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Panel regressions

- “Clustering is a Cambridge sickness”
- TuomoVuolteenaho

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

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 doubles

The 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

Examples of FM from Corporate Finance

- Fama French 2002– Testing tradeoff vs. pecking order

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.

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

- Why is the true standard error increasing as we ramp up the firm effect?

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.

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

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.

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

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

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

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