Performing t-test on highly skewed financial data + outlier treatment?

Question

I need some advice on performing statistical tests on financial ratios and highly skewed data. I have gathered a large sample of several financial ratios for two groups. The sample size is + 40,000 (10yr period) for each ratio. As is known in literature, the distribution of most financial ratios is highly skewed and doesn’t follow a normal distribution. To a large degree this is caused by outliers (and partly due to non-negative values). Take i.e. the market to book ratio. If the denominator approaches to zero the ratio get extremely large. In my case the max value is +8.000 (S&P Industry average is 2.54 for the same period). Though mathematically correct, there is no economic meaning. Skewness for the distribution is 86, the median 2.06 and the mean 4.53. For some other ratios, skewness statistics reach values up to 300.

After conducting tukey’s fence (interquartile range rule) for outlier treatment, where I eliminate all values larger than the outer fence (q1 - 3.0xIQR & q3 + 3.0IQR), skewness is 1.63, the median 1.98 and the mean 2.51. I know there is a large debate on how to deal (or not deal) with outliers, but in my case I have to reduce their influence.

So, my questions are as follows:

1. Is the outlier treatment reasonable?
2. Can I conduct a Welch's t-test? Variances are not equal. After treating for outliers, most skewness statistics are close to -1.5 and +1.5 max is 3.5. I found that literature is mixed on that topic.
3. Why are so many studies out there conducting t-tests on financial ratios, without reporting skewness statistics or outlier treatment, though non-normality can be assumed?
4. Are there better ways to deal with the data?

Any help is highly appreciated. Thanks in advance!

Update: I just added two histograms. The first one uses 3*iqr rule and the second is wisorizing at the 5%,95%-percentile level. Skewness after winsorizing is 1.57, mean 2.81 and median is 2.06. So, is winsorizing in that situation the better (more scientific) option?

Answer 1

I agree with @1muflon1(+1) about systematic removal of outliers, for no reason other than size. That is almost never useful. (Of course, outliers obviously due to data entry errors or equipment failure should be removed: A basketball player listed at 9' 6" tall, reports of a temperature -60$^o$F in Hawaii, etc.)

With sample sizes is the thousands, t tests on data that are not exactly normal can be useful. That is especially true for looking at the difference of two distributions with skewness in the same direction.

In the simulated data below the samples x1 and x2 are both large samples $n_1 = n_2 = 40\,000$ from right-skewed gamma distributions. However, a two-sample t test is useful in detecting the relatively small difference between them.

set.seed(2019)
x1 = rgamma(40000, 5, .1)
x2 = rgamma(40000, 5.1, .1)
t.test(x1, x2)

        Welch Two Sample t-test

data:  x1 and x2
t = -7.2698, df = 79983, p-value = 3.632e-13
alternative hypothesis: 
  true difference in means is not equal to 0
95 percent confidence interval:
 -1.4739377 -0.8479435
sample estimates:
mean of x mean of y 
 49.94401  51.10495 

However, a two-sample Wilcoxon (nonparametric test based on ranks) also finds a difference between population locations.

wilcox.test(x1,x2)

        Wilcoxon rank sum test with continuity correction

data:  x1 and x2
W = 776760000, p-value = 1.119e-12
alternative hypothesis: true location shift is not equal to 0

Answer 2

1. The views on deleting outliers differ but outlier should be some unusual observation that skews the results. If you have a lot of them I am not sure I would call them outliers anymore.

Also, using interquartile range is bit too simplistic and crude for my taste - you also don’t see it often in research - at least not in macro and international economics where I come from. I would recommend to winsorize data or to use even some more sophisticated technique.

2. I would not use Welch’s test in this case. There are so many other options you could do some non-parametric test like wilcoxon test or bootstrap like @Dave suggests. The precise solution depends on specifics of your problem.

3. I will let someone else answer this because I don’t have enough knowledge of the research on comparing financial ratios.

4. Yes as was already mentioned in such case some non-parametric approach would be more appropriate.

Answer 3

Well I checked how many value's have been removed for mtb ratio according to the iqr. It`s 4.6% of more than 40.500. I also considered winsorizing but even the 95%percentile is a mtb ratio of 9.24. Though it reduces skewness, its still pretty high. Which are the advantages of winsorizing other than retaining the values? In what way is it advantages compared to iqr-rule?

According to the definition of Hawking: An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism". When I look at the data, values so far from the median or even mean doesn't really make economically sense. Intuitively, it would make more sense for me to delete them. There are authors which by (expert) judgment, set limits i.e. no values above 5 and exclude all other values. This approach is not really systematic, imho.

Why is the Mann-Whitney U-Test preferable to Welch t-test?

Fagerland (2012): “Non-parametric tests are most useful for small studies. Research authors that use non-parametric tests in large studies may provide answers to the wrong question, thus confusing readers. For large studies, t-tests and their corresponding confidence intervals can and should be used even for heavily skewed data.” https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3445820/

I’m still puzzled what to do?!