Functional Lagged Regression with Sparse Noisy Observations
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A (lagged) time series regression model involves the regression of scalar response time series on a time series of regressors that consists of a sequence of random functions (curves), also known as a functional time series. In practice, the underlying regressor curve time series are not always directly accessible, and often need to be treated as latent processes that are observed (sampled) only at discrete measurement locations. In this paper, we consider the so-called sparse observation scenario where only a relatively small number of measurement locations have been observed, indeed locations that may be different for each curve. The measurements can be further contaminated by additive measurement error. A spectral approach to the estimation of the model dynamics is considered. The spectral density of the regressor time series and the cross-spectral density between the regressors and response time series are estimated by kernel smoothing methods from the sparse observations. The estimation of the impulse response regression coefficients of the lagged regression model is regularised by means of the ridge regression approach (Tikhonov regularisation) or the PCA regression (truncation regularisation). The latent functional time series are then recovered by means of prediction, conditioning on all the observed observed data. A detailed simulation study investigates the performance of the methodology in terms of estimation and prediction for a wide range of scenarios.
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