A Course in Introductory Econometrics

"Under Construction" scheduled to be complete late June 2013

Textbook: Introductory Econometrics (Any Edition) by Jeffrey Wooldridge

  • Review out of class notes (Week 1)

  • Chapter 1 The Nature of Econometrics and Economic Data (Week 1)

  • Chapter 2 The Simple Regression Model (Week 1 & Week 2)

  • Chapter 3 Multiple Regression Analysis: Estimation (Week 2)

  • Chapter 4 Multiple Regression Analysis: Inference (Week 2 & Week 3)

  • Chapter 5 Multiple Regression Analysis: OLS Asymptotics (Week 3 & Week 4)

  • Chapter 6 Multiple Regression Analysis: Further Issues (Week 4)

  • Chapter 7 Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables (Week 5)

  • Chapter 8 Heteroskedasticity (Week 5)

"The Basics" - Section Week 1 - April 4th 2012

Topics: Estimating the Mean, When is an estimate of mean unbiased? Unbiased estimator. Variance of a random variable,

- Week 1 Section Notes (gDoc pdf) by Jacopo

Installing and loading data with R software. Basic summary stats and histogram.

- How to Load a File into R Software (VIDEO)

- Sign-Up: receive updates about my sections, notes, etc.

Section Week 2 - April 11th 2012

Topics: Population Variance, Sample Variance, Standard deviation, Estimating the variance. Adjusting for using the sample mean. Variance of estimate of mean (mu-hat). T-distribution. Inference using t-distribution

- gDoc pdf: Week 2 Section Notes by Kun

Central limit theorem, t-test (small samples), test-statistics (larger samples). Hypothesis testing. T-distribution table.

- Curtis' Section Supplement (pdf, R-Code).

Homework 1 - due April 11th 2012

- pdf: questions

Expectations, mean, variance, and effect in non-systematic reporting error. R-softtware: basics, summary statistics, subsetting.

- pdf: solutions (by Dobkin)

- pdf: solutions (by Curtis)

Section Week 3 - April 18th 2012

Topics: Inference about the location of the sample mean. T-statistic. Asymptotic results - the law of large numbers. The Central Limit Theorem

- gDoc pdf: Week 3 Section Notes by Curtis

large sample test-statistics, hypothesis testing in R software.

Homework 2 - Due April 18th 2012

- pdf: questions

Standard error of the mean, determining bias of an estimate, small and large sample hypothesis testing, t-distribution and z-distribution, central limit theorem in action,

In R-software: taking a sample of data, small and large sample hypothesis testing, taking multiple samples.

- pdf: solutions (by Dobkin)

- .r file: solutions (by Curtis)

- pdf: #6 Solution. Effect on Variance of Measurement error independent of X (by Curtis)

Mid-Term One Review - April 18th 2012

Review of all material leading up to mid-term one.

There will be errors, so be vigilant!

- pdf: Econ113 Mid-Term One Review {S2012 Dobkin}

Section Week 4 - April 26th 2012

Topics: Derivation of beta-hat-1 and beta-hat-nought. Simple regression model example.

- .csv: section and class attendance and exam scores data.

- .R: R code from today's section.

- .pdf: Section Notes (by Jean Paul) (classdata.csv)

P-values. Regression, Ordinary Least Squares (OLS)

in R software.

Section Week 5 - May 2nd 2012

Topics: Conditions under which OLS gives an unbiased estimate. "Proof" that OLS is unbiased under Simple Regression (SR) Assumptions. Assessing the assumptions.

- pdf: Section Notes (by David) (Rcode, data_journals.csv, cigars.csv)

Regression, Ordinary Least Squares (OLS) in R software. Pretty useful notes.

Homework 3 - due May 4th

- pdf: questions (attendance.csv data needed for #4)

Calculate a P-Value. "Prove" Ordinary Least Squares (OLS) regression coefficients (beta-hat-nought and beta-hat-one) minimize the squared distance between the dependent variable and the regression line. Plot, calculate OLS coefficients by hand, and interpret. Plot, calculate OLS coefficients in R, and interpret.

- pdf solutions (by Curtis) (R Code)

- Question: The Ordinary Least Squares (Univariate OLS) regression minimizes the sum of the squared vertical distances from al the observations in the sample to the regression line. Derive the OLS coefficient estimates. (web solutions)

Section Week 6 - May 10nd 2012

Topics: Calculating the bias. Variance of beta-hat-one. Variance of beta-hat-1 and inference.

- pdf: Section Notes (by Jae) (data, Rcode)

Multivariate Regression in R. Normality assumption. R's linear model regression summary. Hypothesis testing with regression results. Finding and interpreting a regression's P-Value.

Homework 4 - due May 11th

- pdf: questions (attendance.csv data)

Derive variance of OLS intercept estimate. Regression, hypothesis testing and interpretation. Examination of regression assumptions required for inference.

- pdf: solutions (by Dobkin)

Section Week 7 - May 17nd 2012

Topics: TSS: total sum of squares. ESS: explained sum of squared. RSS: sum of squared residuals. R-squared, goodness of fit. Motivation for multivariate regression. Estimation and interpretation.

- pdf: Section Notes (by Curtis)

Homework 5 - due May 18th

- pdf: questions (attendance.csv data)

Derive multivariate coefficients. Multivariate regression in R. P-Values, R-squared. Derive effect of mean zero measurement error (attenuation bias). Measurement error in Data, example.

- pdf: solutions (by Jacopo)

- solutions (by Curtis)

Problem 1 (pdf) Derive the formula for the regression coefficients in the multivariate (two variable) case.

Problem 2 (R Code)

Problem 3 (pdf) The effect of mean zero measurement error in one independent on multivariate coefficient regression estimates (two variable case).

Homework 6 - due June 8th

- pdf: questions (NLSY Extract.csv data)

Multiple-variable regression. Level-level vs log-level regression. Interpretation, logging variables. Joint significance test, the F-test, plus interpretation. Normalizing variables and interpretation.

- solutions (pdf by Curtis)

Tip: F-Test critical value in R:

qf(ConfLevel,dof_Numerator,dof_Denominator)

CritVal <- qf(0.95,2,4852)

The Basics and Review

Table of Contents

1.0 Introduction

    • What is econometrics

    • Types of data

1.1 Definitions

  • Random variable:

  • Sample vs population:

  • Probability distribution:

    • Expected value of probability distribution:

    • Variance of probability distribution:

  • Expected value: (expected value theory)

  • Sample statistics and measures:

    • Unbiased:

    • Consistent:

1.2 Intuitive characterization of a distribution:

1.3 Measures of Center

  • Population mean

  • Sample mean:

    • Estimate of population mean.

    • Unbiased?

    • Median

    • Mode

    • Histogram - visualizing a center of a distribution

  • Assumptions

    • 1A The sample is random

    • 2A No systematic error

      • Proof of unbiased sample mean

      • Effect of adding systematic measurement error

      • Effect of adding unsystematic measurement error

1.4 Measures of dispersion

  • Range:

  • Variance

    • Population variance:

      • Discrete vs. continuous

    • Sample variance

      • Sample variance when population mean is known

      • Sample variance when population mean is unknown

    • Bias in estimates of variance

      • (bigger the sample, the lower the bias in the estimate for variance)

      • What determines the level of precision?

      • Effect of measurement error on variance

        • Effect of systematic measurement error on variance estimate

        • Effect of unsystematic measurement error on variance estimate

  • Standard deviation:

    • Histogram - visualizing the dispersion of a distribution

1.5 Describing the relatedness of two variables.

    • Covariance

    • Correlation

    • Scatterplot - visualizing the relatedness of two variables.

      • Examples of correlations via scatterplot

1.6 Odds and ends

  • Normal distribution.

  • Continuous random variable

  • Distribution of sample mean

  • Standard error of sample mean

  • Visualization

    • Histogram:

    • Frequency table:

    • Scatter plot:

    • R Software for Statistical Analysis

    • ...link to overview website....

Regression Analysis

  • Ordinary Least Squares (OLS)

      • Derive the simple regression model

      • Simple Regression Analysis

    • Regressing one variable on one other variable.

  • Assumptions needed for ...particular interpretations

      • Assumption SR1:

      • Assumption SR2:

      • Assumption SR3:

      • Assumption SR4:

      • Assumption SR5:

    • Unbiased Estimate - "Proof" that if ASR1 thru ASR5 hold, then we have a causal estimate, E[\beta\hat_1 | x] = \beta_1 (plus lots of intuition)

    • Measurement Error:

    • Goodness of Fit - how well does the regression fit?

      • Total Sum of Squares (SST)

      • Residual Sum of Squares (SSR)

      • Explained Sum of Squares (SSE)

      • R-Squared (1):

    • Changing Units of Measurement (e.g. switching from thousands to millions....)

    • Nonlinear Effect (e.g. log-level, level-log, log-log, etc. Exponential growth)

    • Proof of OLS unbiasedness.

  • Variance of OLS Estimator

      • Variance of Beta\hat_0

      • Varaince of Beta\hat_1

Hypothesis Testing (Part 1)

Inference

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Problems and Applications

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Problems and Applications

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Outline and example of problem (hypothesis about mean height)

Is our guess of a mean correct? compare estimate mean to ....

Central Limit Theorem

Law of large numbers

Assume x is distributed normally

Significance level:

Critical value:

Test-statistic: (t-stat)

Student's t distribution.

Large vs. small sample

Looking up critical value

degrees of freedom.

  • P-Value for a t-test

    • Computing P-Value for a t-test.

    • Interpreting P-value.

Multiple Regression Analysis

  • Introduction

  • Example with two independent variables

  • With k independent variable.

  • Interpreting multivariate repression results.

  • Comparing simple vs. multivariate regression estimates. (leads to omitted variable bias... )

  • R-Squared (Part 2) with multiple variables:

    • Regression with extraneous variables.

    • Mistake 1: including an irrelevant variable.

    • Omitted Variable Bias.

    • Mistake 2: failing to including a relevant variable.

      • Positive bias example:

      • Negative bias example:

      • Zero bias example:

  • Simple Regression Assumption 6 (SR6): Homoskedasticity

    • Homoskedasticity and inference.

    • Variance of multivariate regression parameter estimates.

      • Standard Error, se(\beta\hat)

      • Components of variance of parameter estimates

      • ~ Error variance sigma^squared

      • ~ Total Sample Variation

      • ~ R-squared from regression x_j on all other independent variables.

      • Multicollinearity.

      • Variance of miss-specified model.

Problems and Applications

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Gauss-Markov Theorem.

Simple Regression Assumption 7 (SR7): Normality.

Theorem Normal Sampling Distributions.

  • Seven Assumptions

  • Distribution of parameters if the errors are normal.

  • Normal sampling distribution.

Hypothesis Testing (Part 2)

  • t-distribution and hypothesis testing with regression estimates.

  • statistical significance.

  • One-sided test

  • Two-sided test

  • Other hypotheses: (not testing if beta is zero)

  • Interpreting regression output:

    • Examples

  • P-Value for a t-test

    • Computing P-Value for a t-test.

    • Interpreting P-value.

  • Practical significance. i.e. vs. statistical significance.

  • Confidence Intervals

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F-Test

  • Testing linear restrictions on parameters.

    • Example: Compare the return of a year of JC to the return on a year of univeristy education.

  • Testing multiple linear restrictions.

  • F-Test. Overall significance of a regression.

  • Testing general linear restrictions.

  • Adjusted R-Squared: (R-squared 3)

    • Limitation of R-squared

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More "Types" of Regression Models

    • Effects of Scaling on Regression estimates

      • Simple example

    • Standardizing coefficients -

  • Models with logs

      • Level level regression

      • Log level regression

      • Exponential

      • Log log regression

      • Level log regression

  • Models with quadratics. (diminishing returns)

  • Models with interaction terms

  • Models with dummy variables.

More on Multiple Regression Models.

  • Predictions and their standard errors

      • Prediction of value of a regression at at particular point.

      • What is our confidence at this point? i.e. what is our prediction and what is our standard error for this prediction?

  • Residual analysis

  • Heteroskedasticity and robust standard errors.

  • Testing for Heteroskedasticity.

    • Breusch-Pagan test for Heteroskedasticity

    • White test for Heteroskedasticity

    • ~ potential issues

    • Adjusting for Heteroskedasticity

    • Weighted least squares.

    • Robust standard errors

  • Measurement Error revisited.

      • Measurement error in independent/explanatory variable

      • Classical error in variables

    • Effect on estimate

    • Effect on variance

    • Measurement error in dependent variable

    • Effect on estimate

    • Effect on variance

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