1.      What is the mean ì of the ARMA model of first differences?

2.      What is è₁ of the ARMA model of first differences?

3.      What is the forecasted value of the ARIMA(0,1,1) process for Period 51?

 

Show the derivations of the parameters and the forecast.

 

 

 

When working out the 2 equations I am getting the mean equal to -.2 and Theta equal to .3 .  Am I even close or my equations completely wrong?  Thanks

To my knowledge, you're equations are completely wrong.  I went about this problem by setting up a table:

Period            ARIMA(0,1,1)                  ARMA(0,1)

           Forcast  Actual    Residual     Forcast       Actual

48     Y-hat₄₇(1)   Y₄₈        e₄₈           --              --

49     Y-hat₄₈(1)   Y₄₉        e₄₉   deltaY-hat₄₈(1)  deltaY₄₉

50     Y-hat₄₉(1)   Y₅₀        e₅₀   deltaY-hat₄₉(1)  deltaY₅₀

51     Y-hat₅₀(1)    --         --    deltaY-hat₅₀(1)    --

 

Note that the residuals are shared between the two processes.  The assignement is fairly straighforward from there.  Let me know if you have any further questions.

RDH

Question for NEAS:  Since the homework specifies that these are "one period ahead" forecasts, does this mean that for calculating the forecasts for the MA(1) process we are suppose to take the forecast in period t minus the actual in period t-1?  For example, deltaY(49)-hat = 72 - 71.5 = .5, or is it 72 - 70.5 = 1.5?

[NEAS: Don't mix periods. One period ahead forecast means we forecast Period T from the values in Period T-1. The residual uses the forecast and the actual values for the same period.]

I don't understand what you mean by "don't mix periods."  You say "One period ahead forecast means we forecast Period T from the values in Period T-1" - All I was asking is if the value in Period T-1 is the actual or forcasted value for that period.

I was asking about the first difference forecasts.  According to the "TS Module 16: ARIMA forecasting: values residuals forecasts" handout on the last line of the first page in the pdf file, we would calculate deltaY(49)-hat = 72 - 71.5 = .5. Right?

I'm getting dizzy reading the last post.  =P

The forecast for the ARMA(0,1) of period 49 uses the forecast of the ARIMA(0,1,1) period 49 (which is calculated from period 48) less the actual ARIMA of period 48.  So, indeed, your deltaY-hat₄₈(1) = 72 - 71.5 = 0.5.

"Don't mix periods" means what it sounds like:  If you're attempting to find forecasts for one period ahead, say, t=2, then only use values derived from the preceeding time period, t=1.  Let me know if you have further questions.

RDH

I guess I don't understand what I'm doing here.  I can't seem to see how the residuals, forecasts, and actual values that I derived fit into either equation 9.3.30 or 9.3.32.  I guess I'm confused because we're supposed to find mu and theta_1, but neither of these equations contains both variables.  And how do we solve this without know phi?

Can somebody steer me right?

We are given the equation (9.3.21) since ARIMA(0,1,1)'s first difference is MA(1):

deltaY-hat_t(1) = mu - theta(e_t)

We know two of the ARMA forecasts and two of the residuals for t = 48 and 49.  This will produce two equations with two unknowns, simple algebra to solve.

RDH

This is very important module, I am getting mean = 2, theta = 1.5, ARIMA forecast for period 51 = 74.1.

Did anyone get the same thing?

That's what I got.

I got what you got too, but I am still a little unsure about that.
An ARMA model:
W(t)=mu+e(t)-theta*e(t-1), is right?
I am not sure if we use W(t)=y(t)-y(t-1), or W(t)=y-hat(t)-y-hat(t-1).
Here, I used W(t)=y(t)-y(t-1) to solve mu and theta.

but, in the moduel 18, this problem appears again, I am a little confused.