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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Instrumental Variables and Frontdoor Criterion\n",
"\n",
"by Jonas Peters, Niklas Pfister, 04.04.2019\n",
"\n",
"This notebook aims to give you a basic understanding of the instrumental variable approach and the frontdoor criterion and when they can be used to infer causal relations."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Instrumental Variable Model\n",
"\n",
"In the following, let all variables have \n",
"* zero mean, \n",
"* finite second moments, and\n",
"* their joint distribution is absolutely continuous with respect to Lebesgue.\n",
"\n",
"The goal of the instrumental variable approach is to estimate the causal effect of a predictor variable $X$ on a target variable $Y$ if the effect from $X$ to $Y$ is confounded. The idea is to account for this confounding by considering an additional variable $I$ called an instrument. Although there exist numerous extensions, here, we focus on the classical case. We provide two definitions."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First, assume the following SEM\n",
"\\begin{align}\n",
"I &:= N_I\\\\\n",
"H &:= N_H\\\\ \n",
"X &:= I \\gamma + H \\delta_X + N_X\\\\\n",
"Y &:= X \\beta + H \\delta_Y + N_Y.\\\\\n",
"\\end{align}\n",
"(All variables except $Y$ could be multi-dimensional, in which case, they should be written as row vectors: $1 \\times d$.) If all variables are $1$-dimensional, the corresponding DAG looks as follows.\n",
"\\begin{align}\n",
" &\\phantom{0}\\\\\n",
" &\\begin{array}{ccc}\n",
" & & &H & \\\\\n",
" & &\\phantom{abcdefgh}\\overset{\\delta_X}{\\swarrow} & & \\overset{\\delta_Y}{\\searrow}\\phantom{abcdefgh}\\\\\n",
" & & & & \\\\\n",
" I &\\overset{\\gamma}{\\longrightarrow} &X & \\overset{\\beta}{\\longrightarrow} & Y\\\\\n",
" \\end{array}\\\\\n",
" &\\phantom{0}\n",
"\\end{align}\n",
"Here, $I$ is called an instrumental variable for the causal effect from $X$ to $Y$. It is essential that $I$ affects $Y$ only via $X$ (and not directly), and that $I$ and $H$ are independent.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Second, it is possible to define instrumental variables without SEMs, too. Let us therefore write\n",
"\\begin{equation}\n",
"Y = X \\beta + \\epsilon_Y\n",
"\\end{equation}\n",
"(this can always be done). Here, $\\epsilon_Y$ is allowed to depend on $X$ (if there is a confounder $H$ between $X$ and $Y$, this is usually the case). In this linear setting, we then call a variable $I$ an instrumental variable if it satisfies the following two conditions:\n",
"1. $\\operatorname{cov}(X,I)$ is of full rank (relevance)\n",
"2. $\\operatorname{cov}(\\epsilon_Y,I)=0$ (exogenity).\n",
"\n",
"Informally speaking, these conditions again mean that $I$ affects $Y$ ''only through its effect on $X$''."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Estimation\n",
"\n",
"We now want to illustrate how the existence of an instrumental variable $I$ can be used to estimate the causal effect $\\beta$ in the model above. Let us therefore assume that we have received data in matrix form\n",
"* $\\mathbf{Y}$ - the target variable $n \\times 1$ \n",
"* $\\mathbf{X}$ - the covariates $n \\times d$\n",
"* $\\mathbf{I}$ - the instruments $n \\times m$\n",
"\n",
"where $n > \\max(m, d)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We now assume that $I$ is a valid instrument (we come back to this question in Exercise 2 below). To estimate the causal effect of $X$ on $Y$, there are several options of writing down the same estimator. \n",
"\n",
"OPTION 1: The following estimator is sometimes called the generalized methods of moments (GMM)\n",
"$$\n",
"\\hat{\\beta}^{GMM}_n := (\\mathbf{X}^t \\mathbf{I} (\\mathbf{I}^t \\mathbf{I})^{-1} \\mathbf{I}^t \\mathbf{X})^{-1} \\, \\mathbf{X}^t \\mathbf{I} (\\mathbf{I}^t \\mathbf{I})^{-1} \\mathbf{I}^t \\mathbf{Y}\n",
"$$\n",
"\n",
"OPTION 2: \n",
"we can use a so-called 2-stage least squares (2SLS) procedure. Step 1: Regress $X$ on $I$ and compute the corresponding fitted values $\\hat{X}$. Step 2: Regress $Y$ on $\\hat{X}$. Use the regression coefficients from step 2.\n",
"\n",
"The following four exercises go over some of the details of the 2SLS and apply it to a real data set."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 1\n",
"Assume that the data are i.i.d. from the following two structural assignments \n",
"\\begin{align*}\n",
"Y &:= X \\cdot \\beta + \\epsilon_Y \\\\\n",
"X &:= I \\cdot \\gamma + \\epsilon_X,\n",
"\\end{align*}\n",
"where $X$ and $I$ are written as $1 \\times d$ and $1 \\times m$ vectors, respectively. Here, $\\epsilon_X$ and $\\epsilon_Y$ are not necessarily independent, but the instrument $I$ is assumed to satisfy the assumptions 1. and 2. above. \n",
"\n",
"a) Write down conditions on $d$ and $m$ that guarantee that $\\hat{\\beta}^{GMM}_n$ is well-defined (with probability one).\n",
"\n",
"b) Prove that under these conditions, the GMM method is consistent, i.e., $\\hat{\\beta}^{GMM}_n \\rightarrow \\beta$ in probability.\n",
"\n",
"c) Assume $d = m$. Prove that the methods 2SLS and GMM provide the same estimate. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Solution 1"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### End of Solution 1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For illustration, we use the <tt>CollegeDistance</tt> data set from [1] available in the R package <tt>AER</tt>."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"library(AER)\n",
"# load CollegeDistance data set\n",
"data(\"CollegeDistance\")\n",
"# read out relevant variables\n",
"Y <- CollegeDistance$score\n",
"X <- CollegeDistance$education\n",
"I <- CollegeDistance$distance"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This data set consists of $4739$ observations on $14$ variables from high school student survey conducted by the Department of Education in $1980$, with a follow-up in $1986$. In this notebook, we only consider the following variables:\n",
"* $Y$ - base year composite test score. These are achievement tests given to high school seniors in the sample.\n",
"* $X$ - number of years of education.\n",
"* $I$ - distance from closest 4-year college (units are in 10 miles).\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 2\n",
"\n",
"Argue whether the variable $I$ can be used as an instrumental variable to infer the causal effect of $X$ on $Y$. Are there arguments, why it might not be a valid instrument? Hint: You can perform a regression in order to test if there is significant correlation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Solution 2"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### End of Solution 2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 3\n",
"Use 2SLS to estimate the causal effect of $X$ on $Y$ based on the instrument $I$. Compare your results with a standard OLS regression of $Y$ on $X$ (that includes an intercept). What happens to the correlation between $X$ and the residuals in both methods? Which attempt yields smaller variance of residuals?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Solution 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### End of Solution 3"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A slightly different approach to 2SLS is to use the formula\n",
"\n",
"OPTION 3:\n",
"\\begin{equation} \\tag{1}\n",
"\\hat{\\beta}_n = (\\mathbf{I}^t \\mathbf{X})^{-1} \\mathbf{I}^t \\mathbf{Y}.\n",
"\\end{equation}\n",
"\n",
"This formula can be shown to be the same as OPTIONS 1 and 2 if $d = m$ (try proving it). "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 4\n",
"Apply the above estimator (1) to <tt>CollegeDistance</tt> data and compare your result with the one from Exercise 3. (If you have included intercepts in the 2SLS, you need to replace the product moments by sample covariances.)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Solution 4"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### End of Solution 4"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Frontdoor Criterion\n",
"\n",
"Similar to the instrumental variable approach this method aims to estimate the causal effect of a predictor variable $X$ on a target variable $Y$ if the effect from $X$ to $Y$ is confounded. Instead of an instrumental variable, the frontdoor criterion resolves the true causal effect based on a variable $Z$ that lies causally between $X$ and $Y$, also called a mediator. The frontdoor criterion is due to [2] and is commonly stated in terms of a DAG model."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"More precisely, assume we are given a the following DAG\n",
"\\begin{align}\n",
" &\\phantom{0}\\\\\n",
" &\\begin{array}{ccc}\n",
" & &H & & \\\\\n",
" &\\swarrow & & \\searrow & \\\\\n",
" & & & & \\\\\n",
" X &\\longrightarrow &Z & \\longrightarrow & Y\\\\\n",
" \\end{array}\\\\\n",
" &\\phantom{0}\n",
"\\end{align}\n",
"Here, $Z$ is a mediator for the causal effect from $X$ to $Y$. It is essential that confounding $H$ does not directly affect $Z$. \n",
"\n",
"More formally, the frontdoor criterion requires that \n",
"1. $Z$ blocks all directed paths from $X$ to $Y$\n",
"2. There are no unblocked backdoor paths from $X$ to $Z$\n",
"3. $X$ blocks all backdoor paths from $M$ to $Y$\n",
"\n",
"If $Z$ satisfies the frontdoor criterion, the interventional density $p^{do(X:=x)} (y)$ can be computed based on observed quantities as follows\n",
"\\begin{equation*}\n",
" p^{do(X:=x)} (y)=\\int_{z} p(z|x) \\int_{\\tilde{x}}p(y|\\tilde{x}, z) p(\\tilde{x}) \\, d\\tilde{x}\\, dz\n",
"\\end{equation*}\n",
"This formula is also referred to as the *frontdoor adjustment* formula.\n",
"\n",
"The following exercise aims to give some intution on the frontdoor criterion."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Exercise 5\n",
"\n",
"We are interested in determining whether dietary cholesterol has a postive causal effect on the risk of atherosclerosis (narrowing of the artery due to the build up of plaque). One might argue that there is a genetic factor which affects a person's risk of atherosclerosis while at the same time increasing that persons appetite for fatty food. In order to account for this, we plan to use a person's body fat content as a mediating variable.\n",
"\n",
"Assume we are given data from a large observational study consisting of the following measurements:\n",
"\n",
"* Does the person consume large amounts of dietary cholesterol? (yes: $x=1$, no: $x=0$)\n",
"* Did the person get atherosclerosis? (yes: $y=1$, no: $y=0$)\n",
"* Does the person have a high body fat content? (yes: $z=1$, no: $z=0$)\n",
"\n",
"The data is summarized in the following table\n",
"$$\n",
"\\begin{array}{r|c|c}\n",
" & p(x=\\cdot, z=\\cdot) & p(y=1|x=\\cdot, z=\\cdot)\\\\\\hline\n",
"x=0, z=0 & 0.16 & 0.05\\\\\\hline\n",
"x=0, z=1 & 0.04 & 0.1\\\\\\hline\n",
"x=1, z=0 & 0.45 & 0.4\\\\\\hline\n",
"x=1, z=1 & 0.35 & 0.6\\\\\n",
"\\end{array}\n",
"$$\n",
"\n",
"a) Is the body fat content $Z$ a suitable mediating variable that satisfies the frontdoor criterion? Give reasons for and against.\n",
"\n",
"b) Apply the frontdoor criterion to compute $p^{do(X:=1)} (y=1)$ and $p^{do(X:=0)} (y=1)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Solution 5"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### End of Solution 5"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## References\n",
"\n",
"[1] Kleiber, C., A. Zeileis (2008). Applied Econometrics with R. Springer-Verlag New York.\n",
"\n",
"[2] Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82(4):669–710.\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
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