Functional instrumental variable regression with an application to estimating the impact of immigration on native wages

Abstract

Functional linear regression gets its popularity as a statistical tool to study the relationship between function-valued response and exogenous explanatory variables. However, in practice, it is hard to expect that the explanatory variables of interest are perfectly exogenous, due to, for example, the presence of omitted variables and measurement errors, and this in turn limits the applicability of the existing estimators whose essential asymptotic properties, such as consistency, are developed under the exogeneity condition. To resolve this issue, this paper proposes new instrumental variable estimators for functional endogenous linear models, and establishes their asymptotic properties. We also develop a novel test for examining if various characteristics of the response variable depend on the explanatory variable in our model. Simulation experiments under a wide range of settings show that the proposed estimators and test perform considerably well. We apply our methodology to estimate the impact of immigration on native wages.

Tendencies for Sub-optimal or Post-growth Pathways

“…In many earlier articles, k T is directly chosen by researchers and thus regarded as deterministic. However, as pointed out by Seong and Seo (2021) in their development of functional endogenous linear models, even in this case, it is generally not recommended to choose k T arbitrarily without taking the eigenvalues λj into account. Therefore, it is natural to view k T as random from a practical point of view.…”

“…We set τ T = 0.01 C XX S 1 T γ and υ T = 0.5T γ for some γ > 0, where note that τ T is designed to reflect the scale of X t as proposed by Seong and Seo (2021), who consider a choice of k T depending on the eigenvalues of a sample operator as in (3.2). We then computed the empirical MSPE associated with A, introduced in (3.3).…”

“…In particular, we show that a standard post dimension-reduction estimator, obtained by (i) first reducing the dimensionality of the predictive variable X t using the principal directions of its sample covariance and then (ii) applying the standard least square method to estimate the linear relationship between the resulting lower dimensional predictive variable and Y t , is effective for linear prediction. This post dimension-reduction estimator has been widely used in practice due to their simplicity, and its statistical properties have also been studied under technical assumptions, which are typically challenging to be verified in practice, particularly concerning the eigenstructure of the covariance of X t ; the reader is referred to e.g., Imaizumi and Kato (2018) and Seong and Seo (2021) where the assumptions of Hall and Horowitz (2007) are adopted for function-on-function regression models. We show that, without such assumptions, a naive use of this simple post dimension-reduction estimator can be justified as a way to obtain a solution which asymptotically minimizes the MSPE.…”

Testing Exogeneity in the Functional Linear Regression Model

“…The (joint) weak stationarity of (X, W ) ensures, that both covariance operators as well as the cross covariance operator have the same exponential system of eigenfunctions, which we denote by (φ k ) k∈N . While this assumption of (joint) second order stationarity seems to be quite restrictive, it is a common assumption when deriving consistency and especially the convergence rates of the slope function estimators, see Johannes (2008), Johannes (2009), Johannes (2016) or Seong and Seo (2021). On the other hand, it is well known, that the scalar product in Hilbert spaces is independent of the chosen basis.…”

Functional principal component analysis for cointegrated functional time series

“…Functional principal component analysis (FPCA) has been a central tool for functional time series (FTS) in various contexts encompassing FTS regression (Park and Qian, 2012;Seong and Seo, 2022), prediction (Hyndman and Ullah, 2007;Aue et al, 2015) and long memory FTS (Li et al, 2020) to name a few. In the recent work by Chang et al (2016), FPCA is applied to non-stationary cointegrated FTS.…”