INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Introduction and Background

Introduction and Background

• Introduction to panel data methods

• Issues with the OLS model in the presence of panel data
• Panel data: Least squares dummy variable (LSDV) method
• Panel data: First difference (FD) estimators
• Panel data: Fixed effects (FE) estimators
• Panel data: Random effects (RE) estimators
• Panel data: FE vs. RE estimators
• Summary and concluding remarks
 INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Introduction to Panel Data Methods

Introduction to Panel Data Methods

Relationship between security returns 𝑟𝑖𝑡 and order imbalance 𝑂𝐼𝐵𝑖𝑡

𝐵𝑢𝑦𝑉𝑜𝑙𝑢𝑚𝑒 −𝑆𝑒𝑙𝑙𝑉𝑜𝑙𝑢𝑚𝑒
• Here OIBit =
𝐵𝑢𝑦𝑉𝑜𝑙𝑢𝑚𝑒 +𝑆𝑒𝑙𝑙𝑉𝑜𝑙𝑢𝑚𝑒

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 + 𝑣𝑡 + α𝑖 + µ𝑖𝑡

• Assume 10 years and 100 securities
• 𝑣𝑡 + α𝑖 + µ𝑖𝑡 are our error terms; let us discuss them one by one
• 𝑣𝑡 (‘t’ from 1…T) is solely time dependent term, e.g., broad market-wide changes
• These time-dependent terms don’t vary across the city, and can be accounted for
by ‘n-1’ (10-1 = 9) dummy variables [i.e., least square dummy variable estimation]
Introduction to Panel Data Methods

Relationship between security returns 𝑟𝑖𝑡 and order imbalance 𝐎𝐈𝐁𝐢𝐭

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 + 𝑣𝑡 + α𝑖 + µ𝑖𝑡
• α𝑖 (‘i’ from 1…n) is the security-specific variable like firm size, firm beta,
industry, etc., and not changing frequently overtime
• Usually, T: number of periods is small, and N: number of individual entities is
large
• So, accounting for α𝑖 through dummy variable method can make the model
extremely inefficient
• So, what if we do not account for this α𝑖
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Issues with Ordinary Least Square

(OLS) Model
Fitting OLS Through Scattered Data Points
Fitting OLS with Panel Data

ONGC
HDFC
JSW
ICICI
TATA
NTPC
INFY
WIPRO
RELIANCE
TCS
L&T
Return ADANI
ITC
BPCL

OIB
Pooled OLS Estimation with Panel Data

Relationship between security returns 𝑟𝑖𝑡 and order imbalance 𝑂𝐼𝐵𝑖𝑡

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 + 𝑣𝑡 + 𝛼𝑖 + µ𝑖𝑡
• 𝑛𝑖𝑡 = 𝛼𝑖 + µ𝑖𝑡 [𝛼𝑖 : Unobserved heterogeneity]
• Cov(𝑛𝑖𝑡 , 𝑂𝐼𝐵𝑖𝑡 ) ≠ 0 [Problem of endogeneity]
• Cov(𝑛𝑖𝑡 ,𝑛𝑖𝑡+1 )=Cov(𝑢𝑖𝑡 +𝛼𝑖 ,𝑢𝑖𝑡+1 +𝛼𝑖 ) ≠ 0 [Problem of
autocorrelation]
• Pooled OLS estimates will be biased and inconsistent
 INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Lease Squares Dummy Variable

(LSDV) Estimators
LSDV Estimators

Relationship between security returns 𝑟𝑖𝑡 and order imbalance 𝑂𝐼𝐵𝑖𝑡

• Assuming that time-varying effects can be modeled using time-dummies
• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 + 𝛼𝑖 + µ𝑖𝑡 (1)
• Include ‘N-1’ dummy variable for ‘N’ securities (𝑆2 , 𝑆3 , … , 𝑆𝑁 ) as follows
• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 + σ𝑁
𝑛=2 𝑎𝑛 𝑆𝑛 + µ𝑖𝑡 (2)
• Here, 𝑆2 is a dummy variable that takes a value of 1 for security 2, and 0
otherwise; and so on for securities 3, 4,…, N
• Thus, we are explicitly accounting for the unobserved heterogeneity for each
security individually
LSDV Estimators

𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 + σ𝑁
𝑛=2 𝑎𝑛 𝑆𝑛 + µ𝒊𝒕

• The estimates of 𝑎𝑛 are consistent under the following conditions

• Cov(𝑢𝑖𝑡 , 𝑂𝐼𝐵𝑖𝑡 ) = 0 ; no serial correlation in errors; homoscedasticity in error
terms
• Theoretically, under these assumptions, the estimates from LSDV are the
same as fixed-effects (FE) estimate
• However, dummy variables allow estimating this 𝛼𝑖 explicitly, unlike FE
estimators
• However, the model is not parsimonious with large N
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First Difference (FD) Estimators

First Differences Estimators

Relationship between security returns 𝑟𝑖𝑡 and order imbalance 𝑂𝐼𝐵𝑖𝑡

• Assuming that time-varying effects can be modeled using time-dummies
• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 + 𝛼𝑖 + µ𝑖𝑡 (1)
• 𝑟𝑖𝑡−1 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡−1 + 𝛼𝑖 + µ𝑖𝑡−1 (2)
• Subtract (1) - (2)
• 𝛥𝑟𝑖𝑡 = 𝑎1 𝛥𝑂𝐼𝐵𝑖𝑡 + 𝛥µ𝑖𝑡 (3)
• Cov(𝛥µ𝑖𝑡 ,𝛥𝑂𝐼𝐵𝑖𝑡 ) =0
• This model can be estimated with OLS estimation
First Differences (FD) Estimators

Relationship between security returns 𝑟𝑖𝑡 and order imbalance 𝑂𝐼𝐵𝑖𝑡

• 𝛥𝑟𝑖𝑡 = 𝑎1 𝛥𝑂𝐼𝐵𝑖𝑡 + 𝛥µ𝑖𝑡 (3)
• 𝐶𝑜𝑣 ∆𝑢𝑖𝑡 , ∆𝑢𝑖𝑡−1 = 𝐶𝑜𝑣 𝑢𝑖𝑡 − 𝑢𝑖𝑡−1 , 𝑢𝑖𝑡−1 − 𝑢𝑖𝑡−2
• However, these is an issue of auto-correlation due to first differencing
• Differencing leads to small variation in variables and therefore considerable
increase in standard error of estimates
• Loss of observation
• Time independent factors can not be estimated
First Differences (FD) Estimators

Relationship between security returns 𝑟𝑖𝑡 and order imbalance 𝑂𝐼𝐵𝑖𝑡

• 𝛥𝑟𝑖𝑡 = 𝑎1 𝛥𝑂𝐼𝐵𝑖𝑡 + 𝛥µ𝑖𝑡 (3)
• All those terms with no variance across time will be eliminated;
so, we need the dependent and independent variables to have
some variation across time and city
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Fixed-Effects (FE) Estimator

Fixed-Effects Estimators

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 +𝜶𝒊 +µ𝒊𝒕 (1)

1 𝑇
• Time-demean equation (1) σ 𝑟 ∀ i’s = 1, 2, 3…N
𝑇 𝑡=1 𝑖𝑡

• 𝑟ഥ𝑖 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖 +𝜶𝒊 +𝑢ഥ𝑖 (2)

• Substract (1) - (2)
• 𝑟𝑖𝑡 -𝑟ഥ𝑖 = 𝑎1 (𝑂𝐼𝐵𝑖𝑡 -𝑂𝐼𝐵𝑖 )+(µ𝒊𝒕 -𝑢ഥ𝑖 )
• 𝑟𝑖𝑡 ෪ 𝑖𝑡 + 𝑈𝑖𝑡 ;
ǁ = 𝑎1 ∗ 𝑂𝐼𝐵
• Here, Cov(𝑂𝐼𝐵
෪ 𝑖𝑡 , 𝑈𝑖𝑡 ) = 0, and pooled OLS estimates will be consistent
Fixed-Effects Estimators

෪ 𝑖𝑡 + 𝑈𝑖𝑡 ;
• 𝑟෤𝑖𝑡 = 𝑎1 ∗ 𝑂𝐼𝐵
• Fixed effects remove any time-constant terms
• Fixed effects are costly (due to transformation of original data)
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Fixed-Effects (FE) vs.

First Difference (FD) Estimators
Fixed-Effects vs. First Difference Estimators

• 𝑟𝑖𝑡 ෪ 𝑖𝑡 + 𝑈𝑖𝑡 ;
ǁ = 𝑎1 ∗ 𝑂𝐼𝐵
• For T = 2, FD = FE
• For T > 2, FD ≠ FE
• With the assumptions that (a) large sample N→ ∞(b) error term (µ𝒊𝒕 ) is
uncorrelated with the independent variable (e.g., 𝑂𝐼𝐵𝑖𝑡 ) (c) sample is
random, and (d) sufficient variance in variables, the following is held

• 𝐸𝑎 ෞ1 𝐹𝐸 = 𝑎1 (both FE and FD estimates of 𝑎1 are unbiased)

ෞ1 𝐹𝐷 = 𝐸 𝑎
𝑝 𝑝
ෞ1 𝐹𝐷 → β ; 𝑎
• 𝑎 ෞ1 𝐹𝐸 → 𝑎1 (both the estimators are consistent )
Fixed-Effects vs. First Difference Estimators

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 +𝛼𝑖 +µ𝑖𝑡

• (A) 𝐶𝑜𝑣 µ𝑖𝑡 , µ𝑖𝑡−1 = 0: error terms (µ𝑖𝑡 ) are serially uncorrelated
• First differencing introducing serial correlation in error terms
• Due to this, the standard error of estimates for FE estimators are lower (more
efficient) than FD estimators: 𝑠𝑒 𝑎ො1𝐹𝐸 < 𝑠𝑒 𝑎ො1𝐹𝐷
• (B) µ𝑖𝑡 = µ𝑖𝑡−1 + 𝑒𝑖𝑡 : i.e., 𝛥µ𝑖𝑡−1 = 𝑒𝑖𝑡
• Random walk structure in error term or strong autocorrelation in errors
• 𝑠𝑒 𝑎ො1𝐹𝐸 > 𝑠𝑒 𝑎ො1𝐹𝐷
Fixed-Effects vs. First Difference Estimators

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 +𝛼𝑖 +µ𝑖𝑡

• (C) µ𝑖𝑡 =𝜌µ𝑖𝑡−1 + 𝑒𝑖𝑡 : AR(1) structure in error terms
• 𝜌 is close to ‘1,’ then the FD estimator is more efficient
• 𝜌 is close to ‘0,’ then the FE estimator is more efficient
• One solution is to examine the autocorrelation structure in FD errors
• If FD errors have a negative autocorrelation, that indicates original errors
have no autocorrelation; hence FE is more appropriate
• If FD errors have a very small correlation, that indicates original errors have
random walk; hence FD estimator is more appropriate
Fixed-Effects vs. First Difference Estimators

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 +𝛼𝑖 +µ𝑖𝑡

One solution is to examine the autocorrelation structure in FD errors
• For scenarios in between, one can estimate both FD and FE and
compare
For non-stationary process, first differences are more useful
For small sample sizes, FD is more appropriate
For data with large time dimension FE estimators are more appropriate
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Random Effects Estimator: Part 1

Random Effects (RE) Estimators

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 +𝛼𝑖 +µ𝑖𝑡

• Recall that the model would have an issue of endogeneity if the unobserved
heterogeneity (𝛼𝑖 ) is correlated with one of the independent variables:
𝐶𝑜𝑣 𝑂𝐼𝐵𝑖𝑡 , 𝛼𝑖 ≠0
• Thus, pooled OLS is not effective, and we used FD/FE methods to remove
𝛼𝑖 from the model
• However, if 𝐶𝑜𝑣 𝑂𝐼𝐵𝑖𝑡 , 𝛼𝑖 is reasonably close to ‘0’ then, we need not apply
FD/FE as they involve a heavy transformation in data
• E.g., FE leads to loss of observations (T-1 periods instead of T)
Random Effects (RE) Estimators
• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 +𝛼𝑖 +µ𝑖𝑡 ;
• 𝐶𝑜𝑣 𝑂𝐼𝐵𝑖𝑡 , 𝛼𝑖 = 0 ; is a reasonable assumption in following cases
• All the relevant variables are accounted for
• 𝜶𝒊 is very small relative to other variables
• In this scenario, pooled OLS provides consistent estimates
• However, the errors may still be serially correlated: 𝐶𝑜𝑣 𝛼𝑖 + µ𝒊𝒕 , 𝛼𝑖 + µ𝒊𝒔 ≠ 0
• This serial correlation can be corrected through RE estimation without
putting a heavy cost of data (as in FD/FE)
• RE is more efficient than Pooled OLS and FE
Random Effects (RE) Estimators

• If you believe that sufficient variables have been entered in the model and
𝐶𝑜𝑣 𝑂𝐼𝐵𝑖𝑡 , 𝛼𝑖 ≠0 [Problem of Endogeneity] has been resolved
• Then RE is better than FE and OLS
• 𝑟𝑖𝑡 -λഥ
𝑟𝑖 = 𝑎0 1 − λ + 𝑎1 (𝑂𝐼𝐵𝑖𝑡 - λ 𝑂𝐼𝐵𝑖 )+(𝑛𝑖𝑡 -λ𝑛ഥ𝑖 ), where 𝑛𝑖𝑡 = 𝛼𝑖 + µ𝑖𝑡
• The above random effect is the pooled estimation of the above
transformation
• λ=0, then RE ≈ Pooled OLS
• λ =1, then RE ≈ FE
 INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Random Effects Estimator: Part 2

Random Effects (RE) Estimators

• 𝑟𝑖𝑡 -λഥ
𝑟𝑖 = 𝑎0 1 − λ + 𝑎1 (𝑂𝐼𝐵𝑖𝑡 - λ 𝑂𝐼𝐵𝑖 )+(𝑛𝑖𝑡 -λ𝑛ഥ𝑖 ), where 𝑛𝑖𝑡 = 𝛼𝑖 + µ𝑖𝑡
• Typically, 0≤ λ≤1, hence RE is somewhere between pooled OLS and FE
• What is λ?
0.5
𝜎𝑢2
• λ=1- ; here 𝜎𝑢2 is the variance of error term, 𝜎𝑎2 is the variance of 𝛼𝑖
𝜎𝑢2 +𝑇𝜎𝑎2

• If 𝜎𝑎2 = 0, then λ = 0; that is 𝛼𝑖 is insignificant/not important: RE converges to pool

• 𝑇𝜎𝑎2 >>> 𝜎𝑢2 , λ = 1, RE converges to FE
• Thus, unlike FE (fully time-demean) RE is quasi time-demean
• RE also allows to estimate time-constant terms
Random Effects (RE) Estimators

• 𝑟𝑖𝑡 = 𝑎0 + 𝑎1 𝑂𝐼𝐵𝑖𝑡 +𝛼𝑖 +µ𝑖𝑡 (1)

• 𝑟𝑖𝑡 -λഥ
𝑟𝑖 = 𝑎0 1 − λ + 𝑎1 (𝑂𝐼𝐵𝑖𝑡 - λ 𝑂𝐼𝐵𝑖 )+(𝑛𝑖𝑡 -λ𝑛ഥ𝑖 ) (2)
• where 𝑛𝑖𝑡 = 𝛼𝑖 + µ𝑖𝑡
0.5
𝜎𝑢2
• λ=1-
𝜎𝑢2 +𝑇𝜎𝑎2
መ this requires estimation of Eq. (1) through FE or
1. First step is to estimate 𝜆:
pooled OLS methods
2. Then estimate the transformed Eq. (2) using 𝜆መ using pooled OLS
• The combined system set-up is RE method of estimation
 INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Random Effects Estimator: Part 3

Assumptions of RE

The following assumptions are made for RE estimators to be consistent, i.e.,

𝑝
ෞ1 R𝐸 → 𝑎1 (as N → ∞)
𝑎

• 𝐶𝑜𝑣 𝑂𝐼𝐵𝑖𝑡 , 𝛼𝑖 =0
• Each cross section is randomly sampled
• 𝐸[𝑢𝑖𝑡 |𝑋𝑖𝑡 , 𝛼𝑖 ]=0
• No perfect multicollinearity
• The last three assumptions are applicable to FE/FD also
• Only the first assumption is specific to RE
Estimating Time Constant Variables with
RE
Recall the transformed model
• 𝑟𝑖𝑡 -λഥ
𝑟𝑖 = 𝑎0 1 − λ + 𝑎1 (𝑂𝐼𝐵𝑖𝑡 - λ 𝑂𝐼𝐵𝑖 )+ 𝑛𝑖𝑡 -λ𝑛ഥ𝑖 )

• In this model let us assume a time constant term 𝑆𝑖𝑧𝑒𝑖 , then the resulting model

• rit -λഥ
ri = 𝑎0 1 − λ + 𝑎1 (OIBit - λ OIBi ) + 𝑎2 ∗ Sizei (1- λ) + (nit -λnഥi )

• As long as λ≠0, we can estimate 𝑎2 , the effect of time constant variable 𝑆𝑖𝑧𝑒𝑖

• However, for these estimates to remain consistent, the assumption pertaining to

RE need to be held [e.g., 𝐶𝑜𝑣 𝑆𝑖𝑧𝑒𝑖 , 𝑂𝐼𝐵𝑖𝑡 ] = 0
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Fixed Effects (FE) vs. Random Effects

(RE)
FE vs. RE

Cov(𝛼𝑖 , 𝑋𝑖𝑡 )=0

• Both FE and RE estimates are consistent
• SE (RE estimate) < SE (FE estimate): Efficiency
• RE effect estimation allows for the effect of time-constant variables on
dependent variables (For FE, that is not possible)

Cov(𝛼𝑖 , 𝑋𝑖𝑡 )≠ 0
• Only the FE estimate is consistent
• SE (RE estimate) < SE (FE estimate)
• Hausman test can be employed to select between the two
Hausman Test

Hausman test statistic tests this hypothesis

• Null H0 => Cov(𝜶𝒊 , 𝑋𝑖𝑡 ) = 0 We should be able to use RE

෣
𝛽 ෣ 2
𝐹𝐸 −𝛽𝑅𝐸
• Estimate W = is distributed as chi-square with one df
𝑉𝑎𝑟(𝛽෣ ෣
𝐹𝐸 ) −𝑉𝑎𝑟(𝛽𝑅𝐸 )

• If H0 is true, the numerator is small (both estimates are consistent), but the denominator is
large, the statistic W is close to 0: Fail to reject the null, use RE estimator

• If null is false, the numerator is large, W is away from zero [Cov(𝜶𝒊 , 𝑋𝑖𝑡 )≠0]: reject the null,
use fixed effect estimator

• Essentially this estimator compares consistency (in numerator) relative to efficiency (in
denominator)
 INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Summary and Concluding Remarks

Summary and Concluding Remarks

• As compared to simple cross-sectional or time-series data, panel data

with longitudinal properties is rich in granularity, and the information it
offers
• However, it also entails a number of issues related to the estimation
• One of the important issues is the role of unobserved heterogeneity
• That is, the individual-specific time-invariant effects
• Though there are also only time-varying effects, they can be easily
modeled by applying ‘T-1’ time dummies for T time periods
Summary and Concluding Remarks

• Usually, there are many individual units as compared to time periods; therefore,
accounting for these units explicitly through dummies can make the model
extremely non-parsimonious

• One simple approach is to estimate such models using the FD method, which is
simply differencing the series by one lag and then applying pooled OLS

• A more evolved FE approach is to estimate time-demeaned series with pooled OLS

• Both FD and FE methods, though useful, put a heavy cost on data due to the
extreme nature of data transformation

• A less exacting approach is that of the RE method, which is a compromise between

two extremes of FE and pooled OLS approach
Summary and Concluding Remarks

• RE method is more appropriate when the assumption that unobserved

heterogeneity is not correlated with the dependent variable [Cov(𝜶𝒊 , 𝑋𝑖𝑡 )=0]
can be held
• Cov(𝜶𝒊 , 𝑋𝑖𝑡 )=0: Both RE and FE are consistent by RE is more efficient
• Cov(𝜶𝒊 , 𝑋𝑖𝑡 )≠ 0: Only FE is consistent
• The decision to select FE vs. RE is taken through Hausman test (HT) statistic
• HT statistic essentially is a test of gains in consistency at the cost of
efficiency while choosing FE vs. RE method
Thanks!
 INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Lesson 2: Panel Data: Application

Introduction

• Application of Panel data algorithm in prediction of security prices

• Prediction of Broad marketwide returns
• Data Visualization
• Pool vs. Panel models (Fixed and Random effects)
• Interpret the output from different models
• Examine the cross-sectional/serial correlation in errors
• Obtain coefficients using robust standard errors
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Case Study: Prediction of Broad

marketwide returns
Case Study: Index Return Prediction

• Broad market wide indices are known to reflect the growth of

economy and are often correlated with macroeconomic factors
such as GDP
• This strategy to invest in market-index based on forecasts related
to factors such as GDP has become a very prevalent strategy
known as factor investing
• However, this exercise may be vitiated by unobserved
heterogeneities such as country specific factors
Case Study: Index Return Prediction

• In this case study, we will employ panel data methods to forecast

the market index prices
Sr Country Year GDP Return
1 A 1990 21.01801 27.8%
2 A 1991 -21.3649 32.1%
3 A 1992 -16.2345 36.3%
4 A 1993 21.69623 24.6%
5 A 1994 21.82465 42.5%
6 A 1995 21.89562 47.7%
7 A 1996 21.73732 50.0%
8 A 1997 21.74277 5.2%
9 A 1998 21.94626 36.6%
10 A 1999 17.49863 39.6%
11 B 1990 -22.5041 -8.2%
12 B 1991 -20.3831 10.6%
Case Study: Index Return Prediction

• The data includes, information about the country, year, GDP, log
scaled mean deviated form, and returns
• Using different panel data methods, the relationship between GDP
and returns to be modeled
• The subsequent slides provide the problem statement
Case Study: Index Return Prediction

The following tasks need to be performed:

Visualize the data
• Plot year wise market returns for each country
• Plot year wise GDP for each country
• Using the box plot show the heterogeneity in GDP and returns
across countries and years
• What do we infer
Case Study: Index Return Prediction

The following tasks need to be performed:

• Model the relationship of GDP with index returns using simple
pooled OLS: what are the problems with this approach
• Through visual approach show the fitted line with actual data
• Follow the LSDV approach and model the relationship after
adding countrywide dummies
Case Study: Index Return Prediction

The following tasks need to be performed:

• Convert the data into panel format with country and year as the
data identifiers (index)
• Model the data using fixed effect, using individual, time, and both
the effects
• Perform the tests of poolability to show whether these fixed
effects are significant
Case Study: Index Return Prediction

The following tasks need to be performed:

• Using the modeled fixed effect object from the panel data, extract
the time and individual fixed effects
• Next model the data with random effects, examine the output and
comment on individual and idiosyncratic heterogeneity
• Comment whether the random effects transformation is closer to
pool and fixed effect
• Conduct the tests to examine whether random or fixed effects
method is more appropriate
Case Study: Index Return Prediction

The following tasks need to be performed:

• Examine the cross-sectional and serial correlation in errors for
pool, fixed, and random effects method
• Using robust standard errors, estimate coefficients that are robust
to autocorrelation and heteroscedasticity in the model
 INDIAN INSTITUTE OF TECHNOLOGY KANPUR

Summary and Concluding Remarks

Summary and Concluding Remarks

• Broad marketwide returns are modelled with GDP factor

• First, data is visualized to see the individual and time effects
• Fixed and random effects models are estimated
• We performed residual diagnostics for cross-sectional and serial
correlation in errors
• Estimate coefficients using robust standard errors
Thanks!