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lectures/dynamic_programming/smoothing.md

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* relabeling consumption as tax collections and nonfinancial income as government expenditures
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* relabeling the consumer's debt as the government's *assets*
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For elaborations on this theme, please see {doc}`Optimal Savings II: LQ Techniques <dynamic_programming/perm_income_cons>` and later parts of this lecture.
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For elaborations on this theme, please see {doc}`Optimal Savings II: LQ Techniques <perm_income_cons>` and later parts of this lecture.
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We'll consider two closely related alternative assumptions about the consumer's
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exogenous nonfinancial income process (or in the tax-smoothing
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### Relationship to Other Lectures
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This lecture can be viewed as a followup to {doc}`Optimal Savings II: LQ Techniques <dynamic_programming/perm_income_cons>` and a warm up for a model of tax smoothing described in {doc}`opt_tax_recur <../dynamic_programming_squared/opt_tax_recur>`.
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This lecture can be viewed as a followup to {doc}`Optimal Savings II: LQ Techniques <perm_income_cons>` and a warm up for a model of tax smoothing described in {doc}`opt_tax_recur <../dynamic_programming_squared/opt_tax_recur>`.
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Linear-quadratic versions of the Lucas-Stokey tax-smoothing model are described in {doc}`lqramsey <../dynamic_programming_squared/lqramsey>`.
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lectures/more_julia/testing.md

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We can see ours by opening it in Atom
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```{code-block} julia
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```
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# Documentation: http://docs.travis-ci.com/user/languages/julia/
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language: julia
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os:

lectures/time_series_models/classical_filtering.md

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\, X_t
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```
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This formula is useful in solving stochastic versions of problem 1 of lecture {doc}`Classical Control with Linear Algebra <time_series_models/lu_tricks>` in which the randomness emerges because $\{a_t\}$ is a stochastic
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This formula is useful in solving stochastic versions of problem 1 of lecture {doc}`Classical Control with Linear Algebra <lu_tricks>` in which the randomness emerges because $\{a_t\}$ is a stochastic
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process.
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The problem is to maximize
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### Implementation
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Code that computes solutions to LQ control and filtering problems using the methods described here and in {doc}`Classical Control with Linear Algebra <time_series_models/lu_tricks>` can be found in the file [control_and_filter.jl](https://github.com/QuantEcon/QuantEcon.lectures.code/blob/master/lu_tricks/control_and_filter.jl).
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Code that computes solutions to LQ control and filtering problems using the methods described here and in {doc}`Classical Control with Linear Algebra <lu_tricks>` can be found in the file [control_and_filter.jl](https://github.com/QuantEcon/QuantEcon.lectures.code/blob/master/lu_tricks/control_and_filter.jl).
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Here's how it looks
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lectures/tools_and_techniques/kalman.md

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```{index} single: Kalman Filter; Programming Implementation
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```
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The [QuantEcon.jl]() package is able to implement the Kalman filter by using methods for the type `Kalman`
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The [QuantEcon.jl](http://quantecon.org/quantecon-jl) package is able to implement the Kalman filter by using methods for the type `Kalman`
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* Instance data consists of:
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* The parameters $A, G, Q, R$ of a given model

lectures/tools_and_techniques/numerical_linear_algebra.md

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The list of specialized packages for these tasks is enormous and growing, but some of the important organizations to
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look at are [JuliaMatrices](https://github.com/JuliaMatrices) , [JuliaSparse](https://github.com/JuliaSparse), and [JuliaMath](https://github.com/JuliaMath)
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*NOTE*: As this section uses advanced Julia techniques, you may wish to review multiple-dispatch and generic programming in {doc}`introduction to types <../getting_starting_julia/introduction_to_types>`, and consider further study on {doc}`generic programming <../more_julia/generic_programming>`.
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*NOTE*: As this section uses advanced Julia techniques, you may wish to review multiple-dispatch and generic programming in {doc}`introduction to types <../getting_started_julia/introduction_to_types>`, and consider further study on {doc}`generic programming <../more_julia/generic_programming>`.
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The theme of this lecture, and numerical linear algebra in general, comes down to three principles:
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### QR Decomposition
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{ref}`Previously <qr_decomposition>`, we learned about applications of the QR decomposition to solving the linear least squares.
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Previously, we learned about applications of the QR decomposition to solving the linear least squares.
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While in principle the solution to the least-squares problem
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The QR decomposition is a decomposition $A = Q R$ where $Q$ is an orthogonal matrix (i.e., $Q'Q = Q Q' = I$) and $R$ is
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an upper triangular matrix.
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Given the {ref}`previous derivation <qr_decomposition>`, we showed that we can write the least-squares problem as
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Given the previous derivation, we showed that we can write the least-squares problem as
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the solution to
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$$
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## Continuous-Time Markov Chains (CTMCs)
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In the previous lecture on {doc}`discrete-time Markov chains <mc>`, we saw that the transition probability
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In the previous lecture on {doc}`discrete-time Markov chains <finite_markov>`, we saw that the transition probability
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between state $x$ and state $y$ was summarized by the matrix $P(x, y) := \mathbb P \{ X_{t+1} = y \,|\, X_t = x \}$.
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As a brief introduction to continuous time processes, consider the same state space as in the discrete

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