Characteristic-Space Metrics in Factor Models: Identification and Inference

Why two factor models with the same fitted returns can split them differently, and how to report standard errors when the rule that splits them is estimated.

Author

Affiliation

Mingyang Liu

Imperial Business School, Imperial College London

Version

September 2026

Paper (PDF) · Online supplement (PDF) · Slides (PDF) · SSRN · BibTeX

SSRN preprint · September 2026 · DOI 10.2139/ssrn.7445120 (listed on SSRN under Yang Liu)

Two factor models can give every stock exactly the same fitted return and still disagree about how much of it comes from hidden factors, from observed factors such as the market, and from an intercept that no factor explains. This paper shows that a convention sets the split: the ruler, which decides when two characteristics, such as two value signals, count as overlapping. Because the ruler is estimated from the same data, its estimation error belongs in the standard errors. In a stylised simulation with 960 time periods, built to isolate this mechanism, leaving that error out makes intervals meant to contain the true value 95 times in 100 contain it only about 89–92 times (Table II, p. 34).

1 The question

In the agenda, this paper follows Geometric Framework and asks: once the ruler is chosen, what does it decide, and how precisely? The next paper, Interpreting Pricing Errors, asks what a smaller alpha then means.

When a factor model built on firm characteristics fits returns, what decides how much of the fit it credits to hidden factors, to observed factors such as the market, and to the intercept? And how precisely can that split be measured?

The model has three parts. Besides hidden factors (common drivers it infers from the data) and observed factors such as the market, there is the intercept: the part of expected return that no factor in this model explains (the home page’s “alpha” is a close relative, measured against a stated benchmark model). You can move part of the fit from the intercept or the market into the hidden factors, adjust the hidden factors to compensate, and leave every fitted return exactly as it was. Every such split fits equally well, so the fit itself cannot choose among them. A convention decides it: the ruler. Characteristics also differ in scale and overlap, even after each is standardised (Section 1, p. 2), so the choice of ruler matters.

Why a model needs a ruler. To say how much of a stock’s expected return each driver explains, a model must first decide when two combinations of characteristics are really different, and how large each one is. The rule it uses for that is the ruler. Give each characteristic a weight and add them up: every stock gets a score. Such a weighted combination is called a direction (it is not a portfolio). The plain ruler (the Euclidean or identity metric) judges a direction by its weights alone. The characteristic ruler (the Gram metric) judges it by the scores it gives real stocks: how large they typically are, and whether two directions’ scores line up across stocks, high on the same stocks and low on the same stocks. So the ruler is the convention a model uses to measure how large a direction is and how much two directions overlap. Two value signals that pick out the same cheap stocks count as overlapping under the characteristic ruler but as unrelated under the plain ruler.

2 Why it matters

When researchers report how closely the hidden factors align with a benchmark, how large the intercept is, or which combinations of characteristics the hidden factors represent, each number depends on the split, not only on the fit (Proposition 1, p. 7).

Economic. R² cannot tell these splits apart, so a smaller reported intercept may reflect the convention, not new explanatory power (Corollary 1, p. 7).
Statistical. The characteristic ruler is estimated from the same data. Treating it as known leaves out the ruler’s own sampling error and the way that error moves with the noise in returns (Theorem 1, p. 16); in the paper’s simulation, this makes standard errors too small (Section 8.2, p. 33). State the ruler; if it is estimated, include its error.

3 The argument at a glance

Same fit, different split. Two models can give every stock the same fitted return yet credit it differently to hidden factors, observed factors and an intercept (Proposition 1, p. 7). → A simple example
The fit cannot decide. Because the fits are identical, R² and residuals cannot choose between the splits (Corollary 1, p. 7). → Why it matters
A convention decides. So the choice falls to a convention: the ruler (Section 3.1, pp. 6–7). → The framework
The two rulers differ in U.S. data. The characteristic ruler counts overlap by the stocks two characteristics pick out, and in U.S. characteristics (no returns used) it differs sharply from the plain ruler (Table I, p. 31). → Design and main results
Its error counts. Because this ruler is estimated from the same data, its error flows into every reported split, so the paper derives standard errors that include it and do not depend on how the model is parameterised (Theorems 1–2, pp. 16, 20). → The framework
Evidence. In a stylised simulation with 960 time periods, leaving that error out makes 95% intervals contain the true value only about 89–92 times in 100; including it restores about 94–95 (Table II, p. 34). → Design and main results
Near a cap. Some models, such as the Geometric Framework paper’s NA-IPCA (no-arbitrage instrumented principal component analysis, named for this cap), limit how much return the intercept may carry. When the estimate sits against that limit, the cap “binds”; below it, the cap is “slack”. The cap is measured with the same estimated ruler, so the ruler’s error moves the cap too; the paper builds an interval that holds either way, at the cost of extra width (Theorem 4, p. 29). → Design and main results

4 The framework

At each date, stock i has characteristics z_{i,t} and a next-period excess return. The model has three blocks (equation 2.1, p. 4):

r_{i,t+1} = z_{i,t}'\Gamma_\beta f_{t+1} + z_{i,t}'\Gamma_\delta g_{t+1} + z_{i,t}'\Gamma_\alpha + \varepsilon_{i,t+1}.

Here f are hidden (latent) factors, g are observed factors and \Gamma_\alpha sets the intercept from characteristics. The characteristic ruler (the Gram metric) measures two directions by the scores they give real stocks (equation 2.5, p. 5):

\langle a,b\rangle_{Q_Z} = a'Q_Z\,b = \mathbb{E}\!\left[N_t^{-1}(Z_t a)'(Z_t b)\right].

The ruler “specifies a convention” for separating the blocks (p. 2): which ruler to use is declared, not discovered, but the characteristic ruler’s values are estimated from the data. The theory makes three claims.

Same fit, different split. For any ruler, a matched change rescales the hidden-factor loadings, removes the other blocks’ overlap with them, and offsets both changes in the factor series, so every fitted return is unchanged (Proposition 1, p. 7). A finite cap on how much the intercept may carry is different: it is not a change of ruler, and it can exclude models and change fitted returns (pp. 2, 7).
Inference with an estimated ruler. The estimate’s large-sample expansion adds a ruler-estimation term to the usual return term, including their covariance (Theorem 1, p. 16). The result is one standard-error formula that includes the ruler’s own error and does not depend on how the model is parameterised (Theorem 2, p. 20).
Near an estimated cap. The estimated ruler moves the cap as well as the estimate, so in large samples the estimate behaves like a normal draw pushed back onto a moving boundary, and its distribution is generally not symmetric (Theorem 3, p. 26). Intervals built from worst-case quantiles have at least nominal coverage in large samples at any distance from the cap, and may be conservative (Theorem 4, p. 29). The guarantee excludes reports whose critical quantile sits at an atom, such as the intercept’s own size when the cap binds exactly (p. 28).

Technical statement

Target. A loading class without hidden-factor rotations, identified by local curvature (Proposition 2, p. 8); covariance alignment, with local regularity and a strictly slack cap, suffices to make it the structural class even with a fixed cross section (Proposition 3, p. 12).
Estimator. Exactly profiled (each date’s factors solved exactly by least squares for any candidate loadings, dates combined with fixed weights, only loadings searched), on a regular local branch (Theorem 1, p. 16).
Reports. Each smooth report has a chart-invariant influence function (Theorem 2, p. 20); HAC estimation gives valid intervals (Proposition 5, p. 23). For one constraint and a scalar report, the slack and binding endpoints suffice (Proposition 6, p. 28).

5 A simple example

palette = ({
  naipca_iab: "#18557f", complete: "#18557f",
  qz_ipca: "#8a6a1a",
  ipca: "#6f9cc0", rp_pca: "#7a7a7a", ff5: "#b3b3b3",
  naipca_beta_only: "#8a8378", beta_only: "#8a8378",
  band: "rgba(0,0,0,0.06)", muted: "rgba(0,0,0,0.6)", rule: "rgba(0,0,0,0.1)"
})
LANG = (document.documentElement.lang || "en").startsWith("zh") ? "zh" : "en"
STR = ({ en: { view: "View", cost: "Cost (bp per traded dollar)", universe: "Universe", sample: "Sample (months)", log: "Log scale", models: "Models", statistic: "Statistic", theme: "Theme", sort: "Sort by", method: "Method", scenario: "Scenario", block: "Block (months)", metric: "Metric", intervals: "Intervals", cost_bp: "One-way cost (bp)", failed: "Interactive data failed to load. The static figure is shown instead." },
        zh: { view: "视图", cost: "成本（每交易一美元，基点）", universe: "股票池", sample: "样本（月）", log: "对数坐标", models: "模型", statistic: "统计量", theme: "主题", sort: "排序", method: "方法", scenario: "情景", block: "块长（月）", metric: "指标", intervals: "区间", cost_bp: "单边成本（基点）", failed: "交互数据加载失败，改为显示静态图。" } })
t = (k) => STR[LANG][k] ?? STR.en[k] ?? k
fmt2 = (x) => (x == null || Number.isNaN(x)) ? "n/a" : x.toFixed(2)
fmt1 = (x) => (x == null || Number.isNaN(x)) ? "n/a" : x.toFixed(1)
fmtBp = (x) => (x == null || Number.isNaN(x)) ? "> 500" : `${Math.round(x)} bp`
parseMonth = (s) => new Date(`${s}-01T00:00:00Z`)
// Sharpe, annualised, sample std (ddof = 1): matches the release convention
sharpe = (r) => {
  const n = r.length, m = r.reduce((a, b) => a + b, 0) / n
  const v = r.reduce((a, b) => a + (b - m) ** 2, 0) / (n - 1)
  const vol = Math.sqrt(v) * Math.sqrt(12)
  return vol > 1e-12 ? (m * 12) / vol : NaN
}
netSeries = (r, to, bp) => r.map((x, i) => x - to[i] * bp / 1e4)
wealthPath = (r) => { let w = 1; return r.map((x) => (w *= 1 + x)) }
nberMarks = (nber) => Plot.rectX(nber, { x1: (d) => parseMonth(d.start), x2: (d) => parseMonth(d.end), fill: palette.band })

An analogy. Three departments share one office, and the company’s total profit is fixed. Each department’s reported profit depends on how shared overhead is allocated. Change the rule and every department’s number changes, but the total does not, so checking the total can never tell you which rule is right. Here the departments are the hidden factors, the observed factors and the intercept; the overhead is the overlap between characteristics that pick out the same stocks.

Try this. 1. Set ρ, the overlap between the two value signals, to 0: both rulers agree. 2. Drag ρ to 0.9: the hidden-factor part grows, the observed-factor and intercept parts shrink, and the dashed total never moves. Why: the observed factor and the intercept load on the second signal, and the more it overlaps book-to-market, where the hidden factor loads, the more of it the characteristic ruler credits to the hidden factor. 3. Switch to stock B: two parts are negative and grow more negative as ρ rises, yet the total stays at 0.20. A negative part means that source is credited with pulling the stock’s fitted return down. What to notice: the fit cannot tell you which bar is right; only the ruler decides.

Figure 1: One fitted return, two rulers. Illustrative example, not paper data.

// Illustrative example (not paper data). Two characteristics, one hidden (latent) factor, one
// observed factor and an intercept. The start is the plain-ruler section (identity metric); the
// characteristic ruler (Gram metric) M = [[1, rho], [rho, 1]] is applied with the matched
// transformation of Proposition 1 (paper eqs. 3.2-3.4). The fitted value z'h is identical under
// both rulers by construction: 1.00 for stock A and 0.20 for stock B.
viewof csm_toy_rho = Inputs.range([0, 0.95], { value: 0.4, step: 0.05, label: LANG === "zh" ? "重叠度 ρ" : "Overlap ρ of the two value signals" })

viewof csm_toy_stock = Inputs.radio(new Map(LANG === "zh"
  ? [["股票 A：两个价值信号都高", "A"], ["股票 B：账面市值比高、另一个价值信号低", "B"]]
  : [["Stock A: high on both value signals", "A"], ["Stock B: high book-to-market, low second signal", "B"]]), { value: "A", label: LANG === "zh" ? "股票" : "Stock" })

csm_toy_split = (rho, z) => {
  // plain-ruler section: hidden-factor loading on characteristic 1, retained loadings on characteristic 2
  const b = [1, 0], d = [0, 0.6], a = [0, 0.4], f = 1, g = 1
  const M = [[1, rho], [rho, 1]]
  const Mv = (x) => [M[0][0] * x[0] + M[0][1] * x[1], M[1][0] * x[0] + M[1][1] * x[1]]
  const dot = (x, y) => x[0] * y[0] + x[1] * y[1]
  const R = Math.sqrt(dot(b, Mv(b)))                  // eq. 3.2: normalize the latent loading
  const bM = b.map((x) => x / R)
  const cd = dot(bM, Mv(d)), ca = dot(bM, Mv(a))      // eq. 3.3: overlap of retained loadings with the latent span
  const dM = d.map((x, i) => x - bM[i] * cd), aM = a.map((x, i) => x - bM[i] * ca)
  const fM = R * f + cd * g + ca                      // eq. 3.4: offsetting factor adjustment
  return { latent: dot(z, bM) * fM, observed: dot(z, dM) * g, intercept: dot(z, aM), exposure: dot(aM, Mv(aM)) }
}
csm_toy_z = csm_toy_stock === "A" ? [0.5, 0.5] : [0.5, -0.3]
csm_toy_I = csm_toy_split(0, csm_toy_z)
csm_toy_G = csm_toy_split(csm_toy_rho, csm_toy_z)
csm_toy_parts = LANG === "zh"
  ? [["latent", "潜在因子"], ["observed", "可观测因子"], ["intercept", "截距"]]
  : [["latent", "Hidden factor"], ["observed", "Observed factor"], ["intercept", "Intercept"]]
// Short axis labels (two lines); the table rows carry the full names.
csm_toy_rows = [[LANG === "zh" ? "普通尺子" : "Plain ruler", csm_toy_I], [LANG === "zh" ? `特征尺子\nρ = ${csm_toy_rho.toFixed(2)}` : `Characteristic ruler\nρ = ${csm_toy_rho.toFixed(2)}`, csm_toy_G]]
  .flatMap(([ruler, s]) => csm_toy_parts.map(([k, lab]) => ({ ruler, part: lab, value: s[k] })))

csm_toy_chart = Plot.plot({
  width: Math.min(width, 640), height: 190, marginLeft: width < 500 ? 120 : 150, marginRight: 20, marginBottom: 46, style: { fontFamily: "inherit", fontSize: 12 },
  ariaLabel: LANG === "zh" ? "图 1：同一个拟合收益，两把尺子（示意）" : "Figure 1: One fitted return, two rulers (illustrative)",
  ariaDescription: LANG === "zh"
    ? "上下两根堆叠条形分别用普通尺子和特征尺子，把同一个拟合收益拆成潜在因子、可观测因子和截距三部分；两把尺子下的总数相同。"
    : "Stacked bars split the same fitted return into hidden-factor, observed-factor and intercept parts under the plain ruler and under the characteristic ruler. The total is the same for both rulers.",
  x: { label: LANG === "zh" ? "对拟合收益的贡献（示意单位）" : "Contribution to the fitted return (illustrative units)", labelAnchor: width < 500 ? "right" : "center", labelOffset: 40, grid: true },
  y: { label: null, domain: csm_toy_rows.map((d) => d.ruler).filter((v, i, a) => a.indexOf(v) === i) },
  color: { domain: csm_toy_parts.map((p) => p[1]), range: [palette.complete, palette.qz_ipca, palette.ipca], legend: true },
  marks: [
    Plot.barX(csm_toy_rows, Plot.stackX({ x: "value", y: "ruler", fill: "part", order: csm_toy_parts.map((p) => p[1]) })),
    Plot.ruleX([0], { stroke: palette.muted }),
    Plot.tickX([{ x: csm_toy_I.latent + csm_toy_I.observed + csm_toy_I.intercept }], { x: "x", stroke: "black", strokeWidth: 2, strokeDasharray: "4 2" }),
    Plot.tip(csm_toy_rows, Plot.pointerY(Plot.stackX({ x: "value", y: "ruler", title: (d) => `${d.ruler}\n${d.part}: ${fmt2(d.value)}` })))
  ]
})

csm_toy_table = {
  const zh = LANG === "zh"
  const r = (s) => ({ lat: fmt2(s.latent), obs: fmt2(s.observed), int: fmt2(s.intercept), tot: fmt2(s.latent + s.observed + s.intercept), exp: s.exposure.toFixed(3) })
  const I = r(csm_toy_I), G = r(csm_toy_G)
  return html`<div class="table-scroll"><table class="table table-sm"><thead><tr><th></th><th>${zh ? "潜在因子" : "Hidden factor"}</th><th>${zh ? "可观测因子" : "Observed factor"}</th><th>${zh ? "截距" : "Intercept"}</th><th>${zh ? "拟合收益" : "Fitted return"}</th><th>${zh ? "截距承载量（平方）" : "How much the intercept carries (squared size)"}</th></tr></thead>
  <tbody><tr><td>${zh ? "普通尺子" : "Plain ruler (treats the two signals as unrelated)"}</td><td>${I.lat}</td><td>${I.obs}</td><td>${I.int}</td><td><strong>${I.tot}</strong></td><td>${I.exp}</td></tr>
  <tr><td>${zh ? `特征尺子（ρ = ${csm_toy_rho.toFixed(2)}）` : `Characteristic ruler (counts shared stocks; overlap ρ = ${csm_toy_rho.toFixed(2)})`}</td><td>${G.lat}</td><td>${G.obs}</td><td>${G.int}</td><td><strong>${G.tot}</strong></td><td>${G.exp}</td></tr></tbody></table></div>`
}

How to read it. The top bar splits a stock’s fitted return under the plain ruler; the bottom bar uses the characteristic ruler, under which the two signals overlap by ρ. The dashed tick marks the fitted return: 1.00 for stock A and 0.20 for stock B, under both rulers. The table’s last column, the same for both stocks, measures with each ruler how much the intercept carries overall, which is what a cap on the intercept limits; it can shrink while one stock’s intercept part grows. The default ρ = 0.40 is already high: the median real overlap is 0.06, and one pair in twenty exceeds 0.38 (p. 31). Source: illustrative calculation using the matched transformation in the paper’s Proposition 1 and equations 3.2 to 3.4; no data file.

Where the analogy differs: the characteristic ruler, once chosen, is computed from the characteristics, so its marks are drawn from the same data being measured.

6 Design and main results

The evidence has three parts (Section 8, pp. 30–38). A U.S. stock panel with 153 monthly characteristics, 1962–2025, measures how much real characteristics overlap (Section 8.1, p. 30). A stylised simulation (L = 12 characteristics, N = 100 stocks, 2,000 replications at T = 240 and 960 time periods) holds the point estimate fixed and compares ways of computing standard errors (pp. 32–33). A local experiment, a stylised simulation of the large-sample approximation that does not re-estimate a factor panel (p. 38), places the true value at set distances from an estimated cap on the intercept and compares intervals near it (Section 8.3, pp. 35–36).

Table 1: How 153 U.S. characteristics spread and overlap. The characteristic ruler, averaged over January 1962–December 2025 and trace-normalised, against the plain ruler (Table I, p. 31).

Diagnostic	Characteristic ruler (Gram metric)	Plain ruler (identity)
Share of variation in the largest direction	0.132	0.007
Share in the five largest directions	0.379	0.033
Directions needed for 90% of variation	68	138
Mean absolute off-diagonal entry (not the cosine used below)	0.114	0.000
Largest ÷ smallest eigenvalue	2,339	1

What this shows. Under the plain ruler, which by construction treats the 153 as unrelated, 138 directions are needed to reach 90% of the variation. Measured by the stocks they pick out, 68 suffice (Table I, p. 31). On a scale where 0 means two signals’ stock scores do not line up at all and 1 means they line up perfectly (an absolute cosine, not necessarily a correlation), most pairs barely overlap (median 0.06), but one pair in twenty exceeds 0.38 (p. 31). No returns are used: this shows where the split of Proposition 1 can move, not how far, and it describes exposure overlap, not the number of priced risks (p. 32).

Figure 2: Treating the estimated ruler as known makes standard errors too small. Same point estimate, five ways of computing standard errors, stylised simulation.

csm_cov_data = FileAttachment("/data/csm_coverage.json").json().catch(() => null)
csm_cov_failed = csm_cov_data === null ? html`<p class="figcaption"><strong>${t("failed")}</strong></p><img src="/assets/img/csm_coverage.png" alt="${LANG === "zh" ? "T = 960 时各标准误算法下 95% 区间的覆盖率（两个所报告的指标，简化模拟）：视尺子为已知时为 89.05% 和 91.75%，计入尺子误差的完整方法为 94.25% 和 94.85%" : "Coverage of 95% intervals by standard-error method at T = 960 for the two reported quantities in a stylised simulation: treating the ruler as known gives 89.05% and 91.75%; the complete method that includes the ruler's error gives 94.25% and 94.85%"}" style="width:100%">` : html``

viewof csm_cov_view = Inputs.radio(new Map(LANG === "zh"
  ? [["95% 区间覆盖率", "coverage95"], ["标准误 ÷ 实际离散度", "se_sd"], ["尾部校准（T = 960）", "qq"]]
  : [["Intervals containing the true value", "coverage95"], ["Standard error ÷ actual spread", "se_sd"], ["Tail calibration (T = 960)", "qq"]]), { value: "coverage95", label: t("view") })

viewof csm_cov_T = Inputs.radio(new Map(LANG === "zh" ? [["T = 240", 240], ["T = 960", 960]] : [["T = 240", 240], ["T = 960", 960]]), { value: 960, label: LANG === "zh" ? "样本长度" : "Sample length", disabled: csm_cov_view === "qq" })

csm_cov_zh = ({ moderate: "中等敏感度指标", high: "高敏感度指标", mc_oracle_scale: "蒙特卡洛实际离散度", oracle_complete_hac: "理想影响函数 HAC", feasible_fixed_metric_hac: "视尺子为已知", feasible_no_cross: "计入尺子误差、不含交叉项", feasible_complete_hac: "完整得分 HAC" })
// EN display names: the paper's Table II labels, with "metric" written as "ruler" (the page's term).
csm_cov_en = ({ moderate: "Moderate-sensitivity quantity", high: "High-sensitivity quantity", feasible_fixed_metric_hac: "Ruler treated as known", feasible_no_cross: "Ruler error, no cross term" })
csm_cov_name = (id, en) => LANG === "zh" ? (csm_cov_zh[id] ?? en) : (csm_cov_en[id] ?? en)
csm_cov_reports = csm_cov_data ? new Map(csm_cov_data.reports.map((r) => [r.id, csm_cov_name(r.id, r.label)])) : new Map()
csm_cov_methods = csm_cov_data ? new Map(csm_cov_data.methods.map((m) => [m.id, csm_cov_name(m.id, m.label)])) : new Map()
csm_cov_order = csm_cov_data ? csm_cov_data.methods.map((m) => csm_cov_name(m.id, m.label)) : []
csm_cov_rows = csm_cov_data ? csm_cov_data.rows.filter((r) => r.T === csm_cov_T).map((r) => ({ ...r, label: csm_cov_methods.get(r.method) ?? r.label, reportLabel: csm_cov_reports.get(r.report), value: r[csm_cov_view] })) : []
csm_cov_known = csm_cov_name("feasible_fixed_metric_hac", "Metric treated as known")
csm_cov_complete = csm_cov_name("feasible_complete_hac", "Complete-score HAC")
csm_cov_qq = csm_cov_data ? csm_cov_data.quantiles.map((q) => ({ ...q, reportLabel: csm_cov_reports.get(q.report), methodLabel: q.method === "feasible_fixed_metric_hac" ? csm_cov_known : csm_cov_complete })) : []

csm_cov_chart = {
  if (!csm_cov_data) return html``
  const repColor = { domain: [csm_cov_reports.get("moderate"), csm_cov_reports.get("high")], range: [palette.complete, palette.qz_ipca], legend: false }
  if (csm_cov_view === "qq") {
    return Plot.plot({
      width: Math.min(width, 640), height: 340, marginLeft: 56, marginBottom: 46, style: { fontFamily: "inherit", fontSize: 12 },
      ariaLabel: LANG === "zh" ? "图 2：T = 960 时标准化指标的尾部校准" : "Figure 2: Tail calibration of the standardised quantities at T = 960",
      ariaDescription: LANG === "zh"
        ? "两个所报告指标的标准化经验分位数与标准正态分位数的对比。视尺子为已知时，两侧尾部都比完整得分曲线离零更远；完整得分曲线贴近对角线。"
        : "Empirical quantiles of the standardised report against standard-normal quantiles for the two reports. With the ruler treated as known, both tails lie farther from zero than the complete-score curves, which stay close to the diagonal.",
      fx: { label: null, domain: [csm_cov_reports.get("moderate"), csm_cov_reports.get("high")] },
      x: { label: LANG === "zh" ? "标准正态分位数" : "Normal quantile", domain: [-2.5, 2.5], labelAnchor: "center", labelOffset: 40 },
      y: { label: LANG === "zh" ? "标准化分位数" : "Standardised quantile", grid: true, domain: [-3.2, 3.2], labelAnchor: "center" },
      color: { domain: [csm_cov_known, csm_cov_complete], range: [palette.qz_ipca, palette.complete], legend: true },
      marks: [
        Plot.line(csm_cov_qq, { fx: "reportLabel", x: "normal", y: "studentized", stroke: "methodLabel", strokeWidth: 1.8 }),
        Plot.line([[-2.4, -2.4], [2.4, 2.4]], { stroke: palette.muted, strokeWidth: 1, strokeDasharray: "4 3" }),
        Plot.tip(csm_cov_qq, Plot.pointer({ fx: "reportLabel", x: "normal", y: "studentized", title: (d) => LANG === "zh" ? `${d.methodLabel}\n概率 ${d.p.toFixed(2)}\n正态 ${fmt2(d.normal)}，标准化 ${fmt2(d.studentized)}` : `${d.methodLabel}\nprobability ${d.p.toFixed(2)}\nnormal ${fmt2(d.normal)}, standardised ${fmt2(d.studentized)}` }))
      ]
    })
  }
  const cov = csm_cov_view === "coverage95"
  return Plot.plot({
    width: Math.min(width, 640), height: 270, marginLeft: width < 500 ? 165 : 185, marginRight: 20, marginBottom: 46, style: { fontFamily: "inherit", fontSize: 12 },
    ariaLabel: LANG === "zh"
      ? `图 2：T = ${csm_cov_T} 时${cov ? " 95% 区间实际包含真值的比例" : "平均标准误与实际离散度之比"}`
      : `Figure 2: ${cov ? "Share of 95% intervals that contain the true value" : "Mean standard error divided by the actual spread across replications"} at T = ${csm_cov_T}`,
    ariaDescription: LANG === "zh" ? "每行一种标准误算法，每个点一个所报告的指标，虚线标出目标值。" : "One row per standard-error method, one point per report, with the target marked by a dashed line.",
    x: { label: cov ? (LANG === "zh" ? "95% 区间实际包含真值的比例（%）" : "Share of 95% intervals containing the true value (%)") : (LANG === "zh" ? "平均标准误 ÷ 各次重复间的实际离散度" : "Mean standard error ÷ actual spread across replications"), domain: cov ? [86, 97] : [0.78, 1.06], labelAnchor: width < 500 ? "right" : "center", labelOffset: 40, grid: true },
    y: { label: null, domain: csm_cov_order },
    color: repColor, symbol: { domain: repColor.domain, range: ["circle", "square"], legend: true },
    marks: [
      Plot.ruleX([cov ? 95 : 1], { stroke: palette.muted, strokeDasharray: "4 3" }),
      ...["moderate", "high"].map((k, i) => Plot.dot(csm_cov_rows.filter((d) => d.report === k), { x: "value", y: "label", fill: "reportLabel", symbol: "reportLabel", r: 5, dy: i ? 4 : -4 })),
      Plot.tip(csm_cov_rows, Plot.pointer({ x: "value", y: "label", title: (d) => `${d.label}, ${d.reportLabel}\nT = ${d.T}\n${cov ? (LANG === "zh" ? "覆盖率 " + d.value.toFixed(2) + "%" : d.value.toFixed(2) + "% coverage") : (LANG === "zh" ? "标准误／实际离散度 " : "standard error ÷ spread ") + d.value.toFixed(3)}` }))
    ]
  })
}

How to read it. Rows are ways of computing standard errors (Table II, p. 34); circles and squares are two reported quantities, each measuring how closely a fixed combination of characteristics lines up with the hidden factors (true value 0.5). The top two rows (“Monte Carlo scale”, “Oracle-influence HAC”) need information available only in a simulation. Compare “Ruler treated as known” with “Complete-score HAC” (HAC: heteroskedasticity- and autocorrelation-consistent), the paper’s full method, which includes the ruler’s error. The dashed line is the target: intervals that contain the true value 95 times in 100, or standard errors equal to the actual spread across replications. “Tail calibration” compares quantiles of the standardised estimate with normal quantiles at T = 960; the dashed grey diagonal is a perfect fit. Source: paper Table II and Figure 2, and Section 8.2 for the paired gains; rebuilt_interior_coverage.csv, paired_coverage_gains.csv, studentized_quantiles.csv. Data

What this shows. With 960 time periods, treating the estimated ruler as known makes standard errors too small, so 95% intervals contain the true value only about 89 and 92 times in 100. Including the ruler’s error restores about 94–95; the paired gains, 5.20 and 3.10 percentage points, have Monte Carlo standard errors of 0.50 and 0.39 (Table II, p. 34). Most of the shortfall is the ruler’s own variance (Section 8.2, p. 33); with 240 time periods even the complete and oracle methods fall slightly short, at about 93–94 times, from finite-sample error in estimating the covariance (p. 33).

Near an estimated cap, the paper’s stylised local experiment (960 time periods) again targets 95% intervals. One calibrated as if the cap binds contains it as few as about 84 times (83.80%) when the cap is in fact slack. One that holds whether or not the cap binds (the paper’s drift-robust interval) contains it about 96–99 times (96.30–98.75%) at every tested distance (Table III, p. 36). The price is width: when the cap binds, it is about 25% wider (3.241 against 2.584) than a benchmark that knows how far the true value sits from the cap, which no one can compute in practice (Table III, p. 36).

7 How it relates to prior work

Factor-model inference usually fixes rotation and scale by a normalisation. This paper asks what that normalisation does to the split when the ruler behind it is estimated from the data.

Prior work	What it established	What this paper adds
Bai (2003); Bai and Ng (2013)	Identification and inference for large approximate factor models under normalisation restrictions.	Loadings identified up to hidden-factor rotations for each ruler, keeping observed-factor and intercept blocks; theory for a fixed cross section.
Kelly et al. (2019)	IPCA: characteristics set how strongly each stock moves with hidden factors, with an intercept and optional observed factors; rotation is fixed with the plain ruler, Γβ′Γβ = I.	Treats that normalisation as one choice of ruler, shows the split depends on it, and carries an estimated characteristic ruler into inference.
Newey (1994); Newey and McFadden (1994); Andrews and Guggenberger (2009); Andrews and Soares (2010)	First-step estimation corrections; uniformly valid inference when inequality constraints may bind.	Applies the two-step correction to a ruler that both selects the split and enters the report; near one smooth cap, uniform intervals from two endpoint calculations.

The angle is measurement, not a new predictor. The tools are known: geometry on matrix manifolds, two-step adjustment and constrained projection. What is new is their combination for one object, a split of fitted returns whose ruler is estimated from the same data and may also set the cap on the intercept.

For specialists: further related work

Matrix-manifold geometry (Edelman et al. 1998) removes hidden-factor rotations; boundary limit laws (Geyer 1994; Andrews 1999; Shapiro 2000) are extended to a boundary learned through the same ruler; and where Li (2025) bootstraps constrained estimators, this paper’s route is analytic and uniform over local slack.

Where it fits. Geometric Framework, an earlier and broader version of this paper, argued for taking the ruler from the data; this paper proves that the ruler decides how a fixed fit is split and shows how to do inference when the ruler is estimated. Characteristic Geometry cites it as its companion theory paper and follows a change of ruler into a portfolio; Interpreting Pricing Errors cites it for the representation identities (p. 4) and holds the ruler fixed to read alphas. The job market paper, which also cites it as a companion paper, asks a parallel question about the library and the ridge penalty. Changing the library is not a change of ruler in this paper’s sense.

8 Scope

Better uncertainty measurement, not better fitted returns (Section 8.2, p. 33).
The simulation is stylised, not calibrated to U.S. data (pp. 32–33).
Reading the intercept as a pricing error needs a specified pricing model (Section 2, p. 5).
The ruler is a declared convention; no ruler is shown to be best (p. 2).
Local, well-identified theory; near a cap, one smooth constraint and one reported number only (Sections 7.3 and 9, pp. 28, 38).

References

Andrews, Donald W. K. 1999. “Estimation When a Parameter Is on a Boundary.” Econometrica 67 (6): 1341–83. https://doi.org/10.1111/1468-0262.00082.

Andrews, Donald W. K., and Patrik Guggenberger. 2009. “Validity of Subsampling and ‘Plug-in Asymptotic’ Inference for Parameters Defined by Moment Inequalities.” Econometric Theory 25 (3): 669–709. https://doi.org/10.1017/S0266466608090257.

Andrews, Donald W. K., and Gustavo Soares. 2010. “Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection.” Econometrica 78 (1): 119–57. https://doi.org/10.3982/ECTA7502.

Bai, Jushan. 2003. “Inferential Theory for Factor Models of Large Dimensions.” Econometrica 71 (1): 135–71. https://doi.org/10.1111/1468-0262.00392.

Bai, Jushan, and Serena Ng. 2013. “Principal Components Estimation and Identification of Static Factors.” Journal of Econometrics 176 (1): 18–29. https://doi.org/10.1016/j.jeconom.2013.03.007.

Edelman, Alan, Tomás A. Arias, and Steven T. Smith. 1998. “The Geometry of Algorithms with Orthogonality Constraints.” SIAM Journal on Matrix Analysis and Applications 20 (2): 303–53. https://doi.org/10.1137/S0895479895290954.

Geyer, Charles J. 1994. “On the Asymptotics of Constrained M-Estimation.” The Annals of Statistics 22 (4): 1993–2010. https://doi.org/10.1214/aos/1176325768.

Kelly, Bryan T., Seth Pruitt, and Yinan Su. 2019. “Characteristics Are Covariances: A Unified Model of Risk and Return.” Journal of Financial Economics 134 (3): 501–24. https://doi.org/10.1016/j.jfineco.2019.05.001.

Li, Jessie. 2025. “The Proximal Bootstrap for Constrained Estimators.” Journal of Statistical Planning and Inference 236: 106245. https://doi.org/10.1016/j.jspi.2024.106245.

Newey, Whitney K. 1994. “The Asymptotic Variance of Semiparametric Estimators.” Econometrica 62 (6): 1349–82. https://doi.org/10.2307/2951752.

Newey, Whitney K., and Daniel McFadden. 1994. “Large Sample Estimation and Hypothesis Testing.” In Handbook of Econometrics, edited by Robert F. Engle and Daniel L. McFadden, vol. 4. Elsevier. https://doi.org/10.1016/S1573-4412(05)80005-4.

Shapiro, Alexander. 2000. “On the Asymptotics of Constrained Local M-Estimators.” The Annals of Statistics 28 (3): 948–60. https://doi.org/10.1214/aos/1015952006.

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Citation

BibTeX citation:

@report{liu2026metrics,
  author = {Liu, Mingyang},
  title = {Characteristic-Space {Metrics} in {Factor} {Models:}
    {Identification} and {Inference}},
  date = {2026-09},
  url = {https://yl7919.github.io/research/characteristic-space-metrics.html},
  doi = {10.2139/ssrn.7445120},
  langid = {en}
}

For attribution, please cite this work as:

Liu, Mingyang. 2026. Characteristic-Space Metrics in Factor Models: Identification and Inference. SSRN preprint. https://doi.org/10.2139/ssrn.7445120.