Charta is on Google Play: practise on the go. Get the app
Charta

Give the definition of autoregressive moving average (ARMA) processes and describe their properties.

FRM® Part I · Quantitative Analysis · Stationary Time Series · FRM1.QA.10.h

The idea

An ARMA(p,q) process combines autoregressive and moving average parts. It has p lags of the series and q lags of the shock. In the ARMA(1,1) case the process is y(t) = δ + φ × y(t − 1) + ε(t) + θ × ε(t − 1). Stationarity depends only on the AR part, whose characteristic roots must lie outside the unit circle. The MA part never threatens it. The mean ignores the θ coefficients and, for ARMA(1,1), is δ ÷ (1 − φ). Its autocorrelations and partial autocorrelations both decay gradually and neither cuts off cleanly. That separates ARMA from a pure AR, whose partial autocorrelation cuts off, and from a pure MA, whose autocorrelation cuts off. For ARMA(1,1) the autocorrelation at lag 1 is ρ1 = (1 + φθ)(φ + θ) ÷ (1 + 2φθ + θ²). After that it decays at rate φ. If φ + θ = 0 the series is white noise around its mean.

Formulas

Lag-1 autocorrelation of an ARMA(1,1)

ρ(1) = (1 + φθ)(φ + θ) ÷ (1 + 2φθ + θ²)

φ
the AR coefficient
θ
the MA coefficient

Stationary ARMA(1,1). Beyond lag 1 the autocorrelation decays at rate φ; if φ + θ = 0 every autocorrelation is zero.

A practice question

From the question bank, with the reason each wrong option is wrong.

An ARMA(1,1) process is best described as:

  1. y_t = delta + phi y_(t-1) + e_t + theta e_(t-1) - correct
  2. y_t = delta + phi y_(t-1) + theta y_(t-2) + e_tWhy not: Puts two lagged values of y in the equation, which is an AR(2); the MA part uses a lagged shock.
  3. y_t = delta + e_t + theta e_(t-1) + theta e_(t-2)Why not: Has only lagged shocks, which is an MA(2) with no autoregressive part.
  4. y_t = delta + phi y_(t-1) + phi e_(t-1)Why not: Multiplies lagged shocks by the AR coefficient, so there is no separate MA parameter and the ARMA is mis-specified.

ARMA(p,q) combines p autoregressive terms in lagged values of y with q moving average terms in lagged shocks. ARMA(1,1) has one of each, with separate coefficients phi and theta.

A harder one

For an ARMA(1,1) with phi = 0.5 and theta = 0.3, the lag-1 autocorrelation is rho(1) = (1 + phi theta)(phi + theta) / (1 + 2 phi theta + theta²). Its value is closest to:

  1. 0.50Why not: Reports phi as the autocorrelation, ignoring the contribution of the moving average term.
  2. 0.66 - correct
  3. 0.28Why not: Uses the pure MA(1) formula theta / (1 + theta²), ignoring the AR part.
  4. 0.80Why not: Adds phi and theta (the second factor of the numerator) without the other factor or the denominator.

Numerator: (1 + 0.15)(0.8) = 0.92. Denominator: 1 + 0.30 + 0.09 = 1.39. rho(1) = 0.92 / 1.39 = 0.662.

The hardest tier

An ARMA(2,1) is y_t = 1.2 + 0.5 y_(t-1) + 0.3 y_(t-2) + e_t - 0.8 e_(t-1). Given that it is covariance stationary (the AR roots are -2.84 and 1.17), its unconditional mean is:

  1. 1.2Why not: Reports the intercept as the mean, ignoring that the AR terms raise the average level.
  2. 6.0 - correct
  3. -2.0Why not: Subtracts theta in the denominator (1 - 0.5 - 0.3 - 0.8); MA coefficients do not enter the mean.
  4. 0.67Why not: Adds the AR coefficients to 1 in the denominator (1.2 / 1.8), a sign error.

Only AR coefficients enter the mean: mu = 1.2 / (1 - 0.5 - 0.3) = 1.2 / 0.2 = 6.0.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Stationary Time Series