Forecasting

Question 1

What is the purpose of forecasting?

Answer 1

• given data X1,X2,…,Xn we want to:
• -predict a future value Xn+l which is l steps ahead with a forecast Xn(l)
• -have a confidence interval for Xn(l)
• -ensure that the forecast error is minimised

Question 2

Forecast Notation

Answer 2

Xn(l) is a forecast for the future value Xn+l given data X1,X2,…,Xn

Question 3

Forecast Error

Answer 3

en(l) = Xn+l - Xn(l)

Question 4

Minimum Mean Square Error

Answer 4

S = E[ (Xn+l - Xn(l))² ]

Question 5

Minimum Mean Square Error Forecast

Answer 5

Xn(l) = βl εn + βl+1 εn-1 + βl+2 εn-2 + …

Question 6

Minimum Mean Square Error Forecast

Steps

Answer 6

-write Xt as an MA model
-express Xn+l in this form
-suppose Xn(l) can be written as a linear combination of the innovations {εt} up to time n:
Xn(l) = βo’ εn + β1’ εn-1 + …
-calculate the expectation of the square error, recalling that E(X²)=Var(X) when E(X)=0
-minimise this to get the forecast

Question 7

Updating Forecasts

Description

Answer 7

• given data up to time n, we would predict Xn+l with Xn(l)

- one time step later when we now know Xn+1 we would predict Xn+l with Xn+1(l-1)

Question 8

Updating Forecasts

Relationship Between Forecasts

Answer 8

-can show that, with βo=1:

Xn+1(l-1) = Xn(l) + βl-1 [Xn+1 - Xn(l)]

Question 9

Estimating Residuals for AR(p) Processes

Answer 9

-for AR(p) processes:
Xt = Σ αk Xt-k
-the residuals satisfy:
et = Xt - Σ αk^ Xt-k, t≥p
-sums from k=1 to k=p

Question 10

Estimating Residuals for MA(q) Processes

Answer 10

-for an MA(q) process:
Xt = εt + Σ βk εt-k
-sum from k=1 to k=q
-the residuals are computed recursively:
e1 = X1
e2 = X2 - β1^ e1
en = Xn - Σβi^ ei
-sum from i=1 to i=n-1, n≥2

Question 11

Minimum Mean Square Error

Error as an MA Process

Answer 11

en(l) = βoεn+l + β1εn+l-1 + … + βl-1εn+1

• an MA(l-1) process
• this means that en(l) are uncorrelated

Question 12

Minimum Mean Square Error

E[en(l)]

Answer 12

-since E[εt]=0 and en(l) can be written as an MA(l-1) process in terms of εt it follows that:
E[en(l)] = 0

Question 13

Minimum Mean Square Error

Var[en(l)]

Answer 13

Var[en(l)] = σε² Σ βk²

-sum from k=0 to k=l-1

Question 14

Minimum Mean Square Error

Cov[en(l), εn-k]

Answer 14

Cov[en(l), εn-k] = 0

• for all k≥0
• the forecast error is uncorrelated with the forecast itself

Question 15

Minimum Mean Square Error

Cov[en(l), en(l+m)]

Answer 15

Cov[en(l), en(l+m)]
= σε² Σ βk βk+m
-sum from k=0 to k=l-1, valid for m>0

Question 16

Minimum Mean Square Error

Relationship Between Xn(1) and Xn for AR(1) Processes

Answer 16

Xn(1) = α Xn

Question 17

Minimum Mean Square Error

Relationship Between Xn(l) and Xn for AR(1) Processes

Answer 17

Xn(l) = α^l Xn

Question 18

Forecast Error as Conditional Mean

Description

Answer 18

-we can also think of the forecast Xn(l) as the mean of Xn+l given known values X1,…,Xn:
E[Xn+l | X1,…,Xn]

Question 19

Forecast Error as Conditional Mean

E[εt | X1,…,Xn]

Answer 19

E[εt | X1,…,Xn] = {0 for t>n, εt for t≤n

-future values are known and past values are known

Question 20

Forecast Error as Conditional Mean

E[Xn+l, X1,…,Xn]

Answer 20

E[Xn+l, X1,…,Xn] = Xn(l)

Question 21

Forecast Error as Conditional Mean

Var[Xn+l, X1,…,Xn]

Answer 21

Var[Xn+l, X1,…,Xn] = Var[en(l)]

Question 22

Forecast Error as Conditional Mean

Confidence Interval

Answer 22

-assuming Gaussian white noise, the approximate 95% prediction interval limits are:
Xn(l) ± 1.96 * √[βo²+β1²+…βl-1²)σε²]

Question 23

Forecast Error as Conditional Mean

E[Xt | X1,…,Xn]

Answer 23

E[Xt | X1,…,Xn] = Xn (t-n) if t>n or Xt if t≤n

Question 24

Model Based Forecasting

Description

Answer 24

-an alternative approach to forecasting is to consider the model for time t after the last observation

Question 25

Model Based Forecasting

Steps

Answer 25

-imagine having data X1,…,Xn for times t=1,…,n
-assume that a model has been fitted then Xn+1 can be written in the format of the model where the data Xn,Xn-1 etc are known, the parameters α, β etc. are estimated and the value of εt for t>n is not known but the variance σε² can be estimated from the data
-can calculate the conditional expectation and variance of Xn+l using the known data
-if Xn+l is normally distributed then the 95% confidence interval for the estimate is:
E[Xn+l] ± 1.96 σε²

Question 26

Model Based Forecasting

AR(2) - Xn+l

Answer 26

Xn+l = α1 Xn + α2 Xn-1 + εn+1

Question 27

Model Based Forecasting

AR(2) - E[Xn+1]

Answer 27

E[Xn+1 | Xn,Xn-1] = α1 Xn + α2 Xn-1

Question 28

Model Based Forecasting

AR(2) - Var[Xn+l]

Answer 28

Var[Xn+l] = σε²

Question 29

Model Based Forecasting

AR(2) - CI

Answer 29

α1^ Xn + α2^ Xn-1 ± 1.96 σε^²

Question 30

Model Based Forecasting

AR(1) - E

Answer 30

E[Xn+1] = α Xn
E[Xn+2] = α² Xn
…
E[Xn+l] = α^l Xn

Question 31

Model Based Forecasting

AR(1) - Var

Answer 31

Var[Xn+1] = σε²
Var[Xn+2] = (1+α²) σε²
…
Var[Xn+l] = (1+α²+…+α^(2(l-1))) σε²