Developments in statistical learning have fueled the analysis of high-dimensional time series. However, even in low-dimensional contexts the issues arising from ill-conditioned regression problems are well-known. Because linear time series modeling is naturally prone to such issues, I propose to apply ridge regression to the estimation of dense VAR models.Theoretical non-asymptotic results concerning the addition of a ridge-type penalty to the least squares estimator are discussed, while standard asymptotic and inference techniques are proven to be valid under mild conditions on the regularizer. The proposed estimator is then applied to the problem of sieve approximation of VAR(∞) processes under moderately harsh sample sizes. Simulation evidence is used to discuss the small sample properties of the ridge estimator (RLS) when compared to least squares and local projection approaches: I use a Monte Carlo exercise to argue that RLS with a lag-adapted cross-validated regularizer achieve meaningfully better performance in recovering impulse response functions and asymptotic confidence intervals than other common approaches.

When a finite order vector autoregressive model is fitted to VAR(∞) data the asymptotic distribution of statistics obtained via smooth functions of least-squares estimates requires care. Lütkepohl and Poskitt (1991) provide a closed-form expression for the limiting distribution of (structural) impulse responses for sieve VAR models based on the Delta method. Yet, numerical simulations have shown that confidence intervals built in such way appear overly conservative. In this note I argue that these results stem naturally from the limit arguments used in Lütkepohl and Poskitt (1991), that they manifest when sieve inference is improperly applied, and that they can be "remedied" by either using bootstrap resampling or, simply, by using standard (non-sieve) asymptotics.

Macroeconomic forecasting has recently started embracing techniques that can deal with largescale datasets and series with unequal release periods. The aim is to exploit the information contained in heterogeneous data sampled at different frequencies to improve forecasting exercises. Currently, MIxed-DAta Sampling (MIDAS) and Dynamic Factor Models (DFM) are the two main stateof-the-art approaches that allow modeling series with non-homogeneous frequencies. We introduce a new framework called the Multi-Frequency Echo State Network (MFESN), which originates from a relatively novel machine learning paradigm called reservoir computing (RC). Echo State Networks are recurrent neural networks with random weights and trainable readout. They are formulated as nonlinear state-space systems with random state coefficients where only the observation map is subject to estimation. This feature makes the estimation of MFESNs considerably more efficient than DFMs. In addition, the MFESN modeling framework allows to incorporate many series, as opposed to MIDAS models, which are prone to the curse of dimensionality. Our discussion encompasses hyperparameter tuning, penalization, and nonlinear multistep forecast computation. In passing, a new DFM aggregation scheme with Almon exponential structure is also presented, bridging MIDAS and dynamic factor models. All methods are compared in extensive multistep forecasting exercises targeting US GDP growth. We find that our ESN models achieve comparable or better performance than MIDAS and DFMs at a much lower computational cost.