2d:null 2e:["$","title",null,{"children":"Bias and Variance in IV Regression | Martin C. Arnold | Observable"}] 33:T93c,The DGP considered in this interactive example is ${tex.block` \begin{aligned} X_i =&\, \gamma \cdot Z_i + v_i,\\ Y_i =&\, \beta_1 \cdot X_i + u_i, \tag{1} \end{aligned} `} where ${tex`Z_i \sim N(0, .25^2)`}. The error terms ${tex`v_i`} and ${tex`u_i`} are ${tex`N(0,1)`}-distributed and are jointly Gaussian with correlation parameter ${tex`\rho`}, ${tex.block` \begin{aligned} \begin{pmatrix} v_i \\ u_i \end{pmatrix} \sim N \left[ \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix} \right]. \end{aligned} `} We are interested in estimating the parameter ${tex`\beta_1`}, determining the effect effect of ${tex`X`} on ${tex`Y`}, using ${tex`N=${the_data.length}`} observations of ${tex`(X_i, Y_i, Z_i)`}. Note how ${tex`X_i`} is a function of the (exogenous) variable ${tex`Z_i`} and the error term ${tex`v_i`}. If the error terms ${tex`(v_i, u_i)'`} in DGP ${tex`(1)`} are correlated (${tex`\rho\neq0`}), then there is correlation between ${tex`X_i`} and ${tex`u_i`}. This implies that ${tex`X`} is an *endogeneous* regressor in the "naive" regression model ${tex.block` Y_i = \beta_0 + \beta_1 X_i + e_i, \quad i = 1,\dots,N, `} and so the ordinary least squares (OLS) estimator of ${tex`\beta_1`} will be *biased and inconsistent*! If ${tex`Z`} has explanatory power for ${tex`X`}, that is ${tex`\gamma\neq0`} with ${tex`\lvert\gamma\rvert`} not too small (${tex`Z`} is *relevant*), and does impact ${tex`Y`} only through ${tex`X`} (${tex`Z`} is *exogenous*), then instrumental variables regression is a way to estimate ${tex`\beta_1`} consistently. **Two-Stage-Least-Squares (2SLS)** 1. ***First stage***: Estimate the linear regression model ${tex.block` X_i = \alpha_0 + \alpha_1 \cdot Z_i + \epsilon_i `} and compute the fitted values ${tex`\widehat{X}_i`}. This step isolates the exogenous variation in ${tex`X_i`} that is due to ${tex`Z_i`}. 2. ***Second stage***: Estimate the effect of ${tex`X`} on ${tex`Y`} using ${tex`\widehat{X}_i`}, the exogenous part of ${tex`X_i`} from Step 1, via the linear regression ${tex.block` Y_i = \delta_0 + \delta_1 \cdot \widehat{X}_i + \varepsilon_i. `} The OLS estimator ${tex`\widehat{\delta}_1`} of ${tex`\delta_1`} is the consistent *2SLS estimator* of ${tex`\beta_1`}.34:T141d,chart = { // Specify the chart’s dimensions. const width = 640; const height = 400; const marginTop = 25; const marginRight = 20; const marginBottom = 40; const marginLeft = 45; // Step 1: Extract the hat_beta_iv values from the_data let hat_beta_iv_values = the_data.map(d => d.hat_beta_iv); // Step 2: Sort the values hat_beta_iv_values.sort(d3.ascending); // Step 3: Compute Q1 and Q3 let q1 = d3.quantile(hat_beta_iv_values, 0.25); let q3 = d3.quantile(hat_beta_iv_values, 0.75); // Step 4: Compute the IQR let iqr = q3 - q1; // Step 5: Define the bounds for non-outliers let lower_bound = q1 - 1.5 * iqr; let upper_bound = q3 + 1.5 * iqr; // Step 6: Filter out the outliers let data_filtered = the_data.filter(d => d.hat_beta_iv >= lower_bound && d.hat_beta_iv <= upper_bound); // filter outliers let sd_IV = d3.deviation(data_filtered.map(d => d.hat_beta_iv)); let sd_OLS = d3.deviation(data_filtered.map(d => d.hat_beta_ols)); let x_min_iv = d3.min(data_filtered.map(d => d.hat_beta_iv)) - sd_IV; let x_max_iv = d3.max(data_filtered.map(d => d.hat_beta_iv)) + sd_IV; let x_min_ols = d3.min(data_filtered.map(d => d.hat_beta_ols)) - sd_OLS; let x_max_ols = d3.max(data_filtered.map(d => d.hat_beta_ols)) + sd_OLS; let x_min = d3.min([x_min_iv, x_min_ols]); let x_max = d3.max([x_max_iv, x_max_ols]); // KDE let n = data_filtered.length; let bandwidth = silverman(data_filtered.map(d => d.hat_beta_iv)); let thresholds = d3.ticks(x_min, x_max, 50); let density = kde(epanechnikov(bandwidth * 2), thresholds, data_filtered.map(d => d.hat_beta_iv)); let density_ols = kde(epanechnikov(bandwidth * 2), thresholds, data_filtered.map(d => d.hat_beta_ols)); let ols_max = d3.max(density_ols.map(d => d.Y)); let iv_max = d3.max(density.map(d => d.Y)); let y_max = d3.max([ols_max, iv_max]); // Define the horizontal scale. const x = d3.scaleLinear() .domain([x_min, x_max]).nice() .range([marginLeft, width - marginRight]); // Define the vertical scale. const y = d3.scaleLinear() .domain([0, y_max + .1 * y_max]).nice() .range([height - marginBottom, marginTop]); // Create the container SVG. const svg = d3.create("svg") .attr("width", width) .attr("height", height) .attr("viewBox", [0, 0, width, height]) .attr("style", "max-width: 100%; height: auto;"); // Add the axes. svg.append("g") .attr("transform", `translate(0,${height - marginBottom})`) .call(d3.axisBottom(x)) .attr("class", "x-axis"); svg.append("g") .attr("transform", `translate(${marginLeft},0)`) .call(d3.axisLeft(y)) .attr("class", "y-axis"); // line function for KDE var line = d3.line() .x(d => x(d.X)) .y(d => y(d.Y)) .curve(d3.curveBasis); // Draw/update KDE svg.selectAll(".KDE") .data([density]) .join( enter => enter.append("path").attr("d", line), update => update .call(update => update .attr("d", line) ), exit => exit.remove() ) .attr("fill", "none") .attr("stroke", "steelblue") .attr("stroke-width", 2); svg.selectAll(".KDE") .data([density_ols]) .join( enter => enter.append("path").attr("d", line), update => update .call(update => update .attr("d", line) ), exit => exit.remove() ) .attr("fill", "none") .attr("stroke", "red") .attr("stroke-width", 2); // true coefficient line svg.append("line") .style("stroke", "black") .style("stroke-width", 1) .attr("x1", x(beta)) .attr("x2", x(beta)) .attr("y1", y(0)) .attr("y2", y(y_max * 1.1)); // labels svg.append("text") .attr("x", width - marginRight) .attr("y", height - 10) .text(`β̂₁`) .style("font-size", "14px") .attr("alignment-baseline", "middle"); svg.append("text") .attr("transform", `rotate(-90, ${0}, ${marginTop})`) .attr("x", -45) .attr("y", 35) .text("density") .style("font-size", "14px") .attr("alignment-baseline", "middle"); svg.append("text") .attr("x", x(beta) + 10) .attr("y", y(y_max * 1.1)) .text("β₁") .style("font-size", "14px") .attr("alignment-baseline", "middle"); // legend const legend = svg.append("g") .attr("class", "legend") .style("font-size", "10pt") .attr("transform", `translate(${width - 3 * marginRight}, 40)`); legend.append("line") .style("stroke", "steelblue") .attr("x1", 0).attr("y1", 0) .attr("x2", 15).attr("y2", 0) .attr("stroke-width", 2); legend.append("text") .attr("x", 20).attr("y", 5) .text("IV"); legend.append("line") .style("stroke", "red") .attr("x1", 0).attr("y1", 20) .attr("x2", 15).attr("y2", 20) .attr("stroke-width", 2); legend.append("text") .attr("x", 20).attr("y", 25) .text("OLS"); return svg.node(); }35:[{"id":"598775016f72a4f0563a36051a30f5195329e99f8d16911de73a4ca57f410e3b41df87b21223d75169be3d42834892133be763f823759cfd2c837ccd04eb5703","href":"https://static.observableusercontent.com/files/598775016f72a4f0563a36051a30f5195329e99f8d16911de73a4ca57f410e3b41df87b21223d75169be3d42834892133be763f823759cfd2c837ccd04eb5703","name":"DALLE_IV.webp","type":"image/webp","lastModified":1718784451809.819,"size":556024}] 31:{"id":"ba2573c3c6053a90","slug":"bias-and-variance-in-iv-regression","title":"Bias and Variance in IV Regression","thumbnail":"3b606b0414f931834b92a9a8e309f2a45b671051368d3a840464a89de9b83091","access_level":"public","create_time":"$D2024-06-19T08:07:31.809Z","update_time":"$D2024-07-02T20:17:40.686Z","view_time":null,"first_public_time":"$D2024-07-02T15:00:46.596Z","has_unpublished_edits":null,"fork":{"id":"2bb3f4b401aa617b","slug":"ivreg","title":"IV Regression: 2SLS Estimation with WebR","owner":{"id":"a15f09cdacc76604","login":"mca91","name":"Martin C. 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Arnold","avatar_url":"https://avatars-next.observableusercontent.com/32a29503786d2aa1a44c308c128c4358cc9b14b88b44d07021d13aec19546bd4","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Bias and Variance in IV Regression","cells":[{"id":0,"value":"# Bias and Variance in IV Regression","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1154,"value":"## (with `WebR`)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":420,"value":"\n

\n Starting WebR... this takes a few seconds ⏳.\n

\n ✅ WebR is ready 👍!\n

","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"webrinfo","database":"$undefined"},{"id":855,"value":"The following packages are available to the `WebR` session backing this notebook:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":844,"value":"{\n let { output } = await shelter.captureR(\"(.packages()) \", { withAutoprint: true, captureConditions: false });\n await shelter.purge();\n return output.map((line) => line.data).join('\\n');\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":728,"value":"*Note*: All `R` chunks in this notebook *update dynamically* as you adjust the DGP! The same holds for the output chunks, which are actual `R` outputs generated by [`WebR`](https://docs.r-wasm.org/webr/latest/)! ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":765,"value":"html`

\n ${await FileAttachment(\"DALLE_IV.webp\").image({\n style: \"border-radius: 5%; box-shadow: 8px 8px 8px;\",\n width: 350,\n title: \"DALL-E envisions lecture on instrumental variables regression\"\n })}\n

How DALL-E envisions a lecture on instrumental variables regression

`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":511,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":494,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":598,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":686,"value":"viewof weak = Inputs.button(\"Preset: Z is a weak instrument\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":736,"value":"viewof exogeneous = Inputs.button(\"Preset: X is exogenous\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":901,"value":"viewof beta1zero = Inputs.button(\"Preset: Z is a 'good' instrument (and we need it)\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1020,"value":"viewof N = Inputs.number({\n value: 1000,\n min: 100,\n max: 1000,\n step: 1,\n label: tex`N:`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1009,"value":"viewof n = Inputs.number({\n value: 100,\n min: 10,\n max: 1000,\n step: 1,\n label: tex`n:`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":996,"value":"viewof beta = Inputs.number({\n value: beta_val,\n min: -2,\n max: 2,\n step: 0.01,\n label: tex`\\beta_1:`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":526,"value":"viewof gamma = Inputs.number([-10, 10], {label: tex`\\gamma :`, step: .01, value: gamma_val})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":145,"value":"viewof cor = Inputs.number([-.99, .99], {label: tex`\\rho(u,v):`, step: .01, value: cor_val})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":648,"value":"Below is the code for generating the data in `R`, given the above parameter settings.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":164,"value":"`library(faux) # install.packages(\"faux\")`
\n`set.seed(1234)`
\n`n <- ${n}`
\n`errors <- rnorm_multi(`
\n `n, mu = c(0, 0), sd = c(1, 1),`
\n `r = ${cor}, varnames = c(\"u\", \"v\")`
\n`)`
\n`Z <- rnorm(n, sd = .25)`
\n`X <- ${gamma} * Z + errors$v`
\n`Y <- ${beta} * X + errors$u`
\n`the_data <- data.frame(X, Y)`","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"simcode","database":"$undefined"},{"id":930,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":924,"value":"### Technical stuff","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":986,"value":"function kde(kernel, thresholds, data) {\n return thresholds.map(t => ({\n X: t,\n Y: d3.mean(data, d => kernel(t - d))\n }));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":988,"value":"function epanechnikov(bandwidth) {\n return x => Math.abs(x /= bandwidth) <= 1 ? 0.75 * (1 - x * x) / bandwidth : 0;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1089,"value":"function silverman(values) {\n\n values.sort(d3.ascending);\n \n // quantiles\n let q1 = d3.quantile(values, 0.25);\n let q3 = d3.quantile(values, 0.75);\n let iqr = q3 - q1;\n\n // bounds\n let lower_bound = q1 - 1.5 * iqr;\n let upper_bound = q3 + 1.5 * iqr;\n\n // standard deviation\n let mean = d3.mean(values);\n let sd = Math.sqrt(d3.mean(values.map(v => Math.pow(v - mean, 2))));\n\n // bandwidth\n let n = values.length;\n let bandwidth = 0.9 * Math.min(sd, iqr / 1.34) * Math.pow(n, -1 / 5);\n\n return bandwidth;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":138,"value":"mutable the_data = { \n \n const ret = await webR.evalR(`\n beta_iv <- function(Y, X, Z) {\n cov(Z,Y)/cov(Z,X)\n }\n\n the_data <- replicate(n = ${N}, {\n\n n <- ${n}\n \n mu <- c(0, 0)\n sigma <- matrix(c(1, ${cor}, ${cor}, 1), ncol = 2) \n chol_decomp <- chol(sigma)\n errors <- matrix(rnorm(2 * n), ncol = 2) %*% chol_decomp\n colnames(errors) <- c(\"u\", \"v\")\n \n Z <- rnorm(n, sd = 0.25)\n \n X <- ${gamma} * Z + errors[, \"v\"]\n Y <- ${beta} * X + errors[, \"u\"]\n \n return(\n c(beta_iv(Y = Y, X = X, Z = Z), cov(X,Y)/var(X))\n )\n \n }, )\n\n the_data <- as.data.frame(t(the_data))\n\n `)\n\n \n // \n const data = await ret.toJs(); \n const the_data = data.values[0].values.map((_, idx) => {\n return {\n hat_beta_iv: data.values[0].values[idx],\n hat_beta_ols: data.values[1].values[idx]\n }\n });\n \n return the_data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":427,"value":"mutable loaded = false","pinned":false,"hidden":true,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":96,"value":"webR = {\n // Set up webR\n let { WebR, ChannelType } = await import('https://webr.r-wasm.org/latest/webr.mjs');\n let webR = new WebR({ channelType: ChannelType.PostMessage });\n await webR.init();\n await webR.evalR('set.seed(1234)');\n mutable loaded = true;\n return webR;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":98,"value":"shelter = {\n return new webR.Shelter();\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":490,"value":"{\n document.getElementById('loadedContent').style.display = loaded ? 'block' : 'none';\n document.getElementById('loadingContent').style.display = loaded ? 'none' : 'block';\n}","pinned":true,"hidden":true,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":710,"value":"{\n weak;\n mutable beta_val = 0.5; \n mutable gamma_val = .2;\n mutable cor_val = .6;\n}","pinned":true,"hidden":true,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":733,"value":"{\n exogeneous;\n mutable beta_val = 0.5; \n mutable gamma_val = 2;\n mutable cor_val = 0;\n}","pinned":true,"hidden":true,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":904,"value":"{\n beta1zero;\n mutable beta_val = 0; \n mutable gamma_val = 2;\n mutable cor_val = -.8;\n}","pinned":true,"hidden":true,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":678,"value":"mutable beta_val = 0.5;","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":697,"value":"mutable cor_val = -.8;","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":701,"value":"mutable gamma_val = 2;","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 32:"$31:body"