Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Identification, Measurement, and Counterfactual Reasoning/I-09
Measurement, Records, and Time · Chapter I-09

Clock Error and Decision-Relevant Moments

Recorded time W can differ from event time G through displacement U. A decision may require only selected moments of that displacement. Binomial moment recursions recover latent temporal moments, and polynomial losses reveal exactly which clock moments determine an action and its residual loss.

Conceptual map

  1. I-09.01Physical and recorded time
  2. I-09.02Required independence restrictions
  3. I-09.03Triangular moment recursions
  4. I-09.04Decision sufficiency
  5. I-09.05Moment minimality

1. Clock error is an additive latent variable

A timing correction is identified only after the relation between event time, recorded time, and displacement is fixed. Independence converts observed moments into a triangular recursion.

Let physical event time be a scalar random variable \(G\), let clock displacement be \(U\), and let recorded time be \(W=G+U\). Assume \(G\) and \(U\) are independent and possess finite moments through order \(d\). Write \(m_{k}^{V}\)=E[\(V^{k}\)] and set every zeroth moment equal to one.

Proposition 1 · Triangular raw-moment recovery

If the moments of \(W\) and \(U\) are known through order \(d\), the moments of \(G\) are uniquely determined by

\[m_{k}^{G}=m_{k}^{W}-\sum _{j=1}^{k} C(k,j)m_{k-j}^{G}m_{j}^{U}, \quad k=1,\ldots ,d.\](1)

Proof. Expand \((G+U)^{k}\) by the binomial theorem and use independence to factor every mixed moment. The term with \(j=0\) is \(m_{k}^{G}\); all remaining latent moments of \(G\) have order below \(k\). Rearrangement gives (1), and induction gives uniqueness. ∎

2. A three-moment calculation

Let \(G\) equal two or four with equal probability. Let an independent displacement \(U\) equal \(-1\) with probability 1/4 and 1 with probability 3/4. The event-time moments are (3,10,36), and the displacement moments are (1/2,1,1/2). Direct enumeration of \(W=G+U\) gives

\[(m_{1}^{W},m_{2}^{W},m_{3}^{W})=(3.5,14,60.5).\](2)

Recursion (1) returns \(3.5-0.5=3\), then \(14-2(3)(0.5)-1=10\), then \(60.5-3(10)(0.5)-3(3)(1)-0.5=36\). The calculation reports raw moments; central moments require the usual recentering identities.

3. Quadratic scheduling separates action and residual loss

Suppose a planner schedules at recorded time plus a constant shift \(s\) and incurs squared timing loss \([G-(W+s)]^{2}\). Since \(G-W=-U\),

\[L(s)=E[(U+s)^{2}]=\operatorname{Var}(U)+[s+E(U)]^{2}.\](3)

The unique optimal shift is \(s^{*}=-E(U)\), while the minimized loss is \(\operatorname{Var}(U)\). The mean displacement identifies the action; the second moment is additionally required to evaluate residual loss. More generally, the expectation of a degree-\(d\) polynomial loss is determined by displacement moments through order \(d\), subject to any moments involving other random inputs in that loss.

Two symmetric clocks, \(U_{A}\in\{-1\),1} with equal probabilities and \(U_{B}=0\), have the same optimal quadratic shift zero. Their minimized losses are one and zero. Agreement about an action therefore carries less information than agreement about its value.

4. Dependence breaks the marginal-moment recursion

Failure case · Calibration moments used without joint restrictions

Let \(G\) take \(-1\) and 1 equally and set \(U=-G\). Then \(W=0\), while both \(G\) and \(U\) have mean zero and variance one. Applying the independence formula at order two would give \(m_{2}^{G}=m_{2}^{W}-m_{2}^{U}=-1\), an impossibility. The omitted mixed moment \(2\mathbb{E}[GU]=-2\) carries the discrepancy.

Mean independence can support the first-moment identity while leaving higher recursions invalid. Conditional calibration, cohort-specific displacement, censoring at observation-window boundaries, and heaping each require an explicit observation law.

5. Implementation, exercises, and sources

Record whether moments are raw or central, compute (1) from low to high order, and substitute recovered moments back into the forward binomial equations. Report the largest forward residual and the conditioning of any scaled moment system. High-order raw moments can be numerically unstable, so rescaling time and using orthogonal polynomial bases may improve computation without changing identification.

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Exercises

  1. Reproduce all moments in (2) by enumerating the four joint support points.
  2. Derive the fourth-order recursion and identify every mixed-moment restriction supplied by independence.
  3. For absolute timing loss, characterize the set of optimal shifts and identify the required feature of the displacement law.
Partial solutions

1. The joint probabilities are 1/8, 3/8, 1/8, and 3/8 for recorded times 1, 3, 3, and 5. Their first three raw moments are 3.5, 14, and 60.5. 2. Subtract \(4m_{3}^{G}m_{1}^{U}+6m_{2}^{G}m_{2}^{U}+4m_{1}^{G}m_{3}^{U}+m_{4}^{U}\) from \(m_{4}^{W}\). 3. Any negative median of \(U\) is optimal.

  1. William Feller, An Introduction to Probability Theory and Its Applications, Volume II, chapters on characteristic functions and moments.Moments of sums and convolution.
  2. Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian Crainiceanu, Measurement Error in Nonlinear Models.Additive measurement-error assumptions and their consequences.
  3. II-11 · Testing a Clock Model.Sampling and overidentification checks for calibrated clock laws.
  4. Chae-Yeon Xon (2026), “Fuzzy Clock: Decision-Specific Timing Information.”Related scheduling application of clock moments.

6. Audit checkpoint

State the event, recorded-time, and displacement variables; sign convention; support and window; independence or conditional-independence restriction; available moment order; raw or central convention; forward residuals; loss function; action menu; and exact target of the timing claim.

7. Scope boundary

The chapter establishes finite-moment recovery and decision sufficiency for an additive clock. Distributional deconvolution, censoring estimators, and duration models require further assumptions.

Prerequisites