bond market, is empirically tested in section 4.5.
The correlation of stocks and bonds is also considered in Tactical Asset Allocation.
For example, Hartpence and Sikorav (1996) construct a strategy based on long term values of stocks and bonds given by valuation models. They assume that movements of stocks and bonds are correlated in the long term and that current deviations of the prices from their fair values are corrected in the short term. Hence, they use an error correction framework to quantify the expected movements of the asset prices. When they apply their model to the French and international markets to forecast future returns, they achieve a higher return (with a lower or equal risk) than the return of the benchmark portfolio given by Strategic Asset Allocation.
-4 -2 0 2 4 6 8 10
1992 1994 1996 1998 2000 2002 2004 2006
[%]
Slope (10Y - 3M) 3M Interest Rate Corporate Credit Spread
Figure 4.6: German three-month money market rate, interest rate of ten-year government bonds minus three-month money market rate (slope) and interest rate of corporate bonds minus three-month money market rate (corporate credit spread).
to be adjusted for stock splits, has to be diversi…ed over all industries and its time series should be of reasonable length. Furthermore, the DAX index is a performance index, that is dividends paid to the shareholders are theoretically reinvested and therefore included in the time series of the DAX index. The stock market is represented by the total year-on-year return on a monthly frequency in line with the suggestions by DuBois (1992). As the year-on-year return of the DAX index is very volatile, a twelve-month moving average is used. In contrast to interest rates of …xed income securities, the nominal future returns of equities are uncertain. That is why today’s return of the DAX index is taken as the best guess for tomorrow’s return (naive forecast).21
The data for the DAX, interest rates of government bonds, money market rates and corporate bonds is collected from the Deutsche Bundesbank. The Deutsche Bundesbank calculates the interest rate of government bonds with a maturity of ten years based on listed Federal securities with the Svensson method, which is an extended Nelson-Siegel approach. The short term riskfree interest rate is the three-month money market rate between banks in Frankfurt. The data for the interest rate of corporate bonds is the yield of outstanding corporate bonds of domestic …rms. The three time series of interest rates are depicted in …gure 4.6.
21Figure 4.7 shows the transformed return series of the DAX index.
4.3.2 Real-Time Output Gap Estimate
As a measure of the economy, the output gap is used, which has strong feedback e¤ects with …nancial markets. This fact is captured by Diermeier, Ibbotson and Siegel (1984) in their concept of macroconsistency, where they attribute the supply of aggregated
…nancial market returns to real business activity and productivity of …rms. However, the distribution of aggregated returns among investors depends on the demand for the various asset classes. That is why the concept of macroconsistency is often omitted in the daily analysis of …nancial markets.
The output gap measures the ability of the real economy to generate …nancial market returns. The output gap is a well-known concept in Macroeconomics and Financial Economics. It is de…ned as the deviation of the growth rate of the real economy from its potential growth rate. The potential growth rate of the economy is its long term growth rate without economic slack or exogenous shocks. It is determined by structural factors of the economy like the growth of productivity and of employment.22
Even though the concept of the output gap is criticized due to measurement problems, it is often used in the macroeconomic literature. One of the …rst who researched on the real-time estimation of the output gap were Orphanides and van Norden (2002). They state that the real-time measurement of the output gap is important, because the output gap might be revised after its initial release. The magnitude of the revision can have the same size as the output gap itself and is the largest around the turning points of the economy. This is a problem for policy makers, because correct information about the output gap is very important for their decisions during turning points.
The output gap which is used in this analysis is based on a real-time GDP series for Germany from the Deutsche Bundesbank. This real time series consists of initial releases of real GDP, that is revisions are not included. The estimation method is the real-time estimate of Orphanides and van Norden, which is a two step procedure. First, a subperiod of the sample is chosen that starts at the beginning and ends at an arbitrarily chosen date in the middle of the sample. For this subperiod, a real-time estimate of real GDP growth is obtained by detrending the GDP growth time series over this subperiod.
22Another measure of the state of the economy is the Non-Accelerating In‡ation Rate of Unemployment (NAIRU).
This procedure is repeated until the end of the sample is reached, whereas the subperiod is enlarged successively by one period. Second, the last periods of the various subperiods are combined to a new time series of real-time output gap estimates. For the analysis of the amount of total revision of the estimated output gap, Orphanides and van Norden compare the real-time estimate of the output gap with the …nal estimate. The …nal estimate is the detrended historical time series of real GDP growth up to the most current release, whereas it contains historical revisions in real GDP growth.
In order to forecast the trend of the growth of the output, it is necessary to divide the time series of GDP growth into a trend and a cyclical component. It is common to apply the Hodrick-Prescott (1997) Filter (HP-Filter) for this division. For example, Bouthevillain et al. (2001) use the HP-Filter in their research on cyclically adjusted budget balances. The HP-Filter is simple, transparent and yields useful results. Never-theless, the HP-Filter is not able to detect a trend shift shortly after the structural break has occurred.23 A further drawback of the HP-Filter is the end of sample problem.24 It is due to the fact that the estimated trend component is a weighted average, so that the trend component is mainly in‡uenced by recent values. To reduce this bias, it is com-mon to run the HP-Filter over an enlarged sample which consists of the historical time series and an out-of-sample forecast. Therefore, in this analysis, the historical sample of the yearly rate of change of real GDP growth is enlarged by forecasts. The forecasts are generated by an AR(4) process, which is appropriate according to Döpke (2004).
The ratio of the number of available observations of real GDP growth to the number of periods forecasted by the AR(4) process is held constant at a ratio of 4:1, because the number of available observations is time-varying.25
The output gap in this chapter is calculated as the actual value of GDP growth minus the trend component of GDP growth (potential growth). A positive (negative) output gap indicates that the economy grows above (below) its potential growth rate.
23A two-sided moving-average is also not able to detect a structural break quickly.
24Orphanides and van Norden (2002) depict the end of sample problem as the main problem, because real-time estimates for the potential growth rate of GDP are very unreliable at the end of the sample.
25The time-varying number of available observations of the sample is due to the di¤erent base dates in the data set of real time GDP provided by the Deutsche Bundesbank.
-8 -6 -4 -2 0 2 4 6
1992 1994 1996 1998 2000 2002 2004 2006
[%]
Output Gap DAX, yoy [12M-Moving Average]
Figure 4.7: German output gap and twelve-month moving average of year-on-year DAX re-turns.
Furthermore, an increase (decrease) in the actual GDP growth or a decrease (increase) in the potential growth rate increases (decreases) the output gap. Figure 4.7 shows the real-time estimate of the output gap and the year-on-year return of the DAX index.
Both of them are stationary (I(0)) and can be used in the following VAR analysis. Due to the fact that GDP is available at a quarterly frequency and the …nancial market variables have a monthly frequency, the estimate of the real-time output gap is used for three successive months.