Outline how AR, MA and ARMA processes are applied, and use model selection criteria to pick the time series model that fits best.

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

The idea

AR models suit persistent, mean-reverting series such as interest rates, spreads or volatility. MA models suit series where a shock fades after a fixed number of periods. ARMA mixes both with fewer parameters. To choose among candidates, estimate each, then record the residual variance σ̂² from T observations with k estimated coefficients. Compute the information criteria for every candidate. The Akaike criterion is AIC = ln(σ̂²) + 2k ÷ T. The Schwarz criterion, also called the Bayesian criterion, is BIC = ln(σ̂²) + k × ln(T) ÷ T. Lower is better. The first term rewards fit, and the second penalises extra parameters. BIC penalises more heavily whenever ln(T) exceeds 2, which is for T of 8 or more, so it favours smaller models and is consistent, while AIC tends to pick larger ones. Choosing by the smallest residual variance alone favours the biggest model, which is the classic error.

Formulas

Akaike information criterion

AIC = ln(σ̂²) + 2k ÷ T

σ̂²: the estimated residual variance
k: the number of estimated coefficients
T: the number of observations

Lower is better, comparing models on the same data. It penalises parameters more lightly than BIC and so tends to pick larger models.

Bayesian information criterion

BIC = ln(σ̂²) + k × ln(T) ÷ T

σ̂²: the estimated residual variance
k: the number of estimated coefficients
T: the number of observations

Lower is better. The penalty beats AIC's once ln(T) exceeds 2, so BIC favours smaller models and is consistent.

A practice question

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

When comparing ARMA models estimated on the same data using AIC or BIC, the preferred model is the one with:

the lowest criterion value - correct
the highest criterion valueWhy not: Reverses the rule; both criteria are penalised measures of residual variance, so lower is better.
the largest number of parametersWhy not: More parameters always reduce residual variance but raise the penalty; the criteria exist to prevent choosing the biggest model.
the highest in-sample R²Why not: R² never penalises extra parameters, so it favours the largest model, unlike AIC or BIC.

AIC = ln(sigma-hat²) + 2k/T and BIC = ln(sigma-hat²) + k ln(T)/T. Both fall with better fit and rise with parameters, and the model with the smallest value is selected.

A harder one

A risk manager studies the monthly change in a corporate bond spread. The sample autocorrelations decay geometrically from lag 1, and the sample partial autocorrelation is significant at lag 1 only. Which model is the most natural choice?

MA(1)Why not: An MA(1) would show an autocorrelation that cuts off after lag 1 and a decaying partial autocorrelation, the opposite of what is observed.
AR(1) - correct
MA(2)Why not: An MA(2) would have an autocorrelation cutting off after lag 2; geometric decay of the autocorrelation points to AR.
White noiseWhy not: White noise would show no significant autocorrelation; geometric decay indicates persistence.

A geometrically decaying ACF with a PACF that cuts off after lag 1 is the signature of an AR(1), which suits a persistent, mean-reverting spread change.

The hardest tier

With T = 500 observations, adding one estimated parameter to an ARMA model will lower the BIC only if the residual variance falls by at least about:

1.2% - correct
0.4%Why not: Uses the AIC penalty 2/T = 0.004; the BIC penalty is ln(T)/T.
6.2%Why not: Uses ln(500) = 6.21 as a percentage, forgetting to divide by T.
2.5%Why not: Doubles the BIC penalty to 2 ln(T)/T, mixing the AIC factor of 2 with the BIC formula.

BIC change = ln(new variance / old variance) + ln(500)/500 must be negative. ln(500)/500 = 0.01243, so variance must drop by 1 - exp(-0.01243) = 0.0124, about 1.2%.

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

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