I translate familiar concepts of discrete-time time-series to contnuous-time equivalent. I cover lag operators, ARMA models, the relation between levels and differences, integration and cointegration, and the Hansen-Sargent prediction formulas

We propose a Bayesian procedure for exploiting small, possibly long-lag linear predictability in the innovations of a finite order autoregression. We model the innovations as having a log-spectral density that is a continuous mean-zero Gaussian process of order 1/√T. This local embedding makes...

This paper measures option-implied skewness for individual firms and the overall stock market between 1980 and 2021, giving real-time measures of conditional micro and macro skewness. There are three key results: 1. Micro skewness is significantly procyclical, while macro skewness is acyclical;...

We model dividend and consumption growth rates as containing a small long-run predictable component and economic uncertainty (i.e., growth rate volatility) as being time-varying. The magnitudes of the predictable variation and changing volatility in growth rates, as in the data, are quite small....

How do Bitcoin prices evolve? What are the consequences for monetary policy? We answer these questions in a novel, yet simple endowment economy. There are two types of money, both useful for transactions: Bitcoins and Dollars. A central bank keeps the real value of Dollars constant, while...

This paper studies asymmetry in economic activity over the business cycle. It develops a tractable multisector model of the economy in which complementarity across inputs causes aggregate activity to be left skewed with countercyclical volatility. We then examine implications of the model...

Climate policy is complicated by the considerable compounded uncertainties over the costs and benefits of abatement. We don't even know the probability distributions for future temperatures and impacts, making cost-benefit analysis based on expected values challenging to say the least. There are...