ACF and PACF vs Ljung Box test

Question

I have a time series with realized sales prices on monthly basis in a large European city which comes as an index and I would like to do 1 period ahead forecasting.

I have run ADF and KPSS for unit root / stationarity as well as Ljung-box for white noise, I get the following results:

Detrended series:

• ADF p-value: 0.826796
• KPSS p-value: < 0.01
• Ljung-box p-value: all 12 lags has p-value of basically 0

1st difference series:

• ADF p-value: 0.001352
• KPSS p-value: > 0.1
• Ljung-box p-value: all 12 lags has p-value of basically 0
• ACF shows lags 1,2,3,4 being significant
• PACF shows lags 1,2,3,4 being significant

2nd difference series:

• ADF p-value: 0.003257
• KPSS p-value: > 0.1
• Ljung-box p-value: all 12 lags has p-value of basically 0
• ACF shows lags 1,2,4,5,6 being significant.
• PACF shows lags 1,2,3,6 being significant

From what I can tell, this is just a plain non-stationary process, since the Ljung-box test rejects the p-values of all 12 lags.

Am i doing something wrong? As I understand it, it should be possible to forecast a housing market index to some extend, I therefore thought that at least the housing index would be difference stationary.

I might do something wrong, can anyone help me understand?

Answer 1

Based on the ADF and KPSS tests, the first-differenced series is already stationary. (Note that presence of autocorrelation does not imply nonstationarity.) Thus you have no reason to take second differences. Differencing is used for accounting for unit roots, not just for any old autocorrelation. The second-differenced series will suffer from overdifferencing and will have a unit-root moving average (MA) pattern.

Meanwhile, to deal with autocorrelation that is not due to a unit root, you can introduce autoregressive (AR) and MA terms or seasonal terms such as Fourier terms or seasonal dummies. Thus you get an ARIMA model (possibly with external regressors) or a regression with ARIMA errors. For the model to be statistically adequate, the residuals should be close to white noise.