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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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