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Summary Empirical Methods in Finance part 1 and 2
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Samenvatting EMF deel 211.1 Introduction: What are Panel Techniques and Why are They Used?
We have data comprising both time-series and cross-sectional elements, and such a dataset
would be known as a panel of data. A panel of data will embody information across both
time and space.
yit =a+ βxit +uit
The simplest way to deal with such data would be to estimate a pooled regression, which
would involve estimating a single equation on all the data together, so that the dataset for y
is stacked up into a single column containing all the cross-sectional and time series
observations, and similarly all of the observations on each explanatory variable would be
stacked up into single columns in the x matrix. Then, this equation would be estimated in the
usual fashion using OLS.
Limitations:
- Pooling the data in this way implicitly assumes that the average values of the
variables and the relationships between them are constant over time and across all of
the cross-sectional units in the sample.
Advantages:
- We can address a broader range of issues and tackle more complex problems with
panel data than would be possible with pure time series or pure cross-sectional data
alone
- Often of interest to examine how variables, or the relationships between them,
change dynamically 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
11.3 The Fixed Effects Model
uit=μi + vit
yit =a+ βxit + μi +vit
This model could be estimated using Dummy Variables, which would be termed the least
square dummy variable (LSDV) approach:
yit =βxit+ μ 1 D 1i+ μ 2 D 2 i
Where D1 is a dummy variable that takes the value of 1 for all observations on the first entity
in the sample and zero otherwise, D2 is a dummy variable that takes the value of 1 for all
observations on the second entity and zero otherwise, and so on. There is no intercept with
this equation.
,In order to avoid the necessity to estimate so many dummy variable parameters, a
transformation is made to the data to simplify matters. This transformation, known as the
within transformation, involves subtracting the time-mean of each entity away from the
values of the variable:
1
y i= ∗Σ yit
T
This is the time-mean of the observations on y for cross-sectional unit i, and similarly
calculate the means of all of the explanatory variables. Then, we can subtract the time-
means from each variable and obtain a regression containing demeaned variables only.
yit − y i=β ( xit−x i )+uit −u i
An alternative to this demeaning would be to simply run a cross-sectional regression on the
time-averaged values of the variables, which is known as the between estimator.
11.4 Time-fixed Effects Models
It is also possible to have a time-fixed effects model rather than an entity-fixed effects model.
We would use such a model where we thought that the average value of yit changes over
time but not cross-sectionally. Hence, with time fixed-effects, the intercept would be allowed
to vary over time but would be assumed to be the same across entities at each given point in
time.
yit =a+ βxit + λt +vit
Where Lambda is a time-varying intercept that captures all of the variables that affect yit and
that vary over time but are constant cross-sectionally.
Time variation in the intercept terms can be allowed for in exactly the same way as with
entity-fixed effects. That is, a least-squares dummy variables model could be estimated. D1,
denotes a dummy variable that takes the value of 1 for the first time period and zero
elsewhere, and so on. The only difference now is that the dummy variables capture time
variation rather than cross-sectional variation.
Finally, it is possible to allow for both entity-fixed effects and time-fixed effects within the
same model. Such a model would be termed a two-way error component model.
, 11.7 The Random Effects Model
An alternative to the fixed effects model described above is the random effects model, which
is sometimes also known as the error components model. As with fixed effects, the random
effects approach proposes different intercept terms for each entity and again these
intercepts are constant over time, with the relationships between the explanatory variables
and explained variables assumed to be the same both cross-sectionally and temporally.
However, the difference is that under the random effects model, the intercepts for each
cross-sectional unit are assumed to arise from a common intercept a (which is the same for
all cross-sectional units and over time), plus a random variable ϵi that varies cross-sectionally
but is constant over time. ϵi measures the random deviation of each entity’s intercept term
from the ‘global’ intercept term a. We can write the random effects panel model as:
yit =a+ βxit + ϖit , ϖit=ϵi+vit
The parameters are estimated consistently but inefficiently by OLS, and the conventional
formulae would have to be modified as a result of the cross-correlations between error
terms for a given cross-sectional unit at different points in time. Instead, a generalized least
squares procedure is usually used. The transformation involved in this GLS procedure is to
subtract a weighted mean of the yit over time.
Section 2.4.1 Time-series Data
Time-series data, as the name suggests, are data that have been collected over a period of
time on one or more variables. Time-series data have associated with them a particular
frequency of observation or frequency of collection of data points. The frequency is simply a
measure of the interval over, or the regularity with which, the data are collected or recorded.
It is also generally a requirement that all data used in a model be of the frequency of
observation. The data may be quantitative or qualitative.
Problems that could be tackled using time-series data:
- How the value of a country’s stock index has varied with that country’s
macroeconomic fundamentals
- How the value of a company’s stock price has varied when it announced the value of
its dividend payment
- The effect on a country’s exchange rate of an increase in its trade deficit.
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