Foundations and Trends® in Finance > Vol 6 > Issue 3

Continuous-Time Linear Models

John H. Cochrane, University of Chicago Booth School of Business and NBER, USA,
Suggested Citation
John H. Cochrane (2012), "Continuous-Time Linear Models", Foundations and Trends® in Finance: Vol. 6: No. 3, pp 165-219.

Published: 19 Nov 2012
© 2012 J. H. Cochrane
Time series analysis
G3 Corporate FinanceC58 financial econometrics
Dynamic corporate financeStructural empirical methodsDynamic capital structure models

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In this article:
1 Introduction
2 Linear Models and Lag Operators
3 Moving Average Representation and Moments
4 ARMA Models
5 Differences
6 Impulse-response Function
7 Hansen–Sargent Formulas
8 Integration and Cointegration
9 Summary


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

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Table of contents:
Linear Models and Lag Operators
Moving Average Representation and Moments
ARMA Models
Impulse-response Function
Hansen-Sargent formulas
Integration and Cointegration

Continuous-Time Linear Models

Discrete-time linear ARMA processes and lag operator notation are convenient for lots of calculations. Continuous-time representations often simplify economic models, and can handle interesting nonlinearities as well. But standard treatments of continuous-time processes typically don't mention how to adapt the discrete-time linear model concepts and lag operator methods to continuous time. Continuous-Time Linear Models attempts that translation and exposits the techniques to make the translation from familiar discrete-time ideas. The concluding section of this monograph collects the important formulas in one place. The author assumes a basic knowledge of discrete-time time-series representation methods and continuous-time representations.