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I have recently been thrown into the deep end with time-series econometrics. The first thing I have learned is that in order to avoid the spurious correlation trap, I need to ensure that all the variables I am working with are stationary. I have just finished coding the Augmented Dickey-Fuller test. My take on stationarity and the ADF test is that if a variable can largely (and spuriously) be explained/predicted by itself lagged, then it is not stationary. The next 200 pages of my book are devoted to ARMA (and other lagged correlation type models) time series modelling, and its use in predicting time series variables. But reading the rest of the book seems futile, I have to first prove (via ADF test) that a variable does not display significant autocorrelation (is stationary), and then I must engage with this modelling approach which is built on autocorrelation. This all seems contradictory, what am I missing in my understanding of time series econometrics?

Andrew Beaven
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    Hi: When testing for a unit root using the DF test, you are not testing to see if there's autocorrelation so there's no contradiction between doing a unit root test and arima modelling. The unit root test is a specific test for non-stationarity of the mean ( and variance ) so nothing to do with autocorrelation. The augmented DF adds lags to the test in order to get rid of autocorrelation in THAT model ( it's not an ARIMA model ) but the key component of the test is the first lag. The other lags are used only for increasing the power of the test and one way is to get rid of autocorrelation. – mlofton May 07 '22 at 12:47
  • I thought that evaluating whether or not a time series has a unit root was achieved by considering the magnitude of the roots of the polynomial characteristic equation? In the MATLAB source code for the ADF test, they conduct a unit root test using the coefficients of lagged difference terms (excluding the first AR(1) coefficient, not sure why). This is clearly auxiliary to the ADF-test, as it will generate a warning about the validity of the ADF-test not an error. Maybe I am confused about testing for stationarity versus testing for a unit root. I have been thinking they were the same thing. – Andrew Beaven May 07 '22 at 13:16
  • https://stats.stackexchange.com/questions/410379/the-theory-behind-fitting-an-arimax-model/410396#410396 might help you understand that time series causal model identification can be enhanced by filtering the observed time series . – IrishStat May 07 '22 at 12:37
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    Re: "if a variable can largely (and spuriously) be explained/predicted by itself lagged, then it is not stationary:" this is neither the definition of stationarity nor is it equivalent. Start, then, by reviewing whatever definitions and characterizations of stationarity have been provided to you. – whuber May 07 '22 at 13:51
  • @mlofton makes a good point. Nonzero autocorrelation is not the same as presence of a unit root. – Richard Hardy May 07 '22 at 13:52
  • I understand the definition and meaning of stationarity and find it intuitive, I guess I will have to read and try and figure out why the significance of the slope coefficient of an AR(1) model is pivotal in testing for a unit root (mlofton). The ADF test does appear in my book in the section about stationarity and testing for it. I am still stuck with my question though, from a practical point of view. – Andrew Beaven May 07 '22 at 15:42
  • Hi Andrew: If you want to understand the details of the DF unit root test, I'm pretty sure hamilton explains it. If not, the original dickey fuller paper is your best bet. I'm not sure what your confusion still is or I'd try to address it ? Testing for a unit root is a specific type of non-stationarity test and it is pretty unrelated to the autocorrelation framework that the ARIMA framework uses. I'm not sure what else to say. – mlofton May 08 '22 at 17:52
  • Note that " evaluating whether or not a time series has a unit root was achieved by considering the magnitude of the roots of the polynomial characteristic equation", is sort of incorrect. After an ARIMA model is estimated, the roots of the characteristic equation can be checked and, if any of the roots are greater than 1 ( for the AR piece ), then a non-stationary ARIMA model has RESULTED during estimation. But, that's not the same as testing for a unit root. Most estimation algorithms that I know of have built in ways of avoiding estimates where the roots of the CE are greater than 1. – mlofton May 08 '22 at 17:57
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    The rule is to check for stationarity BEFORE any estimation step is carried out. This way, if there is evidence of say a unit root, then differencing might help. So, the characteristic equation stuff is really a piece of the estimation algorithm. AFAIK, you'll rarely encounter an estimation algoririthm result in roots of the CE that imply non-stationarity. The algorithm generally insures that. Box-Jenkins is kind of the bible for ARIMA modelling ( although I'm not a big fan of it. I think something easier like chatfield should be read first ) so maybe check that out or something by chafield. – mlofton May 08 '22 at 18:01

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