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.

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.

In this Supplement we examine testing linearity against commonly applied STAR models and also provides simulation evidence of our methodology. We also demonstrate how Hansen's (1996) weighted bootstrap is applied to enhance the applicability of our methodology. Finally, we provide the proofs of the theoretical results in the paper A.1 Monte Carlo Experiments and Application of the Weighted Bootstrap A.1.1 Monte Carlo Experiments Using the ESTAR ModelTo simplify our illustration, we assume that for all t = 1, 2, . . ., u t ∼ IID N (0, σ 2 * ) and y t = π * y t−1 + u t with π * = 0.5. Under this DGP, we specify the following first-order ESTAR model:and γ ∈ Γ}. The model does not contain an intercept, and the transition variable is y t−1 . The nonlinear function f t. Thus, we conclude that QLR n ⇒ sup γ G 2 (γ), which agrees with Theorem 3.The null limit distribution can be approximated numerically by simulating a distributionally equivalent Gaussian process. To do this we present the following lemma: Lemma A. 1. If {z k : k = 0, 1, 2, . . .} is an IID sequence of standard normal random variables, G(•) d = G(•),

This paper considers an endogenous binary response model with many weak instruments.We in the current paper employ a control function approach and a regularization scheme to obtain better estimation results for the endogenous binary response model in the presence of many weak instruments. Two consistent and asymptotically normally distributed estimators are provided, each of which is called a regularized conditional maximum likelihood estimator (RCMLE) and a regularized nonlinear least square estimator (RNLSE) respectively. Monte Carlo simulations show that the proposed estimators outperform the existing estimators when many weak instruments are present. To illustrate the usefulness of our estimation method, we study the effect of family income on college completion.