Durbin-Watson autocorrelation test calculator OLS residuals

The Durbin-Watson test calculator determines the presence of first-order (lag-1) autocorrelation in OLS linear regression residuals, per the test developed by James Durbin and Geoffrey S. Watson 1950-1951 Biometrika vol. 37 and vol. 38 "Testing for Serial Correlation in Least Squares Regression I & II" (London School of Economics and Cambridge). The test statistic: DW = Σ_{t=2}^{n} (e_t - e_{t-1})² / Σ_{t=1}^{n} e_t², with e_t OLS residuals, n sample size.

Range: DW ∈ [0, 4].

Approximation: DW ≈ 2(1 - ρ), ρ = lag-1 estimated autocorrelation; thus ρ ≈ 1 - DW/2.

Interpretation: DW ≈ 2 no autocorrelation (i.i.d. residuals null H0); DW < 2 positive autocorrelation (similar consecutive residuals, "momentum"); DW > 2 negative autocorrelation (opposite consecutive residuals, "mean reversion").

Critical thresholds Durbin-Watson tables depend on n and k (regressor count): for n=100, k=2 → dL≈1.63 and dU≈1.72. Two-tailed test α=5%: reject H0 if DW

4-dL; uncertain zone dL < DW < dU; do not reject dU < DW < 4-dU.

Asymptotic approximation Savin-White 1977 for n>30: dL ≈ 2 - 2·z·√((n-k-1)/(n²+2k+1)), dU ≈ 2 + 2·z·√((n-k-1)/(n²+2k+1)), with z=1.96 at 5%.

Simplified decision: DW < 1 strong positive autocorr (reject); 1 ≤ DW < 1.5 moderate positive; 1.5 ≤ DW ≤ 2.5 no evidence (DW≈2 optimal); 2.5 3 strong negative (reject).

5 applications:

• Linear/multivariate OLS regression model validation (cross-section + time series)
• Econometric time series (ARMA Box-Jenkins, GARCH Engle 1982 residual model checking)
• Bank of Italy 1893 macroeconometrics (quarterly GDP, monthly HICP inflation, BTP yields, ECB rates)
• FTSE MIB 1998 stock returns models (CAPM Sharpe 1964, Fama-French 3-factor 1992)
• Pre-GARCH residual tests Engle 1982 ARCH-LM Bocconi Mario Forni Modena 1985. Italian applications:

• Bank of Italy Rome 1893 macroeconometric models Quarterly Economic Bulletin (GDP, inflation, BTP yields)
• ISTAT 1926 demographic/economic time series (Census, HICP, employment, OECD)
• Bocconi Milan 1902 econometrics Mario Forni Modena 1985 dynamic factor models + Carlo Favero macroeconomics
• Politecnico Milan 1863 + Politecnico Turin 1859 financial engineering + signals
• SISSA Trieste 1978 + ICTP Trieste 1964 data science statistical physics
• Bologna University 1088 + Cattolica Sacro Cuore 1921 economics + sociology school
• Bruno de Finetti Trieste 1931 subjective probability Italian Bayesian framework
• INFN AI Atlas/CMS LHC time series Higgs MC validation
• CINECA Bologna 1969 Marconi100 30 PFlops + Leonardo pre-exascale 250 PFlops 2023 Monte Carlo econometric simulations
• Consob Rome 1974 market supervision + IVASS Rome 2012 Solvency II insurance returns

Italian Durbin-Watson series tables

Italian series	Frequency	n	k	DW observed	ρ estimated	Interpretation	Source
BTP 10y daily yield	daily	2,500	3	1.85	0.075	Slight positive autocorr	Bank of Italy 1893
FTSE MIB daily returns	daily	2,500	3	1.98	0.01	No autocorr (efficient)	Borsa Italiana 1998
ISTAT monthly HICP inflation	monthly	240	2	0.60	0.70	Strong positive autocorr	ISTAT 1926
Italy quarterly GDP	quarterly	120	4	0.80	0.60	Strong positive autocorr	Bank of Italy + ISTAT
ISTAT monthly unemployment	monthly	360	3	0.45	0.775	Strong positive autocorr	ISTAT Labor Force
ECB policy rate	monthly	300	2	0.30	0.85	Strong positive autocorr	ECB 1998
VIX FTSE MIB volatility	daily	1,500	2	1.60	0.20	Moderate positive autocorr	Borsa Italiana
BTP-Bund 10y spread	daily	2,500	3	1.90	0.05	Slight positive autocorr	Bank of Italy
Eni CAPM FTSE MIB residuals	monthly	240	2	2.05	-0.025	No autocorr	Bocconi 1902
Fama-French 3F residuals	monthly	360	4	2.02	-0.01	No autocorr	Bocconi Forni

(Source: Durbin & Watson 1950-1951 Biometrika + Savin-White 1977 Econometrica + Bank of Italy Economic Bulletin + ISTAT 1926 + Bocconi Milan 1902).

Italian Durbin-Watson test examples

• BTP 10y daily yield Bank of Italy 1893 n=2,500 k=3 (3-factor CAPM): OLS residuals regression spread vs Bund-ECB rate-VIX. Σ(e_t-e_{t-1})² = 0.925, Σe_t² = 0.500 → DW = 1.85. ρ ≈ 1 - 1.85/2 = 0.075 (slight positive autocorr). dL Savin-White ≈ 1.93, dU ≈ 2.07. DW=1.85 < dL → slightly in rejection zone, near boundary
• ISTAT monthly HICP inflation 1926 n=240 k=2 (trend+seasonality): ARIMA(1,0,0) model residuals. DW = 0.60 → ρ ≈ 0.70 strong positive autocorrelation (reject H0). Diagnosis: insufficient model, need higher-order ARIMA(p,d,q) or GARCH (Engle 1982 ARCH-LM)
• FTSE MIB daily returns Borsa Italiana 1998 n=2,500 k=3 (Fama-French market+SMB+HML): market efficiency (Fama 1970): DW = 1.98 → ρ ≈ 0.01 no autocorrelation (weak-form efficient market). Consistent with EMH
• Italy quarterly GDP Bank of Italy + ISTAT n=120 k=4 (consumption+investment+government+net exports): DW = 0.80 → ρ ≈ 0.60 strong positive autocorr. Need Sims 1980 vector VAR
• Eni FTSE MIB CAPM residuals Bocconi 1902 n=240 k=2 (market return): reduced regression R_Eni = α + β·R_MIB + ε. DW = 2.05 → ρ ≈ -0.025 no autocorrelation (well-specified CAPM)
• Fama-French 3-factor stock residuals Bocconi Forni 1985 n=360 k=4: DW = 2.02 → no autocorr; superior to CAPM
• ECB policy rate monthly 1998 n=300 k=2: DW = 0.30 → ρ ≈ 0.85 very strong autocorr (Taylor rule smoothing). Persistent monetary policy
• Didactic test example DW=2.0 n=100 k=2: ρ ≈ 0, σ_DW = 2·√((100-2-1)/(10000+4+1)) = 2·√(97/10005) = 0.1969, dL ≈ 2 - 1.96·0.1969 ≈ 1.614, dU ≈ 2.386. DW=2 in "no rejection" zone
• DW=1.0: ρ ≈ 0.5 moderate positive autocorr
• DW=0.5: ρ ≈ 0.75 very strong autocorr (reject), classic mis-specified model signal

Italian econometrics and data science research

Italian econometrics:

• Bank of Italy Rome 1893 macroeconometrics Quarterly Economic Bulletin + Research Department Econometric Models (Banca d'Italia Working Papers Temi di Discussione); BIQM Banca d'Italia Quarterly Model + ITEM (Italian Economic Macromodel) Bayesian DSGE
• ISTAT National Statistics Institute Rome 1926 time series (GDP, HICP inflation, employment, demographics, decennial censuses); X-13 ARIMA-SEATS seasonal adjustment
• Bocconi Milan 1902 + Mario Forni Modena 1985 dynamic factor models; Carlo Favero DSGE macroeconomics; Claudio Tebaldi quantitative finance; IGIER Innocenzo Gasparini Institute for Economic Research
• Politecnico Milan 1863 + Politecnico Turin 1859 stochastic financial engineering (MOX Modelling Computing Math); fintech
• SISSA Trieste 1978 + ICTP Abdus Salam Trieste 1964 data science statistical physics; Matteo Marsili statistical physics complex systems; Stefano Allesina ecology biostatistics
• Bologna University 1088 (oldest in the world) + Cattolica Sacro Cuore Milan 1921 applied econometrics; CRENoS Cagliari 1993 regional economics
• Bruno de Finetti Trieste 1931 subjective Bayesian probability "La prévision: ses lois logiques" Annales IHP 1937 — founder of Italian Bayesian framework (de Finetti theorem exchangeability)
• INFN National Institute of Nuclear Physics 1951 AI Atlas/CMS LHC Higgs MC validation Geant4 simulations time series; INFN-CNAF Bologna Tier-1 + Rome1 Sapienza
• CINECA Casalecchio Bologna 1969 Marconi100 30 PFlops + Leonardo pre-exascale 250 PFlops 2023 (5th worldwide TOP500) Monte Carlo econometric simulations, machine learning autocorrelation neural networks LSTM/Transformers time series
• Consob National Commission for Companies and Stock Exchange Rome 1974 market abuse supervision + IVASS Italian Insurance Supervision Rome 2012 Solvency II returns, technical reserves average MTPL EUR 420 in 2026, TPL claims. Italian Physics Nobel Prizes: Marconi 1909, Fermi 1938, Segrè 1959, Carlo Rubbia 1984 (CERN W/Z bosons), Riccardo Giacconi 2002 (X-ray astronomy Uhuru). Italian R&D tax bonuses 2026: R&D Tax Credit 25% L. 160/2019 art. 1 c. 200; Patent Box 50% L. 178/2020 super-deduction qualifying intangible costs; PNRR Mission 4 Component 2 data science research EUR 6.9B (National HPC, Big Data, Quantum Computing Centers); 50% renovation deductions and 36% capital goods. Disclaimer: DW test is a statistical diagnostic tool, does not replace complete econometric analysis

Updated July 2026