5. Data
5.8 Possible econometrics problems
69 Self-selection bias is believed to have become even a bigger problem than survivorship bias, but is still very difficult to detect and correct for. In regards to our paper, we have not been able to correct for self-selection bias for several reasons. First it is extremely hard to detect it and secondly we did not have enough resources to attempt to identify the cases where it might occur, which might lead our results to be biased.
70 Therefore when dealing with stock or index returns the assumption of no autocorrelation is frequently violated, and it is therefore important to test for it (Gujarti and Porter, 2009).
To investigate if our observations are affected by autocorrelation we have applied the Durbin-Watson (1951) test defined below;
𝒅 =∑𝒕=𝒏𝒕=𝟐∑(𝒖�𝒕− 𝒖�𝒖�𝒕−𝟏)𝟐
𝒕𝟐
𝒕=𝒏𝒕=𝟏 Formula 5.2
This describes the ratio of the sum of squared differences in successive residuals to the RSS.
The number in the numerator of the d-statistics is the number of observations n-1, because one observation is lost when taking successive differences (Gujarti and Porter, 2009).
A point of reference is that the d number in the Durbin Watson test is always between 0 and 4.
A value of 0 indicates a strong positive autocorrelation, and 4 indicates a strong negative autocorrelation. Another way to check for this is to see if the d number lies within the right interval in the Durbin Watson Statistics table. We decided to use this table since it will give us a more accurate measurement. The Durbin Watson upper and lower limits are collected from a Stanford University table8
Before presenting the result it is important to know the underlying assumptions for the Durbin Watson test. First, the explanatory variables in the test are non stochastic. Second, the error terms are assumed to be normal distributed. Third, the regression models do not include the so called lagged values of the regression. And last, only the first order serial correlation is measured, which means that we are not able to measure higher order autocorrelation with the Durbin Watson test (Gujarti and Porter, 2009).
. The lower and upper limits are estimated on a 5% significance level by taking N number of observations and N numbers of explanatory variables.
8 http://www.stanford.edu/~clint/bench/dwcrit.htm
71 Table 5.2: The results from the Durbin Watson test can be found in the table below:
US #OBS Variables d d (4-D) DL95% DU 95%
Jensen Alpha 1304 1 1.978 2.022 1.90729 1.91036
Fama French 261 3 1.925 2.075 1.77344 1.8201
Carhart 4-factor 60 4 ***1.597 2.403 1.44427 1.72735
Europe
Jensen Alpha 1305 1 2.396 ***1.604 1.90729 1.91036
Fama French 261 3 2.482 ***1.518 1.77344 1.8201
Carhart 4-factor 60 4 2.198 1.802 1.44427 1.72735
Scandinavia
Jensen Alpha 1264 1 2.004 1.996 1.90729 1.91036
Fama French 261 3 1.934 2.066 1.77344 1.8201
Carhart 4-factor 60 4 2.008 1.992 1.44427 1.72735
*** = Problems found
To test for positively autocorrelation:
If d < DL, there is evidence of positively autocorrelation
If d > DU, there is evidence that there is NOT positive autocorrelation If DL < d < DU, the test is inconclusive
To test for negatively autocorrelation:
If (4-d) < DL, there is evidence of negatively autocorrelation
If (4-d) > DU, there is evidence that there is NOT negative autocorrelation If DL < (4- d) < DU, the test is inconclusive
In the Carhart 4-factor model regression on the US market the d number is placed within the DL and DU interval. The autocorrelation test is therefore inconclusive, and unfortunately we are unable to decide if autocorrelation exists or not.
In the regression performed using the Jensen Alpha model on the European market, the d (4-d) number lies 0.30329 below the DL number. Thus, there is an indication of a small negative autocorrelation
In the regression using the Fama-French 3-factor model on the European market, the d (4-d) number lies 0.25544 below the DL number, and it is therefore evidence of a small negative autocorrelation.
72 Even though we found some problems with autocorrelation in 3 of the regression series, the numbers are quite small and only express some small evidence of it. We therefore assume that these results will not bias our results much, and we feel certain that the numbers we present in our analysis are not misleading.
5.8.2 Hetroscedasticity
A very important assumption in the linear regression model is that the variance of each disturbance term 𝑢𝑖, conditional on the chosen values of the explanatory variables, is a constant number equal to the 𝜎2(Gujarti and Porter, 2009). This assumption is known as the assumption of homoscedasticity, or equal variance. Hetroscedasticity is the opposite i.e. that the variance is not equal. A consequence of hetroscedasticity is that the OSL estimators are not BLUE, which as previously mentioned might provide inaccurate regression results and misleading results. Symbolically, hetroscedasticity can be expressed as the equation below:
𝐸(𝑢𝑖2) = 𝜎2 Formula 5.3 Hetroscedasticity can be hard to describe in simple terms. However it tends to occur in situations with a large difference in the size of the observations. Nevertheless, we will try to describe it with this classic example: If we look at income versus expenditure on meals. When the income for a person increases the variability of the food that person will consume will also increase. A poor person will spend a constant amount of money on food by eating cheaper food all the time. A wealthy person might from time to time buy expensive food and other times cheap food. Therefore people with high income will display a greater variability in food consumption (Gujarti and Porter, 2009).
To test for hetroscedasticity we have applied the White test. This test is easily implemented in SAS and does not rely on normality assumptions. The results from it are presented in the table below. The table should be read as follows; if there is hetroscedasticity present, the test value should be below 0.05. On the opposite, if there is no hetroscedasticity present the test value should be above 0.05 (Gujarti and Porter, 2009). (Gujarti and Porter, 2009).
73 Table 5.4: Results from hetroscedasticity test:
Market Chi-Square PR > ChiSq
US daily 7.14 0.0246***
US weekly 8.19 0.5147
US monthly 31.74 0.0044***
EU daily 9.04 0.0109***
EU weekly 13.45 0.1433
EU monthly 13.88 0.459
SCN daily 14.53 0.0007***
SCN weekly 12.00 0.2134
SCN monthly 16.22 0.2998
*** = Hetroscedasticity
As one can see from table 6 there are some evidences of hetroscedasticity. The problem is present in all of the three daily series we have used in the regression. However, this is something one could expect looking at the time horizon we have analyzed. The financial crisis during the past couple of years has brought volatility to markets and returns have been far from being ordinary. Therefore, we expect to have some extreme values in the data series, which most likely are the sources for some occurrence of hetroscedasticity in our regressions.
Further, Gyjarti and Porter (2000) state that it is important not to overreact to hetroscedasticity, and that it has never been a reason to dismiss a regression with the OLS method. They also state, that it is only worth to correct for hetroscedasticity if the problems are very severe. However, for our results there are only some minor problems, which we assume stems from the volatile time period of our investigations.
74