plot_lm()

Matrices: Application

Labour market transitions

Let $P_{js}$ denote the probability of being in state $s$ in $t+1$ given that you were in state $j$ in $t$, where possible states are $\{e,u,n\}$. Then given estimates of all $P_{js}$ the dynamic evolution of the distribution of individuals across states can be described by means of the system of equations

$$\begin{align}E_{t+1}&=P_{ee}E_t+P_{ue}U_t+P_{ne}N_t\\&\\U_{t+1}&=P_{eu}E_t+P_{uu}U_t+P_{nu}N_t\\&\\N_{t+1}&=P_{en}E_t+P_{un}U_t+P_{nn}N_t\end{align}\color{white}{\quad\Rightarrow\quad \begin{pmatrix}E_{t+1}\\ \\U_{t+1}\\ \\N_{t+1}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}}$$

Matrices: Application

Labour market transitions

Let $P_{js}$ denote the probability of being in state $s$ in $t+1$ given that you were in state $j$ in $t$, where possible states are $\{e,u,n\}$. Then given estimates of all $P_{js}$ the dynamic evolution of the distribution of individuals across states can be described by means of the system of equations

$$\begin{align}E_{t+1}&=P_{ee}E_t+P_{ue}U_t+P_{ne}N_t\\&\\U_{t+1}&=P_{eu}E_t+P_{uu}U_t+P_{nu}N_t\\&\\N_{t+1}&=P_{en}E_t+P_{un}U_t+P_{nn}N_t\end{align}\quad\Rightarrow\quad \begin{pmatrix}E_{t+1}\\ \\U_{t+1}\\ \\N_{t+1}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}$$

Matrices: Application

Labour market transitions

Further, assuming that the transition probabilities do not change over time

$$\begin{pmatrix}E_{t+2}\\ \\U_{t+2}\\ \\N_{t+2}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}\begin{pmatrix}E_{t+1}\\ \\U_{t+1}\\ \\N_{t+1}\end{pmatrix}$$

Matrices: Application

Labour market transitions

Further, assuming that the transition probabilities do not change over time

$$\begin{pmatrix}E_{t+2}\\ \\U_{t+2}\\ \\N_{t+2}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}$$

Matrices: Application

Labour market transitions

Further, assuming that the transition probabilities do not change over time

$$\begin{pmatrix}E_{t+2}\\ \\U_{t+2}\\ \\N_{t+2}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}^2\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}$$

Matrices: Application

Labour market transitions

Further, assuming that the transition probabilities do not change over time

$$\begin{pmatrix}E_{t+3}\\ \\U_{t+3}\\ \\N_{t+3}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}\begin{pmatrix}E_{t+2}\\ \\U_{t+2}\\ \\N_{t+2}\end{pmatrix}$$

Matrices: Application

Labour market transitions

Further, assuming that the transition probabilities do not change over time

$$\begin{pmatrix}E_{t+3}\\ \\U_{t+3}\\ \\N_{t+3}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}^2\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}$$

Matrices: Application

Labour market transitions

Further, assuming that the transition probabilities do not change over time

$$\begin{pmatrix}E_{t+3}\\ \\U_{t+3}\\ \\N_{t+3}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}^3\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}$$

Matrices: Application

Labour market transitions

Further, assuming that the transition probabilities do not change over time

$$\begin{pmatrix}E_{t+n}\\ \\U_{t+n}\\ \\N_{t+n}\end{pmatrix}=\begin{pmatrix}P_{ee}& &P_{ue}& &P_{ne}\\& & & &\\P_{eu}& & P_{uu}& &P_{nu}\\& & & &\\P_{en}& &P_{un}& &P_{nn}\end{pmatrix}^n\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}$$

Matrices: Numerical example

Labour market transitions

Many economists have estimated transition probabilities between labour market states. For example, Krusell, Mukoyama, Rogerson and Sahin (2011, JET) obtained estimates for average monthly transition probabilities in the US from 1994 to 2007 as follows

$$\begin{align}E_{t+1}&=0.962 E_t+0.276 U_t+0.044 N_t\\&\\U_{t+1}&=0.013 E_t+0.501 U_t+0.027 N_t\\&\\N_{t+1}&=0.025 E_t+0.223 U_t+0.929 N_t\end{align}\quad\Rightarrow\quad \begin{pmatrix}E_{t+1}\\ \\U_{t+1}\\ \\N_{t+1}\end{pmatrix}=\begin{pmatrix}0.962&0.276&0.044\\ & &\\0.013&0.501&0.027\\& &\\ 0.025&0.223&0.929\end{pmatrix}\begin{pmatrix}E_{t}\\ \\U_{t}\\ \\N_{t}\end{pmatrix}$$

Matrices: Numerical example

Labour market transitions

Correspondingly, let

$$\mathbf{A}\equiv\begin{pmatrix}0.962&0.276&0.044\\0.013&0.501&0.027\\ 0.025&0.223&0.929\end{pmatrix}\quad\text{ and }\quad \mathbf{x_0}=\begin{pmatrix}E_0\\U_0\\N_0\end{pmatrix}=\begin{pmatrix}0.5\\0.1\\0.4\end{pmatrix}$$

A = [0.962 0.276 0.044; 0.013 0.501 0.027;0.025 0.223 0.929];

x0=[1/2; 1/10; 4/10];

function xt(A,x0,t)
    return round.((A^t)*x0; digits=3)
end

latexify("[E_1,U_1,N_1]=Ax_0=$A*$x0=$(xt(A,x0,1))")

$\left[ \begin{array}{c} E_{1} \\ U_{1} \\ N_{1} \\ \end{array} \right] = Ax_{0} = \left[ \begin{array}{ccc} 0.962 & 0.276 & 0.044 \\ 0.013 & 0.501 & 0.027 \\ 0.025 & 0.223 & 0.929 \\ \end{array} \right] \cdot \left[ \begin{array}{c} 0.5 \\ 0.1 \\ 0.4 \\ \end{array} \right] = \left[ \begin{array}{c} 0.526 \\ 0.067 \\ 0.406 \\ \end{array} \right]$

latexify("[E_2,U_2,N_2]=(A^2)x_0=$A^2*$x0=$(xt(A,x0,2))")

$\left[ \begin{array}{c} E_{2} \\ U_{2} \\ N_{2} \\ \end{array} \right] = A^{2} \cdot x_{0} = \left[ \begin{array}{ccc} 0.962 & 0.276 & 0.044 \\ 0.013 & 0.501 & 0.027 \\ 0.025 & 0.223 & 0.929 \\ \end{array} \right]^{2} \cdot \left[ \begin{array}{c} 0.5 \\ 0.1 \\ 0.4 \\ \end{array} \right] = \left[ \begin{array}{c} 0.543 \\ 0.052 \\ 0.406 \\ \end{array} \right]$

latexify("[E_3,U_3,N_3]=(A^3)x_0=$A^3*$x0=$(xt(A,x0,3))")

$\left[ \begin{array}{c} E_{3} \\ U_{3} \\ N_{3} \\ \end{array} \right] = A^{3} \cdot x_{0} = \left[ \begin{array}{ccc} 0.962 & 0.276 & 0.044 \\ 0.013 & 0.501 & 0.027 \\ 0.025 & 0.223 & 0.929 \\ \end{array} \right]^{3} \cdot \left[ \begin{array}{c} 0.5 \\ 0.1 \\ 0.4 \\ \end{array} \right] = \left[ \begin{array}{c} 0.554 \\ 0.044 \\ 0.402 \\ \end{array} \right]$

latexify("[E_12,U_12,N_12]=(A^12)x_0=$A^12*$x0=$(xt(A,x0,12))")

$\left[ \begin{array}{c} E_{12} \\ U_{12} \\ N_{12} \\ \end{array} \right] = A^{12} \cdot x_{0} = \left[ \begin{array}{ccc} 0.962 & 0.276 & 0.044 \\ 0.013 & 0.501 & 0.027 \\ 0.025 & 0.223 & 0.929 \\ \end{array} \right]^{12} \cdot \left[ \begin{array}{c} 0.5 \\ 0.1 \\ 0.4 \\ \end{array} \right] = \left[ \begin{array}{c} 0.601 \\ 0.036 \\ 0.363 \\ \end{array} \right]$