🔬 Unit root test
ADF test · Stationarity check · Recommendation for d
Stationarität: Mean, variance and autocovariance constant over time. Non-stationary series must be differenced (→ d in ARIMA).
Original time series
1. difference
ADF-Test: H₀ = unit root. p < 0.05 → stationary.
Critical values: −3.43 (1%) · −2.86 (5%) · −2.57 (10%)
Critical values: −3.43 (1%) · −2.86 (5%) · −2.57 (10%)
🧪 Ljung-Box / ARCH-LM Test
Checks for autocorrelation and ARCH effects prior to modeling.
Ljung-Box: H₀ = No autocorrelation in the residuals/data. Rejection → ARIMA is appropriate.
ARCH-LM Test: H₀ = no conditional heteroskedasticity. Rejection → GARCH is appropriate..
ARCH-LM Test: H₀ = no conditional heteroskedasticity. Rejection → GARCH is appropriate..
Parameter
ACF der Daten
ARCH-LM statistics by lag
📐 ARIMA-Modell
Autoregressive Integrated Moving Average
Parameter
ARIMA(p,d,q): p = AR order, d = differences, q = MA order. Auto-ARIMA selects via AIC minimisation (grid p,q ≤ 3).
Model equation
–
Observed vs. Adjusted
Residues
ACF of the residuals
Residual histogram
➕ ARIMAX-Modell
ARIMA with exogenous regressors (transfer function model)
ARIMAX(p,d,q): Extends ARIMA with external regressors X.
ΔdYₜ = c + AR(p) + MA(q) + β·Xₜ + εₜ
Application: GDP forecast using interest rates, sales volume using advertising expenditure, inflation using oil prices..
ΔdYₜ = c + AR(p) + MA(q) + β·Xₜ + εₜ
Application: GDP forecast using interest rates, sales volume using advertising expenditure, inflation using oil prices..
Parameter
Modellgleichung
–
Observed vs. Adjustedt (ARIMAX)
Residues
Exogenous variable X + Y (normalized)
ARIMA vs. ARIMAX Comparison
🌊 GARCH-Modell
Generalized Autoregressive Conditional Heteroskedasticity
Parameter
GARCH(p,q): σ²ₜ = ω + Σαᵢε²ₜ₋ᵢ + Σβⱼσ²ₜ₋ⱼ · Models time-varying volatility. Application: financial risk, Value-at-Risk.
Variance equation
–
Returns & Conditional Volatility ±σ
Conditional variance σ²
α: ARCH-Effekt (short-term reaction). β: GARCH persistence. α+β close to 1 = highly persistent volatility.
VaR (95%): Maximum loss with 95% probability = −1.645·σ.
VaR (95%): Maximum loss with 95% probability = −1.645·σ.
🔮 Forecast
Forward forecast based on ARIMA/ARIMAX with confidence band
Settings
⚠ Based on the model most recently estimated in the ARIMA tab. Please estimate it there first.
Time series + forecast with confidence band
| Schritt | forecast | Lower limit | Upper limit | Standard errorr |
|---|
🔄 Rolling Forecast
Out-of-sample validation with a rolling estimation window
Rolling Window: The model is repeatedly estimated using a fixed window. After each step, a one-step-ahead forecast is made and compared with the actual value → realistic forecast quality without look-ahead bias.
Parameter
Observed vs. Rolling Forecast (Test-Set)
Forecast errors over time
RMSE (Root Mean Squared Error): Quadratic average error – penalises large errors more heavily.
MAE (Mean Absolute Error): Average absolute error – more robust against outliers.
MAPE (Mean Absolute Percentage Error): Relative error in percent.
MAE (Mean Absolute Error): Average absolute error – more robust against outliers.
MAPE (Mean Absolute Percentage Error): Relative error in percent.
📊 Model Comparison-Grid search over ARIMA orders · AIC · BIC · RMSE
Grid-Suche
AIC: Rewards goodness of fit, penalises complexity moderately. BIC: Penalises more → more parsimonious models. Optimal: lowest value.
AIC / BIC nach Modell
| Modell | AIC | BIC | Log-Lik. | RMSE | Rang |
|---|
🔗 Kointegration
Engle-Granger two-step · Long-run equilibrium relationships
Kointegration: Two I(1) series share a stationary linear combination → long-run equilibrium. Basis for VECM and pairs trading.
Select time series
Y und X (normiert)
Spread (Residuen)
cointEquation
–
Scatter plot + regression line
ACF of the residuals
Critical values (EG, n≥100): 1% = −3.90 · 5% = −3.34 · 10% = −3.04. H₀: no cointegration.
Heatmap ADF-Statistik aller Paare
| Y | X | ADF-Stat. | β | R² | Result |
|---|
⚡ Granger causality
Testet ob X die Prognose von Y verbessert
Granger causality: X "Granger-causes" Y if past values of X improve the forecast of Y (beyond past Y values).
Method: F-Test: Vergleich restriktiertes AR(p) vs. unrestrikiertes VAR(p)-Modell.
Note: Granger causality ≠ true causality – only predictive usefulness.
Method: F-Test: Vergleich restriktiertes AR(p) vs. unrestrikiertes VAR(p)-Modell.
Note: Granger causality ≠ true causality – only predictive usefulness.
Parameter
F-statistic by lag order
| Y | X | Lags | F-Statistik | p-value | Result (5%) |
|---|
🔀 VAR-Modell
Vector Autoregression · Simultaneous modeling of multiple time series
VAR(p): Each variable is affected by its own lags und explained by the lags of all other variables.
Yₜ = c + A₁Yₜ₋₁ + … + AₚYₜ₋ₚ + εₜ
Application: Macroeconomics (GDP, inflation, interest rates), financial market interactions.
Yₜ = c + A₁Yₜ₋₁ + … + AₚYₜ₋ₚ + εₜ
Application: Macroeconomics (GDP, inflation, interest rates), financial market interactions.
variables & Parameter
Adjusted values – variable 1
Adjusted values – variable 2
Residual cross-correlation (selected variables)
Impulse-response functions (IRF) and variance decomposition are advanced extensions of the VAR model and are planned for future versions.