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.
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.
