TS Module 15: Forecasting basics HW


TS Module 15: Forecasting basics HW

Author
Message
NEAS
Supreme Being
Supreme Being (5.7K reputation)Supreme Being (5.7K reputation)Supreme Being (5.7K reputation)Supreme Being (5.7K reputation)Supreme Being (5.7K reputation)Supreme Being (5.7K reputation)Supreme Being (5.7K reputation)Supreme Being (5.7K reputation)Supreme Being (5.7K reputation)

Group: Administrators
Posts: 4.2K, Visits: 1.2K

TS Module 15: Forecasting basics HW

 

(The attached PDF file has better formatting.)

 

Homework assignment: ARIMA(1,1,0) forecasts

 

An ARIMA(1,1,0) process has 40 observations yt, t = 1, 2, …, 40, with y40 = 60 and y39 = 50.

 

This time series is not stationary, but its first differences are a stationary AR(1) process.

 

The parameter è0 of the stationary AR(1) time series of first differences is 5.

 

The 1 period ahead forecast ŷ40(1) is 60.

 

We determine the 2 period ahead forecast ŷ40(2).

 


A.     What is the most recent value of the autoregressive model of first differences? Derive this value from the most recent two values of the ARIMA(1,1,0) process.

B.     What is the one period ahead forecast of the first differences? Derive this value from the the one period ahead forecast of the ARIMA(1,1,0) process.

C.    What is the parameter ö1 of the AR(1) process of first differences? Derive this parameter from the 1 period ahead forecast.

D.    What is the two periods ahead forecast of the AR(1) process of first differences? Use the parameter of the AR(1) process.

E.     What is the two periods ahead forecast of the ARIMA(1,1,0) process? Derive this from the two periods ahead forecast of the AR(1) process.


 

 

 

 


Attachments
Tom McNamara III
Forum Newbie
Forum Newbie (6 reputation)Forum Newbie (6 reputation)Forum Newbie (6 reputation)Forum Newbie (6 reputation)Forum Newbie (6 reputation)Forum Newbie (6 reputation)Forum Newbie (6 reputation)Forum Newbie (6 reputation)Forum Newbie (6 reputation)

Group: Forum Members
Posts: 6, Visits: 1

I am confused as to the relationship between a non-stationary ARIMA(1,1,0) [integrated autoregressive moving average (p,d,q)] and an ARMA.

1st off this question says the process is an ARIMA(1,1,0).  Is the zero necessary.  Isn't this just an ARI(1,1)?

[NEAS: Correct; ARI is an abbreviation for an ARIMA process with q = 0]

An ARIMA(p,d,q) process can be rewritten as am ARMA(p+1,q) process (Cryer Chan page 92).

[NEAS: No, and Cryer Chan do not say this. The first differences of an ARIMA process with d = 1 are an ARMA process of the same order.]

So how can this non-stationary ARIMA(1,1,0) be put into terms of an AR(1)?  Shouldn't it be an AR(2)?

[NEAS: The first differences of ARIMA(1,1,0) are an AR(1) process.]

Also, no where in the Module 15 reading does it mention 1st differences.  I have come across modules like this before where the reading does not match up with the homework assignments very well.  I came across this with the QQ plots in module 14.  I am growing frustrated with doing readings where the corresponding homeworks come out of left field.

[NEAS: Cryer Chan discuss first differences throughout their text, beginning in chapter 5.]

 

 



Tom McNamara III
sbenidt
Forum Newbie
Forum Newbie (2 reputation)Forum Newbie (2 reputation)Forum Newbie (2 reputation)Forum Newbie (2 reputation)Forum Newbie (2 reputation)Forum Newbie (2 reputation)Forum Newbie (2 reputation)Forum Newbie (2 reputation)Forum Newbie (2 reputation)

Group: Forum Members
Posts: 2, Visits: 1
Hi Tom,

First off, ARIMA(1,1,0) and ARI(1,1) are exactly the same process. Either notation should be acceptable to write in the course of a problem. The ARIMA(1,1,0) process can indeed be written in terms of a nonstationary AR(2) process. To achieve stationary, you would need to take first differences of the ARIMA(1,1,0) process which would result in an AR(1) process. The concept of first differences came up in an earlier module. I've found that writing up the explicit equation for both processes helps here in seeing how to derive the one and two period ahead forecasts.
Lidong Wang
Forum Newbie
Forum Newbie (1 reputation)Forum Newbie (1 reputation)Forum Newbie (1 reputation)Forum Newbie (1 reputation)Forum Newbie (1 reputation)Forum Newbie (1 reputation)Forum Newbie (1 reputation)Forum Newbie (1 reputation)Forum Newbie (1 reputation)

Group: Forum Members
Posts: 1, Visits: 1

Can NEAS can explain this problem?

[NEAS: Here is the outline of the solution. Many exam problems are similar. Once you get the intuition, the problems are straight-forward.

The first differences are an AR(1) model: Δyt = 5 + φ1 Δyt-1 + εt.

 


        Step 1: Determine the most recent value of the autoregressive model from the most recent two values of the original time series: Äy40 = y40 – y39 = 60 – 50 = 10

        Step 2: Convert forecasts of the time series to forecasts of the first differences for the one period ahead forecast. The forecast for period 41 is 60, so the forecasted first difference is 60 – 60 = 0.

        Step 3: Find the parameter ö1 from the 1 period ahead forecast.  Äy41 = 5 + ö1 × 10 = 0   ö1 = –0.5. è0 = 5 is not the mean of the AR(1) process; it is the constant term. The AR(1) process can be written two ways: Yt = è0 + ö1 × Yt-1 or Ytì = ö1 × (Yt-1ì). The textbook uses both formats, but è0 and ì are different values.

        Step 4: Solve for the two period ahead forecast from the autoregressive equation.  Äy42 = 5 – 0.5 × 0 = 5

        Step 5: Convert the forecast of the first differences for two periods to the initial time series.  y42 = y41 + 5 = 65


 

 

The textbook has formulas for forecasts and variances of ARIMA processes. 

 


 

        If you understand the intuition, the formulas are easy to recall and provide a good check on your work.

        If you do not understand the intuition, you will mix up the formulas for the various ARIMA processes.  Focus on the intuition.  After a few problems, it is easy.

 


 


Attachments
Woody
Forum Newbie
Forum Newbie (3 reputation)Forum Newbie (3 reputation)Forum Newbie (3 reputation)Forum Newbie (3 reputation)Forum Newbie (3 reputation)Forum Newbie (3 reputation)Forum Newbie (3 reputation)Forum Newbie (3 reputation)Forum Newbie (3 reputation)

Group: Forum Members
Posts: 3, Visits: 1



For forecasting, why don't you use equation 9.3.6 Δyt = 5 + φ1 [Δyt-1 - 5] .

Step 3 will be 0 = 5+ φ1 [10 - 5], φ1 = -1, not .5???

[NEAS: 5 is the constant term (theta-0), not the mean.]


Luke Grady
Junior Member
Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)

Group: Forum Members
Posts: 12, Visits: 1

So basically, the ARIMA(1,1,0) could have been any type of process - we don't care about what type of process it is because all we really know about is the type of process the first difference is. Is that correct?

[NEAS: An ARIMA(1,1,0) process is the integration (summation) of an AR(1) process. If the first difference is AR(1), the process is ARIMA(1,1,0).]


benjaminttp
Forum Newbie
Forum Newbie (7 reputation)Forum Newbie (7 reputation)Forum Newbie (7 reputation)Forum Newbie (7 reputation)Forum Newbie (7 reputation)Forum Newbie (7 reputation)Forum Newbie (7 reputation)Forum Newbie (7 reputation)Forum Newbie (7 reputation)

Group: Forum Members
Posts: 6, Visits: 1
The first differences are an AR(1) model: Δyt = 5 + φ1 Δyt-1 + εt-1

why the last term is et-1 but not et?
AJB1011
Forum Newbie
Forum Newbie (4 reputation)Forum Newbie (4 reputation)Forum Newbie (4 reputation)Forum Newbie (4 reputation)Forum Newbie (4 reputation)Forum Newbie (4 reputation)Forum Newbie (4 reputation)Forum Newbie (4 reputation)Forum Newbie (4 reputation)

Group: Forum Members
Posts: 3, Visits: 1
Can someone clarify how the first difference was derived? I think I'm getting confused with the subscripts.

It's given that the first difference is AR(1) --> Yt = 5 + φΔΥt-1 + et-1

Here's how I began to derive the first difference:
ΔΥt = Yt – Yt-1
ΔΥt = φΥt-1 + et – θet-1 – [φΥt-2 + et-1 – θet-2]


Also, for Part B: I believe the formula being used is 9.3.26 --> Y_hat(L) = Yt + (θ-zero)*L
Does "L" = 0 in this case, then? That way Y41 = Y40.



RayDHIII
Forum Member
Forum Member (43 reputation)Forum Member (43 reputation)Forum Member (43 reputation)Forum Member (43 reputation)Forum Member (43 reputation)Forum Member (43 reputation)Forum Member (43 reputation)Forum Member (43 reputation)Forum Member (43 reputation)

Group: Forum Members
Posts: 39, Visits: 138

AJB, pages 96-97 explain your dilema here.  Also, the AR(1) process is the first difference of the ARIMA(1,1,0) process, so that

ARIMA(1,1,0): unknown, and AR(1): deltaYt = 5 + (phi)deltaYt-1 + et-1.  Where the 5 is accommidating for a nonzero constant mean by theta = mu(1 - phi), also explained on page 97.

Part B wants to know the AR(1) one period ahead forecast, which is the ARIMA(1,1,0) one period ahead forecast (given) less the ARIMA(1,1,0) actual (given).

RDH


Luke Grady
Junior Member
Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)Junior Member (13 reputation)

Group: Forum Members
Posts: 12, Visits: 1

So is that what the d means in the ARIMA(p,d,q)? Does that just mean that the d-th difference of the time series is an ARMA(p,q) process?

Also, is a second difference the difference between Y_t and Y_{t-2}?

[NEAS: No, the second difference is the first difference of the first differences.]


GO
Merge Selected
Merge into selected topic...



Merge into merge target...



Merge into a specific topic ID...





Reading This Topic


Login
Existing Account
Email Address:


Password:


Social Logins

  • Login with twitter
  • Login with twitter
Select a Forum....











































































































































































































































Neas-Seminars

Search