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<meta name="twitter:description" content="Beginners with little background in statistics and econometrics often have a hard time understanding the benefits of having programming skills for learning and applying Econometrics. ‘Introduction to Econometrics with R’ is an interactive companion to the well-received textbook ‘Introduction to Econometrics’ by James H. Stock and Mark W. Watson (2015). It gives a gentle introduction to the essentials of R programming and guides students in implementing the empirical applications presented throughout the textbook using the newly aquired skills. This is supported by interactive programming exercises generated with DataCamp Light and integration of interactive visualizations of central concepts which are based on the flexible JavaScript library D3.js." />
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<li><center><img src="images/logo.png" alt="logo" width="50%" height="50%"style="margin: 15px 0 0 0"></center></li>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a></li>
<li class="chapter" data-level="1" data-path="1-introduction.html"><a href="1-introduction.html"><i class="fa fa-check"></i><b>1</b> Introduction</a>
<ul>
<li class="chapter" data-level="1.1" data-path="1.1-colophon.html"><a href="1.1-colophon.html"><i class="fa fa-check"></i><b>1.1</b> Colophon</a></li>
<li class="chapter" data-level="1.2" data-path="1.2-a-very-short-introduction-to-r-and-rstudio.html"><a href="1.2-a-very-short-introduction-to-r-and-rstudio.html"><i class="fa fa-check"></i><b>1.2</b> A Very Short Introduction to <tt>R</tt> and <em>RStudio</em></a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="2-pt.html"><a href="2-pt.html"><i class="fa fa-check"></i><b>2</b> Probability Theory</a>
<ul>
<li class="chapter" data-level="2.1" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html"><i class="fa fa-check"></i><b>2.1</b> Random Variables and Probability Distributions</a>
<ul>
<li class="chapter" data-level="" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html#probability-distributions-of-discrete-random-variables"><i class="fa fa-check"></i>Probability Distributions of Discrete Random Variables</a></li>
<li class="chapter" data-level="" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html#bernoulli-trials"><i class="fa fa-check"></i>Bernoulli Trials</a></li>
<li class="chapter" data-level="" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html#expected-value-mean-and-variance"><i class="fa fa-check"></i>Expected Value, Mean and Variance</a></li>
<li class="chapter" data-level="" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html#probability-distributions-of-continuous-random-variables"><i class="fa fa-check"></i>Probability Distributions of Continuous Random Variables</a></li>
<li class="chapter" data-level="" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html#the-normal-distribution"><i class="fa fa-check"></i>The Normal Distribution</a></li>
<li class="chapter" data-level="" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html#the-chi-squared-distribution"><i class="fa fa-check"></i>The Chi-Squared Distribution</a></li>
<li class="chapter" data-level="" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html#thetdist"><i class="fa fa-check"></i>The Student t Distribution</a></li>
<li class="chapter" data-level="" data-path="2.1-random-variables-and-probability-distributions.html"><a href="2.1-random-variables-and-probability-distributions.html#the-f-distribution"><i class="fa fa-check"></i>The F Distribution</a></li>
</ul></li>
<li class="chapter" data-level="2.2" data-path="2.2-RSATDOSA.html"><a href="2.2-RSATDOSA.html"><i class="fa fa-check"></i><b>2.2</b> Random Sampling and the Distribution of Sample Averages</a>
<ul>
<li class="chapter" data-level="" data-path="2.2-RSATDOSA.html"><a href="2.2-RSATDOSA.html#mean-and-variance-of-the-sample-mean"><i class="fa fa-check"></i>Mean and Variance of the Sample Mean</a></li>
<li class="chapter" data-level="" data-path="2.2-RSATDOSA.html"><a href="2.2-RSATDOSA.html#large-sample-approximations-to-sampling-distributions"><i class="fa fa-check"></i>Large Sample Approximations to Sampling Distributions</a></li>
</ul></li>
<li class="chapter" data-level="2.3" data-path="2.3-exercises-2.html"><a href="2.3-exercises-2.html"><i class="fa fa-check"></i><b>2.3</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="3-arosur.html"><a href="3-arosur.html"><i class="fa fa-check"></i><b>3</b> A Review of Statistics using R</a>
<ul>
<li class="chapter" data-level="3.1" data-path="3.1-estimation-of-the-population-mean.html"><a href="3.1-estimation-of-the-population-mean.html"><i class="fa fa-check"></i><b>3.1</b> Estimation of the Population Mean</a></li>
<li class="chapter" data-level="3.2" data-path="3.2-potsm.html"><a href="3.2-potsm.html"><i class="fa fa-check"></i><b>3.2</b> Properties of the Sample Mean</a></li>
<li class="chapter" data-level="3.3" data-path="3.3-hypothesis-tests-concerning-the-population-mean.html"><a href="3.3-hypothesis-tests-concerning-the-population-mean.html"><i class="fa fa-check"></i><b>3.3</b> Hypothesis Tests concerning the Population Mean</a>
<ul>
<li class="chapter" data-level="" data-path="3.3-hypothesis-tests-concerning-the-population-mean.html"><a href="3.3-hypothesis-tests-concerning-the-population-mean.html#the-p-value"><i class="fa fa-check"></i>The p-Value</a></li>
<li class="chapter" data-level="" data-path="3.3-hypothesis-tests-concerning-the-population-mean.html"><a href="3.3-hypothesis-tests-concerning-the-population-mean.html#calculating-the-p-value-when-the-standard-deviation-is-known"><i class="fa fa-check"></i>Calculating the p-Value when the Standard Deviation is Known</a></li>
<li class="chapter" data-level="" data-path="3.3-hypothesis-tests-concerning-the-population-mean.html"><a href="3.3-hypothesis-tests-concerning-the-population-mean.html#SVSSDASE"><i class="fa fa-check"></i>Sample Variance, Sample Standard Deviation and Standard Error</a></li>
<li class="chapter" data-level="" data-path="3.3-hypothesis-tests-concerning-the-population-mean.html"><a href="3.3-hypothesis-tests-concerning-the-population-mean.html#calculating-the-p-value-when-the-standard-deviation-is-unknown"><i class="fa fa-check"></i>Calculating the p-value When the Standard Deviation is Unknown</a></li>
<li class="chapter" data-level="" data-path="3.3-hypothesis-tests-concerning-the-population-mean.html"><a href="3.3-hypothesis-tests-concerning-the-population-mean.html#the-t-statistic"><i class="fa fa-check"></i>The t-statistic</a></li>
<li class="chapter" data-level="" data-path="3.3-hypothesis-tests-concerning-the-population-mean.html"><a href="3.3-hypothesis-tests-concerning-the-population-mean.html#hypothesis-testing-with-a-prespecified-significance-level"><i class="fa fa-check"></i>Hypothesis Testing with a Prespecified Significance Level</a></li>
<li class="chapter" data-level="" data-path="3.3-hypothesis-tests-concerning-the-population-mean.html"><a href="3.3-hypothesis-tests-concerning-the-population-mean.html#one-sided-alternatives"><i class="fa fa-check"></i>One-sided Alternatives</a></li>
</ul></li>
<li class="chapter" data-level="3.4" data-path="3.4-confidence-intervals-for-the-population-mean.html"><a href="3.4-confidence-intervals-for-the-population-mean.html"><i class="fa fa-check"></i><b>3.4</b> Confidence Intervals for the Population Mean</a></li>
<li class="chapter" data-level="3.5" data-path="3.5-cmfdp.html"><a href="3.5-cmfdp.html"><i class="fa fa-check"></i><b>3.5</b> Comparing Means from Different Populations</a></li>
<li class="chapter" data-level="3.6" data-path="3.6-aattggoe.html"><a href="3.6-aattggoe.html"><i class="fa fa-check"></i><b>3.6</b> An Application to the Gender Gap of Earnings</a></li>
<li class="chapter" data-level="3.7" data-path="3.7-scatterplots-sample-covariance-and-sample-correlation.html"><a href="3.7-scatterplots-sample-covariance-and-sample-correlation.html"><i class="fa fa-check"></i><b>3.7</b> Scatterplots, Sample Covariance and Sample Correlation</a></li>
<li class="chapter" data-level="3.8" data-path="3.8-exercises-3.html"><a href="3.8-exercises-3.html"><i class="fa fa-check"></i><b>3.8</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="4-lrwor.html"><a href="4-lrwor.html"><i class="fa fa-check"></i><b>4</b> Linear Regression with One Regressor</a>
<ul>
<li class="chapter" data-level="4.1" data-path="4.1-simple-linear-regression.html"><a href="4.1-simple-linear-regression.html"><i class="fa fa-check"></i><b>4.1</b> Simple Linear Regression</a></li>
<li class="chapter" data-level="4.2" data-path="4.2-estimating-the-coefficients-of-the-linear-regression-model.html"><a href="4.2-estimating-the-coefficients-of-the-linear-regression-model.html"><i class="fa fa-check"></i><b>4.2</b> Estimating the Coefficients of the Linear Regression Model</a>
<ul>
<li class="chapter" data-level="" data-path="4.2-estimating-the-coefficients-of-the-linear-regression-model.html"><a href="4.2-estimating-the-coefficients-of-the-linear-regression-model.html#the-ordinary-least-squares-estimator"><i class="fa fa-check"></i>The Ordinary Least Squares Estimator</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="4.3-measures-of-fit.html"><a href="4.3-measures-of-fit.html"><i class="fa fa-check"></i><b>4.3</b> Measures of Fit</a>
<ul>
<li class="chapter" data-level="" data-path="4.3-measures-of-fit.html"><a href="4.3-measures-of-fit.html#the-coefficient-of-determination"><i class="fa fa-check"></i>The Coefficient of Determination</a></li>
<li class="chapter" data-level="" data-path="4.3-measures-of-fit.html"><a href="4.3-measures-of-fit.html#the-standard-error-of-the-regression"><i class="fa fa-check"></i>The Standard Error of the Regression</a></li>
<li class="chapter" data-level="" data-path="4.3-measures-of-fit.html"><a href="4.3-measures-of-fit.html#application-to-the-test-score-data"><i class="fa fa-check"></i>Application to the Test Score Data</a></li>
</ul></li>
<li class="chapter" data-level="4.4" data-path="4.4-tlsa.html"><a href="4.4-tlsa.html"><i class="fa fa-check"></i><b>4.4</b> The Least Squares Assumptions</a>
<ul>
<li class="chapter" data-level="" data-path="4.4-tlsa.html"><a href="4.4-tlsa.html#assumption-1-the-error-term-has-conditional-mean-of-zero"><i class="fa fa-check"></i>Assumption 1: The Error Term has Conditional Mean of Zero</a></li>
<li class="chapter" data-level="" data-path="4.4-tlsa.html"><a href="4.4-tlsa.html#assumption-2-independently-and-identically-distributed-data"><i class="fa fa-check"></i>Assumption 2: Independently and Identically Distributed Data</a></li>
<li class="chapter" data-level="" data-path="4.4-tlsa.html"><a href="4.4-tlsa.html#assumption-3-large-outliers-are-unlikely"><i class="fa fa-check"></i>Assumption 3: Large Outliers are Unlikely</a></li>
</ul></li>
<li class="chapter" data-level="4.5" data-path="4.5-tsdotoe.html"><a href="4.5-tsdotoe.html"><i class="fa fa-check"></i><b>4.5</b> The Sampling Distribution of the OLS Estimator</a>
<ul>
<li class="chapter" data-level="" data-path="4.5-tsdotoe.html"><a href="4.5-tsdotoe.html#simulation-study-1"><i class="fa fa-check"></i>Simulation Study 1</a></li>
<li class="chapter" data-level="" data-path="4.5-tsdotoe.html"><a href="4.5-tsdotoe.html#simulation-study-2"><i class="fa fa-check"></i>Simulation Study 2</a></li>
<li class="chapter" data-level="" data-path="4.5-tsdotoe.html"><a href="4.5-tsdotoe.html#simulation-study-3"><i class="fa fa-check"></i>Simulation Study 3</a></li>
</ul></li>
<li class="chapter" data-level="4.6" data-path="4.6-exercises-4.html"><a href="4.6-exercises-4.html"><i class="fa fa-check"></i><b>4.6</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="5-htaciitslrm.html"><a href="5-htaciitslrm.html"><i class="fa fa-check"></i><b>5</b> Hypothesis Tests and Confidence Intervals in SLR Model</a>
<ul>
<li class="chapter" data-level="5.1" data-path="5.1-testing-two-sided-hypotheses-concerning-the-slope-coefficient.html"><a href="5.1-testing-two-sided-hypotheses-concerning-the-slope-coefficient.html"><i class="fa fa-check"></i><b>5.1</b> Testing Two-Sided Hypotheses concerning the Slope Coefficient</a></li>
<li class="chapter" data-level="5.2" data-path="5.2-cifrc.html"><a href="5.2-cifrc.html"><i class="fa fa-check"></i><b>5.2</b> Confidence Intervals for Regression Coefficients</a>
<ul>
<li class="chapter" data-level="" data-path="5.2-cifrc.html"><a href="5.2-cifrc.html#simulation-study-confidence-intervals"><i class="fa fa-check"></i>Simulation Study: Confidence Intervals</a></li>
</ul></li>
<li class="chapter" data-level="5.3" data-path="5.3-rwxiabv.html"><a href="5.3-rwxiabv.html"><i class="fa fa-check"></i><b>5.3</b> Regression when X is a Binary Variable</a></li>
<li class="chapter" data-level="5.4" data-path="5.4-hah.html"><a href="5.4-hah.html"><i class="fa fa-check"></i><b>5.4</b> Heteroskedasticity and Homoskedasticity</a>
<ul>
<li class="chapter" data-level="" data-path="5.4-hah.html"><a href="5.4-hah.html#a-real-world-example-for-heteroskedasticity"><i class="fa fa-check"></i>A Real-World Example for Heteroskedasticity</a></li>
<li class="chapter" data-level="" data-path="5.4-hah.html"><a href="5.4-hah.html#should-we-care-about-heteroskedasticity"><i class="fa fa-check"></i>Should We Care About Heteroskedasticity?</a></li>
<li class="chapter" data-level="" data-path="5.4-hah.html"><a href="5.4-hah.html#computation-of-heteroskedasticity-robust-standard-errors"><i class="fa fa-check"></i>Computation of Heteroskedasticity-Robust Standard Errors</a></li>
</ul></li>
<li class="chapter" data-level="5.5" data-path="5.5-the-gauss-markov-theorem.html"><a href="5.5-the-gauss-markov-theorem.html"><i class="fa fa-check"></i><b>5.5</b> The Gauss-Markov Theorem</a>
<ul>
<li class="chapter" data-level="" data-path="5.5-the-gauss-markov-theorem.html"><a href="5.5-the-gauss-markov-theorem.html#simulation-study-blue-estimator"><i class="fa fa-check"></i>Simulation Study: BLUE Estimator</a></li>
</ul></li>
<li class="chapter" data-level="5.6" data-path="5.6-using-the-t-statistic-in-regression-when-the-sample-size-is-small.html"><a href="5.6-using-the-t-statistic-in-regression-when-the-sample-size-is-small.html"><i class="fa fa-check"></i><b>5.6</b> Using the t-Statistic in Regression when the Sample Size Is Small</a></li>
<li class="chapter" data-level="5.7" data-path="5.7-exercises-5.html"><a href="5.7-exercises-5.html"><i class="fa fa-check"></i><b>5.7</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="6-rmwmr.html"><a href="6-rmwmr.html"><i class="fa fa-check"></i><b>6</b> Regression Models with Multiple Regressors</a>
<ul>
<li class="chapter" data-level="6.1" data-path="6.1-omitted-variable-bias.html"><a href="6.1-omitted-variable-bias.html"><i class="fa fa-check"></i><b>6.1</b> Omitted Variable Bias</a></li>
<li class="chapter" data-level="6.2" data-path="6.2-tmrm.html"><a href="6.2-tmrm.html"><i class="fa fa-check"></i><b>6.2</b> The Multiple Regression Model</a></li>
<li class="chapter" data-level="6.3" data-path="6.3-mofimr.html"><a href="6.3-mofimr.html"><i class="fa fa-check"></i><b>6.3</b> Measures of Fit in Multiple Regression</a></li>
<li class="chapter" data-level="6.4" data-path="6.4-ols-assumptions-in-multiple-regression.html"><a href="6.4-ols-assumptions-in-multiple-regression.html"><i class="fa fa-check"></i><b>6.4</b> OLS Assumptions in Multiple Regression</a>
<ul>
<li class="chapter" data-level="" data-path="6.4-ols-assumptions-in-multiple-regression.html"><a href="6.4-ols-assumptions-in-multiple-regression.html#multicollinearity"><i class="fa fa-check"></i>Multicollinearity</a></li>
<li class="chapter" data-level="" data-path="6.4-ols-assumptions-in-multiple-regression.html"><a href="6.4-ols-assumptions-in-multiple-regression.html#simulation-study-imperfect-multicollinearity"><i class="fa fa-check"></i>Simulation Study: Imperfect Multicollinearity</a></li>
</ul></li>
<li class="chapter" data-level="6.5" data-path="6.5-the-distribution-of-the-ols-estimators-in-multiple-regression.html"><a href="6.5-the-distribution-of-the-ols-estimators-in-multiple-regression.html"><i class="fa fa-check"></i><b>6.5</b> The Distribution of the OLS Estimators in Multiple Regression</a></li>
<li class="chapter" data-level="6.6" data-path="6.6-exercises-6.html"><a href="6.6-exercises-6.html"><i class="fa fa-check"></i><b>6.6</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="7-htaciimr.html"><a href="7-htaciimr.html"><i class="fa fa-check"></i><b>7</b> Hypothesis Tests and Confidence Intervals in MR Models</a>
<ul>
<li class="chapter" data-level="7.1" data-path="7.1-hypothesis-tests-and-confidence-intervals-for-a-single-coefficient.html"><a href="7.1-hypothesis-tests-and-confidence-intervals-for-a-single-coefficient.html"><i class="fa fa-check"></i><b>7.1</b> Hypothesis Tests and Confidence Intervals for a Single Coefficient</a></li>
<li class="chapter" data-level="7.2" data-path="7.2-an-application-to-test-scores-and-the-student-teacher-ratio.html"><a href="7.2-an-application-to-test-scores-and-the-student-teacher-ratio.html"><i class="fa fa-check"></i><b>7.2</b> An Application to Test Scores and the Student-Teacher Ratio</a>
<ul>
<li class="chapter" data-level="" data-path="7.2-an-application-to-test-scores-and-the-student-teacher-ratio.html"><a href="7.2-an-application-to-test-scores-and-the-student-teacher-ratio.html#another-augmentation-of-the-model"><i class="fa fa-check"></i>Another Augmentation of the Model</a></li>
</ul></li>
<li class="chapter" data-level="7.3" data-path="7.3-joint-hypothesis-testing-using-the-f-statistic.html"><a href="7.3-joint-hypothesis-testing-using-the-f-statistic.html"><i class="fa fa-check"></i><b>7.3</b> Joint Hypothesis Testing using the F-Statistic</a></li>
<li class="chapter" data-level="7.4" data-path="7.4-confidence-sets-for-multiple-coefficients.html"><a href="7.4-confidence-sets-for-multiple-coefficients.html"><i class="fa fa-check"></i><b>7.4</b> Confidence Sets for Multiple Coefficients</a></li>
<li class="chapter" data-level="7.5" data-path="7.5-model-specification-for-multiple-regression.html"><a href="7.5-model-specification-for-multiple-regression.html"><i class="fa fa-check"></i><b>7.5</b> Model Specification for Multiple Regression</a>
<ul>
<li class="chapter" data-level="" data-path="7.5-model-specification-for-multiple-regression.html"><a href="7.5-model-specification-for-multiple-regression.html#model-specification-in-theory-and-in-practice"><i class="fa fa-check"></i>Model Specification in Theory and in Practice</a></li>
</ul></li>
<li class="chapter" data-level="7.6" data-path="7.6-analysis-of-the-test-score-data-set.html"><a href="7.6-analysis-of-the-test-score-data-set.html"><i class="fa fa-check"></i><b>7.6</b> Analysis of the Test Score Data Set</a></li>
<li class="chapter" data-level="7.7" data-path="7.7-exercises-7.html"><a href="7.7-exercises-7.html"><i class="fa fa-check"></i><b>7.7</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="8-nrf.html"><a href="8-nrf.html"><i class="fa fa-check"></i><b>8</b> Nonlinear Regression Functions</a>
<ul>
<li class="chapter" data-level="8.1" data-path="8.1-a-general-strategy-for-modelling-nonlinear-regression-functions.html"><a href="8.1-a-general-strategy-for-modelling-nonlinear-regression-functions.html"><i class="fa fa-check"></i><b>8.1</b> A General Strategy for Modelling Nonlinear Regression Functions</a></li>
<li class="chapter" data-level="8.2" data-path="8.2-nfoasiv.html"><a href="8.2-nfoasiv.html"><i class="fa fa-check"></i><b>8.2</b> Nonlinear Functions of a Single Independent Variable</a>
<ul>
<li class="chapter" data-level="" data-path="8.2-nfoasiv.html"><a href="8.2-nfoasiv.html#polynomials"><i class="fa fa-check"></i>Polynomials</a></li>
<li class="chapter" data-level="" data-path="8.2-nfoasiv.html"><a href="8.2-nfoasiv.html#logarithms"><i class="fa fa-check"></i>Logarithms</a></li>
</ul></li>
<li class="chapter" data-level="8.3" data-path="8.3-interactions-between-independent-variables.html"><a href="8.3-interactions-between-independent-variables.html"><i class="fa fa-check"></i><b>8.3</b> Interactions between Independent Variables</a></li>
<li class="chapter" data-level="8.4" data-path="8.4-nonlinear-effects-on-test-scores-of-the-student-teacher-ratio.html"><a href="8.4-nonlinear-effects-on-test-scores-of-the-student-teacher-ratio.html"><i class="fa fa-check"></i><b>8.4</b> Nonlinear Effects on Test Scores of the Student-Teacher Ratio</a></li>
<li class="chapter" data-level="8.5" data-path="8.5-exercises-8.html"><a href="8.5-exercises-8.html"><i class="fa fa-check"></i><b>8.5</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="9-asbomr.html"><a href="9-asbomr.html"><i class="fa fa-check"></i><b>9</b> Assessing Studies Based on Multiple Regression</a>
<ul>
<li class="chapter" data-level="9.1" data-path="9.1-internal-and-external-validity.html"><a href="9.1-internal-and-external-validity.html"><i class="fa fa-check"></i><b>9.1</b> Internal and External Validity</a></li>
<li class="chapter" data-level="9.2" data-path="9.2-ttivomra.html"><a href="9.2-ttivomra.html"><i class="fa fa-check"></i><b>9.2</b> Threats to Internal Validity of Multiple Regression Analysis</a></li>
<li class="chapter" data-level="9.3" data-path="9.3-internal-and-external-validity-when-the-regression-is-used-for-forecasting.html"><a href="9.3-internal-and-external-validity-when-the-regression-is-used-for-forecasting.html"><i class="fa fa-check"></i><b>9.3</b> Internal and External Validity when the Regression is used for Forecasting</a></li>
<li class="chapter" data-level="9.4" data-path="9.4-etsacs.html"><a href="9.4-etsacs.html"><i class="fa fa-check"></i><b>9.4</b> Example: Test Scores and Class Size</a></li>
<li class="chapter" data-level="9.5" data-path="9.5-exercises-9.html"><a href="9.5-exercises-9.html"><i class="fa fa-check"></i><b>9.5</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="10-rwpd.html"><a href="10-rwpd.html"><i class="fa fa-check"></i><b>10</b> Regression with Panel Data</a>
<ul>
<li class="chapter" data-level="10.1" data-path="10.1-panel-data.html"><a href="10.1-panel-data.html"><i class="fa fa-check"></i><b>10.1</b> Panel Data</a></li>
<li class="chapter" data-level="10.2" data-path="10.2-PDWTTP.html"><a href="10.2-PDWTTP.html"><i class="fa fa-check"></i><b>10.2</b> Panel Data with Two Time Periods: “Before and After” Comparisons</a></li>
<li class="chapter" data-level="10.3" data-path="10.3-fixed-effects-regression.html"><a href="10.3-fixed-effects-regression.html"><i class="fa fa-check"></i><b>10.3</b> Fixed Effects Regression</a>
<ul>
<li class="chapter" data-level="" data-path="10.3-fixed-effects-regression.html"><a href="10.3-fixed-effects-regression.html#estimation-and-inference"><i class="fa fa-check"></i>Estimation and Inference</a></li>
<li class="chapter" data-level="" data-path="10.3-fixed-effects-regression.html"><a href="10.3-fixed-effects-regression.html#application-to-traffic-deaths"><i class="fa fa-check"></i>Application to Traffic Deaths</a></li>
</ul></li>
<li class="chapter" data-level="10.4" data-path="10.4-regression-with-time-fixed-effects.html"><a href="10.4-regression-with-time-fixed-effects.html"><i class="fa fa-check"></i><b>10.4</b> Regression with Time Fixed Effects</a></li>
<li class="chapter" data-level="10.5" data-path="10.5-tferaaseffer.html"><a href="10.5-tferaaseffer.html"><i class="fa fa-check"></i><b>10.5</b> The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression</a></li>
<li class="chapter" data-level="10.6" data-path="10.6-drunk-driving-laws-and-traffic-deaths.html"><a href="10.6-drunk-driving-laws-and-traffic-deaths.html"><i class="fa fa-check"></i><b>10.6</b> Drunk Driving Laws and Traffic Deaths</a></li>
<li class="chapter" data-level="10.7" data-path="10.7-exercises-10.html"><a href="10.7-exercises-10.html"><i class="fa fa-check"></i><b>10.7</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="11-rwabdv.html"><a href="11-rwabdv.html"><i class="fa fa-check"></i><b>11</b> Regression with a Binary Dependent Variable</a>
<ul>
<li class="chapter" data-level="11.1" data-path="11.1-binary-dependent-variables-and-the-linear-probability-model.html"><a href="11.1-binary-dependent-variables-and-the-linear-probability-model.html"><i class="fa fa-check"></i><b>11.1</b> Binary Dependent Variables and the Linear Probability Model</a></li>
<li class="chapter" data-level="11.2" data-path="11.2-palr.html"><a href="11.2-palr.html"><i class="fa fa-check"></i><b>11.2</b> Probit and Logit Regression</a>
<ul>
<li class="chapter" data-level="" data-path="11.2-palr.html"><a href="11.2-palr.html#probit-regression"><i class="fa fa-check"></i>Probit Regression</a></li>
<li class="chapter" data-level="" data-path="11.2-palr.html"><a href="11.2-palr.html#logit-regression"><i class="fa fa-check"></i>Logit Regression</a></li>
</ul></li>
<li class="chapter" data-level="11.3" data-path="11.3-estimation-and-inference-in-the-logit-and-probit-models.html"><a href="11.3-estimation-and-inference-in-the-logit-and-probit-models.html"><i class="fa fa-check"></i><b>11.3</b> Estimation and Inference in the Logit and Probit Models</a></li>
<li class="chapter" data-level="11.4" data-path="11.4-application-to-the-boston-hmda-data.html"><a href="11.4-application-to-the-boston-hmda-data.html"><i class="fa fa-check"></i><b>11.4</b> Application to the Boston HMDA Data</a></li>
<li class="chapter" data-level="11.5" data-path="11.5-exercises-11.html"><a href="11.5-exercises-11.html"><i class="fa fa-check"></i><b>11.5</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="12-ivr.html"><a href="12-ivr.html"><i class="fa fa-check"></i><b>12</b> Instrumental Variables Regression</a>
<ul>
<li class="chapter" data-level="12.1" data-path="12.1-TIVEWASRAASI.html"><a href="12.1-TIVEWASRAASI.html"><i class="fa fa-check"></i><b>12.1</b> The IV Estimator with a Single Regressor and a Single Instrument</a></li>
<li class="chapter" data-level="12.2" data-path="12.2-TGIVRM.html"><a href="12.2-TGIVRM.html"><i class="fa fa-check"></i><b>12.2</b> The General IV Regression Model</a></li>
<li class="chapter" data-level="12.3" data-path="12.3-civ.html"><a href="12.3-civ.html"><i class="fa fa-check"></i><b>12.3</b> Checking Instrument Validity</a></li>
<li class="chapter" data-level="12.4" data-path="12.4-attdfc.html"><a href="12.4-attdfc.html"><i class="fa fa-check"></i><b>12.4</b> Application to the Demand for Cigarettes</a></li>
<li class="chapter" data-level="12.5" data-path="12.5-where-do-valid-instruments-come-from.html"><a href="12.5-where-do-valid-instruments-come-from.html"><i class="fa fa-check"></i><b>12.5</b> Where Do Valid Instruments Come From?</a></li>
<li class="chapter" data-level="12.6" data-path="12.6-exercises-12.html"><a href="12.6-exercises-12.html"><i class="fa fa-check"></i><b>12.6</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="13-eaqe.html"><a href="13-eaqe.html"><i class="fa fa-check"></i><b>13</b> Experiments and Quasi-Experiments</a>
<ul>
<li class="chapter" data-level="13.1" data-path="13.1-poceaie.html"><a href="13.1-poceaie.html"><i class="fa fa-check"></i><b>13.1</b> Potential Outcomes, Causal Effects and Idealized Experiments</a></li>
<li class="chapter" data-level="13.2" data-path="13.2-threats-to-validity-of-experiments.html"><a href="13.2-threats-to-validity-of-experiments.html"><i class="fa fa-check"></i><b>13.2</b> Threats to Validity of Experiments</a></li>
<li class="chapter" data-level="13.3" data-path="13.3-experimental-estimates-of-the-effect-of-class-size-reductions.html"><a href="13.3-experimental-estimates-of-the-effect-of-class-size-reductions.html"><i class="fa fa-check"></i><b>13.3</b> Experimental Estimates of the Effect of Class Size Reductions</a>
<ul>
<li class="chapter" data-level="" data-path="13.3-experimental-estimates-of-the-effect-of-class-size-reductions.html"><a href="13.3-experimental-estimates-of-the-effect-of-class-size-reductions.html#experimental-design-and-the-data-set"><i class="fa fa-check"></i>Experimental Design and the Data Set</a></li>
<li class="chapter" data-level="" data-path="13.3-experimental-estimates-of-the-effect-of-class-size-reductions.html"><a href="13.3-experimental-estimates-of-the-effect-of-class-size-reductions.html#analysis-of-the-star-data"><i class="fa fa-check"></i>Analysis of the STAR Data</a></li>
</ul></li>
<li class="chapter" data-level="13.4" data-path="13.4-qe.html"><a href="13.4-qe.html"><i class="fa fa-check"></i><b>13.4</b> Quasi Experiments</a>
<ul>
<li class="chapter" data-level="" data-path="13.4-qe.html"><a href="13.4-qe.html#the-differences-in-differences-estimator"><i class="fa fa-check"></i>The Differences-in-Differences Estimator</a></li>
<li class="chapter" data-level="" data-path="13.4-qe.html"><a href="13.4-qe.html#regression-discontinuity-estimators"><i class="fa fa-check"></i>Regression Discontinuity Estimators</a></li>
</ul></li>
<li class="chapter" data-level="13.5" data-path="13.5-exercises-13.html"><a href="13.5-exercises-13.html"><i class="fa fa-check"></i><b>13.5</b> Exercises</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="14-ittsraf.html"><a href="14-ittsraf.html"><i class="fa fa-check"></i><b>14</b> Introduction to Time Series Regression and Forecasting</a>
<ul>
<li class="chapter" data-level="14.1" data-path="14.1-using-regression-models-for-forecasting.html"><a href="14.1-using-regression-models-for-forecasting.html"><i class="fa fa-check"></i><b>14.1</b> Using Regression Models for Forecasting</a></li>
<li class="chapter" data-level="14.2" data-path="14.2-tsdasc.html"><a href="14.2-tsdasc.html"><i class="fa fa-check"></i><b>14.2</b> Time Series Data and Serial Correlation</a>
<ul>
<li class="chapter" data-level="" data-path="14.2-tsdasc.html"><a href="14.2-tsdasc.html#notation-lags-differences-logarithms-and-growth-rates"><i class="fa fa-check"></i>Notation, Lags, Differences, Logarithms and Growth Rates</a></li>
</ul></li>
<li class="chapter" data-level="14.3" data-path="14.3-autoregressions.html"><a href="14.3-autoregressions.html"><i class="fa fa-check"></i><b>14.3</b> Autoregressions</a>
<ul>
<li class="chapter" data-level="" data-path="14.3-autoregressions.html"><a href="14.3-autoregressions.html#autoregressive-models-of-order-p"><i class="fa fa-check"></i>Autoregressive Models of Order <span class="math inline">\(p\)</span></a></li>
</ul></li>
<li class="chapter" data-level="14.4" data-path="14.4-cybtmpi.html"><a href="14.4-cybtmpi.html"><i class="fa fa-check"></i><b>14.4</b> Can You Beat the Market? (Part I)</a></li>
<li class="chapter" data-level="14.5" data-path="14.5-apatadlm.html"><a href="14.5-apatadlm.html"><i class="fa fa-check"></i><b>14.5</b> Additional Predictors and The ADL Model</a>
<ul>
<li class="chapter" data-level="" data-path="14.5-apatadlm.html"><a href="14.5-apatadlm.html#forecast-uncertainty-and-forecast-intervals"><i class="fa fa-check"></i>Forecast Uncertainty and Forecast Intervals</a></li>
</ul></li>
<li class="chapter" data-level="14.6" data-path="14.6-llsuic.html"><a href="14.6-llsuic.html"><i class="fa fa-check"></i><b>14.6</b> Lag Length Selection using Information Criteria</a></li>
<li class="chapter" data-level="14.7" data-path="14.7-nit.html"><a href="14.7-nit.html"><i class="fa fa-check"></i><b>14.7</b> Nonstationarity I: Trends</a></li>
<li class="chapter" data-level="14.8" data-path="14.8-niib.html"><a href="14.8-niib.html"><i class="fa fa-check"></i><b>14.8</b> Nonstationarity II: Breaks</a></li>
<li class="chapter" data-level="14.9" data-path="14.9-can-you-beat-the-market-part-ii.html"><a href="14.9-can-you-beat-the-market-part-ii.html"><i class="fa fa-check"></i><b>14.9</b> Can You Beat the Market? (Part II)</a></li>
</ul></li>
<li class="chapter" data-level="15" data-path="15-eodce.html"><a href="15-eodce.html"><i class="fa fa-check"></i><b>15</b> Estimation of Dynamic Causal Effects</a>
<ul>
<li class="chapter" data-level="15.1" data-path="15.1-the-orange-juice-data.html"><a href="15.1-the-orange-juice-data.html"><i class="fa fa-check"></i><b>15.1</b> The Orange Juice Data</a></li>
<li class="chapter" data-level="15.2" data-path="15.2-dynamic-causal-effects.html"><a href="15.2-dynamic-causal-effects.html"><i class="fa fa-check"></i><b>15.2</b> Dynamic Causal Effects</a></li>
<li class="chapter" data-level="15.3" data-path="15.3-dynamic-multipliers-and-cumulative-dynamic-multipliers.html"><a href="15.3-dynamic-multipliers-and-cumulative-dynamic-multipliers.html"><i class="fa fa-check"></i><b>15.3</b> Dynamic Multipliers and Cumulative Dynamic Multipliers</a></li>
<li class="chapter" data-level="15.4" data-path="15.4-hac-standard-errors.html"><a href="15.4-hac-standard-errors.html"><i class="fa fa-check"></i><b>15.4</b> HAC Standard Errors</a></li>
<li class="chapter" data-level="15.5" data-path="15.5-estimation-of-dynamic-causal-effects-with-strictly-exogeneous-regressors.html"><a href="15.5-estimation-of-dynamic-causal-effects-with-strictly-exogeneous-regressors.html"><i class="fa fa-check"></i><b>15.5</b> Estimation of Dynamic Causal Effects with Strictly Exogeneous Regressors</a></li>
<li class="chapter" data-level="15.6" data-path="15.6-orange-juice-prices-and-cold-weather.html"><a href="15.6-orange-juice-prices-and-cold-weather.html"><i class="fa fa-check"></i><b>15.6</b> Orange Juice Prices and Cold Weather</a></li>
</ul></li>
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<h1><span class="header-section-number">5</span> Hypothesis Tests and Confidence Intervals in SLR Model<a href="5-htaciitslrm.html#htaciitslrm" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>This chapter continues our treatment of the simple linear regression model. The following subsections discuss how we may use our knowledge about the sampling distribution of the OLS estimator in order to make statements regarding its uncertainty.</p>
<p>These subsections cover the following topics:</p>
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<li><p>Testing Hypotheses regarding regression coefficients.</p></li>
<li><p>Confidence intervals for regression coefficients.</p></li>
<li><p>Regression when <span class="math inline">\(X\)</span> is a dummy variable.</p></li>
<li><p>Heteroskedasticity and Homoskedasticity.</p></li>
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<p>The packages <tt>AER</tt> <span class="citation">(<a href="#ref-R-AER">Christian Kleiber and Zeileis 2008</a>)</span> and <tt>scales</tt> <span class="citation">(<a href="#ref-R-scales">Wickham and Seidel 2022</a>)</span> are required for reproduction of the code chunks presented throughout this chapter. The package <tt>scales</tt> provides additional generic plot scaling methods. Make sure both packages are installed before you proceed. The safest way to do so is by checking whether the following code chunk executes without any errors.</p>
<div class="sourceCode" id="cb106"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb106-1"><a href="5-htaciitslrm.html#cb106-1" tabindex="-1"></a><span class="fu">library</span>(AER)</span>
<span id="cb106-2"><a href="5-htaciitslrm.html#cb106-2" tabindex="-1"></a><span class="fu">library</span>(scales)</span></code></pre></div>
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<div id="ref-R-AER" class="csl-entry">
Kleiber, Christian, and Achim Zeileis. 2008. <em>Applied Econometrics with <span>R</span></em>. New York: Springer-Verlag. <a href="https://CRAN.R-project.org/package=AER">https://CRAN.R-project.org/package=AER</a>.
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<div id="ref-R-scales" class="csl-entry">
Wickham, Hadley, and Dana Seidel. 2022. <em><span class="nocase">scales: Scale Functions for Visualization</span></em> (version 1.2.1). <a href="https://scales.r-lib.org">https://scales.r-lib.org</a>.
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