class: right, middle, inverse, title-slide .title[ # Lecture 1 - Linear Regression ] .subtitle[ ## <br> Econometrics 1 ] .author[ ### Vincent Bagilet ] .date[ ### 2024-09-17 ] --- class: right, middle # Introduction ??? I introduce myself, they introduce themselves (ask them what they'd like to do later) --- class: titled, middle # Objectives - Equip you with tools to quantitatively explore relevant social science questions - Focus on intuitions - Actual data sets, replications and simulated data ??? - Simulated data enable to know how the data is generated and therefore to study the influence of various factors and assumptions on the outcome of our analysis - Provide you with tools to check hypotheses yourself --- class: right, middle, inverse # Taking Theory to Data --- class: titled, middle # Theory - **Theory**: the output `\(Y_i\)` of a firm `\(i\)` is a function `\(F\)` of the capital `\(K_i\)` and labor `\(L_i\)` it employs - `\(\forall i \in \{1, .., n\}\)`, `$$Y_i = F(K_i, L_i) = \beta_0K^{\beta_{1}}L^{\beta_{2}}$$` - `\(\beta_0\)`: total factor of productivity - `\(\beta_1\)` and `\(\beta_2\)`:output elasticity of capital and labor --- class: titled, middle # To the Data - Goals: - Assess the validity of this theory - Quantify its parameters using observational data (**sign** and **magnitude**) - Then, make general claims about the population : **inference** --- class: titled, middle # Steps of From Theory to Conclusion Theory `\(\longrightarrow\)` Modelling `\(\longleftrightarrow\)` Data `\(\longrightarrow\)` Estimations `\(\longleftrightarrow\)` Corrections `\(\longrightarrow\)` Interpretation --- class: titled, middle # From theory to a model - `\(F\)` can be seen as a simplification of an unknown relationship called the **data generating process** - All possible data generated by this relationship: the **population** - For practical reasons, we only observe a subset of the population: a **sample** - We want the sample to be drawn **randomly** so that it is **representative** of the population --- class: titled, middle layout:true # An econometric model --- - Based on our theory, we write an econometric model, `\(\forall i\)`: `$$Y_i = f(K_i, L_i) + \epsilon_i = \beta_0 K_i^{\beta_1} L_i^{\beta_2} + \epsilon_i$$` - `\(\epsilon\)` is an **error** term: our model does not perfectly explain the output - We can linearize this model by "taking the log": $$ log(Y_i) = \beta_0 + \beta_1 \log(K_i) + \beta_2 \log(L_i) + e_i \qquad \forall i \in \{1, .., n\}$$ --- class: titled, middle $$ log(Y_i) = \beta_0 + \beta_1 \log(K_i) + \beta_2 \log(L_i) + e_i \qquad \forall i \in \{1, .., n\}$$ <br> - The **logarithm** transformation provides meaningful economic interpretation (*ie* elasticities) - The **linearity** allows a simple interpretation of the parameters (*ie* unit-increase) - The **additivity** ensures that parameters can be interpreted separately (*ie* ceteris paribus) --- class: titled, middle layout: false # Getting data - We collect **proxy variables** for the more abstract theoretical variables: - `\(Y \to y\)`: value added - `\(K \to k\)`: value of the capital stock - `\(L \to l\)`: number of workers - Imperfect measures - Scale does not matter for estimation, only for interpretation --- class: titled, middle layout: false # What's in the error? - What cannot be explained by the model - **Misspecification**: *eg* increasing marginal returns, non linear relationship, heterogeneity - **Missing inputs**: *eg* sector effects, non-economic factors and non-observables - **Measurement error**: *eg* number of workers `\(\neq\)` labor productivity --- class: titled, middle # Estimation - We have a model, we want to *estimate* a value for the parameters - Find parameters values that **minimize the error**: - *ie* the difference between what the model predicts and the observed values - *ie* precisely: the sum of squares of the residuals - Here, use Ordinary Least Squares (OLS) --- class: titled, middle # Interpretation and inference - For instance, `\(\forall i \in \{1, .., n\}\)` we find: `$$\log(\hat{y_i}) = \hat{\beta_0} + \hat{\beta_1} \log(k_i) + \hat{\beta_2}\log(l_i) = 0.45 + 0.7\log(k_i) + 1.2\log(l_i)$$` - Estimated parameters do not hold for every single firm but are average over the sample - The explanatory power of the model can be computed using the estimated residuals --- # Summary - Econometrics `\(\simeq\)` methods to **approximate and interpret the data generation process**: -- 1. Choose proxy variables for our theoretical quantities and collect a sample of observations -- 1. Assume a functional form for the empirical model -- 1. Fit the model to a sample of observations -- 1. Interpret the sign and magnitude of the estimated parameters and draw conclusions --- class: right, middle, inverse # Logistics --- # Website <iframe src="https://vincentbagilet.github.io/metrics_m1_2024/" width="100%" height="450px" data-external="1"></iframe> <center> https://vincentbagilet.github.io/metrics_m1_2024/ </center> --- class: titled, middle # Structure - I will teach the 4 first lectures (<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M64 112c-8.8 0-16 7.2-16 16v22.1L220.5 291.7c20.7 17 50.4 17 71.1 0L464 150.1V128c0-8.8-7.2-16-16-16H64zM48 212.2V384c0 8.8 7.2 16 16 16H448c8.8 0 16-7.2 16-16V212.2L322 328.8c-38.4 31.5-93.7 31.5-132 0L48 212.2zM0 128C0 92.7 28.7 64 64 64H448c35.3 0 64 28.7 64 64V384c0 35.3-28.7 64-64 64H64c-35.3 0-64-28.7-64-64V128z"/></svg> [vincent.bagilet@ens-lyon.fr](vincent.bagilet@ens-lyon.fr)) - Gaetan Bakalli: 4 last ones (<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M64 112c-8.8 0-16 7.2-16 16v22.1L220.5 291.7c20.7 17 50.4 17 71.1 0L464 150.1V128c0-8.8-7.2-16-16-16H64zM48 212.2V384c0 8.8 7.2 16 16 16H448c8.8 0 16-7.2 16-16V212.2L322 328.8c-38.4 31.5-93.7 31.5-132 0L48 212.2zM0 128C0 92.7 28.7 64 64 64H448c35.3 0 64 28.7 64 64V384c0 35.3-28.7 64-64 64H64c-35.3 0-64-28.7-64-64V128z"/></svg> [bakalli@em-lyon.com](bakalli@em-lyon.com)) - Lucile Laugrette: a R TD (<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M64 112c-8.8 0-16 7.2-16 16v22.1L220.5 291.7c20.7 17 50.4 17 71.1 0L464 150.1V128c0-8.8-7.2-16-16-16H64zM48 212.2V384c0 8.8 7.2 16 16 16H448c8.8 0 16-7.2 16-16V212.2L322 328.8c-38.4 31.5-93.7 31.5-132 0L48 212.2zM0 128C0 92.7 28.7 64 64 64H448c35.3 0 64 28.7 64 64V384c0 35.3-28.7 64-64 64H64c-35.3 0-64-28.7-64-64V128z"/></svg> [lucile.laugerette@gmail.com ](lucile.laugerette@gmail.com )) --- class: titled, middle # Outline .pull-left[ ### My section 1. Linear regression 1. OLS properties 1. Model specification 1. Covariates selection ] .pull-right[ ### Gaetan Bakalli's section 1. Convergence in probability 1. Asymptotic normality 1. Hypothesis testing - Theory 1. Hypothesis testing - Application ] --- class: titled, middle # A typical lecture 1. Introduce concepts, intuition 1. Exercise and application in R 1. A bit of maths --- # Grading .pull-left[ ### Weekly Quizzes (20%) - Every week, 10 min - Easy => do not aim to penalize you ### Final Exam (40%) Two parts: 1. Computer-based 1. Written test ] .pull-right[ ### Final Project (40%) - In pairs - Research question of your choice - Short written report + notebook with your code ] --- class: right, middle, inverse # R: Why and How? --- class: titled, middle # THE statistical analysis software - Open source - Can do anything linked to data: wrangling, cleaning, analyzing, visualizing, communicating - Huge online community - Packages for anything --- # Massive capabilities ### Websites ([Quarto](https://quarto.org/)) and slides ([Quarto](https://quarto.org/), [Xaringan](https://slides.yihui.org/xaringan/#1)) <iframe src="https://vincentbagilet.github.io/metrics_m1_2024/" width="100%" height="450px" data-external="1"></iframe> --- ### Awesome graphs ([ggplot](https://ggplot2.tidyverse.org/)) .pull-left[ <img src="data:image/png;base64,#images/ggplot_ex_small_multiples.png" width="60%" style="display: block; margin: auto;" /> ] .pull-right[ <img src="data:image/png;base64,#images/rayshader.jpeg" width="2560" /> ] --- 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3.6"],"type":"scatter","mode":"markers","marker":{"autocolorscale":false,"color":"rgba(11,43,81,1)","opacity":1,"size":5.6692913385826778,"symbol":"circle","line":{"width":1.8897637795275593,"color":"rgba(11,43,81,1)"}},"hoveron":"points","showlegend":false,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null}],"layout":{"margin":{"t":34.897637795275585,"r":18.897637795275585,"b":56.091494167214165,"l":49.117646097475671},"paper_bgcolor":"rgba(254,254,241,1)","font":{"color":"rgba(6,22,41,1)","family":"Lato","size":18.596928185969286},"xaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[10.4,45.600000000000001],"tickmode":"array","ticktext":["20","30","40"],"tickvals":[20,30,40],"categoryorder":"array","categoryarray":["20","30","40"],"nticks":null,"ticks":"outside","tickcolor":"rgba(6,22,41,1)","ticklen":4.6492320464923216,"tickwidth":0.74387712743877132,"showticklabels":true,"tickfont":{"color":"rgba(6,22,41,1)","family":"Lato","size":13.947696139476964},"tickangle":-0,"showline":false,"linecolor":null,"linewidth":0,"showgrid":false,"gridcolor":null,"gridwidth":0,"zeroline":false,"anchor":"y","title":{"text":"hwy","font":{"color":"rgba(6,22,41,1)","family":"Lato","size":18.596928185969286}},"hoverformat":".2f"},"yaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[1.3300000000000001,7.2699999999999996],"tickmode":"array","ticktext":["2","3","4","5","6","7"],"tickvals":[2,3,4,5,6,7],"categoryorder":"array","categoryarray":["2","3","4","5","6","7"],"nticks":null,"ticks":"","tickcolor":null,"ticklen":4.6492320464923216,"tickwidth":0,"showticklabels":true,"tickfont":{"color":"rgba(6,22,41,1)","family":"Lato","size":13.947696139476964},"tickangle":-0,"showline":false,"linecolor":null,"linewidth":0,"showgrid":true,"gridcolor":"rgba(199,203,207,1)","gridwidth":0.37193856371938566,"zeroline":false,"anchor":"x","title":{"text":"displ","font":{"color":"rgba(6,22,41,1)","family":"Lato","size":18.596928185969286}},"hoverformat":".2f"},"shapes":[{"type":"rect","fillcolor":null,"line":{"color":null,"width":0,"linetype":[]},"yref":"paper","xref":"paper","x0":0,"x1":1,"y0":0,"y1":1}],"showlegend":false,"legend":{"bgcolor":null,"bordercolor":null,"borderwidth":0,"font":{"color":"rgba(6,22,41,1)","family":"Lato","size":14.505603985056041}},"hovermode":"closest","barmode":"relative","plot_bgcolor":"transparent"},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"d47345211ec":{"x":{},"y":{},"type":"scatter"}},"cur_data":"d47345211ec","visdat":{"d47345211ec":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ### Interactive maps ([Leaflet](https://rstudio.github.io/leaflet/) and [Mapview](https://r-spatial.github.io/mapview/)) <div class="leaflet html-widget html-fill-item" id="htmlwidget-875a517cd178256b69d9" style="width:100%;height:504px;"></div> <script type="application/json" data-for="htmlwidget-875a517cd178256b69d9">{"x":{"options":{"minZoom":1,"maxZoom":52,"crs":{"crsClass":"L.CRS.EPSG3857","code":null,"proj4def":null,"projectedBounds":null,"options":{}},"preferCanvas":false,"bounceAtZoomLimits":false,"maxBounds":[[[-90,-370]],[[90,370]]]},"calls":[{"method":"addProviderTiles","args":["CartoDB.Positron","CartoDB.Positron","CartoDB.Positron",{"errorTileUrl":"","noWrap":false,"detectRetina":false,"pane":"tilePane"}]},{"method":"addProviderTiles","args":["CartoDB.DarkMatter","CartoDB.DarkMatter","CartoDB.DarkMatter",{"errorTileUrl":"","noWrap":false,"detectRetina":false,"pane":"tilePane"}]},{"method":"addProviderTiles","args":["OpenStreetMap","OpenStreetMap","OpenStreetMap",{"errorTileUrl":"","noWrap":false,"detectRetina":false,"pane":"tilePane"}]},{"method":"addProviderTiles","args":["Esri.WorldImagery","Esri.WorldImagery","Esri.WorldImagery",{"errorTileUrl":"","noWrap":false,"detectRetina":false,"pane":"tilePane"}]},{"method":"addProviderTiles","args":["OpenTopoMap","OpenTopoMap","OpenTopoMap",{"errorTileUrl":"","noWrap":false,"detectRetina":false,"pane":"tilePane"}]},{"method":"createMapPane","args":["point",440]},{"method":"addCircleMarkers","args":[45.73360071410955,4.833428242327759,6,null,"ENS",{"crs":{"crsClass":"L.CRS.EPSG3857","code":null,"proj4def":null,"projectedBounds":null,"options":{}},"pane":"point","stroke":true,"color":"#333333","weight":1,"opacity":0.9,"fill":true,"fillColor":"#6666FF","fillOpacity":0.6},null,null,"<div class='scrollableContainer'><table class=mapview-popup id='popup'><tr class='coord'><td><\/td><th><b>Feature ID <\/b><\/th><td>1 <\/td><\/tr><tr><td>1<\/td><th>geometry <\/th><td>sfc_POINT <\/td><\/tr><\/table><\/div>",{"maxWidth":800,"minWidth":50,"autoPan":true,"keepInView":false,"closeButton":true,"closeOnClick":true,"className":""},"1",{"interactive":false,"permanent":false,"direction":"auto","opacity":1,"offset":[0,0],"textsize":"10px","textOnly":false,"className":"","sticky":true},null]},{"method":"addScaleBar","args":[{"maxWidth":100,"metric":true,"imperial":true,"updateWhenIdle":true,"position":"bottomleft"}]},{"method":"addHomeButton","args":[4.833428242327759,45.73360071410955,4.833428242327759,45.73360071410955,true,"ENS","Zoom to ENS","<strong> ENS 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$(document.createElement('div'));\n // append at end of leaflet htmlwidget container\n $(el).append(newDiv);\n //provide ID and style\n newDiv.addClass('lnlt');\n newDiv.css({\n 'position': 'relative',\n 'bottomleft': '0px',\n 'background-color': 'rgba(255, 255, 255, 0.7)',\n 'box-shadow': '0 0 2px #bbb',\n 'background-clip': 'padding-box',\n 'margin': '0',\n 'padding-left': '5px',\n 'color': '#333',\n 'font': '9px/1.5 \"Helvetica Neue\", Arial, Helvetica, sans-serif',\n 'z-index': '700',\n });\n return newDiv;\n }\n\n\n // check for already existing lnlt class to not duplicate\n var lnlt = $(el).find('.lnlt');\n\n if(!lnlt.length) {\n lnlt = addElement();\n\n // grab the special div we generated in the beginning\n // and put the mousmove output there\n\n map.on('mousemove', function (e) {\n if (e.originalEvent.ctrlKey) {\n if (document.querySelector('.lnlt') === null) lnlt = addElement();\n lnlt.text(\n ' lon: ' + (e.latlng.lng).toFixed(5) +\n ' | lat: ' + (e.latlng.lat).toFixed(5) +\n ' | zoom: ' + map.getZoom() +\n ' | x: ' + L.CRS.EPSG3857.project(e.latlng).x.toFixed(0) +\n ' | y: ' + L.CRS.EPSG3857.project(e.latlng).y.toFixed(0) +\n ' | epsg: 3857 ' +\n ' | proj4: +proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +no_defs ');\n } else {\n if (document.querySelector('.lnlt') === null) lnlt = addElement();\n lnlt.text(\n ' lon: ' + (e.latlng.lng).toFixed(5) +\n ' | lat: ' + (e.latlng.lat).toFixed(5) +\n ' | zoom: ' + map.getZoom() + ' ');\n }\n });\n\n // remove the lnlt div when mouse leaves map\n map.on('mouseout', function (e) {\n var strip = document.querySelector('.lnlt');\n if( strip !==null) strip.remove();\n });\n\n };\n\n //$(el).keypress(67, function(e) {\n map.on('preclick', function(e) {\n if (e.originalEvent.ctrlKey) {\n if (document.querySelector('.lnlt') === null) lnlt = addElement();\n lnlt.text(\n ' lon: ' + (e.latlng.lng).toFixed(5) +\n ' | lat: ' + (e.latlng.lat).toFixed(5) +\n ' | zoom: ' + map.getZoom() + ' ');\n var txt = document.querySelector('.lnlt').textContent;\n console.log(txt);\n //txt.innerText.focus();\n //txt.select();\n setClipboardText('\"' + txt + '\"');\n }\n });\n\n }\n ).call(this.getMap(), el, x, data);\n}","data":null},{"code":"function(el, x, data) {\n return (function(el,x,data){\n var map = this;\n\n map.on('keypress', function(e) {\n console.log(e.originalEvent.code);\n var key = e.originalEvent.code;\n if (key === 'KeyE') {\n var bb = this.getBounds();\n var txt = JSON.stringify(bb);\n console.log(txt);\n\n setClipboardText('\\'' + txt + '\\'');\n }\n })\n }).call(this.getMap(), el, x, data);\n}","data":null}]}}</script> --- ### Interactive apps ([Shiny](https://shiny.posit.co/)) <iframe src="https://kaplanas.shinyapps.io/living_in_the_lego_world/?showcase=0" width="100%" height="480px" data-external="1"></iframe> --- layout: true # Literate programming --- - Combine code and natural language <iframe src="https://vincentbagilet.github.io/causal_exaggeration/RDD.html" width="100%" height="450px" data-external="1"></iframe> --- - R Package: [Quarto](https://quarto.org/) - We will use this in this class - Helpful for economic research. Allows to: - Clearly describe **why** your are doing what you are doing - Store details and information for future-self (data sources, data structure, etc) - Analyse your results - Communicate --- class: titled, middle layout: false .pull-left[ <img src="data:image/png;base64,#images/r4ds.jpg" width="450" style="display: block; margin: auto;" /> ] .pull-right[ <br><br><br><br><br> - Install R and RStudio. - Instructions [here](https://r4ds.hadley.nz/intro#prerequisites). - A great recitation taught by Lucile Laugrette ] --- class: right, middle, inverse # Estimation --- class: titled, middle # Example setting - A version of Mincer's equation: link between education and income - Assume we know the true DGP - Never the case in actual setting but helpful to understand the estimation procedure - I generated variables for education, income (defined by the DGP) and add an error term --- layout: false <img src="data:image/png;base64,#slides_1_linear_regression_files/figure-html/sim_data-1.png" width="70%" style="display: block; margin: auto;" /> --- <img src="data:image/png;base64,#slides_1_linear_regression_files/figure-html/plot_sample_1-1.png" width="70%" style="display: block; margin: auto;" /> --- <img src="data:image/png;base64,#slides_1_linear_regression_files/figure-html/plot_sample_2-1.png" width="70%" style="display: block; margin: auto;" /> --- <img src="data:image/png;base64,#slides_1_linear_regression_files/figure-html/plot_sample_3-1.png" width="70%" style="display: block; margin: auto;" /> --- <img src="data:image/png;base64,#slides_1_linear_regression_files/figure-html/plot_sample_4-1.png" width="70%" style="display: block; margin: auto;" /> --- <img src="data:image/png;base64,#slides_1_linear_regression_files/figure-html/plot_sample_large-1.png" width="70%" style="display: block; margin: auto;" /> --- <img src="data:image/png;base64,#slides_1_linear_regression_files/figure-html/plot_sample_selection-1.png" width="70%" style="display: block; margin: auto;" /> --- class: right, middle, inverse # Maths on the board --- class: right, middle, inverse # Lecture summary --- class: titled, middle # What did we learn today? - How theory and applied research can weave together - Usefulness of regression and its basics - Econometrics `\(\simeq\)` methods to **approximate and interpret the data generation process** --- class: right, middle, inverse # Thanks!