I am unable to improve the (MAProcess) time series model

Question

I am a time series novice and have been learning the art/trade from examples while being governed by the NIST handbook (ch.6, sec.4).

My question is regarding:

• How to improve the estimated time series process? Currently the ACF of the residuals of my estimated process is not entirely Normally-distributed noise.
• Is this a time series problem at all? Should I use a distribution fit instead?

I have a dataset (x) fluctuating between discrete values 1 - 5. The raw data and the "differenced" data are shown. The CSV data is available via this Pastebin link. It is comma-separated numeric data.

I converted the data to TemporalData and performed a UnitRootTest.

xtd = TemporalData[Differences[#], {Length[Differences[#]]}] & /@ x;

UnitRootTest[xtd[[1]], 1, "HypothesisTestData", 
 SignificanceLevel -> .05]

(*The null hypothesis that the model of order 1 without a constant offset and no deterministic trend contains a unit root is rejected at the 5. percent level based on the Dickey-Fuller F test.*)

The differenced data is stationary, as indicated by the small p-value from the UnitRootTest and the rejection of the $H_0$.

I used the CorrelationFunction to visualize the autocorrelation of the resulting time series. There is one spike (negative) at the first lag and then a rapid decline to within the 95% confidence interval.

conflevel = 0.95;
samplesize = xtd[[1]]["PathLengths"][[1]];
{minconf, maxconf} = 
 Quantile[NormalDistribution[], {(1 - conflevel)/2, 
   1 - (1 - conflevel)/2}]/Sqrt[samplesize]

ListPlot[N@CorrelationFunction[xtd[[1]]["PathStates"], {100}], 
 PlotRange -> {All, {-1, 1.1}}, Filling -> Axis,
 GridLines -> {None, {-0.05, 0.05}}, 
 PlotStyle -> {Red, PointSize[0.015]}]

Based on my limited knowledge, it would appear that I may need an MAProcess with one parameter since I have a spike at lag 1 along with a quick diminishing of the ACF to within the confidence interval (See Box-Jenkins identification from the NIST Handbook).

I estimate the MAProcess parameters:

est = EstimatedProcess[xtd[[1]]["PathStates"], MAProcess[{a}, v]]

(*MAProcess[{-1.}, 0.78754]*)

Then I compute the residuals of the estimated process with respect to the original data hoping that the residuals would be normally distributed noise (their ACF would be within their 95% confidence interval in their ACF plot)... but the residuals do not stay within the confidence intervals!

So I assume that my estimate is not a good fit.

residuals = 
  Rest[xtd["PathStates"] - 
    KalmanFilter[est, xtd[[1]]]["PathStates"]];

clev = 0.95;
samplesize = 
  TemporalData[residuals, {Length@ residuals}]["PathLengths"][[1]];
{minconf, maxconf} = 
 Quantile[NormalDistribution[], {(1 - clev)/2, 
   1 - (1 - clev)/2}]/Sqrt[samplesize]

DiscretePlot[CorrelationFunction[residuals, n], {n, 0, 10}, 
 ExtentSize -> 1/2, PlotRange -> {All, {-1, 1.1}}, PlotStyle -> Black,
  Frame -> True, FrameStyle -> Black, FrameLabel -> {"lag", "ACF"}, 
 GridLines -> {None, {-0.05, 0.05}}]

• What parameters should I add to my MAProcess? I have tried an ARMAProcess as suggested by TimeSeriesModelFit but that has a worse residual ACF.
• Is this a timeseries problem?