I assume that your data shows a lot of random variation, so that there is no clear trend when you just plot the data. First, you have to check for seasonal fluctuations (monthly, quarterly). Do the time series look better when you have removed these fluctuations? If not, try to fit a time series model based on well-known principles (Random Walk with Drift or any other simple model) and check whether trend parameters are significantly different between the two data sets.

But remember, it may very well be the case that there is no systematic difference between the two time series.

Dear Nico, Thank you so much for the response. The data does not show any seasonal trend, As you said the fluctuations are very random. We fitted ARIMA models for both the areas separately. The parameters were different between the two areas, but we do not know how to test for statistically significant difference between the two models. We tried interrupted simple linear regression and the area B shows a greater negative slope than that of area A. But we do not know how to conclude a difference by using time series models. Can you please explain on that? Any article or resource on that?

You want to test whether slope estimate A (SA) is significantly larger (less negative) than slope estimate B (BA). Test whether (SA-SB) is significantly larger than 0, by computing the variance of (SA-SB). This variance equals the sum of squared standard errors of SA and SB. Here you have ignored the covariance between the two regions. This is reasonable as long as you have reasons to assume that the two regions are independent. When this is not a reasonable assumption, for instance because the same health care facility provides services to both regions, you have to estimate a two-dimensional time series model for the two regions jointly (VAR), and you will obtain estimates of the covariance between your SA and SB, in addition to the usual standard errors.    

1) I think you need to transform your data first since it has a bounded domain (we are talking about percentages here, right?)

2) let x[1],x[2],...,x[70] be data from series A, and y[1],y[2],...,y[70] be data from series B. Now, once you've found a model for A and for B, you can find trend estimates for x and for y. Let us denote them as x^[t] and y^[t] for each t from 1 to 70.

Once we have x^ and y^, we can evaluate how steep the trend was by finding the following coefs:

coefA =  x^[70] / x^[1]

coefB =  y^[70] / y^[1]

If coefA>coefB, you have a greater decline for area A.

But please note that x^ and y^ are only estimates, so refer to CI for ratios if you want to testt hypotheses.

Dear Nico, I understand you point completely. But ARIMA models do not give a single slope if I am right. If it was a linear or poisson regression then we can compare the slopes as you say. But I am completely unaware of the multidimensional time series. I have attached my dataset below. Also, what software does that?? 

Dear Andrey, Thanks a lot for your response. I do not understand what you mean by trend estimates for each datapoint??

Dataset for the neonatal mortality rate.

Dear Andrey, Yes Yes, I got your point x^, y^ are the predicted values from the model. I will try that and present the results. Thanks a lot. But I fell that it seems a bit oversimplification because we are ignoring all the points in between the 1 and 70th values. please advise.

When you have ARIMA model, you have some estimates of trend for each of your points. Of course, if there are some local trends, they can be analysed additionally.

Another approach is to use ARIMA with drift. Then estimates of the drift parameter will indicate the level of decline.

But first I would suggest applying Mann-Kendall test to see if there's any type of decline at all.

I think the best way is just take the fist difference of your series. And then compare the means of those first differences.

It will be just an average decrease per single period.

P.S. This will be equivalent of some estimate of average first-order derivative of trend, but much easier to present to readers.

To keep it simple for your readers, I would suggest the following test for the average decline

Let x -series for area A, y - series for area B

1) Compute

xd[t] = x[t]-x[t-1]

yd[t] = y[t]-y[t-1]

diff[t] = xd[t]-yd[t]

2) Test diff[t] against zero using standard statistical tests (take care of distributions). Or just model diff[t] to see if it was below/above zero during the period of interest.

Dear Andrey, If I am not mistaken is this also called a difference in difference technique?

I would keep it without noting the DID technique, as it is not quite the same.

You can apply ARIMA to diff[t].

But I think you do not have pronounced trends in your data.

You need to apply the Mann-Kendall test first. See this question:

https://www.researchgate.net/post/Statistical_advice_on_analyzing_the_trend_of_cases_over_time/1

P.S.

Applying ARMA with constant to dx[t] or dy[t]  will show you periods with pronounced local trends (if they exist).

 Dear Andrey, Results of MannKendall test 

Group A tau = -0.673, 2-sided pvalue =0.0012178

GRoup B tau = -0.1, 2-sided pvalue =0.39835

There is a strong evidence for group A for the presence of trend. (You can also try 1-sided test for stronger result).

There is no evidence that you have a trend for group B.

So, now how to compare these two with a single comparative test?

Do you mean to compare the level of decline?

Yes.

Try this:

1) Compute

xd[t] = x[t]-x[t-1]

yd[t] = y[t]-y[t-1]

diff[t] = xd[t]-yd[t]

2) Test diff[t] against zero using standard statistical tests (take care of distributions). 

Ok will do this and upload results.