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Summary Empirical methods in Finance part 1 and 2
- Instelling
- Tilburg University (UVT)
Summary of Empirical methods in Finance part 1 and 2.
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Samenvatting Empirical Methods in FinanceChapter 1 The Nature of Econometrics and Economic Data
1.1 What is Econometrics?
Econometrics is based upon the development of statistical methods for estimating economic
relationships, testing economic theories and evaluating and implementing government and
business policy. A common application of econometrics is the forecasting of such important
macroeconomic variables such as interest rates, inflation rates and GDP.
Econometrics has evolved as a sperate discipline from mathematical statistics because the
former focuses on the problems inherent in collecting and analyzing nonexperimental
economic data. Nonexperimental data are not accumulated through controller experiments
on individuals, firms or segments of the economy. Experimental data are often collected in
laboratory environments in the natural sciences, but they are more difficult to obtain in the
social sciences.
1.2 Steps in Empirical Economic Analysis
An empirical analysis uses data to test a theory or to estimate a relationship. The first step in
an empirical analysis is careful formulation of the question of interest. In some cases,
especially those that involve the testing of economic theories, a formal economic model is
constructed. An economic model consists of mathematical equations that describe various
relationships.
After we specify an economic model, we need to turn it into what we call an econometric
model. An econometric model uses “BETA” in the equation. Once an econometric model has
been specified, various hypotheses of interest can be stated in terms of the unknown
parameters.
An empirical analysis by definition requires data. After data on the relevant variables have
been collected, econometric methods are used to estimate the parameters in the
econometric model and to formally test hypotheses of interest.
1.3 The structure of Economic Data
1.3.1 Cross-sectional data
A cross-sectional data set consists of a sample of individuals, households, firms, cities, states,
countries or a variety of other units, taken at a given point in time. An important feature of
cross-sectional data is that we can often assume that they have been obtained by random
sampling from the underlying population.
1.3.2 Time Series Data
A time series data set consists of observations on a variable or several variables over time. A
key feature of time series data that makes them more difficult to analyze than cross-sectional
data is that economic observations can rarely, if ever, be assumed to be independent across
time. Most economic and other time series are related to their recent histories.
Another feature of time series data that can require special attention is the data frequency
at which data are collected. In economics, the most common frequencies are daily, weekly,
monthly, quarterly and annually.
,1.3.3 Pooled Cross Sections
Some data sets have both cross-sectional and time series features. To increase our sample
size, we can form a pooled cross section by combining the two years.
Pooling cross sections from different years is often an effective way of analyzing the effects of
a new government policy.
1.3.4 Panel of Longitudinal Data
A panel data set consists of a time series for each cross-sectional member in the data set.
The key feature of panel data that distinguishes them from a pooled cross section is that the
same cross-sectional units are followed over a given period of time.
1.4 Causality, Ceteris Paribus and Counterfactual Reasoning
In most tests of economic theory, and certainly for evaluating public policy, the economist’s
goal is to infer that one variable has a causal effect on another variable. Simply finding an
association between two or more variables might be suggestive, but unless causality can be
established, is it rarely compelling.
The notion of ceteris paribus (all other relevant factors being equal) plays an important role
in causal analysis. The notion of ceteris paribus can also be described through counterfactual
reasoning, which has become an organizing theme in analyzing various interventions, such as
policy changes. The idea is to imagine an economic unit, such as an individual or a firm, in
two of more different states of the world. By considering these counterfactual outcomes, we
easily “hold others factors fixed” because the counterfactual though experiment applies to
each individual separately.
Chapter 2 The Simple Regression Model
The simple regression model can be used to study the relationship between two variables.
2.1 Definition of the Simple Regression Model
Much of applied econometric analysis begins with the following premise: y and x are two
variables, representing some population and we are interested in explaining y in terms of x.
In writing down a model that will ‘explain y in terms of x’, we must confront three issues:
1. Because there is never an exact relationship between two variables, how do we allow
for other factors to affect y?
2. What is the functional relationship between y an x?
3. How can we be sure we are capturing a ceteris paribus relationship between y and x?
Y = β 0+ β 1 x+u
The equation above defines the simple linear regression model.
Y is called the dependent variable; x is called the independent variable.
The variable u, called the error term in the relationship represents factors other than x that
affect y. A simple regression analysis effectively threats all factors affecting y other than x as
being unobserved.
,If the other factors in u are held fixed, so that the change in u is zero, ∆ u = 0, then x has a
linear effect on y:
∆ y = β 1 ∆ x if ∆ u=0
Thus, the change in y is simply β 1 multiplied by the change in x. this means that β 1 is the
slope parameter in the relationship between y and x. the intercept parameter β 0
sometimes called the constant term, also has it uses, although it is rarely central to an
analysis.
Before we state the key assumption about how x and u are related, we can always make one
assumption about u. as long as the intercept β 0 is included in the equation, nothing is lost
by assuming that the average value of u in the population is zero (E(u) = 0).
We now turn the crucial assumption regarding how u and x are related. A natural measure of
the association between two random variables is the correlation coefficient. If u and x are
uncorrelated, then, as random variables, they are not linearly related. Assuming that u and x
are uncorrelated goes a long way toward defining the sense in which u and x should be
unrelated.
The crucial assumption is that the average value of u does not depend on the value of x. We
can write this assumption as:
E (u|x) = E(u)
When the equation above holds, we stay that u is mean independent of x. When we
combine mean independence with assumption E(u) = 0, we obtain the zero conditional
mean assumption, E(u|x) = 0.
The zero conditional mean assumption gives β 1 another interpretation that is often useful.
Taking the expected value of the simple linear regression, conditional on x and using E(u|x) =
0 gives:
E(y|x) = β 0+ β 1 x
The equation above shows that the population regression function (PRF), is a linear function
of x. The linearity means that a one-unit increase in x changes the expected value of y by the
amount of β 1 .
2.2 Deriving the Ordinary Least Squares Estimates
1. yi = β 0+ β 1 xi+ ui
2. E(u) = 0
3. Cov (x,u) = E(xu) = 0
4. E(y- β 0 - β 1 x ) = 0
, 5. E [x (y- β 0 – β 1 x ¿ ¿=0
6. n−1 Σ ( yi− ^β 0− β^ 1 xi ) =0
7. n−1 Σxi ( yi− β^ 0− ^β 1 xi ) =0
8. y= ^β 0+ β^ 1 x
9. ^β 0= y − ^β 1 x
10. Σ ( xi−x )2> 0
11. ^β 1=Σ ( xi−x )∗¿ ¿
The estimates given in 9 and 11 are called the Ordinary Least Squares (OLS) estimates of
β 0∧β 1. To justify this name, for any ^β 0 and ^β 1 define a fitted value for y when x = xi as:
1. y i= ^
β 0+ ^
β 1 xi
This is the value we predict for y when x = xi for the given intercept and slope. The residual
value for observation i is the difference between the actual yi and its fitted value:
2. u^i= yi− ^y i= yi− ^
β 0+ ^
β 1 xi
Again, there are n such residuals. Now suppose we choose ^
β 0∧ ^
β 1 to make the sum of
squared residuals:
3. Σ u^ i 2=Σ( yi− ^
β 0+ ^
β 1 xi)2
Equation 6 and 7 are often called the first order conditions for the OLS estimates, a term that
comes from optimization using calculus.
Once we have determined the OLS intercept and slope estimates, we form the OLS
regression line:
4. ^y = β^ 0+ β^ 1 x
The equation above is also called the Sample Regression Function (SRF) because it is the
estimated version of the population regression function: E(y|x) = β 0+ β 1 x
In most cases, the slope estimate, which we can write as:
5. ^β 1=∆ ^y /∆ x
Is primary interest. It tells us the amount by which ^y changes when x increases by one unit.
6. ∆ ^y = ^β 1∗∆ x
2.2.1 A Note on Terminology
In most cases, we will indicate the estimation of a relationship through OLS by writing an
equation. Sometimes, we will often indicate that equation has been obtained by OLS in
saying that we run the regression of y on x, or simply that we regress y on x.
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