# Covariance non-zero mean AR(1)

Why when I compute the autocovariance function of a non-zero mean AR(1), X(t)-u=Φ(X(t-1)-u)+ε the presence of the mean does not change my result and so the formula should be the same of a zero-mean AR(1)? I've tried to do that, but i obtain a extra-term (the yellow one), so there is something wrong!

• Could you show where the "$Variance(X_{t-1})$ term comes from? I believe that in introducing it you have double-counted a term equal to $-\phi u^2.$ – whuber Dec 19 '18 at 17:28
• @whuber The variance of X(t-1) is the result of the expectation of X(t-1) squared that I can write as variance – White Noise Dec 19 '18 at 17:34
• Haven't you assumed $E[X_{t-1}]=u$?? Unless $u=0,$ the expectation of $X_{t-1}^2$ is not its variance. – whuber Dec 19 '18 at 17:37
• @whuber You re perfectly right. And so, what does X(t-1) squared stand for? – White Noise Dec 19 '18 at 17:41