Table 5.1 shows that the real annual return for holding the MSCI stock portfolio in Denmark is 7.3% calculated as the sum of the quarterly returns, as shown in section 4.1. The real return for a 3 year period is 21.6% and the 5 year return is 39.0%. The fact that quarterly returns cannot just be multiplied by the number of quarters is due to the method of calculation of the returns over the different periods. The sum of the quarterly returns is taken before the average is calculated and therefore very high returns will impact the sum longer for the longer periods and therefore also impact the average more. The same is true for very low returns, but from the numbers it can be seen, that the impact of the high returns over the long period has been stronger than over the short periods. The standard deviation of the returns increases more from 4 quarters to 12 quarters than it does between 12 and 20 quarters. The standard deviation for the 12 and 20 quarter returns are very similar, which can be due to the flattening effect of summing over both 12 and 20 quarters. When summing over many observations such as 20 and then just changing one observation at the time, the effect on the variation will be less profound since 19 of the 20 observations are the same from one period to the next. In contrast the one new observation will potentially have more impact on the total variation if the number of summing observations is only 4. On the other hand, one would expect the variation to increase with the horizon as the average returns increase.
Additionally, it can be seen from table 5.1 that the py-ratio is larger than the px- and pz-ratios in absolute terms. This is due to the fact that the GDP is larger than the export and import, and this should always be the case. Generally, it can be seen that the px and pz- ratios are almost at the same level and have similar standard deviations, whereas the py-ratio have a higher standard deviation. This is due to the higher value of the py-ratio. Furthermore, the ratios are less volatile than the returns, when accounting for the respected values.
From the correlations, one can see that the ratios are quite correlated, but the px- and pz-ratios are extremely correlated with a coefficient of 0.97. The high correlations between the ratios can also be seen in figure 5.1.1, where the ratios do move very similar. The correlation between the px- and pz-ratios are very high because the export and import often move together, whereas the GDP also depends on consumption, investments, and government spendings, which do not vary as much with the export and import as they do with each other.
The correlation between the returns is highest between the periods closed to each other, but generally not very high. This can be seen in figure 5.1.2, where the different returns seem to move somewhat differently. Lastly, the correlations between the ratios and the returns increase with time, in that the 20 quarter returns have higher correlations with the ratios than the 4 quarter returns. This is what would be expected since the predictability is expected to increase over longer periods.
Table 5.1
Denmark py px pz 4 Quarterly returns
12 Quarterly returns
20 Quarterly returns
Average -4,608 -3,481 -3,400 0,073 0,216 0,390
Standard Deviation 0,495 0,320 0,301 0,264 0,387 0,404
py 1
px 0,69 1
pz 0,76 0,97 1
4 Quarterly returns 0,27 0,37 0,44 1
12 Quarterly returns 0,41 0,50 0,58 0,50 1
20 Quarterly returns 0,52 0,57 0,67 0,28 0,62 1
DF tau -0,77 -2,11 -2,24 -3,85 -2,40 -2,16
p-value (DF) 0,825 0,240 0,193 0,003 0,142 0,220
ADF tau, 1 lag -1,47 -2,95 -3,14 -6,14 -3,59 -3,23
p-value (ADF 1 lag) 0,547 0,042 0,026 <,0001 0,007 0,021
ADF tau, 2 lag -1,62 -3,20 -3,39 -7,71 -3,93 -3,68
p-value (ADF 2 lag) 0,469 0,022 0,013 <,0001 0,002 0,006 Correlations
Unit root test
From the figure 5.1.1 below, where the ratios are plotted against the time, it seems that the ratios move somewhat smoothly, indicating that they may have a unit root. The px- and pz-ratios are stable at the same level though out the period, whereas the py-ratio increases over the period. It is possible that the py-ratio has a structural break around 1997, as the line seems to be at a generally higher level after 1997 than before. This will be investigated further in the section with robustness tests. For testing the stationarity of the time series the Dickey-Fuller and the Augmented Dickey-Fuller test for a random walk with drift will be used, since it
allows for the time series to have a mean different from zero, when they are stationary, which seems to be the case. The test of stationarity in the variables can be seen in the bottom of table 5.1. The idea behind the Dickey-Fuller test for a random walk without drift is that if ρ in the equation At =ρAt−1+ut is 1, Ytis nonstationary131, whereas if ρ is less than one in absolute terms, when At is stationary132. If At−1 is subtracted on both sides and rearranged, the equation is as follow
( )
t t t
t t t
t
t t t t
t
u A A
u A A
A
u A A A
A
+
=
∆
+
−
=
−
+
−
=
−
−
−
−
−
−
−
1
1 1
1 1 1
1 δ
ρ ρ
Where δ =
(
ρ−1)
For the Dickey-Fuller test for a random walk with drift, the equation to be tested will be the following, taken into account the drift133: ∆At =β1+δAt−1+ut
The Dickey-Fuller test runs this equation under the null hypothesis that δ =0, and therefore that ρ =1, which is that the time series are nonstationary134. The test statistics of this is the τ (tau) statistic and can be seen in the first line of the unit root test in table 5.1. The critical value for this test is -1.95 at the 5% significance level, and it can be seen that for the ratios, especially the py-ratio, the δ is far from being significantly different from zero and the time series have a unit root. For the py-ratio, the Augmented Dickey-Fuller test also concludes that the time series has a unit root. However, the Augmented Dickey-Fuller test with one lag rejects the null hypothesis of a unit root for the px- and pz-ratios. The Dickey-Fuller test assumes that the error term ut is uncorrelated. If this is not the case, one can use the
Augmented Dickey-Fuller test, which includes the lagged value of the dependent variable and one should include enough lags so that the error term is no longer correlated. Therefore, the Dickey-Fuller equation ∆At =β1+δAt−1+ut is tested for autocorrelation using the Breusch-Godfrey (LM) test135 and the AR(1) scheme has a value of 22.6 which is highly significant136. Therefore, the error term of the Dickey-Fuller equation is correlated and the Augmented Dickey-Fuller should be used137. This has the following equation for the random walk with
131 A will represent an undefined time series
132 Gujarati, 2003, page 802
133 Gujarati, 2003, page 815
134 Gujarati, 2003, page 814
135 This test will be discussed further in the autocorrelation section
136 All tests can be found in appendix B
137 The right test for stationarity is marked in the table
drift138: t
m
i
i t i t
t A A
A =β +δ + α ∆ +µ
∆
∑
=
−
− 1 1
1 . One should estimate this equation with one lag and test for correlation in the error term. If correlation is present, one should add another lag and the procedure should be repeated. Lastly, from figure 5.1.2 it can be seen that the Autocorrelation Coefficient between the lags for the py-ratio, the ACF, is high up to the lags of 12 or 13, which is a strong indicator of the time series being nonstationary. The Partial Correlation, the PACF, is very high in lag one and just over the level of significance in lag two, which again is a sign of a unit root139. The implications of the unit root in the py-ratio and the correction will be discussed in section 5.5 about estimated ratios.
Figure 5.1.1
Denmark - Ratios
-6,00 -5,00 -4,00 -3,00 -2,00 -1,00 0,00
Q2 1970 Q2 1972 Q2 1974 Q2 1976 Q2 1978 Q2 1980 Q2 1982 Q2 1984 Q2 1986 Q2 1988 Q2 1990 Q2 1992 Q2 1994 Q2 1996 Q2 1998 Q2 2000 Q2 2002 Q2 2004 Q2 2006 Q2 2008 Q2 2010
Ratios Time
py px pz
138 Gujarati, 2003, page 817
139 The figures for the other ratios can be seen in appendix B
Figure 5.1.2
From the figure 5.1.3 below, where the returns are plotted against the time, it can be seen that the returns are fluctuating and seem stationary. The unit root test of the returns can be found in table 5.1, and δ is significantly different from zero at a 1% level for the 4 quarter return and it can therefore be concluded that this time series is stationary. However, δ is not significantly different from zero at a 10% level for the 12 quarter return and the 20 quarter return using the normal Fuller test for a random walk with drift. The AR(1) scheme for the Dickey-Fuller test for 12 quarter return is 26.1, which again is highly significant, and therefore one should use the Augmented Dickey-Fuller test. When testing for autocorrelation in the Augmented Dickey-Fuller test with one lag the AR(1) scheme is 4.06, which is significantly different from zero at a 5% level, hence one should use the Augmented Dickey-Fuller test with two lags, and on the basis of this it can be concluded that the 12 quarter return is stationary. For the 20 quarter return the Augmented Dickey-Fuller test with two lags should be used and it can be concluded that this is also stationary.
Figure 5.1.3
Denmark - Quarterly returns
-1 -0,5 0 0,5 1 1,5
Q2 1970 Q2 1972 Q2 1974 Q2 1976 Q2 1978 Q2 1980 Q2 1982 Q2 1984 Q2 1986 Q2 1988 Q2 1990 Q2 1992 Q2 1994 Q2 1996 Q2 1998 Q2 2000 Q2 2002 Q2 2004 Q2 2006 Q2 2008 Q2 2010
Quarterly returns Time
4Q 12Q 20Q