ARIMA model output

The first plot is the plot of the original series: prices versus time. We clearly see the series is not stationary, since its mean values change over time. This suggests integrating the original series to make it stationary and then fitting a (possibly seasonal) ARIMA model. Seasonality and negative autocorrelation seem to be present, judging by the up-and-down cyclic behavior in the series.

The second plot shows the residuals of the model (after fitting an ARIMA(0,1,3)(2,1,0) model) versus time. The residuals are always centered around zero and don't exhibit a particular trend in levels across time. They do exhibit periods of increased volatility which is not accounted for in the model. One would have to consider fitting a model with non-constant volatility.

The autocorrelation plot exhibits negative autocorrelations. We observe significant autocorrelations of order 24, 48 and 72. Because of parsimony, one might be inclined to ignore such high order autocorrelations, or include a yearly seasonal effect to the model.

The last plot compares a histogram of the residuals with the density of a normal distribution. It seems the residuals have lighter tails than those of a normal distribution. This is likely a consequence of the non-constant variance in the residuals. It is clear that the residuals are centered and symmetric over zero, so inference for the parameters of the model should be valid.

No obvious departure from the assumptions of an ARIMA model are observed, but adding a yearly seasonal effect, or considering a time series model with non-constant volatility (like an EGARCH), would probably improve the fit of the model.