160 B
. Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
crash when the Hessian becomes singular, or it will stop with a biased estimate of b when the cost in
terms of the implied coefficients a and a which also depend on b becomes too large.
1 2
3. Empirical findings
We illustrate the point made in the previous section in an analysis of consumption data of 26 OECD countries. The data are yearly and range from 1960 to 1996, they are taken from the OECD System of
National Accounts. We confine the analysis here to univariate methods, since the inclusion of further conditioning variables hardly changes the qualitative properties of the lag polynomial in consumption.
For extended results see Gaggermeier and Lucke 1999. We add a constant for regressions in first differences to account for the fact that real world data are generated by growing economies.
We initially test the unit root property of log consumption which is a common feature of both the Euler equation and the solution to the full system of first order conditions. We apply standard
augmented Dickey–Fuller tests ADF-tests, where the regression equation contains a linear trend in order to account for the upward trend in consumption under the alternative. The results are suppressed
here, but available on request. They indicate that the unit root hypothesis cannot be rejected at the 5 level of significance for any of the countries.
Next we test whether EEP 2 or FSP 2 are supported by the data. We impose the unit root property to exploit the fact that any integrated ARp process
p
ln C 5 a 1
O
a ln C 1
´
t i
t 2i t
i 51
can be rewritten as an ARp21 process in first differences:
p 21
Dln C 5 a 1
O
f Dln C 1
´
t i
t 2i t
i 51 p
where f 5 2
o a . This transformation yields an equation in stationary variables so that
i j 5i 11
j
inference in OLS regressions is standard. Note that in first differences EEP 2 implies a lag polynomial of even order, while FSP 2 implies
odd order, unless the unit root is the only unstable root, i.e. the lag order in first differences is zero. We begin by evaluating the Schwarz 1978 information criterion SC for the first differences of log
consumption to get an impression of the appropriate lag orders, cf. Table 1. With the notable exception of Germany, all suggested lag lengths are either zero or odd. While we
find evidence for only one country in which the appropriate lag length may be even, precisely half of the countries show evidence for odd and nonzero lag lengths, which are incompatible with EEP 2.
To complement this rather mechanical investigation, we estimate simple autoregressions by OLS. To focus on the main issue, we again neglect conditioning variables like interest rates and labor and
estimate an autoregression in consumption only. For extended results, the interested reader is referred to our discussion paper Gaggermeier and Lucke, 1999. Starting from a total of five lags and
successively deleting insignificant regressors we aim at finding the most parsimonious representation with white noise residuals. The estimation output of this exercise is presented in Table 2.
We find odd lag orders in 17 out of 26 cases, while we find an even lag order in only one case,
B . Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
161 Table 1
Schwarz information criterion for consumption growth rates Lag length
1 2
3 4
5 Preferred
lag order Australia
28.62 28.55
28.46 28.38
28.30 28.33
Austria 27.91
27.81 27.71
27.64 27.55
27.46 Belgium
27.76 27.89
27.77 27.78
27.77 27.66
1 Canada
27.54 27.75
27.63 27.52
27.41 27.32
1 Denmark
27.11 27.03
26.97 26.89
27.01 27.04
Finland 26.70
26.85 26.77
26.68 26.70
26.59 1
France 28.48
28.73 28.64
28.64 28.53
28.42 1
Germany 27.79
28.19 28.32
28.29 28.20
28.08 2
Greece 27.38
27.58 27.53
27.59 27.48
27.39 3
Iceland 25.31
25.23 25.25
25.14 25.04
24.95 Ireland
26.78 26.76
26.68 26.62
26.51 26.40
Italy 27.45
27.64 27.53
27.55 27.53
27.42 1
Japan 27.11
27.18 27.08
27.07 26.96
27.85 1
Luxembourg 27.82
27.98 27.87
27.79 27.70
27.59 1
Mexico 26.37
26.32 26.21
26.10 26.99
26.87 Netherlands
27.64 28.20
28.11 28.00
27.92 27.89
1 New Zealand
26.99 26.97
26.95 26.84
26.77 26.67
Norway 27.30
27.24 27.13
27.07 27.00
26.88 Portugal
25.73 25.69
25.58 25.55
25.56 25.58
South Korea 27.00
27.27 27.22
27.12 27.01
27.05 1
Spain 27.28
27.77 27.67
27.56 27.47
27.43 1
Sweden 27.65
27.72 27.61
27.50 27.43
27.32 1
Switzerland 28.16
28.55 28.52
28.45 28.33
28.22 1
Turkey 25.70
25.60 25.51
25.47 25.49
25.43 United Kingdom
27.37 27.43
27.39 27.30
27.25 27.14
1 USA
28.17 28.14
28.10 27.99
27.87 27.76
which is again Germany. In eight cases the appropriate lag order is zero, which indicates the absence of explosive roots in the Euler equation, i.e. the general solution of the Euler equation does not violate
the transversality condition. Since we imposed the unit root and accordingly deleted the linear trend from the regressions, it is
not too surprising to find some cases in which the lag structure of the growth rate autoregressions is in disaccord with the lag structure of the ADF-regressions. There are discrepancies in eight out of 26
cases: five of these concern decisions of whether lag order zero or lag order one is appropriate. Hence, either way the specifications are compatible with the maintained hypothesis that observed data reflect
the full set of first order conditions. In the other three cases Belgium, Greece, and Italy, both the ADF test and the growth rate autoregression suggests nonzero and odd lag orders, all of which are
compatible with FSP 2, but incompatible with EEP 2.
Along the same lines, observe that in only four cases Ireland, Italy, New Zealand, and USA the lag structure found in the growth rate autoregressions differs from the lag structure suggested by the
Schwarz criterion. This is due to the fact that the autoregressions based on SC-lag lengths did not result in white residuals. However, we again conclude that both the SC-lag lengths and the regression
lag lengths are compatible with FSP 2.
162 B
. Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163 Table 2
a
Growth rate autoregressions
Country a
f f
f P-value
Lag
1 2
3
Q4 order
Q8 Australia
0.021 0.644
Zero 0.002
0.551 Austria
0.028 0.287
Zero 0.003
0.399 Belgium
0.014 0.436
0.124 Odd
0.005 0.157
0.315 Canada
0.012 0.483
0.972 Odd
0.004 0.147
0.671 Denmark
0.020 0.366
Zero 0.005
0.141 Finland
0.013 0.494
0.361 Odd
0.006 0.147
0.605 France
0.008 0.645
0.355 Odd
0.004 0.127
0.170 Germany
0.017 0.791
0.392 0.303
Even 0.005
0.166 0.164
0.357 Greece
0.005 0.395
– 0.405
0.871 Odd
0.006 0.151
0.153 0.697
Iceland 0.030
0.273 Zero
0.011 0.254
Ireland 0.018
0.306 0.457
Odd 0.007
0.168 0.738
Italy 0.009
0.371 –
0.247 0.445
Odd 0.008
0.151 0.150
0.267 Japan
0.022 0.446
0.431 Odd
0.008 0.149
0.077 Luxembourg
0.015 0.438
0.833 Odd
0.005 0.160
0.873 Mexico
0.014 0.558
Zero 0.006
0.844 Netherlands
0.016 0.425
0.604 Odd
0.005 0.150
0.527 New Zealand
0.009 0.295
0.630 Odd
0.005 0.166
0.846 Norway
0.024 0.192
Zero 0.004
0.175 Portugal
0.034 0.169
Zero 0.009
0.180 South Korea
0.040 0.308
0.871 Odd
0.012 0.198
0.638 Spain
0.009 0.651
0.542 Odd
0.005 0.118
0.272 Sweden
0.006 0.452
0.838 Odd
0.004 0.149
0.900 Switzerland
0.004 0.660
0.416 Odd
0.003 0.126
0.247 Turkey
0.019 0.352
Zero 0.009
0.474 United Kingdom
0.013 0.395
0.213 Odd
0.005 0.160
0.648 United States
0.015 0.328
0.760 Odd
0.004 0.162
0.965
a
Standard errors in brackets.
The only lag order in violation of FSP 2 is hence lag order two for Germany. But note that even in this case the regression results are in accord with FSP 3, while they are incompatible with EEP 3: note
that for all countries with lag order larger than zero we find the coefficient of the highest lag to be
B . Lucke, C. Gaggermeier Economics Letters 71 2001 155 –163
163
smaller than one and significantly so. Thus EEP 3 is not at all recognizable in the data, and any attempt to take advantage of such a Euler equation property in estimation is sure to fail. The evidence
thus suggests that neglect of the transversality condition as it is inherently done in many attempts to estimate Euler-equations fundamentally misspecifies the dynamics of observed real world time series
8
even if the underlying model is true. .
4. Conclusions
