Unit Root Test

The Dickey Fuller Test based on linear regression in which case the Augmented

Dickey Fuller test (ADF) was used.

  t j t p j j t t Y Y Y   _ p _   1 4 (  1)

The Philllips-Perron (PP) Test which is a modification of the Dickey Fuller test was

There are various ways of carrying out unit root test, however the common ones

include; carrying out unit root test through informal approach and carrying out unit

root test through formal unit root testing.

There are two types of unit root tests under informal approach namely; unit root test

under time plot which may suggest the presence of unit root (non stationary) that the

mean is not constant over time, and another type is the unit root test which can be

inspected through the empirical correlograms for indicating the decaying time of

time series. Under non stationarity time series empirical correlograms reveals or

exposes the quick decay than in stationary time series. Notwithstanding these

techniques are said to be very weak test for unit root testing, hence this study will

mainly focus in the formal test for unit root testing.

Under formal ways there are several unit root tests which can be employed in

investigating unit root, preferably the study may choose ‘Dickey- Fuller (DF) test, Augmented Dickey- Fuller (ADF) test and Phillip Perron (PP) test (Watson and

Teelucksingh, 2002; Greene, 2003; Gujarat, 2009). However in the contemporary

econometrics the frequently employed techniques for testing unit root include DF,

ADF test and PP test. According to Gujarat (2009), ADF test for unit root is the most

powerful test, based on this fact this study prefers the ADF test.

Very often the economic series data are non-stationary which suggest that its mean

does not fluctuate around its fixed mean. Whenever stationary time series mean

from its fixed mean it returns quickly to the fixed mean. Moreover non stationary

time series usually become a stationary series after a first difference. Formally

written as I (1) means that time series are non-stationary becomes stationary after

differentiating once. Thus, stationary time series is integrated of order zero and

written as I (0). However, other time series sometimes require two differencing

before becoming stationary, as such those series are integrated of order two and

written as I (2) (Watson and Teelucksingh, 2002; Gujarat, 2009).

It is suggested to test unit root in order avoid ‘spurious regressions,’ because it is

obvious that regression of non-stationary time series on another non- stationary time

series gives spurious regression implying meaningless results, for this reason unit

root test is absolutely important. In order to obtain realistic and meaningful results

employing unit root test in time series data is inescapable or a grave mistake to

ignore it. Moreover the test for unit root is essential in order to know if variables in

the study are integrated in the same order or not. When variables are integrated in

the same order, I(0) means stationary time series and its regression output will not be

spurious. Any regression of time series of non-stationary on the stationary time

series will produce spurious regression or meaningless results.

According to Watson and Teelucksingh, 2002; Gujarat, 2009; and Greene, 2003)

spurious results cannot be used in predictions because variables are not integrated in

the same order. Hence, unit root testing under formal approach particularly Dickey-

Fuller (DF) test is conducted using the random walk model.

i. Random walk without deterministic and stochastic trends (constant and trend)

Xt=σXt-1+et

ii. Random walk with deterministic (constant)

iii. Xt=α1+σXt-1+et

iv. Random walk with both deterministic and stochastic trends (constant and

trends)

v. Xt=α1+α2T+σXt-1+et

Where T is trend from time series data under this tests the H0 is σ = 0; which means

the unit root implies non stationary. H1 (σ<0) meaning that time series is stationary.

Notwithstanding, Dickey –Fuller test in testing unit root assumes error term (et) is

uncorrelated, in case if ‘error term is correlated’ DF test cannot be applied. Thus for correlated error term DF develop the powerful which is ADF.

ADF is done by summing equations above by adding the lagged values of the

dependent variables. Hence ADF test is estimated with the following equation:

Xt=α1α2δXt-1

Where εt is pure white noise error term. ADF test include more terms in order to

make error term in equation (3) above be ‘serially uncorrelated.’ Similarly Augmented Dickey Fuller (ADF) test whether Ô=0 as such Augmented Dickey

Fuller is not different from DF hence follows the same asymptotic distribution, thus

the same critical values are utilized (Gujarat, 2009, Greene, 2003).

Decision criteria: According to Zivot (2012), rho-1 =ô, where ô is regression

stationary and if rhois<1then the variable is stationary. Gujarat points out that’ if the computed absolute value of the tau statistic exceeds the DF critical values reject the

null hypothesis (non stationary) then variables are stationary. If the computed

absolute value does not exceed the critical tau values do not reject the null

hypothesis as such variables are non-stationary (Gujarat, 2009p 816).

After establishing that variables are stationary, the study examines if the variables

are co integrated or not. This is a very important stage in regression analysis as

mentioned or explained earlier as it prevents the study from doing meaningless

regression or spurious regression. This study adheres to the rule that co integration

analysis is inevitable for this study. Gujarat (2009) and Greene (2003) insist the

importance of testing for co integration by stating that ‘regression of time series variable on one or more time series variables often can give no sense or spurious

results. This kind of phenomenon is popularly referred to as spurious regression. One

of the means to guard against it is to find out if the time series variables are co

integrated (Greene, 2003; Gujarat, 2009). Therefore after unit root test the study

tested for co integration on time series variables.