{
 "bundle_of": "C1",
 "schema_version": "0.42",
 "paper": {
  "id": "entrylib:2026.000006",
  "version": 1,
  "title": "Staying Positive: Challenges and Solutions in Using Pure Multiplicative ETS Models",
  "licence": "CC-BY-4.0",
  "status": "active",
  "date": "2026-11-12"
 },
 "claims": [
  {
   "id": "C1",
   "statement": "In general, the conventional multi-step point forecast $\\hat{y}_{t+h}$ of a pure multiplicative [[D1:ETS(M,*,*)]] model does not coincide with the conditional arithmetic expectation $\\mu_{y,t+h}$ or the conditional geometric expectation $\\tilde{y}_{t+h}$, satisfying $\\hat{y}_{t+h} \\le \\tilde{y}_{t+h} \\le \\mu_{y,t+h}$ when smoothing parameters are non-zero.",
   "about": {
    "subject": "D1",
    "relation": "produces point forecasts bounded above by conditional geometric and arithmetic expectations",
    "object": null,
    "comparator": null,
    "conditions": [
     "non-zero smoothing parameters",
     "horizon $h > 1$"
    ]
   },
   "type": "formal",
   "role": "principal",
   "novelty": "extension",
   "planning": "exploratory",
   "scope": {
    "conditions": [
     "pure multiplicative ETS models with non-zero smoothing parameters and forecast horizon $h > 1$"
    ]
   },
   "relations": [
    {
     "type": "extends",
     "target": "978-3-540-71918-2",
     "id_type": "isbn",
     "locator": "Chapter 2, pp. 21-23",
     "note": "Derives the explicit ordering between point forecasts, geometric conditional expectations, and arithmetic conditional expectations using logarithmic state space recursion."
    }
   ],
   "support": [
    {
     "kind": "formal_proof",
     "evidence": "E1"
    },
    {
     "kind": "cited",
     "reference": "B3",
     "locator": "pp. 21-23",
     "self": false
    }
   ],
   "assumptions": [
    "C12",
    "C13"
   ],
   "limitations": [
    "L1"
   ],
   "verification": "unchecked",
   "provenance": "author_stated",
   "verifiability": "open",
   "verifiability_flags": []
  },
  {
   "id": "C12",
   "statement": "The multiplicative error term $(1+\\epsilon_t)$ in pure multiplicative ETS models is independent and identically distributed with unit expectation $E(1+\\epsilon_t) = 1$.",
   "type": "formal",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "statistical",
   "check": "unchecked",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  },
  {
   "id": "C13",
   "statement": "Smoothing parameters $\\alpha, \\beta, \\gamma$ in pure multiplicative ETS models lie within $[0, 1]$ so that state update terms $(1 + g\\epsilon_t)$ remain strictly positive whenever $1 + \\epsilon_t > 0$.",
   "type": "formal",
   "role": "supporting",
   "support": [
    {
     "kind": "assumption"
    }
   ],
   "assumption_category": "methodological",
   "check": "tested",
   "check_evidence": "E1",
   "provenance": "author_stated",
   "verifiability": "not_applicable",
   "verifiability_flags": []
  }
 ],
 "definitions": [
  {
   "id": "D1",
   "term": "ETS(M,*,*)",
   "kind": "method",
   "definition": "Pure multiplicative exponential smoothing state space model family written in logarithmic state space form as $$\\begin{aligned} \\log y_t &= w' \\log(v_{t-l}) + \\log(1 + \\epsilon_t) \\\\ \\log v_t &= F \\log v_{t-l} + \\log(1_k + g \\epsilon_t) \\end{aligned}$$ with conventional multi-step point forecast $\\hat{y}_{t+h} = \\exp\\left((w_1' F_1^{h-1} + w_m' F_m^{\\lceil h/m \\rceil - 1}) \\log v_t\\right)$.",
   "source": "B7"
  }
 ],
 "evidence": [
  {
   "id": "E1",
   "profile": "formal",
   "description": "Analytical derivation of recursive conditional expectations and point forecasts for pure multiplicative ETS models in logarithmic state space form (Section 2, Eqs. 2.1-2.11).",
   "foundation": "classical_probability",
   "assumptions": [
    "C12",
    "C13"
   ],
   "steps": [
    {
     "id": "E1.1",
     "statement": "Reformulate pure multiplicative ETS in logarithmic state space form: $\\log y_t = w' \\log(v_{t-l}) + \\log(1 + \\epsilon_t)$ and $\\log v_t = F \\log v_{t-l} + \\log(1_k + g \\epsilon_t)$.",
     "by": "Section 2, Eq. (2.1)",
     "rule": "logarithmic_transformation"
    },
    {
     "id": "E1.2",
     "statement": "Expand $h$-steps-ahead recursion into non-seasonal and seasonal components and take conditional expectation $E(\\log y_{t+h}|t)$.",
     "by": "Section 2, Eqs. (2.5)-(2.6)",
     "rule": "recursive_substitution"
    },
    {
     "id": "E1.3",
     "statement": "Apply Jensen's inequality and non-negativity of $g$ to show point forecast $\\hat{y}_{t+h} \\le \\tilde{y}_{t+h} \\le \\mu_{y,t+h}$, with equality only at $h=1$, $h \\le m$ without trend, or for $ETS(M,N,N)$.",
     "by": "Section 2, Eqs. (2.8)-(2.11)",
     "rule": "jensens_inequality"
    }
   ],
   "availability": {
    "level": "public"
   }
  }
 ],
 "limitations": [
  {
   "id": "L1",
   "category": "model_specification",
   "statement": "Conditional expectations for pure multiplicative ETS models other than $ETS(M,N,N)$ lack closed-form analytical expressions and require Monte Carlo simulation.",
   "affects": [
    "C1"
   ],
   "impact": "reduces_transferability",
   "mitigation": "none"
  }
 ],
 "references": [
  {
   "id": "B3",
   "target": "978-3-540-71918-2",
   "id_type": "isbn"
  },
  {
   "id": "B7",
   "target": "10.1201/9781003452652",
   "id_type": "doi"
  }
 ]
}