Are Seasonal Anomalies Real?

Lakonishok, Josef, and Seymour Smidt, “Are Seasonal Anomalies Real?” The Review of Financial Studies, Vol 1, No 4 (1988), 403-425.

If researchers analyze data using 100 different hypotheses, then formulate a theory based on results and then test the theory using the same data, they are very likely to get significant results for the theory.  This problem frequently arises due to the limited scope of stock return data (only a few standard sources).  Phenomena that are actually just noise get reported as asset-pricing anomalies.

In addition, rational efficient-market economists don’t want to publish or read papers that claim the market is efficient and investors are rational.  Therefore, a type of selection bias can occur when the majority of publications show anomalies, even if the majority of evidence argues against them.

This paper studies anomalies using “new” data to avoid the first problem.  The data are the daily Dow Jones Industrial Average returns, from January 1, 1897 to June 11, 1986.  Recent anomalies studies were done with post-1962 or post-1927 data; thus, using the DJIA since its inception adds 30-65 years of new data.

The 30 firms in the DJIA compose almost 25% of the entire NYSE.  The stocks of these very large firms are highly liquid, and so are unlikely to suffer from issues of nonsynchronous trading, which makes the DJIA a good measure of short-term market activity.  However, using the DJIA means that this study cannot test the January effect, which is observed in small stocks.


  • Monday returns are significantly negative (-0.14%).
  • Turn-of-month price increases are greater than the price increase for the entire month.
  • Prices increase 1.5% between Christmas and New Year’s.
  • Rates of return before holidays is 20x the normal rate of return.
  • Most anomalies are quite small in magnitude.
  • There is no consistent monthly pattern in stock returns.
  • There is no significant evidence that returns in the first part of month are different from returns in the last part.

Models of Stock Returns–A Comparison

Kon, Stanley J., “Models of Stock Returns–A Comparison,” The Journal of Finance, Vol 39, No 1 (1984), 147-165.

Purpose:  To explain the observed kurtosis (fat tails) and positive skewness in the distribution of stock returns.

Findings:  The discrete mixture of normal distributions proposed in this paper explains these moments better than existing models, including the Student-t.

Motivation:  Mean-variance portfolio theory, option pricing models, and empirical tests of capital asset pricing models and efficient markets make assumptions about the distribution of stock returns.  It has been shown that stock returns cannot be approximated using a single normal distribution, but the normal is just what the theoretical models assume.

Methods:  Returns may be driven by multiple normal distributions–one for random shocks, one for firm-specific information, one for macroeconomic information, etc.  In other words, it is not necessary that each observation of stock return be drawn from the same distribution as all others.  This paper uses a mix of up to five normal distributions and likelihood ratio tests to model the returns of the 30 Dow Jones stocks.

  • Let  r_t = \alpha_i + u_{it}, where r_t is the observed return for period t, and u_{it} is normally distributed with mean zero and variance \sigma_i^2.
  • Let \underline{\gamma}_i be a normal distribution with mean \alpha_i and mean \sigma_i^2.
  • Let \lambda_i = T_i/T be the observations associated with information set I_i over total observations, or the proportion of total observations that are associated with information set I_i.
  • Let \underline{\theta} = \{\alpha_i, ..., \alpha_N, \sigma_i^2, ..., \sigma_N^2, \lambda_i, ..., \lambda_{N-1}\} be a vector of parameters.
  • Let \underline{r} be the vector of returns.
  • The vector of parameters, given a vector of returns, can be found by solving

max \ell(\underline{\theta}/\underline{r}) = \Pi_{t=1}^T \left[\sum_{i=1}^N \lambda_i p(r_t|\underline{\gamma_i})\right].


  • All 30 Dow Jones stocks can be explained by a mixture of 2, 3, or 4 normals.
  • Returns of the S&P500, value-weighted, and equal-weighted market indices were also explained by mixtures of normals.
  • The mixture of normals describes 27 of the 30 stocks, and all three indices, better than does the student-t distribution, which also has fat tails and has been used to model stock returns.
  • Differences in the means of the normal distributions can explain the skewness of observed stock returns.
  • Differences in the variance of the normals can explain the kurtosis of stock returns.
  • Stock returns do appear to be normally distributed, but the distribution parameters are time-varying, and the timing of parameter shifts also varies across stocks.

Mean Reversion in Stock Prices?

Kim, Myung Jig, Charles R. Nelson, and Richard Startz, “Mean Reversion in Stock Prices?  A Reappraisal of the Empirical Evidence,” The Review of Economic Studies, Vol 58, No 3 (1991), 515-528.

Background:  In the 1970s and -80s, stock returns were thought to follow a random walk.  Researchers in the late 1980s began to question this view, and used a variance ratio method to show that autocorrelation did exist in stock returns.  Define the “variance ratio” as the return over K periods divided by the product of the return over one period and K.  If returns follow a random walk, this ratio must equal 1.

However, this assumption is not borne out by the data.  The variance ratio is higher than 1 for periods shorter than a year (positive autocorrelation) and is less than one for periods longer than a year (negative autocorrelation).  A common interpretation of this negative autocorrelation over longer periods is to say that returns are mean-reverting.

Fama & French’s approach is to regress the returns from period t to t+k on the return from period t-1 to t:

r_{k,t+K} = \alpha_K + \beta_Kr_{K,t} + \varepsilon_{K,t+K}

In this model, a negative beta indicates mean-reversion, and a zero beta, a random walk.  This model is also better suited to predicting future returns

Purpose:  This paper re-examines the data and finds no evidence of mean reversion after WWII.  Stock returns in the post-war period are actually mean-averting, meaning that disturbances are too persistent to support a mean-reversion theory. Furthermore, indicators of post-WWII mean-aversion are as statistically significant as indicators of mean-reversion for the whole 1926-1986 period.  The comparison of pre- and post-war returns do not support the random-walk hypothesis, but point to a fundamental change occurring at the end of the war.

Method:  Use statistical methods that do not assume returns are normally distributed.


  • Returns are only mean-reverting pre-WWII.
  • Post-war returns are, if anything, mean-averting.
  • The change may have accompanied the resolution of uncertainties surrounding the duration of the Great Depression, the outcome of WWII, and fears of another post-war depression.

Good Morning Sunshine: Stock Returns and the Weather

Hirshleifer, David, and Tyler Shumway, “Good Morning Sunshine:  Stock Returns and the Weather,” The Journal of Finance, Vol 58, No 3 (2003), 1009-1032.

  • Weather (days of sunshine) is a truly exogenous variable
  • Sunshine in the cities of 26 countries’ largest stock exchanges strongly and statistically significantly predicts stock returns between 1982 and 1997
  • After controlling for days of sunshine, other weather patterns such as rain and snow have no effect.
  • Trading based on the weather would be optimal for a trader facing very low transaction costs, but even moderate costs would generally prohibit such a strategy.
  • These results are consistent with a theory of mood affecting behavior, but are not consistent with the theory of rational actors.


  • Table III:  The joint betas are significant.  This might be driven by the large but insignificant betas for Buenos Aires and Rio de Janeiro (Sao Paolo; see below)
  • Hirshleifer & Shumway mistakenly collected weather data for Rio de Janeiro, while their return data comes from the Brazil’s largest stock exchange (Sao Paolo).