Problem Set 4: VAR models

Exercise 1

Consider the following VAR

\[\begin{split}\begin{align} y_t &= (1 + \beta) y_{t-1} - \beta \alpha x_{t-1} + \epsilon_{1t}\\ x_t &= \gamma y_{t-1} + (1 - \gamma\alpha) x_{t-1} + \epsilon_{2t} \end{align}\end{split}\]

Show that this VAR is non-stationary.

To show the former statement, we will first rewrite the system of equations in matrix notation.

\[\begin{split}\begin{align} \begin{pmatrix} y_t \\ x_t \end{pmatrix} = \begin{pmatrix}1 + \beta & -\alpha\beta\\\gamma & 1 - \alpha\gamma\end{pmatrix} \begin{pmatrix}y_{t-1}\\x_{t-1}\end{pmatrix} + \begin{pmatrix}\epsilon_{1t}\\\epsilon_{2t}\end{pmatrix} \end{align}\end{split}\]

For stationarity, it has to hold that

\[\det(A(z)) = \det(I_2 - A_1 z) \neq 0\]

Inserting the values from the previous formula yields

\[\begin{split}\begin{align} \det(A(z)) &= \det(I_2 - A_1 z)\\ &= \det\begin{pmatrix} 1 - (1 + \beta)z & \alpha\beta z\\ -\gamma z & 1 - (1 - \alpha\gamma)z \end{pmatrix}\\ &= (1 - (1 + \beta)z)(1 - (1 - \alpha\gamma)z) + \alpha\beta\gamma z^2\\ &= 1 - (1 - \alpha\gamma) z - (1 + \beta)z + (1 + \beta)(1 - \alpha\gamma)z^2 + \alpha\beta\gamma z^2\\ &= 1 - z + \alpha\gamma z - z - \beta z + (1 - \alpha\gamma + \beta - \alpha\beta\gamma)z^2 + \alpha\beta\gamma z^2\\ &= 1 + (\alpha\gamma - 2 - \beta)z + (1 - \alpha\gamma + \beta)z^2 \end{align}\end{split}\]

TODO: Finish proving non-stationarity

Exercise 2

Consider a stationary vector-autoregressive process \(A(L)X_t = \epsilon_t\) and its corresponding moving average representation \(X_t = C(L)\epsilon_t\), where \(C(L) = \sum^\infty_{i=0} C_i L^i\).

1. Find the moving average coefficients for a \(VAR(1)\) process.

Given a \(VAR(1)\) with \(X_t = \nu + A_1 X_{t-1} + \epsilon_t\), we can rewrite the process the following way:

\[\begin{split}\begin{align} X_t &= \nu + A_1 X_{t-1} + \epsilon_t\\ (I_K - A_1 L) X_t &= \nu + \epsilon_t\\ \end{align}\end{split}\]

Then, for the Wold representation we need that \(C(L)(I_K - A_1 L) = I_K\) with \(C(L) = \sum^\infty_{i=0} C^i L^i\). Premultiply yields:

\[\begin{split}\begin{align} I_K &= \sum^\infty_{i=0} C_i L^i (I_K - A_1 L)\\ &= C_0 + C_1 L + C_2 L^2 + \dots\\ &- C_0 A_1 L - C_1 A_1 L^2 + \dots\\ \end{align}\end{split}\]

Matching by the lag operator yields the following relation

\[\begin{split}\begin{align} L = 0: && C_0 &= I_K\\ L = 1: && C_1 &= C_0 A_1 = A_1\\ L = 2: && C_2 &= C_1 A_1 = A^2_1\\ \vdots && \vdots\\ L = i && C_i &= A^i_1 \end{align}\end{split}\]

2. Show that the moving average coefficients for a \(VAR(2)\) can befound recoursively by \(C_0 = I\), \(C_1 = A_1\), and \(C_i = A_1 C_{i-1} + A_2 C_{i-2}\) for \(i = 2, \dots\).

First, let us define a \(VAR(2)\) process as \(X_t = \nu + A_1 X_{t-1} + A_2 X_{t-2} + \epsilon_t\). Then, rewrite the process with the lag operator and apply the Wold representation.

\[\begin{split}\begin{align} X_t &= \nu + A_1 X_{t-1} + A_2 X_{t-2} + \epsilon_t\\ (I_K - A_1 L - A_2 L^2) X_t &= \nu + \epsilon_t\\ \end{align}\end{split}\]

The Wold representation requires that \(C(L)(I_K - A_1 L - A_2 L) = I_K\).

\[\begin{split}\begin{align} I_K &= \sum^\infty_{i=0} C_i L^i (I_K - A_1 L - A_2 L^2)\\ &= C_0 + C_1 L + C_2 L^2 + C_3 L^3 + \dots\\ &- C_0 A_1 L - C_1 A_1 L^2 - C_2 A_1 L^3 - \dots\\ &- C_0 A_2 L^2 - C_1 A_2 L^3 - \dots \end{align}\end{split}\]

Matching by the lag operator yields the following relation

\[\begin{split}\begin{align} L = 0: && C_0 &= I_K\\ L = 1: && C_1 &= C_0 A_1 = A_1\\ L = 2: && C_2 &= C_1 A_1 + C_0 A_2\\ L = 3: && C_3 &= C_2 A_1 + C_1 A_2\\ \vdots && \vdots\\ L = i && C_i &= C_{i-1} A_1 + C_{i-2} A_2 \end{align}\end{split}\]

TODO: The Wold presentation is on the wrong side of matrix \(A_i\). It has to be \(A_1 C_0\) instead of \(C_0 A_1\), but I want to know the reason for this position change, except that it is obvious as the multiplication would not work without it.

Exercise 3

Data Collection

Get the data from the data folder or go to https://fred.stlouisfed.org and download the following quarterly data series (code in parentheses) for the years 1947Q1-2015Q4: - Real GDP (gdpc96) - GDP Deflator (GDPDEF) - Government consumption (GCE) - Population (B230RC0Q173SBEA) - Personal Consumption: Nondurable Goods (PCND) - Personal Consumption: Services (PCESV)

Construct the three time series: real per capita GDP Y , real per capita government consumption G, and real per capita private consumption C (consisting of nondurables and services consumption).

using LinearAlgebra, XLSX

xf = XLSX.readxlsx("problem_set_4_data/us_data.xlsx")

gdp_real = xf["Tabelle1!B8:B283"]
gdp_deflator = xf["Tabelle1!D8:D283"]
nom_gov_cons_inv = xf["Tabelle1!F8:F283"]
nom_priv_cons_ndg = xf["Tabelle1!J8:J283"]
nom_priv_cons_services = xf["Tabelle1!L8:L283"]
nipa_pop = xf["Tabelle1!N8:N283"]

close(xf)

Y = gdp_real ./ nipa_pop
G = nom_gov_cons_inv ./ nipa_pop ./ gdp_deflator
C = (nom_priv_cons_ndg + nom_priv_cons_services) ./ nipa_pop ./ gdp_deflator

timeline = (1947:0.25:2015.75);

VAR Estimation

Use the following function to estimate a \(VAR(4)\) on the vector of observables \(x_t = [log(G_t), log(Y_t), log(C_t)]\) via OLS. Also include a constant and a linear time trend.

function olsvar(y, p, trend_dummy)

    t, K = size(y)
    y = y'

    # create lags
    Z = y[:, p:t-1]
    for i = 1:p-1
        Z = [Z; y[:, p-i:t-i-1]]
    end

    if trend_dummy == 1
        Z = [ones(1, t-p); (1:t-p)'; Z]
    else
        Z = [ones(1, t-p); Z]
    end

    Y = y[:, p+1:end]
    # Lecture 8, Equations 2 and following
    A = (Y * Z') / (Z * Z')
    U = Y - A * Z
    Σ = U * U' / (t - p - p*K - 1)
    V = A[:, 1:1+trend_dummy]
    A = A[:, 2+trend_dummy:K*p+1+trend_dummy]

    np = length(A[:])
    Σ_ML = (t - p - p*K - 1) / (t - p) * Σ
    AIC = logdet(Σ_ML) + 2 * np / (t - p)
    BIC = logdet(Σ_ML) + log(t - p) * np / (t - p)
    HQ  = logdet(Σ_ML) + 2 * log(log(t - p)) * np / (t - p)


    return A, Σ, V, U, Y, Z, AIC, BIC, HQ
end;

The function for the OLS estimation of the \(VAR(4)\) requires three inputs where y is the matrix containing the log series of the three enitities, govermnent spending, GDP and private consumption. p is the number of lags, 4, and trend_dummy indicates whether the model should contain a trend dummy.

x = [log.(G) log.(Y) log.(C)]
lags = 4
trend = 1;

A, Σ, V, U, Y, Z, AIC, BIC, HQ = olsvar(x, lags, trend);

lag_maximum = 8

aic_array = Vector{Float64}(undef, lag_maximum)
bic_array = Vector{Float64}(undef, lag_maximum)
hq_array = Vector{Float64}(undef, lag_maximum)

for i = 1 : lag_maximum
    _, _, _, _, _, _, aic_array[i], bic_array[i], hq_array[i] = olsvar(x[lag_maximum + 1 - i:end, :], i, 1)
end

argmin(aic_array), argmin(bic_array), argmin(hq_array)

(2, 2, 2)