negative fitted value for positive original value auto.arima

Question

I am trying to forecast a time series using auto. arima. My command is,

auto.arima(X, stationary = F, ic = "aic", stepwise = T, trace = T, test = "adf", allowdrift = F, allowmean = T, lambda = BoxCox.lambda(prd.xts, method = "loglik"), biasadj = T);

My original time series has 36 observations (monthly for 3 years). When plotting the fitted vs original values, I found the fitted value to be a negative one. Please find below the original and fitted:

X:

 [1]  5200  4420  5297  6815  8385  8000  5700  6610  5810  5680  4100  4750  2205  4748  5170  8050  8900  7050  6810
[20]  7030  5890  7160  6405  5370  5360  7649  7730  9090 10174  7775

Fitted:

 [1]  4932.4310  4935.5587  4003.1725  5045.8954  6690.9964  8315.6047  7920.8435  5495.5020  6474.9251  5616.1591
[11]  5473.4794  3585.5492  4409.2808  -716.4241  4406.8788  4901.2397  7972.2124  8841.2718  6937.2260  6685.7413
[21]  6916.3253  5703.4412  7052.0103  6257.4684  5128.3367  5117.0722  7559.2477  7642.8563  9034.6446 10133.6204

Looking at these two closely, we can observe, 14th element in original series is: 4748 & in fitted (ARIMA modelled) is -716.4241.

I am a novice in Time series forecasting. The best model auto.arima spits out is arima(0,1,0) with an AIC of 964.095.

My questions: 1) Is the model correct? Is it alright, if the fitted value is negative? 2) Would it be reasonable to have just 36 observations to do a monthly forecasting?

Any help would be much appreciated.

Answer 1

ARIMA assumes normally distributed innovations, and it can definitely go below zero. (If you take differences, you will often end up with negative values.)

In your specific case, I suspect that your X is not identified as monthly data, i.e., that it has a frequency of 12. Once we do that and create a seasonplot, seasonality is rather evident:

library(forecast)
foo <- c(5200, 4420, 5297, 6815, 8385, 8000, 5700, 6610, 5810, 5680, 
4100, 4750, 2205, 4748, 5170, 8050, 8900, 7050, 6810, 7030, 5890, 
7160, 6405, 5370, 5360, 7649, 7730, 9090, 10174, 7775)

X <- ts(foo,frequency=12)
seasonplot(X)

Now auto.arima() fits a seasonal model, and neither fits nor forecasts are negative:

model <- auto.arima(X)
plot(forecast(model,h=12))
lines(fitted(model),col="red")