296 I.A. Moosa International Review of Economics and Finance 8 1999 293–304
g
j ,t
5 g
j ,t
2
1
cosl
j
1 g
j ,t
2
1
sinl
j
1 k
j ,t
8 g
j ,t
5 g
j ,t
2
1
sinl
j
1 g
j ,t
2
1
cosl
j
1 k
j ,t
9 where j 5 1, . . . , s2 2 1, l
j
5 2pjs and
g
j ,t
5 2g
j ,t
2
1
1 k
j ,t
, j 5 s2 10
where k
j ,t
z NID0, s
2
k
and k
j ,t
z NID0, s
2
k
. Again, the assumption s
2
k
5 s
2
k
is imposed. One advantage of this specification is that it allows for smoother changes
in the seasonals Harvey Scott, 1994, p. 1328. The extent to which the trend, seasonal, and cyclical components evolve over time
depends on the values of s
2
h
, s
2 z
, s
2
k
, s
2
v
, u, and r which are known as the hyperparamet- ers. The hyperparameters as well as the components can be estimated by maximum
likelihood using the Kalman filter to update the state vector. This procedure requires writing the model is state space form. Related smoothing algorithms can be used to
obtain the estimates of the state vector at any point in time within the sample period Harvey, 1989, Chapter 4.
The relationship between the cyclical components of unemployment and output, u
c
and y
c
, may be written in a stochastic form as m
c t
5 a y
c t
1 m
t
11 Eq. 11 assumes that the relationship is totally contemporaneous, which may not be
plausible theoretically. It may also be inadequate empirically owing to the omission of short-run dynamics. Following Hendry, Pagan, and Sargan 1984 the dynamic
model used here is the ADL model
m
c t
5
o
m i
5
1
d
i
m
c t
2
i
1
o
m i
5
a
i
y
c t
2
i
1 y
t
12 where the contemporaneous impact or short-run effect of output on unemployment
is measured by the coefficient a while the long-run effect is measured by calculating
a function of the coefficients, v, which is given by Eq. 13: v 5
o
m i
5
a
i
1 2
o
m i
5
1
d
i
13 The question that arises here is whether Okun’s coefficient is a
or v. The tendency is to define Okun’s coefficient as measuring the long-run effect, because the relationship
between unemployment and output is not necessarily contemporaneous. Hence, Okun’s coefficient is taken to be v.
5
3. Data and empirical results
The methodology outlined in the previous section is applied to a sample of U.S. quarterly data covering the period 1947:1–1992:2 as reported in Gordon 1993, Table
I.A. Moosa International Review of Economics and Finance 8 1999 293–304 297
Table 1 The estimated parameters of cyclical components
Parameter Unemployment
Output s
2
h
0.060 0.000
s
2
§
0.000 2.11 3 10
2
6
s
2
v
0.059 5.58 3 10
2
5
u 0.355
0.328 r
0.950 0.911
s˜ 0.380
0.0098 R
2
0.95 0.99
R
2 s
0.21 0.109
N 33.38
22.89 H
0.54 0.66
r 1
0.51 0.11
r 12
2 0.08
2 0.12
DW 0.98
1.78 Q
72.04 17.02
A-2. The output variable is real GDP measured at 1987 prices. The cyclical compo- nents of unemployment and output are extracted by the Structural Time Series Ana-
lyser, Modeller and Predictor STAMP version 5.0. The procedure requires the estima-
tion of the univariate time series model represented by Eq. 1 for the level of unemployment and the logarithm of output. For details, see Koopman, Harvey, Door-
nik, and Shephard 1995. The maximum likelihood estimates of the parameters are reported in Table 1. Also
reported are some measures of goodness of fit and diagnostic test statistics. The goodness of fit measures include the coefficient of determination R
2
and the modified coefficient of determination R
2 s
. The latter is calculated on the basis of the seasonal mean. Also reported is the standard error s˜, which is calculated as the square root
of the one-step ahead predictive error variance. Diagnostic statistics are reported for normality, heteroscedasticity, and serial correlation. The diagnostic for normality N
is the Bowman-Shenton 1975 test for normality, which is based on the departure of the third and fourth moments from their expected values under normality. It is distrib-
uted as x
2
2. The diagnostic for heteroscedasticity H is calculated as the ratio of the squares of the last h residuals to the squares of the first h residuals where h is
the closet integer to one-third of the sample size. It is distributed as Fh,h. The autocorrelation coefficients of the residuals with lags one and 12 are also reported.
For serial correlation, the diagnostics include the DW statistic and the Ljung-Box Q
-statistic Ljung Box, 1978 calculated on the basis of the first 12 autocorrelation coefficient. It is distributed as x
2
7. The results show that the frequency of the unemployment and output cycles are rather close and so are the damping factors on
the amplitude of the two cycles. These estimates imply that the cycle period for unemployment is 17.66 4.416 years while the cycle period for output is 19.15 4.78
years. Moreover, both models seem to be well determined and pass the diagnostics.
298 I.A. Moosa International Review of Economics and Finance 8 1999 293–304
Fig. 1. Unemployment rate actual and trend.
Fig. 2. Logarithm of output actual and trend.
I.A. Moosa International Review of Economics and Finance 8 1999 293–304 299
Fig. 3. Cyclical unemployment.
Finally, the estimated value of s
2
h
is high for unemployment and zero for output, implying a volatile trend for the former and a smooth trend for the latter.
The estimated trends and cyclical components are shown in Fig. 1–4. Fig. 1 and 2 show the actual time series and the extracted trends for unemployment and the
logarithm of output respectively. Figs. 3 and 4 show the extracted cyclical components of the two series. Notice, however, that the cyclical component of output as extracted
from the logarithmic univariate model is scaled by multiplying it by 100. This is because unemployment is measured in percentage terms while output is measured in natural
logarithms. Since the cyclical component of output is measured as a fraction, it requires this kind of adjustment to be measured in percentage terms see Gordon, 1993, pp.
322–323.
Now that observations on the cyclical components are available, it is possible to proceed with the estimation of Okun’s coefficient. When Eq. 11 is estimated by
OLS, the following results are obtained m
c t
5 2 0.379y
c t
226.31 R
2
5 0.79
DW 5 0.60
Although Okun’s coefficient is statistically significant and correctly signed, the fitted equation suffers from serial correlation, most likely due to the omission of short-run
dynamics. It is widely accepted now that serial correlation should not be “corrected,”
300 I.A. Moosa International Review of Economics and Finance 8 1999 293–304
Fig. 4. Cyclical output.
but rather the equation should be re-specified.
6
Thus, the estimate of the coefficient on the basis of a static regression is not reliable.
Because it is more appropriate to estimate the short-run and long-run effect from a properly specified dynamic model, Eq. 12 is estimated by OLS. The equation is
estimated for a number of large lengths to find out if the results are robust with respect to the lag selection. The choice here is to use lags between 2 and 5.
7
The estimation results are reported in Table 2 which also reports the diagnostics for serial
correlation, S, functional form, F, heteroscedasticity, H, and normality, N.
8
These test statistics are distributed as x
2
with 4, 1, 1, and 2 degrees of freedom respectively. The marginal significance levels are given in parentheses underneath the estimated values
of the test statistics. The first thing to notice about these results is the absence of serial correlation as
a result of introducing short-run dynamics, even if the lag is two. Moreover, the results are robust with respect to the lag length, implying that a lag of 2 is sufficient to capture
all of the dynamics. The estimate of the short-run impact coefficient is 20.160, which is substantially lower than the value obtained by Okun 1962 or that obtained from
a static regression. As previously pointed out, Okun’s coefficient is more appropriately defined as the long-run coefficient. The estimated value of the long-run coefficient is
around 20.38, which is, perhaps surprisingly, what is obtained from the static regres- sion. It is interesting to note that the estimates reported in Table 2 are close to
the estimates obtained by Weber from his static and cointegrating regressions. The estimates obtained by Weber from the dynamic models are much smaller because he
I.A. Moosa International Review of Economics and Finance 8 1999 293–304 301
Table 2 OLS estimates of the ADL model [Eq. 12]
a
m a
v R
2
S 4
F 1
H 1
N 2
2 2
0.160 2
0.366 0.97
7.61 2.28
2.09 0.21
212.21 212.10
0.10 0.13
0.12 0.90
3 2
0.162 2
0.373 0.97
7.90 2.03
2.47 0.56
212.03 213.06
0.09 0.15
0.14 0.76
4 2
0.162 2
0.386 0.97
6.69 1.79
2.42 0.95
212.31 214.59
0.23 0.18
0.14 0.62
5 2
0.160 2
0.385 0.97
2.54 1.51
2.43 0.62
212.07 215.40
0.63 0.22
0.15 0.73
a
The t statistics are given in parentheses underneath the estimated coefficients a and v. The marginal
significance levels are given in parentheses underneath the diagnostic statistics S, F, H, and N.
defined Okun’s coefficient to exclude the impact coefficient. This explains the differ- ence between Weber’s two sets of results.
9
We now turn to the issue of structural stability to examine whether there is a structural break in the relationship following the oil shock of 1973. To start with , the
model with five lags seems to be structurally stable as it passes the CUSUM test as shown in Fig. 5. The model is then subjected to the Chow test and the predictive
failure test Chow’s second test taking the fourth quarter of 1973 as the breakpoint. The model passes both of these tests with test statistics of 1.21 and 0.97, distributed
Fig. 5. Cumulative sum of recursive residuals.
302 I.A. Moosa International Review of Economics and Finance 8 1999 293–304
as F11,155 and F74,92 respectively. Hence, the evidence shows that there is no structural break in the relationship between cyclical output and cyclical unemployment.
4. Conclusion
