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Increasingly, knowledge about causal inference is expected throughout the professional world. It is no longer simply something that academics sit around and debate. It is crucial knowledge for making business decisions as well as for interpreting policy. Finally, this book is written for people very early in their careers, be they undergraduates, graduate students, or newly minted PhDs.

Yet so often people make this kind of mistake when naively interpreting simple correlations. But weirdly enough, sometimes there are causal relationships between two things and yet no observable correlation. Now that is definitely strange.

How can one thing cause another thing without any discernible correlation between the two things? Consider this example, which is illustrated in Figure 1. A sailor is sailing her boat across the lake on a windy day. As the wind blows, she counters by turning the rudder in such a way so as to exactly offset the force of the wind. Back and forth she moves the rudder, yet the boat follows a straight line across the lake.

Hers is broken! Figure 1. Just because there is no observable relationship does not mean there is no causal one. Imagine that instead of perfectly countering the wind by turning the rudder, she had instead flipped a coin—heads she turns the rudder left, tails she turns the rudder right. What do you think this man would have seen if she was sailing her boat according to coin flips?

If she randomly moved the rudder on a windy day, then he would see a sailor zigzagging across the lake. Why would he see the relationship if the movement were randomized but not be able to see it otherwise? Because the sailor is endogenously moving the rudder in response to the unobserved wind. This sounds like a silly example, but in fact there are more serious versions of it.

Consider a central bank reading tea leaves to discern when a recessionary wave is forming. Seeing evidence that a recession is emerging, the bank enters into open-market operations, buying bonds and pumping liquidity into the economy. Insofar as these actions are done optimally, these open-market operations will show no relationship whatsoever with actual output.

In fact, in the ideal, banks may engage in aggressive trading in order to stop a recession, and we would be unable to see any evidence that it was working even though it was!

Human beings engaging in optimal behavior are the main reason correlations almost never reveal causal relationships, because rarely are human beings acting randomly. And as we will see, it is the presence of randomness that is crucial for identifying causal effect. Optimization Makes Everything Endogenous Certain presentations of causal inference methodologies have sometimes been described as atheoretical, but in my opinion, while some practitioners seem comfortable flying blind, the actual methods employed in causal designs are always deeply dependent on theory and local institutional knowledge.

It is my firm belief, which I will emphasize over and over in this book, that without prior knowledge, estimated causal effects are rarely, if ever, believable. Prior knowledge is required in order to justify any claim of a causal finding.

And economic theory also highlights why causal inference is necessarily a thorny task. Let me explain. The latter is also sometimes called observational data. Experimental data is collected in something akin to a laboratory environment.

In a traditional experiment, the researcher participates actively in the process being recorded. In many observational studies, you collect data about what happened previously, as opposed to collecting data as it happens, though with the increased use of web scraping, it may be possible to get observational data closer to the exact moment in which some action occurred.

But regardless of the timing, the researcher is a passive actor in the processes creating the data itself. She observes actions and results but is not in a position to interfere with the environment in which the units under consideration exist. This is the most common form of data that many of us will ever work with. Economic theory tells us we should be suspicious of correlations found in observational data.

In observational data, correlations are almost certainly not reflecting a causal relationship because the variables were endogenously chosen by people who were making decisions they thought were best. In pursuing some goal while facing constraints, they chose certain things that created a spurious correlation with other things. And we see this problem reflected in the potential outcomes model itself: a correlation, in order to be a measure of a causal effect, must be based on a choice that was made independent of the potential outcomes under consideration.

Yet if the person is making some choice based on what she thinks is best, then it necessarily is based on potential outcomes, and the correlation does not remotely satisfy the conditions we need in order to say it is causal. To put it as bluntly as I can, economic theory says choices are endogenous, and therefore since they are, the correlations between those choices and outcomes in the aggregate will rarely, if ever, represent a causal effect.

Now we are veering into the realm of epistemology. Identifying causal effects involves assumptions, but it also requires a particular kind of belief about the work of scientists. Credible and valuable research requires that we believe that it is more important to do our work correctly than to try and achieve a certain outcome e.

The foundations of scientific knowledge are scientific methodologies. True scientists do not collect evidence in order to prove what they want to be true or what others want to believe. That is a form of deception and manipulation called propaganda, and propaganda is not science.

Rather, scientific methodologies are devices for forming a particular kind of belief. Scientific methodologies allow us to accept unexpected, and sometimes undesirable, answers. They are process oriented, not outcome oriented. And without these values, causal methodologies are also not believable.

Example: Identifying Price Elasticity of Demand One of the cornerstones of scientific methodologies is empirical analysis. The first step in conducting an empirical economic analysis is the careful formulation of the question we would like to answer.

In some cases, we would like to develop and test a formal economic model that describes mathematically a certain relationship, behavior, or process of interest. Those models are valuable insofar as they both describe the phenomena of interest and make falsifiable testable predictions.

A prediction is falsifiable insofar as we can evaluate, and potentially reject, the prediction with data. One clear issue we immediately face is regarding the functional form of the model, or how to describe the relationships of the variables we are interested in through an equation. Another important issue is how we will deal with variables that cannot be directly or reasonably observed by the researcher, or that cannot be measured very well, but which play an important role in our model.

A generically important contribution to our understanding of causal inference is the notion of comparative statics. Comparative statics are theoretical descriptions of causal effects contained within the model. If they were changing, then they would be correlated with the variable of interest and it would confound our estimation.

Policy-makers and business managers have a natural interest in learning the price elasticity of demand because knowing it enables firms to maximize profits and governments to choose optimal taxes, and whether to restrict quantity altogether [Becker et al.

But the problem is that we do not observe demand curves, because demand curves are theoretical objects. More specifically, a demand curve is a collection of paired potential outcomes of price and quantity. We observe price and quantity equilibrium values, not the potential price and potential quantities along the entire demand curve. Only by tracing out the potential outcomes along a demand curve can we calculate the elasticity. The price elasticity of demand is the ratio of percentage changes in quantity to price for a single demand curve.

Yet, when there are shifts in supply and demand, a sequence of quantity and price pairs emerges in history that reflect neither the demand curve nor the supply curve. In fact, connecting the points does not reflect any meaningful or useful object. Figure 2. Figure from Wright, P. The Tariff on Animal and Vegetable Oils. The Macmillan Company. The price elasticity of demand is the solution to the following equation: But in this example, the change in P is exogenous.

For instance, it holds supply fixed, the prices of other goods fixed, income fixed, preferences fixed, input costs fixed, and so on. In order to estimate the price elasticity of demand, we need changes in P that are completely and utterly independent of the otherwise normal determinants of supply and the other determinants of demand.

Otherwise we get shifts in either supply or demand, which creates new pairs of data for which any correlation between P and Q will not be a measure of the elasticity of demand. The problem is that the elasticity is an important object, and we need to know it, and therefore we need to solve this problem.

So given this theoretical object, we must write out an econometric model as a starting point. First, we need numerous rows of data on price and quantity. Second, we need for the variation in price in our imaginary data set to be independent of u. We call this kind of independence exogeneity. Without both, we cannot recover the price elasticity of demand, and therefore any decision that requires that information will be based on stabs in the dark.

Conclusion This book is an introduction to research designs that can recover causal effects. But just as importantly, it provides you with hands-on practice to implement these designs. Implementing these designs means writing code in some type of software. I have chosen to illustrate these designs using two popular software languages: Stata most commonly used by economists and R most commonly used by everyone else.

The book contains numerous empirical exercises illustrated in the Stata and R programs. The data needed for the latter have been made available to you at Github.

DTA is the name of a particular data set. For R users, it is a somewhat different process to load data into memory. This is partly based on a library called haven, which is a package for reading data files.

It is secondly based on a set of commands that create a function that will then download the data directly from Github. I encourage you to use this opportunity to invest in learning one or both of these languages. It is beyond the scope of this book to provide an introduction to these languages, but fortunately, there are numerous resources online. Stata is popular among microeconomists, and given the amount of coauthoring involved in modern economic research, an argument could be made for investing in it solely for its ability to solve basic coordination problems between you and potential coauthors.

But a downside to Stata is that it is proprietary and must be purchased. And for some people, that may simply be too big of a barrier— especially for anyone simply wanting to follow along with the book.

R on the other hand is open-source and free. Using this time to learn R would likely be well worth your time. Perhaps you already know R and want to learn Stata. Or perhaps you know Stata and want to learn R. Then this book may be helpful because of the way in which both sets of code are put in sequence to accomplish the same basic tasks.

But, with that said, in many situations, although I have tried my best to reconcile results from Stata and R, I was not always able to do so. Ultimately, Stata and R are different programming languages that sometimes yield different results because of different optimization procedures or simply because the programs are built slightly differently.

This has been discussed occasionally in articles in which authors attempt to better understand what accounts for the differing results. I was not always able to fully reconcile different results, and so I offer the two programs as simply alternative approaches. You are ultimately responsible for anything you do on your own using either language for your research. I leave it to you ultimately to understand the method and estimating procedure contained within a given software and package.

In conclusion, simply finding an association between two variables might be suggestive of a causal effect, but it also might not. Before we start digging into the causal methodologies themselves, though, I need to lay down a foundation in statistics and regression modeling.

Buckle up! This is going to be fun. I could imagine living without poetry, so I took his advice and quit. Interestingly, when I later found economics, I went back to Rilke and asked myself if I could live without it.

So I stuck with it and got a PhD. This book is my best effort to explain causal inference to myself. I felt that if I could explain causal inference to myself, then I would be able to explain it to others too. Not thinking the book would have much value outside of my class, I posted it to my website and told people about it on Twitter. I was surprised to learn that so many people found the book helpful. But economics favors formalism and deductive methods.

That is a strength, not a weakness. No more and no less. Thus, when we invoke ceteris paribus, we are implicitly invoking covariate balance—both the observable and the unobservable covariates. Often the URL was simply too long for the margins of the book otherwise.

Probability and Regression Review Numbers is hardly real and they never have feelings. But you push too hard, even numbers got limits. Mos Def Basic probability theory. In practice, causal inference is based on statistical models that range from the very simple to extremely advanced. A random process is a process that can be repeated many times with different outcomes each time.

The sample space is the set of all the possible outcomes of a random process. We distinguish between discrete and continuous random processes Table 1 below. Discrete processes produce, integers, whereas continuous processes produce fractions as well. We define independent events two ways. The first refers to logical independence. For instance, two events occur but there is no reason to believe that the two events affect each other.

Table 1. Examples of discrete and continuous random processes. The second definition of an independent event is statistical independence. For a deck of 52 cards, what is the probability that the first card will be an ace? There are 52 possible outcomes in the sample space, or the set of all possible outcomes of the random process.

Of those 52 possible outcomes, we are concerned with the frequency of an ace occurring. Assume that the first card was an ace. Now we ask the question again. If we shuffle the deck, what is the probability the next card drawn is also an ace? It is no longer 13 1 because we did not sample with replacement. We sampled without replacement. To make the two events independent, you would have to put the ace back and shuffle the deck.

So two events, A and B, are independent if and only if: An example of two independent events would be rolling a 5 with one die after having rolled a 3 with another die. The two events are independent, so the probability of rolling a 5 is always 0. In , the Golden State Warriors were 3—1 in a best-of-seven playoff. What had to happen for the Warriors to lose the playoff? The Cavaliers had to win three in a row. If the unconditional probability of a Cleveland win is 0. For independent events, to calculate joint probabilities, we multiply the marginal probabilities: where Pr A,B is the joint probability of both A and B occurring, and Pr A is the marginal probability of A event occurring.

Now, for a slightly more difficult application. What is the probability of rolling a 7 using two six-sided dice, and is it the same as the probability of rolling a 3? In Table 2 we see that there are six different ways to roll a 7 using only two dice. Table 3 shows that there are only two ways to get a 3 rolling two six-sided dice.

So, no, the probabilities of rolling a 7 and rolling a 3 are different. Table 2. Total number of ways to get a 7 with two six-sided dice. Table 3. Total number of ways to get a 3 using two six-sided dice. Events and conditional probability. Let A be some event. And let B be some other event. For two events, there are four possibilities. A and B: Both A and B occur. Probability tree. We can represent these two events in a probability tree.

Probability trees are intuitive and easy to interpret. Second, at every branching off from a node, we can further see that the probabilities associated with a given branch are summing to 1.

The joint probabilities are also all summing to 1. Venn diagrams and sets. A second way to represent multiple events occurring is with a Venn diagram. Venn diagrams were first conceived by John Venn in They are used to teach elementary set theory, as well as to express set relationships in probability and statistics. This example will involve two sets, A and B. After several mediocre seasons, his future with the school is in jeopardy.

But if they do, then he likely will be rehired. A and B are events, and U is the universal set of which A and B are subsets. Let A be the probability that the Longhorns get invited to a great bowl game and B be the probability that their coach is rehired.

The complement means that it is everything in the universal set that is not A. The same is said of B. Therefore: We can rewrite out the following definitions: Whenever we want to describe a set of events in which either A or B could occur, it is: A B.

Any element that is in either set A or set B, then, is also in the new union set. That is, only things inside both A and B get added to the new set. First, you can look at all the instances where A occurs with B. But then what about the rest of A that is not in B?

A similar style of reasoning can help you understand the following expression. Now it is just simple addition to find all missing values. Then we have: When working with sets, it is important to understand that probability is calculated by considering the share of the set for example A made up by the subset for example A B. When we write down that the probability that A B occurs at all, it is with regards to U.

Twoway contingency table. I left this intentionally undefined on the left side so as to focus on the calculation itself. This is: Notice, these conditional probabilities are not as easy to see in the Venn diagram. We are essentially asking what percentage of a subset—e.

This reasoning is the very same reasoning used to define the concept of a conditional probability. Contingency tables. Another way that we can represent events is with a contingency table. Contingency tables are also sometimes called twoway tables. Table 4 is an example of a contingency table. We continue with our example about the worried Texas coach. Note that to calculate conditional probabilities, we must know the frequency of the element in question e.

Given two events, A and B: Using equations 2. And this is the formula for joint probability. Given equation 2. We will now decompose this equation more fully, though, by substituting equation 2. Substituting equation 2. A is Texas making a great bowl game, and B is the coach getting rehired. We can make each calculation using the contingency tables. Or formally, Pr A B. We can use the Bayesian decomposition to find this probability. Check this against the contingency table using the definition of joint probability: So, if the coach is rehired, there is a 63 percent chance we made a great bowl game.

This is a fun one, because most people find it counterintuitive. It even is used to stump mathematicians and statisticians. Behind one of the doors is a million dollars. Behind each of the other two doors is a goat. Monty Hall, the game-show host in this example, asks the contestants to pick a door. After they pick the door, but before he opens the door they picked, he opens one of the other doors to reveal a goat.

Therefore, why switch? But what exactly did he say? Assume that you chose door 1, D1. We will call that event A1. We will call the opening of door 2 event B. We will call the probability that the million dollars is behind door i, Ai. We now write out the question just asked formally and decompose it using the Bayesian decomposition.

We are ultimately interested in knowing what the probability is that door 1 has a million dollars event A1 given that Monty Hall opened door 2 event B , which is a conditional probability question. There are basically two kinds of probabilities on the right side of the equation. We call this the prior probability, or prior belief. It may also be called the unconditional probability. The conditional probability, Pr B Ai , requires a little more careful thinking.

Take the first conditional probability, Pr B A1. And then the last conditional probability, Pr B A3. Each of these conditional probabilities requires thinking carefully about the feasibility of the events in question.

If the money is behind door 2, how likely is it for Monty Hall to open that same door, door 2? Keep in mind: this is a game show. So that gives you some idea about how the game-show host will behave. Do you think Monty Hall would open a door that had the million dollars behind it?

He only opens a door if the door has a goat. What then is Pr B A1? That is, in a world where you have chosen door 1, and the money is behind door 1, what is the probability that he would open door 2? There are two doors he could open if the money is behind door 1—he could open either door 2 or door 3, as both have a goat behind them. What about the second conditional probability, Pr B A2? Under our assumption that he never opens the door if it has a million dollars, we know this probability is 0.

And finally, what about the third probability, Pr B A3? What is the probability he opens door 2 given that the money is behind door 3? The only door, therefore, he could open is door 2. Thus, this probability is 1. Have your beliefs about that likelihood changed now that door 2 has been removed from the equation? This new conditional probability is called the posterior probability, or posterior belief.

And it simply means that having witnessed B, you learned information that allowed you to form a new belief about which door the money might be behind. Summation operator. The tools we use to reason about causality rest atop a bedrock of probabilities. We are often working with mathematical tools and concepts from statistics such as expectations and probabilities. One of the most common tools we will use in this book is the linear regression model, but before we can dive into that, we have to build out some simple notation.

The Greek letter the capital Sigma denotes the summation operator. Let x1,x2,. We can compactly write a sum of numbers using the summation operator as: The letter i is called the index of summation.

Other letters, such as j or k, are sometimes used as indices of summation. The subscript variable simply represents a specific value of a random variable, x. The numbers 1 and n are the lower limit and the upper limit, respectively, of the summation. The first property is called the constant rule.

Say we are given: We can apply both of these properties to get the following third property: Before leaving the summation operator, it is useful to also note things which are not properties of this operator. First, the summation of a ratio is not the ratio of the summations themselves.

Second, the summation of some squared variable is not equal to the squaring of its summation. We can use the summation indicator to make a number of calculations, some of which we will do repeatedly over the course of this book.

For instance, we can use the summation operator to calculate the average: where x is the average mean of the random variable xi. The sum of the deviations from the mean is always equal to 0: Table 5. Sum of deviations equaling 0. You can see this in Table 5.

This is because One result that will be very useful throughout the book is: An overly long, step-by-step proof is below. Note that the summation index is suppressed after the first line for easier reading. A more general version of this result is: Or: Expected value. The expected value of a random variable, also called the expectation and sometimes the population mean, is simply the weighted average of the possible values that the variable can take, with the weights being given by the probability of each value occurring in the population.

Suppose that the variable X can take on values x1,x2,. Note that X2 takes only the values 1, 0, and 4, with probabilities 0. And the third property is that if we have numerous constants, a1,.

It refers to the whole group of interest, not just to the sample available to us. Its meaning is somewhat similar to that of the average of a random variable in the population. Some additional properties for the expectation operator can be explained assuming two random variables, W and H.

But in large samples, this degree-offreedom adjustment has no practical effect on the value of S2 where S2 is the average after a degree of freedom correction over the sum of all squared deviations from the mean. First, the variance of a line is: And the variance of a constant is 0 i. The last part of equation 2. The covariance measures the amount of linear dependence between two random variables.

We represent it with the C X,Y operator. They could have a nonlinear relationship. The covariance between two linear functions is: The two constants, a1 and a2, zero out because their mean is themselves and so the difference equals 0. Interpreting the magnitude of the covariance can be tricky. For that, we are better served by looking at correlation. We define correlation as follows. A positive negative correlation indicates that the variables move in the same opposite ways.

Population model. We begin with cross-sectional analysis. We will assume that we can collect a random sample from the population of interest. Assume that there are two variables, x and y, and we want to see how y varies with changes in x. One, what if y is affected by factors other than x? How will we handle that? Two, what is the functional form connecting these two variables?

Three, if we are interested in the causal effect of x on y, then how can we distinguish that from mere correlation? This model is assumed to hold in the population. Equation 2.

For models concerned with capturing causal effects, the terms on the left side are usually thought of as the effect, and the terms on the right side are thought of as the causes.

This equation also explicitly models the functional form by assuming that y is linearly dependent on x. These describe a population, and our goal in empirical work is to estimate their values. We never directly observe these parameters, because they are not data I will emphasize this throughout the book.

What we can do, though, is estimate these parameters using data and assumptions. To do this, we need credible assumptions to accurately estimate these parameters with data. We will return to this point later. In this simple regression framework, all unobserved variables that determine y are subsumed by the error term u. First, we make a simplifying assumption without loss of generality. Let the expected value of u be zero in the population. If we normalize the u random variable to be 0, it is of no consequence.

Mean independence. An example might help here. Because people choose how much schooling to invest in based on their own unobserved skills and attributes, equation 2. Ordinary least squares. Plug any observation into the population equation: where i indicates a particular observation.

We observe yi and xi but not ui. We just know that ui is there. We talked about the first condition already. The second one, though, means that the mean value of x does not change with different slices of the error term. And again, note that the notation here is population concepts. Recall the properties of the summation operator as we work through the following sample properties of these two equations.

We begin with equation 2. We have already shown that the first equation equals zero equation 2. This gives So the equation to solve is12 The previous formula for is important because it shows us how to take data that we have and compute the slope estimate.

The estimate, , is commonly referred to as the ordinary least squares OLS slope estimate. Cunningham is the author of several books on Wicca and various other alternative religious subjects.

The book Earth Power by Scott Cunningham is a very simple to read and easy to follow book that covers the basics of magic and witchcraft. The author provides several easy to follow spells and rituals that enable a new seeker to be able to read the book and start practicing magic right away.

Book of shadows scott cunningham book of shadows pdf download Book Of Shadows. His work Wicca: A Guide for the Solitary Practitioner , is one of the most successful books on Wicca ever published; [1] he was a friend of notable occultists and Wiccans such as Raymond Buckland, and was a member of the Serpent Stone Family, and received his Third Degree Initiation as a member of that coven. The doctors in Royal Oak declared the mild climate in San Diego ideal for her.

Outside of many trips to Hawaii, Cunningham lived in San Diego all his life. We do not guarantee that these techniques will work for you. Some of the techniques listed in Cunninghams Encyclopedia of Magical Herbs Llewellyns Sourcebook Series may require a sound knowledge of Hypnosis, users are advised to either leave those sections or must have a basic understanding of the subject before practicing them.

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Loved each and every part of this book. I will definitely recommend this book to religion, wicca lovers.