# What are some recommended Graduate-Level Probability and/or Statistics textbooks for an incoming Econ PhD Student?

Title covers most of it. I'm an undergrad who will be graduating in a few weeks and will be starting a PhD Program in Economics. I'm interested in Econometric theory. I've taken undergrad level probstats, real analysis, abstract algebra, etc but wanted to work through a grad level textbook this summer to better prepare myself.

My analysis class didn't go too deeply into measure theory and I know I'll need that, so maybe a probability theory book that deals with that could help. "Measure Theory" by Doob has been floated by my Professors. Also, Royden's "Real Analysis" has been said to have a good amount of measure theory (not sure about the probability aspects in this one). I've used Baby Rudin, Abbott, and Apostle's analysis, so that's the background I'm coming from if that helps.

Let me know if this question is better suited for math.stackexchange.com instead too! First time posting here.

• stats.stackexchange.com/a/6191/9330 might be helpful Commented Mar 3, 2022 at 5:40
• Beyond standard econometrics books, I would recommend Machine Learning: A Probabilistic Perspective by Kevin Murphy. A lot of modern Statistics and "Machine Learning" in the econometrics literature today. Commented Mar 10, 2022 at 22:36

Here are a couple of grad-level books that cover probability and stats at a high level of rigor as a foundation for econometrics:

The first is

Bierens, Herman J.. Introduction to the Mathematical and Statistical Foundations of Econometrics (Themes in Modern Econometrics). Spain: Cambridge University Press, 2004.

It covers

1. Probability and Measure
2. Borel Measurability, Integration, and Mathematical Expectations
3. Conditional Expectations
4. Distributions and Transformations
5. The Multivariate Normal Distribution and its Application to Statistical Inference
6. Modes of Convergence Dependent
7. Laws of Large Numbers and Central Limit Theorems
8. Maximum Likelihood Theory

It also contains three math appendices on linear algebra, misc practical math, and complex analysis (useful for characteristic functions). You can look at a draft here.

Probability and Statistics for Economists by Bruce E. Hansen is the second, albeit less technical, intro. It uses calculus rather than measure theory but covers more terrain. It introduces:

1. Basic Probability Theory
2. Random Variables
3. Parametric Distributions
4. Multivariate Distributions
5. Normal and Related Distributions
6. Sampling
7. Law of Large Numbers
8. Central Limit Theory
10. Maximum Likelihood Estimation
11. Method of Moments
12. Numerical Optimization
13. Hypothesis Testing
14. Confidence Intervals
15. Shrinkage Estimation
16. Bayesian Methods
17. Nonparametric Density Estimation
18. Empirical Process Theory

There is a draft here.

Hansen's second volume, Econometrics, is also quite good. The two would be at the right level for your first-year coursework.

• Dimitriy, could you add more to what the material covers, the features etc.? Commented Jun 11 at 6:01