We propose a statistical procedure to determine the dimension of the nonstationary subspace of cointegrated functional time series taking values in the Hilbert space of square-integrable functions defined on a compact interval. The procedure is based on sequential application of a proposed test for the dimension of the nonstationary subspace. To avoid estimation of the long-run covariance operator, our test is based on a variance ratio-type statistic. We derive the asymptotic null distribution and prove consistency of the test. Monte Carlo simulations show good performance of our test and provide evidence that it outperforms the existing testing procedure. We apply our methodology to three empirical examples: age-specific U.S. employment rates, Australian temperature curves, and Ontario electricity demand.

我们提出了一种统计程序来确定协整函数时间序列的非平稳子空间的维数，该时间序列在紧区间上定义的平方可积函数的希尔伯特空间中取值。该过程基于对非平稳子空间维度的建议测试的顺序应用。为了避免估计长期协方差算子，我们的测试基于方差比型统计。我们推导出渐近零分布并证明测试的一致性。蒙特卡罗模拟显示了我们测试的良好性能，并提供了它优于现有测试程序的证据。我们将我们的方法应用于三个实证示例：特定年龄的美国就业率、澳大利亚温度曲线和安大略省的电力需求。