Standard Errors in CF

Robin GreenwoodEmpirical Topics in Corporate Finance

March 2011

Panel regressions• “Clustering is a Cambridge sickness”– Tuomo Vuolteenaho

Panel Regressions• Hard to disentangle issues associated with

specification design with issues related to standard errors

• Today all about SEs

Lang, Ofek, Stulz (1995)• Leverage and Investment (panel regression)– OLS, p-values, no firm fixed effects

Opler, Pinkowitz, Stulz, Williamson (1999)• Corporate Cash Holdings• White SE, or Fama Macbeth, plus FE regs

Stulz and Williamson (2003)• Culture, Openness, and Finance• Back to OLS country regressions

Helwege, Pirinsky, and Stulz• Why do firms become widely held? • Pooled OLS: Depvar = 5% drop in ownership

Stulz and Fahlenbrach (2009)• Changes in q and changes in ownership, cluster

Doidge, Karolyi, and Stulz• Why has IPO activity picked up everywhere but for

the US?

Fama-Macbeth

• Workhorse empirical method in modern finance• Used to deal with panels where there is high degree

of cross-sectional correlation, but not much time correlation

• Makes sense to use this when describing returns– Mostly random– Correlated across firms– Some time-dependence, however, at least in the expected

return component

• Vastly overused – Together with “portfolio” approaches

Panel Analysis

←N→

↑T↓

←N→100

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Xit

Yitit t t itY a b X

1 2

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Panel Analysis

←N→

↑T↓

←N→100

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Yitit t t itY a b X

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/tb

SE T

Watch out for persistence

• Good scenario for bts:

• Bad scenario:

Watch out for persistence

• Good scenario for bts:

• Bad scenario:

Watch out for persistence

• Good scenario for bts:

• Bad scenario: Simple fix:Modify using Newey-West

In my experience,Approximately doublesThe SEs

Fama & Macbeth• Original use in asset pricing

• Stage 1: Estimate betas• Stage 2: Estimate cross-sectional relation between returns

and betas• Stage 3: Collect your estimates and get t-stat

• Benefit: Flexible parameters, not memory intensive• These benefits are less apparent today, yet method still

popular because it’s hard to game• Main benefit: Weights PERIODS equally• Can get close to this by running panel and weighting by

1/N(t), but people will be suspicious

Still mostly used in Asset Pricing• Pontiff and Woodgate, Share issuance and cross-

sectional returns• Table VI

Gong, Louis, Sun“Earnings management following open-market

repurchases”

Examples of FM from Corporate Finance• Fama French 2002– Testing tradeoff vs. pecking order

Main table 1965-1999

Estimating Standard Errors in Finance Panel Data Sets: Comparing Approaches

Mitch PetersenRFS 2009

Papers Contribution• Examines a variety of approaches to estimating

standard errors and statistical significance in panel data sets

• Also looks at a variety of papers published from 2001-2004:– Only 42% of papers adjusted standard errors for possible

dependence in residuals.• Many different approaches.• Which are correct under what circumstances.

• The bar for you will be much higher

Overview• OLS standard errors are unbiased when residuals are

independent and identically distributed.• Residuals in panel data may be correlated by firm-

specific effects that are correlated across time.– Firm effect.

• Residuals of a given year may be correlated across different firms (cross sectional dependence)– Time effect.

Paper’s Approach• Simulate data that has either firm effect or a time

effect.• Test various estimation techniques.• See how they deal with the simulated data.• Then takes regression approaches to actual data and

compares them.

Firm Fixed Effects• Assumption of OLS is that cross product matrix has

only non-zero numbers on the diagonal.• Figure 1 – Example of a firm effect.– Cluster standard errors by firm.

Time for other firms

Time for this firm

Time for other firmsTime for this firm Firm 1, date 1

Firm 1, date 2…

Firm 2, date 1Firm 2, date 2

OLS vs. Clustering by Time vs. FM with Firm Effect• Simulate 5000 samples with 5000 observations.

– 500 firms and ten years of observations.• Let the residual and independent variable variance due to the firm effect

vary between 0 and 75%.How do you do this?

X_g = normrnd(0,1,[NUM_FIRMS 1]);X_i = normrnd(0,1,[NUM_BOTH 1]);E_g = normrnd(0,2,[NUM_FIRMS 1]);E_i = normrnd(0,2,[NUM_BOTH 1]);X(i) = sqrt(variation_X)*X_g(c_f) + sqrt(1-

variation_X)*X_i(i);E(i) = sqrt(variation_E)*E_g(c_f) + sqrt(1-

variation_E)*E_i(i);

• 500 clusters by firm.

OLS vs. Clustering on Firm vs. FM with Firm Effect

• Table 1– Compare average coefficients, st. dev. of coefficient

estimates, % significant, average SE clustered and % significant with clustered SE.

– Vary how much of the independent variable variation is due to firm effect and how much of the residual variation is due to firm effect.

• Figure 2 – Compare OLS, Clustered by firm, and Fama-McBeth.

• Table 2- Fama-MacBeth

Table 1

Table 1• Why is the true standard error increasing as we ramp

up the firm effect?

1 ( 1)

1 9

1 9*0.5*0.5

1 9*0.5*0.5

1.8

(0.0508 / 0.0283) 1.8!

X

X

T

Table 2

OLS vs. Clustering by Time vs. FM with Time Effect

• Simulate 5000 samples with 5000 observations.• Let the residual and independent variable variance

due to the time effect vary between 0 and 75%.• Not this is the situation that FM developed FM for.• Clustering will be by the 10 years.

OLS vs. Clustering by Time vs. FM with Time Effect

• Table 3 – Compare OLS and Clustering by time.• OLS does pretty poor job.• Table 4 – Using FM to estimate.

Table 3

Table 4

Lit Review• Petersen points out many papers which have

persistent firm characteristics on other persistent firm characteristics. Both OLS and FM will be biased here– Fama and French 2001 (DivPayer on M/B, size, etc)– M/B on firm chars

• Pastor and Veronesi, Kemsley and Nissim

– Capital structure regressions• Baker and Wurgler 2002; Fama and French 2002; Johnson 2003

Lit Review• Obnoxious• Wu (2004) “FM method accounts for the lack of

independence because of multiple yearly observations per company”

• Denis, Denis, Yost (2002” “pooling of cross-sectional and TS data in our tests creates a lack of independence in the regression models…..to address the importance of this bias, we estimate the regression model separately for each of the 14 years…”

• Choe, Bong-Chan, and Stulz (2005) “The FM regressions take into account the cross-correlations and the serial correaltion in the error term, so that the t-stats are more conservative”

OLS vs. Clustering by Time vs. FM with Firm and Time Effect

• In many typical examples, could have both a firm and time effect.

• Figure 6, typical structure with both.• Can cluster by firm and time together.– See Samuel Thompson’s 2006 working paper for math.– We’ll cover this later today

Figure 6

OLS vs. Clustering by Time vs. FM with Firm and Time Effect

• Simulate 5000 samples with 5000 observations.• Let the residual and independent variable variance

due to firm and time effect vary• Table 5 – Compare OLS, with and without firm

dummies, Clustered by firm and time, GLS, and FM.

Real Data• Table 6 – Look at asset pricing application.– Equity returns on asset tangibility.– Different methods matter.– OLS and firm clusters do poorly.– Time and firm clustering and FM work well.– Seems to say that for returns may be more affected by a

time effect.

Real Data• Table 7 – Capital structure regressions.– OLS, clustering by time, and FM do poorly.– Clustering by firm and clustering by firm and time do well.– Says that within corporate finance a lot of the effects

seem to have firm level persistence.

Table 6

Table 7

Recommendations• Think about the structure of the panel data structure.• What is the likely source of dependence.• Comparing different methods may provide additional

information about the research question.

• Starting point should probably be double clustering by firm and year

Samuel Thompson• Simple formulas for standard errors that cluster by

both firm and time (JFE 2011)

• Basic formula:

• This means you can do it in STATA• Email Sam Hanson or go to Mitch Petersen’s website,

there is pre-packaged code to do this

,0 ,0ˆ ˆ ˆ ˆ( ) firm time whiteVar V V V

Firm effects, time effects, and persistent common shocks

• Firm effects: arbitrary correlation across time for a given firm

• Time effects: errors have arbitrary correlation across firms at a moment in time

• Persistent common shocks: correlation between firms, but these shocks die out over time

( | ) 0it ik it ikE x x for t k

( | ) 0it jt it jtE x x for i j

( | ) 0 & | |it jt it jtE x x for i j t k L

Single vs. Double Cluster• Bias largest when the time and firm dimensionality of

the dataset is approximately the same• If you have ten firms and 1000 time period, biggest

bias reduction?– Cluster by firm

When does double clustering work?• Monte Carlos suggest that double clustering pretty

good for N & T greater than 25• To allow for persistent common shocks, need T>100

Application to industry Profitability• Hypothesis: Profitability is higher in more

concentrated industries• Unit of observation: Industry-year• 434 industries, 43 years

Application to industry Profitability• Hypothesis: Profitability is higher in more concentrated

industries• Unit of observation: Industry-year• 434 industries, 43 years

• “Clustering makes a big difference when both the error and the regressor are correlated within the clustering dimension”

Application to industry Profitability• Average ROA varies across time but not across

industries• HHI varies a lot between industries, but not much

over time within an industry

• Double clustering gives you a conservative estimate in both cases

Thompson- summary• More and more papers are using this double

clustering, probably will become the de facto standard