Compression is the reason to use this model. The evidence makes a Tucker-factored vector HAR look like an inexpensive way to forecast daily variance for a couple of hundred stocks together. Its results are almost indistinguishable from those of the low-rank model it aims to improve on.
Letting the data replace 1, 5 and 22
The classic HAR forecasts tomorrow's realized variance from three averages: yesterday's, the past 5 days' and the past 22 days'. Those averages fix the weights on earlier days before estimation. Yuan, Lu, Guo and the two Lis let the data pick the weights instead.
Their starting point is an unrestricted vector autoregression of daily variance for N stocks, with P lags and P = 22 or 66. They collect its N×N coefficient matrices into an N×N×P tensor. A Tucker decomposition then compresses that tensor along its separate dimensions, using ranks (r1, r2, r3). The ranks have different roles. The second, r2, reduces the N predictor stocks to a small set of factors, a step already taken by VHARI, Cubadda et al.'s vector HAR index model. The first, r1, limits the directions available to tomorrow's cross-section of variances. The third, r3, supplies the new element: it determines the width of U3, a P×r3 loading matrix. Each column holds weights across the past P days. The heterogeneous components are therefore weighted sums of lagged variances learned from the data.
This saves a lot of parameters. Unrestricted vector HAR (VHAR) requires 3N²; MLR-HAR, the authors' name for their multilinear low-rank model, requires r1r2r3 + Nr1 + Nr2 + Pr3. Our arithmetic puts the comparison at 4,946 against 157,323 when N = 229, P = 22 and the rank grid reaches (10, 10, 3). The authors estimate by least squares over the low-rank set. Projected gradient descent solves the problem, applying a truncated SVD mode by mode after each step.
A foundation in levels, forecasts in logs
The paper also offers a continuous-time foundation through a vector HAR-Itô diffusion. Spot variance in this model depends on the current day's accumulated integrated variance and lagged daily integrated variances. Proposition 1 requires the spectral radius of α(1) to be below one. Daily integrated volatility then follows a VAR(P) in levels with a lagged jump-variation term and i.i.d. innovations. Remove jumps and it becomes a pure vector HAR. The authors believe this is the first high-frequency foundation for HAR-type models, a contribution that matters for inference.
Their forecasts use centered log RV, outside the coverage of Proposition 1. There is a separate issue with the error bound, sqrt(dM/T) + T^(1+2δ)/m^(1/4). The theorem requires m^(1/4) ≳ T^(1+2δ), implying that m must grow faster than T⁴. In the application, m = 78 and T ranges between 502 and 1,760. The theorem supplies an estimator rate; its hypotheses fail at m = 78.
Can a desk rebuild the MLR-HAR forecasts?
Mostly yes.
The sample consists of the 150 or 229 S&P 500 constituents with the highest trading volume over 2017 to 2024. The authors calculate variance from five-minute returns during 9:30 to 4:00, giving 78 observations per day and excluding overnight returns. They use both RV and medRV. The latter is a median-based realized variance measure that dampens jumps.
The exercise has three rolling settings. Setting I fits on 2017 to 2018 (T = 502) and forecasts 2019. Setting II fits on 2020 to 2023 (T = 1,006); Setting III fits on 2017 to 2023 (T = 1,760). Both forecast 2024. In each case the estimation window advances one day at a time, producing one-step-ahead forecasts over M = 252 days.
A BIC with tuning constant 0.5×10⁻⁴ selects ranks from r1, r2 ≤ 10 and r3 ≤ 3. Those ranks stay fixed throughout the forecast year. A desk holding minute bars can reconstruct five-minute RV, although the authors use WRDS trade data.
The universe selection affects every result. Volume is measured over a window containing 2019 and 2024, so the choice of stocks uses knowledge of the evaluation years.
VHARI leaves little room: at most 0.05 points
Unrestricted VHAR accounts for the largest improvements in the tables. It is an easy benchmark to beat. In Setting I, for RV with N = 229, the paper's diagonal-risk criterion drops from 1.79% for VHAR to 1.00% for VHARI. Both MLR-HAR(22) and MLR-HAR(66) record 0.99%. Almost the entire gain comes from introducing a rank restriction of any kind. With a longer estimation window, the unrestricted model loses less ground: Setting III (T = 1,760) gives VHAR 1.30 and VHARI 1.21. That is a gap of 0.09 points, compared with 0.79 in 2019. Short windows give rank restriction its greatest value.
The RV comparisons with VHARI are much tighter. For N = 150, VHARI records 1.21, 1.60 and 1.47 across the three settings; MLR-HAR(22) records 1.19, 1.59 and 1.46. Setting II at N = 229 has all three reduced-rank models tied at 1.31. The widest separation occurs in Setting III at N = 229: MLR-HAR(66) delivers 1.16 versus VHARI's 1.21. MLR-HAR(22) also records 1.21 there. The authors' cited 4.1% improvement thus belongs to the longer lag order in that single cell. medRV tells much the same story. For N = 229, the triples (VHARI / MLR-HAR(22) / MLR-HAR(66)) are 0.93/0.92/0.92, 1.24/1.23/1.24 and 1.13/1.13/1.12.
The authors acknowledge the narrow margins. They write that "several comparisons are tied at the reported precision, and increasing the lag order does not consistently improve performance." In their conclusion, reductions in diagonal risk relative to VHARI are "generally modest". They suggest that restricting responses and learning lags may help where cross-asset dynamics are genuinely low-dimensional and fixed 1/5/22 aggregation misses the relevant pattern. They also allow that VHARI may already approximate this setting adequately, explaining the near-ties.
That is a reasonable defence. The paper offers no experiment that separates those explanations. Its largest table gap is 0.05 points (1.16 vs 1.21, RV, N = 229, Setting III, P = 66). Without a significance test, that gap cannot be distinguished from noise. We did not find a Diebold-Mariano or model-confidence-set test. Two of the three settings use the same 252 days of 2024. QLIKE, the quasi-likelihood loss, is presented only in boxplots: MLR-HAR has lower medians, but no figures are supplied.
The lag result is the more interesting null.
Moving from P = 22 to P = 66 produces no systematic improvement. The authors interpret this as limited information beyond one month. Give a model designed to discover heterogeneous horizons three times the lag window, and it finds little more. The longer window does produce its one sizeable advantage over VHARI (1.16 vs 1.21), in a single cell.
What does the allocation test reward?
The allocation exercise assigns inverse-variance weights. After forecast variances are exponentiated back to levels, stocks receive weights proportional to 1/ẑ. A diagonal matrix of realized variances supplies the risk score. The authors themselves identify both limitations of this design. AVol "should not be interpreted as the annualized volatility of realized portfolio returns," since the calculation excludes cross-stock covariances. Multiplying every forecast on a given date by the same constant also leaves the weights unchanged.
Only the relative sizes of stock variance forecasts matter to this criterion. It discards the level that determines sizing for a vol-targeting book, along with the correlation structure that drives realized portfolio risk. We found no turnover or cost figures for this exercise.
MLR-HAR is therefore a defensible input for relative per-name weighting across 229 stocks. At the top of the rank grid it needs at most about 4,946 parameters, compared with 157,323 for VHAR. I would be persuaded by an evaluation that separates it from VHARI, through either a full-covariance realized-volatility test or a significance test on QLIKE. On Table 1 as published, choosing VHARI would have cost between zero and five hundredths of a percentage point on the diagonal-risk criterion, and usually one hundredth.