FULL SUMMARY OF PART 2 EMPERICAL METHODS IN FINANCE

Emperical Methods of Finance

Part 2 – Block 2: (More on) Panel data

Panel data: have both a time series and cross-sectional dimension
 It measures the same collection of people, firms (etc.) over several periods.

Simplest setup = 𝑦𝑖𝑡 = 𝛼 + 𝛽1𝑥1,𝑖𝑡 + ⋯ + 𝛽𝑘𝑥𝑘,𝑖𝑡 + 𝑢𝑖t
 Yit could be the stock return of firm i in year t
 The explanatory variables could be:
o firm-level variables (such as earnings growth of firm i in year t)
o but also pure time-series variables such as GDP growth
o or pure cross-sectional variables (such as industry dummies)

Advantages of using panel data:
 Can address broader range of issues and tackle more complex problems with panel
data than with pure time series or pure cross-sectional data alone; like for example:
o Difference-in-difference (as discussed in part 1)
o Regression discontinuity (as discussed in part 1)
 Is often of interest to examine how the relationships between objects change over
time;
 By structuring the model in an appropriate way, we can remove the impact of certain
forms of omitted variables bias in regression results.

Estimation models/methods for panel data: (*Most important methods)
1. Pooled OLS
2. Seemingly unrelated regressions (SUR)
3. Fixed effects estimator
4. Random effects estimator
5. Fama-MacBeth estimator (see point II.)

1)Pooled OLS
 This approach is advisable in case of a small sample (both N and T small).
 Estimate a single, pooled regression on all the observations together;
 Pooling the data in this way assumes that the constant term and slope coefficients do
not vary across i and t.
 Regression: = 𝑦𝑖𝑡 = 𝛼 + 𝛽1𝑥1,𝑖𝑡 + ⋯ + 𝛽𝑘𝑥𝑘,𝑖𝑡 + 𝑢𝑖t

3) Fixed effects model
Slope coefficients the same across i (firms), but constant term allowed to differ across i.
 By including fixed effects you control for unobservable differences across the units
(firms, individuals) you analyze; and prevent omitted variables.
 Can think of 𝛼𝑖 as capturing all (omitted) variables that affect 𝑦𝑖𝑡 cross-sectionally but
do not vary over time.
 Fixed effects models:
a. One approach: incorporate N dummy variables
b. The Within Transformation;
c. Time Fixed Effects Models;


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a) One approach
 Regression: 𝑦𝑖𝑡 = 𝛼1𝐷1𝑖 + 𝛼2𝐷2𝑖 ... + 𝛼𝑁𝐷𝑁𝑖 + 𝛽1𝑥1,𝑖𝑡 + ⋯ + 𝛽𝑘𝑥𝑘,𝑖𝑡 + 𝑢𝑖𝑡
 D1 is a dummy variable equal to 1 for observations on the first entity in
the sample and zero otherwise.
 Not practical if N is large (so use ‘’within’’ transformation)

b) Within transformation
T
 Take the time-series mean of each entity (firm): Yi=∑ Y ¿ /T
t=1
 Subtract this (step above) from the values of the variable 𝑦𝑖1, ... , 𝑦𝑖𝑇
 Do this for all entities i (firms) and for all explanatory variables
 The model containing the demeaned variables is:
Y ¿ −Y i=β ( X ¿ −X i ) +u ¿−ui

c) Time fixed effects model
 If the average value of yit changes over time but not cross-sectionally
 By including time fixed effects, you control for any common time-series variation
in the variables.
 Time-fixed effects model: 𝑦𝑖𝑡 = 𝛼 + t + 𝛽1𝑥1,𝑖𝑡 + ⋯ + 𝛽𝑘𝑥𝑘,𝑖𝑡 + 𝑢𝑖𝑡
o t = a time-varying intercept
 Also possible to allow for both entity fixed effects and time fixed effects within the
same model. Such a model would contain both cross-sectional and time dummies.

4) Random effects model
 Under the random effects model, the intercepts for each cross-sectional unit are
assumed to arise from:
o a common intercept  (the same for all cross-sectional units and over time); and
o plus, a random variable i that varies cross-sectionally but is constant over time.
 Random effects panel model: y ¿ =α + β x ¿ +❑¿
 ❑¿=ε i +V ¿:
 Heterogeneity (variation) in the cross-sectional dimension occurs via the i terms (and
not via dummies). So, this framework requires that the i has:
o Zero mean and constant variance;
o Is independent of the individual observation error term v ¿;
o Is independent of the explanatory variables.
 Advantages of the random effects model:
 Less parameters to be estimated, compared to fixed effects.
 One can still estimate the effect of variables that are constant over time.

o Standard errors
 Often applicable issue:
o error term across firms, households, etc, are positively correlated; and
o error term over time positively correlated (persistence)
 In these cases, if you neglect these positive correlations, standard errors are too low.
 Correct the standard errors for such patterns by using clustered standard errors.



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 Define clusters: Error terms are then allowed to be correlated within each cluster
but assumed not to be correlated across clusters.

 If you cluster standard errors by firm: each firm is a cluster;
– So error terms within each firm are allowed to be correlated across time:
Cov ( u is , u jt ) ≠ 0
– But no correlation across firms: Cov ( u is , u¿ ) =¿0

 Can also cluster by firm and year (double or two-way clustering):
– Cov ( u is , u jt ) ≠ 0 : correlation across firms in a year
– Cov ( u is , u¿ ) ≠0 : correlation across years in a firm
– But Cov ( u is , u jt ) =¿0: no correlation across year and firm
 Usually, two-way clustering is used.

Panel data of asset returns:
Methods of panel data of asset returns:
I. The cross-sectional approach to test the CAPM
II. Additional test of the CAPM: An extended second-stage regression
III. Fama-Macbeth procedure.

I. The cross-sectional approach to test the CAPM
 CAPM says: E ( Ri ) =Rf + β i [ E ( Rm ) −Rf ]
 Risk premium = β i [E ( R m )−R f ]
 Steps of testing the CAPM:
1. Estimating the stocks beta;
2. Analyze if average returns indeed increase with stock beta (in the cross-section).

Step 1: Calculating the beta:
e e
Cov( R i , Rm )
 Approach 1: Calculate it directly => β i=
Var ¿ ¿
 Approach 2: Run time-series regression of the excess portfolio returns on the
e e
excess market returns => Ri , t=ai+ βi Rm , t +ui , t
o Do this separately for each portfolio, and the slope estimate will be the
beta.

Step 2: Single cross-sectional regression of average portfolio return (over time) on a
constant and the betas:
- Ri = λ0 + λ1 ^β i+ v i
e


- The number of observations for this cross-sectional regression is the size of the
cross-section.
- The regression will provide estimates for Lambda 0 en lambda 1.
- According to CAPM:
o λ 0=0
o λ 1=E ¿
 CAPM-predicted value for: E( Ri ) - R f = β i R m
o Rm =Sample average of the excess market return
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