PACF and ACF for AR(q) and MA(q)
ACF:
AR -> Tails off gradually
MA -> Cuts off after lag q
PACF:
AR -> Cuts off after lag p
MA -> Tails off gradually
Estimating AR(p) process methods and step
MLE and Yule-Walker equations:
- Model: X_t = φ₁X_{t−1} + … + φₚX_{t−p} + ε_t
- Compute sample autocovariances:
γ̂(h) = (1/n) Σ_{t=h+1}^{n} (X_t − X̄)(X_{t−h} − X̄) - Set up Yule–Walker equations:
γ̂(h) = Σ_{k=1}^p φ_k γ̂(h−k), h = 1,…,p - Solve for φ:
In matrix form → Γ_p φ = γ_p - Estimate noise variance:
σ̂² = γ̂(0) − Σ_{k=1}^p φ_k γ̂(k)
- Compute sample autocovariances:
MLE conditional vs unconditional
Unconditional: accounts for whole likelihood, more complex, more efficient but no closed form answer
Conditional: Condition on first P observations, use MLE on remaining, has closed form OLS estimator.
Why YW only for AR
No closed form, non linear equations. We use for AR because autocovriances are determined by AR coefficients entirely.
Assumptions of YW
Process is weakly stationary, Errors are white noise
Aic and bic formula
Akaike Information Criterion:
2k - 2ln(L),
• k = number of estimated parameters
• L = maximum value of the likelihood function
Bayesian Info C:
k * ln(n) - 2ln(L)
Aic and bic meaning
Both AIC and BIC are used for model selection (like deciding the order of AR, MA, or ARIMA models). Both measure the trade-off between model fit and model complexity.
AIC and BIC characteristics
Aic and bic - the smaller the better.
Aic tends to choose more complex models.
ARIMA parameters and meaning
p = AR order, d = number of differences, q = MA order. Handles non stationary by differencing.
SARIMA meaning and parameters
Arima with seasonal patterns.
SARIMA(p, d, q)(P, D, Q)m
m → number of periods per season
How to identify Seasonality? When does SARIMA perform poorly?
Check ACF and PACF and look spikes at regular intervals.
When seasonality is changing over time.
ARCH formula and meaning
Autoregressive Conditional Heteroskedasticity
ε_t = σ_t * z_t
σ_t² = α₀ + α₁ε_{t-1}² + α₂ε_{t-2}² + … + α_qε_{t-q}²
It models time-varying volatility — variance changes over time
What is GARCH formula and meaning
Generalised ARCH
σ_t² = α₀ + α₁ε_{t-1}² + … + α_qε_{t-q}² + β₁σ_{t-1}² + … + β_pσ_{t-p}²
ARCH meaning and formula
Autoregressive Conditional Heteroskedasticity.
Variance changes over time depending on past squared shocks.
ε_t = σ_t * z_t
σ_t² = α₀ + α₁ε_{t-1}² + α₂ε_{t-2}² + … + α_qε_{t-q}²
GARCH meaning and formula
Generalised ARCH.
σ_t² = α₀ + α₁ε_{t-1}² + … + α_qε_{t-q}² + β₁σ_{t-1}² + … + β_pσ_{t-p}²
α terms → short-term shocks
β terms → long-term persistence in volatility
Why use ARCH/GARCH for modelling Volatility?
- Volatility Clustering
- Finance is leptokurtic (fat tails)
- Heteroskedasticity
