Skip to main content
Cross Validated

Return to Answer

clarify non-stationarity of AR(1)
Source Link
user5957401
user5957401
  • 489
  • 3
  • 8

@Stats, @mlofton, and @asdf are all correct. Let me try to summarize.

Non-stationarity is not a single concept. Rather it is the lack of a single concept -- stationarity. This means that there are (as @Stats mentions) an infinite number of non-stationary structures. There is a non-stationary model where the truth is an ARIMA model (with your specified p,d, and q), but where the coefficients associated with the p and q change every 10 periods. Could the ARIMA model do a good job of modeling that? Making predictions in that setting? Learning coefficients under those conditions? No.

But there are some non-stationary structures the ARIMA model can handle. An $AR(1)$ model with $\phi=1$ is non-stationary in one sense -- the distribution ofdiffers for each time period is dependent on the previous one. But for data which is truly $AR(1)$, we can still model it successfully with an ARIMA (under some conditions). Similarly, data which follows the ARIMA structure more generally is inherently non-stationary. However as long as it sticks to that type of non-stationarity -- namely some finite number of differences being modeled by an ARMA process -- things are manageable.

This is why there is some disagreement. It is true that there are types of non-stationarity that ARIMA models can handle, but it is also true that there are many types that it is totally worthless in the face of.

@Stats, @mlofton, and @asdf are all correct. Let me try to summarize.

Non-stationarity is not a single concept. Rather it is the lack of a single concept -- stationarity. This means that there are (as @Stats mentions) an infinite number of non-stationary structures. There is a non-stationary model where the truth is an ARIMA model (with your specified p,d, and q), but where the coefficients associated with the p and q change every 10 periods. Could the ARIMA model do a good job of modeling that? Making predictions in that setting? Learning coefficients under those conditions? No.

But there are some non-stationary structures the ARIMA model can handle. An $AR(1)$ model is non-stationary in one sense -- the distribution of each time period is dependent on the previous one. But for data which is truly $AR(1)$, we can model it (under some conditions). Similarly, data which follows the ARIMA structure is inherently non-stationary. However as long as it sticks to that type of non-stationarity -- namely some finite number of differences being modeled by an ARMA process -- things are manageable.

This is why there is some disagreement. It is true that there are types of non-stationarity that ARIMA models can handle, but it is also true that there are many types that it is totally worthless in the face of.

@Stats, @mlofton, and @asdf are all correct. Let me try to summarize.

Non-stationarity is not a single concept. Rather it is the lack of a single concept -- stationarity. This means that there are (as @Stats mentions) an infinite number of non-stationary structures. There is a non-stationary model where the truth is an ARIMA model (with your specified p,d, and q), but where the coefficients associated with the p and q change every 10 periods. Could the ARIMA model do a good job of modeling that? Making predictions in that setting? Learning coefficients under those conditions? No.

But there are some non-stationary structures the ARIMA model can handle. An $AR(1)$ model with $\phi=1$ is non-stationary in one sense -- the distribution differs for each time period. But we can still model it successfully with an ARIMA (under some conditions). Similarly, data which follows the ARIMA structure more generally is inherently non-stationary. However as long as it sticks to that type of non-stationarity -- namely some finite number of differences being modeled by an ARMA process -- things are manageable.

This is why there is some disagreement. It is true that there are types of non-stationarity that ARIMA models can handle, but it is also true that there are many types that it is totally worthless in the face of.

Source Link
user5957401
user5957401
  • 489
  • 3
  • 8

@Stats, @mlofton, and @asdf are all correct. Let me try to summarize.

Non-stationarity is not a single concept. Rather it is the lack of a single concept -- stationarity. This means that there are (as @Stats mentions) an infinite number of non-stationary structures. There is a non-stationary model where the truth is an ARIMA model (with your specified p,d, and q), but where the coefficients associated with the p and q change every 10 periods. Could the ARIMA model do a good job of modeling that? Making predictions in that setting? Learning coefficients under those conditions? No.

But there are some non-stationary structures the ARIMA model can handle. An $AR(1)$ model is non-stationary in one sense -- the distribution of each time period is dependent on the previous one. But for data which is truly $AR(1)$, we can model it (under some conditions). Similarly, data which follows the ARIMA structure is inherently non-stationary. However as long as it sticks to that type of non-stationarity -- namely some finite number of differences being modeled by an ARMA process -- things are manageable.

This is why there is some disagreement. It is true that there are types of non-stationarity that ARIMA models can handle, but it is also true that there are many types that it is totally worthless in the face of.