Is this data really detrended? I have a time-series vector for a macroeconomic indicator, which clearly has trends, when I apply the detrend method of matlab, I see a change in the autocorrelation plot, but I suspect it's still not really decorrelated.
This is the original acf plot:

and this is the plot after detrend :

the autocorrelation definitely decreased for bigger lags, but still most of them are in the reject region, so I'm not sure if I'm doing something correct.
AFAIK, the detrend method computes the linear trend with simple least-squares fitting, and subtracts the trend from each data point. so maybe I need a more advanced detrending method. can I use PCA for this purpose, or any other alternatives?
In the end, I want to obtain a measure of how the value changed with respect to last time point (month), so I'm actually expecting big autocorrelation values in lags 12, 24 etc.
Thanks for any help!
Edit
The data I'm using is a time-series vector for a macroeconomic indicator, namely the consumer price index. It is monthly data which starts at January 1990. Here is the link for the data: http://pastebin.com/e31TahHj
 A: Autocorrelated series show exponential decay in ACF. The second ACF falls into this pattern pretty clearly. So, it may belong to AR(p) process $$x_t=c+\sum_{i=1}^p\theta_ix_{t-i}+e_t$$
On the other hand, the first ACF doesn't look like exponential decay. It could belong to many kind of processes, including MA(q) process, such as 
$$x_t=c+e_t+\sum_{i=1}^q\phi_ie_{t-i}$$
In fact, the first ACF may also be ARMA(p,q), you don't have PACF plot to get a better idea of the lag structure of the process.
Note, how I'm avoiding the discussion of time or other trends, because even without trends pure ARIMA(p,d,q) can generate both of your ACF plots. Hence, ACF plots are not the good tool to study trends. Actually, it's hard to tell the trend from autocorrelation from differences in short series in practice.
UPDATE
On an economic side, CPI is usually considered to be exponentially growing series. So, you better take the log of CPI before detrending it with MATLAB detrend function, because this guy is for linear trends only. Also, economists usually work with inflation rather than CPI, which is either simple or continuous growth rate of CPI. Unless you use CPI to deflate assets or output.
For instance, here's the CPI of energy cost in Canada. It looks exponentially growing, generally, but with periods of deflation. 

It looks nothing like a nice linear or even smooth exponential though. It's quite rough, which can be seen better on the plot with inflation/deflation that follows next.

