11.4 Example: A Dynamic Latent Variable Model for German Share Prices


11.4.1 The General Path Model

The main purpose of DPLS in XploRe is the construction of general variables which optimally represent the dynamics of corresponding set of numerous indicators with their individual but similar dynamics. Indicators or manifest variables of this sort are share prices. Share price indexes are weighted sums of individual share prices. Weights are e.g. returns or quantities of stocks purchased. But by usage of dynamic path models the weights will be estimated as coefficients within the latent variable model. Such a coefficient represents a sort of importance of the individual share price within the construct of the latent variable in the context of the whole model.

The question to be answered by this specific model is, whether a certain construct of share prices can show the dynamic dependence of share prices on economic indicators. This construct could then be used as a kind of new share price index.

The challenge is to specify a dynamic model that represents the most important economic relationship of the index as a stable representant of the set of share prices under consideration.

Besides the latent share price variable (SP), the model under consideration contains the LV labour market (LM), money market (MM), domestic economic performance (DP) and foreign market (FM).

The next table shows the manifest variable belonging to these LVs. The next figure shows the relationships selected: The latent share price variable SP is assumed to statistically depend on LM, MM, DP, and FM. Furthermore a first-order auto-regression of SP and a first-order lagged dependency on LM is supposed. This selection is the result of some previous empirical pilot studies eliminating further lagged relationships.


11.4.2 Manifest Variables and Sources of Data

Domestic Performance:

Nr: Domestic Performance: Unit:
1 Incoming Orders (Processing Business) 1991=100
2 Incoming Orders (Construction Industry) 1991=100
3 Production (Manufacturing Business) 1991=100
4 Production (Processing Business) 1991=100
5 Commodity Trade (Exportation) Billion DM

Origin of Data: Monthly Reports DBBK without Table 4 and Zahlungsbilanzstatistik DBBK


Foreign Market:

Nr.: Foreign Market: Unit:
6 Dow Jones Industrial Average Index
7 Commodity Trade (Importation) 1991=100    
8 US$ against 18 Industrial Countries 1991=100
9 Incoming Orders from Foreign Countries 1991=100
10 Discount Rate USA % p.A.

Origin of Data: Monthly Reports DBBK without Table 4 and Zahlungsbilanzstatistik DBBK


Money Market:

Nr.: MoneyMarket:: Unit:
11 MoneySuppley(M3) BillionDM
12 DiscountRateDBBK                                         %p.A.
13 DM against US$ 1972=100
Origin of Data: Monthly Reports DBBK without Table 4


Labour Market:

Nr.: Labour Market Unit:  
14 Gross Earnings Quantity  
15 Number of Unemploymed Quantity  
16 Vacancies Zahlungsbilanzstatistik                 Quantity  
17 Short-Time Workers Quantity  
Origin of Data: Zahlungsbilanzstatistik DBBK


German Stock Prices:

Nr.: German Stock Prices Unit:  
18 ALV DM  
19 BAS DM  
20 BAY DM  
21 BMW DM  
22 CBK DM  
23 DAI DM  
24 DBK DM  
25 DGS DM  
26 DRB DM  
27 HEN3 DM  
28 HOE DM  
29 KAR DM  
30 LHA DM  
31 LIN DM  
32 MAN DM  
33 MMW DM  
34 PRS DM  
35 RWE DM  
36 SCH DM  
37 SIE DM  
38 THY DM  
39 VEB DM  
40 VIA DM  
41 VOW DM  
Origin of Data: Deutsche Börse


11.4.3 Empirical Results

Both models with levels and with differences had been estimated. Transformations such as differences or logarithms can easily added within the XploRe environment. Despite the models on level base have produced some significant path coefficients and high redundancy we have finally decided for a model on first difference base.

The Figure 11.3 shows the path coefficients of the latent variables and the weights of the manifest variables. Dotted arrows indicate lagged dependencies. The differentiated latent share-price variable has only a very low autoregressive component (0.08). This is only 1/10 of the amount found for levels. The strongest dependency is that on the latent foreign-market variable (0.33). The weakest one is that on the domestic economic performance (-0.06). We have found a medium degree dependency on the latent money-market variable (0.26). Furthermore there is a significant first order lagged relationship with the latent labour market (0.18).

Figure 11.3: A Model with German share prices

The weights can be interpreted as the partial contribution of the individual manifest variable within the block of indicators belonging to a latent variable. In this meaning, it is easily to see that some of the MVs can be dropped because of their negligible contribution, e.g. importation, gross earnings and each of the share prices taken alone.