{
 "bundle_of": "C3",
 "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": "C3",
   "statement": "For [[D13:ETS(M,Md,N)]] simulated on [[D14:Simulation experiment 1 synthetic series]] with an [[D6:Inverse Gaussian error distribution]], the [[D8:percentage difference (PD)]] between the conditional expectation and the point forecast increases with the damping parameter $\\phi$, the trend smoothing parameter $\\beta$, the forecast horizon $h$, the scale parameter $\\sigma$, and the level smoothing parameter $\\alpha$, with $\\phi$ having the largest effect and $\\alpha$ having the smallest effect among the scaled predictors.",
   "about": {
    "subject": "D13",
    "relation": "exhibits increasing forecast bias with respect to model parameters and horizon",
    "object": "D8",
    "comparator": null,
    "conditions": [
     "D14",
     "D6"
    ]
   },
   "type": "descriptive",
   "role": "principal",
   "novelty": "no_prior_found",
   "planning": "exploratory",
   "scope": {
    "conditions": [
     "$ETS(M,Md,N)$ simulated over $10,000$ iterations with Inverse Gaussian errors across parameter grids"
    ]
   },
   "relations": [
    {
     "type": "contradicts",
     "target": "978-3-540-71918-2",
     "id_type": "isbn",
     "locator": "Chapter 2, p. 23",
     "note": "Shows that the difference between point forecasts and conditional expectations is non-negligible and grows substantially with horizon and parameter values."
    }
   ],
   "support": [
    {
     "kind": "simulation",
     "evidence": "E2",
     "artefacts": [
      "A3"
     ],
     "results": [
      "E2.T1"
     ]
    }
   ],
   "assumptions": [
    "C12",
    "C13"
   ],
   "limitations": [
    "L2"
   ],
   "assessments": [
    {
     "by": "author",
     "strength": "high",
     "rationale": "Based on 10,000 Monte Carlo iterations across a broad parameter grid with 534,884 degrees of freedom and $R^2 = 0.986$."
    }
   ],
   "provenance": "author_stated",
   "verifiability": "unverifiable",
   "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"
  },
  {
   "id": "D6",
   "term": "Inverse Gaussian error distribution",
   "kind": "concept",
   "definition": "Probability distribution on $(0, \\infty)$ applied to $1+\\epsilon_t$ with location $\\mu = 1$ and scale $\\sigma^2$ so that $E(1+\\epsilon_t) = 1$ and $V(1+\\epsilon_t) = \\sigma^2$."
  },
  {
   "id": "D8",
   "term": "percentage difference (PD)",
   "kind": "variable",
   "unit": "1",
   "definition": "Relative deviation of the conventional point forecast $\\hat{y}_{t+h}$ from the conditional expectation $\\mu_{y,t+h}$, defined as $$PD = \\frac{\\mu_{y,t+h} - \\hat{y}_{t+h}}{\\mu_{y,t+h}}$$ lying in $[0, 1)$."
  },
  {
   "id": "D13",
   "term": "ETS(M,Md,N)",
   "kind": "method",
   "broader": [
    "D1"
   ],
   "definition": "Pure multiplicative exponential smoothing model with damped multiplicative trend and no seasonality.",
   "source": "B3"
  },
  {
   "id": "D14",
   "term": "Simulation experiment 1 synthetic series",
   "kind": "data",
   "definition": "10,000 iterations of 100-observation time series simulated from $ETS(M,Md,N)$ with Inverse Gaussian errors across parameter grids $\\sigma \\in \\{0.01, 0.05, 0.1, 0.5, 1\\}$, $\\alpha, \\beta \\in \\{0, 0.1, 0.2, 0.5\\}$, $\\phi \\in \\{0.5, 0.9, 0.95, 1\\}$, and horizons $h \\in \\{10, 50, 100\\}$."
  },
  {
   "id": "D19",
   "term": "Box-Cox odds regression procedure",
   "kind": "design",
   "definition": "Transformation of percentage difference $PD$ into odds $PD/(1-PD)$ followed by Box-Cox transformation ($\\lambda = 0.146$) and linear regression on scaled parameters $\\sigma$, horizon, $\\alpha$, $\\beta$, and $\\phi$ using `greybox::alm` (`distribution='dbcnorm'`)."
  }
 ],
 "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"
   }
  },
  {
   "id": "E2",
   "profile": "simulation",
   "description": "Monte Carlo simulation of $10,000$ iterations of $100$ observations from $ETS(M,Md,N)$ with Inverse Gaussian errors using `smooth::sim.es()`, followed by Box-Cox odds regression (`greybox::alm` with `distribution='dbcnorm'`) of percentage difference $PD$ on scaled parameters $\\sigma$, horizon, $\\alpha$, $\\beta$, and $\\phi$ (Section 3, Table 1).",
   "setup": {
    "studied": [
     "D13",
     "D6"
    ],
    "cases": [
     "D14"
    ],
    "procedure": [
     "D19"
    ],
    "lens": [
     "D8"
    ]
   },
   "model": "$ETS(M,Md,N)$ with Inverse Gaussian errors via `smooth::sim.es` v3.2.0",
   "runs": 10000,
   "tables": [
    {
     "id": "T1",
     "description": "Box-Cox odds regression output for percentage difference $PD$ in Simulation Experiment 1 ($\\lambda = 0.146$, $R^2 = 0.986$, $\\text{df} = 534,884$).",
     "artefact": "A3",
     "row_labels": [
      "variable"
     ],
     "columns": [
      "estimate",
      "std_error",
      "lower_0.05pct",
      "upper_99.95pct"
     ],
     "units": [
      "1",
      "1",
      "1",
      "1"
     ],
     "rows": [
      {
       "id": "r1",
       "label": [
        "(Intercept)"
       ],
       "values": [
        -9.222,
        0.006,
        -9.242,
        -9.202
       ]
      },
      {
       "id": "r2",
       "label": [
        "sigma"
       ],
       "values": [
        2.755,
        0.003,
        2.745,
        2.765
       ]
      },
      {
       "id": "r3",
       "label": [
        "horizon"
       ],
       "values": [
        3.656,
        0.003,
        3.646,
        3.667
       ]
      },
      {
       "id": "r4",
       "label": [
        "alpha"
       ],
       "values": [
        1.578,
        0.007,
        1.555,
        1.6
       ]
      },
      {
       "id": "r5",
       "label": [
        "beta"
       ],
       "values": [
        3.739,
        0.007,
        3.716,
        3.762
       ]
      },
      {
       "id": "r6",
       "label": [
        "phi"
       ],
       "values": [
        6.167,
        0.006,
        6.148,
        6.187
       ]
      }
     ]
    }
   ],
   "artefacts": [
    "A3"
   ],
   "availability": {
    "level": "unavailable",
    "reason": "other"
   }
  }
 ],
 "limitations": [
  {
   "id": "L2",
   "category": "model_specification",
   "statement": "Simulation experiment 1 evaluates point forecast bias exclusively on $ETS(M,Md,N)$ with Inverse Gaussian errors and known true parameters.",
   "affects": [
    "C3"
   ],
   "impact": "reduces_transferability",
   "mitigation": "none"
  }
 ],
 "artefacts": [
  {
   "id": "A3",
   "kind": "table",
   "uri": "https://entrylib.org/p/2026.000006/v1/files/files/simulation-results.csv",
   "version": "sha256:c12d5bad7a350e03b8f6ef8606cb1ad7f12b8baca1b4daa50a482b99570f2296",
   "licence": "CC-BY-4.0",
   "availability": {
    "level": "public"
   },
   "description": "Full simulation results tables (Table 1 from Section 3 and Tables 4-5 from Section 5) exported as CSV."
  }
 ],
 "references": [
  {
   "id": "B3",
   "target": "978-3-540-71918-2",
   "id_type": "isbn"
  },
  {
   "id": "B7",
   "target": "10.1201/9781003452652",
   "id_type": "doi"
  }
 ]
}