Diagnostics

euler_equation_error(m, cₜ, logSDFxR, 𝔼_quadrature, zₜ = m.z;
    c_init = NaN, return_soln = false, kwargs...)
euler_equation_error(m, cₜ, logSDFxR, 𝔼_quadrature, shock_matrix, zₜ = m.z;
    c_init = NaN, summary_statistic = x -> norm(x, Inf), burnin = 0,
    return_soln = false, kwargs...)

calculates standard Euler equation errors, as recommended by Judd (1992). The first method calculates the error at some state zₜ, which defaults to the stochastic steady state. The second method simulates the state vector from an initial state zₜ (defaults to stochastic steady state) given a sequence of drawn shocks, evaluates the Euler equation errors, and returns some summary statistic of the errors specified by the keyword summary_statistic.

The Euler equation is

where $m_{t + 1} = \log(M_{t + 1})$ is the log stochastic discount factor and $r_{t + 1} = \log(R_{t + 1})$ is the risk free rate.

Inputs

• m::RiskAdjustedLinearization: A solved instance of a risk-adjusted linearization
• cₜ::Function: a function of (m, zₜ) that calculates consumption at state zₜ, given the state-space representation implied by m.
• logSDFxR::Function: a Function evaluating $m_{t + 1} + r_{t + 1}$. The Function must take as input (m, zₜ, εₜ₊₁, c), where m is a RiskAdjustedLinearization, zₜ is a state vector at which to evaluate, εₜ₊₁ is a draw from the distribution of exogenous shocks, and c is a guess for consumption at zₜ implied by the conditional expectation in the Euler equation when calculated with a quadrature rule. Note that c can be either the consumption level or some transformation (e.g. log consumption), but the user should be consistent in the definition of the cₜ function with the guess c, i.e. both should return the same transformation of consumption (e.g. both should return the level).
• 𝔼_quadrature::Function: a quadrature rule whose single input is a Function with a single input, which is a shock εₜ₊₁.
• zₜ::AbstractVector: a state at which to evaluate the Euler equation error
• shock_matrix::Abstractmatrix: a Nε × T matrix of shocks drawn from the distribution of exogenous shocks.

Keywords

• c_init::Number: an initial guess to be used when solving the "true" consumption policy using quadrature. The default is the consumption policy according to the RiskAdjustedLinearization
• summary_statistic::Function: a Function used to compute a summary statistic from the ergodic set of Euler equation errors. The default is the maximum absolute error.
• burnin::Int: number of periods to drop as burn-in
• return_soln::Bool: if true, return the solution to the nonlinear equation isntead of the error
• kwargs: Any keyword arguments for nlsolve can be passed, too, e.g. ftol or autodiff since nlsolve is used to calculate the "true" consumption policy.

dynamic_euler_equation_error(m, cₜ, logSDFxR, 𝔼_quadrature, endo_states, n_aug,
    shock_matrix, zₜ = m.z; c_init = NaN, summary_statistic = x -> norm(x, Inf),
    burnin = 0, return_soln = false, kwargs...)

calculates dynamic Euler equation errors, as proposed in Den Haan (2009). The Euler equation is

$math \begin{aligned} 0 = \log \mathbb{E}_t \exp(m_{t + 1} + r_{t + 1}) = \log \mathbb{E}_t[M_{t + 1} R_{t + 1}], \end{aligned}$

where $m_{t + 1} = \log(M_{t + 1})$ is the log stochastic discount factor and $r_{t + 1} = \log(R_{t + 1})$ is the risk free rate.

The dynamic errors are computed according the following algorithm.

1. Simulate according to the risk-adjusted linearization time series for the state variables
2. Using the time series from 1, compute time series for consumption and some state variable (usually capital) that can ensure budget constraints hold and markets clear when computing consumption by applying quadrature.
3. Generate a second "implied" time series for consumption and the "capital" state variable, starting from the same initial state as 2. Repeat the following steps at each time period. (i) Compute the conditional expectation in the Euler equation using quadrature to obtain implied consumption. (ii) Use budget constraint/market-clearing to compute implied capital.

By default, dynamic_euler_equation_error returns some summary statistic of the errors specified by the keyword summary_statistic.

Inputs

• m::RiskAdjustedLinearization: A solved instance of a risk-adjusted linearization
• cₜ::Function: a function of (m, zₜ) that calculates consumption at state zₜ, given the state-space representation implied by m.
• logSDFxR::Function: a Function evaluating $m_{t + 1} + r_{t + 1}$. The Function must take as input (m, zₜ, εₜ₊₁, cₜ), where m is a RiskAdjustedLinearization, zₜ is a state vector at which to evaluate, εₜ₊₁ is a draw from the distribution of exogenous shocks, and cₜ is the a guess for consumption at zₜ implied by the conditional expectation in the Euler equation when calculated with a quadrature rule.
• 𝔼_quadrature::Function: a quadrature rule whose single input is a Function with a single input, which is a shock εₜ₊₁.
• endo_states::Function: augments the state variables in the risk-adjusted linearization, usually with one additional variable, which represents capital or assets.
• n_aug::Int: number of extra state variables added by endo_states (usually 1).
• zₜ::AbstractVector: a state at which to evaluate the Euler equation error
• shock_matrix::Abstractmatrix: a Nε × T matrix of shocks drawn from the distribution of exogenous shocks.

Keywords

• c_init::Number: an initial guess to be used when solving the true consumption policy using quadrature. The default is the consumption policy according to the RiskAdjustedLinearization
• summary_statistic::Function: a Function used to compute a summary statistic from the ergodic set of Euler equation errors. The default is the maximum absolute error.
• burnin::Int: number of periods to drop as burn-in
• kwargs: Any keyword arguments for nlsolve can be passed, too, e.g. ftol or autodiff since nlsolve is used to calculate the "true" consumption policy.

gausshermite_expectation(f, μ, σ, n = 10)
gausshermite_expectation(f, μ, Σ, n = 10)
gausshermite_expectation(f, μ, Σ, ns)

calculates the expectation of a function of a Gaussian random variable/vector. The first method evalulates $\mathbb{E}[f(X)]$ where $X \sim N(\mu, \sigma)$, while the other two methods evaluate $\mathbb{E}[f(X)]$ where $X \sim \mathcal{N}(\mu, \Sigma)$ and $\Sigma$ is diagonal. The latter two methods differ in that the first assumes the same number of quadrature points in every dimension while the second does not.

Inputs

• f::Function: some function of a random variable. If f(x) = x, then gausshermite_expectation(f, μ, σ) calculates the mean of $N(\mu, \sigma)$ using 10-point Gauss-Hermite quadrature.
• μ::Number or μ::AbstractVector: mean of the Gaussian random variable/vector.
• σ::Number: standard deviation of the Gaussian random variable.
• Σ::AbstractVector: diagonal of the variance-covariance matrix of the Gaussian random vector.
• n::Int: number of quadrature points to use
• ns::AbstractVector{Int} or ns::NTuple{N, Int} where N: number of quadrature points to use in each dimension of the Gaussian random vector.