How to interpret mean of this estimated AR(1) process

Question

I estimated an AR(1) process, my data looks like this:

Making usual unit root test, they suggest that an estimated AR(1) from this data is stationary. Estimating the AR(1) over this data, these are the results:

z test of coefficients:

           Estimate Std. Error z value  Pr(>|z|)    
ar1        0.728652   0.085601  8.5121 < 2.2e-16 ***
intercept 20.176618   0.809543 24.9235 < 2.2e-16 ***

The purpose of this estimation is to get the stationary mean of the process, i.e.:

$x_t=c+\phi x_{t-1}+\varepsilon_t$

$E\{x_t\}=\frac{c}{1-\phi}$

With the estimated values this turns out to be, given the significance of estimators:

$E\{x_t\}=\frac{20.176618}{1-0.728652}\approx74.3569$

Which clearly is outside the range the values data take, and is much larger than expected. What is wrong? Maybe I misinterpreted the unit root tests or something similar?

Answer 1

I made the same mistake several years ago. When in doubt read the documentation or do a simulation. Here is a simulation

T <- 10000
y <- rep(0,T)
c <- 5.6
phi <- 0.72
y[1] <- rnorm(1) + c/(1-phi)
for (t in 2:T)
    {
        y[t] <- phi*y[t-1] + c + rnorm(1) 
    }

arima(y,order=c(1,0,0))

for which arima call return 20 as intercept

Coefficients: ar1 intercept 0.7213 20.0361 s.e. 0.0069 0.0358

so the intercept reported is

$$\mu = \frac{c}{1-\phi}$$