Download Unit roots and seasonal unit roots in macroeconomic time series

Economics Letters
North-Holland
35 (1991) 273-277
273
Unit roots and seasonal unit roots
in macroeconomic time series
Canadian evidence
Hahn Shik Lee
Bureau of the Census, Washington,
DC 20233, USA
Pierre L. Siklos
Wilfrid Laker
Received
Accepted
Uniuersity,
15 August
2 October
Waferloo, Ont. N2L 3C5, Canada
1990
1990
Unit root tests are conducted
to determine whether the unit root found in seasonally adjusted data may be due to seasonal
adjustment
filters typically apptied by government
agencies (i.e., ARIMA X-11). We find that a unit root exists in both the
raw and adjusted data (except for the unemployment
rate). Second, we also find that the commonly used seasonal adjustment
filter introduced
by Box and Jenkins (1970) is generally inappropriate.
1. Introduction
The vast literature on unit roots attests to the somewhat strong belief that shocks to many
aggregate times series tend to be permanent in nature, although any such consensus has shown signs
of weakening [e.g., see Cochrane (1988) and Perron (1989) but see also Kormendi and Meguire
(1990)]. Part of the interest in testing for unit roots stems from the work of Box and Jenkins (1970)
who suggested that many time series are difference stationary.
While the focus of recent research has been about the time series properties of seasonally adjusted
time series, the focus of attention has lately reverted back to the implications of using seasonally
adjusted versus unadjusted data after some neglect over the past few years. For example, Barsky and
Miron (1989) suggest that the routine elimination of the seasonal component of time series by well
known methods, such as ARIMA X-11 [see Hylleberg (1986, ch. 5) for a survey], results in the
omission of an important source of macroeconomic fluctuations. Furthermore, since economists
prefer testing their theories using data expressed in real terms, it is not apparently the case that the
common practice of using seasonally adjusted time series, which are then deflated by the appropriate
price index, has non-neutral consequences on estimates from econometric models [Ghysels and
Karangwa (1990)].
Moreover, the unit root commonly found in postwar U.S. GNP may, in part, arise from the use of
seasonally adjusted data [Ghysels (1990)]. The problem with U.S. data, in particular, is the relative
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H.S. Lee, P.L. S~klos / Unit roots and seasonal unit roots
274
difficulty in obtaining high quality data in both seasonally adjusted and unadjusted
forms [e.g., for
U.S. GNP see Ghysels (1990, Appendix A) and Barsky and Miron (1989, Appendix A). Finally, there
is considerable
interest in whether aggregate time series contain a unit root at some of the seasonal
frequencies, as well as a unit root at the long-run (or zero) frequency found in seasonally adjusted
data.
The present study considers several postwar quarterly Canadian macroeconomic
series, which are
widely available both in seasonally adjusted and unadjusted
forms to examine the sensitivity of the
unit root result in seasonally adjusted data to seasonal adjustment
procedures
(i.e., ARIMA X-11)
applied by government
agencies.
Some of our results can be summarized
as follows. We first find that the unit root at the zero
frequency found in seasonally unadjusted
data is also present in seasonally adjusted data, confirming
that seasonal adjustment
via the ARIMA
X-11 does not eliminate
the unit roots at the zero
frequency found in the original time series data. We also find evidence for the presence of roots at
some seasonal frequencies in seasonally unadjusted
data, while the same is not true for seasonally
adjusted data, indicating
that the X-11 method does remove seasonal unit roots, if any, in the raw
data. The presence of unit roots at seasonal frequencies
found in seasonally
unadjusted
data,
however, turns out to be sensitive to the removal of deterministic
seasonality.
Also, there are some
differences
in the test results between real and nominal
magnitudes.
For example, seasonally
unadjusted real per capita GDP possibly has a unit root at the biannual frequency, while nominal per
capita GDP does not. We also find that most series, with the exception of the narrow money stock
(Ml) and GDP, do not contain a seasonal unit root at the annual frequency.
2. Testing procedure
Tests and issues surrounding
the existence of unit roots in seasonally adjusted data are so well
known that there is little need to dwell on the main points. A common test procedure is to generate
the so-called augmented Dickey-Fuller
(DF) test by estimating the following regression:
P
Ax, = (Y+ ,f?t + 7x,-i
+ c a,Ax,_, + ~1,
j=l
(1)
where x is a time series, often expressed in the logarithm of the levels, t is a time trend, and A is the
difference operator (i.e., Ax, - x,-i). Under the null hypothesis, series x is postulated to have a unit
root with no drift, acceptance
of which requires that the t-statistic
on the coefficient
7 not be
statistically
different from zero (as well as /? if a trend is included). Critical values for this ‘t’ test
may be found in Fuller (1976, Table 8.5.2.). Since the order of the autoregression
is unknown it was
chosen on the basis of the well known Akaike and Schwarz information
criteria. Drawbacks with the
DF test include the possible presence of a moving average term in E, in eq. (l), and the failure of the
autoregressive correction to increase with sample size [see Schwert (1987)]. Accordingly,
the unit root
test suggested by Stock and Watson (1986) was also implemented.
For the seasonally unadjusted
series, the procedure developed by Hylleberg, Engle, Granger and
Yoo (1990; hereafter HEGY) was used, which consists in estimating the following regression:
p is a constant, t a time trend, ‘k(L) is a polynominal
autoregression
correction terms as in eq. (1)
difference
D, are the deterministic
seasonal dummies and A4x, = x, - x,_~, that is, the fourth-order
215
H.S. Lee, P. L. Siklos / Unit roots and seasonal unit roots
of series x,. The terms Y,,, to Y,,, reflect
decomposed as follows, namely:
(1 - L)(l
the fact that
+ L + L2 + L3)x, = (1 - L)Y,,,,
a fourth-order
difference
may be further
(3)
(1-L)(1-L+L2-L3)X,=(1+L)Y2,r,
(4)
(1 + L2)(1 - LZ)x, = (1 + L*)Y3,r.
(5)
Eqs. (3) to (5) define the existence of four possible unit roots in seasonally unadjusted
data. Eq. (3)
represents that part of x, free of seasonal unit roots, while expressions (4) and (5) suggests that a unit
root can exist at the biannual
and annual frequencies,
respectively. Therefore, in estimating eq. (2)
the unit root found in seasonally adjusted data signifies accepting the null that ZZ, is zero. A finding
that II2 is zero implies the existence of a unit root at the twice yearly seasonal cycle, while, if both
II3 and II, are zero, there is a unit root at the annual seasonal cycle. HEGY (1990) give critical
Miron (1990) have recently
values for the foregoing test based on quarterly data, while Beaulieu-and
extended the procedure to monthly data.
3. Empirical results
Given the large volume of test results only a summary is provided in table 1. The detailed statistics
are available upon request. All the data are from CANSIM (Canadian
Socio-Economic
Information
Management
System), and are also available
upon request. The table indicates
for the series
considered
whether they are integrated
of order one, that is Z(l), at the zero and seasonal
frequencies. The results for seasonally unadjusted
data may be influenced
by whether or not some
seasonality is deterministic
in nature. Accordingly,
eq. (2) was estimated with and without seasonal
dummies. If seasonality indeed arises regularly, it may be better captured by the practice of adding
deterministic
seasonal dummies, when no unit roots appear at the seasonal frequencies.
The unit root tests for all the series listed in table 1 were applied to data in the logarithm of the
levels, except for the unemployment
rate and interest rate series which, following previous convention, are in levels. Whenever inference was sensitive to either the type of unit root test conducted, or
to the lag length of the autoregressive
correction factors in eqs. (1) and (2) an asterisk appears next
to the outcome of the test. Finally, all inference is based on the 5% significance
level.
A comparison of columns (l), (2), and (5) in table 1 reveals that the unit root finding in seasonally
adjusted data is not generally affected by seasonal adjustment
at the source. Thus, a series x, remains
Z(1) at the zero frequency whether the data are in raw or adjusted forms. One notable exception,
however, is the unemployment
rate which is Z(1) at the zero frequency in seasonally adjusted data
but is Z(0) for unadjusted
data. By contrast, and unlike Ghysels findings based on US data, ’ GDP
in its various representations
is consistently
Z(1).
The apparent difference in the behaviour
of GDP versus the unemployment
rate is interesting
because it suggests that while shocks to GDP are permanent,
shocks to the unemployment
rate may
be transitory. Blanchard and Quah (1989) have recently built and tested a model with such features
by supposing that aggregate supply shocks are transmitted
through the unemployment
rate while
aggregate demand side shocks influence output.
’ Ghysels
does not, however, use the HEGY test to arrive at his conclusions.
For the unemployment
series, one would
conclude the series to be I(1) in the case where no seasonal dummies are included and when the autoregressive
correction
term is not of order one.
276
H.S. Lee, P. L. Siklos / Unii roots and seasonal unit roots
Table 1
Unit roots and seasonal
unit roots in Canadian
Series
frequency:
Seasonally
adjusted
(1)
Nominal GDP
I(I)
Real GDP
I(I)
Per cap nom GDP
I(1)
Real per cap GDP
I(1)
Implicit price deflator
I(1)
Imports
I(1)
Real imports
I(1)
Exports
I(1)
Real exports
I(1)
Money supply (Ml)
I(1)
Real Ml
I(l)
Unemployment
rate
I(1)
Stock prices
I(1)
Treasury bill rate (TB)
I(1)
Ex post real TB
I(0)
Long term bond rate (LTBRJ(1)
Ex post real LTBR
I(0)
Exchange rate
I(I)
Real exchange rate
I(I)
macroeconomic
time series. a
Seasonally unadjusted
no seasonal dummies
Seasonally unadjusted
seasonal dummies
Zero
Biannual
Annual
Zero
Biannual
Annual
(2)
(3)
(4)
(5)
(6)
(7)
I(1)
I(I)
I(1)
I(1)
I(1)
l(I)
I(0) b
I(1)
I(1)
I(1)
I(1)
I(0) h
I(1)
I(1)
I(0)
I(1)
I(0)
I(I)
I(1)
I(O)
I(l)
I(0)
I(1)
I(0)
I(1)
I(I)
I(1)
I(0)
I(1)
I(1)
Ill)
I(O)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0) b
I(I)
I(1)
I(1)
I(1)
I(0)
I(0)
I(O)
I(O)
I(0)
I(1)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0)
I(l)
I(I)
I(1)
I(I)
I(1)
I(1)
I(O)
I(I)
I(I)
I(1)
I(1)
I(0)
I(I)
I(I)
I(0)
I(1)
I(0)
I(I)
I(I)
I(0) h
I(1)
I(0)
I(1)
I(0)
I(0)
I(0)
I(0)
I(I)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0)
I(O)
I(I) b
I(0) b
I(I)
I(I)
l(I)
I(0)
I(0)
I(0)
I(0)
I(0)
I(0)
l(0)
l(0)
l(0)
I(0)
I(0)
l(0)
l(O)
I(0)
I(0)
Sample is 1947 (I)-1984 (IV), except for the money supply
unadjusted;
1966 (I)-1987 (Iv) adjusted], the exchange
Prices [Standard
and Poor’s 500 Index, 1956 (I)-1987
Government
of Canada bonds, 1948 (Q-1987 (IV)]. In all
in logarithms of the levels, except for the unemployment
levels,
Signifies that inference is sensitive to either the lag length
type of unit root test conducted.
b
b
b
b
b
[1953 (I)-1987 (IV)], the unemployment
rate [1962 (I)-1987 (IV),
rate [closing spot versus U.S. dollar, 1970 (ll)-1987
(Iv)], Stock
(Iv), and the long-term
bond rate (yield on 10 years and over
cases inference is based on a 5% level of significance. All series are
rate, the Treasury bill rate, the long-term bond rate, which are in
specification,
based on the Akaike
and Schwarz
criteria,
or to the
A glance down columns (4) and (7) suggests that, regardless whether one believes that some
seasonality
is deterministic
in nature, there are relatively few macroeconomic
time series which
appear to process a unit root at the annual seasonal cycle. Hence, the A4x, filter recommended
by
Box and Jenkins (1970) appears not to be warranted.
A few more series display a unit root at the
biannual cycle (column (3)) but this result is, as expected, sensitive to the removal of deterministic
seasonality
(column (6)). Note also that, as one would expect, interest rates and the nominal
exchange rate possess no seasonal unit roots, while ex post real interest rates 2 are Z(0).
Finally, there are some differences, at the seasonal frequencies,
between time series expressed in
nominal and real terms. For example, nominal and real GDP, either in log levels or in per capita
forms, are Z(0) and Z(l), respectively,
at the biannual
seasonal cycle. Thus, using nominal series
which are then routinely deflated may have consequences
in inferences about economic relationships
[see also Ghysels and Karangwa (1989)].
* Computed
as the nominal
interest
rate level less the log first difference
in the consumer
price index
H.S. Lee, P.L. Sikh
/ Unit roots and seasonal unit roots
277
4. Conclusions
The use of seasonally unadjusted
data, with one possible exception, does not influence the finding
of a unit root in seasonally adjusted data. However, there are few series which exhibit a seasonal unit
root at the annual cycle thereby casting doubts on the appropriateness
of a commonly
used data
filter for unadjusted
data advocated by Box and Jenkins (1970). Nevertheless,
in some instances,
most notably GDP and the money supply (Ml), a seasonal unit root is present, suggesting perhaps
the possibility that some seasonal unit roots may be common. In other words, some series may be
cointegrated
at some of the seasonal frequencies.
This line of research was initiated
by Engle,
Granger, and Hallman (1989) and HEGY (1990) and is currently being pursued by, among others,
Lee (1989) Lee and Siklos (1990), and Ghysels, Lee, and Siklos (1990).
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