What is a white noise
A series of uncorrelated random variables with: Mean = 0, Variance = σ², Cov(Xₜ, Xₛ) = 0 for t ≠ s.
If normal ⇒ White Gaussian Noise
What is weak and strong stationarity
Weak: mean, variance, and autocovariance depend only on lag, not time.
Strong: full distribution of the process is time-invariant.
AR(1) process equation
X_t = φ * X_{t-1} + ε_t
What is the condition for an MA(q) process to be invertible?
The roots of the MA polynomial theta(B) = 0 must lie outside the unit circle.
General ARMA(p,q) equation
X_t = φ₁X_{t-1} + φ₂X_{t-2} + … + φ_pX_{t-p} + ε_t + θ₁ε_{t-1} + … + θ_qε_{t-q}
What does the autocorrelation function (ACF) measure? Properties also.
Linear dependence between Xₜ and Xₜ₋ₖ.
ACF is always 1 at h=0
If ACF stays flat, non stationary
Autocovariance (γ) and autocorrelation (ρ) formula
γ_k = Cov(X_t, X_{t-k})
ρ_k = γ_k / γ_0
Autocovariance measures covariance between time-lagged values; autocorrelation is its normalized form (between -1 and 1).
What is the lag operator and how is it used?
L^k X_t = X_{t-k}
What’s the difference operator Δ?
ΔX_t = X_t - X_{t-1} = (1 - L)X_t
Used to difference non-stationary series.
What’s the purpose of the Box–Pierce and Ljung–Box tests? Which is better
They test whether a series’ residuals are white noise (i.e., no autocorrelation). Ljung box is better for small samples also.
What does the Durbin–Watson test check?
Detects first-order autocorrelation in regression residuals.
If 0 -> +ve corr
If 2 -> no corr
If 4 -> -ve corr
For AR(1) tell if stationary or non stationary |φ| ≥ 1, |φ| < 1.
If |φ| ≥ 1, shocks explode and variance infinite so non stationary.
Else stationary
Breusch Godfrey test
Tests for higher order (upto lag q) autocorrelation of residuals in a regression. More general and robust version of Durbin Watson test.
What are unit roots and characteristic equation of AR and MA models
AR(p) is stationary if all roots of
1 - φ₁z - φ₂z² - … - φₚzᵖ = 0
lie outside the unit circle (i.e., |z| > 1).
MA(q) is invertible if all roots of
1 + θ₁z + θ₂z² + … + θ_qz^q = 0
lie outside the unit circle (i.e., |z| > 1).
Stationarity Tests (DF/ADF steps)
Augmented Dickey Fuller Test:
1) Start with AR(1)
2) Difference operator
3) Choose p lags to make residuals white noise
4) Test hypothesis H0: unit roots
H1: stationary
All stationarity tests, null or alternate is stationary?
Augmented Dickey Fuller (alternate), Phillips Perron (alternate) and KPSS (null)
What is partial corr?
At lag k, measures direct linear relationship between Yt and Yt-k after removing effect of all intermediate lags.
PACF(k) = Corr(Yt, Yt-k | Yt-1,…,Yt-k+1)
