Time Series

Time Series Analysis Skill

This skill provides guidance for univariate and multivariate time series analysis in empirical economics. It covers stationarity testing, ARIMA modeling, VAR/VECM systems, cointegration, and Granger causality.

Analysis Workflow

Step 1: Exploratory Inspection

• Plot the series (level, first difference, log)
• Examine ACF and PACF plots to identify dependence structure
• Check for obvious trends, seasonality, structural breaks

Step 2: Stationarity Testing

Always test for unit roots before modeling. Preferred approach:

1. ADF test (H₀: unit root present)
2. KPSS test (H₀: series is stationary)
3. If ADF rejects AND KPSS fails to reject → stationary (I(0))
4. If both suggest non-stationary → take first difference, retest

ADF vs KPSS Conflict Resolution: When ADF fails to reject (suggests unit root) but KPSS also fails to reject (suggests stationary), the tests disagree. Recommended approach: (a) check for structural breaks — a break can make a stationary series look like it has a unit root; (b) use Zivot-Andrews test to allow for one structural break; (c) examine the series visually and consider economic theory. When in doubt, err on the side of differencing to avoid spurious regressions.

Step 3: Model Selection

• Univariate: ARIMA(p,d,q) based on AIC/BIC and ACF/PACF
• Multivariate, no cointegration: VAR in differences
• Multivariate, with cointegration: VECM (VAR in error-correction form)

Step 4: Estimation and Diagnostics

• Check residual white noise (Ljung-Box test)
• Test for parameter stability (CUSUM)
• Impulse response functions for structural interpretation

Step 5: Forecasting (if applicable)

• Report RMSE and MAE on holdout sample
• State forecast horizon and confidence intervals

Quick Code Templates

Stationarity Tests

# Python
from statsmodels.tsa.stattools import adfuller, kpss

# ADF test
adf_result = adfuller(series, autolag='AIC')
print(f"ADF stat: {adf_result[0]:.4f}, p-value: {adf_result[1]:.4f}")
print(f"Critical values: {adf_result[4]}")

# KPSS test
kpss_result = kpss(series, regression='c', nlags='auto')
print(f"KPSS stat: {kpss_result[0]:.4f}, p-value: {kpss_result[1]:.4f}")

# R
library(tseries); library(urca)
adf.test(series)
kpss.test(series, null = "Level")

* Stata
dfuller series, lags(4) regress
kpss series

ARIMA Model

# Python — auto order selection
from statsmodels.tsa.arima.model import ARIMA
import itertools

# Manual
model = ARIMA(series, order=(1, 1, 1)).fit()
print(model.summary())

# Forecast
forecast = model.forecast(steps=12)

# R
library(forecast)
model <- auto.arima(series, ic = "aic")
summary(model)
forecast(model, h = 12)

* Stata
arima series, arima(1,1,1)
predict yhat, xb

VAR Model

# Python
from statsmodels.tsa.api import VAR

model = VAR(data_matrix)
lag_order = model.select_order(maxlags=8)
print(lag_order.summary())

results = model.fit(lag_order.aic)
print(results.summary())

# Granger causality
results.test_causality('y1', ['y2'], kind='f')

# Impulse response
irf = results.irf(periods=10)
irf.plot(orth=True)

# R
library(vars)
VARselect(data_matrix, lag.max = 8, type = "const")
var_model <- VAR(data_matrix, p = 2, type = "const")
causality(var_model, cause = "y2")
irf(var_model, impulse = "y2", response = "y1", n.ahead = 10)

Key Decision Rules

ARIMA Order Selection Guide

• AR(p): PACF cuts off after lag p, ACF decays
• MA(q): ACF cuts off after lag q, PACF decays
• ARMA(p,q): Both ACF and PACF decay gradually
• Use AIC for forecasting, BIC for parsimony/inference
• Always check residual ACF for remaining autocorrelation

Residual Diagnostics (Ljung-Box White Noise Test)

# Python — Ljung-Box test for residual autocorrelation
from statsmodels.stats.diagnostic import acorr_ljungbox

# After fitting ARIMA model:
residuals = model.resid
lb_result = acorr_ljungbox(residuals, lags=[10, 20], return_df=True)
print(lb_result)
# If p-value > 0.05 at all lags → residuals are white noise (good)

# R — Ljung-Box test
Box.test(residuals(model), lag = 10, type = "Ljung-Box")
# p > 0.05 → no remaining autocorrelation

* Stata — Ljung-Box (portmanteau) test
* After arima estimation:
wntestq residuals, lags(10)
* Q-stat with p > 0.05 → white noise residuals

ARCH/GARCH Models (Volatility Clustering)

Use when residuals exhibit time-varying variance (heteroskedasticity that clusters over time). Common in financial and macroeconomic data.

# Python — GARCH(1,1) using arch library
from arch import arch_model

# First, fit mean model (e.g., AR(1)) and extract residuals
# Then model the conditional variance:
garch_model = arch_model(series, vol='GARCH', p=1, q=1, dist='normal')
garch_result = garch_model.fit(disp='off')
print(garch_result.summary())

# Extract conditional volatility
cond_vol = garch_result.conditional_volatility
# ARCH LM test for remaining ARCH effects:
from arch.unitroot import VarianceRatio
from statsmodels.stats.diagnostic import het_arch
lm_stat, lm_pval, _, _ = het_arch(garch_result.resid)
print(f"ARCH LM test p-value: {lm_pval:.4f}")  # should be > 0.05

# R — GARCH(1,1) using rugarch
library(rugarch)

spec <- ugarchspec(
  variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
  mean.model     = list(armaOrder = c(1, 0), include.mean = TRUE),
  distribution.model = "norm"
)
fit <- ugarchfit(spec = spec, data = series)
show(fit)

# Test for remaining ARCH effects
# ArchTest(residuals(fit), lags = 10)  # from FinTS package

* Stata — ARCH/GARCH
* First run mean model:
arima series, arima(1,0,0)
predict resid_mean, residuals

* ARCH LM test:
estat archlm, lags(1 2 5)

* Fit GARCH(1,1):
arch series, arch(1) garch(1) ar(1)

For cointegration tests (Johansen, Engle-Granger), VECM specification, and structural break tests, see references/time-series-reference.md.

Common Pitfalls

• Regressing non-stationary series on each other: Produces spurious regression — always test for unit roots first
• Using VAR in levels when series are I(1) without cointegration: Leads to invalid inference — difference the data or use VECM
• Wrong lag length: Too few lags → autocorrelated residuals; too many → overfitting. Use information criteria
• Confusing Granger causality with true causality: Granger causality is about predictive content, not causal mechanisms
• Ignoring structural breaks: A break can mimic a unit root — use Zivot-Andrews test, which allows for one endogenous break under the alternative:

# R — Zivot-Andrews test (allows break under stationarity alternative)
library(urca)
za <- ur.za(series, model = "both", lag = 4)
summary(za)
# If test statistic < critical value → reject unit root even with break
# za@teststat gives statistic; za@cval gives 1%, 5%, 10% critical values

# Python — Zivot-Andrews test
from statsmodels.tsa.stattools import zivot_andrews

za_stat, pval, cvdict, bpindex, baselag = zivot_andrews(series, maxlag=4, regression='ct')
print(f"ZA stat: {za_stat:.4f}, p-value: {pval:.4f}, break at index: {bpindex}")
# pval < 0.05 → series is stationary with one structural break