CARACTERIZACIÓN DE LOS RECUBRIMIENTOS

This phase investigates whether the distributions of IBES analysts’ earnings forecasts are non-normal. Attached to the usage of IBES consensus mean/ median is the assumption that IBES analysts’ earnings forecasts are a priori normally distributed. This can be claimed because the use of the mean or the median assumes a normal distribution for the most accurate point estimate descriptor of the population consensus (Kreyszig, 1993). For instance, Collins and Hopwood, 1980, have assumed multivariate normality52 in their linear regression analysis based on the fact that their use of sufficiently large sample sizes satisfies the multivariate central limit theorem criterion53.

4.2.1.1 Hypothesis 1 Proposal

This section draws from the statistical background theory of Chapter 3 to discuss the theoretical basis why sample distributions of IBES analysts’ earnings forecasts are non- normal, as opposed to the normal distribution implicitly assumed in literature which employs the IBES consensus (reviewed in Chapter 2).

One may argue, with the central limit theorem (CLT) in mind, that distributions of IBES analysts’ earnings forecasts can be assumed, a priori, to be normal54 _{because there is a}

sufficiently55 _{large sample of IBES analysts’ earnings forecasts per period per firm. In}

reality, the sample size is low on a per firm basis and acts as one of the reasons why the CLT may not hold56_.

52 _{Multivariate normality is the assumption that all variables and all combinations of the variables are}

normally distributed. When the assumption is met the residuals are normally distributed and independent, the differences between predicted and obtained scores (the errors) are symmetrically distributed around a mean of zero and there is no pattern to the errors.

53 _{See Anderson, 1958 p. 2, Harris, 1975 p. 232 and Ito, 1969.} 54 _{A definition of the normal distribution can be found in 3.4.1.}

55 _{The matter of sample size sufficiency is a subject of debate. For example, 30 or greater was}

suggested by Mansfield (1986), p. 241.

56 _{The CLT presumption is also bounded by the Lyapunov condition (defined in Wolfram Research,}

2005 and refer to 3.4.1 for its mathematical form). This condition must hold for all sample random variables for the CLT to be valid. However, it will fail if the moments (see 3.4.3 for its formal definition) of the population distribution of analysts’ earnings forecasts do not exist. One example is

Additionally, the CLT breaks down in cases where IBES analysts’ earnings forecast samples are neither independently nor identically distributed (refer to 3.4.2 for a formal definition of IID). This month by month breakdown of the assumption of IID is evident from research examining analysts’ underreaction to past earnings information as a potential explanation for post earnings announcement drifts (Lys and Sohn, 1990, Mendenhall, 1991, Ali et al., 1992). Serial correlations of individual IBES analysts’ forecast revisions have been shown to exist in these studies implying past IBES analysts’ earnings forecasts have a significant role to play in determining future IBES analysts’ earnings estimates. This outcome suggests IBES analysts’ earnings forecasts are not independently distributed from one period to the next, which implies a breakdown of the CLT and the existence of non-normality.

Furthermore, the existence of lower and upper bounds in earnings figures introduces the possibility of skewness into analysts’ earnings estimates distributions. Lower bounds cause positive skewness and upper bounds cause negative skewness (Gu and Wu, 2003). Data that have a lower bound are often skewed right while data that have an upper bound are often skewed left57_{. In cases where upper bounds exist, the source of constraint lies in the prior}

growth rate of the actual reported earnings stream58_.

Skewness can also result from start-up firm effects. For example, initial IBES analysts’ earnings forecasts for a firm may contain a large number of upper outliers (failures given disproportionate start-up firm’s expected profits) causing right skewness in the distribution. However, as the forecast horizon59 _{shortens, the accuracy of the distribution of IBES}

analysts’ earnings forecasts improves and forecasts tend to cluster around the actual reported earnings figure. Collins and Hopwood (1980) point out this improvement in the accuracy of IBES analysts’ earnings forecasts over 4 quarters prior to the announcement date of actual earnings.

the case of the earnings forecast population following a Cauchy distribution, which does not have a mean and has an infinite variance.

57 _{By skewed left (negatively skewed), it means that the left tail is heavier than the right tail.}

58 _{This is based on the assumption of stable earnings streams of large-cap firms. For small-cap firms,}

earnings streams will be more volatile and bounds may not exist.

59 _{This is the period of time from the date the earnings forecast is made to the next earnings}

Hence both theoretical grounds and empirical evidence suggest that the distributions of IBES analysts’ earnings forecasts are non-normal. This conjecture, if shown to be true empirically, means literature which uses the IBES consensus measure will need to be reviewed and further points towards the possibility of a more accurate surrogate consensus to be developed. The latter implication is examined as part of the hypotheses development in Phase 3 (refer to 4.2.3).

Based on the rationale presented in the previous section, Hypothesis 1 may be formally stated as:

Hypothesis 1: Distributions of IBES analysts’ earnings forecasts are significantly non-normal, with:

H1null: Distributions of IBES analysts’ earnings forecasts are normal.

H1alternative: Distributions of IBES analysts’ earnings forecasts are non-normal.

The process of testing Hypothesis 1 is described in 6.2.

4.2.1.2 Application of the Bias and Accuracy Metrics in Relation to Research Aims The examination of distribution normality as part of research aim (1) reiterated in the introduction of this section requires the construction of appropriate distributions. Two measures used in literature, the bias and accuracy of IBES analysts’ earnings forecasts, are employed in this thesis. The notion of the bias of analysts’ earnings forecasts is used to develop the distributions of IBES analysts’ earnings forecasts for goodness of fit testing (in Phase 1 and Phase 2). On the other hand, the concept of the accuracy of analysts’ earnings forecasts is used to compare the forecast performance of the surrogate consensus against the IBES consensus (in Phase 3).

The forecast error metrics employed in the literature surrounding the accuracy of analysts’ earnings forecasts include Theil’s U inequality index60 _{(Theil, 1966), which was used in the}

1960s by Malkiel and Cragg (1968) and Brown and Rozeff (1978). However, the emergence

of the absolute percentage forecast errors (APFE) in the 1980s and its usage by Brown et al. (1987), Brown and Rozeff (1980) and Collins and Hopwood (1980) helped the APFE gain popularity amongst researchers of analysts’ earnings forecasts. The minor notable difference between the APFE and Theil’s U is that Theil’s U is based on the correlations (sums of squares of the error deflated by the sum of the square of the realised value) versus the APFE which is based upon a percentage of the absolute error relative to a deflationary component. In essence, both benchmarks translate the deviation of the forecast from the realised actual into a quantifiable value. The applicability of the accuracy and bias metrics are discussed in relation to the aims of this thesis.

The two-fold benefits of the bias measure have been set out in 2.5.1. The bias measure has been used extensively to proxy the bias of analysts’ earnings forecasts in research examining patterns of analyst optimistic behaviour (Abarbanell, 1991; Abarbanell and Bernard, 1992; Ali et al., 1992; Lys and Sohn, 1990 and Mendenhall, 1991). In Phases 1 and 2 of this thesis, the bias measure is designed to compare each individual IBES analyst’s earnings forecast to the IBES consensus in the same period. An additional advantage of employing the bias metric in this manner is that the distributions of IBES analysts’ earnings forecasts across different firms can be standardised and tested for goodness of fit to specific distribution types.

Notwithstanding the ability of the accuracy metric to also standardise IBES analysts’ earnings forecasts between firms using the realised actual earnings, the distribution type discovered using such metric bears no meaning in the context of Phase 1 (Hypothesis 1). As a result, the best estimators of the distributions of IBES analysts’ earnings forecasts based on accuracy become irrelevant to Phases 1 and 2. Thus, the bias measure is selected as the metric of choice to construct the distributions of IBES analysts’ earnings forecasts. Figure 4- 1 serves to clarify the differences between the use of the accuracy and bias metrics to construct the distributions of IBES analysts’ earnings forecasts.

Figure 4-1 is a 3-dimensional diagram representing the cross-sectional distributions of IBES analysts’ earnings forecasts at any particular Forecast Horizon period (m months prior t). The x-axis represents the Forecast Horizon. The y-axis represents the values of IBES individual analysts’ earnings forecasts. The imaginary z-axis sits orthogonal to the x-y plane and is indicated by each dashed line, which also represents the y-axis of a particular accuracy/ bias distribution. The y-axis of a particular accuracy/ bias distribution intersects each distribution’s x-axis at zero. The top distribution depicts an instance of the distribution of IBES individual analysts’ earnings forecasts based on the accuracy metric at Forecast Horizon m for firm n. The bottom distribution depicts an instance of the distribution of IBES individual analysts’ earnings forecasts based on the bias metric at Forecast Horizon m for firm n. Each Forecast Horizon period may consist of n number of these accuracy/ bias distributions, with one accuracy/ bias distribution per firm. By definition, the accuracy distribution’s observations will only exist in the positive region of the x-y plane whilst the

Figure 4-1. Accuracy versus Bias as the Means of Constructing Distributions of Analysts' Earnings Forecasts

Actual Earnings Announcement Date t m months prior t Firm n Analysts’ Ear nings Forecasts Forecast Horizon Bias Distributions (Forecast 1 – Consensus) Zero (Forecast f – Consensus)

Accuracy (Error) Distributions

Firm

n

|Forecast 1 – Actual| Zero |Forecast f – Actual|

bias distribution’s observations may exist across both the positive and negative regions of the x-y plane.

The next section discusses the hypothesis development of Phase 2.

4.2.2 Phase 2: Fitting Distributions of IBES Analysts’ Earnings Forecasts to