Solved – negative fitted value for positive original value auto.arima

arimartime series

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:

 [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.

Best Answer

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")

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Solved – Ljung-Box Statistics for ARIMA residuals in R: confusing test results

You've interpreted the test wrong. If the p value is greater than 0.05 then the residuals are independent which we want for the model to be correct. If you simulate a white noise time series using the code below and use the same test for it then the p value will be greater than 0.05.

m = c(ar, ma)
w = arima.sim(m, 120)
w = ts(w)
plot(w)
Box.test(w, type="Ljung-Box")

Solved – Ljung-Box always significant for ARIMA models – what now

A note on terminology: commonly we fit a model to the data rather than fit the data to a model.

I can do step 1, but don't know how to relate that to step 2. Am I using the remainder from stl analysis for ARIMA modeling? If not, what's the point of step 1?

From STL you obtain three components: trend, seasonal and remainder. You could remove the seasonal component and use the sum of trend and remainder for further modelling with ARIMA.

But I can't get past the diagnostics. My Ljung-Box values are ALWAYS significant for ALL lags. Okay, so that means my residuals are correlated (I think). And since I want to use the residuals for cross-correlation, I assume that's bad.

Yes, having significant autocorrelations for ALL lags is clearly a problem. I would generally agree with the comment by @Glen_b, but in a case where all lags are significant the problem seem hard to deny. Curiously, the ACF plot does not immediately suggest that the autocorrelations are a really big problem (only a few lags stick outside the confidence interval by much) and the latter only becomes evident from the Ljung-Box test. I would not stop there and I would not accept a model with such a terrible Ljung-Box picture. Instead, I would look for other models.

One caveat: if you use STL and remove the seasonal component before estimating ARIMA models on trend+seasonal, you should not allow for a seasonal component in the ARIMA model (making it a SARIMA model); use option seasonal=FALSE in function auto.arima. Perhaps making this change will help you find better models.

Note also that after taking the 24-hour difference, the ACF and PACF still have significant 24-hour lags. This may indicate that taking the 24-hour difference was not such a good idea. Normally you would expect the lag at which you have differenced the data to not have significant ACF or PACF value.

Does this mean my time series doesn't fit an ARIMA model?

The model you showed us indeed does not seem to fit the data well as evidenced by the poor Ljung-Box statistics. If I were you, I would try some other model instead.

Best Answer

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