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Models for Non-Stationary Time Series. The ARIMA( p,d,q ) time series. The ARIMA( p,d,q ) time series. Many non-stationary time series can be converted to a stationary time series by taking d th order differences. Let { x t |t  T } denote a time series such that

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models for non stationary time series

Models for Non-Stationary Time Series

The ARIMA(p,d,q) time series

the arima p d q time series
The ARIMA(p,d,q) time series

Many non-stationary time series can be converted to a stationary time series by taking dth order differences.

slide3
Let {xt|t  T} denote a time series such that

{wt|t T} is an ARMA(p,q) time series where

wt= Ddxt

= (I – B)dxt

= the dth order differences of the series xt.

Then {xt|t T} is called an ARIMA(p,d,q) time series (an integrated auto-regressive moving average time series.)

slide4
The equation for the time series {wt|t  T} is:

b(B)wt = d+ a(B)ut

The equation for the time series {xt|t  T} is:

b(B)Ddxt = d+ a(B)ut

or

f(B)xt =d+ a(B)ut..

Where

f(B) =b(B)Dd =b(B)(I - B) d

slide5
Suppose that d roots of the polynomialf(x)are equal to unity thenf(x) can be written:

f(B)= (1 -b1x - b2x2-... -bpxp)(1-x)d.

and f(B) could be written:

f(B) = (I -b1B - b2B2 -... -bpBp)(I-B)d= b(B)Dd.

In this case the equation for the time series becomes:

f(B)xt = d+ a(B)ut

or

b(B)Ddxt =d+ a(B)ut..

comments
Comments:
  • The operator

f(B) =b(B)Dd= 1 -f1x-f2x2 -... - fp+dxp+d

is called the generalized autoregressive operator. (d roots are equal to 1, the remaining p roots have |ri| > 1)

2.The operatorb(B) is called the autoregressive operator. (p roots with |ri| > 1)

3.The operatora(B) is called moving average operator.

example arima 1 1 1
Example – ARIMA(1,1,1)

The equation:

(I – b1B)(I – B)xt = d+ (I + a1)ut

(I – (1 + b1) B+b1B2)xt = d+ ut + a1 ut - 1

xt – (1 + b1) xt-1+b1xt-2 = d+ ut + a1 ut – 1

or

xt= (1 + b1) xt-1–b1xt-2+d+ ut + a1 ut – 1

slide9
If a time series, {xt: t  T},that is seasonal we would expect observations in the same season in adjacent years to have a higher auto correlation than observations that are close in time (but in different seasons.

For example for data that is monthly we would expect the autocorrelation function to look like this

the ar 1 seasonal model
The AR(1) seasonal model

This model satisfies the equation:

The autocorrelation for this model can be shown to be:

This model is also an AR(12) model with

b1 = … = b11 = 0

the ar model with both seasonal and serial correlation
The AR model with both seasonal and serial correlation

This model satisfies the equation:

This model is also an AR(13) model.

The autocorrelation for this model will satisfy the equations:

slide20
The Difference equation Form:

xt =f1xt-1 + f2xt-2+... +fp+dxt-p-d

+ d + ut +a1ut-1+ a2ut-2+...+ aqut-q

  • b(B)Ddxt = d+a(B)ut
slide21
The Random Shock Form:

xt =m(t) + ut+y1ut-1 + y2ut-2 +y3ut-3 +...

xt=m(t) +y(B)ut

slide22
The Inverted Form:

xt =p1xt-1 +p2xt-2 +p3xt-3+ ...+ t + ut

p(B)xt =t+ ut

example
Example

Consider the ARIMA(1,1,1) time series

(I – 0.8B)Dxt= (I + 0.6B)ut

(I – 0.8B) (I –B) xt= (I + 0.6B)ut

(I – 1.8B + 0.8B2) xt= (I + 0.6B)ut

xt = 1.8 xt - 1 - 0.8 xt - 2 + ut+ 0.6ut -1

Difference equation form

slide24

The random shock form

(I – 1.8B + 0.8B2) xt= (I + 0.6B)ut

xt= (I – 1.8B + 0.8B2)-1(I + 0.6B)ut

xt= (I + 2.4B + 3.52B2 +

4.416B3 + 5.1328B4 + … )ut

xt= ut+ 2.4 ut - 1 + 3.52 ut - 2 + 4.416 ut - 3+ 5.1328 ut - 4+ …

slide25

The Inverted form

(I – 1.8B + 0.8B2) xt= (I + 0.6B)ut

(I + 0.6B)-1(I – 1.8B + 0.8B2)xt= ut

(I - 2.4B + 2.24B2 – 1.344 B3 + 0.8064B4 +… )xt= ut

xt= 2.4 xt - 1 - 2.24 xt - 2 + 1.344 xt - 3- 0.8064 xt - 4+ … +ut

forecasting an arima p d q time series
Forecasting an ARIMA(p,d,q) Time Series
  • Let PT denote {…, xT-2, xT-1, xT} = the “past” until time T.
  • Then the optimal forecast of xT+l given PT is denoted by:
  • This forecast minimizes the mean square error
three different forms of the forecast
Three different forms of the forecast
  • Random Shock Form
  • Inverted Form
  • Difference Equation Form

Note:

random shock form of the forecast
Random Shock Form of the forecast

xt =m(t)+ ut+y1ut-1+y2ut-2 +y3ut-3 +...

Recall

or

xT+l =m(T + l)+uT+l +y1uT+l-1 +y2uT+l-2+y3uT+l-3+...

Taking expectations of both sides and using

slide29

Note:

xt =m(t) +ut +y1ut-1+y2ut-2+y3ut-3+...

To compute this forecast we need to compute

{…, uT-2, uT-1, uT} from {…, xT-2, xT-1, xT}.

Thus

and

Which can be calculated recursively

slide30
The Error in the forecast:

The Mean Sqare Error in the Forecast

Hence

prediction limits for forecasts
Prediction Limits for forecasts

(1 – a)100% confidence limits for xT+l

slide32
The Inverted Form:

p(B)xt =t+ utor

xt =p1xt-1+p2xt-2+p3x3+ ...

+ t + ut

where

p(B) = [a(B)]-1f(B)=[a(B)]-1[b(B)Dd]

= I -p1B -p2B2-p3B3-...

the inverted form of the forecast
The Inverted form of the forecast

Note:

xt =p1xt-1 +p2xt-2 +... +t+ ut

and for t = T+l

xT+l = p1xT+l-1+p2xT+l-2+...+t + uT+l

Taking conditional Expectations

the difference equation form of the forecast
The Difference equation form of the forecast

xT+l =f1xT+l-1+f2xT+l-2+ ...+ fp+dxT+l-p-d

+ d+ uT+l +a1uT+l-1+a2uT+l-2+... + aquT+l-q

Taking conditional Expectations

example arima 1 1 2
Example: ARIMA(1,1,2)

The Model:

xt - xt-1 = b1(xt-1 - xt-2) + ut +a1ut + a2ut

or

xt = (1 + b1)xt-1 - b1xt-2 + ut+ a1ut + a2ut

or

f(B)xt = b(B)(I-B)xt = a(B)ut

where

f(x) = 1 - (1 + b1)x + b1x2 = (1 - b1x)(1-x) and

a(x) = 1 + a1x + a2x2 .

the random shock form of the model
The Random Shock form of the model:

xt =y(B)ut

where

y(B) = [b(B)(I-B)]-1a(B) = [y(B)]-1a(B)

i.e.

y(B) [f(B)] = a(B).

Thus

(I + y1B + y2B2 + y3B3 + y4B4 + ... )(I - (1 + b1)B + b1B2)

= I + a1B + a2B2

Hence

a1 = y1 - (1 + b1) or y1 = 1 + a1 + b1.

a2 = y2 - y1(1 + b1) + b1 or y2 =y1(1 + b1) - b1 + a2.

0 = yh - yh-1(1 + b1) + yh-2b1

or yh = yh-1(1 + b1) - yh-2b1 for h ≥ 3.

the inverted form of the model
The Inverted form of the model:

p(B)xt = ut

where

p(B) = [a(B)]-1b(B)(I-B) = [a(B)]-1f(B)

i.e.

p(B) [a(B)] = f(B).

Thus

(I - p1B - p2B2 - p3B3 - p4B4 - ... )(I + a1B + a2B2)

= I - (1 + b1)B + b1B2

Hence

-(1 + b1) = a1 - p1 or p1 = 1 + a1 + b1.

b1 = -p2 - p1a1 + a2 or p2 = -p1a1 - b1 +a2.

0 = -ph - ph-1a1 - ph-2a2 or ph= -(ph-1a1 + ph-2a2) for h ≥ 3.

slide38
Now suppose that b1 = 0.80, a1 = 0.60 and a2 = 0.40 then the Random Shock Form coefficients and the Inverted Form coefficients can easily be computed and are tabled below:
computation of the random shock series one step forecasts
Computation of the Random Shock Series, One-step Forecasts

One-step Forecasts

Random Shock Computations

computation of the mean square error of the forecasts and prediction limits
Computation of the Mean Square Error of the Forecasts and Prediction Limits

Mean Square Error of the Forecasts

Prediction Limits