# 59. Additive Functionals#

Co-authored with Chase Coleman and Balint Szoke

## 59.1. Overview#

Some time series are nonstationary.

For example, output, prices, and dividends are typically nonstationary, due to irregular but persistent growth.

Which kinds of models are useful for studying such time series?

Hansen and Scheinkman [HS09] analyze two classes of time series models that accommodate growth.

They are:

**additive functionals**that display random “arithmetic growth”**multiplicative functionals**that display random “geometric growth”

These two classes of processes are closely connected.

For example, if a process \(\{y_t\}\) is an additive functional and \(\phi_t = \exp(y_t)\), then \(\{\phi_t\}\) is a multiplicative functional.

Hansen and Sargent [HS17] (chs. 5 and 6) describe discrete time versions of additive and multiplicative functionals.

In this lecture we discuss the former (i.e., additive functionals).

In the next lecture we discuss multiplicative functionals.

We also consider fruitful decompositions of additive and multiplicative processes, a more in depth discussion of which can be found in Hansen and Sargent [HS17].

## 59.2. A Particular Additive Functional#

This lecture focuses on a particular type of additive functional: a scalar process \(\{y_t\}_{t=0}^\infty\) whose increments are driven by a Gaussian vector autoregression.

It is simple to construct, simulate, and analyze.

This additive functional consists of two components, the first of which is a **first-order vector autoregression** (VAR)

Here

\(x_t\) is an \(n \times 1\) vector,

\(A\) is an \(n \times n\) stable matrix (all eigenvalues lie within the open unit circle),

\(z_{t+1} \sim {\cal N}(0,I)\) is an \(m \times 1\) i.i.d. shock,

\(B\) is an \(n \times m\) matrix, and

\(x_0 \sim {\cal N}(\mu_0, \Sigma_0)\) is a random initial condition for \(x\)

The second component is an equation that expresses increments of \(\{y_t\}_{t=0}^\infty\) as linear functions of

a scalar constant \(\nu\),

the vector \(x_t\), and

the same Gaussian vector \(z_{t+1}\) that appears in the VAR (59.1)

In particular,

Here \(y_0 \sim {\cal N}(\mu_{y0}, \Sigma_{y0})\) is a random initial condition.

The nonstationary random process \(\{y_t\}_{t=0}^\infty\) displays
systematic but random *arithmetic growth*.

### 59.2.1. A linear state space representation#

One way to represent the overall dynamics is to use a linear state space system.

To do this, we set up state and observation vectors

Now we construct the state space system

This can be written as

which is a standard linear state space system.

To study it, we could map it into an instance of LSS from QuantEcon.jl.

We will in fact use a different set of code for simulation, for reasons described below.

## 59.3. Dynamics#

Let’s run some simulations to build intuition.

In doing so we’ll assume that \(z_{t+1}\) is scalar and that \(\tilde x_t\) follows a 4th-order scalar autoregession

Let the increment in \(\{y_t\}\) obey

with an initial condition for \(y_0\).

While (59.3) is not a first order system like (59.1), we know that it can be mapped into a first order system

for an example of such a mapping, see this example

In fact this whole model can be mapped into the additive functional system definition in (59.1) – (59.2) by appropriate selection of the matrices \(A, B, D, F\).

You can try writing these matrices down now as an exercise — the correct expressions will appear in the code below.

### 59.3.1. Simulation#

When simulating we embed our variables into a bigger system.

This system also constructs the components of the decompositions of \(y_t\) and of \(\exp(y_t)\) proposed by Hansen and Scheinkman [HS09].

All of these objects are computed using the code below.

```
using Distributions, LaTeXStrings, Plots, QuantEcon
using LinearAlgebra, Statistics
```

```
function AMF_LSS_VAR(A, B, D, F = nothing; upsilon = nothing)
if B isa AbstractVector
B = reshape(B, length(B), 1)
end
# unpack required elements
nx, nk = size(B)
# checking the dimension of D (extended from the scalar case)
if ndims(D) > 1
nm = size(D, 1)
if D isa Union{Adjoint, Transpose}
D = convert(Matrix, D)
end
else
nm = 1
D = reshape(D, 1, length(D))
end
# set F
if isnothing(F)
F = zeros(nk, 1)
elseif ndims(F) == 1
F = reshape(F, length(F), 1)
end
# set upsilon
if isnothing(upsilon)
upsilon = zeros(nm, 1)
elseif ndims(upsilon) == 1
upsilon = reshape(upsilon, length(upsilon), 1)
else
throw(ArgumentError("upsilon must be column vector!"))
end
if size(upsilon, 1) != size(D, 1)
error("The size of upsilon is inconsistent with D!")
end
# construct BIG state space representation
lss = construct_ss(A, B, D, F, upsilon, nx, nk, nm)
return (; A, B, D, F, upsilon, nx, nk, nm, lss)
end
AMF_LSS_VAR(A, B, D) = AMF_LSS_VAR(A, B, D, nothing, upsilon = nothing)
function AMF_LSS_VAR(A, B, D, F, upsilon)
AMF_LSS_VAR(A, B, D, [F], upsilon = [upsilon])
end
function construct_ss(A, B, D, F, upsilon, nx, nk, nm)
H, g = additive_decomp(A, B, D, F, nx)
# auxiliary blocks with 0's and 1's to fill out the lss matrices
nx0c = zeros(nx, 1)
nx0r = zeros(1, nx)
nx1 = ones(1, nx)
nk0 = zeros(1, nk)
ny0c = zeros(nm, 1)
ny0r = zeros(1, nm)
ny1m = I + zeros(nm, nm)
ny0m = zeros(nm, nm)
nyx0m = similar(D)
# build A matrix for LSS
# order of states is: [1, t, xt, yt, mt]
A1 = hcat(1, 0, nx0r, ny0r, ny0r) # transition for 1
A2 = hcat(1, 1, nx0r, ny0r, ny0r) # transition for t
A3 = hcat(nx0c, nx0c, A, nyx0m', nyx0m') # transition for x_{t+1}
A4 = hcat(upsilon, ny0c, D, ny1m, ny0m) # transition for y_{t+1}
A5 = hcat(ny0c, ny0c, nyx0m, ny0m, ny1m) # transition for m_{t+1}
Abar = vcat(A1, A2, A3, A4, A5)
# build B matrix for LSS
Bbar = vcat(nk0, nk0, B, F, H)
# build G matrix for LSS
# order of observation is: [xt, yt, mt, st, tt]
G1 = hcat(nx0c, nx0c, I, nyx0m', nyx0m') # selector for x_{t}
G2 = hcat(ny0c, ny0c, nyx0m, ny1m, ny0m) # selector for y_{t}
G3 = hcat(ny0c, ny0c, nyx0m, ny0m, ny1m) # selector for martingale
G4 = hcat(ny0c, ny0c, -g, ny0m, ny0m) # selector for stationary
G5 = hcat(ny0c, upsilon, nyx0m, ny0m, ny0m) # selector for trend
Gbar = vcat(G1, G2, G3, G4, G5)
# build LSS type
x0 = hcat(1, 0, nx0r, ny0r, ny0r)
S0 = zeros(length(x0), length(x0))
lss = LSS(Abar, Bbar, Gbar, zeros(nx + 4nm, 1), x0, S0)
return lss
end
function additive_decomp(A, B, D, F, nx)
A_res = \(I - A, I)
g = D * A_res
H = F .+ D * A_res * B
return H, g
end
function multiplicative_decomp(A, B, D, F, upsilon, nx)
H, g = additive_decomp(A, B, D, F, nx)
upsilon_tilde = upsilon .+ 0.5 * diag(H * H')
return H, g, upsilon_tilde
end
function loglikelihood_path(amf, x, y)
(; A, B, D, F) = amf
k, T = size(y)
FF = F * F'
FFinv = inv(FF)
temp = y[:, 2:end] - y[:, 1:(end - 1)] - D * x[:, 1:(end - 1)]
obs = temp .* FFinv .* temp
obssum = cumsum(obs)
scalar = (logdet(FF) + k * log(2pi)) * (1:T)
return -(obssum + scalar) / 2
end
function loglikelihood(amf, x, y)
llh = loglikelihood_path(amf, x, y)
return llh[end]
end
function plot_additive(amf, T; npaths = 25, show_trend = true)
# pull out right sizes so we know how to increment
(; nx, nk, nm) = amf
# allocate space (nm is the number of additive functionals - we want npaths for each)
mpath = zeros(nm * npaths, T)
mbounds = zeros(2nm, T)
spath = zeros(nm * npaths, T)
sbounds = zeros(2nm, T)
tpath = zeros(nm * npaths, T)
ypath = zeros(nm * npaths, T)
# simulate for as long as we wanted
moment_generator = moment_sequence(amf.lss)
# pull out population moments
for (t, x) in enumerate(moment_generator)
ymeans = x[2]
yvar = x[4]
# lower and upper bounds - for each additive functional
for ii in 1:nm
li, ui = 2(ii - 1) + 1, 2ii
if sqrt(yvar[nx + nm + ii, nx + nm + ii]) != 0.0
madd_dist = Normal(ymeans[nx + nm + ii],
sqrt(yvar[nx + nm + ii, nx + nm + ii]))
mbounds[li, t] = quantile(madd_dist, 0.01)
mbounds[ui, t] = quantile(madd_dist, 0.99)
elseif sqrt(yvar[nx + nm + ii, nx + nm + ii]) == 0.0
mbounds[li, t] = ymeans[nx + nm + ii]
mbounds[ui, t] = ymeans[nx + nm + ii]
else
error("standard error is negative")
end
if sqrt(yvar[nx + 2nm + ii, nx + 2nm + ii]) != 0.0
sadd_dist = Normal(ymeans[nx + 2nm + ii],
sqrt(yvar[nx + 2nm + ii, nx + 2nm + ii]))
sbounds[li, t] = quantile(sadd_dist, 0.01)
sbounds[ui, t] = quantile(sadd_dist, 0.99)
elseif sqrt(yvar[nx + 2nm + ii, nx + 2nm + ii]) == 0.0
sbounds[li, t] = ymeans[nx + 2nm + ii]
sbounds[ui, t] = ymeans[nx + 2nm + ii]
else
error("standard error is negative")
end
end
t == T && break
end
# pull out paths
for n in 1:npaths
x, y = simulate(amf.lss, T)
for ii in 0:(nm - 1)
ypath[npaths * ii + n, :] = y[nx + ii + 1, :]
mpath[npaths * ii + n, :] = y[nx + nm + ii + 1, :]
spath[npaths * ii + n, :] = y[nx + 2nm + ii + 1, :]
tpath[npaths * ii + n, :] = y[nx + 3nm + ii + 1, :]
end
end
add_figs = []
for ii in 0:(nm - 1)
li, ui = npaths * (ii), npaths * (ii + 1)
LI, UI = 2ii, 2(ii + 1)
push!(add_figs,
plot_given_paths(T, ypath[(li + 1):ui, :], mpath[(li + 1):ui, :],
spath[(li + 1):ui, :],
tpath[(li + 1):ui, :], mbounds[(LI + 1):UI, :],
sbounds[(LI + 1):UI, :],
show_trend = show_trend))
end
return add_figs
end
function plot_multiplicative(amf, T, npaths = 25, show_trend = true)
# pull out right sizes so we know how to increment
(; nx, nk, nm) = amf
# matrices for the multiplicative decomposition
H, g, upsilon_tilde = multiplicative_decomp(A, B, D, F, upsilon, nx)
# allocate space (nm is the number of functionals - we want npaths for each)
mpath_mult = zeros(nm * npaths, T)
mbounds_mult = zeros(2nm, T)
spath_mult = zeros(nm * npaths, T)
sbounds_mult = zeros(2nm, T)
tpath_mult = zeros(nm * npaths, T)
ypath_mult = zeros(nm * npaths, T)
# simulate for as long as we wanted
moment_generator = moment_sequence(amf.lss)
# pull out population moments
for (t, x) in enumerate(moment_generator)
ymeans = x[2]
yvar = x[4]
# lower and upper bounds - for each multiplicative functional
for ii in 1:nm
li, ui = 2(ii - 1) + 1, 2ii
if yvar[nx + nm + ii, nx + nm + ii] != 0.0
Mdist = LogNormal(ymeans[nx + nm + ii] -
0.5t * diag(H * H')[ii],
sqrt(yvar[nx + nm + ii, nx + nm + ii]))
mbounds_mult[li, t] = quantile(Mdist, 0.01)
mbounds_mult[ui, t] = quantile(Mdist, 0.99)
elseif yvar[nx + nm + ii, nx + nm + ii] == 0.0
mbounds_mult[li, t] = exp.(ymeans[nx + nm + ii] -
0.5t * diag(H * H')[ii])
mbounds_mult[ui, t] = exp.(ymeans[nx + nm + ii] -
0.5t * diag(H * H')[ii])
else
error("standard error is negative")
end
if yvar[nx + 2nm + ii, nx + 2nm + ii] != 0.0
Sdist = LogNormal(-ymeans[nx + 2nm + ii],
sqrt(yvar[nx + 2nm + ii, nx + 2nm + ii]))
sbounds_mult[li, t] = quantile(Sdist, 0.01)
sbounds_mult[ui, t] = quantile(Sdist, 0.99)
elseif yvar[nx + 2nm + ii, nx + 2nm + ii] == 0.0
sbounds_mult[li, t] = exp.(-ymeans[nx + 2nm + ii])
sbounds_mult[ui, t] = exp.(-ymeans[nx + 2nm + ii])
else
error("standard error is negative")
end
end
t == T && break
end
# pull out paths
for n in 1:npaths
x, y = simulate(amf.lss, T)
for ii in 0:(nm - 1)
ypath_mult[npaths * ii + n, :] = exp.(y[nx + ii + 1, :])
mpath_mult[npaths * ii + n, :] = exp.(y[nx + nm + ii + 1, :] -
collect(1:T) * 0.5 *
diag(H * H')[ii + 1])
spath_mult[npaths * ii + n, :] = 1 ./
exp.(-y[nx + 2 * nm + ii + 1, :])
tpath_mult[npaths * ii + n, :] = exp.(y[nx + 3nm + ii + 1, :] +
(1:T) * 0.5 *
diag(H * H')[ii + 1])
end
end
mult_figs = []
for ii in 0:(nm - 1)
li, ui = npaths * ii, npaths * (ii + 1)
LI, UI = 2ii, 2(ii + 1)
push!(mult_figs,
plot_given_paths(T, ypath_mult[(li + 1):ui, :],
mpath_mult[(li + 1):ui, :],
spath_mult[(li + 1):ui, :],
tpath_mult[(li + 1):ui, :],
mbounds_mult[(LI + 1):UI, :],
sbounds_mult[(LI + 1):UI, :],
horline = 1.0, show_trend = show_trend))
end
return mult_figs
end
function plot_martingales(amf, T, npaths = 25)
# pull out right sizes so we know how to increment
(; A, B, D, F, upsilon, nx, nk, nm) = amf
# matrices for the multiplicative decomposition
H, g, upsilon_tilde = multiplicative_decomp(A, B, D, F, upsilon, nx)
# allocate space (nm is the number of functionals - we want npaths for each)
mpath_mult = zeros(nm * npaths, T)
mbounds_mult = zeros(2nm, T)
# simulate for as long as we wanted
moment_generator = moment_sequence(amf.lss)
# pull out population moments
for (t, x) in enumerate(moment_generator)
ymeans = x[2]
yvar = x[4]
# lower and upper bounds - for each functional
for ii in 1:nm
li, ui = 2(ii - 1) + 1, 2ii
if yvar[nx + nm + ii, nx + nm + ii] != 0.0
Mdist = LogNormal(ymeans[nx + nm + ii] -
0.5^2 * t * diag(H * H')[ii],
sqrt(yvar[nx + nm + ii, nx + nm + ii]))
mbounds_mult[li, t] = quantile(Mdist, 0.01)
mbounds_mult[ui, t] = quantile(Mdist, 0.99)
elseif yvar[nx + nm + ii, nx + nm + ii] == 0.0
mbounds_mult[li, t] = ymeans[nx + nm + ii] -
0.5^2 * t * diag(H * H')[ii]
mbounds_mult[ui, t] = ymeans[nx + nm + ii] -
0.5t * diag(H * H')[ii]
else
error("standard error is negative")
end
end
t == T && break
end
# pull out paths
for n in 1:npaths
x, y = simulate(amf.lss, T)
for ii in 0:(nm - 1)
mpath_mult[npaths * ii + n, :] = exp.(y[nx + nm + ii + 1, :] -
(1:T) * 0.5 *
diag(H * H')[ii + 1])
end
end
mart_figs = []
for ii in 0:(nm - 1)
li, ui = npaths * (ii), npaths * (ii + 1)
LI, UI = 2ii, 2(ii + 1)
push!(mart_figs,
plot_martingale_paths(T, mpath_mult[(li + 1):ui, :],
mbounds_mult[(LI + 1):UI, :], horline = 1))
plot!(mart_figs[ii + 1],
title = L"Martingale components for many paths of $y_{ii + 1}$")
end
return mart_figs
end
function plot_given_paths(T, ypath, mpath, spath, tpath, mbounds, sbounds;
horline = 0.0, show_trend = true)
# allocate space
trange = 1:T
# allocate transpose
mpath_T = Matrix(mpath')
# create figure
plots = plot(layout = (2, 2), size = (800, 800))
# plot all paths together
plot!(plots[1], trange, ypath[1, :], label = L"y_t", color = :black)
plot!(plots[1], trange, mpath[1, :], label = L"m_t", color = :magenta)
plot!(plots[1], trange, spath[1, :], label = L"s_t", color = :green)
if show_trend
plot!(plots[1], trange, tpath[1, :], label = L"t_t", color = :red)
end
plot!(plots[1], seriestype = :hline, [horline], color = :black,
linestyle = :dash, label = "")
plot!(plots[1], title = "One Path of All Variables", legend = :topleft)
# plot martingale component
plot!(plots[2], trange, mpath[1, :], color = :magenta, label = "")
plot!(plots[2], trange, mpath_T, alpha = 0.45, color = :magenta, label = "")
ub = mbounds[2, :]
lb = mbounds[1, :]
plot!(plots[2], ub, fillrange = lb, alpha = 0.25, color = :magenta,
label = "")
plot!(plots[2], seriestype = :hline, [horline], color = :black,
linestyle = :dash, label = "")
plot!(plots[2], title = "Martingale Components for Many Paths")
# plot stationary component
plot!(plots[3], spath[1, :], color = :green, label = "")
plot!(plots[3], Matrix(spath'), alpha = 0.25, color = :green, label = "")
ub = sbounds[2, :]
lb = sbounds[1, :]
plot!(plots[3], ub, fillrange = lb, alpha = 0.25, color = :green,
label = "")
plot!(plots[3], seriestype = :hline, [horline], color = :black,
linestyle = :dash, label = "")
plot!(plots[3], title = "Stationary Components for Many Paths")
# plot trend component
if show_trend == true
plot!(plots[4], Matrix(tpath'), color = :red, label = "")
end
plot!(plots[4], seriestype = :hline, [horline], color = :black,
linestyle = :dash, label = "")
plot!(plots[4], title = "Trend Components for Many Paths")
return plots
end
function plot_martingale_paths(T, mpath, mbounds;
horline = 1, show_trend = false)
# allocate space
trange = 1:T
# create the plot
plt = plot()
# plot martingale component
ub = mbounds[2, :]
lb = mbounds[1, :]
#plot!(plt, lb, fillrange = ub, alpha = 0.25, color = :magenta, label = "")
plot!(plt, seriestype = :hline, [horline], color = :black,
linestyle = :dash, label = "")
plot!(plt, trange, Matrix(mpath'), linewidth = 0.25, color = :black,
label = "")
return plt
end
```

```
plot_martingale_paths (generic function with 1 method)
```

For now, we just plot \(y_t\) and \(x_t\), postponing until later a description of exactly how we compute them.

```
phi_1, phi_2, phi_3, phi_4 = 0.5, -0.2, 0, 0.5
sigma = 0.01
upsilon = 0.01 # growth rate
## A matrix should be n x n
A = [phi_1 phi_2 phi_3 phi_4;
1 0 0 0;
0 1 0 0;
0 0 1 0]
# B matrix should be n x k
B = [sigma, 0, 0, 0]
D = [1 0 0 0] * A
F = [1, 0, 0, 0] ⋅ vec(B)
amf = AMF_LSS_VAR(A, B, D, F, upsilon)
T = 150
x, y = simulate(amf.lss, T)
plt_1 = plot()
plt_2 = plot()
plots = [plt_1, plt_2]
# plots = plot(layout = (2, 1))
plot!(plots[1], 1:T, y[amf.nx + 1, :], color = :black, lw = 2, label = "")
plot!(plots[1], title = L"A particular path of $y_t$")
plot!(plots[2], 1:T, y[1, :], color = :green, lw = 2, label = "")
plot!(plots[2], seriestype = :hline, [0], color = :black, lw = 2,
linestyle = :dashdot, label = "")
plot!(plots[2], title = L"Associated path of $x_t$")
# plot(plots)
plot(plots[1], plots[2], layout = (2, 1), size = (700, 500))
```

Notice the irregular but persistent growth in \(y_t\).

### 59.3.2. Decomposition#

Hansen and Sargent [HS17] describe how to construct a decomposition of an additive functional into four parts:

a constant inherited from initial values \(x_0\) and \(y_0\)

a linear trend

a martingale

an (asymptotically) stationary component

To attain this decomposition for the particular class of additive functionals defined by (59.1) and (59.2), we first construct the matrices

Then the Hansen-Scheinkman [HS09] decomposition is

At this stage you should pause and verify that \(y_{t+1} - y_t\) satisfies (59.2).

It is convenient for us to introduce the following notation:

\(\tau_t = \nu t\) , a linear, deterministic trend

\(m_t = \sum_{j=1}^t H z_j\), a martingale with time \(t+1\) increment \(H z_{t+1}\)

\(s_t = g x_t\), an (asymptotically) stationary component

We want to characterize and simulate components \(\tau_t, m_t, s_t\) of the decomposition.

A convenient way to do this is to construct an appropriate instance of a linear state space system by using LSS from QuantEcon.jl.

This will allow us to use the routines in LSS to study dynamics.

To start, observe that, under the dynamics in (59.1) and (59.2) and with the definitions just given,

and

With

we can write this as the linear state space system

By picking out components of \(\tilde y_t\), we can track all variables of interest.

## 59.4. Code#

The type AMF_LSS_VAR mentioned above does all that we want to study our additive functional.

In fact AMF_LSS_VAR does more, as we shall explain below.

(A hint that it does more is the name of the type – here AMF stands for “additive and multiplicative functional” – the code will do things for multiplicative functionals too)

Let’s use this code (embedded above) to explore the example process described above.

If you run the code that first simulated that example again and then the method call you will generate (modulo randomness) the plot

```
plt = plot_additive(amf, T)
plt[1]
```

When we plot multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plot population 95% probability coverage sets computed using the LSS type.

We have chosen to simulate many paths, all starting from the *same* nonrandom initial conditions \(x_0, y_0\) (you can tell this from the shape of the 95% probability coverage shaded areas).

Notice tell-tale signs of these probability coverage shaded areas

the purple one for the martingale component \(m_t\) grows with \(\sqrt{t}\)

the green one for the stationary component \(s_t\) converges to a constant band

### 59.4.1. An associated multiplicative functional#

Where \(\{y_t\}\) is our additive functional, let \(M_t = \exp(y_t)\).

As mentioned above, the process \(\{M_t\}\) is called a **multiplicative functional**.

Corresponding to the additive decomposition described above we have the multiplicative decomposition of the \(M_t\)

or

where

and

An instance of type AMF_LSS_VAR includes this associated multiplicative functional as an attribute.

Let’s plot this multiplicative functional for our example.

If you run the code that first simulated that example again and then the method call

```
plt = plot_multiplicative(amf, T)
plt[1]
```

As before, when we plotted multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plotted population 95% confidence bands computed using the LSS type.

Comparing this figure and the last also helps show how geometric growth differs from arithmetic growth.

### 59.4.2. A peculiar large sample property#

Hansen and Sargent [HS17] (ch. 6) note that the martingale component \(\widetilde M_t\) of the multiplicative decomposition has a peculiar property.

While \(E_0 \widetilde M_t = 1\) for all \(t \geq 0\), nevertheless \(\ldots\).

As \(t \rightarrow +\infty\), \(\widetilde M_t\) converges to zero almost surely.

The following simulation of many paths of \(\widetilde M_t\) illustrates this property

```
plt = plot_martingales(amf, 12000)
plt[1]
```