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<h2 class="hd hd-2 unit-title">Handouts</h2>
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<h4>Handouts:</h4>
<p>The slides used in this tutorial session can be downloaded here: <a href="/assets/courseware/v1/047cd0ffc96e4e45ea75c956b88a03e7/asset-v1:MITx+15.481x+1T2021+type@asset+block/Recitation_7.pdf" target="[object Object]">[SLIDES]</a></p>
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<h2 class="hd hd-2 unit-title">Linear Regression - Overview</h2>
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<h2 class="hd hd-2 unit-title">Linear Regression - OLS</h2>
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<h2 class="hd hd-2 unit-title">Linear Regression - OLS Estimation</h2>
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<h2 class="hd hd-2 unit-title">Linear Regression - Accuracy of the Coefficients</h2>
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<h2 class="hd hd-2 unit-title">Linear Regression - Accuracy of the Model</h2>
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<h3 class="hd hd-2">Multiple Regression - Overview</h3>
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<h2 class="hd hd-2 unit-title">Multiple Regression - OLS Estimation</h2>
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<h2 class="hd hd-2 unit-title">Multiple Regression - Accuracy of the Coefficients</h2>
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<h2 class="hd hd-2 unit-title">Multiple Regression - Accuracy of the Model</h2>
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<h3 class="hd hd-2">Multiple Regression - Accuracy of the Model</h3>
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<h2 class="hd hd-2 unit-title">Challenge Question 1</h2>
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(Ungraded) Challenge Question 1
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Suppose you run the following simple linear regressions:</p>
<p>\(\begin{cases} y_i = \beta x_i + \epsilon_i \\ x_i = \delta y_i + \tilde{\epsilon_i} \end{cases}\)</p>
<p>Can we say that \(\hat\delta = \frac{1}{\hat\beta}\) i.e., that \(\hat\beta \times \hat\delta = 1\)?</p>
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<h3 class="hd hd-2">Challenge Question 1</h3>
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<h2 class="hd hd-2 unit-title">Challenge Question 2</h2>
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(Ungraded) Challenge Question 2
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<p>I collect \(n\) datapoints \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\).</p>
<p>An error in my code duplicates the data. This means I now have the following \(2n\) datapoints: \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n), (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\).</p>
<p>I now run a simple linear regression of \(Y\) on \(X\): I hence estimate the model \(Y = \hat\beta_1 X\).</p>
<p>I then realize my mistake and run a simple linear regression of \(Y\) on \(X\) using the non-duplicated data i.e., I use only the \(n\) datapoints \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\): I hence estimate the model \(Y = \hat\beta_2 X\).</p>
<p>How did the glitch in my code affect the following quantities:</p>
<p>\(\bullet\) The estimate \(\hat\beta\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ed9b2873607649b881c232db048cc6e3_2_1">
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<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_2_1" id="input_ed9b2873607649b881c232db048cc6e3_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ed9b2873607649b881c232db048cc6e3_2_1-choice_0-label" for="input_ed9b2873607649b881c232db048cc6e3_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_2_1"> \(\hat\beta_1 = \hat\beta_2\), so duplicating the data did not affect my estimate of \(\beta\).
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<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_2_1" id="input_ed9b2873607649b881c232db048cc6e3_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ed9b2873607649b881c232db048cc6e3_2_1-choice_1-label" for="input_ed9b2873607649b881c232db048cc6e3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_2_1"> \(\hat\beta_1 &gt; \hat\beta_2\)
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<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_2_1" id="input_ed9b2873607649b881c232db048cc6e3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ed9b2873607649b881c232db048cc6e3_2_1-choice_2-label" for="input_ed9b2873607649b881c232db048cc6e3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_2_1"> \(\hat\beta_1 \lt \hat\beta_2\)
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<p>\(\bullet\) The standard error \(SE[\hat\beta]\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ed9b2873607649b881c232db048cc6e3_3_1">
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<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_3_1" id="input_ed9b2873607649b881c232db048cc6e3_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ed9b2873607649b881c232db048cc6e3_3_1-choice_0-label" for="input_ed9b2873607649b881c232db048cc6e3_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_3_1"> \(SE[\hat\beta_1] = SE[\hat\beta_2]\), so duplicating the data did not affect the standard error on my estimate of \(\beta\).
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<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_3_1" id="input_ed9b2873607649b881c232db048cc6e3_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ed9b2873607649b881c232db048cc6e3_3_1-choice_1-label" for="input_ed9b2873607649b881c232db048cc6e3_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_3_1"> \(SE[\hat\beta_1] &gt; SE[\hat\beta_2]\)
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<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_3_1" id="input_ed9b2873607649b881c232db048cc6e3_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ed9b2873607649b881c232db048cc6e3_3_1-choice_2-label" for="input_ed9b2873607649b881c232db048cc6e3_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_3_1"> \(SE[\hat\beta_1] \lt SE[\hat\beta_2]\)
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<p>\(\bullet\) The t-statistic \(t(\hat\beta)\)?</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ed9b2873607649b881c232db048cc6e3_4_1">
<fieldset aria-describedby="status_ed9b2873607649b881c232db048cc6e3_4_1">
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<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_4_1" id="input_ed9b2873607649b881c232db048cc6e3_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="ed9b2873607649b881c232db048cc6e3_4_1-choice_0-label" for="input_ed9b2873607649b881c232db048cc6e3_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_4_1"> \(t(\hat\beta_1) = t(\hat\beta_2)\), so duplicating the data did not affect the t-statistic on my estimate of \(\beta\).
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<div class="field">
<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_4_1" id="input_ed9b2873607649b881c232db048cc6e3_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ed9b2873607649b881c232db048cc6e3_4_1-choice_1-label" for="input_ed9b2873607649b881c232db048cc6e3_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_4_1"> \(t(\hat\beta_1) &gt; t(\hat\beta_2)\)
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<input type="radio" name="input_ed9b2873607649b881c232db048cc6e3_4_1" id="input_ed9b2873607649b881c232db048cc6e3_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ed9b2873607649b881c232db048cc6e3_4_1-choice_2-label" for="input_ed9b2873607649b881c232db048cc6e3_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ed9b2873607649b881c232db048cc6e3_4_1"> \(t(\hat\beta_1) \lt t(\hat\beta_2)\)
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<h3 class="hd hd-2">Challenge Question 2</h3>
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