ECONOMETRICS II
TUTORIAL VI
The 6th tutorial will be devoted to integration tests of a series. We
will analyse two tests, the ADF (Augmented Dickey and Fuller ) and the
KPSS (Kwiatkowski, Phillips, Schmidt and Shin) tests. To this end, we
will use the data in the workfile ppp.wf1. The example is taken from
Verbeek, chap. 8.
The workfile contains 6 variables:
1. cpif r, consumer price index for France, base 1990 = 100;
2. cpiit, consumer price index for Italy, base 1990 = 100;
3. lnf r, log of cpif r;
4. lnit, log of cpiit;
5. lnp, difference between lnit and lnf r;
6. lnx, log of the nominal exchange rate between Italian Lira and
French Frank.
These variables are observed from January 1981 to June 1996, for a total
of 186 monthly observations.
With these data we can test the validity of the so-called law of one price,
i.e. that nominal exchange rate–if there is no trade restriction–adjusts
to tradeable goods relative price variations.
In formula:
Pt
St = ∗ (1)
Pt
where St is the spot exchange rate (home currency price of a unit of
foreign exchange), Pt is the (aggregate) price in the domestic country
and Pt∗ is the price in the foreign country. Taking logs of (1)
st = pt − p∗t (2)
where st = ln (St ), pt = ln (Pt ) and p∗t = ln (Pt∗ ). Conditions (1) and (2)
are called purchasing power parity and should be valid, at least in the
long run. Therefore, the log of the real exchange rate, given by:
rst ≡ st − (pt − p∗t ) (3)
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should be a stationary series, so that deviations from the equilibrium
should be only transitory.
Before verifying whether the log of the real exchange rate is stationary
or not, we have to assess whether the two series log of price and log of
nominal echange rate are stationary or not.
In our dataset: st = lnx; pt = lnit; p∗t = lnf r and (pt − p∗t ) = lnp.
1 Unit root tests
Graphical analysis Before performing formal tests, we have to plot
the series in levels and in first differences. The graphical analysis allows
one to assess the existence of a constant and/or of a deterministic trend.
Example 1. Generate three variables, given by the first difference of the
log prices (call them dlnf r and dlnit) and of the nominal exchange rate
(call it dlnx) and plot the graph of the series in levels and in differences.
Notice that if we look at the series in levels the log price is not – both for
Italy and for France – stationary and if we look at the series in differences
that a positive constant appears. Same conclusions can be drawn if we
look at the nominal exchange rate series.
NB: to generate the series in differences use the commands: series dln-
fr=d(lnfr), series dlnit=d(lnit) and series dlnx=d(lnx) in the command
window.
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Formal statistical tests Both ADF and KPSS tests are implemented
in EViews through the command:
View...
Unit root test...
from the series menu, by selecting the specific test and the appropriate
options.
Notice that:
1) the DF test supposes that one of the following models is true:
yt = δ + θyt−1 + εt (4)
yt = δ + γt + θyt−1 + εt (5)
and tests
H0 : θ = 1 (6)
H1 : |θ| < 1 (7)
Under the null of unit root (θ = 1), the two models become:
∆yt = δ + εt (8)
∆yt = δ + γt + εt (9)
2) To implement the test, an auxiliary regression of the form:
∆yt = δ + (θ − 1)yt−1 + εt (10)
| {z }
π
∆yt = δ + γt + (θ − 1)yt−1 + εt (11)
| {z }
π
is estimated by OLS and the hypothesis
H0 : π = 0 (12)
H1 : π < 0 (13)
is tested. Notice that the inclusion of the trend and/or of the constant
in the auxiliary regression (10) or (11) must be indicated in the ADF
test menu.
3) the ADF test follows the same logic of the DF test but adds lags of
∆yt among the regressors: the reason is that the test is valid only if εt
is a white noise.
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4) the null hypothesis of the ADF test is the presence of a unit root
whereas the null hypothesis of the KPSS test is that the series is statio-
nary;
5) when performinig the ADF test it is crucial to find the optimal lag,
i.e. the number of lags of ∆yt to include as additional regressors in order
to make the errors white noise. In this respect, three approaches are
possible: include as many lags as possible; use model selection criteria
(AIC or BIC); assess the significance of the lags in ∆yt .
Notice how for both log prices the 12th lag is very significant, whereby
suggesting the presence of seasonality in prices (these are monthly pri-
ces). As for the log of nominal exchange rate, instead, it is not possible
to reject the null hypothesis of unit root for all lags.
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After generating the log real exchange rate (through the command: series
rs = lnx−lnp), notice how for this variable it is not possible to reject the
null hypothesis of existence of a unit root. This evidence contraddicts
the purchasing power parity theory.
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All these conclusions are confirmed by using the KPSS test. For instance,
if we test the null hypothesis of stationarity of the real exchange rate
series, the corresponding output is:
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