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Suppose that the data generated process is the following:

$Y_{t} = 1.2Y_{t-1} + 0.2$

The process is clearly non-stationary.

My question is why we can't fit an AR(1) model and make predictions?

Stochastic
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gnikol
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  • Because the slope is greater than 1. An AR model assumes that the series is stationary and the coefficient is lower than one. – gnikol May 16 '20 at 19:51

1 Answers1

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The short answer is that ARIMA models rely on the assumption that the time series being modeled is stationary. Therefore that assumption needs to hold if you want to use these models.

When we fit an AR(1), we obtain a value for the parameter $\phi_1$. If the value of this is approximately equal to 1 or even close enough to 1, the series is not stationary. This is called the Dickey-Fuller test. You can find some information on that here. In the example here I believe they use a $\rho$ where I used $\phi_1$.

Here is a great post on why time series needs to be stationary

If the coefficient has a value larger than 1, the auto regressive process is said to be explosive

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Stochastic
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  • Thank you very much for your answer. I know that AR assumes that the series is stationary and that the coefficient should be lower than 1. But in the above problem we could fit an OLS regression with covariate the lagged value of the dependent variable and accomplish a perfect fit. An OLS regression with lagged values as covariates is not consider autoregression? – gnikol May 16 '20 at 19:50
  • Well, what you are saying there makes sense to me. I would like to think that it can be considered an auto regressive process. Apparently you do get AR(1) models that have a coefficient larger than 1. The auto regressive process is called explosive then. [Here](https://stats.stackexchange.com/questions/281483/can-autoregressive-coefficient-values-be-greater-than-1) is a post where this is spoken about – Stochastic May 16 '20 at 20:08
  • Thank you very much for the links. Let me modify my question. I remember from my studies that in order to have good predictions and if I remember correctly small standard errors the process should be stationary. In the above non-stationary process an AR(1) model should yield perfect predictions. So, there is a contradiction.. – gnikol May 17 '20 at 15:54
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    If the coefficient is greater than 1.0, then this means that the proces is explosive in the sense that there is no equilbrium in terms of an unconditional mean. So, such a model can't be estimated because the estimated coefficients won't have any meaning. Also, note that a lagged dependent variable model is an AR(1). ( with the appropriate restrictions on the lagged coefficient ). – mlofton Oct 08 '20 at 12:57