Intermediate Econometrics: Stationary Time Series and ARMA Processes

First-order Autoregressive Process: AR(1)

  • Defined by: xt=ϕxt−1+ϵtx_t = \phi x_{t-1} + \epsilon_t where ϕ\phi is the autoregressive parameter and ϵt\epsilon_t is white noise.

  • Stationarity Condition: ∥ϕ∥<1\|\phi\| < 1

  • Mean: E(xt)=0E(x_t) = 0.

  • Variance: V(xt)=σ21−ϕ2V(x_t) = \frac{\sigma^2}{1 - \phi^2}.

  • Autocovariance: γk=ϕkσ21−ϕ2\gamma_k = \phi^k \frac{\sigma^2}{1 - \phi^2}.

  • Autocorrelation: ρk=ϕk\rho_k = \phi^k, showing geometric decay.

Autoregressive Process of Order p: AR(p)

  • Defined as: xt=∑i=1pϕixt−i+ϵtx_t = \sum_{i=1}^p \phi_i x_{t-i} + \epsilon_t.

  • Stationarity requires roots of lag polynomial to be outside the unit circle.

First-order Moving Average Process: MA(1)

  • Defined by: xt=ϵt+θϵt−1x_t = \epsilon_t + \theta \epsilon_{t-1}.

  • Always stationary.

  • Invertibility Condition: ∥θ∥<1\|\theta\| < 1.

Autoregressive Moving Average Process: ARMA(1,1)

  • Combines AR(1) and MA(1): xt−ϕxt−1=ϵt+θϵt−1x_t - \phi x_{t-1} = \epsilon_t + \theta \epsilon_{t-1}.

  • Conditions for stationarity and invertibility are ∥ϕ∥<1\|\phi\| < 1 and ∥θ∥<1\|\theta\| < 1.

Case Study: UK Interest Rate Spread

  • Analyzes the spread between 20-year Gilts and 91-day Treasury bills from 1952 to 2014, fitting various AR and ARMA models with an emphasis on stationarity and significant coefficients.