Durbin-Watson Statistic (d ≈ 2(1 − ρ)) & First-Order AR(1) Autocorrelation Calculator

Convert Durbin-Watson d (0–4) into first-order serial correlation ρ and GLS quasi-differencing weight.

Implied AR(1) Serial Correlation (ρ)
0.38 ρ (−1 to +1)
Effective Independent Sample Size (N_eff) 35.942 obs
Prais-Winsten First-Obs Weight √(1−ρ²) 0.925 weight
Distance from Ideal d = 2.00 -0.76 Δd
Inputs
⚡ Scenarios:
d (0–4)
Durbin-Watson test statistic d (0.0 to 4.0; d=2.0 means zero autocorrelation).
N
Number of time-series observations N.
Results & Graph
Implied AR(1) Serial Correlation (ρ)
0.38 ρ (−1 to +1)
Effective Independent Sample Size (N_eff)
35.942 obs
Prais-Winsten First-Obs Weight √(1−ρ²)
0.925 weight
Distance from Ideal d = 2.00
-0.76 Δd

Implied AR(1) Serial Correlation (ρ)

Graph 1 • Durbin-Watson d & AR(1) ρ • Domain Architecture & Live Telemetry Maps Durbin Watson Statistic D, Time Series Sample Size N through Durbin-Watson d & AR(1) ρ Formula.
Implied AR(1) Serial Correlati: 0.38 ρ (−1 to +1) Durbin-Watson d & AR(1) ρ • Invariants & Metrics Implied AR(1) Serial Correla 0.38 ρ (−1 to +1) Effective Independent Sample 35.942 obs Prais-Winsten First-Obs Weig 0.925 weight Distance from Ideal d = 2.00 -0.76 Δd
Graph 2 • Implied AR(1) Serial Correlation (ρ) Across Durbin Watson Statistic D Sweep (40% → 160%) Evaluates Implied_ρ ≈ 1 − (d / 2); EffectiveIndependentN ≈ N × (1 − ρ) / (1 + ρ) across 7 operating points (current input highlighted).
0 ρ (−1 0.550 ρ (−1 1.10 ρ (−1 0.752 0.496 0.628 0.744 0.504 0.992 0.380 1.24 0.256 1.49 0.132 1.74 0.00800 1.98
Graph 3 • Joint Sensitivity: Durbin Watson Statistic D × Time Series Sample Size N Compares Implied AR(1) Serial Correlation (ρ) when Time Series Sample Size N is shifted +25% (solid) vs. −25% (dashed).
0 ρ (−1 0.550 ρ (−1 1.10 ρ (−1 --- Time Series Sample Size N @ 75% 0.566 -30% 0.473 -15% 0.380 +0% 0.287 +15% 0.194 +30%
Summary: A Durbin-Watson statistic of d = 1.24 implies first-order autocorrelation ρ ≈ 0.38, reducing 80 nominal observations to ~35.9 effective independent observations.
Summary: A Durbin-Watson statistic of d = 1.24 implies first-order autocorrelation ρ ≈ 0.38, reducing 80 nominal observations to ~35.9 effective independent observations.
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Formula & Mathematical Methodology: Durbin-Watson d & AR(1) ρ Formula

Implied_ρ ≈ 1 − (d / 2)
EffectiveIndependentN ≈ N × 1 − ρ1 + ρ
Implied_ρ ≈ 1 − (d / 2); EffectiveIndependentN ≈ N × (1 − ρ) / (1 + ρ)
  • durbinWatsonStatisticD: Durbin Watson Statistic D (d (0–4))
  • timeSeriesSampleSizeN: Time Series Sample Size N (N)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Durbin-Watson d & AR(1) ρ

Convert Durbin-Watson d (0–4) into first-order serial correlation ρ and GLS quasi-differencing weight.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: Implied_ρ ≈ 1 − (d / 2); EffectiveIndependentN ≈ N × (1 − ρ) / (1 + ρ).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Durbin-Watson d & AR(1) ρ via Implied_ρ ≈ 1 − (d / 2); EffectiveIndependentN ≈ N × (1 − ρ) / (1 + ρ).

Frequently Asked Questions

How does the Durbin-Watson Statistic (d ≈ 2(1 − ρ)) & First-Order AR(1) Autocorrelation Calculator compute the solution?

It evaluates Implied_ρ ≈ 1 − (d / 2); EffectiveIndependentN ≈ N × (1 − ρ) / (1 + ρ) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Econometrics & Regression

Durbin-Watson Statistic (d ≈ 2(1 − ρ)) & First-Order AR(1) Autocorrelation Calculator: inputs, method, and how to read the result

Convert Durbin-Watson d (0–4) into first-order serial correlation ρ and GLS quasi-differencing weight.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Durbin Watson Statistic D: Durbin-Watson test statistic d (0.0 to 4.0; d=2.0 means zero autocorrelation).
  • Time Series Sample Size N: Number of time-series observations N.

What the calculation includes

With the default values shown, Implied AR(1) Serial Correlation (ρ) is 0.38 ρ (−1 to +1). The calculation follows Durbin-Watson d & AR(1) ρ Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Durbin Watson Statistic D from 1.24 d (0–4) to 1.36 d (0–4) changes Implied AR(1) Serial Correlation (ρ) from 0.38 ρ (−1 to +1) to 0.32 ρ (−1 to +1). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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