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The Dickey-Fuller test tests an AR(1) series for stationarity.

An AR(1) series can be written as:

$x_t = \phi x_{t-1} + \epsilon_t $

with $\phi$ constant and $\epsilon_t $ white noise.

The series is stationary only if $\phi<1$.

This series can be written as:

$\Delta x_t = \beta x_{t-1} + \epsilon_t $

with $\beta = \phi - 1$

The null and alternative hypothesis of the DF test are respectively:

H0: $\beta=0$ (i.e. $\phi=1$) -> non-stationary

H1: $\beta<0$ (i.e. $\phi<1$) -> stationary

The test statistic is

$t = \hat{\beta} / SE(\hat{\beta})$

It looks like the exact same hypothesis test used to test for linearity.

I do not understand how is this Dickey-Fuller test different from testing for linearity?

I think I have read that the test statistics is different... how is it possible?

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  • $\begingroup$ Testing for linearity is a fairly more complex issue because the alternative (against the null of linearity) can take variety of forms (because non-linearity can take many forms). Unit root (which you are naming as stationarity test) is completely different. What exactly makes you think that the two are related, let alone same? $\endgroup$
    – Dayne
    Mar 15, 2021 at 8:10
  • $\begingroup$ What makes me think they are the same is that the hypothesis test and the test statistic are the same. (H0: beta=0, H1: beta<0 (or beta different form 0), test statistic: beta/SE(beta)... are all the same). Maybe there is something deep I do not understand about statistic. I am trying to find out what I do not understand with your help... $\endgroup$ Mar 15, 2021 at 10:11
  • $\begingroup$ Hypothesis test is a procedure and test statistic is a number. What do you mean when you say they are the same. And how is 'linearity' coming into picture here? $\endgroup$
    – Dayne
    Mar 15, 2021 at 10:19
  • $\begingroup$ H0: beta=0 means constant (slope=0). In this sense you are testing if 'there is a linear relationship'. For me linear relationship means slope different from 0. Basically the question I am answering is 'if I fit the data with a line, how significant is beta to say that actually it is different from 0? $\endgroup$ Mar 15, 2021 at 18:25
  • $\begingroup$ Your definition of linearity is wrong and hence the confusion. In DF test the process is linear in both null and alternative. To get a better understanding of nonlinear models and related tests search for TAR or SETAR models for example $\endgroup$
    – Dayne
    Mar 16, 2021 at 1:32

1 Answer 1

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Well Dickey-Fuller test is the test where we try to disapprove the null hypothesis that a unit root is present in an autoregressive model.

The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term.

You may consider DF test as special case of Autoregressive–moving-average model (ARMA).

Testing for linearity is totally different thing. It is just one of 5-6 or even more types of test where we try to detect if regression meat the assumptions. I will name few:

  • linear relationship
  • multivariate normality
  • multicollinearity test
  • No auto-correlation
  • homoscedasticity

So DF kind of test is type of Hypothesis testing.


Addition regression assumptions are particularly made for us when we would like to understand if we can relay on these numbers marked with gold:

enter image description here

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  • $\begingroup$ How would you test if there is a linear relationship between $\Delta x_t$ and $x_{t-1}$? Isn't it exactly testing for the hypothesis which is in the DF test? I don't see any difference. $\endgroup$ Mar 11, 2021 at 19:10
  • $\begingroup$ @randomwalker there is conceptual difference a linearity test is one of the test where we try to detect if we can relay on regression calculation for parameters in gold. $\endgroup$
    – Good Luck
    Mar 11, 2021 at 19:18
  • $\begingroup$ Thanks. How would you test for 'linear relationship'? $\endgroup$ Mar 11, 2021 at 19:30
  • $\begingroup$ The linearity assumption can best be tested with scatter plots when we remove the outliers, if we can prove that features influence the target linearly the assumption holds. $\endgroup$
    – Good Luck
    Mar 11, 2021 at 19:40
  • $\begingroup$ Don't you test the linearity like this: The null and alternative hypothesis are respectively: H0: $\beta=0$ (the slope) -> there is no linear relationship H1: $\beta \neq 0$ -> there is linear relationship The test statistic is $t = \hat{\beta} / SE(\hat{\beta})$ $\endgroup$ Mar 11, 2021 at 20:34

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