Drawing Inferences from Statistics based on Multiyear Asset Returns

Motivation(s)

Random walk finance model of stock prices using multiyear returns has recently gained traction due to promising attempts of uncovering low-frequency correlations in the data (e.g. slowly mean-reverting component of stock prices). Statistical findings such as long-run negative correlation in stock returns have generally been assessed using asymptotic theory (e.g. checking whether the estimates lie within two standard errors of the null). This theory seems to provide a poor approximation to the sampling distribution according to the latest Monte Carlo experiments, which brings up the question whether the long-term trends and patterns are valid.

Proposed Solution(s)

The major practical value of asymptotic theory is to provide a robust approximation to the small-sample distribution of the statistics; determining which theory is most appropriate reduces to an empirical question. Monte Carlo simulations of the multiyear statistics demonstrate that the proposed theory provides a much better approximation to the finite-sample distributions.

The proposed alternative asymptotic distribution theory treats the overlap in the data (denoted as \(J\)) as tending to a fixed nonzero fraction (\(\delta\)) of the sample size \(T\), whereas the conventional theory treats \(J\) as fixed so that \(\frac{J}{T} \rightarrow 0\). The fixed \(J\) theory implies statistics such as variance ratios and autocorrelations of multiyear returns are consistent and have asymptotic normal distributions; this may be a poor approximation since the number of non-overlapping observations can still be small despite a large sample size. In contrast, the \(\frac{J}{T} \rightarrow \delta\) theory imply the statistics are not consistent and have limiting distributions that are typically functionals of Brownian motion.

Evaluation(s)

The numerical evaluation of nonstandard asymptotic distributions by a sequence of Monte Carlo simulations is increasingly common. The empirical results show that the Monte Carlo percentiles converge rapidly for the proposed theory, which verifies that the reduced number of rejections in statistical tests is accurate. In contrast, the fixed \(J\) theory failed to capture the skewness of the Monte Carlo distributions. This authors assert that their usage of asymptotic \(p\)-values for statistics with nonnormal limits are more appropriate than the typical two-standard errors. However, they admitted the conclusions drawn from using these values are weaker than conventional standard errors. The statistics considered in this paper cannot provide decisive evidence for economic theories in which mean reversion at long horizons plays a central role.

Future Direction(s)

  • Is this indicator still reliable (e.g. prediction of 2008 crash)?

Question(s)

  • What background knowledge is needed to understand the derivations?

Analysis

Finance models need to take into account overlapping data and finite-sample distributions. While the idea of using Monte Carlo percentiles to verify the simulations’ accuracy is very intriguing idea, the usage of \(p\)-values is not so convincing. It seems the authors favored statistical over economic significance.

Notes

Variance-Ratio Statistics

  • If returns are serially uncorrelated, the variance of the \(J\)-period return will increasely linearly with \(J\).

  • If there is mean reversion, where the returns are negatively correlated, the variance of the \(J\)-period return will increase less than linearly.

Multiyear Autocorrelation Statistics

  • The \(F\)-test provides a simple way to check whether the restrictions on multiperiod correlations apply simultaneously.

  • If there is mean reversion in stock prices over a long horizon, the following statistics are heuristics to detect that: sum of multiyear correlations, maximum correlation, minimum correlation, or maximum absolute correlation.

  • The use of overlapping data will induce a moving average process in the error term of a regression model.

References

RS89

Matthew Richardson and James H Stock. Drawing inferences from statistics based on multiyear asset returns. Journal of Financial Economics, 25(2):323–348, 1989.