,Chapter 1
Econometrics is about how we can use theory and data from economics,
business, and the social sciences, along with tools from statistics, to answer “how
much” questions. It is about how to best estimate economic parameters given
the data we have.
In an econometric model we must first realize that economic relations are not
exact. Economic theory does not claim to be able to predict the specific
behaviour of any individual or firm, but rather describes the average or
systematic behaviour of many individuals or firms.
A key aspect of experimental data is that the values of the explanatory variables
can be fixed at specific values in repeated trials of the experiment.
An example of nonexperimental data is survey data. Data on all variables are
collected simultaneously, and the values are neither fixed nor repeatable.
Micro-data collected on individual economic decision-making units such as
individuals, households, and firms.
Macro-data resulting from a pooling or aggregating over individuals, households,
or firms at the local, state, or national levels.
Flow-outcome measures over a period of time.
Stock-outcome measured at a particular point in time.
A time-series is data collected over discrete intervals of time.
A cross-section of data is collected across sample units in a particular time
period.
A panel of data has observations on individual micro-units who are followed over
time.
Chapter 2
A simple regression function is called simple regression not because it is easy,
but because there is only one explanatory variable on the right-hand side of the
equation.
E(y l x) = β1 + β2x
The unknown regression parameters β1 and β2 are the intercept and slope of
the regression function, respectively.
Assumptions are the ‘if’ part of an ‘if-then’ type statement. If the assumptions we
make are true, then certain things follow. And, as importantly, if the assumption
do not hold, then the conclusions we draw may not hold. Part of the challenge of
econometric analysis is making realistic assumptions and then checking that they
hold.
The constant variance assumption var(y l x) = σ2 implies that at each level of
income x we are equally certain about how far values of y might fall from their
mean value, E(y l x) = β1 + β2x, and the certainty does not depend on income or
anything else. Data satisfying this condition are said to be homoskedastic. If
this assumption is violated the data are said to be heteroskedastic.
, Mathematicians spend their lives trying to prove the same theorem with weaker
and weaker sets of assumptions. This mindset spills over to econometrician to
some degree. Consequently, econometric models often make an assumption that
is weaker than statistical independence.
In order to carry out a regression analysis is to measure the effect of changes in
one variable, x, on another, y. In order to do this x must take at least two values
within the sample of data. Secondly, we will assume that the x-values are given,
and not random. All our results will be conditional on the given x-values.
Assumption of the simple linear regression model – I
- The mean value of y, for each value of x, is given by the linear
regression function: E(y l x) = β1 + β2x
- For each value of x, the values of y are distributed about their mean value,
following probability distributions that all have the same variance: var(y l x) = σ2
- The sample values of y are all uncorrelated and have zero covariance implying
that there is no linear association among them, cov(yi, yj) = 0. This assumption
can be made stronger by assuming that he values of y are all statistically
independent.
- The variable x is not random and must take at least two different value.
- (optional) the values of y are normally distributed about their mean for each
value of x.
The essence of regression analysis is that any observation on the dependent
variable y can be decomposed into two parts: a systematic component and a
random component. The systematic component of y is its mean, E(y l x) = β1 +
β2x. The random component of y is the difference between y and its conditional
mean value E(y l x). This is called a random error term, and it is defined as:
e = y - E(y l x) = y - β1 + β2x
Y = β1 + β2x + e
The dependent variable y is explained by a component that varies systematically
with the independent variable x and by the random error term e.
It is customary in econometrics to state the assumptions of the regression model
in terms of the random error e. For future reference the assumptions are named
SR1-SR6, ‘SR’ denoting ‘simple regression.’
Assumptions of the simple linear regression model – II
SR1. The value of y, for each value of x, is Y = β1 + β2x + e
SR2. The expected value of the random error e is: E(e) = 0, which is equivalent
to assuming that
E(y) = β1 + β2x
SR3. The variance of the random error is var(e) = σ2 = var(y)
The random variables y and e have the same variance because they differ only
by a constant.
SR4. The covariance between any pair of random errors ei and ej is cov(ei, ej) =
cov(yi, yj) = 0
SR5. The variable x is not random and must take at least two different values.
SR6. (optional) The values of e are normally distributed about their mean if the
values of y are normally distributed, and vice versa. e N(0, σ2)
Econometrics is about how we can use theory and data from economics,
business, and the social sciences, along with tools from statistics, to answer “how
much” questions. It is about how to best estimate economic parameters given
the data we have.
In an econometric model we must first realize that economic relations are not
exact. Economic theory does not claim to be able to predict the specific
behaviour of any individual or firm, but rather describes the average or
systematic behaviour of many individuals or firms.
A key aspect of experimental data is that the values of the explanatory variables
can be fixed at specific values in repeated trials of the experiment.
An example of nonexperimental data is survey data. Data on all variables are
collected simultaneously, and the values are neither fixed nor repeatable.
Micro-data collected on individual economic decision-making units such as
individuals, households, and firms.
Macro-data resulting from a pooling or aggregating over individuals, households,
or firms at the local, state, or national levels.
Flow-outcome measures over a period of time.
Stock-outcome measured at a particular point in time.
A time-series is data collected over discrete intervals of time.
A cross-section of data is collected across sample units in a particular time
period.
A panel of data has observations on individual micro-units who are followed over
time.
Chapter 2
A simple regression function is called simple regression not because it is easy,
but because there is only one explanatory variable on the right-hand side of the
equation.
E(y l x) = β1 + β2x
The unknown regression parameters β1 and β2 are the intercept and slope of
the regression function, respectively.
Assumptions are the ‘if’ part of an ‘if-then’ type statement. If the assumptions we
make are true, then certain things follow. And, as importantly, if the assumption
do not hold, then the conclusions we draw may not hold. Part of the challenge of
econometric analysis is making realistic assumptions and then checking that they
hold.
The constant variance assumption var(y l x) = σ2 implies that at each level of
income x we are equally certain about how far values of y might fall from their
mean value, E(y l x) = β1 + β2x, and the certainty does not depend on income or
anything else. Data satisfying this condition are said to be homoskedastic. If
this assumption is violated the data are said to be heteroskedastic.
, Mathematicians spend their lives trying to prove the same theorem with weaker
and weaker sets of assumptions. This mindset spills over to econometrician to
some degree. Consequently, econometric models often make an assumption that
is weaker than statistical independence.
In order to carry out a regression analysis is to measure the effect of changes in
one variable, x, on another, y. In order to do this x must take at least two values
within the sample of data. Secondly, we will assume that the x-values are given,
and not random. All our results will be conditional on the given x-values.
Assumption of the simple linear regression model – I
- The mean value of y, for each value of x, is given by the linear
regression function: E(y l x) = β1 + β2x
- For each value of x, the values of y are distributed about their mean value,
following probability distributions that all have the same variance: var(y l x) = σ2
- The sample values of y are all uncorrelated and have zero covariance implying
that there is no linear association among them, cov(yi, yj) = 0. This assumption
can be made stronger by assuming that he values of y are all statistically
independent.
- The variable x is not random and must take at least two different value.
- (optional) the values of y are normally distributed about their mean for each
value of x.
The essence of regression analysis is that any observation on the dependent
variable y can be decomposed into two parts: a systematic component and a
random component. The systematic component of y is its mean, E(y l x) = β1 +
β2x. The random component of y is the difference between y and its conditional
mean value E(y l x). This is called a random error term, and it is defined as:
e = y - E(y l x) = y - β1 + β2x
Y = β1 + β2x + e
The dependent variable y is explained by a component that varies systematically
with the independent variable x and by the random error term e.
It is customary in econometrics to state the assumptions of the regression model
in terms of the random error e. For future reference the assumptions are named
SR1-SR6, ‘SR’ denoting ‘simple regression.’
Assumptions of the simple linear regression model – II
SR1. The value of y, for each value of x, is Y = β1 + β2x + e
SR2. The expected value of the random error e is: E(e) = 0, which is equivalent
to assuming that
E(y) = β1 + β2x
SR3. The variance of the random error is var(e) = σ2 = var(y)
The random variables y and e have the same variance because they differ only
by a constant.
SR4. The covariance between any pair of random errors ei and ej is cov(ei, ej) =
cov(yi, yj) = 0
SR5. The variable x is not random and must take at least two different values.
SR6. (optional) The values of e are normally distributed about their mean if the
values of y are normally distributed, and vice versa. e N(0, σ2)
