earnings uncertainty and - city university of new york

Firms’ Relative Operational Efficiency and Analysts’ Earnings Forecasts

Donal Byard Stan Ross Department of Accounting

Zicklin School of Business Baruch College -- City University of New York

One Bernard Baruch Way, Box B 12-225 New York, NY 10010-5585

(646) 312-3187 [email protected]

Fatma Cebenoyan* Department of Economics

Hunter College -- City University of New York 695 Park Avenue

New York, NY 10021 (212) 772-5393

[email protected]

December 2002

* Corresponding author. We thank Sinan Cebenoyan, Neal Galpin, Hongtao Guo, Ying Li, Devra Golbe, Kevin Sachs, Ping Zhou, and two anonymous referees for their comments. We also gratefully acknowledge the contribution of IBES International Inc. for providing earnings per share forecast data, available through the Institutional Brokers’ Estimate System. These data have been provided as part of a broad academic program to encourage earnings expectation research.

mailto:[email protected]


Abstract Relatively more efficient firms tend to maintain more stable levels

of output and operating performance compared to their industry peers

(Mills and Schumann 1985). Such firms also tend to have more

sustainable performance (Berger et al. 1993). Given their more stable and

sustainable levels of operating performance, we test the related prediction

that financial analysts find the earnings of relatively more efficient firms

to be more predictable. Using a stochastic frontier approach to measure

firms’ relative operational efficiency, we find that analysts face less

earnings uncertainty for relatively more efficient firms, compared to other

firms in the same industry. We find that this broader more encompassing

measure of firms’ relative operational efficiency yields stronger results

than comparable accounting ratios (ROA and ROE). These results

indicate that analysts behave as if they factor into their forecasts an

understanding of the underlying economics of a business of an industry.

Furthermore, the users of these forecasts can also benefit from this

information in terms of assessing the level of reliability of forecasts.

Keywords: Relative Operational Efficiency, Return on Assets, Return

on Equity, Analysts’ Earnings Forecasts, Earnings Uncertainty.

Data Availability: All data are available from public sources.

Biographies Donal Byard is an Assistant Professor of Accounting at the Zicklin School

of Business of Baruch College, City University of New York. His

research focuses on the role of financial analysts as information

intermediaries in financial markets. Professor Byard’s research focuses on

financial analysts’ forecasts for high-tech firms with relatively large

amounts of intangible assets, financial analysts’ reactions to earnings

announcements, and analysts’ role as processors of public disclosures.

Fatma Cebenoyan is an Assistant Professor of Accounting in the

Department of Economics of Hunter College, City University of New

York. Her research interests include the information content of reported

accounting earnings and its components. Professor Cebenoyan is also

investigating the determinants of takeover probability, and the effects of

deregulatory and technical changes in the banking industry.


1. Introduction Relatively more efficient firms tend to have more stable levels of output compared to

other firms within the same industry (Mills and Schumann 1985). These firms also tend to have

more sustainable levels of operating performance, as indicated by the lower firm-specific time-

series standard deviations of their profitability ratios (Berger et al. 1993; Berger and Mester

1997). Given their more stable and sustainable levels of performance, we expect that more

efficient firms will also have relatively more predictable levels of performance compared to their

industry peers. We test this prediction using analysts’ forecasts of annual earnings. In this

study, we examine the relationship between analysts’ level of earnings uncertainty, defined as

the average squared error in the analysts’ forecasts for a firm, and a proxy for firms’ relative

operational efficiency, calculated using a stochastic frontier estimation technique.

Due to their intuitive interpretation as measures of efficiency, and their ease of

measurement, we also conduct our analysis using two accounting ratios, Return on Assets (ROA)

and Return on Equity (ROE), as alternative proxies for firms’ operational efficiency. We

compare the performance of our measure of firms’ relative operational efficiency to that of these

two accounting ratios. Because the measure of relative efficiency is based on a broader set of

data (including separate data on sales output, cost of goods sold, general and administration

expenses, and physical capital cost) regarding a firm’s inputs and outputs than either ROA or

ROE, and involves a more sophisticated estimation controlling for randomness in performance,

we expect it to proxy for a broader and more sophisticated level of knowledge regarding a firm’s

relative efficiency within its industry.

The more efficient firms within an industry have, perhaps through better management,

achieved a competitive advantage vis-à-vis their rivals. This competitive advantage results in a

more sustainable level of profitability, and more stable earnings for these firms. A sophisticated

level of knowledge regarding operational efficiencies within an industry is, thus, useful when

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predicting earnings for firms within that industry. Analysts who possess such knowledge will be

able to identify firms with relatively more stable (and predictable) earnings within the industry

segment they follow.

We adopt the Barron et al. (1998) measure of individual analysts’ earnings uncertainty as

a proxy for the average level of uncertainty individual analysts face when forecasting earnings

for a firm. This variable is essentially the average squared error in the individual forecasts of

analysts following a firm. We model a firm’s relative operational efficiency using a stochastic

frontier methodology, which is used to develop a measure of firm-specific relative operational

efficiency. This relative operational efficiency measure is the independent variable in our study.

We perform rank regressions of our measure of the quality of individual analysts’ information on

this explanatory variable, with control variables for firm size and industry effects. For

comparison, we also perform the same analysis using two accounting ratios, ROA and ROE, as

alternative proxies for firms’ efficiency level. We then compare the performance of these two

ratios to that of the broader and more encompassing relative efficiency measure.

Consistent with our expectations, results indicate that, controlling for a firm’s industry

and size (market capitalization), the average level of uncertainty regarding earnings is lower for

firms that are relatively more efficient that their competitors. . This association is significantly

stronger for the measure of relative operational efficiency than for the two comparable

accounting ratios, ROA and ROE. These results indicate that analysts behave as if they factor

into their forecasts an understanding of the underlying economics of specific industries.

Furthermore, this useful within-industry knowledge regarding operational efficiencies that

analysts seem to factor into their forecasts appears to be broader than just the information

reflected in accounting ratios. These results, thus, provide indirect empirical evidence

supporting one potential reason why individual analysts specialize in only following firms from a

very limited number of industries. Specifically, by specializing by industry, analysts develop in-

depth knowledge regarding within-industry operational efficiencies, which are, in turn, useful in

forecasting earnings more accurately for the firms within that industry. These results may also

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be of benefit to the users of forecasts. Investors who rely on analysts’ earnings forecasts could

use an analysis of operational efficiency to identify the firms within particular industries that are

likely to have more reliable earnings forecasts, resulting from the reduced uncertainty in

analysts’ forecasting process for these firms.

The paper is organized as follows. Section 2 discusses the background, our measures of

firms’ operational efficiency and analysts’ level of uncertainty, and our expectations. Section 3

outlines our sample selection procedures, and the variables and empirical models used in the

study. This is followed by section 4, which describes our results, including tests of robustness.

The paper concludes with a discussion in section 5.

2. Background 2.1 Theoretical Background

Relative operational efficiency is viewed in both the Industrial Organization and Strategic

Management literatures as the product of firm-specific factors such as: management skill,

innovation, cost control, and market share, which determine current firm performance, and

critically, the sustainability of this level of performance (McWilliams and Smart 1993). In

essence, relative operational efficiency characterizes a firms’ competitive strength relative to its

competitors.

Prior studies indicate that more efficient firms have more stable performance. For

example, several studies within the banking industry report a negative correlation between

banks’ level of inefficiency and the stability of their level of profitability over time (Berger et al.

1993; Berger and Mester 1997). Berger et al. (1993) and Kwan and Eisenbeis (1996) both argue

that this relationship is consistent with the idea that more efficient firms avoid riskier projects in

order to avoid the possibility of jeopardizing their relatively more profitable position within their

industry. Similarly, Mills and Schumann (1985) conclude that more efficient firms have more

stable output levels compared to less efficient firms. In their examination of the effects of

demand fluctuations on firms, they argue that firms become relatively more efficient through

cost-minimizing strategies. This cost-minimization approach results in more long-term stability

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in the level of output for relatively more efficient firms, but at the cost of a reduction in their

flexibility to respond to short-term changes in demand. In a recent study, McGahan and Porter

(1999) find evidence that highly profitable firms (profitable due to their efficient operations)

have more persistent performance than less profitable firms within the same industries.

Collectively, these studies suggest that relatively more efficient firms have more stable earnings

and maintain their performance through higher risk avoidance, and/or through the organization

of their production processes to yield more stable levels of output.

We estimate firm efficiency using a stochastic frontier methodology, which gives a

relative industry-based performance measure. The frontier concept builds on the original work

of Farrell (1957), who first suggested the use of industry best practice as the benchmark to

evaluate firm performance. After its operationalization by Aigner et al. (1977), the concept of

relative efficiency estimated using a frontier approach has become a frequently employed firm

performance measure in both the Finance and Economics literatures (Allen and Rai 1996;

Merger and Mester 1997; Rogers 1998; Wheelock and Wilson 2000, among others). The

widespread use of the frontier approach to measure firms’ performance results from these

measures conceptual appeal, as they capture firms’ relative performance within their industries.

In effect, this type of analysis yields a comprehensive benchmarking of a firm’s performance

relative to its competitors.1 In its estimation this methodology also removes non-controllable

random factors such as luck, climate, and machine performances from the observed level of

corporate performance, providing a direct firm-specific measure of the systematic factor(s)

affecting a firm’s performance, such as managerial effectiveness (Hughes et al. 2002; see

Appendix A for technical details).

Since analysts care about forecast accuracy (Mikhail et al. 1999), they have an incentive

to use any knowledge they may have regarding the relative efficiency of firms within the

1 In fact, both individual firms and industry consultants have used this within-industry relative performance

evaluation approach to quantify objective performance rankings within industries (Berger and Humphrey 1997).

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industry sector they follow to generate more accurate forecasts for these firms.2 As a result,

because more efficient firms have more stable earnings, we expect analysts with sophisticated

knowledge regarding the firms in the industry they follow to identify relatively more efficient

firms and generate more accurate earnings forecasts for these firms. In other words, we expect

there to be a negative association between analysts’ level of earnings uncertainty and a measure

of relative operational efficiency.

As an additional analysis, we also compare the performance of our measure of relative

operations efficiency to that of two simple accounting ratios (i.e., ROE and ROA), which may be

viewed as two alternative measures of firm efficiency. Efficiency is a performance measure, and

return on investment (ROI) measures such as return-on-assets (ROA) or return-on-equity (ROE)

are frequently employed as the “best-available” criteria to evaluate business performance

(Jacobson, 1987). Our measure of relative efficiency, however, captures more information

regarding a firm’s relative performance than these two accounting ratios. This results from the

fact that unlike the ratios ROA and ROE, our measure of relative operational efficiency is based

on a separate analysis of each industry, it considers multiple decision variables (multiple inputs),

and it adjusts for the effects of random shocks. As a result, our measure of relative operational

efficiency is a conceptually more thorough performance measure than either ROE or ROA. We

thus expect that if more efficient firms have more predictable earnings, then there will be a

stronger association between analysts’ earnings uncertainty and relative operational efficiency,

than between analysts’ earnings uncertainty and either ROA or ROE. On the other hand, if ROA

and/or ROE work equally as well as measures of firms’ operational efficiency, then we would

expect the empirical association between these ratios and analysts’ earnings uncertainty to be as

strong as the association between firms’ relative operational efficiency and analysts’ earnings

uncertainty.

2 Annual rankings of analysts use forecast accuracy as one of the criteria for choosing the best analysts (see

Institutional Investor 2000; and Wall Street Journal 2001). In fact, the Wall Street Journal’s annual survey ranking of the best analysts is based only on analysts’ earnings forecast accuracy.

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More broadly, our study investigates whether analysts can use extensive within-industry

knowledge regarding production processes and relative operational efficiencies to forecast

earnings more accurately for firms within the industry in which they specialize. Our study, thus,

provides some empirical evidence regarding one potential reason why individual analysts tend to

specialize in only following firms from a very limited number of industries.3 Consistent with the

observed industry concentration of individual analysts, surveys provide evidence of analysts’

possessing some from of “industry knowledge” which they use as part of their forecasting

activities.4 Our measure of relative operational efficiency, because of its industry-based nature,

is thus likely to capture knowledge similar to that used by analysts when forecasting earnings for

the firms within the industry sector they specialize in following. In addition, users of forecasts

can benefit from an understanding of the relationship between efficiency and analysts earnings

uncertainty in terms of improving their assessment of the reliability of forecasts.

We now outline in more detail our measure of firms’ relative operations efficiency, which

we use as an experimental variable in our analysis. We then outline our measure of the quality

of individual analysts’ information, which we use as the dependent variable in our analysts.

2.2 Operational Efficiency

We model firms’ relative operational efficiency using a stochastic frontier methodology.

Standard microeconomic production theory specifies the input-output relationship in terms of a

firm’s production function (as well as their value-equivalent cost, profit and revenue functions).

This function describes an optimal relationship that achieves a maximum or a minimum, under

constraints imposed by technology and prices. In other words, it describes a frontier tracing the

3 For example, selecting only forecasts of annual earnings for 1997 from the Institutional Brokerage Estimation

System’s (IBES) Detail file, we find that for 1997 IBES tracked 4,753 analysts forecasting for 6,171 firms. The median (mean) number of firms each analyst followed was 10 (12.78). Although these 6,171 different firms were spread over 70 different two-digit SIC codes, the median number of two-digit SIC codes representing the firms each individual analyst followed was just 2. Furthermore, over 75-percent of analysts only followed firms from 4 or less two-digit SIC codes.

4 For example, in surveys of institutional investors, “industry knowledge” is consistently ranked as the most important attribute they ascribe to analysts each year since Institutional Investor magazine began these surveys (Institutional Investor 2000, p. 62). Each year in these surveys investors have ranked an analyst’s “industry knowledge” as the most important attribute from among a set of ten potential attributes of analysts, such as “accessibility,” “independence from corporate finance,” “timely calls and visits,” “written reports,” “financial models,” “earnings estimates,” or “stock selection” (Institutional Investor 2000)

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best practice of a specified objective such as cost minimization or profit maximization (Färe and

Primont 1995). Since the frontier concept is consistent with firms’ optimizing behavior in

microeconomic theory, deviations from the industry-specific best-practice frontier provide an

intuitive measure of a firm’s level of inefficiency relative to its competitors.5

We measure operational efficiency in terms of a firm's revenue (output) generating

ability, given the resources (inputs) it expends relative to its competitors. To do so, we employ a

stochastic frontier methodology that incorporates a two-component error structure. One

component represents random, uncontrollable factors affecting a firm’s relative efficiency,

whereas the second component measures the systematic firm-specific component of a firm’s

relative (in)efficiency. This systematic component of firm-specific error is transformed to a

firm-specific measure of corporate efficiency, EFFIC. EFFIC measures how efficient a firm is

relative to other firms in the same industry (this methodology is described in Section 3.3, and in

more technical detail in Appendix A). As a result, EFFIC captures a wide spectrum of data

relating to a firms’ performance within a given industrial sector.

2.3 The Barron, Kim, Lim, and Stevens (1998) Measure of Analysts’ Uncertainty

We adopt a measure of the quality of individual analysts’ information outlined in the

model of Barron, Kim, Lim and Stevens (1998; hereafter BKLS).6 Specifically, we adopt the

BKLS measure of earnings uncertainty for a given firm, which they define as the average

squared error in the individual analysts’ forecasts for a given firm. BKLS show how individual

analysts’ uncertainty (U) can be expressed in terms of expected dispersion (D) and squared error

in the mean forecast (SE) as follows (see BKLS equation 15, p. 427):7

SEDN

U +

−=

11 , (1)

5 As a result, frontier models have been widely adopted in the economics and finance literatures for the study of

production and cost efficiencies in evaluating firm performance (Färe et al. 1985; and Berger and Humphrey 1997).

6 See Barron et al. (2002a and 2002b), and Botosan and Harris (2000), for examples of recent empirical studies using the BKLS model.

7 The primary assumption underlying the BKLS uncertainty measure (U) is that analysts strive to issue the most accurate forecast possible, or analysts issue unbiased forecasts (Barron et al. 2002b). We examine the robustness of our results to this assumption (see section 4.3).

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where: (2

11∑=−

−

N

aa FF

N)1

=D , (2)

( 2FA −= )SE , (3)

and where: U is the level of uncertainty of an individual analyst (average squared error in an individual analysts’ forecast of a firm);

N is the number of analysts forecasting; Fa is the forecast by analyst a;

F is the mean forecast; and A is the actual earnings realization.

Use of the BKLS uncertainty measure (U) above is similar to the approach used in prior

studies that have also examined observable properties of analysts’ earnings forecasts with a view

to drawing inferences regarding analysts’ underlying information (Brown and Han 1992). BKLS

develop this measure of the quality of individual analysts’ information in terms of expected (ex-

ante) dispersion (D) and squared error (SE) in the mean forecast. We use ex post realizations of

D and SE as substitutes, which introduces measurement error into our estimates of the BKLS

uncertainty measure (Barron et al. 2002b). This measurement error will, however, be

ameliorated when firm-years are average over time, as are used here.

2.4 Control Variables

We include size (market capitalization deflated to constant 1993 CPI-dollars) and

industry dummy variables as control variables. Firm size has been shown to be an important

determinant of the demand for information regarding a firm. Furthermore, analysts may have an

incentive to expend more effort forecasting for larger firms, as this is likely to generate more

trading volume for their brokerage firm employers. As a result, analysts may have an incentive

to expend more effort forecasting for larger firms. Consistent with the greater demand for

analysts’ information for larger firms, prior studies demonstrate an association between firm size

and analysts’ information and forecast accuracy (Lys and Soo 1995).

The focus of our analysis is on within-industry differences in the level of earnings

uncertainty of individual analysts (U) forecasting earnings. Hence, our inclusion of industry

dummy (control) variables.

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3. Study Design and Sample Selection 3.1 Sample Selection Procedures

We use the Active and Research files from the 1999 edition of the Compustat database to

identify our sample firms. Our final sample includes all firms from selected industries (see

below) with sufficient data to estimate industry-specific production-functions in any year of our

five-year sample period (1993-1997). We use a five-year sample period as prior studies indicate

that efficiency scores vary through time, and as a result, DeYoung (1997) recommends the use of

a sample period of about six years in efficiency studies.8

We exclude financial services and other regulated industries, such as transportation and

utilities. We exclude these regulated industries to avoid pooling firms with different operating

environments (e.g., regulated versus competitive), which may affect firms’ behavioral goals.

The selection process for the remainder of the potential sample industries is restricted by the

availability of the relevant data to estimate the production frontier for that industry in that year.

If there are not sufficient data for enough firms to estimate the efficient production frontier for

that industry in that year, then all firms from that industry in that year are excluded from the

sample. At the frontier estimation stage, the data is tested for possible outliers using the

standardized residuals method, and observations with standardized residuals in excess of two are

deleted from the final set (Belsley et al. 1980). Data for firm-years with the required efficiency

scores (and firm size) are then matched with data for the BKLS uncertainty measure, which is

calculated using forecasts of annual earnings sampled over the period 1994 through 1998. The

efficiency score is lagged one year before the BKLS uncertainty measure derived from analyst

forecasts (see below).

3.2 Data

Calculation of our measure of individual analyst uncertainty (the average squared error in

individual forecasts) requires the selection of a point in time at which to gather our forecast data. 8 If too long a sample period is selected, the concept of "average firm efficiency" looses its meaning. This arises

because other factors such as management, technology, and/or regulatory environment may change over time and also affect the stability of the efficiency measures averaged. DeYoung (1997) shows that using a sample period of about six year alleviates this concern.

9

We base our analysis on forecasts of annual earnings for year t made during a 30-day forecast

window immediately after the announcement of first quarter earnings for year t. This choice of

forecast period ensures that the individual forecasts for each firm are conditioned on the same

publicly available information, in this case a public earnings announcement. The short 30-day

forecast period controls for the potential confounding effects of stale forecasts (Brown and Han

1992). In addition, to target active analysts, we only select forecasts that are updates of previous

forecasts, issued in the 60-day period before the announcement of first quarter earnings in year t

by the same individual analysts, because these analysts are less likely to be “herding” (Barron

and Stuerke 1998).

Our efficiency measure (EFFICij,t-1) is calculated using accounting data for year t-1,

while our BKLS uncertainty measure is based on forecasts of earnings for year t. This is done to

avoid a spurious correlation between our accounting data-based efficiency measure (EFFICij,t-1)

and the BKLS measure of earnings uncertainty (Uijt), because actual earnings is a component of

the calculation of both variables. Our sample of firm-years, thus, draws on the IBES forecasts of

annual earnings for fiscal year t, and Compustat data for fiscal year t-1, where the sample years

are drawn from the period 1994 through 1998. Our sample firm-years meet the following four

requirements:

1) The quarterly earnings announcement date for first quarter earnings for year t is available from either the Active or Research Compustat quarterly files;

2) At least two forecasts of year t annual earnings from two different individual analysts are issued in the 30-day period immediately following the first quarter earnings announcement date who are updating a previous forecast, made in the preceding 60 day period;9

3) Actual EPS data for year t annual earnings are available from the IBES Actual Earnings file; and

4) The necessary Compustat data needed to calculate the efficiency score, and data for ROA, ROE, and control variables, is available for firm i in year t-1.10

9 In rare cases where multiple forecasts are available from the same analyst in the 30-day forecast window after the

first quarter earnings announcement, only the last forecast is selected. 10 In addition, we only include firms from industries with sufficient data to calculate the efficiency score needed.

Some industries in certain years could not be included due to the inability of the industry-specific production function estimates to converge, especially when there is a small number of firms in the industry. Instead of overriding the stringent converging rules, all non-converging industries are left out of the sample to avoid possible model specification problems.

10

As can be seen in Panel A of Table 1, 19,985 firm-years have the required Compustat

data for ROA and ROE, and the Compustat data needed to calculate the firm-specific efficiency

score for that firm-year. When this sample of firm-years is matched with forecast data drawn

from the IBES database, it yields a final sample of 1,839 firm-years, representing 907 separate

firms. Panel B of Table 1 shows the industry composition of this sample of 907 firms. These

firms are spread over 34 different industries, with concentrations in Oil and Gas Extraction (9-

percent of sample), Computer Services (mainly software) (8.2-percent), Apparel and Accessory

Stores (7.3-percent), Computer and Office Equipment (6.6-percent), and Electronic Components

and Accessories (6.2-percent).

(Insert Table 1 About Here)

3.3 Measuring Firms’ Relative Efficiency

For the measure of relative production efficiency, we calculate individual indices of

operating inefficiency for each firm-year separately by industry-year. For each industry-year, we

employ a stochastic frontier methodology based on a translog production function. The firm-

specific objective function employed in this study can be expressed as:

( )vw,X,fY = , (4)

where: Y is sales revenue, X is a vector of operational inputs generating this revenue, w

represents firm-specific deviations from the efficient frontier due to factors under managerial

control, and v represents random uncontrollable factors that affect firm’s performance (see

Berger and Mester 1997). To separately estimate this production relation using data for each

industry-year, we employ a standard translog function with more than two inputs.11 In the case

of one output (Y) and three inputs (X’s), the translog function can be expressed as follows (e.g.,

see Bairam 1994), which are estimated for each industry-year:

11 The translog specification is used in preference to the Cobb-Douglas specification. Unlike the Cobb-Douglas

specification, the translog specification is flexible in that it allows for non-uniform scale characteristics, and a degree of substitution among inputs that is not limited to unity.

11

( ) ( )( ) ( )

( ) ( ) ( ) εβββββ

βββββ

lnlnXnXllnXnXllnXnXllnXlnX

lnXlnXlnXlnXYln 1

++++++

++++=

322331132112

2333

2222

2111332210

...5.05.0

5.0

(5)

This is followed by the decomposition of the error term from the estimates of this equation into

its two components (v and w), as defined in equation (4). The decomposition is achieved by

specifying the following distributional assumptions (see Jondrow et al. 1982):

( )2,0 vNiidv σ≈ , and ( ) |,0| 2wNw σ≈ .

This specification of error terms is based on the intuition that the performance of firms differs as

a result of : 1) random fluctuations such as luck, climate, machine performance etc (captured by

v); and 2) a firm-specific components (w), capturing a firms’ ability to follow the industry best

practice. The firm-specific inefficiency scores obtained through this error decomposition (w) are

further transformed into the more appealing efficiency scores (EFFIC) using Battese and Coelli’s

(1988) algorithm. The specific details of the decomposition and the transformation are given in

Appendix A.

In our selection of input and output variables used to estimate equation (5) we follow the

prior efficiency studies with some modification. We use the dollar value of net sales (Compustat

item number A12) as a measure of output. The quantity of output, rather than its dollar value,

has traditionally been used in efficiency studies. However, this approach has been criticized.

Kolari and Zardkoohi (1987) among others, argue that firms compete to increase their market

share, which is measured as their share of dollar sales in the market, as opposed to the quantity

sold. Also, when there are (even slight) differences in the quality of products among firms in the

same industry, this leads to a distortion in the measurement of relative efficiency across firms if

one uses only a quantity measure of output. This distortion results from the fact that higher

quality producers, who will tend to have higher production costs, will appear to be systematically

less efficient. If higher quality producers can pass-on their higher production costs to consumers

in the form of higher prices, then the dollar value of their output(s) will be a more appropriate

measure of their output.

12

The three inputs used in our specification of firm’ production function are: Cost of

Goods Sold, Selling, General and Administrative Expenses, and Physical Capital Cost (that

consists of rent and depreciation expense) (Compustat item numbers A41, A189, and the total of

A47+A14-A163, respectively). Our choice of these three input categories also follows the choice

of different types of costs in the literature regarding the evaluation of input choices in the

production process, such as variable, semi-variable, and fixed costs (van den Broeck 1988;

Bairam 1994). We scale all the variables in the frontier estimation by total assets to avoid

heteroskedasdicity and scale bias (Berger and Mester 1997).12

3.4 Measuring Individual Analysts’ Uncertainty

Using our sample of earnings forecasts, we calculate ex-post realized squared error in the

mean forecast (denoted SE^ ) and dispersion (denoted D^ ). Similar to equations (2) and (3) in

Section 2.2, we calculate the ex-post realized squared error in the mean forecast and dispersion

for firms as follows:

( 2itSE itit FA −=

∧ ) (6)

(2

111ˆ ∑

=

−−

=N

aitait

jtit FF

ND ) , (7)

where is the estimated squared error in the mean forecast for firm i in year t; is the

sample variance of the forecasts for firm i in year t; A

itSE∧

itD̂

it is the actual earnings for firm i in year t;

Fait is the forecast of earnings from analyst a for firm i in year t; itF is the mean of the forecasts

for firm i in year t; and Nit is the observed number of forecasts for firm i in year t. We scale

these estimates of the squared error in the mean forecast (SE^ ) and dispersion ( D^ ) by the absolute value of actual EPS ( itA ). Scaled SE^ and , together with the observed number of D^

12 Since the costs and revenues of large firms are expected to be larger than for small firms, the random errors of

larger firms would have larger variances without any normalization. This is an important consideration since the (in)efficiency terms are derived from the combined residuals. Without an appropriate normalization, this effect may cause the variances in these terms to depend on firm size.

13

analysts forecasting (N), are substituted into equation (1) to calculate BKLS uncertainty (Uit) for

firm i in year t.13

3.5 Empirical Models

Important data and econometric issues arise in estimating the above equations. First, our

data consists of observations for 1,839 firm-years, which represent 907 separate firms. Our

sample consists of an unbalanced set of panel data; some firms have more yearly observations

than others. In addition, firm efficiency scores (and firm size) tend to be correlated across

sample years (Kwan and Eisenbeis, 1996), introducing the possibility of cross-sectional

dependence affecting a pooled cross-sectional time-series analysis of the data. As a result, we

replace the firm-year observations with firm-specific mean values, computed across all the firm-

years available for each firm. Using such across-panel means is a widely adopted procedure for

such unbalanced panel datasets (Greene 2000; p. 567).14

In addition, we cannot predict linear relations between the dependent and independent

variables. This arises from the fact that BKLS uncertainty, our dependent variable, is based upon

squared error in individual forecasts. Thus, similar to Lang and Lundholm (1996), we use non-

parametric rank regressions (with the average squared error in individual forecasts (U) as our

dependent variable).15

We test the association between BKLS uncertainty and firm’ relative operational

efficiency using the following equation:

Model 1: . (8) i

J

jjjiii IDmSIZEmEFFICmU εγψψψ ∑

=

++++=2

210

13 We scale by the absolute value of actual EPS ( itA ) to ensure that SE^ and D^ are comparable across firm-years.

We also remove observations with absolute values of itA less than 10 cents from the sample as a control for extreme observations induced by the choice of scaling variable. We also conduct our analysis using unscaled

and D and scaling by price. Our inferences are unchanged using these alternative specifications. SE^ ^

14 We also use a Weighted Least Squared (WLS) estimator (see Section 4.3) using the number of firm-year observations per firm as a weighting variable. Our inferences are unchanged using this alternative specification.

15 Use of ranks relaxes the linearity assumption, and assumes only a monotonic relation (Conover 1980; and Lang and Lundholm 1996). As a result, the rank regression specification is more efficient than ordinary least squares using the untransformed data. Results from a comparison of the regression residuals for the untransformed and rank-transformed specifications are consistent with an improved specification using the rank transformation.

14

mX indicates that we are taking the mean value of the variable X for firm i across all available

years for our sample period. We also estimate Model 1 using both ROA and ROE in the place of

EFFIC.

Where: mUi is the (rank of) average level of uncertainty across the analysts forecasting earnings for firm i;

mEFFICi is the (rank of) relative efficiency score for firm i; mSIZEi is (rank of) market capitalization of firm i; and IDj are a series of industry dummy variables equal to one for firms in industry j.

Model 1 tests the association between a firms’ relative production efficiency (EFFIC) and

individual analysts’ earnings uncertainty, controlling for firm size and industry effects. We

predict that: ψ1<0. As an alternative, we also estimate Model 1 using either ROA or ROE as

alternative experimental variables proxying for firms’ relative operational efficiency. Again, for

these alternative estimates of Model 1, we predict that: ψ1<0. In summary, to test our first

conjecture that more efficient firms have earnings that are easier for analysts to predict, we

estimate three variants of Model 1 above, using EFFIC, ROA, or ROE as three alternative

proxies for firms’ relative efficiency.

Our second hypothesis relates to the relative strength of the relationship between earnings

uncertainty and within-industry variation in EFFIC versus the ratios ROA or ROE. We test the

directional hypothesis that there is a stronger relationship between EFFIC and U then between

either ROE or ROA and U. We use a Vuong test, which is based upon a comparison of non-

nested models, to test this hypothesis. This test is based upon a comparison of the R2 statistics of

the different models estimated (i.e., Model 1 with EFFIC vs. Model 1 with ROA; and Model 1

with EFFIC vs. Model 1 with ROE).


Table 2, presents descriptive statistics for the sample of 907 firms. Consistent with

concurrent research (Barron et al. 2002a and 2002b), the data for BKLS uncertainty (mUi)

indicates that the distribution of this variable is highly skewed -- for example, the mean of the

sample is 0.4992, whereas the standard deviation is 1.5607. This is not surprising, since BKLS

uncertainty is a measure of the average squared error in the individual forecasts of analysts

15

following a firm, and is calculated using dispersion (forecast variance) and squared error in the

mean forecast. Our sample firms are quite large, with a mean market capitalization of

approximately $2.3 billion.

4. Empirical Results As the first step in our analysis, a production frontier is estimated for each industry-year

in our sample period (1993 through 1997) using the Translog equation (5) setout in section 3.3.

The residuals from these industry-specific efficient frontier estimates are then decomposed into

their random (v) and firm-specific components (w) for each firm-year. The firm-specific

component (w) is, in turn, used to calculate the efficiency score, EFFIC (see Appendix A for

details of the transformation). This efficiency score data is then matched with IBES earnings

forecast data to produce the final dataset of 1,839 firm-years representing 907 firms described in

Tables 1 and 2.

4.1 Pairwise Correlation Analysis

Table 3 presents Spearman (Rank) correlations between our variables of interest. As can

be seen in Table 3, consistent with our expectation, we find a negative association between

firms’ relative production efficiency (EFFIC) and BKLS uncertainty (U) (correlation coefficient

equals –0.14; significant at p<0.01, one-tailed test). In addition, analysts’ earnings uncertainty is

also negatively related to both the return on assets (ROA) and return on equity (ROE) ratios --

correlation coefficients are –0.19 and –0.20, both significant at p<0.01, one-tailed. Thus, the

strength of the association between both ratios and analysts’ earnings uncertainty seems, in fact,

to be stronger than the association between relative production efficiency and individual

analysts’ earnings uncertainty. Consistent with prior research (Lys and Soo 1995), we also find a

negative association between firm size (SIZE) and BKLS uncertainty (correlation coefficient

equals –0.07; significant at p<0.01, one-tailed test).


The pairwise evidence presented in Table 3 suggests that BKLS uncertainty (U), is

negatively associated with both firms’ relative production efficiency, and the ratios ROA and

16

ROE. These analyses are incomplete however, as we have not controlled for other known

determinants of analysts’ information environment, and therefore, they do not constitute

evidence of a marginal effect of EFFIC controlling for size and industry effect. We address this

question next.

4.2 Multivariate Regression Analysis and Vuong Test Results

Table 4 presents our results from our Ordinary Least Squares (OLS) rank regression

estimates of models based on equation (8). Note, we do not display the coefficient estimates for

the 33 industry dummy variables also included in the estimation. Consistent with our

expectations, the OLS rank regression analysis of equation (8) confirms a negative relationship

between firms’ relative production efficiency, as proxied by EFFIC, and BKLS uncertainty (U) -

- see Model 1a. Consistent with our first general expectation, controlling for firm size and

industry, there is a significant negative association between within-industry variations in

analysts’ earnings uncertainty and the relative production efficiency of those firms (p<0.01, one-

tailed).


Table 4 also shows the results of two alternative estimated models based on equation (8) -

- Models 1b and 1c, where we use either ROA or ROE as alternative experimental variables.

Consistent with our expectation, we find a negative association between either ROA or ROE and

individual analysts’ earnings uncertainty (p<0.01, one-tailed, for both models). Again, consistent

with prior research indicating that analysts have more precise information, as reflected in smaller

forecast errors for larger firms (see Lys and Soo 1995), in all three estimates of Model 1 we find

a significantly negative association between BKLS uncertainty and firm size (p<0.05, one

tailed).16

The R2’s reported in Table 4 suggest that efficiency explains more of the variation in

analysts’ uncertainty than either ROA or ROE. However, to compare Model 1a formally with

16 The results on SIZE are marginally significant (at the 0.05, one-tailed test). In part this is due to the industry

control variables, which capture variations in firm size across industries. Running the analysts without the industry control variables yields stronger results for SIZE.

17

models 1b and 1c in terms of their explanatory powers, we employed Vuong test for nonnested

models. Other nonnested model tests such as J-test by Davidson and Mackinnon (1981) can give

inconclusive results when the models include variables with incremental explanatory variables as

may be the case here (Dechow 1994). The Vuong test on the other hand is a more powerful test,

and allows a directional test to determine the more powerful model for explaining a given

dependent variable.17 Table 4 includes the results of the Vuong tests comparing Model 1a with

both Models 1b and 1c. The z-statistics (and corresponding one-tail p-values) comparing these

models are shown on the right hand side of Table 4. Since positive and significant test statistics

implies residuals from models 1b (ROA) and 1c (ROE) are larger than model 1a (EFFIC),

consistent with our argument, we find that the model with EFFIC is a better model in explaining

the variation in analyst’ earnings uncertainty (p=0.02 and 0.03, one-tailed test).

As an additional test to compare the relative strength of the association between

individual analysts’ level of earnings uncertainty (U) and relative operational efficiency

(EFFIC), compared to either the ROA or ROE ratio, we also estimate nested models and use a

Wald test (Greene 2000, p. 273) to compare coefficient estimates. Specifically, we estimate two

models; first including both EFFIC and ROA, and second including both EFFIC and ROE as

independent variables. We then use a Walt test to compare the magnitude of the coefficient on

EFFIC with that of either ROA or ROE. Untabulated results indicate that the coefficient on

EFFIC is statistically larger (p<0.01) for both models.

EFFIC, thus, seems to capture more information associated with within-industry

variations in the quality of analysts’ information about earnings than either ROA or ROE. Since

17 Following Dechow (1994), we obtained mi (mean of likelihood ratio, LR) for each observation using:

−+

=

eff

ieff

roeroa

iroeiroa

effic

roeroai RSS

eRSSen

RSSRSS

m2

,

2,, )()(

2log

21

where eiroa or iroe and eieff residuals, and RSSroa, roe and RSSeff are the residual sum of squares from models 1b, 1c and 1a respectively. Next, we calculated the standard deviation of LR by regressing mi on unity. The coefficient in this regression is equal to the first term in mi above giving us the mean difference in explanatory power between the efficiency scores and ROA or ROE. The standard error from this regression provides the significance of this difference. The z-statistic to evaluate the significance, then, is calculated using the t-statistic from this regression multiplied by (( . 2/1)/)1 nn −

18

EFFIC is a broader and more encompassing proxy for relative efficiency, which has been shown

to be related to the firm’s performance stability, this indicates that analysts act as if they use

extensive within-industry knowledge regarding production processes and operational efficiency

to forecast earnings more accurately. This may partly help explain why individual analysts

specialize so much by industry in selecting the firms they tend to follow.

4.3 Sensitivity Analysis

We performed several sensitivity tests to establish the robustness of the results. Unless

otherwise outlined below, our inferences from our results on Tables 4 and 5 are unchanged using

any of these alternative specifications:

Weighted Least squares: We re-estimated all the regressions reported in this paper using

Weighted Least Squares (WLS), where the number of observations available for each firm in

each quarterly regression forms the weighting variable (Kmenta 1997, p. 368). Our results for

firm size are more significant using the WLS estimator. This is not surprising, as the WLS

estimator places relatively more weighting on the observations for larger firms that tend to have

a greater number of firm-year observations.

Scaling Variable: In our analysis D^ and SE^ are scaled by the absolute value of actual Earnings

Per Share (EPS). We also conduct our analysis using stock price at the end of the fiscal year as a

scaling variable, and using unscaled D^ and SE^ .

Average Bias-Adjusted Estimator: The principal assumptions underlying the BKLS uncertainty

measure is that analysts strive to issue the most accurate forecast possible, or that forecasts are

unbiased (see Barron et al. 2002b). We test the possibility that the results of our hypotheses tests

could be affected by bias. Following Barron et al. (2002b), as a diagnostic measure, we re-

calculate the squared error in the mean forecast as follows:

( ) (2

11*

ijtSE

−∑

=−−=

∧ I

iIFA ijtijtijtijt FA ) , (12)

where is the squared error in the mean forecast for firm i in industry j in year t, adjusted for

the average level of error across all analysts forecasting for all analysts forecasting for all firm-

*

ijtSE∧

19

years in the sample. Thus, we adjust the squared errors in the mean forecasts of firms for the

average level of “optimism” in all forecasts for all firm-years in our sample.

Different Specifications of ROA: We also use a number of different specifications of ROA (for

example scaling by the average of total assets between the start and end of the fiscal year).

Alternative Forecast Sample Periods: We also conduct our analysts using samples of forecast

revisions made after the second and third quarterly earnings announcements.

5. Conclusion Evidence indicates that relatively more efficient firms tend to have more stable levels of

output compared to other firms within the same industries (Mills and Schumann 1985). These

firms also tend to have more sustainable levels of operating performance, as indicated by the

lower firm-specific time-series standard deviations of their profitability ratios (ROA’s) (Berger

et al. 1993; Berger and Mester 1997). Given their more stable and sustainable level of

performance, we test the related prediction that these firms have more predictable earnings. We

test this prediction using analysts’ forecasts of annual earnings. In this study, we examine the

relationship between the quality of individual analysts’ information regarding annual earnings

(analysts’ earnings uncertainty), and a proxy for firms’ relative operational efficiency.

We estimate firm efficiency using a stochastic frontier approach using data on multiple

firm inputs. This approach provides a broad measure of a firm’s relative operational efficiency

within its industry. First, we test if this measure of a firm’s relative efficiency is related to the

quality of analysts’ information regarding earnings, controlling for firm size and industry effects.

Results indicate that analysts’ earnings uncertainty (the quality of their information) is lower

(higher) for more efficient firms. Second, we compare the performance of this variable to that of

two related accounting ratios, Return on Assets (ROA) and Return on Equity (ROE), which we

use as two alternative proxies for firms’ level of relative operational efficiency. The results of

this analysis indicate a stronger association between analysts’ earnings uncertainty and the

measure of relative operational efficiency, than the accounting ratios ROA and ROE.

20

These results indicate that analysts can use extensive within-industry knowledge

regarding production processes and operational efficiencies to forecast earnings more accurately

for firms within the industry in which they specialize. Furthermore, this useful within-industry

knowledge regarding operational efficiencies would appear to be broader than simply the

information reflected in some accounting ratios (ROA and ROE). More broadly, our tests

provide empirical evidence supporting one potential reason why individual analysts specialize in

only following firms from a very limited number of industries. That is, through industry

specialization analysts can use knowledge regarding the underlying economics of a business to

identify firms with more stable earnings. In addition, our results indicate that the users of

forecasts can benefit (in terms of improving their assessment of the reliability of forecasts) from

an understanding of the relationship between efficiency and analysts’ earnings uncertainty

outlined here.

21

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24

Appendix A Stochastic Frontier Methodology

The stochastic frontier methodology is derived from the econometric efficiency model

developed by Aigner et al. (1977) and improved upon by Jondrow et al. (1982). This

methodology takes any objective function, estimates a best practice frontier based on the chosen

variables, and then provides the firm-specific inefficiencies from a decomposed residual term.

The objective function can be rewritten, in short form, as:

eXfY += );( β , (A1)

where Y is the maximum attainable output from the input vector of X’s, β is the vector of

parameters estimated, and a composite error term made up of two components. The

justification for this specification of the error term is that firms differ from each other in terms of

their objective due to: 1) random fluctuations such as luck, climate, machine performance, etc,

and 2) their ability to follow the best practice.

e

18 Hence e decomposes as follows:

wve += , (A2)

where v represents the disturbances due to uncontrollable events, and w the deviations caused by

the controllable factors. Let Y* be the maximum output (sales in this study) that can be

produced given the inputs of X1, X2, and X3 (cost of goods sold, general and administrative

expenses, and physical capital, respectively). Y* is the frontier objective function of which its

parameters, vector β , are estimated from the given observations of Y, and X’s. Since the

frontier function is assumed to be stochastic, rather than deterministic, equation (A1) becomes:

Y wY += * , (A3) where: ( ) vXfY += β;* , (A4)

and where v assumes random fluctuations (both negative and positive), but w assumes only

negative values. Since the external events can be both favorable and unfavorable, they can

increase or decrease the output and are consequently assumed to be drawn from a two-sided 18 This is the biggest advantage of stochastic frontier approach. None of the other technologies make any

accommodations for (random) non-controllable factors. Furthermore, this approach is stochastic which allows the researcher to make statistical inference based on the results.

25

distribution (usually normal). On the other hand, inefficiencies only decrease the output or

increase the costs and are assumed to drawn from a one-sided distribution (usually half-normal).

Since the inefficiency component, w, cannot be observed directly, it has to be obtained from the

estimated e, which obviously contains information on w. The suggested solution considers the conditional distribution of w, given e , ( )wvw + . To do this, first, the joint density of w and v is

written as the product of their individual densities. Since e is defined as the sum of these two

error terms, this joint density is transformed initially into the joint density of e and w, and

subsequently into the density of e by integrating w. The conditional distribution of w, then, is

given as the ratio of the joint density of e and w to the density of e (Maddala 1977; see also

Appendix of Jondrow et al. 1982, for detailed calculations). The mean or the mode of this

distribution is used as a point estimate of firm-specific inefficiency measure w. Jondrow et al.

(1982) gives expressions for these point estimates assuming half-normal for wi as follows:19

=

− + 2

1

( ) ( )( )**

**** 1

|σµ

σµσµ−−

−+

FfewE , (A5)

where: 22* σσµ ew= , 222

vw σσσ = , and 222* σσσσ vw= ,

and f and F represent the standard normal density and cumulative density functions respectively.

Since the likelihood function used to decompose the error term evaluates f and F at the point

where λ = σw / σv , -µ* / σ* in A5 becomes eλ/σ, we thus get:

−

−=

σλ

σλσλσ e

eFefewE

)/()/()|( * . (A6)

The production function, defined in equation (5) is estimated for each industry, for all

years in our sample period to obtain firm-specific residuals (e’s estimated with (5)). These errors

are subsequently decomposed into firm-specific (w) and random (v) components following

maximum likelihood estimation and computation of residual using the equation (A6) above. The

firm-specific inefficiency measure (the w) is then transformed into a more appealing firm-

19 These distributional assumptions can be relaxed if one can use panel data to estimate the inefficiency measures for

each producer (Greene 1993).

26

specific efficiency measure (which we label EFFIC). This transformation is accomplished using

Battese and Coelli’s (1988) algorithm for the case of logarithmic production functions, as is used

here. Specifically, for the case of logarithmic production functions, Battese and Coelli specify

the firm-specific efficiency measure (EFFIC) as:

)exp()exp(

ii

iiii vx

wvxEFFIC

+−+

=β

β (A7)

where a firm’s estimated production function at its own inefficient state is compared to the firm’s

estimated production function if the firm-specific level of inefficiency ( ) is zero. This

measure, by design, has values between zero and one. If a firm’s efficiency measure is 0.70,

then this implies that the firm realizes 70 percent of the production possible for a fully efficient

firm operating under the same conditions (Battese and Coelli, 1988).

iw

27

Table 1

Sample Selection and Industry Composition of Sample Panel A: Sample Selection

Year-Years Firms Number of observations with required Compustat Data for ratios

and to calculate efficiency scores for usable industriesa 19,985 5,315

Less: 1) Firms (firm-years) missing Compustat announcement date for first quarter earnings

2) Firms (firm-years) missing required forecast updates from IBES detail file

(17,912) (4,772)

Total Available Sample 1,839 907

Panel B: Industry Composition of Sample Industry

SIC Code

Firms

Industry

SIC Code

Firms

Oil and Gas Extraction 13 82 Measuring & Controlling Devices

382 20

Construction 15 7 Medical Instruments & Supplies 384 32 Food 20 26 Communications 48 29 Textiles 22 28 Durable Goods 50 22 Wood Products 24 10 Wholesale Trade 51 6 Paper & Allied Products 26 9 General Merchandise 53 12 Printing & Publishing 27 15 Food Retailers 54 9 Chemicals & Allied Products 28 45 Apparel & Accessory Stores 56 66 Drugs 283 17 Eating & Drinking Places 58 32 Rubber, Leather and Glass 30 17 Miscellaneous Retail 59 8 Primary Metal Industries 33 36 Personal Services 72 4 Industrial Machinery 35 46 Business Services 73 32 Computer & Office Equipment 357 60 Entertainment Services 78 17 Electronic Equipment 36 13 Health Services 80 9 Communications Equipment 366 34 Educational Services 82 3 Electronic Components &

Accessories 367 56 Engineering & Management

Services 87 13

Transport Equipment 37 9 Computer Services 737 74 Total Number of Firms

907

a Usable industries are those where the efficiency score estimates converge.

27

Table 2

Descriptive Statistics (Sample Size = 907 Firms)

Variable

Mean Standard

Deviation 25th

Percentile

Median 75th

Percentile

mUi 0.4992 1.5607 0.0822 0.0537 0.3191 mEFFICi 0.8921 0.0866 0.8643 0.9158 0.9455 mROAi 5.2095 11.6967 2.6180 5.8547 9.8000 mROEi 11.6891 18.1007 6.0335 12.3500 18.9833 mSIZEi 2,267 6,965 112 424 1,519

imU = Is the mean of Uit for firm i across all available years t. Uit is the measure of individual analyst uncertainty from the Barron et al. (1998) model and is calculated for firms i in year t. Uit is the average squared error in the individual forecasts of the analysts forecasts for firm i in year t.

mEFFICi = Is the mean of EFFICi,t-1 (within industry relative efficiency) for firm i across all available years t-1. EFFICi,t-1 is the relative production efficiency of firm i in year t-1.

mROAi = Is the mean of ROAi,t-1 for firm i across all available years t-1. ROAi,t-1 is the return on assets for firm i in year t.

mROEi = Is the mean of ROEi,t-1 for firm i across all available years t-1. ROEi,t-1 is the return on equity for firm i in year t.

mSIZEi = Is the mean of SIZEi,t-1 for firm i across all available years t-1. SIZEi,t-1 is the market capitalization of firm i at the end of fiscal year t-1, deflated to (constant) 1993 CPI dollars.

28

Table 3 Spearman (Rank) Correlation Analysis

(Sample Size = 907 Firms)

imU mEFFICi mROAi mROEi mSIZEi

imU (Pred. Sign) Corr. Coeff. (p-value)a

(-) -0.14 (<0.01)

(-) -0.19 (<0.01)

(-) -0.20 (<0.01)

(-) -0.07 (0.01)

mEFFICi (+)

0.31 (<0.01)

(+) 0.28 (<0.01)

(?) 0.12 (<0.01)

mROAi (+)

0.78 (<0.01)

(?) 0.09 (<0.01)

mROEi (?)

0.12 (<0.01)

mSIZEi (Pred. Sign)

Corr. Coeff. (p-value)a






a p-value for one-tailed tests of significance where sign is predicted, two-tailed otherwise.

29

Table 4 OLS Rank Regression Analysis

(Sample Size = 907 Firms)

i

J

j

jjiii IDmSIZEmEFFICmUaModel εγψψψ ∑=

++++=2

210:1

i

J

j

jjiii IDmSIZEmROAmUbModel εγψψψ ∑=

++++=2

210:1

i

J

jjjiii IDmSIZEmROEmUcModel εγψψψ ∑

=

++++=2

210:1

Model

Intercept

Prediction Coefficient

t-stat. (p-value)a

EFFIC, ROA, or ROE

Prediction

Coefficient t-stat.

(p-value)a

SIZE

Prediction Coefficient

t-stat. (p-value)a

Adjusted R2 (%)

Vuong Testb Comparison

with Model 1a

Vuong Model Z statistic

Comparison (p-value)a Model 1a

(?) 832.16 22.48

(<0.01)

(-) -0.34 6.96

(<0.01)

(-) -0.08 2.18

(0.02)

15.61

Model 1b

(?) 799.04 21.63

(<0.01)

(-) -0.17 4.79

(<0.01)

(-) -0.09 2.48

(0.01)

13.21

Model 1a

Vs. Model 1b

1.93

(0.03)

Model 1c (?)

794.23 21.52

(<0.01)

(-) -0.16 4.45

(<0.01)

(-) -0.08 2.35

(0.01)

12.89 Model 1a

Vs. Model 1c

2.05

(0.02)






a p-value for one-tailed tests of significance where sign is predicted. b The Vuong test provides a test for comparing the explanatory power of nonnested models. See Dechow

(1994).

31

earnings uncertainty and - city university of new york

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