0.1 MIDAS Models
Standard time series regressions require all variables at a common sampling frequency, forcing either aggregation of high-frequency regressors (discarding information) or interpolation of low-frequency ones (fabricating information). Mixed Data Sampling (MIDAS) regressions (Ghysels, Santa-Clara & Valkanov, 2004) instead regress a low-frequency variable directly on many lags of a high-frequency regressor, using a parsimonious weighting scheme to avoid parameter proliferation.
To forecast a low-frequency variable \(y_t\) (e.g. quarterly GDP growth, or monthly realized volatility) using a high-frequency regressor \(x^{(m)}\) sampled \(m\) times per low-frequency period, \[ y_{t} = \beta_0 + \beta_1 \sum_{j=0}^{J} w(j;\boldsymbol\theta)\, x^{(m)}_{t - j/m} + \varepsilon_t, \] where \(w(j;\boldsymbol\theta)\) is a weighting function with a small number of hyperparameters \(\boldsymbol\theta\) (rather than \(J\) free coefficients), e.g.: - Exponential Almon lag: \(w(j;\theta_1,\theta_2) \propto \exp(\theta_1 j + \theta_2 j^2)\), - Beta lag weighting: weights from a Beta density shape, flexible hump/decay patterns with just 2 parameters. This reduces a potentially huge lag-coefficient vector to 2-3 parameters, estimated by nonlinear least squares.
Why not just aggregate? Aggregating high-frequency data (e.g. averaging daily data to monthly) before regressing discards the within-period dynamics and timing of the high-frequency series, which MIDAS shows carries genuine forecasting content — especially valuable for nowcasting (using partially released high-frequency indicators to forecast the current low-frequency period before it is fully observed).
Applications. - Forecasting GDP growth from monthly/weekly indicators (industrial production, financial conditions indices). - MIDAS volatility models (e.g. GARCH-MIDAS): decompose volatility into a short-run daily GARCH component and a slow-moving long-run component driven by MIDAS-weighted lags of a lower-frequency macro/RV variable — a direct competitor to HAR-RV for long-horizon volatility forecasting. - Mixed-frequency VARs and mixed-frequency factor models as alternatives requiring explicit state-space treatment of the mismatch (see State-Space Models and the Kalman Filter).