The
Dickey** **and
Fuller (1979) test is widely used to diagnose whether a time series has a unit
root or not. The Dickey and Fuller is a regression-based test conducted on
transformed versions of the AR(1) model, assuming the following models:

1)- **Random Walk**

2)- Random walk with drift

3)- **Random walk with drift and trend**

Random walk is a time series process where next period's
value is obtained as the previous period's value, plus an unpredictable random
error. We focus the second model here. Consider the following equation of **AR(1)** **model**.

x_{t} = b_{0} + b_{1}x_{t-1} + ε_{t}

Subtracting x_{t-1} from both sides gives

x_{t} - x_{t-1} = b_{0} + (b_{1} - 1)x_{t-1} + ε_{t}

_{or}

x_{t} - x_{t-1} = b_{0} + 𝛿x_{t-1} + ε_{t}

_{where, }𝛿 = (b_{1} - 1)

If b1 = 1 then 𝛿 = 0

Hence, a test of 𝛿 = 0 is a test of b_{1} = 1

We can formulate the following set of hypothesis.

H_{0} : 𝛿 = 0 versus H_{a} : 𝛿 ≠ 0

The null hypothesis of the test is that the time
series contains a unit root and is nonstationary—and the alternative hypothesis
is that the time series does not contain a unit root and is stationary.

In short, we regress the first-differenced time series (dependent variable) on the first lag of that time series (independent variable), where constant is nonzero.

The slope t-statistic obtained from the regression output is a statistic of the Dickey Fuller test, which is the basis for deciding whether or not to reject the null hypothesis, comparing it with the Dickey Fuller critical points rather than conventional t-critical values. The Dickey Fuller critical values are different for the above three models.

The comparison values we choose are based on the level of significance selected. The level of significance reflects how much sample evidence we require to reject the null. We can use three conventional significance levels to conduct hypothesis tests: 0.10, 0.05, and 0.01.

If we find that the calculated value of the test statistics is less than the Dickey Fuller t-critical value, we reject the null hypothesis.

### Example: Starbucks Corporation's Quarterly Sales

Suppose, we decide to model the log of Starbucks Corporation's quarterly sales using an AR (1) model.

Sales_{t} = b_{0} + b_{1} ln Sales_{t-1} + ϵ_{t}.

Before, we estimate this regression, we should use the Dickey–Fuller test to determine whether there is a unit root in the log of SBUX’s quarterly sales. SBUX is the trade name of Starbucks Corporation on the New York Stock Exchange.

We use a short sample of quarterly data
on SBUX’s sales from the first quarter of 2000 through the fourth quarter of
2007, as shown by data range A4:B35 in the Figure-1. We take the natural log (ln)
of each observation using the cells C4:C35. The first lag of log sales appears
in cells D5:D35. Note that the log differenced sales in column E are calculated
by placing the formula = C5 - D5

**Source : SEC filing.**

We use the Excel 'Data Analysis Tools' to run a regression, by selecting log differenced sales (E4:E35) as the Y data range and the first lag of log sales (D4:D35) as the X data range. We start a range from fourth row so that to include labels, as shown in the Figure-2.

The output from the
regression analysis is shown below:

**Conclusion : **

The bottom part of the Figure-3 clearly shows that because the computed Dickey–Fuller test statistic (cell J18) is more extreme than the Dickey-Fuller rejection point (cell J23), so we fail to reject the null hypothesis that there is a unit root in the log of SBUX’s quarterly sales.