Explanation

For the Dickey-Fuller test, the regression specification is:

$\Delta x_t = \gamma x_{t-1} + \varepsilon_t$

Where:

• $\Delta x_t = x_t - x_{t-1}$ (first difference, dependent variable)
• $x_{t-1}$ (first lag of the time series, independent variable)

Key points:

• The Dickey-Fuller test examines whether $\gamma = 0$ in this regression
• If $\gamma = 0$, then $x_t = x_{t-1} + \varepsilon_t$, which is a random walk (unit root)
• If $\gamma < 0$, the series is stationary

Why option A is correct:

• The independent variable is simply the first lag $x_{t-1}$
• No transformation like subtracting 1 is needed
• The test directly examines the coefficient on the lagged level

Why other options are incorrect:

• Option B: No need to subtract 1 from the lagged variable
• Option C: This would be testing for higher-order unit roots, not the basic Dickey-Fuller test