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L39 Causality Tests and Further

Casuality test \(\implies\) asses whether one TS can predict or cause change in another.

  • Granger causality test is used
    • Does past value of \(X\) contain some information about future values of \(Y\)

Understanding

  • \(X\) "Granger causes" \(Y\) \(\implies\) knowing all past \(X\) values improves the prediction of \(Y\)

Granger Causality \(\neq\) true causality

Mathematical Foudnation

  • Model 1 (unrestricted)… lagged value of \(Y\) and \(X\) both
  • Model 2 (restricted)… only lagged values of \(Y\) (not \(X\))

$$
Y_{t} = \alpha + \sum_{i=1}^{p} \beta_{i}Y_{t-i} + \sum_{j=1}^{p} \gamma_{j} X_{t-j} + \epsilon_{t}
$$
And the restricted model

\[ Y_{t} = \alpha + \sum_{i=1}^{p} \beta_{i} Y_{t-i} + \epsilon_{t} \]
\[ \begin{align} H_{0} & : \gamma_{j} = 0\ \forall j, X \text{ does not Granger-cause } Y \\ H_{1} & : \gamma_{j} \neq 0 \text{ for at least one } j, X \text{ Granger-causes } Y \end{align} \]
  • Use \(F\)-test (for small samples)
  • Wald test (large samples)
  • If statistically significant \(\implies\) Reject \(H_{0}\), \(X\) Granger-causes \(Y\).

Assumptions of Granger Causality

  • Stationarity: both \(X\) and \(Y\) should be stationary (transform if not by differencing)
  • Appropriate Lag Length: choose \(p\) (it's crucial) using AIC or BIC
  • No Cointegration (for VAR models)
    • If so, use ECM or VECM (more appropriate)

Extensions of Granger Causality

  1. VECM causality
    • If cointegrated
    • Include both ECT and lagged differences
  2. Nonlinear Granger Causality
    • Requiring neural networks or kernel based methods
  3. Instantaneous Causality
    • If change in \(X\) at \(t\) immediately affects \(Y\) at \(t\)
    • Simultaneous Equation Model (SEM) framework

The Haugh-Pierce Test

  • context: Two stationary TS. How? Examine their cross-correlation

  • IDENTIFY if \(X_{t-i}\) can improve prediction for \(Y_{t+k}\) without the need to estimate a model

  • Procedure

    1. Pre-whiten Each Series: Fit an ARIMA model to each TS. Remove auto correlations (thus, isolates residuals for each series)
    2. Compute Cross-Correlation of Residuals. CCF represents relationship between unexplained changes in one series and past of another.
\[ \begin{align} H_{0} & : \text{No Causality, cross-correlation} = 0 \\ H_{1} & : \text{Causality exists. CC insignificant} \\ \end{align} \]
  • Test statistic
    • Sum of Cross-Correlations up to specified lag \(K\) (maximum lag of interest)
    • \(Q = T \sum_{k=1}^{N}\hat{\rho}_{XY}(k)^{2}\)
      • \(T\) is number of observations
      • \(\hat{\rho}_{XY}(k)\) is the cross-correlation at lag \(k\)
      • follows \(\chi^{2}\) distribution with \(K\)-d.f.

Assumptions

  • Stationarity
  • Model assumptions: appropriateness of A

Hsiao Procedure

"H" is silent… "Siao"

Cheng Hsiao in 1981.

  • To overcome some limitations of traditional Granger causality tests
  • Especially useful for lag selection in TS data
  • A more adaptable way to determine causal relationships
  • Combines Akaike's Final Prediction Error (FPE) criterion + traditional Granger causality test

Motivation

  • Systematically determines best lag structure
  • FPE criterion balances model fit and complexity \(\implies\) accurate and parsimonious