`time-series`¶

Pack: 100xOS shared skills

Category: analysis

Field: economics

License: private (curator-owned)

Updated: 2026-05-20

Stages: data-analysis

Curator-private skill — copy text from 100xOS/shared/skills/econometrics/time-series.md.

↗ view SKILL.md on source

Time Series Econometrics¶

Overview¶

Time series methods model the temporal dependence structure in financial and economic data. The core challenge is distinguishing genuine predictability from spurious correlation induced by persistence, non-stationarity, or structural breaks.

Stationarity and Unit Root Tests¶

A time series is (weakly) stationary if its mean, variance, and autocovariances are constant over time. Most financial prices are non-stationary (unit root); returns are typically stationary.

Testing for Unit Roots¶

Augmented Dickey-Fuller (ADF): - H0: Unit root (non-stationary). H1: Stationary. - Include sufficient lags to whiten residuals (use AIC/BIC for lag selection). - Include intercept; add trend only if the series has a visible trend under the alternative. - Low power against near-unit-root alternatives.

Phillips-Perron (PP): - Non-parametric correction for serial correlation (Newey-West). - Same null as ADF but more robust to heteroskedasticity. - Can over-reject in small samples with negative MA components.

KPSS (Kwiatkowski-Phillips-Schmidt-Shin): - H0: Stationary. H1: Unit root. - Use jointly with ADF: if ADF rejects and KPSS does not reject, conclude stationarity. If both reject or neither rejects, the series may be fractionally integrated.

Zivot-Andrews / Lee-Strazicich: - Unit root tests allowing for structural breaks. - Use when you suspect a regime change (policy shift, crisis).

Practical Rules¶

Always plot the series first. Visual inspection catches most issues.
Test in levels and first differences.
For financial prices: work in log returns (continuously compounded).
For macro series: test whether first-differencing or detrending is appropriate.

ARIMA Models¶

ARIMA(p, d, q): AutoRegressive Integrated Moving Average. - p: AR order (number of lagged values of y) - d: Integration order (number of differences to achieve stationarity) - q: MA order (number of lagged error terms)

Identification¶

Check stationarity (unit root tests). Set d accordingly (usually d=0 for returns, d=1 for prices).
Examine ACF and PACF of the stationary series:
ACF cuts off at lag q, PACF decays → MA(q)
PACF cuts off at lag p, ACF decays → AR(p)
Both decay → ARMA(p,q)
Use information criteria (AIC, BIC) to select p, q. BIC penalizes complexity more.

Estimation¶

Python

from statsmodels.tsa.arima.model import ARIMA

model = ARIMA(y, order=(p, d, q))
results = model.fit()
print(results.summary())

Diagnostics¶

Ljung-Box test on residuals (H0: no autocorrelation). Test at multiple lag lengths.
Jarque-Bera test for normality of residuals.
Plot residual ACF — should show no significant spikes.
Check for ARCH effects in residuals (Engle's LM test) → if present, use GARCH.

GARCH Models¶

Generalized Autoregressive Conditional Heteroskedasticity. Models time-varying volatility — the defining feature of financial return series (volatility clustering).

GARCH(1,1)¶

The workhorse model:

r_t = mu + epsilon_t epsilon_t = sigma_t * z_t, z_t ~ N(0,1) or t-distributed sigma_t^2 = omega + alpha * epsilon_{t-1}^2 + beta * sigma_{t-1}^2

omega > 0, alpha >= 0, beta >= 0
alpha + beta < 1 for stationarity (unconditional variance = omega / (1 - alpha - beta))
alpha captures shock impact; beta captures persistence
Typical financial data: alpha ~ 0.05-0.10, beta ~ 0.85-0.95

Python

from arch import arch_model

model = arch_model(returns, vol='Garch', p=1, q=1, dist='t')
results = model.fit(disp='off')
print(results.summary())

## Conditional volatility
vol = results.conditional_volatility

## Standardized residuals
std_resid = results.std_resid

Extensions¶

EGARCH (Nelson, 1991): Captures leverage effect (negative returns increase volatility more than positive returns of equal magnitude).

log(sigma_t^2) = omega + alpha * (|z_{t-1}| - E|z_{t-1}|) + gamma * z_{t-1} + beta * log(sigma_{t-1}^2)

gamma < 0 indicates leverage effect.
Log specification ensures sigma^2 > 0 without parameter constraints.

GJR-GARCH (Glosten-Jagannathan-Runkle, 1993): Alternative leverage specification.

sigma_t^2 = omega + (alpha + gamma * I_{t-1}) * epsilon_{t-1}^2 + beta * sigma_{t-1}^2

I_{t-1} = 1 if epsilon_{t-1} < 0. gamma > 0 implies leverage.

GARCH-in-Mean: Adds conditional variance to the mean equation.

r_t = mu + delta * sigma_t^2 + epsilon_t

Tests risk-return tradeoff. delta > 0 means higher volatility commands higher expected return.

Python

## EGARCH
model = arch_model(returns, vol='EGARCH', p=1, q=1)

## GJR-GARCH
model = arch_model(returns, vol='Garch', p=1, o=1, q=1)  # o=1 adds asymmetry

Distribution Assumptions¶

Normal: default, but financial returns have fat tails.
Student-t: dist='t' — recommended default for financial data.
Skewed-t: dist='skewt' — captures both fat tails and asymmetry.
GED: dist='ged' — Generalized Error Distribution.

GARCH Diagnostics¶

Ljung-Box on squared standardized residuals (should show no remaining ARCH effects).
QQ-plot of standardized residuals against assumed distribution.
News impact curve (plot sigma^2 as function of epsilon_{t-1}).
Compare AIC/BIC across GARCH variants.

VAR Models¶

Vector Autoregression: models multiple time series jointly. Each variable is regressed on its own lags and lags of all other variables.

Y_t = c + A_1 * Y_{t-1} + ... + A_p * Y_{t-p} + u_t

where Y_t is a K-dimensional vector.

Identification and Estimation¶

Test each series for unit roots. If I(1), consider VECM instead.
Select lag order p using AIC/BIC/HQ on the VAR in levels (or differences if non-stationary).
Estimate by OLS equation-by-equation (equivalent to SUR when all equations have the same regressors).

Python

from statsmodels.tsa.api import VAR

model = VAR(data[['y1', 'y2', 'y3']])
results = model.fit(maxlags=8, ic='bic')
print(results.summary())

Granger Causality¶

X Granger-causes Y if past values of X improve prediction of Y beyond Y's own past.
Test via F-test on excluded lags in the Y equation.
NOT causal in the structural sense — purely predictive.

Python

results.test_causality('y1', causing='y2', kind='f')

Impulse Response Functions (IRFs)¶

Trace the dynamic response of each variable to a one-unit shock in another.
Ordering matters for Cholesky decomposition: variable ordered first is assumed contemporaneously exogenous.
Report confidence bands (bootstrap or asymptotic).

Python

irf = results.irf(periods=20)
irf.plot(orth=True)  # Orthogonalized (Cholesky) IRFs

Structural VAR (SVAR)¶

Imposes economic restrictions on the contemporaneous impact matrix.
Short-run restrictions (Cholesky or custom zero restrictions).
Long-run restrictions (Blanchard-Quah: permanent vs transitory shocks).
Sign restrictions (Uhlig): identify shocks by the sign of responses.

Forecast Error Variance Decomposition (FEVD)¶

Decomposes the forecast error variance of each variable into contributions from each shock.
Complements IRFs: shows relative importance of shocks.

Cointegration¶

Two or more I(1) series are cointegrated if a linear combination is I(0). The series share a common stochastic trend and cannot drift apart indefinitely.

Engle-Granger Two-Step¶

Estimate cointegrating regression: y_t = alpha + beta * x_t + u_t (OLS)
Test residuals u_t for unit root (ADF with Engle-Granger critical values, NOT standard ADF tables)
If cointegrated, estimate ECM: Delta(y_t) = gamma * u_{t-1} + lagged differences + e_t
gamma < 0 is the error correction speed.

Limitations: only finds one cointegrating relationship; normalization matters; poor small-sample properties.

Johansen Test¶

Tests for number of cointegrating vectors among K variables.
Trace test and maximum eigenvalue test.
Can find 0, 1, ..., K-1 cointegrating relationships.
Specify deterministic components (intercept/trend in cointegrating equation vs short-run).

Python

from statsmodels.tsa.vector_ar.vecm import coint_johansen

result = coint_johansen(data, det_order=0, k_ar_diff=2)
print(result.trace_stat)
print(result.trace_stat_crit_vals)  # 90%, 95%, 99%

VECM (Vector Error Correction Model)¶

If cointegration exists, estimate VECM rather than VAR in differences:

Delta(Y_t) = alpha * beta' * Y_{t-1} + Gamma_1 * Delta(Y_{t-1}) + ... + u_t

beta: cointegrating vectors (long-run equilibrium)
alpha: adjustment coefficients (speed of correction)
Gamma: short-run dynamics

Python

from statsmodels.tsa.vector_ar.vecm import VECM

model = VECM(data, k_ar_diff=2, coint_rank=1)
results = model.fit()

HAC Standard Errors¶

With time series data, OLS standard errors are inconsistent under heteroskedasticity and autocorrelation. Use Heteroskedasticity and Autocorrelation Consistent (HAC) estimators.

Newey-West: The standard choice for time series regressions.

Python

import statsmodels.api as sm

model = sm.OLS(y, X).fit(cov_type='HAC', cov_kwds={'maxlags': L})

Bandwidth selection: Newey-West with automatic bandwidth (Andrews 1991) or fixed rule L = floor(4 * (T/100)^(2/9)).

When to use: Always in time series regressions unless you have strong reasons to believe errors are homoskedastic and serially uncorrelated (rare in finance).

Forecast Evaluation¶

In-Sample vs Out-of-Sample¶

In-sample fit (R^2, AIC) overstates predictive ability.
Use expanding or rolling window out-of-sample forecasts.
Compare against a benchmark (random walk, historical mean).

Metrics¶

RMSE, MAE, MAPE for point forecasts.
Diebold-Mariano test: H0 equal predictive accuracy between two models.
Clark-West test: for nested model comparisons (adjusts for the fact that the null imposes restrictions).
Mincer-Zarnowitz regression: regress realized on forecast. Test alpha=0, beta=1 jointly.

Density Forecasts¶

Probability integral transform (PIT): if model is correct, PIT values are U(0,1).
Log predictive score for comparing density forecasts.

Structural Breaks and Regime Switching¶

Chow test: Known break date. F-test for parameter stability.

CUSUM / CUSUM-of-squares: Sequential tests for parameter instability.

Bai-Perron: Multiple unknown structural breaks. Estimates break dates and tests for number of breaks.

Markov-switching models: Regime-dependent parameters with transition probabilities.

Python

from statsmodels.tsa.regime_switching.markov_regression import MarkovRegression

model = MarkovRegression(y, k_regimes=2, trend='c', switching_variance=True)
results = model.fit()

Practical Checklist¶

Plot the series. Check for trends, breaks, volatility clustering, outliers.
Test for unit roots (ADF + KPSS). Transform to stationarity if needed (log returns, first differences).
Choose model class: univariate (ARIMA/GARCH) vs multivariate (VAR/VECM).
Select lag order via information criteria.
Estimate and check residual diagnostics (Ljung-Box, ARCH-LM, normality).
For GARCH: use Student-t or skewed-t distribution. Test for leverage (EGARCH/GJR).
For VAR: test Granger causality, compute IRFs with confidence bands, report FEVD.
For cointegrated systems: use VECM, not VAR in differences.
Always use HAC (Newey-West) standard errors for time series regressions.
Evaluate forecasts out-of-sample against a naive benchmark. Use Diebold-Mariano or Clark-West.
Test for structural breaks if the sample spans crises or policy changes.
Report unconditional and conditional moments, persistence measures (half-life), and economic interpretation.

Key References¶

Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press.
Luetkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer.
Tsay, R.S. (2010). Analysis of Financial Time Series, 3rd ed. Wiley.
Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics.
Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors. Econometrica.
Newey, W.K. and West, K.D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica.
Diebold, F.X. and Mariano, R.S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics.

time-series¶

Time Series Econometrics¶

Overview¶

Stationarity and Unit Root Tests¶

Testing for Unit Roots¶

Practical Rules¶

ARIMA Models¶

Identification¶

Estimation¶

Diagnostics¶

GARCH Models¶

GARCH(1,1)¶

Extensions¶

Distribution Assumptions¶

GARCH Diagnostics¶

VAR Models¶

Identification and Estimation¶

Granger Causality¶

Impulse Response Functions (IRFs)¶

Structural VAR (SVAR)¶

Forecast Error Variance Decomposition (FEVD)¶

Cointegration¶

Engle-Granger Two-Step¶

Johansen Test¶

VECM (Vector Error Correction Model)¶

HAC Standard Errors¶

Forecast Evaluation¶

In-Sample vs Out-of-Sample¶

Metrics¶

Density Forecasts¶

Structural Breaks and Regime Switching¶

Practical Checklist¶

Key References¶

`time-series`¶