Statistics
Courses in Statistics fall into two categories: statistics and actuarial science.
COR1-GB.1305
3 credits
This course is designed to achieve an understanding of fundamental notions of data presentation and analysis and to use statistical thinking in the context of business problems. The course deals with modern methods of data exploration (designed to reveal unusual or problematic aspects of databases), the uses and abuses of the basic techniques of inference, and the use of regression as a tool for management and for financial analysis.
STAT-GB.3321
3 credits
Prerequisites: STAT-GB.3301
This is an introductory course in stochastic processes. The course places emphasis on probabilistic thinking and on learning how to model the real-life phenomena, which evolve over time. It presents classes of stochastic processes which are widely used as modeling tools in diverse fields of applications including finance, economics, accounting, marketing and actuarial science. It covers basic theory and applications of discrete and continuous-time Markov chains; discrete and continuous time martingales; and Brownian motion and its generalizations. The introduction to Ito stochastic calculus is presented with a view towards financial applications. The course also discusses some statistical aspects of considered processes.
Statistics courses cover techniques relating to the application of the theory of probability to decisions that must be made in the face of uncertainty. Statistical theory and methods are used in a variety of applications, such as
- sampling
- data analysis
- design of market research studies
- quantitative methods in cost accounting
- statistical quality control of manufactured products
- economic forecasting
- financial modeling
Statistical computing algorithms are used for analyzing data and statistical estimation.
Actuarial science courses prepare students for an actuarial career, applying probability and statistics to the fields of insurance and pensions. The courses in actuarial science and related fields of probability, statistics, economics, and finance cover all of the material that appears in the first two examinations jointly sponsored by the Society of Actuaries and the Casualty Actuarial Society. Other courses at Stern cover portions of examinations three and four.
The program offers students the theory and techniques to solve business problems. Each course emphasizes the application of statistical research methods to actual business problems. The applied courses make extensive use of computers.
A student considering statistics courses beyond the core course should speak to a faculty member about prerequisites and career objectives before registering. In certain instances, instructors may waive prerequisites for an advanced course.
Graduate-level courses are also offered for students who have less formal mathematical backgrounds. These are generally, but not exclusively, computer-intensive courses that develop skills in quantitative techniques. For the most part, the only prerequisite for these courses is the MBA core course, Statistics and Data Analysis, COR1-GB.1305 (B01.1305). These courses are open to students regardless of their areas of specialization. Follow the links to the left for a description of Statistics courses.
MBA Core in Statistics
Statistics and Data AnalysisCOR1-GB.1305
3 credits
This course is designed to achieve an understanding of fundamental notions of data presentation and analysis and to use statistical thinking in the context of business problems. The course deals with modern methods of data exploration (designed to reveal unusual or problematic aspects of databases), the uses and abuses of the basic techniques of inference, and the use of regression as a tool for management and for financial analysis.
MBA Electives - Stochastic Processes
Introduction to Stochastic ProcessesSTAT-GB.3321
3 credits
Prerequisites: STAT-GB.3301
This is an introductory course in stochastic processes. The course places emphasis on probabilistic thinking and on learning how to model the real-life phenomena, which evolve over time. It presents classes of stochastic processes which are widely used as modeling tools in diverse fields of applications including finance, economics, accounting, marketing and actuarial science. It covers basic theory and applications of discrete and continuous-time Markov chains; discrete and continuous time martingales; and Brownian motion and its generalizations. The introduction to Ito stochastic calculus is presented with a view towards financial applications. The course also discusses some statistical aspects of considered processes.
MBA Electives - All Specializations
Statistics group faculty members offer courses for MBA students interested in applications of quantitative methods to various aspects of business activity. These courses may also be taken to meet requirements for a specialization in statistics. The only prerequisite for these courses is Statistics and Data Analysis (COR1-GB.1305), with the exception of Mathematics of Investment, STAT-GB.2309, which requires one semester of undergraduate calculus or the instructor's permission. The STAT-GB.2XXX courses emphasize applications and present the theory at the level of intuitive arguments. The STAT-GB.3XXX courses emphasize the theory and methodology, using the applications as illustrations.
Regression and Multivariate Data Analysis
STAT-GB.2301
3 credits
Prerequisites: COR1-GB.1305
This is a data-driven, applied statistics course focusing on the analysis of data using regression models. It emphasizes applications to the analysis of business and other data and makes extensive use of computer statistical packages. Topics include simple and multiple linear regression, residual analysis and other regression diagnostics, multicollinearity and model selection, autoregression, heteroscedasticity, regression models using categorical predictors, and logistic regression. All topics are illustrated on real data sets obtained from financial markets, market research studies, and other scientific inquiries.
STAT-GB.2301
3 credits
Prerequisites: COR1-GB.1305
This is a data-driven, applied statistics course focusing on the analysis of data using regression models. It emphasizes applications to the analysis of business and other data and makes extensive use of computer statistical packages. Topics include simple and multiple linear regression, residual analysis and other regression diagnostics, multicollinearity and model selection, autoregression, heteroscedasticity, regression models using categorical predictors, and logistic regression. All topics are illustrated on real data sets obtained from financial markets, market research studies, and other scientific inquiries.
Forecasting Time Series Data
STAT-GB.2302
3 credits
Prerequisites: COR1-GB.1305
Presented in this course are practical time series forecasting techniques with emphasis on the Box-Jenkins ARIMA (autoregressive integrated moving average) method and conditional volatility ARCH (autoregressive conditional heterogeneity) and GARCH (generalized autoregressive conditional heterogeneity) models. The course gives a mix of practical data analysis along with an introduction to the relevant theory. The ARIMA models are used to forecast series like interest spreads, while ARCH models are used in estimating and forecasting the volatility of series like stock returns and exchange rate returns. Students analyze data sets of their own choice in projects. Additional topics of interest covered in the course are methods of testing for nonstationary (Dickey-Fuller tests) as well as models for capturing seasonality as seen, for example, in series of monthly sales figures. The low-cost forecasting method of exponential smoothing is discussed, and its connection to the RiskMetricsTM methods of J. P. Morgan and GARCH models is explored. If time permits, we also study methods of forecasting multivariate time series, where information from several series is pooled to forecast a single series. The concept of co-integration or co-movement of multivariate series is discussed (interest rates being a prime example), along with their implications for forecasts. Other potential topics in the course include the use of ARCH models in value at risk (VAR) analysis and in option pricing.
STAT-GB.2302
3 credits
Prerequisites: COR1-GB.1305
Presented in this course are practical time series forecasting techniques with emphasis on the Box-Jenkins ARIMA (autoregressive integrated moving average) method and conditional volatility ARCH (autoregressive conditional heterogeneity) and GARCH (generalized autoregressive conditional heterogeneity) models. The course gives a mix of practical data analysis along with an introduction to the relevant theory. The ARIMA models are used to forecast series like interest spreads, while ARCH models are used in estimating and forecasting the volatility of series like stock returns and exchange rate returns. Students analyze data sets of their own choice in projects. Additional topics of interest covered in the course are methods of testing for nonstationary (Dickey-Fuller tests) as well as models for capturing seasonality as seen, for example, in series of monthly sales figures. The low-cost forecasting method of exponential smoothing is discussed, and its connection to the RiskMetricsTM methods of J. P. Morgan and GARCH models is explored. If time permits, we also study methods of forecasting multivariate time series, where information from several series is pooled to forecast a single series. The concept of co-integration or co-movement of multivariate series is discussed (interest rates being a prime example), along with their implications for forecasts. Other potential topics in the course include the use of ARCH models in value at risk (VAR) analysis and in option pricing.
Applied Stochastic Processes for Financial Models
STAT-GB.2308
3 credits
Prerequisites: COR1-GB.1305
In this class we study stochastic models for the financial markets mostly in a discrete time setting. We shall discuss the concept of martingales and risk-neutral probability measures, and derive the general pricing formula for contingent claims. We shall study the binomial model and derive the price of a European call option on this model, called the binomial Black-Scholes (BS) formula. We study put options using the put-call parity. We shall compare the binomial BS formula to the continuous time BS formula, and analyze the latter via the “Greeks”. We shall also look at exotic options such as the lookback and the knockout option. Additionally, American options, forward and future contracts, and fixed income models will be included as well.
STAT-GB.2308
3 credits
Prerequisites: COR1-GB.1305
In this class we study stochastic models for the financial markets mostly in a discrete time setting. We shall discuss the concept of martingales and risk-neutral probability measures, and derive the general pricing formula for contingent claims. We shall study the binomial model and derive the price of a European call option on this model, called the binomial Black-Scholes (BS) formula. We study put options using the put-call parity. We shall compare the binomial BS formula to the continuous time BS formula, and analyze the latter via the “Greeks”. We shall also look at exotic options such as the lookback and the knockout option. Additionally, American options, forward and future contracts, and fixed income models will be included as well.
Mathematics of Investment
STAT-GB.2309
3 credits
Prerequisites: one semester of undergraduate calculus or permission of the instructor
The course discusses mathematical and technical aspects of investments. Topics include measurement of interest and discount rates, accumulated value and present value, annuities, sinking funds, amortization of debt, and determination of yield rates on securities. Applications include bond evaluation, mortgages, capital budgeting, depreciation methods, and insurance.
STAT-GB.2309
3 credits
Prerequisites: one semester of undergraduate calculus or permission of the instructor
The course discusses mathematical and technical aspects of investments. Topics include measurement of interest and discount rates, accumulated value and present value, annuities, sinking funds, amortization of debt, and determination of yield rates on securities. Applications include bond evaluation, mortgages, capital budgeting, depreciation methods, and insurance.
MBA Electives - Statistics and Actuarial Science
These electives are open to all students (MBA and PhD) who have the required mathematical prerequisites (two semesters of calculus and one semester of matrix algebra at the graduate or undergraduate level). Typically, students who majored in engineering or mathematics would automatically satisfy this requirement.
Introduction to the Theory of Probability
STAT-GB.3301
3 credits
Prerequisites: two semesters of undergraduate calculus or permission of instructor.
This course covers the basic concepts of probability. Topics include the axiomatic definition of probability; combinatorial theorems; conditional probability and independent events; random variables and probability distributions; expectation of functions of random variables; special discrete and continuous distributions, including the chi-square, t, and F distributions; joint distributions with emphasis on the bivariate normal distribution; law of large numbers, central limit theorem; and moment generating functions. The theory of statistical estimation is introduced with a discussion on maximum likelihood estimation.
STAT-GB.3301
3 credits
Prerequisites: two semesters of undergraduate calculus or permission of instructor.
This course covers the basic concepts of probability. Topics include the axiomatic definition of probability; combinatorial theorems; conditional probability and independent events; random variables and probability distributions; expectation of functions of random variables; special discrete and continuous distributions, including the chi-square, t, and F distributions; joint distributions with emphasis on the bivariate normal distribution; law of large numbers, central limit theorem; and moment generating functions. The theory of statistical estimation is introduced with a discussion on maximum likelihood estimation.
Statistical Inference and Regression Analysis
STAT-GB.3302
3 credits
Prerequisites: STAT-GB.3301 plus one semester of linear algebra
The course has two distinct components: statistical inference and regression analysis. Topics included in statistical inference are principles of statistical estimation and inference, Neyman-Pearson Lemma, testing of means, variances, tests of independence, and nonparametric methods. Regression analysis focuses on the general linear regression model, least squares estimation, departures from standard assumptions, autocorrelation, multicollinearity, analysis of residuals, choice of variables, and nonlinear models.
STAT-GB.3302
3 credits
Prerequisites: STAT-GB.3301 plus one semester of linear algebra
The course has two distinct components: statistical inference and regression analysis. Topics included in statistical inference are principles of statistical estimation and inference, Neyman-Pearson Lemma, testing of means, variances, tests of independence, and nonparametric methods. Regression analysis focuses on the general linear regression model, least squares estimation, departures from standard assumptions, autocorrelation, multicollinearity, analysis of residuals, choice of variables, and nonlinear models.
Multivariate Statistical Analysis
STAT-GB.3303
3 credits
Prerequisites: STAT-GB.3302
This course covers multivariate distributions. It focuses on the multivariate normal, geometric principle of sampling, multivariate asymptotics, principles of multivariate inference, tests of the mean vector for one and several populations leading to Hotelling's T2 statistic and MANOVA (multiple analysis of variance), techniques of multiple comparisons, multivariate linear regression models, principal components, factor analysis, canonical correlations, discrimination and classification, clustering, and graphical displays of multivariate data.
STAT-GB.3303
3 credits
Prerequisites: STAT-GB.3302
This course covers multivariate distributions. It focuses on the multivariate normal, geometric principle of sampling, multivariate asymptotics, principles of multivariate inference, tests of the mean vector for one and several populations leading to Hotelling's T2 statistic and MANOVA (multiple analysis of variance), techniques of multiple comparisons, multivariate linear regression models, principal components, factor analysis, canonical correlations, discrimination and classification, clustering, and graphical displays of multivariate data.
Advanced Theory of Statistics
STAT-GB.3304
3 credits
Prerequisites: STAT-GB.3302
The course covers topics in statistical estimation and hypothesis testing. Topics on estimation include sufficiency, exponential family, Pitman Koopman Theorem, criteria for choice of estimators, lower bounds for variance of estimators, sufficiency and completeness, maximum likelihood estimation, theorems on limiting distributions, and robust estimation. Topics on hypothesis testing include theory of optimum tests, Neyman- Pearson Lemma, M.P. (most powerful) and U.M.P. (uniformly most powerful) tests, unbiased tests, composite hypotheses, Neyman structure, likelihood principle, and likelihood ratio tests.
STAT-GB.3304
3 credits
Prerequisites: STAT-GB.3302
The course covers topics in statistical estimation and hypothesis testing. Topics on estimation include sufficiency, exponential family, Pitman Koopman Theorem, criteria for choice of estimators, lower bounds for variance of estimators, sufficiency and completeness, maximum likelihood estimation, theorems on limiting distributions, and robust estimation. Topics on hypothesis testing include theory of optimum tests, Neyman- Pearson Lemma, M.P. (most powerful) and U.M.P. (uniformly most powerful) tests, unbiased tests, composite hypotheses, Neyman structure, likelihood principle, and likelihood ratio tests.
Bayesian Inference and Statistical Decision Theory
STAT-GB.3305
3 credits
Prerequisites: STAT-GB.3302
This course has two components: statistical decision theory and the Bayesian paradigm for statistical inference. Statistical decision theory is concerned with the problem of making decisions in the presence of relevant statistical knowledge. Topics include decision rules, utility, risk functions, admissibility, consistency, expected loss, randomized decision rules, minimax decision rules, Bayes decision rules, and game theory. Both Frequentist and Bayesian concepts are considered. The Bayesian paradigm is the approach to statistics that formally seeks to utilize prior information. Topics include the notion of subjective probability, the specification of prior information, credibility sets, predictive distributions, empirical and hierarchical Bayes analysis, Bayesian robustness, and computation. Comparisons are made with the classical approaches to typical problems. Business case studies are used to illustrate both components.
STAT-GB.3305
3 credits
Prerequisites: STAT-GB.3302
This course has two components: statistical decision theory and the Bayesian paradigm for statistical inference. Statistical decision theory is concerned with the problem of making decisions in the presence of relevant statistical knowledge. Topics include decision rules, utility, risk functions, admissibility, consistency, expected loss, randomized decision rules, minimax decision rules, Bayes decision rules, and game theory. Both Frequentist and Bayesian concepts are considered. The Bayesian paradigm is the approach to statistics that formally seeks to utilize prior information. Topics include the notion of subjective probability, the specification of prior information, credibility sets, predictive distributions, empirical and hierarchical Bayes analysis, Bayesian robustness, and computation. Comparisons are made with the classical approaches to typical problems. Business case studies are used to illustrate both components.
Time Series Analysis
STAT-GB.3306
3 credits
Prerequisites: STAT-GB.3302
This course presents the Fourier analysis of time series. The frequency domain approach covered here provides a complementary outlook on time series to the usual time domain Box-Jenkins approach. Topics include periodicity (cycles) in time series data, the periodogram and its distribution, linear filters and transfer functions, spectral density, spectral representations of autocovariances and stationary processes, ARMA (autoregressive moving average) models and their spectra, model selection, the linear forecasting problem, and spectral estimation. We also discuss long memory models, including fractional ARIMA (autoregressive integrated moving average) and nonlinear time series, including ARCH (autoregressive conditional heterogeneity) models and chaos, as time permits.
STAT-GB.3306
3 credits
Prerequisites: STAT-GB.3302
This course presents the Fourier analysis of time series. The frequency domain approach covered here provides a complementary outlook on time series to the usual time domain Box-Jenkins approach. Topics include periodicity (cycles) in time series data, the periodogram and its distribution, linear filters and transfer functions, spectral density, spectral representations of autocovariances and stationary processes, ARMA (autoregressive moving average) models and their spectra, model selection, the linear forecasting problem, and spectral estimation. We also discuss long memory models, including fractional ARIMA (autoregressive integrated moving average) and nonlinear time series, including ARCH (autoregressive conditional heterogeneity) models and chaos, as time permits.
Categorical Data
STAT-GB.3307
3 credits
Prerequisites: STAT-GB.3302
Discrete random variables are the subject of this course, with most of the emphasis going to the bivariate and multivariate situations. The major topics are the chi-squared statistic, Fisher's exact test, odds ratio estimates and intervals, sets of tables, the log-linear model, model fitting, and logit analysis. The fundamental paper by Leo Goodman in the 1970 issue of the Journal of the American Statistical Association is discussed. M.B.A. and undergraduate students registering for this course are evaluated primarily on their ability to formulate and analyze data-based problems. All other students are evaluated primarily on their understanding of methodological and theoretical issues associated with the analysis of categorical data.
STAT-GB.3307
3 credits
Prerequisites: STAT-GB.3302
Discrete random variables are the subject of this course, with most of the emphasis going to the bivariate and multivariate situations. The major topics are the chi-squared statistic, Fisher's exact test, odds ratio estimates and intervals, sets of tables, the log-linear model, model fitting, and logit analysis. The fundamental paper by Leo Goodman in the 1970 issue of the Journal of the American Statistical Association is discussed. M.B.A. and undergraduate students registering for this course are evaluated primarily on their ability to formulate and analyze data-based problems. All other students are evaluated primarily on their understanding of methodological and theoretical issues associated with the analysis of categorical data.
Sampling Techniques
STAT-GB.3308
3 credits
Prerequisites: STAT-GB.3302
The course considers commonly used sampling schemes, such as simple, random, stratified, multistage, and double sampling. The efficiencies of these plans are discussed in detail. Also included are methods of estimation, including ratio and regression. Other topics include poststratification, multivariate surveys, analytic studies, problems of nonresponse, nonsampling errors, and randomized response technique. Theory is illustrated with examples from diverse fields.
STAT-GB.3308
3 credits
Prerequisites: STAT-GB.3302
The course considers commonly used sampling schemes, such as simple, random, stratified, multistage, and double sampling. The efficiencies of these plans are discussed in detail. Also included are methods of estimation, including ratio and regression. Other topics include poststratification, multivariate surveys, analytic studies, problems of nonresponse, nonsampling errors, and randomized response technique. Theory is illustrated with examples from diverse fields.
Experimental Design
STAT-GB.3309
3 credits
Prerequisites: STAT-GB.3302
This course develops the analysis of variance model in detail through the “one-way” and “two-way” designs, including partitioning sums of squares, orthogonal polynomials, interactions, multiple comparisons, and fixed and random effects. The concepts of randomization and blocking lead to discussions of design strategy. Further topics, covered if time permits, are the higher-order designs, split-plot designs, and fractional factorials. The material of this course is vital to those performing designed experiments, and the information can also be helpful in observational studies.
STAT-GB.3309
3 credits
Prerequisites: STAT-GB.3302
This course develops the analysis of variance model in detail through the “one-way” and “two-way” designs, including partitioning sums of squares, orthogonal polynomials, interactions, multiple comparisons, and fixed and random effects. The concepts of randomization and blocking lead to discussions of design strategy. Further topics, covered if time permits, are the higher-order designs, split-plot designs, and fractional factorials. The material of this course is vital to those performing designed experiments, and the information can also be helpful in observational studies.
Statistical Computing and Sampling Methods with Applications to Finance
STAT-GB.3314
3 credits
Prerequisites: STAT-GB.3302
This course covers most of the classical and modern Monte Carlo methods for statistical estimation. In particular, the fast growth of Monte Carlo Markov Chain (MCMC) methods has enabled the use of Bayesian inference in many applied fields. Methodologies are illustrated with financial applications such as estimation of implied volatility and risk measures. Examples are drawn from published research and survey papers in current literature (Risk magazine, J. P. Morgan’s Risk Metrics). The course integrates three basic components of statistical analysis in financial areas: (1) modeling and inference (with emphasis on Bayesian methodology), (2) computing and sampling methods for statistical estimation (with emphasis on MCMC), and (3) applications to financial data (with emphasis on volatility and risk). The focus is placed on the second component, bridging the gap between what can be said in theory (first component) and what can be done in practice (third component). The goals of the course are modest so that a full treatment of all major topics can be achieved.
STAT-GB.3314
3 credits
Prerequisites: STAT-GB.3302
This course covers most of the classical and modern Monte Carlo methods for statistical estimation. In particular, the fast growth of Monte Carlo Markov Chain (MCMC) methods has enabled the use of Bayesian inference in many applied fields. Methodologies are illustrated with financial applications such as estimation of implied volatility and risk measures. Examples are drawn from published research and survey papers in current literature (Risk magazine, J. P. Morgan’s Risk Metrics). The course integrates three basic components of statistical analysis in financial areas: (1) modeling and inference (with emphasis on Bayesian methodology), (2) computing and sampling methods for statistical estimation (with emphasis on MCMC), and (3) applications to financial data (with emphasis on volatility and risk). The focus is placed on the second component, bridging the gap between what can be said in theory (first component) and what can be done in practice (third component). The goals of the course are modest so that a full treatment of all major topics can be achieved.
Advanced Theory of Probability
STAT-GB.3352
3 credits
Prerequisites: STAT-GB.3301 plus two semesters of undergraduate calculus
The aim of the course is to establish a comprehensive foundation of the theory of probability. The topics covered are basic measure theory, random variables, and induced measures and distributions; independence of random variables; integration in a probability space with emphasis on conditional expectation and martingales; modes of convergence of random variables, including almost sure convergence, convergence in LP, convergence in probability, and convergence in distribution; characteristic functions and the inversion formula; and the central limit theorem for independent identically distributed random variables and also for martingale differences. If time permits, additional topics will include functional central limit theorems and their applications.
STAT-GB.3352
3 credits
Prerequisites: STAT-GB.3301 plus two semesters of undergraduate calculus
The aim of the course is to establish a comprehensive foundation of the theory of probability. The topics covered are basic measure theory, random variables, and induced measures and distributions; independence of random variables; integration in a probability space with emphasis on conditional expectation and martingales; modes of convergence of random variables, including almost sure convergence, convergence in LP, convergence in probability, and convergence in distribution; characteristic functions and the inversion formula; and the central limit theorem for independent identically distributed random variables and also for martingale differences. If time permits, additional topics will include functional central limit theorems and their applications.
Frequency Domain Time Series Analysis
STAT-GB.3383
3 credits
Frequency Domain Time Series is an advanced course on foundations and applications of time Series. Methods involving periodograms and spectral densities are emphasized. Linear filtering and spectral representations (stochastic integrals) for stationary time series are used as unifying themes. The second half of the course considers GARCH models, fractals, long memory and fractional cointegration. Again, emphasis is on insights gained from the frequency domain viewpoint.
STAT-GB.3383
3 credits
Frequency Domain Time Series is an advanced course on foundations and applications of time Series. Methods involving periodograms and spectral densities are emphasized. Linear filtering and spectral representations (stochastic integrals) for stationary time series are used as unifying themes. The second half of the course considers GARCH models, fractals, long memory and fractional cointegration. Again, emphasis is on insights gained from the frequency domain viewpoint.
The mathematics used in the course is Fourier analysis, a useful tool for all technically-oriented students. All mathematical results are presented in a self-contained manner.
The course grades are based on homework assignments (70% of the grade) and an in-class open-book final exam (30% of the grade). Homeworks can be re-submitted for further credit, at any time.
There is a clear need for advanced students in statistics, finance and economics to have a deep understanding of time series in the frequency domain. Increasingly, frequency domain methods and models are being used by practitioners. If time permits, we will discuss some of these methods along with papers that have appeared in the literature. Examples include:
- "The distribution of realized exchange rate volatility" (Anderson, Bollerslev, Diebold and Labys, Journal of the American Statistical Association, 2001).
- "Unit root tests in ARMA models with data-dependent methods for the selection of the truncation lag" (Ng and Perron, Journal of the American Statistical Association, 1995).
- "The size and power of the variance ratio test in finite samples: A Monte Carlo investigation" (Lo and MacKinlay, J. Econometrics, 1989).
- "Long memory in continuous time stochastic volatility models" (Comte and Renault, Mathematical Finance, 1998).
- "An asymptotic approximation for heteroskedasticity autocorrelation robust tests" (T. Vogelsang, 2002).
- "A fractional cointegration analysis of purchasing power parity" (Cheung and Lai, Journal of Business and Economic Statistics, 1993).
- "On the power of Dickey-Fuller tests against fractional alternatives" (Diebold and Rudebusch, Economics Letters, 1991).
- "Long Memory in Stock-Market Trading Volume" (Lobato and Velasco, Journal of Business and Economic Statistics, 2000).
- "Non-Stationary log-periodogram regression" (Velasco, J. Econometrics, 1999).
- "A bias-reduced log-periodogram regression estimator for the long memory parameter" (D.W.K. Andrews and Patrik Guggenberger, Coes Foundation for Research in Economics, Yale University, 1999).
MBA Electives - Actuarial Science Focus
Students completing the MBA with a focus in actuarial science will be prepared for the actuarial examinations of the Society of Actuaries and the Casualty Actuarial Society. The MBA Program meets the needs of students wishing to focus in actuarial science and to obtain a broad background in applied business areas that interest the actuary (such as information systems, accounting, finance, economics, marketing, and management). A student enrolled in the MBA Program should complete the courses listed below, in addition to the core.
Forecasting Time Series Data
STAT-GB.2302
3 credits
Prerequisites: COR1-GB.1305
Presented in this course are practical time series forecasting techniques with emphasis on the Box-Jenkins ARIMA (autoregressive integrated moving average) method and conditional volatility ARCH (autoregressive conditional heterogeneity) and GARCH (generalized autoregressive conditional heterogeneity) models. The course gives a mix of practical data analysis along with an introduction to the relevant theory. The ARIMA models are used to forecast series like interest spreads, while ARCH models are used in estimating and forecasting the volatility of series like stock returns and exchange rate returns. Students analyze data sets of their own choice in projects. Additional topics of interest covered in the course are methods of testing for nonstationary (Dickey-Fuller tests) as well as models for capturing seasonality as seen, for example, in series of monthly sales figures. The low-cost forecasting method of exponential smoothing is discussed, and its connection to the RiskMetricsTM methods of J. P. Morgan and GARCH models is explored. If time permits, we also study methods of forecasting multivariate time series, where information from several series is pooled to forecast a single series. The concept of co-integration or co-movement of multivariate series is discussed (interest rates being a prime example), along with their implications for forecasts. Other potential topics in the course include the use of ARCH models in value at risk (VAR) analysis and in option pricing.
STAT-GB.2302
3 credits
Prerequisites: COR1-GB.1305
Presented in this course are practical time series forecasting techniques with emphasis on the Box-Jenkins ARIMA (autoregressive integrated moving average) method and conditional volatility ARCH (autoregressive conditional heterogeneity) and GARCH (generalized autoregressive conditional heterogeneity) models. The course gives a mix of practical data analysis along with an introduction to the relevant theory. The ARIMA models are used to forecast series like interest spreads, while ARCH models are used in estimating and forecasting the volatility of series like stock returns and exchange rate returns. Students analyze data sets of their own choice in projects. Additional topics of interest covered in the course are methods of testing for nonstationary (Dickey-Fuller tests) as well as models for capturing seasonality as seen, for example, in series of monthly sales figures. The low-cost forecasting method of exponential smoothing is discussed, and its connection to the RiskMetricsTM methods of J. P. Morgan and GARCH models is explored. If time permits, we also study methods of forecasting multivariate time series, where information from several series is pooled to forecast a single series. The concept of co-integration or co-movement of multivariate series is discussed (interest rates being a prime example), along with their implications for forecasts. Other potential topics in the course include the use of ARCH models in value at risk (VAR) analysis and in option pricing.
Mathematics of Investment
STAT-GB.2309
3 credits
Prerequisites: one semester of undergraduate calculus or permission of the instructor
The course discusses mathematical and technical aspects of investments. Topics include measurement of interest and discount rates, accumulated value and present value, annuities, sinking funds, amortization of debt, and determination of yield rates on securities. Applications include bond evaluation, mortgages, capital budgeting, depreciation methods, and insurance.
STAT-GB.2309
3 credits
Prerequisites: one semester of undergraduate calculus or permission of the instructor
The course discusses mathematical and technical aspects of investments. Topics include measurement of interest and discount rates, accumulated value and present value, annuities, sinking funds, amortization of debt, and determination of yield rates on securities. Applications include bond evaluation, mortgages, capital budgeting, depreciation methods, and insurance.
Introduction to the Theory of Probability
STAT-GB.3301
3 credits
Prerequisites: two semesters of undergraduate calculus or permission of instructor.
This course covers the basic concepts of probability. Topics include the axiomatic definition of probability; combinatorial theorems; conditional probability and independent events; random variables and probability distributions; expectation of functions of random variables; special discrete and continuous distributions, including the chi-square, t, and F distributions; joint distributions with emphasis on the bivariate normal distribution; law of large numbers, central limit theorem; and moment generating functions. The theory of statistical estimation is introduced with a discussion on maximum likelihood estimation.
STAT-GB.3301
3 credits
Prerequisites: two semesters of undergraduate calculus or permission of instructor.
This course covers the basic concepts of probability. Topics include the axiomatic definition of probability; combinatorial theorems; conditional probability and independent events; random variables and probability distributions; expectation of functions of random variables; special discrete and continuous distributions, including the chi-square, t, and F distributions; joint distributions with emphasis on the bivariate normal distribution; law of large numbers, central limit theorem; and moment generating functions. The theory of statistical estimation is introduced with a discussion on maximum likelihood estimation.
Life Contingencies
STAT-GB.3335
3 credits
Prerequisites: STAT-GB.3301
Applies probability and mathematics of investment to problems of premiums and reserves on annuities and insurance policies. Topics include probabilities of mortality, laws of mortality, joint life probabilities and annuities, and multiple decrement theory. Application to pension plans is discussed.
STAT-GB.3335
3 credits
Prerequisites: STAT-GB.3301
Applies probability and mathematics of investment to problems of premiums and reserves on annuities and insurance policies. Topics include probabilities of mortality, laws of mortality, joint life probabilities and annuities, and multiple decrement theory. Application to pension plans is discussed.