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Heterogeneous AutoRegressive (HAR) models and graph signal processing (GSP)-based extensions such as the graph signal processing HAR model (GSPHAR) embed spillover structure via the magnetic Laplacian and graph Fourier transform. However, they still summarize each market\u2019s recent history mainly through multi-scale lag averages, which can miss nonlinear within-window path information. To address this limitation, we develop a signature-enhanced graph spectral HAR framework, Sig-GSPHAR, which incorporates path-signature features extracted from each market\u2019s 22-day volatility window and fuses them with HAR lags in the directed graph spectral domain. An empirical study on 29 stock markets and five realized-volatility proxies shows that, for one-step-ahead forecasting (\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{h = 1}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ), Sig-GSPHAR outperforms econometric and deep learning benchmarks and delivers consistent gains over GSPHAR, with paired one-sided Wilcoxon signed-rank tests (Holm-adjusted) indicating significance across multiple proxies. For long-horizon forecasting (\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\varvec{h = 22}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>22<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ), direct comparisons against GSPHAR indicate that the advantage of Sig-GSPHAR becomes more pronounced. We further show that the static spillover graph is structurally stable over the sample, equip the point forecasts with distribution-free conformal prediction intervals, and provide a detailed diagnostic of when signature features help.\n                  <\/jats:p>","DOI":"10.1007\/s10489-026-07445-1","type":"journal-article","created":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T00:57:32Z","timestamp":1788829052000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Signature-enhanced graph spectral HAR for multivariate realized volatility forecasting"],"prefix":"10.1007","volume":"56","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-2970-8499","authenticated-orcid":false,"given":"Zeyu","family":"Guo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andi","family":"Han","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chao","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junbin","family":"Gao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,9,8]]},"reference":[{"issue":"2","key":"7445_CR1","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1111\/1468-0262.00418","volume":"71","author":"TG Andersen","year":"2003","unstructured":"Andersen TG, Bollerslev T, Diebold FX, Labys P (2003) Modeling and forecasting realized volatility. 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