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Say I have a sequence of stock returns: $X_1, ..., X_n$. Then I make a sample of products: $X_1X_2, X_2X_3, ..., X_iX_{i+1},..., X_{n-1}X_n$

How to test, if the sample mean is significantly different from zero or not?

Should we assume, that the sample have distribution Normal Product Distribution?

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    $\begingroup$ Which mean are you interested in testing for difference from zero - the products or the original stock returns? What's the ultimate purpose of testing that? $\endgroup$
    – Glen_b
    Oct 28, 2014 at 0:27
  • $\begingroup$ I'm interested in the products sample. The purpose of testing is to identify if there is a significant dependence between signs of neighbor returns in a sequence. $\endgroup$
    – romek
    Oct 28, 2014 at 9:32
  • $\begingroup$ If you're interested in dependence in sign, why would you not look at the product of signs? That would require fewer assumptions to hold. $\endgroup$
    – Glen_b
    Oct 28, 2014 at 9:34
  • $\begingroup$ @Glen_b Good idea! Should I use "Bernoulli scheme"? But, I heard about statistics, using products of returns (the aim is to to take into account sizes of returns too). I don't remenber exactly, it's something like sum(X[1:(n-1)]*X[2:n])/sum(X[1:(n-1)]^2). But I don't think it's statistically correct... $\endgroup$
    – romek
    Oct 28, 2014 at 10:25
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    $\begingroup$ Which you should test depends on the precise hypothesis you are interested in. If you're interested in what you said, then I'd be inclined to look at the product of the signs to test it. If you're interested in the mean product, you could test that.That last statistic you mention is effectively an estimate of the lag 1 autocorrelation if that's a zero-mean process, otherwise it's measuring something else. $\endgroup$
    – Glen_b
    Oct 28, 2014 at 11:56

2 Answers 2

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Should we assume, that the sample have distribution Normal Product Distribution?

No. You should not. Ito calculus assumes that both the distribution and the parameters are known. In that framework, where all parameters are known, the assumption of normality can work in the framework of $x_{t+1}=Rx_t+\varepsilon_{t+1}$, but if $R$ is unknown then the maximum likelihood estimator is the least squares estimator but the sampling distribution is the Cauchy distribution. That solution has no statistical power. A sample of one observation has the same statistical power as a sample of one million observations.

If we work with the reward for investing as the return plus one, the following outcomes will answer your question. Returns just shift the solution by one and have no other impact.

Let us ignore mergers, bankruptcies, dividends and liquidity costs. If you are including mergers and so forth, then you need to build a far more complex model.

Let us begin with the assumption that prices have an equilibrium value at every point in time, $t$. If the price at time $t$ is denoted $p_t$, then we could decompose it as $p_t=p_t^*+\epsilon_t,\forall{t}$.

If we define the reward for investing as $$R_t=\frac{p_{t+1}}{p_t}$$ then we can think of the equilibrium reward as $$R^*_t=\frac{p_{t+1}^*}{p_t^*}.$$

We could also think of the reward for investing as $$R_t=R^*_t+\xi_t,\forall{t}.$$ It follows then that $$R_t=\frac{p_{t+1}^*+\epsilon_{t+1}}{p_{t}^*+\epsilon_{t}}=\frac{p_{t+1}^*}{p_t^*}+\xi_t.$$ So $$\xi_t=\frac{p_{t+1}^*+\epsilon_{t+1}}{p_{t}^*+\epsilon_{t}}-\frac{p_{t+1}^*}{p_t^*}$$

Looked at in vector space, the distribution of $R_t$, except for location, depends only on the random vector $$\begin{bmatrix}\epsilon_t\\ \epsilon_{t+1}\end{bmatrix},$$ which in the simple case is the ratio of the errors. If prices are not in equilibrium or close to equilibrium, then that will not be true. In a bubble or after a crash there will be a more complicated case such as $$\begin{bmatrix}\epsilon_t+\alpha\\ \epsilon_{t+1}\end{bmatrix}.$$ In that case $p_t\ne{k_t}\ne{PV}(\delta_1\dots\delta_T),$ the price is no longer equal to the present value of dividends nor the cost of physical capital.

If we assume very many buyers and sellers and that equity securities are sold in a double auction, then there would be no winner's curse. As such, the rational behavior of the actors is to bid their subjectively believed expected value for the security. Errors become appraisal errors. The distribution will tend to normality as the sample size goes to infinity. However, the following outcome only depends on both distributions being elliptical.

The distribution of returns, when neither the distribution nor the parameters are known and where mergers, bankruptcy, dividends and liquidity costs are ignored is $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{R^*}{\gamma}\right)\right]^{-1}\frac{\gamma}{\gamma^2+(R_t-R^*)^2},$$ note that I assumed returns were stationary.

Visually, you can see this in almost all equity securities, though there are a few special exceptions such perpetual preferred securities. Here, for example, is Apple's daily returns since inception.apple I used a kernel density estimate and did not marginalize out the uncertainty. I used the MLE estimates and plugged them in. It would have been better to consider liquidity effects and marginalization of the parameters.

You can also see it with Carnival Cruise Lines carnival

Since the above distribution has no mean, it is not appropriate to use a mean based test. The center of location is the mode. The median is to the right of the mode in this case. On the population of end of day trades in the CRSP universe, they are separated by 2% for annual returns.

The underlying distribution is the Cauchy distribution. Eugene Fama wrote a paper in the sixties showing that the pivotal quantity is normally distributed and that you could condition on an ancillary statistic to get sufficiency for the underlying distribution. It may also be true for the truncated distribution.

You could construct a simulation to get the sampling distribution. If you use the median, you are nearly guaranteed that the product will not be zero because of the skew. If you are including mergers and bankruptcies, then everything above is too simple to work. That is also true for thinly traded securities because the effect of liquidity can become large. It also won't work for firms engaging in liquidating dividends.

Also note, the CAPM, APT, Black-Scholes, Ito calculus methods and the factor models depend on the parameters being known. If they are not known, then, as was observed by Mandelbrot in 1963, the distributions will lack a mean.

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  • $\begingroup$ this is all quite unorthodox you must admit $\endgroup$
    – Aksakal
    Aug 12, 2019 at 18:52
  • $\begingroup$ @Aksakal True, but economics has known since 1963 that the orthodox methods do not work. By '73, Fama and MacBeth had done serious falsification work. Really, none of it has any empirical validity. I do have a paper proposing there is an unnoticed branch of stochastic calculus pending. It does not require the existence of an expectation to work. It also does not require knowledge of the parameters. I proposed a Bayesian and Frequentist calculus. I am also proposing a new options model. $\endgroup$ Aug 12, 2019 at 19:56
  • $\begingroup$ Great, your comment clarifies a lot $\endgroup$
    – Aksakal
    Aug 12, 2019 at 20:00
  • $\begingroup$ @Aksakal I have done the empirical work. Finding an editor willing to publish has been the difficulty. I am hoping that is now overcome. In 1953, von Neumann published a warning note to economists that many things that appear to be proofs may not be proofs. The underlying branch of mathematics had yet to be solved. Shortly after that, he would be elected to the Atomic Energy Commission and then die a few years later, never to return to the question. His intuition was correct. The proofs are wrong. $\endgroup$ Aug 12, 2019 at 20:00
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Yes, model the dataset using the Normal Product Distribution. Now perform t-test and check if the p value is less than 0.05 or 0.01 or 0.001 based on the level of significance you want

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