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Mostly Harmless Econometrics: An Empiricist's Companion
The core methods in today's econometric toolkit are linear regression for statistical control, instrumental variables methods for the analysis of natural experiments, and differences-in-differences methods that exploit policy changes. In the modern experimentalist paradigm, these techniques address clear causal questions such as: Do smaller classes increase learning? Should wife batterers be arrested? How much does education raise wages? Mostly Harmless Econometrics shows how the basic tools of applied econometrics allow the data to speak.
In addition to econometric essentials, Mostly Harmless Econometrics covers important new extensions--regression-discontinuity designs and quantile regression--as well as how to get standard errors right. Joshua Angrist and Jorn-Steffen Pischke explain why fancier econometric techniques are typically unnecessary and even dangerous. The applied econometric methods emphasized in this book are easy to use and relevant for many areas of contemporary social science.
An irreverent review of econometric essentials A focus on tools that applied researchers use most Chapters on regression-discontinuity designs, quantile regression, and standard errors Many empirical examples A clear and concise resource with wide applications
In addition to econometric essentials, Mostly Harmless Econometrics covers important new extensions--regression-discontinuity designs and quantile regression--as well as how to get standard errors right. Joshua Angrist and Jorn-Steffen Pischke explain why fancier econometric techniques are typically unnecessary and even dangerous. The applied econometric methods emphasized in this book are easy to use and relevant for many areas of contemporary social science.
An irreverent review of econometric essentials A focus on tools that applied researchers use most Chapters on regression-discontinuity designs, quantile regression, and standard errors Many empirical examples A clear and concise resource with wide applications
392 pages, Paperback
First published December 15, 2008
382 people are currently reading
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Displaying 1 - 30 of 72 reviews
September 10, 2011
Okay so my brother wrote it and I didn't understand a lot of it. But if you liked Freakonomics and want to geta bit more technical, it's totally for you.
May 16, 2012
A decent book if you already know your econometrics. Don't be fooled by the title and other playful aspects of the book's presentation though--it is a serious econometrics book and not for beginners.
March 18, 2017
The professor of my econometrics class last semester (the first econometrics class I'd ever taken) recommended this book as a supplement to class/our textbook. The book takes a lighthearted (at times humorous) approach, and does a good job explaining the big picture. However, it seems to not really be for people who have taken only one class in the subject, so I spent a lot of time either going back through the textbook or just plain confused. It's clearly a very good resource, but I would have gotten much more out of it if I had more background in the field.
October 15, 2024
3.5/5. Should be called “Slightly Less Harmful Than Hansen But Still Hurtful Econometrics”.
The best part of the book is the focus on empirical applications, what applied microeconomists are actually using and doing and how they are thinking. Coming out of an econometrics class in grad school, I felt like I had fully lost my intuition. This helped bring it back, somewhat.
The worst part of this book is the highly inconsistent level of explanation. In the beginning, the book basically coddles the reader. Past that, though, the book blasts through pages and pages of math (sometimes leaving the derivations completely out of book and referring to papers, which makes me question the math’s inclusion at all) and high-concept niches that can only be followed with pen and paper, and time, and patience. Reading through this book casually, I ended up doing a LOT of skimming, because this isn’t meant to be read like that, I think. It’s best used as a reference. Like, you encounter a problem, then see what this and other books say about it.
I can only recommend this to working or soon-to-be-working professionals. For the lay reader, this might as well be written in a foreign language. I saw some reviews compare this to freakonomics… what? Make no mistake, that is NOT true. This is a textbook, at the end of the day.
The best part of the book is the focus on empirical applications, what applied microeconomists are actually using and doing and how they are thinking. Coming out of an econometrics class in grad school, I felt like I had fully lost my intuition. This helped bring it back, somewhat.
The worst part of this book is the highly inconsistent level of explanation. In the beginning, the book basically coddles the reader. Past that, though, the book blasts through pages and pages of math (sometimes leaving the derivations completely out of book and referring to papers, which makes me question the math’s inclusion at all) and high-concept niches that can only be followed with pen and paper, and time, and patience. Reading through this book casually, I ended up doing a LOT of skimming, because this isn’t meant to be read like that, I think. It’s best used as a reference. Like, you encounter a problem, then see what this and other books say about it.
I can only recommend this to working or soon-to-be-working professionals. For the lay reader, this might as well be written in a foreign language. I saw some reviews compare this to freakonomics… what? Make no mistake, that is NOT true. This is a textbook, at the end of the day.
September 24, 2016
Well, the general framing of this book is just awful. Having no prior knowledge of econometrics and reading it is a pure waste of time as it assumes a great amount of a priori knowledge. Therefore I consider it to be extremely problematic that nowhere in the beginning the target audience is clearly defined. And even if you have a decent amount of knowledge in econometrics this is not going to be easy. To be honest I read this book while working through Woolridge's famous Econometric Analysis of Cross-Section and Panel-Data and for every 10 pages I read in the Woolridge I was just able to understand one in Mostly Harmless Econometrics. Wrapping your head around the conditional-expectation-causal-analysis framework is time-consuming, but on the other hand it describes a perspective on econometrics and empirical research in a totally different way and gives many fascinating insights. But to really appreciate that you need some good foundations in statistics and econometrics. Therefore I consider this book to be everything but not harmless.
May 31, 2012
A good book, but you definitely need to know econometrics before diving into this one. So it's not "pop economics" like Freakonomics. There's a decent review also at the Econometrics Books website.
August 22, 2016
Este es un libro indispensable para todo científico social interesado en el análisis empírico de corte econométrico. De gran claridad al explicar los conceptos y los posibles problemas a los que se enfrentan quienes hacen investigación aplicada, así como las soluciones que existen a dichos problemas. Todo ello con una prosa sumamente agradable y a bastante divertida por la cantidad de referencias a películas y novelas de ciencia ficción.
Sumamente recomendable.
Sumamente recomendable.
October 9, 2017
Pretty good book for econometrics if you want to understand how econometrics experiments work and what are some of the most important checks that you need to do in order to have unbiased results.
Read
August 12, 2019man I read this thing cover to cover so I'm counting it towards my total and you can't stop me
September 15, 2020
Not an introductory book to econometrics for sure, but if you've already taken courses that cover basic asymptotics and have been introduced to stuff like instrumental variables and fixed effects models, you are going to get a lot of mileage out of this book.
Angrist and Pischke put causal identification and inference front-and-center in this book, beginning with Rubin's potential outcomes model, the selection bias problem, and the ideal of a randomized experiment. In practice, we are often stuck with observational data, so a second-best identification strategy will involve seeking out natural or quasi-experiments. This is where the authors really shine, often using classic papers as case studies in, say, differences-in-differences or regression discontinuity design.
At times they skip over the technical details on certain topics, which is why I'm only giving it 4/5, but to be honest their main concern is general ideas and providing the reader some tricks of the trade. If you want to see why a fixed effects estimator is asymptotically more efficient than a first differences estimator, go read Jeff Wooldridge's Econometric Analysis. If you want a great sampling of an experimentalist perspective on econometric issues, this is a must-have.
Angrist and Pischke put causal identification and inference front-and-center in this book, beginning with Rubin's potential outcomes model, the selection bias problem, and the ideal of a randomized experiment. In practice, we are often stuck with observational data, so a second-best identification strategy will involve seeking out natural or quasi-experiments. This is where the authors really shine, often using classic papers as case studies in, say, differences-in-differences or regression discontinuity design.
At times they skip over the technical details on certain topics, which is why I'm only giving it 4/5, but to be honest their main concern is general ideas and providing the reader some tricks of the trade. If you want to see why a fixed effects estimator is asymptotically more efficient than a first differences estimator, go read Jeff Wooldridge's Econometric Analysis. If you want a great sampling of an experimentalist perspective on econometric issues, this is a must-have.
August 4, 2018
This is an excellent book on causal inference for econometrics and related problems (many observations, some unobserved covariates, no randomized experiment or only partial compliance in the experiment). If you want to learn about propensity score matching, discontinuity regression, and a hell of a lot about instrumental variables, this is the book for you.
As a minimum preparation, I think you would need to read and clearly understand parts 1 and 2 of Data Analysis Using Regression and Multilevel/Hierarchical Models before reading this book. It is written in a style which is at times humorous, but don't let this mislead you into thinking that it is for a general audience; this is a highly technical book and at times it's tough going. You'll want a thorough understanding of the linear algebra behind OLS, and this book is often cursory on the derivations.
As a minimum preparation, I think you would need to read and clearly understand parts 1 and 2 of Data Analysis Using Regression and Multilevel/Hierarchical Models before reading this book. It is written in a style which is at times humorous, but don't let this mislead you into thinking that it is for a general audience; this is a highly technical book and at times it's tough going. You'll want a thorough understanding of the linear algebra behind OLS, and this book is often cursory on the derivations.
January 21, 2014
As his other brother I couldn't agree more...
July 21, 2025
Go-to on multivariate regression and causal inference (mainly through IVs).
Some nice pieces on quantile regression and the origin of the term regression: from Francis Galton’s regression to the mean in human height. The fact that the predictor and the target (height gen 1, height gen 2) have the same unconditional distribution results in the ‘regression to the mean’ effect.
Some nice pieces on quantile regression and the origin of the term regression: from Francis Galton’s regression to the mean in human height. The fact that the predictor and the target (height gen 1, height gen 2) have the same unconditional distribution results in the ‘regression to the mean’ effect.
April 2, 2024
Mostly harmless? More like mostly harmful.
December 1, 2025
Great book to learn Microeconometrics. Everything is very well explained and with Empirical Econometrics examples. A must!
May 5, 2017
Not too technical, has plenty of advice for empiricists, quite funny sometimes, too. Works better as a complement, starts out simple but later still assumes quite some basic background in econometrics, naturally so as the problems become harder and bigger. Leaves out a bunch of things to keep the book short, but makes me scratch my head since I want to know in details what was going on behind.
Oh well, in general a decent read, except for the little bit extreme idea that all you need in econometrics is OLS and IV.
Oh well, in general a decent read, except for the little bit extreme idea that all you need in econometrics is OLS and IV.
did-not-finish
November 18, 2021Definitely got catfished by the title/reviews. Was looking for a practical intro book and the first chapters were encouraging, but it rapidly went downhill in chapter 3 when they started proving properties of the conditional expectation function… I’m a simple boy. I want applicable results, not proofs.
Will look around for something aimed at amateurs like me.
Will look around for something aimed at amateurs like me.
December 14, 2012
Great companion to a normal econometrics textbook. I used this in my second semester of econometrics and it was helpful to have explanations and a bit of a narrative in addition to just equations and proofs. Helps make good sense of the subject.
January 2, 2014
This is the book of common prayer for econometrics. Although, to be honest, it's more useful than that.
Seriously, this book is an actually funny to read book which clearly runs through a lot of important concepts in statistics for the social sciences.
Seriously, this book is an actually funny to read book which clearly runs through a lot of important concepts in statistics for the social sciences.
September 19, 2022
Bueno, le dí una leída de corrido. Ya lo había usado en la maestría. Me resultó un embole. Es cierto que son temas difíciles, pero me parece que pone pocos ejemplos y no tiene una lectura amena. Para colmo entercortado, en el subte, en inglés. No sé si fue una buena idea...
December 28, 2018
A nice reference book if you already have a decent background in the field. There are so many references to other parts of the book that it felt jumpy, but it's a pretty comprehensive guide to the tools used by applied microeconomists.
February 2, 2024
I work with applications of statistics and probability to recursive models for time series arising in engineering (where the inference task is called system identification), and I read this book in order to understand methods that germinated in econometrics (like IV) but may be meaningfully translated to other fields. My emphasis on abstraction runs against the grain of this very practical "companion," full of concrete details of real-life studies (many of them at the hand by one of the authors). But if I'd wanted math, I'd have read a math book. MHE helped me understand a few new concepts and better follow econometric ways of thinking (in labor econ at the very least).
This book, by its own admission, plays fast and loose with the theoretical foundations of statistics. Asymptotics are generally taken for granted. There is no discussion on integrability assumptions for the Central Limit Theorem and the Strong Law of Large Numbers, and Nassim Taleb has very memorably reminded us that even CLT convergence can be painfully slow. Maybe I am a curmudgeon in this respect, but let me give an example. I once witnessed a presentation that approximated a normalized sum of i.i.d. random variables with a t-distribution. But the random variables in question were actually quotients of Normal variates, and as such didn't actually have any finite moments.
Two important assumptions made in the first chapter are
1) models have errors-in-variables until proved otherwise, that is, if y = ax + b, both y and x will be measured with random error. But the author doesn't discuss the fact that the EiV linear least squares estimator as well as the maximum likelihood estimator can have heavy tails. There is a passing reference to the problem of regression dilution and a further discussion on how to mitigate dilution with IV.
2) the estimated quantities a and b in such a linear model are not necessary understood, as often in Bayesian or frequentist statistics, as the "truth" of an underlying data-generating process. Rather, statistics derive their validity by converging to a (strong) population limit, which is taken as interesting in itself.
My biggest critique is that this MHE, despite (or rather pursuant to) its title, doesn't adequately caution against multiple comparisons. In fact, on one occasion it encourages them. In practical steps for 2SLS analysis, "Report the F-statistic on the excluded instruments." So far, so good. "Pick your single best instrument and report just-identified estimates using this one only." Uh-oh. (Econometric "experiments" can never be perfectly repeated.) A frequentist would save the day by including "pick your single best instrument" as one step in a statistic that can be bootstrapped against the data. A Bayesian would say the day by proposing a continuous mixture of models governed by a prior. In either case, the day needs to be saved.
Things I learned
In this section, I will write (mostly for myself) short technical expositions of statistical techniques that I hope to remember.
Suppose that our model is the almost-sure law
0 = f(X, W, θ)
where X is a vector of observed random variables, W is a vector of unobserved random variables, and θ is the parameter we are trying to measure. We invert this law for θ using linear regression or MLE or something else. In econometrics, often f is a linear relationship between "cause" and "effect" coordinates of X, W is random effects, and θ is the coefficient vector. Let F be any σ-algebra in this probability space. Let us observe that we may take conditional expectations of both sides of this equation.
0 = E[ f(X, W, θ) | F ]
Why would we do this? One reason would F is a σ-algebra that smooths away (confounding) randomness in X and W while leaving the identifiability of θ intact. The generators of F are called instrumental variables.
As an aside, F can also be taken to be generated by a sufficient statistic of X, with a result similar to Rao-Blackwellization.
From this abstract point of view, forbidden regression is the assumption that conditional expectation commutes with nonlinear function composition, which now sounds obviously false.
Finally, quantile regression is not as scary as it seems. One may be familiar with squared error minimization (least squares) or absolute error minimization (least absolute deviation). Absolute error minimization corresponds to median regression. What if the convex loss function looked like the absolute value, except it were more sloped on one side of 0 than the other? The result is a quantile regression where the discrepancy between positive and negative residual penalty determines what quantile you are trying to fit.
This book, by its own admission, plays fast and loose with the theoretical foundations of statistics. Asymptotics are generally taken for granted. There is no discussion on integrability assumptions for the Central Limit Theorem and the Strong Law of Large Numbers, and Nassim Taleb has very memorably reminded us that even CLT convergence can be painfully slow. Maybe I am a curmudgeon in this respect, but let me give an example. I once witnessed a presentation that approximated a normalized sum of i.i.d. random variables with a t-distribution. But the random variables in question were actually quotients of Normal variates, and as such didn't actually have any finite moments.
Two important assumptions made in the first chapter are
1) models have errors-in-variables until proved otherwise, that is, if y = ax + b, both y and x will be measured with random error. But the author doesn't discuss the fact that the EiV linear least squares estimator as well as the maximum likelihood estimator can have heavy tails. There is a passing reference to the problem of regression dilution and a further discussion on how to mitigate dilution with IV.
2) the estimated quantities a and b in such a linear model are not necessary understood, as often in Bayesian or frequentist statistics, as the "truth" of an underlying data-generating process. Rather, statistics derive their validity by converging to a (strong) population limit, which is taken as interesting in itself.
My biggest critique is that this MHE, despite (or rather pursuant to) its title, doesn't adequately caution against multiple comparisons. In fact, on one occasion it encourages them. In practical steps for 2SLS analysis, "Report the F-statistic on the excluded instruments." So far, so good. "Pick your single best instrument and report just-identified estimates using this one only." Uh-oh. (Econometric "experiments" can never be perfectly repeated.) A frequentist would save the day by including "pick your single best instrument" as one step in a statistic that can be bootstrapped against the data. A Bayesian would say the day by proposing a continuous mixture of models governed by a prior. In either case, the day needs to be saved.
Things I learned
In this section, I will write (mostly for myself) short technical expositions of statistical techniques that I hope to remember.
Suppose that our model is the almost-sure law
0 = f(X, W, θ)
where X is a vector of observed random variables, W is a vector of unobserved random variables, and θ is the parameter we are trying to measure. We invert this law for θ using linear regression or MLE or something else. In econometrics, often f is a linear relationship between "cause" and "effect" coordinates of X, W is random effects, and θ is the coefficient vector. Let F be any σ-algebra in this probability space. Let us observe that we may take conditional expectations of both sides of this equation.
0 = E[ f(X, W, θ) | F ]
Why would we do this? One reason would F is a σ-algebra that smooths away (confounding) randomness in X and W while leaving the identifiability of θ intact. The generators of F are called instrumental variables.
As an aside, F can also be taken to be generated by a sufficient statistic of X, with a result similar to Rao-Blackwellization.
From this abstract point of view, forbidden regression is the assumption that conditional expectation commutes with nonlinear function composition, which now sounds obviously false.
Finally, quantile regression is not as scary as it seems. One may be familiar with squared error minimization (least squares) or absolute error minimization (least absolute deviation). Absolute error minimization corresponds to median regression. What if the convex loss function looked like the absolute value, except it were more sloped on one side of 0 than the other? The result is a quantile regression where the discrepancy between positive and negative residual penalty determines what quantile you are trying to fit.
February 20, 2022
Defining a target audience for this book is difficult: It mixes conversational style with heavy derivations. The authors choice to frame all of their expositions in a less common framework does not help lowering the complexity.
The derivations and difficult language, mixed in with some missing explanations and missing higher-level framing of the different topics make this book fairly unaccessible at an even with some level of knowledge in the area of (micro)econometrics, and should potentially be consumed together with a more textbook-style work, which explains the contents within the context.
For more seasoned practitioners, the authors offer some interesting insights in interpretations of model estimates and potentially some (new?) ideas for model designs. The framing of the models in the conditional expectations framework may be insightful, but requires very close reading. But crucially, this book misses out on many more detailed mathematical/statistical/econometrical explanations. I can see that this book could be used as a starting point to dig deeper into one (or more) of the cited texts, or to reflect on some of the user's practices. The review by A. Gelman at https://doi.org/10.1177/1536867X09009... gives a more insight into more technical issues with the book.
In addition to these, I found the style of writing and layout of the book difficult to follow, which is extremely dense in some areas and very wordy at others, which is a lot to digest when reading the book front-to-back. I do not think this book works good as a reference work either, due to interdependencies between chapters. (The authors seem to have been aware of this, there is some amount of repetition happening.)
This book is only occasionally harmless, and lacks in some areas of its writing. You may be better off buying a "proper" textbook on microeconometrics, such as Wooldridge or Cameron.
The derivations and difficult language, mixed in with some missing explanations and missing higher-level framing of the different topics make this book fairly unaccessible at an even with some level of knowledge in the area of (micro)econometrics, and should potentially be consumed together with a more textbook-style work, which explains the contents within the context.
For more seasoned practitioners, the authors offer some interesting insights in interpretations of model estimates and potentially some (new?) ideas for model designs. The framing of the models in the conditional expectations framework may be insightful, but requires very close reading. But crucially, this book misses out on many more detailed mathematical/statistical/econometrical explanations. I can see that this book could be used as a starting point to dig deeper into one (or more) of the cited texts, or to reflect on some of the user's practices. The review by A. Gelman at https://doi.org/10.1177/1536867X09009... gives a more insight into more technical issues with the book.
In addition to these, I found the style of writing and layout of the book difficult to follow, which is extremely dense in some areas and very wordy at others, which is a lot to digest when reading the book front-to-back. I do not think this book works good as a reference work either, due to interdependencies between chapters. (The authors seem to have been aware of this, there is some amount of repetition happening.)
This book is only occasionally harmless, and lacks in some areas of its writing. You may be better off buying a "proper" textbook on microeconometrics, such as Wooldridge or Cameron.
December 31, 2018
one of the only statistics books I recommend to people because it addresses real-world issues that impact whether real projects get funding or not. It's a question we'd like to get right.
This is one of two books I can think of off the top of my head (Weisberg's is the other) which foregrounds _what_does_regression_actually_mean_. Versus the mathematical process.
On the other hand, maybe statistical measures like lm beta and sd are nothing more than high-dimensional hypotenuse constructions. Most of what people want to know from lm questions could be answered by looking at random subsets of treatment group and non-treatment group, then drilling down with counter-questions that come from real world common sense.
Using a concrete example from a random popular armchair bullshitter https://twitter.com/clairlemon/status..., people have various ideas about why divorced men might be less productive, and subgroups / extra data columns, if they're available, might be used to answer some of these real-world counter questions.
If the gritty details of subscripts and high-dimensional hypotenuses weren't so focussed on, the time used to understand the minutiae of these arguments could instead be used to read the data collection methodology and variable definitions, which in my experience always swamp debates about what the algebra "means".
This is one of two books I can think of off the top of my head (Weisberg's is the other) which foregrounds _what_does_regression_actually_mean_. Versus the mathematical process.
On the other hand, maybe statistical measures like lm beta and sd are nothing more than high-dimensional hypotenuse constructions. Most of what people want to know from lm questions could be answered by looking at random subsets of treatment group and non-treatment group, then drilling down with counter-questions that come from real world common sense.
Using a concrete example from a random popular armchair bullshitter https://twitter.com/clairlemon/status..., people have various ideas about why divorced men might be less productive, and subgroups / extra data columns, if they're available, might be used to answer some of these real-world counter questions.
If the gritty details of subscripts and high-dimensional hypotenuses weren't so focussed on, the time used to understand the minutiae of these arguments could instead be used to read the data collection methodology and variable definitions, which in my experience always swamp debates about what the algebra "means".
November 6, 2019
Great book on econometrics but definitely NOT the first one you should read. There are useful and practical tips for practitioners. It provides a big picture on recent developlemnts in econometrics and empirical research focused on causal questions. I agree with other reviews that "Mostly harmless" in the title is quite painful reading with advanced formal notation. I recommend to start with Mastering 'Metrics: The Path from Cause to Effect or Causal Inference: The Mixtape for modern approach to econometrics, or with a standard texbook Introductory Econometrics.
September 21, 2020
This highly technical book was a challenge for me to read because I am unpracticed in statistics. It is written with attention to data scientists who might utilize programming libraries to do the work than mathematics.
Overall the consideration is on teasing out trends and being able to understand how to model and interpret data as it applies to economics. The majority of this book is about types of models. There is much discussion about the mathematical consequences for various models, but not much discussion about how to select models per data -- the examples are selected in order to align with the mathematics and not the other way around.
I intend to read this again when I have a little more meat on my stats.
Overall the consideration is on teasing out trends and being able to understand how to model and interpret data as it applies to economics. The majority of this book is about types of models. There is much discussion about the mathematical consequences for various models, but not much discussion about how to select models per data -- the examples are selected in order to align with the mathematics and not the other way around.
I intend to read this again when I have a little more meat on my stats.
January 17, 2024
A good theorem-centric view of econometrics models. Written concisely and packed with information. One major gripe is the author's terrible notation style -- matrices are written backwards as are indices in general. Also, it could have been slightly less concise ... the authors have a dumbed down version from 2018 covering most the stuff with more verbal exposition which is helpful to refer to if you have never met these models before. Also, the authors should be less blank-slatist and should have considered the role of genetics as a confounder explicitly, instead of just ignoring genetics as economists tend to do, outside of oblique references to "pre-existing ability", whatever that means.
March 20, 2022
Todo un clásico, uno de los libros indispensables para cualquier científico social que desee hacer trabajo empírico. Es intuitivo y fácil de seguir, incluso en las partes técnicas que requieren algo de álgebra. Libro de cabecera para consultar cuando uno se encuentra con problemas al momento de pensar la estrategia de identificación de la investigación en turno.
Read
August 26, 2022Good addition to a standard introductory text in econometrics, focuses mostly on microeconometrics, less on macroeconometrics and barely any financial econometrics. Basic undergraduate linear algebra, calculus and statistics are required. Contains a lot of real-world examples which help build a strong intuition for applied econometrics.
February 18, 2024
One of the best books I ever read. A perfect guide to understanding research of all kinds, the ideal research set up, and ways around it.
You have to have had a class in regression analysis and be comfortable in math to appreciate it though. The book does not require, however, highly advanced math skills.
You have to have had a class in regression analysis and be comfortable in math to appreciate it though. The book does not require, however, highly advanced math skills.
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