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An Introduction To The Theory Of Point Processes

高等学校計算数学学報 = Numerical mathematics

The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

The commutation matrix: Some properties and applications

https://www.federalreserve.gov/pubs/feds/2014/201447/201447pap.pdf

OccBin: A toolkit for solving dynamic models with occasionally binding constraints easily

This paper presents a fast, simple and intuitive algorithm for simulation of linear dynamic stochastic general equilibrium models with inequality constraints. The algorithm handles both the computation of impulse responses, and stochastic simulation, and can deal with arbitrarily many bounded variables. To illustrate the usefulness and efficiency of this algorithm we provide two applications according to the zero lower bound (ZLB) on nominal interest rates. Our solution principle is much faster than comparable methods. We therefore expect this algorithm to be very helpful also for estimation procedures, and for a wide range of applications apart from monetary policy analysis.

Efficient Simulation of DSGE Models with Inequality Constraints

We present the first necessary and sufficient conditions for there to be a unique perfect-foresight solution to an otherwise linear dynamic model with occasionally binding constraints, given a fixed terminal condition. We derive further conditions on the existence of a solution in such models. These results give determinacy conditions for models with occasionally binding constraints, much as Blanchard and Kahn (1980) did for linear models. In an application, we show that widely used New Keynesian models with endogenous states possess multiple perfect foresight equilibrium paths when there is a zero lower bound on nominal interest rates, even when agents believe that the central bank will eventually attain its long-run, positive inflation target. This illustrates that a credible long-run inflation target does not render the Taylor principle sufficient for determinacy in the presence of the zero lower bound. However, we show that price level targeting does restore determinacy in these situations.

https://www.econstor.eu/bitstream/10419/204472/1/TheoryPaperBody.pdf

Existence and uniqueness of solutions to dynamic models with occasionally binding constraints.

This paper builds a dynamic stochastic general equilibrium (DSGE) model of endogenous growth that is capable of generating substantial degrees of endogenous persistence in productivity. When products go out of patent protection, the rush of entry into their production destroys incentives for process improvements. Consequently, old production processes are enshrined in industries producing non-protected products, resulting in aggregate productivity persistence. Our model also generates sizeable delayed movements in productivity in response to preference shocks, providing a form of endogenous news shock. Finally, if we calibrate our model to match a high aggregate mark-up then we can replicate the negative response of hours to a positive technology shock, even without the inclusion of any frictions.

Products, patents and productivity persistence: A DSGE model of endogenous growth

We present the first necessary and sufficient conditions for the existence of a unique perfect-foresight solution, returning to a given steady-state, in an otherwise linear model with occasionally binding constraints. We derive further conditions on the existence of a solution in such models, and provide a proof of the inescapability of the “curse of dimensionality” for them. We also construct the first solution algorithm for these models that is guaranteed to return a solution in finite time, if one exists. When extended to allow for other non-linearities and future uncertainty, our solution algorithm is shown to produce fast and accurate simulations. In an application, we show that widely used New Keynesian models with endogenous states possess multiple perfect foresight equilibrium paths when there is a zero lower bound on nominal interest rates. However, we show that price level targeting is sufficient to restore determinacy in these situations.

Existence, uniqueness and computation of solutions to dynamic models with occasionally binding constraints

We construct the first algorithm for the perfect foresight solution of otherwise linear models with occasionally binding constraints, with fixed terminal conditions, that is guaranteed to return a solution in finite time, if one exists. We also provide a proof of the inescapability of the “curse of dimensionality” for this problem when nothing is known a priori about the model. We go on to extend our algorithm to deal with stochastic simulation, other non-linearities, and future uncertainty. We show that the resulting algorithm produces fast and accurate simulations of a range of models with occasionally binding constraints.

https://www.econstor.eu/bitstream/10419/144569/1/ComputationalPaper_rev.pdf

Tom Holden

Papers

Efficient Simulation of DSGE Models with Inequality Constraints

Existence and uniqueness of solutions to dynamic models with occasionally binding constraints.

Products, patents and productivity persistence: A DSGE model of endogenous growth

Existence, uniqueness and computation of solutions to dynamic models with occasionally binding constraints

Computation of solutions to dynamic models with occasionally binding constraints.