Manually replicating an ARIMA forecast

Question

I hate to ask this question but I am going insane and other links haven't solved this problem. I have a seasonal ARIMA with just over two years of weekly data ARIMA(0,1,1)(0,1,1)[52]. It's highly seasonal and this model fits well enough for initial forecasting. I'm using Rob Hyndman's forecast package and the Arima and forecast functions within it. However, I am providing the forecasts and the forecasting equation to a client. So far, so good.

However, when I write out the forecasting equation I get numbers close but not equal to the forecast. I have double-checked my equation (here) and cannot figure out what I'm doing wrong. Here are the numbers:

    Y_20_49 <- 791.7044   # One period back
    Y_19_50 <- 516.0694   # One year ago
    Y_19_49 <- 812.9433   # One year and one period ago

    # Residuals acquired with sweep::augment(); 
    # same as residuals(model)
    resid_20_49 <- 1.048402e+01  # Residual one period ago
    resid_19_50 <- -1.865834e+02 # Residual one year ago
    resid_19_49 <- 2.784324e+01  # Residual one year and a period ago.

    ma1  <- -0.8761              # MA1 coefficient from ARIMA
    sma1 <- 1                    # SMA1 coef from ARIMA

    # Manual = 274.6686
    Manual <- Y_20_49 + Y_19_50 - Y_19_49 + 
      ma1 * resid_20_49 + sma1 * resid_19_50 +
      (ma1 * sma1) * resid_19_49
    Actual <- 334.4015           # Result from forecast(h = 1)    

There's something obvious I'm missing but I'd appreciate any help. This model is not transformed but I have the same problem with another similar model that is logged.

UPDATE: The model results report only MA and SMA terms; no constants. Here is a picture of the summary(model) results:

UPDATE2: The manual calculations seem to work when the MA and SMA terms are not close to one (even around abs(0.88) seems OK). I cannot be certain if this is the case but this happened during my quest to reproduce the forecasts.

Answer 1

For example a seasonal model of 12 looks like this .

Check your signs for the ma terms ... you have all positives ..... should be 2 positives and 1 negative .

I used AUTOBOX which has an option to present the model And the forecast in simple terms ( as you are trying to do ) . I used weekly data for 177 periods and HARDWIRED i.e specified a starting model and disabled estimation and/or model revision and obtained the following

I then captured the forecast coming from the model and obtained this result

I then selected this output option to present the model as a pure right-handside ...called rhside.txt presented here

If you wish to contact me offline I would be glad to share the data ….and the residuals from the model so that you could use them to investigate your enigma .

I don't really know how to attach a csv file to this answer and if I knew I would be happy to do just that.

Note that AUTOBOX presents the coefficients in the standard/classical way ….not in the negative as your software does.