Uncertainty propagation: keep covariance between inputs

CoTrading editorial team 2026-10-25 阅读中文全文

Uncertainty in a calculated difference or ratio depends on the covariance between its inputs, not only their separate uncertainties. Shared measurement errors can cancel in a difference and reinforce in a sum. Preserve the dependence assumptions, distinguish standard uncertainty from hard bounds and avoid reporting a precise derived value while discarding how its inputs were measured.

Define what is uncertain

Suppose two estimates are used to compare operating quantities. Their uncertainties concern the estimated inputs, not automatically the volatility of share returns or the range of every future business outcome. State the measurement, unit, period and meaning of each uncertainty before propagating it. A standard deviation and a maximum possible error are different inputs to an analysis.

The NIST explanation of combining uncertainty components includes covariance terms and describes a first-order approximation for nonlinear functions. That supplies a mathematical reference, not evidence for the covariance of a particular company’s data. The following numbers and dependence assumptions are hypothetical.

A hypothetical difference and sum

Assume estimate A is 100 with standard uncertainty 10, while estimate B is 80 with standard uncertainty 8. Suppose their error correlation is 0.75, making covariance 0.75 × 10 × 8 = 60. The difference is 20 and the sum is 180, but their combined uncertainties move in opposite directions when the positive covariance is included.

Derived quantity	Variance calculation	Standard uncertainty
A minus B	100 + 64 - 120 = 44	About 6.63
A plus B	100 + 64 + 120 = 284	About 16.85
Either, incorrectly assuming independence	100 + 64 = 164	About 12.81

For a difference, the covariance term is subtracted twice; for a sum, it is added twice. The independence assumption overstates uncertainty for the difference and understates it for the sum in this example. It is not automatically conservative simply because it ignores a relationship that is difficult to estimate.

Shared exchange-rate conversion, a common survey sample or the same estimated quantity can create linked errors. Different data suppliers can still inherit one underlying input. Conversely, a correlation between the observed economic series does not automatically equal the correlation between their measurement errors. Identify the relevant error-generating mechanism rather than importing an unrelated market correlation.

Keep a covariance or correlation assumption beside each pair of inputs, including how it was estimated or why it remains unknown. Do not assign a convenient zero solely because the source omitted it. When evidence is limited, show sensitivity to a few defensible dependence scenarios and explain that the scenario range is not a confidence interval.

Treat ratios as nonlinear calculations

For R = A / B, a first-order variance approximation contains contributions from the uncertainty of A, the uncertainty of B and their covariance. Under the same example assumptions, R is 1.25 and its approximate standard uncertainty is about 0.088. Ignoring covariance would give about 0.177 under that approximation.

Those values are not guaranteed error limits or automatically 95% intervals. The approximation assumes that a local linear description is suitable. When the denominator is near zero, uncertain in sign or highly variable, ratio behavior can be poorly represented by this calculation. Do not conceal that failure with extra decimal places or a symmetric interval chosen for presentation convenience.

Reconcile assumptions with the research question

Check whether rounding, revisions and model assumptions were counted twice. If the input uncertainties already include a shared conversion effect, adding another independent allowance for the same effect may duplicate it. If an important source of error was excluded, covariance arithmetic cannot restore the missing component.

For nonlinear or constrained cases, an appropriate simulation or another method may be needed, but simulation still requires a defensible joint input distribution. Drawing inputs independently does not make independence true. Preserve the chosen dependence structure, randomization settings and validation checks with any derived results.

Report a conditional conclusion

Show the point estimate, uncertainty meaning, propagation method and dependence assumptions together. If covariance is unknown, say how that limits the claim. A derived comparison can remain useful without pretending its precision is fully established. The purpose is to make uncertainty traceable, not to turn estimated financial quantities into risk-free facts.

FAQ

Is assuming independence always conservative?

No. Its effect depends on the function and the sign of covariance. The sum and difference example shows opposite errors from the same assumption.

Not without a justified link. Economic co-movement and shared estimation error are different relationships.