Unusual few large spikes at pacf of arima residual model I am using shampoo sales dataset which can be obtained from github. I fit the dataset using ARIMA$(5,1,0)$ and plot its residuals. The following are residual plots, acf and pacf. 




Question How to interpret the few large spikes at pacf plot? 

I notice that the acf plot seems quite nice, that is, the residuals are uncorrelated, which is desirable. 
But I cannot make sense of the few large spikes at pacf. It would be good if someone can explain to me the existence of those few large spikes.
 A: The residuals from your model  are not random as one can "see" a change in the mean possibly at year 2 period 4 effectively identifying the need for deterministic structure of some form . This data set requires a hybrid model containing memory (ARIMA) and deterministic structure. I will present the "logic" in identifying this model. Following this discussion Seeking certain type of ARIMA explanation and this paradigm https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf and schemes to identify deterministic structure here http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html. Your data has been studied here https://www.kaggle.com/dromosys/shampoo-sales and here https://books.google.com/books?id=S4wwDwAAQBAJ&pg=PA367&lpg=PA367&dq=shampoo+data+set&source=bl&ots=bV2ddJq_Hk&sig=ACfU3U3wPaDdTODGvGsYo-mkTxphnM-zxA&hl=en&sa=X&ved=2ahUKEwj18Ni2nMjjAhXvs1kKHcwcCq4Q6AEwCXoECAoQAQ#v=onepage&q=shampoo%20data%20set&f=false and here https://machinelearningmastery.com/persistence-time-series-forecasting-with-python/.
Here is a set of residuals of a useful model suggesting not only memory but determinstic structure  with an ACF here  with Actual/Fit and Forecast here  . Two very obvious visual trends in the data trends in the data periods 1-15 (1/1 - 2/3) and periods 16-36 (2/4 -3/12)are discovered/identified and incorporated into the model.
Model identification is not simply done by fitting a trial set of models but rather in an iterative self-checking way validating the Gaussian assumptions of randomness. The initial data's ACF and PACF is here  and here  suggesting some sort of possibly complicated model.
An initial model ( a trial balloon ! ) of (1,0,0) results in  with a residual ACF & PACF here  and residual plot (trending) here 
Following Tsay's suggested approach we identify  the need for two time trends ( one starting at year 2 period 4 )and one pulse indicator suggesting the need to add three dummy series to our model 
Final analysis suggests this hybrid model  with the following statistics  and ACF/PACF here
You say your residuals are uncorrelated , I say the variance of your residuals is bloated by a mean shift ( year 2 period 4)  leading to a downwards estimate of the acf .. This is commonly referred to as the "Alice in Wonderland Effect" How to determine order of sarima? .  While the series is non-stationary, differencing is the wrong remedy in this case.
Finally the unusual spikes at the end are a resultant of estimating toooooo many ACF VALUES . You have 36 observations and you fit a model that implied that the first 6 errors are zero (5,1,0) thus you effectively have 30 non-zero residuals. Estimating auto-correlations up to lag 34 is not suggested thus numerical non-sense. 
Note that a level shift in a first difference model is functionally a time trend in the undifferenced data. The impact of visual hint (level shift/mean shift in the OP's residuals) in the first paragraph of this response is now clearer. 
