It is possible to get a general formula for stationary ARMA(p,q) autocovariance function. Suppose $X_t$ is a (zero mean) stationary solution of an ARMA(p,q) equation:
$$\phi(B)X_t=\theta(B)Z_t$$
Multiply this equation by $X_{t-h}$, $h>q$, take expectations and you will get
$$r(h)-\phi_1r(h-1)-...-\phi_pr(h-p)=0$$
This is a recursive equation, which has a general solution. If all the roots $\lambda_i$ of polynomial $\phi(z)=1-\phi_1z-...-\phi_pz^p$ are different,
$$r(h)=\sum_{i=1}^pC_i\lambda_i^{-h}$$
where $C_i$ are constants which can be derived from the initial conditions. Since $|\lambda_i|>1$ to ensure stationarity it is very clear why the autocorrelation function (which is autocovariance function scaled by a constant) is decaying rapidly (if $\lambda_i$ are not close to one).
I've covered the case of unique real roots of the polynomial $\phi(z)$, all other cases are covered in general theory, but formulas are a bit messier. Nevertheless the terms $\lambda^{-h}$ remain.
Answers to question 2 and 3 more or less follow from this formula. For $AR(1)$ process $r(h)=c\phi_1^h$ and when $\phi_1$ is close to one, i.e. close to non-stationarity, you get the behaviour you describle. The same goes for general formula, if the process is nearly unit-root one of the roots $\lambda_i$ is close to 1 and it dominates other terms, producing the slow decay.
Should I consider these lags for my analysis?
Would the scope of your question of research be changed? If yes, I would say that dropping out these lags is a bad idea.
I am using MATLAB to plot the PACF, would there be any bugs?
It is almost sure that not.
How can the correlation be >+1 or <-1.
Because these values are very likely to come from a regression of your series over its ~550 (simultaneously considered) lagged versions. To me the reason why you get greater-than $1$ (absolute) values is either because:
- your data are not time series, e.g. spatial series with overlapping autocorrelation structures
- your data are temporal but with redundant values. (which is another manner to say that you have overlapping autocorrelation structure).
Did you check your data ? Are you sure that you have no duplicate entries ?
Another possibility is that your series is over/under-differenced. You may want to take a look at
Identifying the order of differencing
Best Answer
this acf suggests non-stationarity which might be remedied by incorporating a daily effect as it appears to evidence structure at lag 24. The daily effect could be either auto-regressive of order 24 or it might be deterministic where 23 hourly dummies might be needed. You could try either of these and assess the results. Further structure appears to be needed. This could be either the need to include level shifts or some form of short-term auto-regressive structure like a differncing operator of lag 1. After identifying and estimating a useful mode, the residuals might suggest further action (model augmentation)to ensure that the signal has fully extracted all information and rendered a noise process that is normal or Gaussian. This will then answer your vague question regarding "stability". Hope this helps !
A slight addition !
The word "suggests" is used as the acf is not the final word on this while the actual data is. In the absence of the actual data the acf is sometimes useful in characterizing the process.