Forecasting future realized variance paths with depth-weighted ridge and conformal diagnostics

Abstract

We studied the problem of forecasting full future realized–variance (FRV) paths yt,1:H over H = 30 trading days. We proposed a depth–weighted ridge (DW–ridge) estimator that (i) enforces the natural monotonicity of cumulative variance via a pool–adjacent violators post–projection and (ii) adapts to market regimes through observation weights derived from a Wasserstein–based curve depth. At the daily frequency, we took squared returns as a practical realized–variance proxy, so that the FRV path is the cumulative sum of next–day squares. Empirically, we used daily data for two liquid U.S. exchange-traded funds (ETFs; XLE and SLV) and two major cryptocurrencies (BTC–USD and ETH–USD) from January 1, 2020, to December 31, 2024, under a 60%/20%/20% train–calibration–test split. On the ETF benchmarks, DW–ridge improved all–horizon pathwise root mean squared error (RMSE) by about 3.1% (XLE) and 2.8% (SLV) relative to a monotone ridge baseline, with statistically significant short–horizon (H1–3/H1–5) mean squared error (MSE) gains under a moving–block bootstrap. On BTC–USD and ETH–USD, all–horizon RMSE reductions were around 6.0% and 6.5%, respectively. A block–conformal diagnostic based on depth–derived nonconformity scores attained near–nominal or conservative coverage on test blocks, so sharper forecasts were not obtained at the expense of reliability. Overall, depth reweighting provided a simple, fast, and empirically effective enhancement to monotone FRV path forecasting across both sector ETFs and major cryptocurrencies.