2c:null 2d:["$","title",null,{"children":"IV Regression: 2SLS Estimation with WebR | Martin C. Arnold | Observable"}] 32: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 ${tex`\beta_1`}.33:T1328,chart = { // Specify the chart’s dimensions. const width = 640; const height = 400; const marginTop = 25; const marginRight = 20; const marginBottom = 35; const marginLeft = 40; // Define the horizontal scale. const x = d3.scaleLinear() .domain([-6, 6]).nice() .range([marginLeft, width - marginRight]); // Define the vertical scale. const y = d3.scaleLinear() .domain([-6.5, 6]).nice() .range([height - marginBottom, marginTop]); // Create the container SVG. const svg = d3.create("svg") .attr("width", width) .attr("height", height) .attr("viewBox", [0, 0, width, height]) .attr("style", "max-width: 100%; height: auto;"); // Add the axes. svg.append("g") .attr("transform", `translate(0,${height - marginBottom})`) .call(d3.axisBottom(x)) .attr("class", "x-axis"); svg.append("g") .attr("transform", `translate(${marginLeft},0)`) .call(d3.axisLeft(y)) .attr("class", "y-axis"); // Append a circle for each data point. svg.append("g") .selectAll("circle") .data(the_data) .join("circle") .filter(d => d.X) .attr("cx", d => x(d.X)) .attr("cy", d => y(d.Y)) .attr("r", 2.5) .attr("fill", "black") .style("fill-opacity", ".7") .style("stroke-width", ".1") .style("stroke-opacity", ".7"); if (showX_hat) { // Illustrate the non-endogenous variation in X svg.append("g") .selectAll(".ivpoints") .data(the_data) .enter().append("circle") .attr("cx", d => x(d.X)) .attr("cy", d => y(d.X_hat)) .attr("r", 2.5) .attr("fill", "orange") .attr("class", "ivpoints") .style("fill-opacity", ".7") .style("stroke-width", ".1") .style("stroke-opacity", ".7"); } let xRange = [-6,6]; // legend const legend = svg.append("g") .attr("class", "legend") .style("font-size", "10pt") .attr("transform", `translate(${width - 3 * marginRight}, 40)`); if (regtype.includes("truth")) { // true relationship let data = d3.range(xRange[0], xRange[1]).map(function(x) { return { x: x, y: beta * x }; }); svg.append("path") .datum(data) .attr("fill", "none") .attr("stroke", "darkgreen") .attr("d", d3.line() .x(function(d) { return x(d.x); }) .y(function(d) { return y(d.y); }) ); legend.append("line") .style("stroke", "darkgreen") .attr("x1", 0).attr("y1", 40) .attr("x2", 15).attr("y2", 40) .attr("stroke-width", 2); legend.append("text") .attr("x", 20).attr("y", 45) .text("truth"); } if (regtype.includes("2SLS")) { // IV regression line let data = d3.range(xRange[0], xRange[1]).map(function(x) { return { x: x, y: beta_1_est_IV * x + beta_0_est_IV }; }); svg.append("path") .datum(data) .attr("fill", "none") .attr("stroke", "blue") .attr("d", d3.line() .x(function(d) { return x(d.x); }) .y(function(d) { return y(d.y); }) ); 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"); } if (regtype.includes("Naive OLS")) { // OLS regression line let data = d3.range(xRange[0], xRange[1]).map(function(x) { return { x: x, y: beta_1_est * x + beta_0_est }; }); svg.append("path") .datum(data) .attr("fill", "none") .attr("stroke", "red") .attr("stroke-width", 1.5) .attr("d", d3.line() .x(function(d) { return x(d.x); }) .y(function(d) { return y(d.y); }) ); 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"); } // labels svg.append("text") .attr("x", width - marginRight) .attr("y", height - 10) .text("X") .style("font-size", "14px") .attr("alignment-baseline", "middle"); svg.append("text") .attr("transform", `rotate(-90, ${0}, ${marginTop})`) .attr("x", -5) .attr("y", 35) .text("Y") .style("font-size", "14px") .attr("alignment-baseline", "middle"); return svg.node(); }34:T48d,mutable the_data = { const ret = await webR.evalR(` set.seed(1234) n <- 1000 mu <- c(0, 0) sigma <- matrix(c(1, ${cor}, ${cor}, 1), ncol = 2) chol_decomp <- chol(sigma) errors <- matrix(rnorm(2 * n), ncol = 2) %*% chol_decomp colnames(errors) <- c("u", "v") Z <- rnorm(n, sd = 0.25) X <- ${gamma} * Z + errors[, "v"] Y <- ${beta} * X + errors[, "u"] # first stage first_stage <- lm(X ~ Z) X_hat <- fitted(first_stage) # second stage second_stage <- lm(Y ~ X_hat) beta0_est_IV <- coefficients(second_stage)[1] beta1_est_IV <- coefficients(second_stage)[2] the_data <- data.frame(X, Y, X_hat) `) await webR.evalR(` # estimate the model by OLS OLS_model <- lm(Y ~ X, data = the_data) beta0_est <- coefficients(OLS_model)[1] beta1_est <- coefficients(OLS_model)[2] `) // const data = await ret.toJs(); const the_data = data.values[0].values.map((_, idx) => { return { X: data.values[0].values[idx], Y: data.values[1].values[idx], X_hat: data.values[2].values[idx], } }); return the_data; }35:[{"id":"598775016f72a4f0563a36051a30f5195329e99f8d16911de73a4ca57f410e3b41df87b21223d75169be3d42834892133be763f823759cfd2c837ccd04eb5703","href":"https://static.observableusercontent.com/files/598775016f72a4f0563a36051a30f5195329e99f8d16911de73a4ca57f410e3b41df87b21223d75169be3d42834892133be763f823759cfd2c837ccd04eb5703","name":"DALLE_IV.webp","type":"image/webp","lastModified":1714733477737.299,"size":556024}] 30:{"id":"2bb3f4b401aa617b","slug":"ivreg","title":"IV Regression: 2SLS Estimation with WebR","thumbnail":"3b606b0414f931834b92a9a8e309f2a45b671051368d3a840464a89de9b83091","access_level":"public","create_time":"$D2024-04-30T14:43:37.977Z","update_time":"$D2024-07-03T11:20:44.195Z","view_time":null,"first_public_time":"$D2024-05-06T08:12:58.656Z","has_unpublished_edits":null,"fork":{"id":"6d7c773a6d3673b0","slug":"d3-scatterplot","title":"D3 scatterplot","owner":{"id":"f35c755083683fe5","login":"observablehq","name":"Observable"}},"stars":1,"starred":false,"owner":{"type":"team","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":1,"collection":null,"type":"notebook","body":{"title":"IV Regression: 2SLS Estimation with WebR","cells":[{"id":0,"value":"# IV Regression: 2SLS Estimation with `WebR`","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$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":"$32","pinned":false,"hidden":false,"mode":"md","author":"$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":"$undefined"},{"id":159,"value":"viewof beta = Inputs.range([-2, 2], {label: tex`\\beta_1:`, step: .01, value: beta_val})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":526,"value":"viewof gamma = Inputs.range([-10, 10], {label: tex`\\gamma :`, step: .01, value: gamma_val})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":145,"value":"viewof cor = Inputs.range([-.99, .99], {label: tex`\\rho(u,v):`, step: .01, value: cor_val})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":375,"value":"viewof showX_hat = Inputs.toggle({label: tex`\\text{Plot } X_i \\text{ vs. } \\widehat{X}_i?`, value: false})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":368,"value":"viewof regtype = Inputs.checkbox([\"truth\", \"2SLS\", \"Naive OLS\"], {label: tex`\\text{Plot lines:}`, value: [\"truth\", \"2SLS\", \"Naive OLS\"]})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$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":"RCodeSim = `\nset.seed(1234)\nn <- 1000\nerrors <- rnorm_multi(\n n, mu = c(0, 0), sd = c(1, 1),\n r = ${cor}, varnames = c(\"u\", \"v\")\n)\nZ <- rnorm(n, sd = .25)\ngamma <- ${gamma}\nbeta <- ${beta}\nX <- gamma * Z + errors$v\nY <- beta * X + errors$u\nthe_data <- data.frame(X, Y)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":633,"value":"### Naive OLS Estimation (dashed regression line)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":277,"value":"Here's what happens in the `R` console when we estimate the model as
\n\n`> mod <- lm(formula = Y ~ X, data = the_data)`
\n\nand then summarize the results using
\n\n`> summary(mod)`:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":252,"value":"{\n mutable the_data\n let { output } = await shelter.captureR(\"summary(OLS_model)\", { withAutoprint: true, captureConditions: false });\n await shelter.purge();\n return output.map((line) => line.data).join('\\n');\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":623,"value":"The naive OLS estimate for ${tex`\\beta_1`} is ${tex`${d3.format(\".5f\")(beta_1_est)}`}. Note that this estimate is close to the true value of ${tex`${beta}`} if ${tex`\\rho\\approx 0`} but is biased for pronounced correlation between ${tex`u_i`} and ${tex`v_i`} (current setting: ${tex`\\rho=${cor}`}). \n\nUnfortunately, the bias also results in invalid p-values, which may even lead us to the conclusion that ${tex`\\beta_1`} is significantly different from zero when ${tex`\\beta_1=0`}!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":806,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":636,"value":"### 2SLS Estimation, step-by-step (purple regression line)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":838,"value":"Here's how to implement the IV regression estimator using 2SLS in `R`.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":638,"value":"`# first stage`
\n`> first_stage <- lm(X ~ Z)`
\n`> X_hat <- fitted(first_stage)`
\n\n`# second stage`
\n`> second_stage <- lm(Y ~ X_hat)`
\n`> summary(second_stage)`","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":751,"value":"{ // OLS\n mutable the_data; // I use a mutable state to re-run the cell on data update\n let rcode = `summary(second_stage)`;\n let { output } = await shelter.captureR(rcode, { withAutoprint: true, captureConditions: false });\n return output.map((line) => line.data).join('\\n'); await shelter.purge();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":801,"value":"Note that the 2SLS estimate of ${tex`\\beta_1`} is close to the true value of ${tex`${beta}`} if ${tex`Z`} is a valid instrument for ${tex`X`}. However, the inferential statistics for the second stage regression reported by `summary()` are ***invalid*** because they do not account for the uncertainty from the estimation of the first stage!\n\nA convenient alternative is `ivreg::ivreg()`. Let's redo the analysis with `ivreg()`:
\n\n`iv_model <- ivreg(Y ~ X | Z)`
\n`summary(iv_model)`","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":797,"value":"{ // with ivreg\n mutable the_data; // I use a mutable state to re-run the cell on data update\n let rcode = `iv_model <- ivreg(Y ~ X | Z); summary(iv_model)`;\n let { output } = await shelter.captureR(rcode, { withAutoprint: true, captureConditions: false });\n return output.map((line) => line.data).join('\\n'); await shelter.purge();\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":916,"value":"`ivreg(Y ~ X | Z)` yields the same parameter estimates as the manual 2SLS implementation above but `summary(iv_model)` gives different, asymptotically valid p-values. It also tests if ${tex`Z`} is a weak instrument (`Weak instruments`) and whether ${tex`X`} is exogenous (`Wu-Hausmann`). ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$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":327,"value":"beta_1_est_IV = {\n mutable the_data; \n const ret = await webR.evalR(\"beta1_est_IV\");\n const data = await ret.toJs();\n const out = data.values[0]\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":324,"value":"// Get dataset from R, convert to d3 style data, and plot\nbeta_0_est_IV = {\n mutable the_data; \n const ret = await webR.evalR(\"beta0_est_IV\");\n const data = await ret.toJs();\n const out = data.values[0]\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":305,"value":"// Get dataset from R, convert to d3 style data, and plot\nbeta_1_est = {\n mutable the_data; \n const ret = await webR.evalR(\"beta1_est\");\n const data = await ret.toJs();\n const out = data.values[0]\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":284,"value":"// Get dataset from R, convert to d3 style data, and plot\nbeta_0_est = {\n mutable the_data; \n const ret = await webR.evalR(\"beta0_est\");\n const data = await ret.toJs();\n const out = data.values[0]\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":138,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$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":"$undefined"},{"id":96,"value":"webR = {\n // Set up webR\n let { WebR, ChannelType } = await import('https://webr.r-wasm.org/latest/webr.mjs');\n let webR = new WebR({ channelType: ChannelType.PostMessage });\n await webR.init();\n await webR.installPackages(['ivreg'])\n await webR.evalR('set.seed(1234); library(ivreg)')\n mutable loaded = true;\n return webR;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$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":"$undefined"},{"id":490,"value":"{\n document.getElementById('loadedContent').style.display = loaded ? 'block' : 'none';\n document.getElementById('loadingContent').style.display = loaded ? 'none' : 'block';\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":710,"value":"{\n weak;\n mutable beta_val = 0.5; \n mutable gamma_val = .2;\n mutable cor_val = .6;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":733,"value":"{\n exogeneous;\n mutable beta_val = 0.5; \n mutable gamma_val = 2;\n mutable cor_val = 0;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":904,"value":"{\n beta1zero;\n mutable beta_val = 0; \n mutable gamma_val = 2;\n mutable cor_val = -.8;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$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"}} 31:"$30:body"