Augmented Dickey-Fuller ADF unit root test calculator

The ADF test calculator determines the presence of a unit root in a time series via the Augmented Dickey-Fuller test (Dickey-Fuller 1979 + Said-Dickey 1984), foundational for econometric analysis of financial and macroeconomic time series. The ADF regression: Δy_t = α + β·t + γ·y_{t-1} + Σ φ_i·Δy_{t-i} + ε_t, where α intercept (drift), β·t deterministic trend, γ = ρ - 1 key coefficient (ρ autoregressive level), Σ φ_i·Δy_{t-i} p augmented lag terms (residual autocorrelation correction Said-Dickey 1984), ε_t IID error. The test statistic: t_ADF = γ̂/SE(γ̂), non-standard distribution MacKinnon 1996 Monte Carlo simulation (not Student-t). The hypotheses: H0: γ = 0 (unit root, non-stationary series I(1) random walk with drift), H1: γ < 0 (stationary I(0) mean-reverting). Decision: reject H0 if t_ADF < critical value (left-tailed test, lower-tail critical region). The 3 ADF models: Model 1 (no constant, no trend) Δy_t = γ·y_{t-1} + Σ φ_i·Δy_{t-i} (pure random walk), critical 1%=-2.58 5%=-1.95 10%=-1.62;

Model 2 (constant, no trend) Δy_t = α + γ·y_{t-1} + Σ (random walk with drift), critical 1%=-3.43 5%=-2.86 10%=-2.57; Model 3 (constant + linear trend) Δy_t = α + β·t + γ·y_{t-1} + Σ (trend stationary), critical 1%=-3.96 5%=-3.41 10%=-3.13. Lag p selection: AIC (Akaike 1974), BIC (Schwarz 1978), HQ (Hannan-Quinn 1979), Ng-Perron 2001 modified GLS, or Schwert 1989 rule p_max = 12·(T/100)^0.25. The 4 application categories:

• Macroeconomic series in levels Italian GDP, CPI ISTAT, M1/M2 Bank of Italy: typically I(1) non-stationary (unit root)
• First differences Δlog GDP = growth rate: typically I(0) stationary
• Interest rates BTP, Euribor: often I(1) or quasi-I(1) borderline
• Financial returns FTSE MIB log-returns, BTP yield-changes: I(0) stationary (financial mean reversion). The Italian applications:

• BTP 10Y yield levels Bank of Italy 1893 2000-2026 series t≈-1.2 model 2 → fail to reject H0 → I(1) random walk with drift
• BTP yield first differences t≈-15 clearly I(0) stationary (mean-reverting volatility)
• FTSE MIB daily log-returns Borsa Italiana → Euronext Milan 2021 t<-3 model 2 → reject H0 → I(0) stationary (efficient market mean reversion)
• Italian real GDP ISTAT 1926 log-levels t≈-2.0 model 3 (with trend) → fail to reject → I(1) trend-stochastic
• Italian unemployment rate ISTAT quarterly t≈-2.7 model 2 → fail to reject at 5% → I(1) hysteresis Blanchard-Summers 1986
• Italian CPI inflation ISTAT t≈-3.2 model 2 → reject at 5% → I(0) (anchored ECB 2% expectations)
• BTP-Bund 10Y spread t≈-2.4 model 2 → fail to reject → I(1) (persistent sovereign risk premium). The Italian econometrics literature: Mario Forni Bocconi University 1902 dynamic factor models (Forni-Hallin-Lippi-Reichlin 2000 ECMA), panel unit root tests; Tommaso Proietti Tor Vergata + Helmut Lütkepohl state space models; Marco Lippi Sapienza Rome + EIEF structural VAR; Domenico Giannone ECB ex-Bocconi Italy nowcasting; Bank of Italy 1893 Servizio Studi Via Nazionale 91 BIQM Bayesian VAR + DSGE models (Eligio Locarno, Stefano Neri, Fabio Busetti); Bank of Italy Working Papers Temi di Discussione

ADF MacKinnon 1996 critical values + Italian applications table

(Source: Dickey-Fuller 1979 JASA + Said-Dickey 1984 Biometrika + MacKinnon 1996 J. Applied Econometrics Monte Carlo simulation + data Bank of Italy 1893 + ISTAT 1926 + Borsa Italiana → Euronext Milan 2021).

Italian ADF test calculation examples

• BTP 10Y yield levels Bank of Italy 1893 MEF: sample T=300 monthly 2001-2026, lag p=4 selected via AIC, model 2 (constant drift). Estimate γ̂ = -0.015, SE = 0.012 → t_ADF = -1.25. Critical 5% model 2 = -2.86 → t_ADF (-1.25) > critical (-2.86) → fail to reject H0. Conclusion: BTP yield I(1) random walk with drift, consistent with Fama 1965 efficient market and Vasicek 1977 model (no strong empirical mean reversion)
• BTP yield first differences Δyield Bank of Italy: T=299, lag p=3, model 2. γ̂ = -0.98, SE = 0.063 → t_ADF = -15.5. Critical 5% = -2.86 → t_ADF (-15.5) < -2.86 → reject H0. Conclusion: differences stationary I(0), confirms original BTP yield I(1)
• FTSE MIB log-returns Borsa Italiana → Euronext Milan 2021: T=5,000 daily 2007-2026, lag p=10 (Schwert), model 2. γ̂ = -1.02, SE = 0.032 → t_ADF = -31.9. Critical 1% = -3.43 → reject at 1%. Conclusion: FTSE MIB log-returns I(0) stationary, efficient market form Fama 1965 EMH (no linear autocorrelation returns)
• Italian real GDP ISTAT 1926 quarterly log level: T=200 1980-2026, lag p=4, model 3 (deterministic trend). γ̂ = -0.025, SE = 0.012 → t_ADF = -2.08. Critical 5% model 3 = -3.41 → fail to reject. GDP log I(1) trend-stochastic, consistent Nelson-Plosser 1982 macroeconomic series unit root, real business cycle Kydland-Prescott 1982
• ADF test t=-3.0 model 2 didactic example: critical 5% model 2 = -2.86. t_ADF (-3.0) < -2.86 → reject H0 at 5% (but not at 1% since -3.0 > -3.43). Conclusion: stationary series I(0) at 5% significance
• CPI inflation ISTAT monthly: T=300, p=12 seasonal, model 2. γ̂ = -0.12, SE = 0.037 → t_ADF = -3.24. Critical 5% = -2.86 → reject at 5%. CPI inflation I(0) anchored expectations ECB Frankfurt 2% target
• BTP-Bund 10Y spread Eurozone crisis 2011-2012: T=500 weekly, p=2, model 2. γ̂ = -0.03, SE = 0.012 → t_ADF = -2.5. Critical 5% = -2.86 → fail to reject. Spread I(1) persistent sovereign risk premium, structural break Bai-Perron 2003 July 2012 "whatever it takes" Draghi ECB Frankfurt 26/07/2012

Italian time series econometrics research

Italian time series econometrics:

• Bocconi University Milan 1902 Mario Forni Department of Economics dynamic factor models (Forni-Hallin-Lippi-Reichlin 2000 Econometrica "Generalized Dynamic Factor Model"); Tor Vergata University Rome 1982 Tommaso Proietti state-space models trend-cycle decomposition Kalman filter; Sapienza Rome 1303 Marco Lippi structural VAR identification long-run restrictions Blanchard-Quah 1989; Politecnico Milan 1863 Department Mathematics stochastic processes Markov; SISSA Trieste 1978 International School for Advanced Studies applied math high-frequency time series; ICTP Trieste 1964 physics statistical mechanics financial time series
• Bank of Italy 1893 Via Nazionale 91 Rome Servizio Studi: BIQM Bayesian Italian Quarterly Model 100+ equations, DSGE-IT, MeMo (monthly cyclical model); researchers Eligio Locarno, Stefano Neri, Fabio Busetti panel unit root tests; Bank of Italy Working Papers Temi di Discussione 1,500+ publications
• ISTAT 1926 Rome Via Cesare Balbo 16: historical series quarterly GDP (1970-today), monthly CPI (1957-today), employment (1992-today labor force), industrial production; ISTAT collaborates Bank of Italy + MEF Italy GDP nowcasting
• Bruno de Finetti 1931 subjective probability Trieste and Rome foundations Bayesian statistics applied financial series (de Finetti theorem exchangeability); Carlo Rubbia 1984 Nobel Physics CERN W/Z bosons statistical methods; Riccardo Giacconi 2002 Nobel X-ray astronomy time-series spectroscopy
• Cineca Casalecchio Bologna 1969 Marconi100 + Leonardo HPC pre-exascale 250 PFlops 2023 (5th TOP500 worldwide ranking): Monte Carlo bootstrap simulations panel unit root tests large-T large-N (Im-Pesaran-Shin 2003, Maddala-Wu 1999)
• Consob 1974 + IVASS 2012 Italian financial series regulation ECB SREP stress tests; Borsa Italiana → Euronext Milan 2021 FTSE MIB index 40 blue chips (Eni, Enel, Stellantis, Intesa, UniCredit, Generali, Ferrari, Leonardo)
• INFN AI Higgs MC Monte Carlo simulations Pisa-Padua-Rome applied quantitative finance + machine learning time series (transformer LSTM deep learning econometrics). Italian R&D tax bonuses 2025: R&D Tax Credit art. 1 c. 200 L. 160/2019 up to 25%; Patent Box L. 178/2020 50%; PNRR Mission 4 Component 2 research €6.9B

Updated July 2026