AR(1) coefficients are too high

Question

I am fitting an AR(1) model and the results I'm getting are too high and don't make sense. Perhaps I'm misunderstanding something, can someone clear this up?

data <- c(0.1287426, 0.1447894, 0.1541330, 0.1481999, 0.1348838, 0.1165681, 0.1140563, 0.1179652, 0.1404826, 0.1418325)
model <- arima(data, c(1,0,0))
model$coef

AR1 = .5582418
Intercept = .1343918

This seems to be saying that the model is

$$ X_t = 0.134 + .558X_{t-1} $$

which fits values closer to .2, much higher than any values in my series. What am I doing wrong here?

EDIT: My residuals are shown below

-0.004686990  0.013551235  0.013936841  0.002787753 -0.007216242 -0.018098339 -0.010385551 -0.005074459  0.015260830  0.004040577

Answer 1

They way arima function works in R can be confusing. The resulting model is not the one that you stated. It's actually $$(X_t-0.134) = 0.558(X_{t−1} - 0.134) + e_t$$

So your prediction of the next value is

$0.1343918 + 0.5582418(0.1418325 - 0.1343918) = 0.1385455$

which is equal to what you get from

predict(model)$pred

Answer 2

Well, maybe nothing. You have shown what the intercept and coefficient for the first term are, but you didn't show what the fit was. You've asked ARIMA to generate an output - and it's generated it. It doesn't mean it's a good fit. Maybe an AR1 model isn't the right choice.