1. df tests

Oct 15, 20121 like793 views

This document discusses the Dickey-Fuller test for unit roots, which tests whether a time series is stationary or nonstationary. It presents the three regression equations used in the Dickey-Fuller test and the corresponding critical values. It then provides steps for performing the Dickey-Fuller test in EViews, including specifying the test type, level/difference of the series, regression model, lag length, and interpreting the test results by comparing the test statistic to the critical values.

Data Analysis & Forecasting Faculty of Development Economics

NONSTATIONARY & UNIT ROOTS
DICKEY-FULLER TESTS

1. DF Test Equations
The simple Dickey-Fuller test is based on the following AR(1) model:
Yt = ρYt-1 + ut (*)
H0: ρ = 1 (Yt has a unit root ~ Yt is nonstationary)
H1: ρ < 1 (Yt does not have a unit root ~ Yt is stationary)
If we subtract Yt-1 from both sides of (*):
Yt – Yt-1 = ρYt-1 – Yt-1 + ut
∆Yt = (ρ - 1)Yt-1 + ut
∆Yt = δYt-1 + ut (1)
H0: ρ = 1 (Yt has a unit root ~ Yt is nonstationary)
H1: ρ < 1 (Yt does not have a unit root ~ Yt is stationary)
Dickey and Fuller (1979) also proposed two alternative regression equations that
can be used for testing for the presence of a unit root.
The first contains a constant in the random walk process as in the following
equation:
∆Yt = α + δYt-1 + ut (2)
The second case is also allow, a non-stochastic time trend in the model, so as
to have:
∆Yt = α + γT + δYt-1 + ut (3)
This test does not have a conventional ‘t’ distribution and so we must use special
critical values which were originally calculated by Dickey and Fuller.
MacKinnon (1991,1996) tabulated appropriate critical values for each of the
three above models and these are presented in Table 1.
Table 1: Critical values for DF test
Model 1% 5% 10%
∆Yt = δYt-1 + ut -2.56 -1.94 -1.62
∆Yt = α + δYt-1 + ut -3.43 -2.86 -2.57
∆Yt = α + γT + δYt-1 + ut -3.96 -3.41 -3.13
Standard critical values -2.33 -1.65 -1.28
Source: Asteriou (2007)

Phung Thanh Binh (2010) 1

Data Analysis & Forecasting Faculty of Development Economics

If the DF statistical value is smaller in absolute terms than the critical value then
we reject the null hypothesis of a unit root and conclude that Yt is a stationary
process.
2. Performing DF Test in Eviews
Step 1 Open the file DF.wf1 by clicking File/Open/Workfile and then
choosing the file name from the appropriate path

Step 2 Let’s assume that we want to examine whether the series named GDP
contains a unit root. Double click on the series named ‘GDP’ to open
the series window and choose View/Unit Root Test …. In the unit-
root test dialog box that appears, choose the type test (i.e., the
Augmented Dickey-Fuller test) by clicking on it.

Step 3 We then specify whether we want to test for a unit root in the level,
first difference, or second difference of the series. We can use this
option to determine the number of unit roots in the series.

Step 4 We also have to specify which model of the three DF models we wish
to use. For the model given by equation (1) click on ‘none’ in the
dialog box; for the model given by equation (2) click on ‘intercept’;
and for the model given by equation (3) click on ‘intercept and
trend’.

Step 5 Type ‘0’ on the ‘user specified’ below the “lag length’ dialog box.

Step 6 Having specified these options, click <OK> to carry out the test.

Step 7 We reject the null hypothesis of a unit root against the one-sided
alternative if the DF statistic is less than (lies to the left of) the critical
value, and we conclude that the series is stationary.

Source: Asteriou (2007)

Phung Thanh Binh (2010) 2

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1. df tests

1. Data Analysis & Forecasting Faculty of Development Economics NONSTATIONARY & UNIT ROOTS DICKEY-FULLER TESTS 1. DF Test Equations The simple Dickey-Fuller test is based on the following AR(1) model: Yt = ρYt-1 + ut (*) H0: ρ = 1 (Yt has a unit root ~ Yt is nonstationary) H1: ρ < 1 (Yt does not have a unit root ~ Yt is stationary) If we subtract Yt-1 from both sides of (*): Yt – Yt-1 = ρYt-1 – Yt-1 + ut ∆Yt = (ρ - 1)Yt-1 + ut ∆Yt = δYt-1 + ut (1) H0: ρ = 1 (Yt has a unit root ~ Yt is nonstationary) H1: ρ < 1 (Yt does not have a unit root ~ Yt is stationary) Dickey and Fuller (1979) also proposed two alternative regression equations that can be used for testing for the presence of a unit root. The first contains a constant in the random walk process as in the following equation: ∆Yt = α + δYt-1 + ut (2) The second case is also allow, a non-stochastic time trend in the model, so as to have: ∆Yt = α + γT + δYt-1 + ut (3) This test does not have a conventional ‘t’ distribution and so we must use special critical values which were originally calculated by Dickey and Fuller. MacKinnon (1991,1996) tabulated appropriate critical values for each of the three above models and these are presented in Table 1. Table 1: Critical values for DF test Model 1% 5% 10% ∆Yt = δYt-1 + ut -2.56 -1.94 -1.62 ∆Yt = α + δYt-1 + ut -3.43 -2.86 -2.57 ∆Yt = α + γT + δYt-1 + ut -3.96 -3.41 -3.13 Standard critical values -2.33 -1.65 -1.28 Source: Asteriou (2007) Phung Thanh Binh (2010) 1

2. Data Analysis & Forecasting Faculty of Development Economics If the DF statistical value is smaller in absolute terms than the critical value then we reject the null hypothesis of a unit root and conclude that Yt is a stationary process. 2. Performing DF Test in Eviews Step 1 Open the file DF.wf1 by clicking File/Open/Workfile and then choosing the file name from the appropriate path Step 2 Let’s assume that we want to examine whether the series named GDP contains a unit root. Double click on the series named ‘GDP’ to open the series window and choose View/Unit Root Test …. In the unit- root test dialog box that appears, choose the type test (i.e., the Augmented Dickey-Fuller test) by clicking on it. Step 3 We then specify whether we want to test for a unit root in the level, first difference, or second difference of the series. We can use this option to determine the number of unit roots in the series. Step 4 We also have to specify which model of the three DF models we wish to use. For the model given by equation (1) click on ‘none’ in the dialog box; for the model given by equation (2) click on ‘intercept’; and for the model given by equation (3) click on ‘intercept and trend’. Step 5 Type ‘0’ on the ‘user specified’ below the “lag length’ dialog box. Step 6 Having specified these options, click <OK> to carry out the test. Step 7 We reject the null hypothesis of a unit root against the one-sided alternative if the DF statistic is less than (lies to the left of) the critical value, and we conclude that the series is stationary. Source: Asteriou (2007) Phung Thanh Binh (2010) 2

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1. df tests

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