AR, MA and ARIMA

Idea: A gap that closes

$2$ share classes trade $14$ basis points apart. Wide gaps narrow over days, never all at once.

• Tomorrow: a constant, plus a fraction of today, plus news
• That is an autoregression with $1$ lag, $\mathrm{AR}(1)$

Formula: The $\mathrm{AR}(1)$ model

$X_t=c+\varphi X_{t-1}+{ϵ}_t$

The shock has mean 0 and is independent of the past. Stationary only when |φ| < 1.

Formula: Mean reversion forecast

$E[X_{t+h}\mid X_t]=\mu +{\varphi }^h(X_t-\mu )$

The mean is μ = c/(1 − φ). Measure today’s distance from it and shrink it by φ each step.

Example: The gap 2 days out

The fit gives $c=3$ and $\varphi =0.7$; today’s gap is $14$. Forecast the gap the day after tomorrow.

1. Mean $\frac{3}{1-0.7}=10$, distance $14-10=4$
2. Tomorrow: $10+0.7\times 4=12.8$
3. Next day: $10+0.49\times 4=11.96$

Formula: $\mathrm{MA}(1)$: memory in the shocks

$X_t=\mu +{ϵ}_t+\theta {ϵ}_{t-1}$

A shock partly reverses on the next print, then is gone. Not a chart’s rolling mean.

Quick check: An $\mathrm{MA}(1)$ forecast

An $\mathrm{MA}(1)$ has $\mu =0$ and $\theta =-0.4$. Today’s forecast error is ${ϵ}_t=5$. What is the forecast for the next step?

Idea: $\mathrm{AR}$ fades, $\mathrm{MA}$ stops dead

An $\mathrm{AR}$ forecast fades a little at every step. An $\mathrm{MA}(q)$ forecast stops after $q$ steps.

• $\mathrm{ARMA}(p,q)$ adds $p$ lagged values and $q$ lagged shocks

Idea: A series with no home

A running total has no level to return to: difference it, then fit $\mathrm{ARIMA}(p,d,q)$.

• Decide first whether there is a level: that sets $d$
• Changes fade to $\frac{0.5}{1-0.6}=1.25$
• The level forecast rises $1.25$ a step, never flat

Example: An $\mathrm{ARIMA}(1,1,0)$ step

$\Delta X_t=0.5+0.6\Delta X_{t-1}+{ϵ}_t$, and the last levels are $200$ and $204$. Forecast the next level.

1. Last change $204-200=4$
2. Next change $0.5+0.6\times 4=2.9$
3. Level $204+2.9=206.9$

Common mistake: The change quoted as the level

Quoting the forecast change $2.9$ as where the series lands.

Instead Start from the last level and add every forecast change up to the horizon: $206.9+2.24=209.14$.