[IrisToolbox] for Macroeconomic Modeling

Balanced growth path models

jaromir.benes@iris-toolbox.com

\[ \newcommand{\ss}{\mathrm{ss}} \notag \]

Key point

Balanced growth models do not need to be stationarized

We can handle (solve, simulate, analyze, etc.) balanced growth path models in their original nonstationary forms.

A very informal definition of a balanced growth path model

A BGP model is a nonstationary model that we are able to manually stationarize so that

a stationary steady state (fixed point) exists for the new system of transformed variables,

and each variable in the transformed system is either of the following

a nontransformed variable from the original system, or
a linear combination of the original variables (or their lags and leads), or
a log-linear combination of the original variables (or their lags and leads)

The key point rephrased

If we are able to manually stationarize the model, we don't actually need to do that...

Digression: Under what conditions does a nonstationary model feature a balanced-growth path?

This is a separate question
Very difficult (and not very insightful) to define general, context-agnostic conditions
Two typical examples of necessary conditions:

Nominal terms in budget constraints and definitions (e.g. households' budget constaint, nominal expenditures in the definition of nominal GDP, etc.) grow at the same rate and

Elasticity of substitution (in production, consumption, etc.) between things that possibly grow at different rates is exactly one in the long run

Practical issues in solving BGP models

How to define, describe and calculate the (nonstationary) steady state of a BGP model?
How to deal with the non-unique levels (infinitely many solutions), aka a singularity in the steady-state system?
How to calculate a local approximation around a nonstationary steady state?

How to describe steady state of nonstationary BGP models?

From the (very informal) definition of BGP, each variables has to be one of the following types:

Type of variable	Behavior in steady state
Stationary	Constant at a fixed unique level
Flat	Not changing over time but an unpredetermined level
Linear growth	Changing over time: fixed first difference
Log-linear growth	Changing over time: fixed rate of change

We need to know what type each variable when writing the model

BGP

Take a snapshot of the variables as they move along their BGP(s) at one arbitrary point of time
Read off two numerical values for each variable:
1. its current level
2. its change over time, i.e. first difference or rate of change
For everything we need to do, it does not matter which particular snapshot we take - any snapshot along the BGP is equally good (and equivalent to any other)

Declare log-variables

We need to know which variable is what type before we start steady-state calculations

Trivial example

Use the following trivial model

\[ \begin{gathered} A_{t} = \alpha \, A_{t-1} \exp \epsilon_t \\[5pt] Y_t = \gamma \, A_{t-1} \end{gathered} \]

to show its solution

the traditional way (manually stationarizing the system)
the direct route (dealing with the variables in their original levels)

Traditional way

Steady state

Define new transformed variables

\[ \begin{gathered} \hat a_t \equiv A_t / A_{t-1} \\[5pt] y_t \equiv Y_t / A_t \end{gathered} \]
Rewrite the original equations

\[ \begin{gathered} A_{t} = \alpha \, A_{t-1} \exp \epsilon_t \quad\longrightarrow\quad \hat a_t = \alpha \exp \epsilon_t \\[5pt] y_t = \gamma A_{t-1} \quad\longrightarrow\quad y_t = \left. \gamma \middle/ \hat a_t \right. \end{gathered} \]
Solve for the fixed point in \(\hat a\), \(y\), this uniquely determines
- the steady-state rate of change in \(a_t\) and \(y_t\)
- the steady-state ratio \(y_t/a_t\)

A point on the BGP

We can create an arbitrary point on the BGP by picking any valid value for \(A_t\), and calculating the corresponding \(Y_t \equiv y_t \, A_t\)
There are infinitely many points we could pick, one for each valid \(A_t\)
This fact leads to a little singularity in the direct approach presented later, but lo and behold, we can easily deal with it...
Note that when going the traditional way, we don't actually need to explicitly express a BGP point for the variables in their original levels

First-order approximation

Given the fixed point \(\hat a_\ss\), \(y_\ss\), calculate first-order Taylor expansion and approximate solution

\[ \begin{bmatrix} \hat a_t - \hat a_\ss \\ y_t - y_\ss \end{bmatrix} = T \, \begin{bmatrix} \hat a_{t-1} - \hat a_\ss \\ y_{t-1} - y_\ss \end{bmatrix} + R \, \epsilon_t \]
In forward-loking models, apply Blanchard-Kahn-Klein procedure using (generalized) eigenvalue decomposition to integrate out future expectations

Go the extra mile

We can alternatively calculate a log approximation, where the log transformation preserves linearity

\[ \begin{gathered} \begin{bmatrix} \log \hat a_t - \log \hat a_\ss \\ \log y_t - \log y_\ss \end{bmatrix} = T^* \, \begin{bmatrix} \log \hat a_{t-1} - \log \hat a_\ss \\ \log y_{t-1} - \log y_\ss \end{bmatrix} + R^* \, \epsilon_t \\[15pt] \log \hat a_t \equiv \log A_t - \log A_{t-1} \\[5pt] \log y_t \equiv \log Y_t - \log A_T \end{gathered} \]
Substitute the definitons of the log transformations back

\[ \begin{bmatrix} \log A_t - \log A_{t-1} - \log \hat a_\ss \\ \log Y_t - \log A_t - \log y_\ss \end{bmatrix} = T^* \, \begin{bmatrix} \log A_{t-1} - \log A_{t-2} - \log \hat a_\ss \\ \log Y_{t-1} - \log A_t - \log y_\ss \end{bmatrix} + R^{*} \, \epsilon_t \]
Rearrange to get

\[ \begin{bmatrix} \log A_t \\ \log A_{t-1} \\ \log Y_t \end{bmatrix} = T^{**} \begin{bmatrix} \log A_{t-1} \\ \log A_{t-2} \\ \log Y_{t-1} \end{bmatrix} + K^{**} + R^{**} \, \epsilon_t \]

Note that

an intercept vector \(K^{**}\) now appears in the system, based on the fixed-point numbers \(\log a_\ss\), \(\log y_\ss\)
the transion matrix \(T^{**}\) now contains a unit root
in forward-looking models, the same Blanchard-Kahn-Klein procedure can be applied (treating unit roots as *stable roots)

Direct route

Absolutely no stationarizing step needed to calculate

a point on the BGP
a first-order approximate solution

Steady state

Write the equations in their steady-state form for the following four unknowns:
Level of \(A_t\), denoted by \(A_\ss\)
Level of \(Y_t\), denoted by \(Y_\ss\)
Rate of change in \(A\), denoted by \(\hat A_\ss\)
Rate of change in \(Y\), denoted by \(\hat Y_\ss\)

\[ \begin{aligned} A_{t} = \alpha \, A_{t-1} \exp \epsilon_t &\quad\longrightarrow\quad A_\ss = \alpha \, A_\ss / \hat A_\ss \\[15pt] Y_t = \gamma \, A_{t-1} &\quad\longrightarrow\quad Y_\ss = \gamma \, A_\ss / \hat A_\ss \end{aligned} \]
Problem: Two equations in four unknowns. Solution: Repeat the system at a different time point

\[ \begin{gathered} A_\ss = \alpha \, A_\ss \, \hat A_\ss^{-1} \\[5pt] Y_\ss = \gamma \, A_\ss \, \hat A_\ss{}^{-1} \\[15pt] A_\ss \, \hat A_\ss{}^k = \alpha \, A_\ss \, \hat A_\ss{}^{k-1} \\[5pt] Y_\ss \, \hat Y_\ss{}^{k} = \gamma \, A_\ss\, \hat A_\ss{}^{k-1} \end{gathered} \]

Singularity problem

The system has infinitely many solutions, the Jacobian is singular
We do have algorithms capable of dealing with the singularity, delivering one (arbitrary) solution, e.g. Levenberg-Marquardt
One arbitrary

First-order approximation

Calculate Taylor expansion of each model equation around the BGP
- w.r.t. to the log of the variables that grow at a constant rate of change
- w.r.t. the other variables
\[ f_k \left( A_t, A_{t-1}, Y_{t}, Y_{t-1} \right) = 0 \]

\[ \begin{gathered} \left. \frac{\partial f_k\left( A_t, A_{t-1}, Y_{t}, Y_{t-1} \right)}{\partial \, \log A_t} \middle|_{A_t=A_\ss,\ A_{t-1}=A_\ss \hat A_\ss{}^{-1}, \ Y_t=Y_\ss, \ Y_{t-1}=Y_\ss \hat Y_\ss{}^{-1}} \right. \\[30pt] \left. \frac{\partial f_k\left( A_t, A_{t-1}, Y_{t}, Y_{t-1} \right)}{\partial \, \log A_{t-1}} \middle|_{A_t=A_\ss,\ A_{t-1}=A_\ss \hat A_\ss{}^{-1}, \ Y_t=Y_\ss, \ Y_{t-1}=Y_\ss \hat Y_\ss{}^{-1}} \right. \\[30pt] \left. \frac{\partial f_k\left( A_t, A_{t-1}, Y_{t}, Y_{t-1} \right)}{\partial \, \log Y_{t}} \middle|_{A_t=A_\ss,\ A_{t-1}=A_\ss \hat A_\ss{}^{-1}, \ Y_t=Y_\ss, \ Y_{t-1}=Y_\ss \hat Y_\ss{}^{-1}} \right. \\[30pt] \left. \frac{\partial f_k\left( A_t, A_{t-1}, Y_{t}, Y_{t-1} \right)}{\partial \, \log Y_{t-1}} \middle|_{A_t=A_\ss,\ A_{t-1}=A_\ss \hat A_\ss{}^{-1}, \ Y_t=Y_\ss, \ Y_{t-1}=Y_\ss \hat Y_\ss{}^{-1}} \right. \\[30pt] \left. \frac{\partial f_k\left( A_t, A_{t-1}, Y_{t}, Y_{t-1} \right)}{\partial \, \epsilon_{t}} \middle|_{A_t=A_\ss,\ A_{t-1}=A_\ss \hat A_\ss{}^{-1}, \ Y_t=Y_\ss, \ Y_{t-1}=Y_\ss \hat Y_\ss{}^{-1}} \right. \end{gathered} \]
Set up the dynamic system

\[ \begin{bmatrix} \log A_t - \log A_\ss \\[5pt] \log A_{t-1} - \log (A_\ss \hat A_\ss{}^{-1}) \\[5pt] \log Y_t - \log Y_\ss \\[5pt] \end{bmatrix} = T^{**} \begin{bmatrix} \log A_{t-1} - \log (A_\ss\hat A_\ss{}^{-1}) \\[5pt] \log A_{t-2} - \log (A_\ss \hat A_\ss{}^{-2}) \\[5pt] \log Y_{t-1} - \log (Y_\ss\hat Y_\ss{}^{-1}) \\[5pt] \end{bmatrix} + R^{**} \, \epsilon_t \]
Rearrange collecting the constant terms

\[ \begin{bmatrix} \log A_t \\[5pt] \log A_{t-1} \\[5pt] \log Y_t \\[5pt] \end{bmatrix} = T^{**} \begin{bmatrix} \log A_{t-1} \\[5pt] \log A_{t-2} \\[5pt] \log Y_{t-1} \\[5pt] \end{bmatrix} + K^{**} + R^{**} \, \epsilon_t \]

Note that

the system matrices \(T^{**}\), \(K^{**}\), \(R^{**}\) are exactly the same as in the traditional approach
the system matrices are independent of the particular snapshot of the BGP because the derivatives are independent of these as well up to scaling/normalization

First-order solution

Two forms of first-order solution in Iris

Triangular representation is used whenever we need to technically separate unit root dynamics from the stable dynamics of the model: stationarity diagnosis, Kalman filtering, autocovariance and power spectrum functions
Nontriangular (aka rectangular) is used in simulations

Non-triangular (rectangular) form

Rectangular transition equation with no forward expansion

\[ \begin{gathered} x_t \equiv \begin{bmatrix} xf_t \\[5pt] xb_t \end{bmatrix} = T \, xb_{t-1} + R \, \epsilon_t + K \\[20pt] T = \begin{bmatrix} T_1 \\[5pt] T_2 \end{bmatrix} \end{gathered} \]

where

\(xf_t\) is an \(nf \times 1\) vector of non-predetermined (forward-looking) variables, including auxiliary leads
\(xb_t\) is an \(nb \times 1\) vector of predetermined (backward-looking) variables, including auxiliary lags
\(\epsilon_t\) is an \(ne \times 1\) vector of shocks
\(T\) is an \(nx \times nb\) transition matrix, \(nx=nf+nb\)
\(T_2\) is a \(nb \times nb\) proper transition block with stable and unit eigenvalues
\(R\) is an \(nx \times ne\) shock impact matrix
\(K\) is an \(nx \times 1\) constant vector

Triangular form

Triangular transition equation with no forward expansion

\[ \begin{gathered} x_t^{\small\Delta} \equiv \begin{bmatrix} xf_t \\[5pt] \alpha_t \end{bmatrix} = T^{\small\Delta} \, \alpha_{t-1} + R^{\small\Delta} \, \epsilon_t + K^{\small\Delta} \\[20pt] T^{\small\Delta} = \begin{bmatrix} T_1^{\small\Delta} \\[5pt] T_2^{\small\Delta} \end{bmatrix} \\[20pt] \alpha = U \, xb_t \end{gathered} \]

where

\(xf_t\) is an \(nf \times 1\) vector of non-predetermined (forward-looking) variables identical to the rectangular solution
\(\alpha_t\) is an \(nb \times 1\) vector of transformed predetermined (backward-looking) variables
\(T^{\small\Delta}\) is an \(nx \times nb\) transformed transition matrix, \(nx=nf+nb\)
\(T_2\) is a \(nb \times nb\) upper triangular proper transition block with unit roots concentrated in the top-left corner
\(R^{\small\Delta}\) is an \(nx \times ne\) transformed shock impact matrix
\(K^{\small\Delta}\) is an \(nx \times 1\) transformed constant vector
\(U\) is an invertible transformation matrix