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futureNotes.html
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<h1>Predictions</h1>
<p>Increasingly, <a href="(https://www.aeaweb.org/articles?id=10.1257/aer.p20151023)">economists are getting involved in the prediction game</a>.<a href="#fn1" class="footnoteRef" id="fnref1"><sup>1</sup></a> Up to this point, we’ve talked about the behavior of the mean—what will happen in expectation—but when we make predictions—estimates for an individual from the population—there is some additional uncertainty. This additional uncertainty is called the <em>prediction error</em>. Let’s (formally) discuss this <em>prediction error</em>.</p>
<p>Take our standard linear model</p>
<p><span class="math display">\[ \mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon} \]</span></p>
<p>We can easily make predictions using this model—replace <span class="math inline">\(\boldsymbol{\boldsymbol{\beta}}\)</span> with its estimator <span class="math inline">\(\mathbf{b}\)</span>, drop <span class="math inline">\(\boldsymbol{\varepsilon}\)</span>, and you get <em>in-sample</em> predictions, <em>i.e.</em>,</p>
<p><span class="math display">\[ \hat{\mathbf{y}} = \mathbf{X}\mathbf{b} \]</span></p>
<p>But if you want <em>out-of-sample</em> predictions—predictions for <span class="math inline">\(y\)</span> that use values of <span class="math inline">\(\mathbf{X}\)</span> not observed in our dataset—then you need to include the individual-to-individual variance.</p>
<p>Let’s think about making a prediction for the data series <span class="math inline">\(y^0,\,\mathbf{X}^0\)</span>.</p>
<p>There</p>
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<h1>TODO</h1>
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<li>Prediction intervals</li>
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<li>Bootstrapped standard errors</li>
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<li id="fn1"><p>Macroeconomists have been in it for a while.<a href="#fnref1">↩</a></p></li>
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