Time series with 24 yearly data points - advice needed I have a dataset containing the prevalence rate of Malaria in Botswana, starting in 1990 and ending in 2014. My task is to verify whether these data can be used in order to make predictions on the future Malaria prevalance rate. I know that 24 data points is probably not enough, but I decided to give it a try.
#plot data
Mal.TS <- ts(Mal$Value, start=1990, end=2014, freq=1) 
plot.ts(Mal.TS)


Now it would have been nice if there was a completely increasing/decreasing trend, but unfortunately the trend was first increasing and later decreasing.
#Test for stationarity
adf.test(Mal.TS)


Because of the first increasing and later decreasing trend and the absence of variance as there is only one data point per year in a limited dataset, the Dickey-Fuller test suggests that the data are stationary.
#Test AFC and PACF
acf(Mal.TS)
pacf(Mal.TS)



#Do  arima
fit <- arima(Mal.TS, order=c(2,0,1))


#Predict
pred <- predict(fit, n.ahead = 5)
ts.plot(Mal.TS,pred$pred, lty = c(1,3))


I know this is a poor analysis because of the lack of datapoints. However, I would like some suggestions on how to interpret the results that I found and reasons why this cannot work. The plot shows that the Malaria Prevalence in Botswana has been decreasing over the last 10 years, so one would expect the data to suggest that in the future the prevalence will keep decreasing. Yet, the model predicts the prevalance to increase. Why is this?
One way to possibly address this contra-intuitive result is by adding exogeneous data using the xreg argument. If I could include time series data on GDP, Health care expenditures,... which probably correlate with the Malaria prevalence data and thus also show a decreasing trend, I might be able to predict the future prevalence to be decreasing. Is this correct?
If the task of predicting the future malaria prevalence rate using the 24 earlier data points is not possible, could you give a clear reasoning why this is the case?
So in short I have following questions:


*

*How come the arima model predicts the prevalance rate to increase in the future

*Can I cause the predictions to be decreasing using exogeneous data like GDP and health care expenditures?

*If the analysis is really hopeless, could you give a clear reasoning why this is the case?

 A: 
  
*
  
*How come the arima model predicts the prevalance rate to increase in the future
  

ARIMA tries to find patterns in historical data and (when forecasting) to extrapolate them into the future. Based on the patterns observed over the 24 data points that you have, the model apparently "thinks" it is time for the series to trend upwards. There is no subject-matter reasoning involved, just pure pattern recognition. The question is, is it sensible to use an ARIMA model to describe this process?

One way to possibly address this contra-intuitive result is by adding exogeneous data using the xreg argument. If I could include time series data on GDP, Health care expenditures,... which probably correlate with the Malaria prevalence data and thus also show a decreasing trend, I might be able to predict the future prevalence to be decreasing. Is this correct?

(Also, question 2.)
Including additional variables could certainly help. Prevalence of malaria should not be too difficult to explain using subject-matter reasoning and having the relevant data. Using pure pattern recognition techniques on your time series could hardly be effective. Therefore, I would certainly try including some relevant variables.

If the task of predicting the future malaria prevalence rate using the 24 earlier data points is not possible, could you give a clear reasoning why this is the case?

(Also, question 3.)
The data generating process might be evolving over time. There are probably a lot of forces at play and the development of the series is far from stable. For example, at times the government might be introducing new programmes for fighting malaria, while at other times they might be cutting the programmes due to insufficient funding. People might be moving in or out of the malaria-infested regions in the country. Gradual changes in natural conditions might make malaria more or less prevalent over time. If you think in terms of an ARIMA model, you could say that the shocks (the errors, the innovations) are far from i.i.d., and even the AR and MA coefficients might be changing over time.
A: Your data is not stationary as evidenced by the spikes in the ACF and PACF. Take first differences of the data, then examine the ACF and PACF to tentatively identify the model. For additional information about time series modeling goto autobox.com
