The Sub-indexes

Individual time series of stock prices and dividends tend to change too much and be too affected by different dividend policies applied by different owners. The time series were therefore put together to form indexes. First all companies were put into an aggregated index, which I then tested. This data set will be denoted A. I then constructed four sub indexes by adding five different companies together in each index. These data sets will be called Xn, where n=1,2,3,4. A test of all the sub indexes together (the expanded test proposition) requires that all the indexes must share the same dividend process. The series was therefor sorted into groups so that the common beta was approximately equal between the groups. By doing this, it was assumed that for equal risk (beta), an equal dividend process should be expected. To sort so few companies into groups so that the beta was approximately equal for each group, and at the same time take into account that the introduction of extended companies had to be spread within each sub-index, was difficult. The betas therefore varied a little between the groups. This should not be a problem, since the estimated beta must be expected to vary some

around the true beta¹³. Two companies had such a high beta that their influence on the sub indexes made it difficult to get the sub-index betas equal. Their influence was therefore reduced by one quarter. The calculated betas where as follows:

Table 5.1: The calculated beta’s for each sub-index

Sub-index 1 Sub-index 2 Sub-index 3 Sub-index 4 0,590465 0,586729 0,666076 0,637035

5.1.3 Stationarity

Due to a small sample, the test for stationarity was not possible to carry out in its full extent. I checked whether the data set was stationary or not by using an Augmented Dickey-Fuller test (ADF). When variables were stationarity for any higher order of serial correlation, they were mostly also stationary for all orders less than this. I did not test which lag length gave the ADF test most power. If such tests had revealed that the best specification of serial correlation of an order less or equal to the highest obtained, there would be no problem. If the order that gave best results was higher, one could claim stationarity due to the loss in degrees of freedom in the small sample. Therefore, when this is the case, the highest order of serial correlation assumed that gave stationarity is indicated in Table 5.2.

For the undifferenced price variable in the aggregate index, this was not the case. This was stationary for the eighth order of serial correlation, but not when any less orders were

assumed. Due to the small sample, and the fact that a autocorrelation order of eight in yearly time series are rare, this variable was discarded as stationary.

If the unit root of a series was outside a 95% confidence interval of unity it was regarded as non stationary. Variables in levels, first difference and logarithmic difference were tested.

Table 5.2: Stationarity

13 Thus, the estimated risk has some variance. Therefore, even if the beta for the groups was estimated as completely equal, we could not be sure that the risk for the different portfolios really was exactly the same.

The highest order serial correlation assumed that gave non-stationarity and was stationary when any lower orders of serial correlation was assumed. The order is indicated by a spot:

\Order of serial-correlation:

Variable:

NONE DF(0) ADF(1) ADF(2) ADF(3) Dividends Aggregated index,

levels



Dividends Sub-index 1, levels



Dividends Sub-index 2, levels



Dividends Sub-index 3, levels



Dividends Sub-index 4, levels



Dividends Aggregate index,

differenced



Dividends Sub-index 1,

differenced



Dividends Sub-index 2,

differenced



Dividends Sub-index 3,

differenced



Dividends Sub-index 4,

differenced



Dividends Aggregate index,

logarithmic difference



Prices Aggregate index, levels



Prices Aggregate index,

differenced



In the further discussion, I will treat series that are stationary when no autocorrelation is assumed as stationary. This is due to the small number of cases which makes it difficult do reject them as this. In addition, the series that are indicated as stationary for any higher order of autocorrelation in figure Table 5.2 will of course be regarded as that. All the differenced series are therefore regarded as stationary.

5.2 Shiller’s variance test

In this test I have used only the aggregated index A, as the data set. As described in the previous chapter, the variance of the price and the variance of the constructed fundamental price is compared to see if the inequality ( 4.3) holds. The hypothesis used in the test are therefore (F is the F-test tests statistic):

( 5.1) H0:

 

Where H₀ is the efficient market hypothesis (EMH). I have used an F-test to compare the two variances. The degrees of freedom are equal to the sample size for both the denominator and the numerator. As in Shiller, the time series was de-trended by dividing an the exponential growth factor (calculated by using the first and last price in the aggregated index sample). The discount factor was found to be 0.973, which yielded a test statistic of F=6.13. This results in a level of significance, the probability that H ₀ is true if it is rejected, of approximately zero.

Thus the EMH is be rejected in this test.

There has to be said though, as I mentioned in the previous chapter, that this test does not take account of possible non stationarity in the dividend or the price process since no differencing at all is done. The dividends and prices was found to be non stationary in levels but not in firs difference in the previously presented DF test. In performing the test, I also noticed that the results proved very sensitive to the discount factor. When the factor is calculated as Shiller suggests, the null is rejected. But if the discount factor is lower than 0,95, for example the ex post real return calculated in 5.3.7, the null hypothesis is not rejected.