Why Normal Returns and a K–S Test Do Not Establish Market Efficiency
Summary
The document asks whether the Kolmogorov–Smirnov test can establish that a time series is normally distributed and whether normality can then establish market efficiency. The responses describe the K–S test as a way to compare a sample with a reference distribution, while noting that substantial data may be needed. One response suggests the Jarque–Bera test as another normality test, but the discussion does not provide a formal comparison of test assumptions or power.
The central lesson is that distributional shape alone does not establish market efficiency. Normally distributed returns can still be predictable when one asset’s price is a lagged version of another. Likewise, individual assets can follow random walks while their shared relationships offer a possible pairs-trading opportunity. The examples illustrate limits of inference rather than a complete efficiency-testing procedure; the document mentions asset-pricing models in passing but does not specify an empirical design for testing efficiency.
Key ideas
- The K–S test compares a sample distribution with a reference distribution, but a test result is not absolute proof of normality.
- A large sample may be needed for the K–S test to be informative.
- Normal returns do not by themselves imply that prices are unpredictable.
- Assets that each appear to follow random walks may still have exploitable relationships.
- Testing market efficiency requires a framework beyond testing a return distribution.
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Full text
# Kolmogorov-Smirnov test
# Kolmogorov-Smirnov test
Is Kolmogorov-Smirnov test self-sufficient to prove normal distribution of a time series? And then test efficiency of a market?
## Answer by Bob Jansen (score 2)
https://quant.stackexchange.com/a/7548
Wikipedia says
> In statistics, the Kolmogorov–Smirnov test (K–S test) is a nonparametric test for the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test).
so yes but also warns that a large number of data points might be required. Why don't you apply the Jarque-Bera test?
I think I've a simpler example to show that a normal distribution does not imply market efficiency:
Suppose security 1 prices are given by the GBM $X(t)$ and the prices of security 2 $X(t-1)$. Then returns are normally distributed but predictable.
## Answer by Akavall (score 1)
https://quant.stackexchange.com/a/7547
I think that knowing that price of every asset is a random walk is not enough to say that the market is efficient. What if prices of asset y and asset z follow the same price as x plus some noise. If x is a random walk, then y and z are also random walks; however, it would be possible to exploit their relationship via pairs trading; hence, markets are not efficient. In my example x could price of oil and y and z could be two oil companies prices of who's shares are closely related to the price of oil.
Here is an illustration in Python:
```
import pylab
import random as rn
def gen_random_walk(n):
x = [100]
for _ in xrange(n):
change = rn.gauss(0, x[-1]/100.0)
x.append(x[-1] + change)
return x
def plot_pair(n):
x = gen_random_walk(n)
noise_y = [rn.gauss(0, 1) for _ in xrange(n)]
noise_z = [rn.gauss(0, 1) for _ in xrange(n)]
y = [ele_x + noise_y for ele_x, noise_y in zip(x, noise_y)]
z = [ele_z + noise_y for ele_z, noise_y in zip(x, noise_z)]
pylab.plot(zip(y,z))
pylab.show()
plot_pair(250)
```
## Answer by 4pie0 (score 1)
https://quant.stackexchange.com/a/7551
no. this test as same as any else can't say for 100% that distribution is normal. and how do you want to use it to test market efficiency? you didn't mention. basically to test market efficiency the CAPM or APT model is used. you can find here more info about it. in short it must holds that higher return correspond to higher $\beta$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.