ARIMA Models
2021
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Nonstationarity
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Nonstationarity
To fit ARMA models we need stationary series, but many series are not
Generally a trend can be seen
Even if no trend there are major shifts in level
While actual measurements are a problem, what about first differences?
For ease of notation I introduce a new operator, the difference operator, ∇
∇Xt =Xt −Xt−1
Notation similar to backshift operator in that ∇k Xt = Xt − Xt−k
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Nonstationarity
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PACF and ACF required
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PACF and ACF required
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ACF of differenced series
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PACF and ACF required
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ARIMA Processes
Time series of differences of the original time series may be an ARMA model
If ∇d{Xt} = {Yt} and {Yt} is an ARMA(p,q) process, then {Xt} is an ARIMA(p,d,q) process.
ARIMA stands for AutoRegressive Integrated Moving Average The d-term is the order of integration
∇dXt is applying ∇ to {Xt} repeatedly, d times;
∇2Xt =∇[∇Xt]
= ∇[Xt − Xt−1]
= (Xt − Xt−1) − (Xt−1 − Xt−2) = Xt − 2Xt−1 + Xt−2
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Differencing to get Stationarity in Series With Trend
Almost always d ≤ 2, and usually d ≤ 1
d = 0 is what you should start from as you start by looking at original data
Fractional differencing is studied (ARFIMA), but too complicated for this course
Differencing may not solve all your problems in getting a good model fit (forecast)
So how do we go about it?
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Retail Trade Quarterly
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ACF of Raw Retail Trade Quarterly
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∇{Xt }
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ACF of ∇{Xt}
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∇4{Xt}
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ACF of ∇4{Xt}
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SARIMA Models
A D difference, spaced say s apart
s is number of measurements over a year
s = 4 if quarterly, s = 12 if monthly, s = 52 if weekly
d is the detrending and D is the deseasonalising
By doing this difference we are aiming for a stationary process (shifts spaced s and 1 apart) as that allows us to fit an ARMA model
SARIMA stands for Seasonal AutoRegressive Integrated Moving Average
Even if we have s = 52
we are modelling the first and the seasonal difference, so there are
two ARMA components, SARIMA model order (p, d, q) × (P, D, Q)s
All the stuff on using ACF and PACF to find p and q, also applies to P and Q, though first on d differenced series and the other on s differenced series
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SARIMA Models – the maths
If ∇Ds {Xt} = {Yt} then
φ(B)Φ(Bs)Yt = θ(B)Θ(Bs)Wt
∇Ds is the Dth order difference every s units apart Bs = Bs is the shift back s units,
φ(B)Φ(Bs),θ(B) and Θ(Bs) are polynomials in B of order p, P, q and Q respectively.
When dealing with SARIMA models – as I do frequently – you find the concise notation is what you report. But you need to be able to expand it so you know what is happening to your data (or what you should do to your data before get R to estimate the parameters.
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SARIMA Models – the formulae
ConsideraSARIMA(1,1,2)×(1,1,1)4 model
Concise representation – and {Xt} are the time series measurements we are using
Yt = ∇4∇1Xt
= ∇4(Xt − Xt−1
= [Xt − Xt−4] − [Xt−2 − Xt−5] =Xt −Xt−2 −Xt−4 +Xt−5
and {Yt} is modelled as
φ(B)Φ(B4)Yt = θ(B)Θ(B4)Wt
with
φ(B) = (1 − φ1B)
Φ(Bs) = (1 − Φ1B4) θ(B) = (1 + θ1B + θ2B2) Θ(Bs) = (1 + Θ1B4
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SARIMA Models – the practice
If you have seasonal variations then you will need (P,D,Q)s term in ARIMA model
However D=1 quite often as usually seasonality is stable Usually (0, 1, 1)s is best to use
Seasonal variation is not generally affected by events, though COVID-19 is doing this
Seasonality is generally a stable term -– hence the need to take it out as nuisance parameter by seasonal adjustment. Stability usually implies the effect is usually close to average, hence use MA(Q)
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Steps to ARIMA Modelling
1 Plot series and look at it. What is it like?
2 Transform series?
3 Use all the series?
4 acf of series as is, and differenced series? This may be iterative as may need seasonal as well. This decides I= and Is =?
5 acf and pacf to decide p, q, P, D as required
Recommend simulate models you think it is to ensure acf and pacf look right
6 Fit possible models
You may have a couple of feasible models, so choosing between them.
7 Diagnostics checks. Residual check
8 Forecast
9 Write up with how certain you are
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