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NEAS
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TS Module 18: Forecast updates and weights HW (The attached PDF file has better formatting.) Homework assignment: ARIMA(0,1,1) forecasts An ARIMA(0,1,1) model for a time series of 100 observations, yt, t = 1, 2, …, 100, has θ1 = 0.4.
The forecast of the next observation, y101, is 25. The actual value of y101 is 26. The forecast of the next observation, y102, is 26. The actual value of y102 is 26.
We continue to use the same ARIMA model. That is, we don’t re-estimate the parameters with the additional data. We forecast y103, the ARIMA value in the next period.
A. From the actual and forecasted values of y101, derive the residual for the ARMA model of the first differences. B. From the actual value of y101 and the forecasted value of y102, derive the forecasted value for Period 102 for the ARMA model of the first differences. C. This forecasted value for Period 102 is a function of ì, è1, and the residual for Period 101. Derive the ì (mean) of the ARMA model of first differences. D. From the actual and forecasted values of y102, derive the residual for the ARMA model of the first differences for Period 102. E. Using this residual, determine the forecasted first difference for the next period. F. From the forecasted first difference, derive the forecasted value of the original time series.
The values of ì and θ1 are the coefficients of the ARMA process for the first differences. (Cryer and Chan use è for an MA(1) process, not è1.)
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Luke Grady
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For C did anyone get mean = 0.4?
I'm not sure what the question is for part F - is it asking for the forecasted value of the first difference of period 101? Is that the "original time series"?
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RayDHIII
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Luke, I got your answer for part C. Part F "From the forecasted first difference, derive the forecasted value of the original series." Original = ARIMA, First Differences = ARMA. I can't recall where my formula came from, but it is logical and reads: the forecast of ARIMA period 103 is the sum of the actual ARIMA period 102 and one-period ahead forecast ARMA period 102. In formula form: Y-hat102(1) = Y102 + deltaY-hat102(1). Hope it helps! RDH
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mchonejd
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shouldn't the mean be -.4. residual = mu + (theta)(et) = 0 = mu + (.4)(1), so mu = -.4? [NEAS: The coefficient is -theta, not theta.]
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flexi127
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Luke, I believe parts E and F are just breaking down the forecast into steps. Part E is asking for the first difference, which would just be mu - theta1 * residual102. Part F is then asking for the forecast of y103, which would be this difference plus the actual value for y102. Look at Question 1.4 in "Forecast updates intuition" to see similar wording.
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letsfinish
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So part A I'm comfortable with, it grades down from there... A. Error = Actual - Expected = 26-25 =1 (this one I'm comfortable with) B. "Derive the forecasted value for period 102 for the ARMA model of the first differences" Ok, so for this one I used equation 9.6.1 on page 207, from that I infered Yhat 101 (1) = Yhat101(2)+ theta ( Y101 - YHat101) = 26 + .4( 26-25) = 26.4 From this, I think I should subtract the original estimate Yhat101 (2) = 26, to get 26.4-26=.4 C. Ok so for this part, I found an equation in exercise 1.1 of this module 18, that says theta= mu*(1- phi) . However I only know theta, so I'm not sure how I would solve for this, unless I'm also suppossed to use the values for 101. Is this even the right approach? D. E102 = Y102-Yhat102= 26-26 =0 E. Should this use equation 9.6.1 again? F. ??? No idea Any help would be greatly appreciated 
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John R
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In order: A) I agree B) What is the forecasted first difference? What is the expected difference between y102 and actual y101, based on the actual value for y101 and the projected value for y102? Should be simple subtraction. C) Using the answers to questions A and B (this gives you the residual and the forecast), find the mean. Use the formula Delta Y(hat) = mu - theta*residual. Solve for mu. D) I agree E) Same as C, except you are solving for Delta Y(hat), and you know mu, theta, and the residual F) E gives you the expected difference between y103 and y102. You know y102 already. What do you expect the answer to be for y103?
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moo5003
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I had a question I was hoping someone could clarify: In the prompt when it tells us the forecast of the next observation, y_102, is 26. Are they referring to: y_100(2) or y_101(1). The later comments lead me to believe that this is y_100(2). My follow up question is then directed for part D: Do we need to use y_100(2) to calculate y_101(1) using 9.6.1 so that we may then calculate e_102 as: y_102 - y_101(1)? I guess I'm just unsure if we can say e_102 = y_102 - y_100(2). Thank you for any assistance. The answers I got are as follows, let me know if you guys differ:
A) 1
B) 0
C) .4
D) 0
E) .4
F) 26.4
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palantathraiel
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At first I thought the forecast of the observation for t=102 is y_100(2), but when I read in Part (C) that "This forecasted value for Period 102 is a function of mu, theta1, and the residual for Period 101," it made me feel pretty sure that it should be y_100(1) instead. Because if it's y_100(2), it would be a function of just mu (remember from Module 16? When l > 1, y_t(l) = mu). Moreover, if it's y_100(2), then the answer for (C) should be 0, and not 0.4.
So in (D), I used the ff. equation:
e_102 = y_102 - y_101(1)
Anyway, I got the same answers that you did for all parts. And essentially, I used the same method that was used for the Module 16 HW.
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scomurphy
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I would like to make sure that I am understanding that equations 9.7.4, 9.7.5, and 9.7.6 from the reading are specifically for IMA(1,1) models, is this correct?
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