Augmented Dickey-Fuller Test

ERROR

Might have errors. I’ll check later TODO

Definition

The Augmented Dickey-Fuller (ADF) test extends the Dickey-Fuller Test to AR(p) models where $p>1$, handling autocorrelation in residuals.

Model

For AR(p):$\nabla Y_t=\pi Y_{t-1}+c_1\nabla Y_{t-1}+c_2\nabla Y_{t-2}+\cdots +c_{p-1}\nabla Y_{t-p+1}+u_t$

where:

• $\pi ={\theta }_1+{\theta }_2+\cdots +{\theta }_p-1={\sum }_{i=1}^p{\theta }_i-1$
• $c_i=-({\theta }_{i+1}+\cdots +{\theta }_p)=-{\sum }_{j=i+1}^p{\theta }_j$
• $u_t$ : White noise

TIP

$u_t$ is white noise — the error term after adding the lagged difference terms $\sum c_i\nabla Y_{t-i}$.

That’s the whole point of ADF over DF: the extra lagged differences are chosen specifically so that $u_t$ is white noise (no remaining autocorrelation in residuals).

Hypotheses

$$\begin{array}{rl}{H}_0& :\pi =0\text{(unit root, non-stationary)}\\ H_1& :\pi <0\text{(stationary)}\end{array}$$

Test Statistic

$\tau =\frac{\hat{\pi }}{\text{se}(\hat{\pi })}$

Decision Rule

Reject $H_0$ if $|\tau |>|{\tau }_{\text{critical}}|$

Lag Selection

The number of lagged difference terms ($p-1$) can be selected using AIC/BIC or by testing residual autocorrelation.

Problem with DF Test

The DF test assumes $u_t$ is uncorrelated (white noise). When residuals are autocorrelated, DF test is invalid.

Model Variants

Three variants of the ADF test depending on the assumed structure of the time series.

Variant 1: No Constant, No Trend

Model: $\nabla Y_t=\pi Y_{t-1}+{\sum }_{i=1}^{p-1}{c}_i\nabla Y_{t-i}+u_t$

Use when:

• Series appears to fluctuate around zero
• No apparent trend

Variant 2: With Constant (Intercept)

Model: $\nabla Y_t={\beta }_1+\pi Y_{t-1}+{\sum }_{i=1}^{p-1}{c}_i\nabla Y_{t-i}+u_t$

Use when:

• Series fluctuates around non-zero mean
• No apparent trend

Variant 3: With Constant and Trend

Model: $\nabla Y_t={\beta }_1+{\beta }_{2}t+\pi Y_{t-1}+{\sum }_{i=1}^{p-1}{c}_i\nabla Y_{t-i}+u_t$

Use when:

• Series shows deterministic trend
• Want to test if trend is stochastic or deterministic

Summary table

Critical Values

Each variant has different critical values. Use the appropriate DF critical value table for the chosen model.

Model	Equation	When to Use
No constant/trend	$\nabla Y_t=\pi Y_{t-1}+{\sum }_{i=1}^{p-1}{c}_i\nabla Y_{t-i}+u_t$	Fluctuates around zero
With constant	$\nabla Y_t={\beta }_1+\pi Y_{t-1}+{\sum }_{i=1}^{p-1}{c}_i\nabla Y_{t-i}+u_t$	Non-zero mean, no trend
With constant & trend	$\nabla Y_t={\beta }_1+{\beta }_{2}t+\pi Y_{t-1}+{\sum }_{i=1}^{p-1}{c}_i\nabla Y_{t-i}+u_t$	Shows trend

Tip

Only difference with Dickey-Fuller Test is that this model adds

Related

• Dickey-Fuller Test
• Weakly Stationary
• Differencing to Achieve Stationarity