4.3. The Data
4.3.2. Predetermined Information Variables
Similarly to Ilmanen (1995) a 36-month window and a smoothing coefficient of 0.9 are used.50 However, we have considered other possibilities (a 60-month and 83- month window and a smoothing coefficient of 0.8) and the results were quite similar.
As we mentioned previously, the term spread is the difference between the yield of a long-term bond and the yield of a short-term bond (a short-term nominal rate) and the real bond yield is the difference between the long-term bond yield and the expected inflation rate over the remaining life of the bond. As the long-term bond yield we considered the yield on a 10-year Government bond (or approximately), information that we obtained through the Central Banks.51 It would be more appropriate to also use a Government bond rate as the short-term rate. However, the majority of the countries considered do not have a liquid Treasury bill market. For the countries which we could not obtain that rate, we used the 3-month Interbank offered rate.52 The inflation rate is the recent year-on-year inflation rate. In assuming this as the expected inflation rate we implicitly assume that the inflation rate follows a random walk. Once again we have taken into account the publication lag for the CPI53. Both variables are in annual rates.
The variable yield spread in relation to the German bond (DM spread) is measured by the difference between the yield on the domestic 10-year Government bond (or approximately) and the yield on the equivalent German bond. Finally, the
50 The smoothing coefficient value of 0.9 was chosen in order to capture business cycle effects, and thus reduce the impact of short-term fluctuations.
51 This is the most commonly used maturity for representing a long-term bond yield. For Portugal we used the yield on Treasury bonds with remaining maturity between 108 and 126 months; for Spain the yield on a 10-year government bond; for Italy the yield on the 10-year BTP (Buoni Poliennali del Tesoro); for France the ”taux de l`emprunt phare a 10 ans”; for Germany the yield on listed Federal securities with a residual maturity of 9-10 years (only bonds eligible as underlying instruments for futures contracts are included) and for UK the yield on a 10-year Government bond.
52 This was the case for Portugal, Italy, UK and Germany. For France, we used the “taux de référence des bons de trésor à 3 mois” from the Bank of France, and for Spain the rate on 34 to 94 days Treasury bill secondary market obtained from the Bank of Spain.
53 The inflation rate of January is used to calculate the real bond yield for the month of February and this will be used to predict excess bond returns for the month of March and so forth.
dummy for the month of January: it takes a value of 1 if the next month is the month of January and 0 otherwise.
Table 4.2 reports the summary statistics for the information variables.54 All the variables have high first-order autocorrelations (AC1), ranging between 0.742 and 0.987.
Table 4.2 – Summary statistics on the predetermined information variables
In this table we report the mean (Mean), standard deviation (StD) and the first-order autocorrelation (AC1) for each of the information variables over the period January 1994 to November 2000. The Term Spread is the difference between the yield on a long-term Government bond and a short-term bond rate (or the 3-month interbank offered rate). The Real Bond Yield is the difference between the yield on the long-term government bond and the inflation rate. The DM spread is the difference between the yield on a domestic long-term government bond and the yield on an equivalent German bond. These three variables are expressed in annual rates. IRW (inverse relative wealth) is the ratio between the exponentially weighted average of past real wealth and current real wealth.
Note: For Portugal, the predetermined information variables are restricted to the period December 1994 to November 2000.
54 Appendix 4.2 presents the graphics with the evolution of these variables for each of the countries. In particular for the Latin countries, we can see that the variables related with bond yields show a clearly decreasing trend during the sample period.
Portugal Spain Italy France Germany UK Information Variables
Term Spread
Mean 1.148 1.626 0.997 1.617 1.804 0.359
StD 0.616 0.930 1.027 0.850 0.859 1.601
AC1 0.890 0.877 0.942 0.850 0.904 0.974
Real Bond Yield
Mean 4.088 4.160 4.682 4.546 4.086 4.010
StD 2.037 1.658 1.825 0.905 0.745 1.525
AC1 0.948 0.955 0.961 0.924 0.799 0.960
IRW
Mean 0.904 0.875 0.892 0.883 0.883 0.923
StD 0.130 0.113 0.109 0.099 0.085 0.056
AC1 0.902 0.849 0.838 0.869 0.798 0.742
DM spread
Mean 1.461 1.666 2.058 0.197 1.048
StD 1.671 1.646 1.979 0.274 0.586
AC1 0.972 0.987 0.984 0.924 0.923
We tested the stationarity of these variables using the Augmented Dickey-Fuller (ADF) test55 and concluded that, in most of the cases, we cannot reject the null hypothesis of a unit root, as shown in Table 4.3. This table reports the ADF test statistic considering three alternative regressions: with no mean, with a mean and with both a mean and trend. We report the results of the three models considering one lagged value of the dependent variable in order to account for serial correlation in the residuals.
Although not reported, we also considered other numbers of lags and the results were similar.
55 It is important to check whether the series are stationary or not before using them in a regression as standard inference procedures do not apply to regressions that contain integrated regressors. The formal method to test that is through the Unit Root Test. We followed the Augmented Dickey-Fuller test (see Dickey and Fuller, 1979). The null hypothesis is that the series is I(1) and the alternative hypothesis is that the series is stationary or I(0). We reject the null hypothesis of the series being I(1) in favour of the alternative hypothesis of a I(0) series if the test statistic has a more negative value than the MacKinnon critical value (MacKinnon, 1991).
Table 4.3 – ADF test statistics for the predetermined information variables
The ADF test statistic is reported for three alternative regressions: with no mean, with a mean and with both a mean and a trend. The results are relative to the models incorporating one lagged value of the dependent variable. At the bottom of the table, we also report the MacKinnon critical values for the 1%, 5% and 10% level of significance. The predetermined information variables are as defined previously.
Note: In shaded we indicate the cases for which we reject, at the 5 percent level, the null hypothesis of a unit root in favour of the alternative hypothesis of a stationary series.
Given these results, we followed the suggestion of Ferson, Sarkissian and Simin (2003b) and use the variables subtracted by the 12-month moving average56, in order to
56 In monthly data, a 12-month moving average is usually adopted in order to remove possible seasonality in some information variables.
Term Spread Real Bond Yield IRW DM spread ADF test statistic ADF test statistic ADF test statistic ADF test statistic Portugal
no mean -1.329 -2.003 -0.346 -2.713
with a mean -1.910 -1.590 -1.863 -1.849
with a mean and trend -1.912 -1.808 -1.795 -0.614
Spain
no mean -0.932 -0.793 0.209 -1.005
with a mean -1.971 -0.480 -2.132 -0.547
with a mean and trend -2.609 -3.773 -2.065 -1.827
Italy
no mean -1.250 -0.815 -0.090 -1.048
with a mean -1.553 -0.978 -2.343 -0.661
with a mean and trend -1.648 -2.716 -2.606 -1.881
France
no mean -1.380 -0.591 -0.045 -1.164
with a mean -3.195 -1.202 -2.190 -1.758
with a mean and trend -3.184 -2.918 -3.103 -2.769
Germany
no mean -0.722 -0.459 0.106
with a mean -1.299 -1.743 -2.240
with a mean and trend -2.882 -3.307 -2.158
UK
no mean -1.326 -0.972 0.271 -0.798
with a mean -1.125 -0.910 -3.111 -0.821
with a mean and trend -1.862 -2.327 -3.093 -3.949
Critical Values (MacKinnon critical values for rejection of the hypothesis of a unit root)
Portugal no mean with a mean with a mean and trend
1% -2.596 -3.525 -4.093
5% -1.945 -2.903 -3.474
10% -1.618 -2.589 -3.164
Other Countries
1% -2.592 -3.512 -4.074
5% -1.944 -2.897 -3.465
10% -1.618 -2.586 -3.159
reduce the problem of spurious regression, a problem that may be found when persistent regressors are used (as mentioned in section 3.4 of Chapter 3). This simple form of stochastic detrending does not require any parameter estimation, so it is an appealing alternative to using time series models or time trends to deal with near non- stationarity.57 After applying this procedure, in general, the new series present relatively lower values for the first-order autocorrelation.
Another issue that can be raised is related with the scale of these variables, which is not specified by theory, but can affect the results. Usually, the solution is to use mean zero variables (see Bernhardt and Jung, 1979), a procedure that we also followed.
Panel A and Panel B of Table 4.4 report the correlations between the variables (excluding the January dummy), considering both the level variables and the stochastically detrended mean zero variables. For the level variables, we find relatively high correlations, namely between the term spread and real bond yield and between the real bond yield and DM spread. When we consider the stochastically detrended and mean zero variables, we find lower correlations, except for the correlation between term spread and real bond yield for all markets (the lowest is for France with 0.47 and for all others it is superior to 0.64).
57 This stochastic detrending procedure is equivalent to a triangular weighted average of changes in the variable, so it is stationary even if there is a unit root in the variable (Campbell, 1996).
Table 4.4 – Correlation between the predetermined information variables
Panel A reports the correlation between the predetermined information variables as defined previously (level variables). Panel B presents the correlations considering the stochastically detrended (by subtracting a 12-month moving average) and mean zero variables.
Note: For Portugal, the predetermined information variables are restricted to the period December 1994 to November 2000.
Term Spread Real Bond Yield IRW DM spread PANEL A - Level Variables
Portugal
Term Spread 1
Real Bond Yield 0.41 1
IRW 0.49 0.22 1
DM spread 0.34 0.90 0.42 1
Spain
Term Spread 1
Real Bond Yield 0.58 1
IRW 0.59 0.40 1
DM spread 0.51 0.91 0.59 1
Italy
Term Spread 1
Real Bond Yield 0.51 1
IRW 0.36 0.52 1
DM spread 0.37 0.90 0.63 1
France
Term Spread 1
Real Bond Yield 0.12 1
IRW -0.10 0.69 1
DM spread -0.20 0.78 0.72 1
Germany
Term Spread 1
Real Bond Yield 0.81 1
IRW 0.02 0.37 1
DM spread ___ ___ ___ ___
UK
Term Spread 1
Real Bond Yield 0.95 1
IRW 0.26 0.23 1
DM spread 0.66 0.66 -0.14 1
Table 4.4 – Correlation between the predetermined information variables (continued)
Note: For Portugal, the predetermined information variables are restricted to the period December 1994 to November 2000.
Term Spread Real Bond Yield IRW DM spread PANEL B - Mean Zero Variables
Portugal
Term Spread 1
Real Bond Yield 0.73 1
IRW 0.31 0.07 1
DM spread 0.52 0.53 0.56 1
Spain
Term Spread 1
Real Bond Yield 0.64 1
IRW 0.44 0.43 1
DM spread 0.35 0.71 0.66 1
Italy
Term Spread 1
Real Bond Yield 0.70 1
IRW 0.36 0.41 1
DM spread 0.33 0.63 0.63 1
France
Term Spread 1
Real Bond Yield 0.47 1
IRW 0.05 0.38 1
DM spread -0.02 0.68 0.54 1
Germany
Term Spread 1
Real Bond Yield 0.84 1
IRW -0.06 0.21 1
DM spread ___ ___ ___ ___
UK
Term Spread 1
Real Bond Yield 0.83 1
IRW 0.34 0.29 1
DM spread 0.52 0.37 0.27 1