### Use QuantEcon Package

Source: https://github.com/quantecon/quantecon.jl/blob/master/docs/src/index.md

Include this command in your Julia session to start using the QuantEcon package after installation.

```julia
using QuantEcon
```

--------------------------------

### LQ Constructor Example

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lq-control.md

Demonstrates how to initialize an LQ type for a simple consumption-savings problem and solve for the infinite horizon stationary values.

```julia
using QuantEcon

# Simple consumption-savings problem
# Minimize E[sum(beta^t * (x_t^2 + u_t^2))]
# Subject to: x_{t+1} = 0.9*x_t + u_t + w_t

Q = 1.0      # cost of control
R = 1.0      # cost of deviation
A = 0.9      # state transition
B = 1.0      # control effect
C = 1.0      # shock effect

lq = LQ(Q, R, A, B, C; bet=0.95)

# Solve for infinite horizon
stationary_values!(lq)
println("Policy F: ", lq.F)
println("Value P: ", lq.P)
```

--------------------------------

### Linear Quadratic Control in Julia

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/README.md

Illustrates the setup and usage of Linear Quadratic (LQ) control problems with QuantEcon.jl. Ensure the QuantEcon package is installed.

```julia
using QuantEcon

lq = LQ(1.0, 1.0, 0.95, 1.0, 0.1; bet=0.95)
P, F, d = stationary_values(lq)

x0 = 1.0
x_path, u_path, w_path = compute_sequence(lq, x0, ts_length=100)
```

--------------------------------

### Install QuantEcon Package

Source: https://github.com/quantecon/quantecon.jl/blob/master/docs/src/index.md

Use this command to install the QuantEcon package using the Julia package manager.

```julia
Pkg.add("QuantEcon")
```

--------------------------------

### DiscreteDP Constructor Example (Dense)

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/discrete-dp.md

Demonstrates how to create a DiscreteDP problem instance using the dense formulation with reward and transition matrices. This is suitable for smaller problems where all state-action transitions are explicitly defined.

```julia
using QuantEcon

# Simple 2-state, 2-action problem
R = [10.0 5.0;   # state 1: action 1 gives 10, action 2 gives 5
     8.0  2.0]   # state 2: action 1 gives 8, action 2 gives 2

Q = zeros(2, 2, 2)
# State 1 transitions:
Q[1, 1, :] = [0.8, 0.2]  # action 1
Q[1, 2, :] = [0.6, 0.4]  # action 2
# State 2 transitions:
Q[2, 1, :] = [0.7, 0.3]  # action 1
Q[2, 2, :] = [0.5, 0.5]  # action 2

ddp = DiscreteDP(R, Q, 0.95)

# Check number of states
println(num_states(ddp))  # 2
```

--------------------------------

### Install QuantEcon Package in Julia

Source: https://github.com/quantecon/quantecon.jl/blob/master/README.md

To install the QuantEcon package, open the Julia package manager (Pkg) and type 'add QuantEcon'. This is the standard method for adding Julia packages.

```julia
add QuantEcon
```

--------------------------------

### LAE Constructor and Estimation Example

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lae.md

Demonstrates how to create a Look-Ahead Estimator (LAE) instance with a custom kernel and observed data, then estimate the stationary density at multiple points. Requires QuantEcon and Distributions packages.

```julia
using QuantEcon
using Distributions

# Univariate AR(1) example
# y_{t+1} = 0.9*y_t + σ*ε_t, ε_t ~ N(0,1)

rho = 0.9
sigma = 0.1

# Stochastic kernel (transition density)
function kernel(x, y)
    # Density of y given x under AR(1)
    mu = rho .* x
    sig = sigma
    return pdf.(Normal.(mu, sig), y)
end

# Simulate data
n = 1000
X = zeros(n)
for i in 2:n
    X[i] = rho * X[i-1] + sigma * randn()
end

# Create estimator
lae = LAE(kernel, X)

# Estimate density at various points
y_vals = -0.5:0.1:0.5
psi_hat = lae_est(lae, y_vals)

println("Estimated density: ", psi_hat)
```

--------------------------------

### Kalman Filter Smoothing Example

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/kalman.md

Demonstrates how to set up a Kalman filter, generate synthetic data, and then smooth the state estimates using the `smooth` function. Ensure the Kalman filter is initialized with prior state and covariance.

```julia
using QuantEcon

# Setup
A = [0.95]'
G = [1.0]'
Q = [0.01]'
R = [0.1]'
kal = Kalman(A, G, Q, R)
set_state!(kal, [0.0], [1.0])

# Generate synthetic data and smooth
T = 100
true_states = randn(1, T)
y = G * true_states + sqrt(0.1) * randn(1, T)

x_sm, logL, Sigma_sm = smooth(kal, y)
println("Log-likelihood: ", logL)
```

--------------------------------

### Example: Integrating exp(x) using do_quad

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/quadrature.md

Demonstrates how to use `do_quad` with nodes and weights generated by `qnwlege` to numerically integrate the exponential function. It also prints the absolute error compared to the true integral.

```julia
using QuantEcon

# Integrate exp(x) from 0 to 1
nodes, weights = qnwlege(20, 0.0, 1.0)
integral = do_quad(exp, nodes, weights)
true_integral = exp(1) - 1
println("Error: ", abs(integral - true_integral))
```

--------------------------------

### Example: Simple Consumption Problem with Robustness

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/robust-lq.md

Demonstrates how to set up and solve a simple consumption problem with robustness using the RBLQ type and the `robust_rule` function.

```julia
using QuantEcon

# Simple consumption problem with robustness
Q = 1.0      # cost of control
R = 1.0      # cost of deviation
A = 0.95     # state persistence
B = 1.0      # control effect
C = 0.1      # shock volatility
beta = 0.99
theta = 0.1  # robustness parameter

rlq = RBLQ(Q, R, A, B, C, beta, theta)

# Solve
F, K, P = robust_rule(rlq)
println("Optimal policy F: ", F)
println("Worst-case K: ", K)
```

--------------------------------

### Demonstrate file handling during branch checkout

Source: https://github.com/quantecon/quantecon.jl/wiki/Git:-Branching

This example shows how Git handles committed and uncommitted files when switching branches. Only committed files are updated; uncommitted files remain.

```bash
qe_website|master ⇒ git checkout -b topic
Switched to a new branch 'topic'

qe_website|topic ⇒ echo "This is file1" > file1.txt

qe_website|topic ⇒ echo "This is file2" > file2.txt

qe_website|topic ⇒ git add file1.txt

qe_website|topic ⇒ git commit -m "Added file1.txt"
[topic 0b07059] Added file1.txt
 1 file changed, 1 insertion(+)
 create mode 100644 file1.txt

qe_website|topic ⇒ git status
On branch topic
Untracked files:
  (use "git add <file>..." to include in what will be committed)

	file2.txt

nothing added to commit but untracked files present (use "git add" to track)

qe_website|topic ⇒ ls file*.txt
file1.txt file2.txt

qe_website|topic ⇒ git checkout master
Switched to branch 'master'
Your branch is ahead of 'origin/master' by 1 commit.
  (use "git push" to publish your local commits)

qe_website|master ⇒ git status
On branch master
Your branch is ahead of 'origin/master' by 1 commit.
  (use "git push" to publish your local commits)

Untracked files:
  (use "git add <file>..." to include in what will be committed)

	file2.txt

nothing added to commit but untracked files present (use "git add" to track)

qe_website|master ⇒ ls file*.txt
file2.txt
```

--------------------------------

### Kalman Filter Example: 1D Local Level Model

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/kalman.md

Demonstrates initializing and using the Kalman filter for a 1D local level model. It shows how to set the initial state estimate and perform an update with an observation.

```julia
using QuantEcon

# 1D example: local level model
A = [1.0]
G = [1.0]
Q = [0.1]
R = [1.0]

kal = Kalman(A, G, Q, R)

# Set initial state estimate
set_state!(kal, [0.0], [1.0])

# Filter observation
y = 0.5
update!(kal, y)
println("Updated estimate: ", kal.cur_x_hat)
```

--------------------------------

### Create and Checkout a New Branch

Source: https://github.com/quantecon/quantecon.jl/wiki/Git:-Branching

Use this command to create a new branch and switch to it in a single step. This is useful for starting new features or fixes in isolation.

```git
git checkout -b <BRANCH NAME>
```

--------------------------------

### Kalman Filter Update Example: Multiple Observations

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/kalman.md

Demonstrates the complete one-step Kalman filter update process using multiple observations. It initializes the filter, sets the initial state, and then iteratively updates the filter with each observation, printing the filtered estimate.

```julia
using QuantEcon

# Create filter
A = [0.9]'
G = [1.0]'
Q = [0.1]'
R = [0.5]'
kal = Kalman(A, G, Q, R)

# Initial state
set_state!(kal, [0.0], [1.0])

# Process observations
observations = [0.1, 0.2, 0.15, 0.25, 0.3]
for y in observations
    update!(kal, [y])
    println("Filtered estimate: ", kal.cur_x_hat[1])
end
```

--------------------------------

### Compute Optimal State and Control Sequences

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lq-control.md

Generates the optimal state and control sequences for a given LQ problem starting from an initial state x0 and a specified time series length.

```julia
function compute_sequence(lq::LQ, x0::ScalarOrArray, ts_length::Integer=100)
```

--------------------------------

### Kalman Filter Initialization and Update in Julia

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/README.md

Demonstrates initializing and updating a Kalman filter for state space models using QuantEcon.jl. Requires the QuantEcon package and observation data.

```julia
using QuantEcon

A = [0.95]'
G = [1.0]'
Q = [0.1]'
R = [0.5]'

kal = Kalman(A, G, Q, R)
set_state!(kal, [0.0], [1.0])

# Process observations
for y in observations
    update!(kal, [y])
end
```

--------------------------------

### Get Period of Markov Chain

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Calculates and returns the period of a Markov chain, which is the GCD of cycle lengths.

```julia
function period(mc::MarkovChain)
end
```

--------------------------------

### Get Number of States in Markov Chain

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Retrieves the total number of states in a given MarkovChain object.

```julia
mc = MarkovChain([0.9 0.1; 0.2 0.8])
n = n_states(mc)  # Returns 2
```

--------------------------------

### Markov Chain Analysis and Simulation in Julia

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/README.md

Demonstrates how to create, simulate, and analyze a Markov chain using QuantEcon.jl. Requires the QuantEcon package.

```julia
using QuantEcon

# Create chain
P = [0.9 0.1; 0.2 0.8]
mc = MarkovChain(P)

# Simulate
path = simulate(mc, 100; init=1)

# Analyze
statdists = stationary_distributions(mc)
println(is_irreducible(mc))
```

--------------------------------

### LSS Constructor and Usage

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lss.md

Demonstrates how to create an LSS instance and simulate a path. The LSS type represents Gaussian linear state space models.

```APIDOC
## LSS Constructor and Usage

### Description

This section shows how to instantiate the `LSS` type and use it to simulate state and observation paths. The `LSS` type is fundamental for working with Gaussian linear state space models in QuantEcon.jl.

### Constructor

```julia
LSS(A, C, G, H, mu_0, Sigma_0=zeros(size(G,2), size(G,2)))
LSS(A, C, G; H=zeros(size(G,1)), mu_0=zeros(size(G,2)), Sigma_0=zeros(size(G,2), size(G,2)))
```

### Parameters

| Parameter | Type | Required | Default |
|---|---|---|---|
| `A` | `Matrix` | Yes | — |
| `C` | `Matrix` | Yes | — |
| `G` | `Matrix` | Yes | — |
| `H` | `Matrix` | No | `zeros(k)` |
| `mu_0` | `Vector` | No | `zeros(n)` |
| `Sigma_0` | `Matrix` | No | `zeros(n,n)` |

### Example

```julia
using QuantEcon

# AR(1) model: x_{t+1} = 0.9*x_t + w_t, y_t = x_t + v_t
A = [0.9]
C = [1.0]
G = [1.0]
H = [0.1]

lss = LSS(A, C, G, H=H)

# Simulate
x, y = simulate(lss, 100)
println("State shape: ", size(x))     # (1, 100)
println("Observation shape: ", size(y))  # (1, 100)
```
```

--------------------------------

### Create and Inspect a MarkovChain

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Demonstrates how to create a MarkovChain instance with custom state values and access its properties.

```julia
using QuantEcon

# Create a 2-state Markov chain
P = [0.9 0.1; 0.2 0.8]
mc = MarkovChain(P, ["Good", "Bad"])

# Access properties
println(n_states(mc))        # 2
println(mc.state_values)     # ["Good", "Bad"]
```

--------------------------------

### Discrete Dynamic Programming Solver in Julia

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/README.md

Illustrates setting up and solving a discrete dynamic programming problem using QuantEcon.jl with the Value Function Iteration (VFI) algorithm. Ensure QuantEcon is imported.

```julia
using QuantEcon

R = [10.0 5.0; 8.0 2.0]
Q = zeros(2, 2, 2)
Q[1, 1, :] = [0.8, 0.2]
Q[1, 2, :] = [0.6, 0.4]
Q[2, 1, :] = [0.7, 0.3]
Q[2, 2, :] = [0.5, 0.5]

ddp = DiscreteDP(R, Q, 0.95)
result = solve(ddp, algorithm=VFI())
```

--------------------------------

### Get Impulse Response of ARMA Process

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/arma.md

Retrieves the impulse response of an ARMA model for a specified horizon length. The impulse_length must be greater than or equal to the number of AR coefficients (p). Includes visualization using Plots.

```julia
arma = ARMA([0.8, 0.1], [0.5, 0.2], 1.0)
psi = impulse_response(arma, impulse_length=50)

# Visualize the response
import Plots
Plots.plot(psi, xlabel="Lag", ylabel="Response", title="Impulse Response")
```

--------------------------------

### compute_fixed_point

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/fixed-point.md

Finds the fixed point of a function `T` by repeatedly applying it starting from an initial guess `v` until convergence or a maximum number of iterations is reached. It supports various convergence tolerances, verbosity levels, and works with different data types for `v`.

```APIDOC
## compute_fixed_point

### Description
Finds the fixed point of a function `T` by repeatedly applying it starting from an initial guess `v` until convergence or a maximum number of iterations is reached. It supports various convergence tolerances, verbosity levels, and works with different data types for `v`.

### Signature
```julia
function compute_fixed_point(T::Function, v::TV; err_tol=1e-4, max_iter=100, verbose=2, print_skip=10) where TV
```

### Parameters
#### Path Parameters
- **T** (Function) - Required - Operator function: v_new = T(v_old)
- **v** (Object) - Required - Initial guess for fixed point

#### Query Parameters
- **err_tol** (Real) - Optional - Convergence tolerance. Default: `1e-4`
- **max_iter** (Integer) - Optional - Maximum number of iterations. Default: `100`
- **verbose** (Integer) - Optional - Verbosity level (0, 1, or 2). Default: `2`
- **print_skip** (Integer) - Optional - Print interval when verbose=2. Default: `10`

### Returns
- **v** (TV) - Fixed point approximation (same type as input)

### Example
```julia
using QuantEcon

# Find fixed point of T(x) = 4μx(1-x) for μ=0.3
mu = 0.3
T(x) = 4.0 * mu * x * (1.0 - x)

x_star = compute_fixed_point(x -> T(x), 0.4)
println("Fixed point: ", x_star)
println("T(x*) - x* = ", T(x_star) - x_star)

# With vector valued function
A = [0.9 0.1; 0.1 0.9]
b = [1.0, 1.0]
T_mat(v) = A*v + b
v_star = compute_fixed_point(T_mat, [0.0, 0.0])
println("Fixed point vector: ", v_star)
```
```

--------------------------------

### ARMA Process Analysis and Simulation in Julia

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/README.md

Shows how to define and analyze an Autoregressive Moving-Average (ARMA) process using QuantEcon.jl. The QuantEcon package must be loaded.

```julia
using QuantEcon

arma = ARMA(0.5, 0.0, 1.0)
w, spect = spectral_density(arma)
acov = autocovariance(arma)
path = simulation(arma, ts_length=1000)
```

--------------------------------

### DiscreteDP Constructor (State-Action)

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/discrete-dp.md

Constructs a DiscreteDP problem using the state-action formulation. This is efficient for problems with sparse or structured state-action spaces.

```APIDOC
## DiscreteDP Constructor (State-Action)

### Description
Constructs a `DiscreteDP` object for a state-action formulation.

### Signature
```julia
DiscreteDP(R::Vector{T}, Q::Matrix{T}, beta::Real, s_indices::Vector, a_indices::Vector)
```

### Parameters
| Parameter | Type | Required | Default | Description |
|-----------|------|----------|---------|-------------|
| `R` | `Vector{T}` | Yes | — | Reward vector (L) |
| `Q` | `Matrix{T}` | Yes | — | Transition probability matrix (L×n) |
| `beta` | `Real` | Yes | — | Discount factor ∈ [0,1] |
| `s_indices` | `Vector` | Yes | — | State indices for each state-action pair |
| `a_indices` | `Vector` | Yes | — | Action indices for each state-action pair |
```

--------------------------------

### simulate!

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Fills a pre-allocated array with sample paths of a Markov chain. This is an in-place operation that can be more memory-efficient for large simulations. It supports specifying initial states for multiple paths.

```APIDOC
## simulate!

### Description

Fill a pre-allocated array with sample paths of a Markov chain. This is an in-place operation.

### Method

`simulate!(X::Union{AbstractVector,AbstractMatrix}, mc::MarkovChain; init=rand(1:n_states(mc), size(X,2)))

### Parameters

#### Path Parameters

- **X** (Matrix) - Required - Pre-allocated n×m matrix to fill with m sample paths of length n
- **mc** (MarkovChain) - Required - Markov chain instance
- **init** (Union{Int, Vector}) - Optional - Random - Initial state(s): scalar or vector of length matching number of paths

### Notes

- If `init` is scalar, same initial state used for all paths
- If `init` is vector with fewer elements than columns in X, init cycles through columns
- State values from mc.state_values appear in X, not indices

### Returns

- **X** (Matrix) - The input array filled with sample paths
```

--------------------------------

### Create and Commit to a New Branch

Source: https://github.com/quantecon/quantecon.jl/wiki/Git:-Branching

Create a new file, add it to staging, and commit it to the current branch. Use 'git status' to observe changes.

```bash
qe_website|olg-basic ⇒ vim website/olg_basic.rst  # then edit file and save

qe_website|olg-basic ⇒ cat website/olg_basic.rst  # display file contents
.. _olg_basic:

************************************
Overlapping Generations (OLG) Models
************************************

Overview
========

The OLG model is the macro workhorse model for exploring life-cycle dynamics
of economic agnets.

qe_website|olg-basic ⇒ git status
On branch olg-basic
Untracked files:
  (use "git add <file>..." to include in what will be committed)

	website/olg_basic.rst

nothing added to commit but untracked files present (use "git add" to track)
qe_website|olg-basic ⇒ git add .

qe_website|olg-basic ⇒ git commit -m "OLG: added stub for basic olg module"
[olg-basic 4e0b68b] OLG: added stub for basic olg module
 1 file changed, 11 insertions(+)
 create mode 100644 website/olg_basic.rst

qe_website|olg-basic ⇒ git status
On branch olg-basic
nothing to commit, working directory clean
```

--------------------------------

### Create and switch to a new branch

Source: https://github.com/quantecon/quantecon.jl/wiki/Git:-Branching

Use the shortcut `git checkout -b <BRANCH NAME>` to create a new branch and immediately switch to it.

```bash
qe_website|olg-basic ⇒ git checkout -b olg-basic
Switched to a new branch 'olg-basic'
```

--------------------------------

### Define and Sample DiscreteRV

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/discrete-rv.md

Demonstrates how to create a DiscreteRV object with given probabilities and draw a single sample or multiple samples from it.

```julia
using QuantEcon

# Create a discrete RV with probabilities [0.2, 0.5, 0.3]
q = [0.2, 0.5, 0.3]
drv = DiscreteRV(q)

# Draw a sample (returns 1-based index)
sample = rand(drv)
println("Drew state: ", sample)

# Draw multiple samples
samples = rand(drv, 1000)
```

--------------------------------

### DiscreteRV Constructor and Usage

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/discrete-rv.md

Demonstrates how to create a DiscreteRV object and draw samples from it. The constructor automatically computes the cumulative distribution function and validates the probabilities.

```APIDOC
## DiscreteRV

Type for discrete random variable with specified probability distribution.

### Signature

```julia
mutable struct DiscreteRV{TV1<:AbstractVector, TV2<:AbstractVector}
    q::TV1              # Probabilities (must sum to 1)
    Q::TV2              # Cumulative probabilities
end
```

### Constructor

```julia
DiscreteRV(q::AbstractVector)
```

Automatically computes cumulative sum. Validates that probabilities sum to 1 (within 1e-10 tolerance).

### Example

```julia
using QuantEcon

# Create a discrete RV with probabilities [0.2, 0.5, 0.3]
q = [0.2, 0.5, 0.3]
drv = DiscreteRV(q)

# Draw a sample (returns 1-based index)
sample = rand(drv)
println("Drew state: ", sample)

# Draw multiple samples
samples = rand(drv, 1000)
```
```

--------------------------------

### Simulate Markov Chain Path into Pre-allocated Array

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Fills a pre-allocated array with sample paths of a Markov chain. This is useful for performance when generating many paths. The `init` parameter can be a scalar or a vector for multiple initial states.

```julia
function simulate!(X::Union{AbstractVector,AbstractMatrix}, mc::MarkovChain; init=rand(1:n_states(mc), size(X,2)))
    # ... implementation details ...
end
```

--------------------------------

### Kalman Filter Constructor

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/kalman.md

Initializes a Kalman filter instance. It sets the state transition matrix (A), observation matrix (G), state noise covariance (Q), and observation noise covariance (R). Initial state estimate and covariance are set to zeros.

```julia
Kalman(A, G, Q, R)
```

--------------------------------

### Simulate Bernoulli Trials

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/discrete-rv.md

Shows how to use DiscreteRV to simulate a Bernoulli random variable and count outcomes from multiple trials.

```julia
using QuantEcon

# Bernoulli(0.7)
drv = DiscreteRV([0.3, 0.7])

# Simulate coin flips
flips = rand(drv, 100)  # 1 = heads (prob 0.3), 2 = tails (prob 0.7)
println("Heads: ", count(x -> x == 1, flips))
```

--------------------------------

### simulate

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Generates one sample path of a Markov chain. The path contains state values. You can specify the initial state or let it be chosen randomly.

```APIDOC
## simulate

### Description

Generate one sample path of a Markov chain. The path contains state values.

### Method

`simulate(mc::MarkovChain, ts_length::Int; init::Int=rand(1:n_states(mc)))

### Parameters

#### Path Parameters

- **mc** (MarkovChain) - Required - Markov chain instance
- **ts_length** (Int) - Required - Length of sample path
- **init** (Int) - Optional - Random - Initial state index (1-based)

#### Response

- **X** (Vector) - Sample path of length ts_length with state values

### Request Example

```julia
mc = MarkovChain([0.9 0.1; 0.2 0.8], ["Good", "Bad"])
path = simulate(mc, 100; init=1)
println(path[1:10])  # First 10 periods
```
```

--------------------------------

### DiscreteDP Constructor (Dense)

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/discrete-dp.md

Constructs a DiscreteDP problem using the dense formulation. This is suitable for problems where the number of states and actions are relatively small.

```APIDOC
## DiscreteDP Constructor (Dense)

### Description
Constructs a `DiscreteDP` object for a dense problem formulation.

### Signature
```julia
DiscreteDP(R::Array{T,2}, Q::AbstractArray{T,3}, beta::Real)
```

### Parameters
| Parameter | Type | Required | Default | Description |
|-----------|------|----------|---------|-------------|
| `R` | `Matrix{T}` | Yes | — | Reward matrix (n×m) |
| `Q` | `AbstractArray{T,3}` | Yes | — | Transition probability tensor (n×m×n) |
| `beta` | `Real` | Yes | — | Discount factor ∈ [0,1] |
```

--------------------------------

### gridmake!

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/utilities.md

An in-place version of `gridmake` that fills a pre-allocated output matrix with all combinations of the input arrays.

```APIDOC
## gridmake!

### Description

In-place version of gridmake.

### Signature

```julia
function gridmake!(out::AbstractMatrix, arrays::Union{AbstractVector,AbstractMatrix}...)
```

### Parameters

#### Parameters
- **out** (`AbstractMatrix`) - Pre-allocated output array
- **arrays** (`Union{Vector, Matrix}...`) - Input arrays

### Returns

- **out** (`AbstractMatrix`) - The filled output array
```

--------------------------------

### In-place Grid Generation with gridmake!

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/utilities.md

An in-place version of `gridmake` that fills a pre-allocated output array with all combinations of input arrays. This can be more memory-efficient for large datasets.

```julia
function gridmake!(out::AbstractMatrix, arrays::Union{AbstractVector,AbstractMatrix}...)
    # ... implementation details ...
end
```

--------------------------------

### LQ Constructor

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lq-control.md

Constructs an LQ type instance for representing and solving linear quadratic optimal control problems. It supports both infinite and finite horizon formulations.

```APIDOC
## LQ Constructor

### Description
Constructs an LQ type instance for representing and solving linear quadratic optimal control problems. It supports both infinite and finite horizon formulations.

### Signature
```julia
LQ(Q, R, A, B, C=fill(zero(eltype(R)), size(R,1)), N=zero(B'A); bet=1.0, capT=nothing, rf=fill(NaN, size(R)...))
```

### Parameters

| Parameter | Type | Required | Default | Description |
|-----------|------|----------|---------|-------------|
| `Q` | `ScalarOrArray` | Yes | — | k×k payoff for control, symmetric nonnegative definite |
| `R` | `ScalarOrArray` | Yes | — | n×n payoff for state, symmetric nonnegative definite |
| `A` | `ScalarOrArray` | Yes | — | n×n state transition coefficient |
| `B` | `ScalarOrArray` | Yes | — | n×k control transition coefficient |
| `C` | `ScalarOrArray` | No | `zeros(n)` | n×j shock transition coefficient |
| `N` | `ScalarOrArray` | No | `zeros(k,n)` | k×n cross product term |
| `bet` | `Real` | No | `1.0` | Discount factor ∈ [0,1] |
| `capT` | `Union{Int, Nothing}` | No | `nothing` | Terminal period; `nothing` means infinite |
| `rf` | `ScalarOrArray` | No | `fill(NaN,...)` | n×n terminal payoff matrix |
```

--------------------------------

### MVNSampler

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/other-functions.md

Initializes a multivariate normal sampler.

```APIDOC
## MVNSampler

### Description
Multivariate normal sampler.

### Signature
```julia
MVNSampler(mu::Vector, Sigma::Matrix)
```

### Methods
- `rand(sampler)` — Draw one sample
- `rand(sampler, n)` — Draw n samples
```

--------------------------------

### LAE Estimation at Single and Multiple Points

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lae.md

Shows how to use the `lae_est` function to estimate the stationary density at a single point and across a grid of values. Includes plotting the estimated density. Requires QuantEcon.

```julia
using QuantEcon

# Create kernel
rho = 0.9
sigma = 0.1
function kernel(x, y)
    mu = rho .* x
    return pdf.(Normal.(mu, sigma), y)
end

# Simulate AR(1)
n = 1000
X = zeros(n)
for i in 2:n
    X[i] = rho * X[i-1] + sigma * randn()
end

lae = LAE(kernel, X)

# Estimate at single point
psi_at_zero = lae_est(lae, 0.0)
println("ψ(0) ≈ ", psi_at_zero)

# Estimate at multiple points
y_grid = range(-1, 1, length=50)
psi_vals = lae_est(lae, y_grid)

using Plots
plot(y_grid, psi_vals, label="Estimated density")
```

--------------------------------

### Simulate and Plot ARMA Process

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/arma.md

Simulates an ARMA process with specified parameters and plots the resulting time series. Ensure the Plots package is imported.

```julia
arma = ARMA(0.7, [0.1, -0.3], 1.0)
path = simulation(arma, ts_length=1000, impulse_length=50)

# Plot the simulated path
import Plots
Plots.plot(path, xlabel="Time", ylabel="Value", label="Simulated ARMA Process")
```

--------------------------------

### Compute Agent's Best Response (K_to_F)

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/robust-lq.md

Calculates agent 1's best response policy (F) and the corresponding value function (P) given the worst-case shock response (K). Use this when determining optimal strategies against an adversary.

```julia
function K_to_F(rlq::RBLQ, K::Matrix)
    # ... implementation details ...
end
```

--------------------------------

### Define ARMA Process

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/arma.md

Creates an ARMA(1,2) process with specified AR and MA parameters and noise standard deviation. Demonstrates accessing the p, q, and sigma fields of the created ARMA instance.

```julia
using QuantEcon

# Create an ARMA(1,2) process
phi = 0.5
theta = [0.0, -0.8]
sigma = 1.0
arma = ARMA(phi, theta, sigma)

# Access fields
println(arma.p)     # 1
println(arma.q)     # 2
println(arma.sigma) # 1.0
```

--------------------------------

### simulate

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lss.md

Simulates one path of the LSS model for a specified length of time.

```APIDOC
## simulate

### Description

Simulates one path of the LSS model, generating sequences of state and observation values over a given time period.

### Signature

```julia
function simulate(lss::LSS, ts_length=100)
```

### Parameters

| Parameter | Type | Required | Default |
|---|---|---|---|
| `lss` | `LSS` | Yes | — |
| `ts_length` | `Int` | No | `100` |

### Returns

- `x::Matrix` — n×ts_length matrix of state values
- `y::Matrix` — k×ts_length matrix of observation values

### Notes

- The initial state is drawn from a multivariate normal distribution defined by `mu_0` and `Sigma_0`.
- Each column in the returned matrices represents a single time step.

### Example

```julia
lss = LSS([0.9], [0.1], [1.0])
x, y = simulate(lss, 500)

using Plots
plot(y', label="Observations", xlabel="Time")
```
```

--------------------------------

### Simulate Markov Chain Path

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Generates a sample path of a Markov chain. Specify the chain, desired path length, and an optional initial state.

```julia
mc = MarkovChain([0.9 0.1; 0.2 0.8], ["Good", "Bad"])
path = simulate(mc, 100; init=1)
println(path[1:10])  # First 10 periods
```

--------------------------------

### Equidistributed Quadrature Nodes and Weights

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/quadrature.md

Generates equidistributed or grid quadrature nodes and weights for a given number of nodes and interval.

```julia
function qnwequi(n::Int, a::Real, b::Real)
```

--------------------------------

### Checkout Master and Create New Branch

Source: https://github.com/quantecon/quantecon.jl/wiki/Git:-Branching

Switch to the main branch and create a new branch for independent development, ensuring it's based on the latest stable code.

```git
git checkout master
git pull origin master
git checkout -b industry-dyn
```

--------------------------------

### Simulate Markov Chain Path (Indices)

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Generates a sample path of a Markov chain, returning state indices instead of state values. Useful when the state names are not needed.

```julia
function simulate_indices(mc::MarkovChain, ts_length::Int; init::Int=rand(1:n_states(mc)))
    # ... implementation details ...
end
```

--------------------------------

### set_state!

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/kalman.md

Sets the initial state estimate (mean and covariance) for the Kalman filter.

```APIDOC
## set_state!

Set the current state estimate of the Kalman filter.

### Signature

```julia
function set_state!(k::Kalman, x_hat, Sigma)
```

### Parameters

| Parameter | Type | Description |
|-----------|------|-------------|
| `k` | `Kalman` | Kalman filter instance |
| `x_hat` | `Vector` | State mean estimate |
| `Sigma` | `Matrix` | State covariance estimate |

### Returns

- `nothing` — Modifies the filter in-place

### Notes

Sets both `k.cur_x_hat` and `k.cur_sigma` to provided values.
```

--------------------------------

### List branches after checkout

Source: https://github.com/quantecon/quantecon.jl/wiki/Git:-Branching

Verify the current branch after using `git checkout` by listing all branches.

```bash
qe_website|olg-basic ⇒ git branch
  david-evans-model
  master
* olg-basic
```

--------------------------------

### backward_induction

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/discrete-dp.md

Solves a finite horizon discrete dynamic programming problem using the backward induction method.

```APIDOC
## backward_induction

### Description
Solve finite horizon DP by backward induction.

### Signature
```julia
function backward_induction(ddp::DiscreteDP)
```

### Parameters
#### Parameters
- `ddp` (DiscreteDP) - Problem instance
```

--------------------------------

### qnwunif

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/quadrature.md

Computes quadrature nodes and weights for the Uniform distribution.

```APIDOC
## qnwunif

### Description
Quadrature nodes and weights for Uniform distribution.

### Signature
```julia
function qnwunif(n::Int, a::Real, b::Real)
```
```

--------------------------------

### replicate

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lss.md

Simulates multiple replications of the LSS model at a fixed time point.

```APIDOC
## replicate

### Description

Simulates multiple independent paths of the LSS model up to a specified time point `t`, allowing for the analysis of distributions at that specific time.

### Signature

```julia
function replicate(lss::LSS, t::Integer, num_reps::Integer=100)
replicate(lss::LSS; t::Integer=10, num_reps::Integer=100)
```

### Parameters

| Parameter | Type | Required | Default |
|---|---|---|---|
| `lss` | `LSS` | Yes | — |
| `t` | `Int` | No | `10` |
| `num_reps` | `Int` | No | `100` |

### Returns

- `x::Matrix` — n×num_reps matrix of state values at time t
- `y::Matrix` — k×num_reps matrix of observation values at time t

### Example

```julia
lss = LSS([0.9], [0.1], [1.0])
x_t, y_t = replicate(lss, t=50, num_reps=1000)

# Compute sample mean at t=50
mean_y = mean(y_t, dims=2)
```
```

--------------------------------

### Generate All Combinations with gridmake

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/utilities.md

Creates all possible combinations of elements from input arrays (Cartesian product). The first array varies fastest. Use for generating all possible states or parameter combinations.

```julia
using QuantEcon

x = [1, 2, 3]
y = [10, 20]
z = [100, 200]

grid = gridmake(x, y, z)
# Result: 12×3 matrix with all combinations
# (1,10,100), (2,10,100), (3,10,100), (1,20,100), ...

# Or with integers:
grid = gridmake((3, 2))  # All combinations of 1:3 and 1:2
```

--------------------------------

### Uniform Distribution Quadrature Nodes and Weights

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/quadrature.md

Computes quadrature nodes and weights for the Uniform distribution.

```julia
function qnwunif(n::Int, a::Real, b::Real)

```

--------------------------------

### RBLQ Constructor

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/robust-lq.md

Constructs an RBLQ object for robust linear quadratic control problems. It automatically reshapes scalar or vector inputs to the appropriate matrix dimensions.

```APIDOC
## RBLQ Constructor

### Signature

```julia
RBLQ(Q::ScalarOrArray, R::ScalarOrArray, A::ScalarOrArray,
     B::ScalarOrArray, C::ScalarOrArray, bet::Real, theta::Real)
```

### Parameters

| Parameter | Type | Description |
|-----------|------|-------------|
| `Q` | `Matrix` | k×k control cost matrix (symmetric, positive definite) |
| `R` | `Matrix` | n×n state cost matrix (symmetric, nonnegative definite) |
| `A` | `Matrix` | n×n state transition |
| `B` | `Matrix` | n×k control coefficient |
| `C` | `Matrix` | n×j shock coefficient |
| `bet` | `Real` | Discount factor ∈ [0,1] |
| `theta` | `Real` | Robustness parameter > 0 |
```

--------------------------------

### Tauchen's Method for AR(1) Approximation

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-approx.md

Approximates an AR(1) process using Tauchen's method. Requires specifying the number of grid points, persistence, and standard deviation of innovation.

```julia
using QuantEcon

# AR(1): y_t = 0.9*y_{t-1} + 0.1*ε_t
mc = tauchen(9, 0.9, 0.1)

# Simulate
path = simulate(mc, 100)
println("Grid points: ", mc.state_values)
println("Transition matrix (3×3):")
println(mc.p[1:3, 1:3])
```

--------------------------------

### Checkout to Existing Branch

Source: https://github.com/quantecon/quantecon.jl/wiki/Git:-Branching

Switch back to a previously worked-on branch to continue development, ensuring the working directory is clean first.

```git
git status
git checkout olg-basic
```

--------------------------------

### simulate_indices!

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/markov-chain.md

Fills a pre-allocated array with state indices from Markov chain simulations. This in-place version is efficient for generating multiple index paths.

```APIDOC
## simulate_indices!

### Description

Fill a pre-allocated array with state indices from Markov chain simulations. This is an in-place operation.

### Method

`simulate_indices!(X::Union{AbstractVector{T},AbstractMatrix{T}}, mc::MarkovChain; init=rand(1:n_states(mc), size(X,2)))) where T<:Integer

### Parameters

#### Path Parameters

- **X** (AbstractMatrix{Int}) - Required - Pre-allocated matrix to fill with indices
- **mc** (MarkovChain) - Required - Markov chain instance
- **init** (Union{Int, Vector}) - Optional - Random - Initial state(s)

### Returns

- **X** (AbstractMatrix{Int}) - The input array filled with state indices
```

--------------------------------

### Backward Induction

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/discrete-dp.md

Solves finite horizon discrete dynamic programming problems using the backward induction method. Requires only the discrete dynamic programming problem instance.

```julia
function backward_induction(ddp::DiscreteDP)

```

--------------------------------

### Brent's Method Variant for Root Finding

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/other-functions.md

A variant of Brent's method for root finding. Tolerances and maximum iterations can be specified.

```julia
function brent(f, a, b; tol=1e-6, max_iter=100)
end
```

--------------------------------

### Brent's Method for Root Finding

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/other-functions.md

Finds the root of a function within a given interval using Brent's method. Tolerances and maximum iterations can be specified.

```julia
function brenth(f, a, b; tol=1e-6, max_iter=100)
end
```

--------------------------------

### List all branches

Source: https://github.com/quantecon/quantecon.jl/wiki/Git:-Branching

Execute `git branch` to see a list of all local branches, with an asterisk indicating the current branch.

```bash
qe_website|master ⇒ git branch
  david-evans-model
* master
  olg-basic
```

--------------------------------

### is_stable

Source: https://github.com/quantecon/quantecon.jl/blob/master/_autodocs/api-reference/lss.md

Tests the stability of a linear state space system.

```APIDOC
## is_stable

### Description
Tests whether a linear state space system is stable by checking if all eigenvalues of the system matrix A have a modulus less than 1.

### Parameters
#### Path Parameters
None

#### Query Parameters
None

#### Request Body
None

### Method
Not applicable (function call)

### Endpoint
Not applicable (function call)

### Parameters
- **lss** (`LSS`) - Required - LSS instance

### Returns
- **stable** (`Bool`) - True if all eigenvalues of A have modulus < 1

### Notes
Removes constant rows/columns before checking eigenvalues.
```