# I. Syllabus:

Testing the predictive power of regressors is often a task in applied econometric work: think of fundamentals supposedly predicting stock returns or of growth regressions. The usual inferential tools (e.g. the t statistic) provide biased inference when the regressor is persistent.
The course sets to characterize the problem and to discuss the solutions available in the literature. To this end, the course provides (not always very technical) knowledge about asymptotics for time series models. It reviews basic notions and tools such as convergence types or Op(•)/op(•) notation, as well as properties of dependent stochastic processes (in particular the martingale difference property). We'll pay more attention to linear processes, covering long memory processes as well, and the theory part is rounded up by some asymptotics for nonstationary processes (functional CLTs, for instance).
After successfully participating, you should be able to understand and analyze the asymptotics of specific solutions to the posed predictive problem, and expand your understanding to more general econometric inferential problems.

• Introduction
• Why bother?
• Optimal forecasts under general loss functions
• A primer on persistence
• Basic tools (Review)
• Limits and convergence
• Useful inequalities
• "Big-Oh" and "Little-Oh" notation (definition, examples)
• Stochastic processes (Review)
• Stationarity, ergodicity and mixing
• Uniform properties
• Uncorrelatedness and martingale difference [md] sequences
• Linear processes
• Existence and properties
• Short vs. long memory
• Fractional integration
• Limiting theorems
• The iid case
• Condition sets for md sequences/arrays
• Linear processes and the Phillips-Solo device
• Asymptotics for nonstationary processes
• Stochastic analysis (Wiener processes, stochastic integrals)
• (Weak) Convergence to functionals of Wiener processes
• Example: the basic problem in predictive regressionsn
• Predictive regressions revisited
• Bonferroni and other approaches
• Extensions

# V. Materials:

• Lecture notes, to be provided.
• Textbooks: White (2001), Davidson (1994).
• The basic problem: Stambaugh (1999), Elliott and Stock (1994), Maynard and Phillips (2001).
• Previous solutions: Campbell and Yogo (2006), Rodrigues and Rubia (2011).
• Another approach: Breitung and Demetrescu (2013).

# VI. References:

• Breitung, J. and M. Demetrescu (2013). Instrumental Variable and Variable Addition Based Inference in Predictive Regressions. Working paper, Department of Economics, University of Bonn.
• Campbell, J. Y. and M. Yogo (2006). Efficient Tests of Stock Return Predictability. Journal of Financial Economics 81 (1), 27-60.
• Davidson, J. (1994). Stochastic Limit Theory. Oxford University Press.
• Elliott, G. and J. H. Stock (1994). Inference in Time Series Regression when the Order of Integration of a Regressor Is Unknown. Econometric Theory 10 (3-4), 672-700.
• Maynard, A. and P. C. B. Phillips (2001). Rethinking an Old Empirical Puzzle: Econometric evidence on the forward discount anomaly. Journal of Applied Econometrics 16 (6), 671-708.
• Rodrigues, P. M. M. and A. Rubia (2011). A Class of Robust Tests in Augmented Predictive Regressions. Working paper, Banco de Portugal: Link
• Stambaugh, R. F. (1999). Predictive Regressions. Journal of Financial Economics 54 (3), 375-421.
• White, H. (2001). Asymptotic Theory for Econometricians. Academic Press New York.