R Arima order(1,0,0) forecast not giving what expected

Question

My understanding of a simple ARIMA(1,0,0) equation is: $$y_t=\phi_1y_{t-1}+c+\varepsilon_t$$

Yet when I build an ARIMA(1,0,0) model on the consumption column of the usconsumption dataset in R, the forecast is not what I expect.

    > library(forecast)
    > library(fpp)
    > data(usconsumption)
    > tail(usconsumption[,1], 1)
    [1] 0.8753521
    > ar1 <- Arima(usconsumption[,1], order=c(1,0,0))
    > forecast(ar1, 1)
            Point Forecast       Lo 80    Hi 80      Lo 95   Hi 95
    2011 Q1      0.7979218 -0.03481309 1.630657 -0.4756363 2.07148
    > summary(ar1)
    Series: usconsumption[, 1] 
    ARIMA(1,0,0) with non-zero mean 

    Coefficients:
             ar1  intercept
          0.3553     0.7552
    s.e.  0.0726     0.0780

    sigma^2 estimated as 0.4222:  log likelihood=-161.06
    AIC=328.13   AICc=328.28   BIC=337.43

So using the forecast function gives 0.7979 as the forecast, but using the equation would give 0.8753521*0.3553+0.7552=1.066213. I'm not understanding the discrepency.

Answer 1

There are multiple (equivalent) parametrizations for ARIMA models. There's the one you quote (sometimes called the ARMAX parametrization):

$y_t = \phi y_{t-1} + c + \epsilon_t$

And the equivalent regression with ARMA errors parametrization:

$(y_t - c') = \phi (y_{t-1} - c') + \epsilon_t$

The forecast package uses the second parametrization, so that the forecast is:

$\hat{y}_{t+1} = \hat{c}' + \hat{\phi} (y_t - \hat{c}') = 0.7552 + 0.3553 * (0.8753521 - 0.7552) = 0.7979$

The package author explains the difference between the two parametrizations and the rationale for choosing this one on his blog. Mostly, when adding other regressors (other than the constant), this version is easier to interpret. Since the function forecast::Arima allows for other regressors, it makes sense to treat them all in the same way.

Answer 2

 y(t+1)=[phi]*y(t) + [1-phi]*C
        .3557*.87535 + .6443*.7552
       = .313 +          .485
       =.7979

The constant that is reported is the Left hand side constant in the model

$[Y(T)-C] = A(T)/[1- \phi * B]$

$[1- B]*Y(T)-[1-\phi * B]*C = A(T)$

or

$Y(T)=\phi*Y(T-1)+ (1-\phi]*C +A(T)$