Write short notes on the following: a) ARCH model b) Granger-causality

a) ARCH Model (Autoregressive Conditional Heteroskedasticity)

The ARCH model, introduced by Robert Engle in 1982, is used to model and forecast time-varying volatility in time series data, especially in financial markets where periods of high and low volatility alternate.

Key Features:

• Conditional heteroskedasticity: The variance of the error term depends on past squared errors.
• Captures volatility clustering — large changes tend to be followed by large changes, and small changes by small changes.

Basic ARCH(q) Model:

Let the model be:

y_t = x_tβ + u_t, where u_t ~ N(0, h_t)

h_t = α₀ + α₁u_t−1² + … + α_qu_t−q²

The conditional variance h_t changes over time based on past squared errors.

Applications:

• Modeling stock market volatility
• Risk management and option pricing

Extensions:

• GARCH (Generalized ARCH): Incorporates past variances as well as past errors for better volatility modeling.

---

b) Granger Causality

Granger causality is a statistical hypothesis test used to determine whether one time series can predict another. It does not imply true causality but only predictive capability.

Definition:

Variable X is said to Granger-cause Y if past values of X contain information that helps predict Y beyond what is contained in past values of Y alone.

Basic Test:

Consider two stationary time series Y_t and X_t. Estimate the following two regressions:

1. Restricted Model (no X): Y_t = α + Σβ_iY_t−i + ε_t
2. Unrestricted Model (with X): Y_t = α + Σβ_iY_t−i + Σγ_jX_t−j + ε_t

If the additional lagged values of X significantly improve the prediction of Y (based on F-test), then X Granger-causes Y.

Assumptions:

• Time series must be stationary or made stationary (via differencing)
• Proper lag length selection is crucial

Applications:

• Macroeconomic forecasting (e.g., does money supply Granger-cause inflation?)
• Financial markets (e.g., do stock returns Granger-cause exchange rates?)

Conclusion

The ARCH model is key to modeling volatility in time series, especially in financial applications, while Granger causality is widely used to assess the predictive relationship between time series variables. Both are fundamental tools in applied econometric research.