I am with you, I get the theory of this stuff but when it comes to application, I get a little fuzzy.
I used as my e^t = actual (delta)yt - predicted (delta)yt, for the first differenced equation. Basically it is actual minus predicted. I had a little trouble calculating e^(t-1) which we need for the MA component of the model. In other words, I need to calculate this in order to get the predicted (delta)yt. What I did was use e^(t-1) = actual (delta)yt-1 - mean. I used the calculated mean of my model. This was just to get the intial value of e^(t-1). Then for all subsequant values, I used actual minus predicted (delta)yt-1 values.
I hope this makes sense. I am really looking for someone to give me an idea of what they did for this. For people whose models have an MA component, how did you calculate the e^(t-1) value to get your predicted values.
Jacob: How do we get the estimated values for an autoregressive model?
Rachel: Suppose the order of the autoregressive process is p. For an AR(1) model, p = 1; for an AR(2) model, p = 2.
For t > 2, yt is estimated from the ARIMA model.
For t = 2, we assume y0 = the mean of the ARIMA model.
For t = 1, we assume y0 and y–1 = the mean of the ARIMA model.
Jacob: If the ARIMA model has a moving average component, how do we determine the estimated values?
The estimate for y1 requires knowledge of ε0, which we don’t know.
The estimate for yt requires knowledge of εt-1, which we don’t know, since we don’t know the estimated value of yt-1.
Unless we have a way of starting, we don’t know the estimates for any values.
Rachel: We assume the residuals for all values before the first observed value are zero.
Jacob: The candidate who posted this message used the actual minus the mean as the residual. Is this wrong?
Rachel: This is not wrong; we don’t know the true residual for the first period. This candidate’s method is as good as any, though the textbook uses the method described above.