Fig 1.
Log-normal fit of the main body part of forecast errors.
We fit the main body part of analyst forecast errors to the log-normal distribution. (The figure is based on non-stale analyst forecast errors on the last trading day of 1993; main body parts from other calendar years fit log-normal distributions as well.) We define the main body part as the bottom 95 percent of analyst forecast errors (129202 observations in 1993). The red line is the fitted log-normal density while blue circles represent the data points. Error bars have been estimated from the deviation of the data. The fitted parameter values are: location parameter μ = −5.46 and scale parameter σ = 1.94.
Fig 2.
Power-law fit of the tail of forecast errors.
We fit the tail part of analyst forecast errors to the power-law distribution. (The figure is based on non-stale analyst forecast errors on the last trading day of 1993; tail parts from other calendar years fit power-law distributions as well). Since the power-law distribution in Eq (3) implies a linear relationship in the logarithmic scale, we use ordinary least squares (OLS) to fit the empirical cumulative distribution. The red line depicts the fitted power-law distribution while blue circles represent the data points. In 1993, the power-law exponent approximates unity (α ≈ 1), which indicates that analyst forecast errors follow Zipf’s law.
Fig 3.
Annual values of the power-law exponent.
Blue dots represent annual estimates of the power-law exponent α from year 1984 to 2012. The annual value of the exponent is estimated by fitting all forecast errors in each calendar year. The black horizontal line represents the power-law exponent of unity (α = 1), which corresponds to Zipf’s law, while two red/green dashed lines indicate one/two standard deviations away from Zipf’s law. The vertical line designates year 2000, the year of “Regulation Fair Disclosure (FD)”, while the shaded periods correspond to NBER recessions.
Table 1.
Regressions of information disparity and the power-law exponent.
Models (1) and (2) test the association of information disparity and economic uncertainty; Models (3) and (4) test how the power-law exponent α deviates from unity due to information disparity. Models (5) and (6) test how collective behavior changed after the Regulation FD was introduced. R takes the value unity for years after Regulation FD was introduced and zero otherwise, I is the proxy for information disparity measured by the scaled forecast dispersion of the market, E is the economic uncertainty measure from Ref. [48] and c is constant. The numbers in parentheses are t-statistics calculated from heteroskedasticity and autocorrelation consistent standard errors according to Newey and West [49]. ** and *** represent significance at 5% and 1% levels, respectively.
Fig 4.
Monthly variations of the power-law exponent by year.
Shown are the data in calendar years 1985, 1989, 1992, 1996, 1999, 2003, 2007, and 2012; these sample years have been chosen for the sake of brevity, but the trends in other calendar years are largely similar to those shown in this figure. The colored circles represent the monthly estimates of the power-law exponent from March to December in each given year, with error bars estimated from the power-law fit. Lines connecting circles are merely guides to the eye. There appears a clear descending pattern of the power-law exponent within a year. Namely, the power-law exponent tends to decrease towards the year-end.
Fig 5.
Monthly cut-off points by year.
For comparison, the same sample years as those in Fig 4 have been selected. We show the cut-off points for the tail part of forecast errors (95th percentile of the observations) from March to December in each given year. The descending trend of cut-off points manifests that the forecast errors converge to zero towards the year-end, which implies that analysts are becoming more precise in their forecasts.
Fig 6.
Monthly evolution of (a) the location parameter μ and (b) the scale parameter σ.
This figure presents the monthly changes in μ and σ of the log-normal fit of the main body part of forecast errors in 1993. As predicted by the model, μ is linear in time and σ is linear with respect to the square root of time (note that the scale of the horizontal axis is given in the square root of time). Blue dots in (a) and (b) are the estimates of the location and the scale parameters, respectively; dashed lines in (a) and (b) have the slopes −0.0032 and 0.0148, respectively, per day.