Stationarity

A strictly or weakly stationary process exhibits a constant mean over time. The data fluctuates around a stable long-run average without trending upwards or downwards.

• Mathematical condition: (E[X_t] = \mu) for all (t).
• Practical check: A rolling average should remain roughly horizontal.
• Violation: If a stock price trends from $10 to $100, the mean is time-dependent, violating stationarity.
• Remedy: Apply differencing (calculating period-over-period changes) to remove the trend and stabilize the mean.

The variance of the series must be finite and constant through time. The spread or volatility of the data points around the mean should not systematically expand or contract.

• Mathematical condition: (Var[X_t] = \sigma^2 < \infty) for all (t).
• Violation: Volatility clustering in financial returns (calm periods followed by turbulent periods) breaks this assumption.
• Impact: Models like ARIMA assume constant variance; non-constant variance leads to unreliable prediction intervals.
• Remedy: Apply a logarithmic or Box-Cox transformation to stabilize the variance.

The covariance between two time points depends solely on the distance (lag) between them, not on the specific point in time.

• Mathematical condition: (Cov[X_t, X_{t+h}] = \gamma(h)) for all (t).
• Interpretation: The relationship between today's value and the value 5 days ago is the same whether you measure it in 2010 or 2020.
• ACF/PACF: This property allows the Autocorrelation Function to be a reliable tool for model identification.
• Violation: Structural breaks in the economy change the underlying data-generating process, altering the autocorrelation structure.

It is critical to distinguish between the theoretical ideal and the practical requirement for most models.

• Strict Stationarity: The entire joint probability distribution is time-invariant. This is a very strong, non-parametric condition rarely met in real data.
• Weak (Covariance) Stationarity: Only the first two moments (mean and variance) and the autocovariance are time-invariant. This is the standard requirement for Box-Jenkins models.
• Gaussian Processes: For a Gaussian process, weak stationarity implies strict stationarity because the distribution is fully defined by the first two moments.

A unit root is a primary cause of non-stationarity. If a process has a unit root, shocks have a permanent effect, and the series does not revert to a mean.

• Random Walk: (X_t = X_{t-1} + \epsilon_t) is the classic unit root process. Its variance increases to infinity over time.
• Testing: The Augmented Dickey-Fuller (ADF) test checks for the presence of a unit root. A p-value > 0.05 suggests non-stationarity.
• Spurious Regression: Regressing two independent random walks on each other often yields high (R^2) and significant t-stats, creating a completely false relationship.
• Solution: Differencing the series ((X_t - X_{t-1})) removes a single unit root, transforming it into an integrated process of order 1, or I(1).

Not all non-stationarity is a unit root. It is vital to distinguish the type to apply the correct transformation.

• Trend-Stationary (TS): The process is stationary around a deterministic time trend. (X_t = \beta t + \epsilon_t). Removing the trend via regression yields a stationary residual.
• Difference-Stationary (DS): The process is a random walk with drift. (X_t = \alpha + X_{t-1} + \epsilon_t). Differencing is required.
• Mis-specification Risk: Detrending a DS process does not make it stationary; it only removes the drift, leaving the stochastic trend intact. Conversely, differencing a TS process introduces a non-invertible moving average error.

A statistical property where a linear combination of two or more non-stationary time series is itself stationary. This reveals a long-run equilibrium relationship, preventing the variables from drifting arbitrarily far apart over time. Unlike correlation, cointegration implies a shared stochastic trend.

• Key Test: Engle-Granger two-step method or Johansen test.
• Application: Pairs trading strategies rely on cointegrated assets to generate mean-reversion signals.
• Contrast with Stationarity: Stationarity applies to a single series; cointegration applies to a system of non-stationary series.

A regression that suggests a statistically significant relationship between two or more independent non-stationary variables when no meaningful causal link exists. High R-squared values and significant t-statistics are misleading artifacts of shared stochastic trends.

• Cause: Violating the classical linear regression assumption of stationary residuals.
• Detection: Check for a Durbin-Watson statistic significantly below 2.0, indicating autocorrelated errors.
• Solution: Difference the data to achieve stationarity or use a Vector Error Correction Model (VECM) if cointegration is present.

An econometric model that captures the linear interdependencies among multiple time series. Each variable is modeled as a linear function of its own past lags and the past lags of all other variables in the system.

• Prerequisite: All variables in a standard VAR must be covariance stationary.
• Structure: A system of equations where the number of coefficients grows quadratically with the number of variables and lags.
• Output: Impulse Response Functions (IRFs) trace the dynamic effect of a shock to one variable on the entire system.

A statistical hypothesis test for determining whether one time series is useful in forecasting another. A variable X 'Granger-causes' Y if past values of X contain information that helps predict Y beyond the information contained in past values of Y alone.

• Stationarity Requirement: The test assumes the underlying time series are stationary. Non-stationary data can produce spurious Granger-causality results.
• Limitation: It measures predictive causality, not true structural causality. It does not account for latent confounding variables.
• Implementation: Typically conducted via an F-test on the lagged coefficients of X in a regression of Y.

A function tracing the dynamic effect of a one-time exogenous shock in one variable on the current and future values of all endogenous variables in a system, such as a VAR.

• Stationarity Implication: In a stationary system, the IRF decays to zero over time—the effect of a shock is transitory.
• Non-Stationary Implication: In a non-stationary system, the effect of a shock can be permanent, and the IRF will not converge to zero.
• Orthogonalization: Cholesky decomposition is often used to isolate the effect of a shock to a single variable by imposing a causal ordering.

A restricted VAR designed for use with cointegrated non-stationary series. It separates the long-run equilibrium relationship (the error correction term) from the short-run dynamic adjustments.

• Mechanism: The error correction term measures the deviation from the long-run equilibrium at the previous period and forces a corrective adjustment.
• Stationarity Connection: A VECM is estimated by differencing non-stationary data to achieve stationarity while explicitly modeling the stationary cointegrating relationship.
• Advantage over VAR: A VAR on differenced data alone omits the long-run equilibrium information, leading to model misspecification.