Approximate factor models with restrictions on the loadings may be interesting both for structural analysis (simpler structures are easier to interpret) and forecasting (parsimonious models typically deliver superior forecasting performances). However, the issue is largely unexplored. In particular, no currently available test is entirely suitable for the empirically important case of non-stationary data. Building on the intuition that de-factoring the data under a correct set of restrictions will lower the number of factors, a bootstrap test based on the comparison of the number of factors selected for the raw and de-factored data is proposed. The test is shown analytically to be asymptotically valid and by simulation to have good small sample properties.

对于结构分析（更简单的结构更易于解释）和预测（简约模型通常可提供出色的预测性能）而言，受载荷限制的近似因子模型可能会很有趣。但是，这个问题在很大程度上尚待探讨。特别是，目前没有可用的测试完全适合非平稳数据在经验上很重要的情况。基于在正确的限制条件下对数据进行解映射将减少因子数量的直觉，基于对原始数据和解因子数据选择的因子数量的比较，提出了一种自举测试。分析证明该测试是渐近有效的，并且通过仿真显示该测试具有良好的小样本属性。