，同时保留线性效应作为特殊情况。在这两种情况下，条件分布函数的归纳类都使用 Cramér-von Mises 型检验统计量进行测试，我们为此推导了理论极限分布并提出了引导方法。第三，导出修改后的检验统计量以提高检验的功效。我们在详细的 MC 研究和两个真实数据示例中强调了我们测试的优点。我们对德国条件收入分配的首次应用表明，以年龄和年份为条件时，不仅东西方之间仍然存在显着差异，而且条件收入分配的分位数之间也存在显着差异。对澳大利亚国家电力市场数据的第二个应用揭示了使用交互效应对以日期、时间和需求为条件的能源价格的高度倾斜和重尾分布进行建模的重要性。