Advances in functional linear regression

信息处理方面的技术革新和存储能力的提高使得在经济、金融、气象和医学等各个领域收集连续时间数据成为可能。研究人员、公司和政府都在寻找利用这些丰富信息的方法。我的长期目标是开发统计工具来处理不断增长的数据量，并确保所有的信息都得到利用。将这些数据视为曲线是解决这个问题的一种优雅方式。本研究议程侧重于回归，其中回归量和响应都是函数（或曲线）。主要目标是在回归量为内生的函数回归中估计因果效应。估计器将基于功能性工具变量的使用，加上一种称为偏最小二乘的正则化技术。举例来说，考虑用电需求作为电价函数的情况。大公司的用电需求可能会影响价格，事实上，他们用电需求的增加应该会推动价格上涨。所以对他们来说，电价不是给定的，而是内生的。为了估计这些公司的价格弹性，需要使用一个变量（称为工具），它影响价格而不影响消费。这种变量的一个例子是风力。的确，一方面，风是外生的，因为没有人能控制风。另一方面，风会影响价格，因为当有风时，风力涡轮机产生更多的电力，从而导致价格下降。这个模型可以用来估计电力需求的价格弹性。我将考虑基本模型的各种扩展，以允许异方差和自相关误差。我计划提出对大范围的异方差和自相关具有鲁棒性的标准误差估计。考虑到异方差和时间依赖性，可以获得更准确的置信区间和检验。我计划开发的方法适用于任何频繁观测时间数据或在多个地点观测空间数据的领域。本研究的结果将有助于理论研究者和应用统计学家获得更可靠的估计和更好的决策。此外，我会为我的方法提供计算机代码，以确保知识的更快传播，并使其更容易应用。

The technological innovations in information processing and the increased storage capability have made possible to collect continuous-time data in various fields such as economics, finance, meteorology and medicine. Researchers, companies, and governments look for ways to exploit this rich information. My long term objective is to develop statistical tools to deal with this ever growing amount of data and to make sure that all the information is taken advantage of. Looking at these data as curves is an elegant way to address this issue. This research agenda focuses on regressions where both the regressor and the response are functions (or curves). The main goal is the estimation of the causal effect in a functional regression where the regressor is endogenous. The estimator will be based on the use of functional instrumental variables coupled with a regularization technique named partial least squares. For illustration, consider the case of the electricity demand as a function of the price of electricity. The demand of large companies who consume a lot of electricity may affect the price, indeed an increase of their demand should push the price upward. So for them, the electricity price is not given but endogenous. To estimate the price elasticity of such companies, one needs to use a variable (called instrument) which affects the price without affecting the consumption. An example of such a variable is the wind power. Indeed, on the one hand, the wind is exogenous because no one can control the wind. On the other hand, the wind affects the price because when it is windy, the wind turbines produce more electricity yielding a decline in the price. This model would allow to estimate the price elasticity of the electricity demand. I will consider various extensions of the basic model to allow for heteroskedastic and auto-correlated errors. I plan to propose estimators of the standard errors robust to a wide range of heteroscedasticity and autocorrelation. Accounting for heteroscedasticity and time-dependence permits to obtain more accurate confidence intervals and tests. The methods, I plan to develop, apply to any area where temporal data are observed very frequently or spatial data are observed for multiple locations. The results derived in this research will help both theoretical researchers and applied statisticians to obtain more reliable estimates and make better decisions. Moreover, I will provide computer codes for my methods to ensure a faster transmission of knowledge and make it easier to apply.