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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?

randomwalker
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  • 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? – Dayne Mar 15 '21 at 08:10
  • 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... – randomwalker Mar 15 '21 at 10:11
  • 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? – Dayne Mar 15 '21 at 10:19
  • 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? – randomwalker Mar 15 '21 at 18:25
  • 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 – Dayne Mar 16 '21 at 01:32

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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

Good Luck
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  • 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. – randomwalker Mar 11 '21 at 19:10
  • @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. – Good Luck Mar 11 '21 at 19:18
  • Thanks. How would you test for 'linear relationship'? – randomwalker Mar 11 '21 at 19:30
  • 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. – Good Luck Mar 11 '21 at 19:40
  • 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 $t = \hat{\beta} / SE(\hat{\beta})$ – randomwalker Mar 11 '21 at 20:34
  • Yes, standard error is used for hypothesis testing, but to check the linearity you don't need hypothesis testing in general case. – Good Luck Mar 11 '21 at 22:01
  • @randomwalker check also [this](https://stats.stackexchange.com/questions/326741/why-is-the-dickey-fuller-test-different-from-a-simple-t-test) and you may use the "at" someone who you trust to ask for details. Guys in here will do anything to help you out. You are allowed to comment below your question :) – Good Luck Mar 11 '21 at 22:04