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I am running a Dickey Fuller test on the average sale price of a home from 1980-2012 in monthly time period to see if it is stationary. Therefore I have the following hypothesis tests: $$H_o: B=0$$ $$H_a: B<0$$

Correct?

In order to solve for this I found the first differences of my data and then ran a regression in excel. Here is a picture. enter image description here

So I essentially ran $\Delta Y_t=BY_{t-1}+\epsilon_t$

Here is a picture of the output: enter image description here

Now if the test statistic is less than the critical value, we reject the null hypothesis and conclude that no unit-root is present. But since -1.08>-1.95 and .278>.05 we fail to reject null. But the coefficient is negative as desired. Is there something wrong with my analysis or test?

Afterwards, proceed to check for Cointegration between the Average Sale Price and other variables I am using? OR do I have check to see if each variable is stationary?

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Two things here:

First, the coefficient should indeed by negative under stationarity, but as with any hypothesis test, the matter is to decide if the estimate is sufficiently far away from the value under the null (here, of nonstatiority, so that $B=0$ under the null). Your results suggest it is not.

Second, as many posts on this site discuss, the t-ratio follows a different distribution under nonstationarity, so you must not employ the p-values reported by standard regression packages. Instead, packages like urca in R report correct p-values, as they "know" that the behavior of the series is different - nonstationary - under the null.

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