AR / MA / ARMA / ARIMA — The Box-Jenkins family

This is the classic family of univariate time-series models that describe and forecast a series from its own past:

• AR(p) — Autoregressive: $Y_t$ depends on its own lagged values.
• MA(q) — Moving Average: $Y_t$ depends on lagged errors (shocks).
• ARMA(p,q) — combines AR and MA for stationary series.
• ARIMA(p,d,q) — adds differencing of order $d$ to handle non-stationary series.

When to use

Use for forecasting a single series (revenue, inflation, prices). Check stationarity first (ADF/KPSS); if non-stationary, difference it (order $d$) ⇒ ARIMA. Seasonal ⇒ SARIMA; volatility clustering ⇒ GARCH.

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Model specification

ARMA(p,q):

$$Y_t = c + \sum_{i=1}^{p} \phi_i Y_{t-i} + \varepsilon_t + \sum_{j=1}^{q} \theta_j \varepsilon_{t-j}$$

ARIMA(p,d,q): apply ARMA(p,q) to the $d$-times differenced series $\Delta^d Y_t$.

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Box-Jenkins workflow

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Running in EcoLab

1. Modeling module → Univariate time series family → ARIMA.
2. Choose the series $Y$; declare $(p,d,q)$ or use auto-ARIMA (AIC/BIC).
3. Run; view residual diagnostics + forecasts with confidence intervals; export the replication code.

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Replication code

Stata

* --- ARIMA(1,1,1) ---
* Declare time variable
tsset time

* Estimate ARIMA(1,1,1) by MLE
arima gdp_growth, arima(1,1,1)

* Post-estimation diagnostics
predict resid, residuals
corrgram resid, lags(20)

* Forecast
tsappend, add(12)
predict yhat, dynamic(.)

R

# --- ARIMA ---
library(forecast)

# Convert to time series object
ts_data <- ts(df$gdp_growth, start = c(1990, 1), frequency = 4)

# Auto-select (p,d,q) by AIC
fit <- auto.arima(ts_data)
summary(fit)

# Residual diagnostics
checkresiduals(fit)

# Forecast 12 periods ahead
fc <- forecast(fit, h = 12)
plot(fc)

Python

from statsmodels.tsa.arima.model import ARIMA
import matplotlib.pyplot as plt

# Estimate ARIMA(1,1,1)
model = ARIMA(df['gdp_growth'], order=(1, 1, 1))
result = model.fit()
print(result.summary())

# Residual diagnostics
result.plot_diagnostics(figsize=(10, 6))
plt.tight_layout(); plt.show()

# Forecast 12 periods
forecast = result.get_forecast(steps=12)
print(forecast.summary_frame())

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Limitations

• Assumes a linear relationship and stable structure; weak under structural breaks.
• Does not model time-varying variance ⇒ use ARCH/GARCH.

Video tutorial

Video Tutorial: Running ARIMA in EcoLab

See also

• SARIMA · GARCH · ARDL · Catalog