From Error Propagation to Newey-West Standard Errors

Ednaldo Silva’s blog post “Measurements and Uncertainties in Transfer Pricing” shows that uncertainty propagates through mathematical operations. This principle is directly relevant for econometric models in which variables are transformed, differenced, and estimated over time. Starting from error propagation, we examine why covariance terms matter and why dynamic regressions with lagged dependent variables require Newey-West standard errors for robust inference.

1. Error propagation

We begin with the algebra of error propagation as the formal basis for understanding how uncertainty moves through mathematical operations.

The general formula “for a statistical measure or value involving two variables m = p x”:

(1)\quad \sigma_m^2
=
(x\sigma_p)^2
+
(p\sigma_x)^2
+
2m\sigma_m

(https://edgarstat.com/blog/measurements-and-uncertainties-in-transfer-pricing/, formular (2))

Where:

• m = (p ± σ_p) × (x ± σ_x) is a calculated variable (e.g. sales).
• p = ∂m/∂x is a measured variable (e.g. price).
• x = ∂m/∂p is another measured variable (e.g. quantity).
• σₘ² = σ_px² is the variance of m and thus the uncertainty of the calculated value.
• σ_p² and σ_x² are the variances (uncertainties) in the measurements of p and x.
• σₘ = σ_px = Cov(p,x) shows how the uncertainties are correlated based on the covariance between p and x.

“Dividing (1) by m² yields to:”

(2)\quad \left(\frac{\sigma_m}{m}\right)^2
=
\left(\frac{\sigma_p}{p}\right)^2
+
\left(\frac{\sigma_x}{x}\right)^2
+
2\frac{\sigma_m}{m}

(https://edgarstat.com/blog/measurements-and-uncertainties-in-transfer-pricing/, formular (3a))

Where (σₘ/m), (σ_p/p) and (σ_x/x) are the coefficients of variation of m, p and x (relative uncertainties).

The importance of covariance:

“Price and quantity may be related through Alfred Marshall’s trope about supply and demand”
(https://edgarstat.com/blog/measurements-and-uncertainties-in-transfer-pricing/, Mathematical Appendix)

If p and x are correlated: σₘ = σ_px = Cov(p,x) ≠ 0, then a covariance term must be taken into consideration: 2 σₘ/m. Without it, the uncertainty is miscalculated.

If σₘ = σ_px = Cov(p,x) ≈ 0 ⇒ x and p are uncorrelated and we obtain:

(3)\quad\left(\frac{\sigma_m}{m}\right)^2
=
\left(\frac{\sigma_p}{p}\right)^2
+
\left(\frac{\sigma_x}{x}\right)^2

(https://edgarstat.com/blog/measurements-and-uncertainties-in-transfer-pricing/, formular (4))

Key message:

• Every mathematical operation (addition, multiplication, adjustment) propagates uncertainty.
• This also applies to time series operations such as difference formation.

(https://edgarstat.com/blog/measurements-and-uncertainties-in-transfer-pricing/)

2. The estimation process using ordinary least squares (OLS), small insert

OLS is a method used in linear regression to find the best line through data by minimizing the sum of squared deviations (errors) between the predicted and actual values.

We define linear models as follows:

Y_t = β₀ + β₁ X_t + ε_t

The residuum is defined as:

ε_t = Y_t – Ŷ_t = Y_t – (β₀ + β₁ X_t)

OLS searches for the values β̂₀ and β̂₁ that minimize the sum of the squared residuals:

S(β₀,β₁) = Σⁿ_t=1 (Y_t – β₀ + β₁ X_t)²

The smaller S is, the better the curve fits the data.

The consequence of the estimation is that in models with delayed dependent variables, residuals may exhibit temporal correlation (connection).

“The discussion builds from a standard identity to an econometric model estimated with Newey-West standard errors”.
(https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL)

Procedure:

We estimate equation (6) of the blog post “Methodological expansion for “Debt-Driven GDP Growth”” with:

Y_t = α Y_t-1 + β₁ ΔA_t + β₂ ΔD_t + β₃ ΔD_t-1 + εₜ,
(https://edgarstat.com/blog/methodological-expansion-for-debt-driven-gdp-growth/)

where α, β₁, β₂ and β₃ true (unknown) parameters are and εₜ is a true unknown error.

We obtain estimated coefficients α̂, β̂₁, β̂₂, β̂₃, where:

• α̂ is the estimated coefficient for Y_t-1.
• β̂₁ is the estimated coefficient for ΔA_t.
• β̂₂ is the estimated coefficient for ΔD_t.
• β̂₃ is the estimated coefficient for ΔD_t-1.

We obtain the estimated residuals ε̂ₜ, the difference between the observed and the predicted value:

ε̂_t = Y_t – (α̂ Y_t-1 + β̂₁ ΔA_t + β̂₂ ΔD_t + β̂₃ ΔD_t-1)

Problem: The estimated residues ε̂ₜ may be correlated over time: Cov(ε̂_t, ε̂_t-1) ≠ 0.

The reason for this can be a delayed dependent variable in the model structure:

If the estimated value α̂ is close to 1, e.g., 0.92, then α̂ < 1 is satisfied, but too close to 1. This means that Y_t depends too strongly on Y_t-1 and the values are not independent at different points in time.

Newey-West standard error corrects the standard errors of the estimated coefficients α̂, β̂₁, β̂₂, β̂₃. The coefficients remain the same, but the standard errors SE(α̂,), SE(β̂₁), SE(β̂₂) and SE(β̂₃) are corrected. The correction takes into account possible temporal correlation in the residuals.

This is important because the standard errors measure the uncertainty about the coefficients. Without correction, we might mistakenly think we are more certain about the coefficients than we actually are.

We have shown why error propagation matters for econometric inference. Covariance terms cannot be ignored, and in dynamic time series models they appear as temporal correlation in the residuals. Newey-West standard errors correct this problem without changing the estimated coefficients.

References

Ednaldo Silva’s LinkedIn post “Debt-Driven GDP Growth” https://www.linkedin.com/posts/ednaldosilva_macroeconomics-gdp-keynesian-activity-7335373909543837718-D8qL

Ednaldo Silva’s blog post “Measurements and Uncertainties in Transfer Pricing” https://edgarstat.com/blog/measurements-and-uncertainties-in-transfer-pricing/

Sara Sardar’s blog post “Methodological expansion for “Debt-Driven GDP Growth””
https://edgarstat.com/blog/methodological-expansion-for-debt-driven-gdp-growth/