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Introductory EconometricsLecture 1

Xiaoxia Shi

University of Wisconsin - Madison

09/02/2010

Lecture (1) Intro Metrics 09/02/2010 1 / 25

What is Econometrics?

What is Econometrics?

Econometrics employs statistical methods

to analyze data

in order to

1 estimate economic relationships,

2 evaluate government and business policies,

3 test economic theories, and

4 make predictions and forecasts.

Lecture (1) Intro Metrics 09/02/2010 2 / 25

What is Econometrics?

What is Econometrics?

Econometrics employs statistical methods

to analyze data

in order to

1 estimate economic relationships,

2 evaluate government and business policies,

3 test economic theories, and

4 make predictions and forecasts.

Lecture (1) Intro Metrics 09/02/2010 2 / 25

What is Econometrics?

What is Econometrics?

Econometrics employs statistical methods

to analyze data

in order to

1 estimate economic relationships,

2 evaluate government and business policies,

3 test economic theories, and

4 make predictions and forecasts.

Lecture (1) Intro Metrics 09/02/2010 2 / 25

What is Econometrics?

What is Econometrics?

Econometrics employs statistical methods

to analyze data

in order to

1 estimate economic relationships,

2 evaluate government and business policies,

3 test economic theories, and

4 make predictions and forecasts.

Lecture (1) Intro Metrics 09/02/2010 2 / 25

What is Econometrics?

What is Econometrics?

Econometrics employs statistical methods

to analyze data

in order to

1 estimate economic relationships,

2 evaluate government and business policies,

3 test economic theories, and

4 make predictions and forecasts.

Lecture (1) Intro Metrics 09/02/2010 2 / 25

What is Econometrics?

What is Econometrics?

Econometrics employs statistical methods

to analyze data

in order to

1 estimate economic relationships,

2 evaluate government and business policies,

3 test economic theories, and

4 make predictions and forecasts.

Lecture (1) Intro Metrics 09/02/2010 2 / 25

What is Econometrics?

What is Econometrics?

Econometrics employs statistical methods

to analyze data

in order to

1 estimate economic relationships,

2 evaluate government and business policies,

3 test economic theories, and

4 make predictions and forecasts.

Lecture (1) Intro Metrics 09/02/2010 2 / 25

Data that Econometricians Use

Four types of Data:

Cross-Sectional Data

Time Series Data

Pooled Cross Sections

Panel Data

Lecture (1) Intro Metrics 09/02/2010 3 / 25

Data that Econometricians Use

Four types of Data:

Cross-Sectional Data

Time Series Data

Pooled Cross Sections

Panel Data

Lecture (1) Intro Metrics 09/02/2010 3 / 25

Data that Econometricians Use

Four types of Data:

Cross-Sectional Data

Time Series Data

Pooled Cross Sections

Panel Data

Lecture (1) Intro Metrics 09/02/2010 3 / 25

Data that Econometricians Use

Four types of Data:

Cross-Sectional Data

Time Series Data

Pooled Cross Sections

Panel Data

Lecture (1) Intro Metrics 09/02/2010 3 / 25

Random Variables

What is a random variable?

In the linear regression:

Y = 0 + 1X + U

are Y ,X ,U random variables?

are 0, 1 random variables?

Are the OLS estimators of 0, 1 random variables?

Lecture (1) Intro Metrics 09/02/2010 4 / 25

Random Variables

What is a random variable?

In the linear regression:

Y = 0 + 1X + U

are Y ,X ,U random variables?

are 0, 1 random variables?

Are the OLS estimators of 0, 1 random variables?

Lecture (1) Intro Metrics 09/02/2010 4 / 25

Random Variables

What is a random variable?

In the linear regression:

Y = 0 + 1X + U

are Y ,X ,U random variables?

are 0, 1 random variables?

Are the OLS estimators of 0, 1 random variables?

Lecture (1) Intro Metrics 09/02/2010 4 / 25

Random Variables

What is a random variable?

In the linear regression:

Y = 0 + 1X + U

are Y ,X ,U random variables?

are 0, 1 random variables?

Are the OLS estimators of 0, 1 random variables?

Lecture (1) Intro Metrics 09/02/2010 4 / 25

Expectation

What is Expectation?

Whats the dierence between expectation and sample average?

Properties of expectation:

E (aX + bY + c) =?

Whats conditional Expectation?

Law of Iterated Expectation:

E [E (Y jX )] = E (Y )

Lecture (1) Intro Metrics 09/02/2010 5 / 25

Expectation

What is Expectation?

Whats the dierence between expectation and sample average?

Properties of expectation:

E (aX + bY + c) =?

Whats conditional Expectation?

Law of Iterated Expectation:

E [E (Y jX )] = E (Y )

Lecture (1) Intro Metrics 09/02/2010 5 / 25

Expectation

What is Expectation?

Whats the dierence between expectation and sample average?

Properties of expectation:

E (aX + bY + c) =?

Whats conditional Expectation?

Law of Iterated Expectation:

E [E (Y jX )] = E (Y )

Lecture (1) Intro Metrics 09/02/2010 5 / 25

Expectation

What is Expectation?

Whats the dierence between expectation and sample average?

Properties of expectation:

E (aX + bY + c) =?

Whats conditional Expectation?

Law of Iterated Expectation:

E [E (Y jX )] = E (Y )

Lecture (1) Intro Metrics 09/02/2010 5 / 25

Variance, Covariance and Correlation

Whats the variance of a random variable X?

Whats the covariance of random variable X and Y ?Properties of variances:

Var (aX + bY + c) =?

Whats the correlation between X and Y ?

(X ,Y ) =??

Lecture (1) Intro Metrics 09/02/2010 6 / 25

Variance, Covariance and Correlation

Whats the variance of a random variable X?

Whats the covariance of random variable X and Y ?

Properties of variances:

Var (aX + bY + c) =?

Whats the correlation between X and Y ?

(X ,Y ) =??

Lecture (1) Intro Metrics 09/02/2010 6 / 25

Variance, Covariance and Correlation

Whats the variance of a random variable X?

Whats the covariance of random variable X and Y ?Properties of variances:

Var (aX + bY + c) =?

Whats the correlation between X and Y ?

(X ,Y ) =??

Lecture (1) Intro Metrics 09/02/2010 6 / 25

Variance, Covariance and Correlation

Whats the variance of a random variable X?

Whats the covariance of random variable X and Y ?Properties of variances:

Var (aX + bY + c) =?

Whats the correlation between X and Y ?

(X ,Y ) =??

Lecture (1) Intro Metrics 09/02/2010 6 / 25

Simple Regression

We want to know the parameters 0 and 1 in the populationequation:

Y = 0 + 1X + U

We have a sample fXi ,Yigni=1. How do we estimate the parameters?Ordinary least square (OLS):

0, 1 minimizesn

i=1(Yi b0 b1Xi )2

What is the formula for the OLS estimators? (hint: take rstderivatives and set to zero)

0 =?, 1 =?

Lecture (1) Intro Metrics 09/02/2010 7 / 25

Simple Regression

We want to know the parameters 0 and 1 in the populationequation:

Y = 0 + 1X + U

We have a sample fXi ,Yigni=1. How do we estimate the parameters?

Ordinary least square (OLS):

0, 1 minimizesn

i=1(Yi b0 b1Xi )2

What is the formula for the OLS estimators? (hint: take rstderivatives and set to zero)

0 =?, 1 =?

Lecture (1) Intro Metrics 09/02/2010 7 / 25

Simple Regression

We want to know the parameters 0 and 1 in the populationequation:

Y = 0 + 1X + U

We have a sample fXi ,Yigni=1. How do we estimate the parameters?Ordinary least square (OLS):

0, 1 minimizesn

i=1(Yi b0 b1Xi )2

What is the formula for the OLS estimators? (hint: take rstderivatives and set to zero)

0 =?, 1 =?

Lecture (1) Intro Metrics 09/02/2010 7 / 25

Simple Regression

We want to know the parameters 0 and 1 in the populationequation:

Y = 0 + 1X + U

We have a sample fXi ,Yigni=1. How do we estimate the parameters?Ordinary least square (OLS):

0, 1 minimizesn

i=1(Yi b0 b1Xi )2

What is the formula for the OLS estimators? (hint: take rstderivatives and set to zero)

0 =?, 1 =?

Lecture (1) Intro Metrics 09/02/2010 7 / 25

Another Way to Derive OLS Estimators

Population Moments: the true parameters (0, 1) solve:

E (Y 0 1X ) = 0EYX 0X 1X 2

= 0

Sample analogue: 0, 1 solves:

n1n

i=1(Yi b0 b1Xi ) = 0

n1n

i=1

YiXi b0Xi b1X 2i

= 0

0 =?, 1 =?

Lecture (1) Intro Metrics 09/02/2010 8 / 25

Another Way to Derive OLS Estimators

Population Moments: the true parameters (0, 1) solve:

E (Y 0 1X ) = 0EYX 0X 1X 2

= 0

Sample analogue: 0, 1 solves:

n1n

i=1(Yi b0 b1Xi ) = 0

n1n

i=1

YiXi b0Xi b1X 2i

= 0

0 =?, 1 =?

Lecture (1) Intro Metrics 09/02/2010 8 / 25

Another Way to Derive OLS Estimators

Population Moments: the true parameters (0, 1) solve:

E (Y 0 1X ) = 0EY

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