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Work with State Transitions

This example shows how to work with transition data from an empirical array of state counts, and create a discrete-time Markov chain (dtmc) model characterizing state transitions. For each quarter in 1857–2022, the example categorizes the state of the US economy as experiencing a recession or expansion by using historical recession periods defined by the National Bureau of Economic Research (NBER). Then, the example forms the empirical transition matrix by counting the quarter-to-quarter transitions of the economic state.

Load and Process Data

Load the historical, NBER-defined recession start and end dates.

load Data_Recessions

The Recessions variable in the workspace is a 34-by-2 matrix of serial date numbers. Rows of Recessions correspond to successive periods of recession, and columns 1 and 2 contain the start and end dates, respectively. For more details, enter Description at the command line.

Because MATLAB® datetime arrays enable you to work with date data efficiently, convert Recessions to a datetime array. Specify the date format yyyy-MM-dd. Visually compare the data types of the first few periods and the last few periods.

NBERRec = datetime(Recessions,'ConvertFrom',"datenum",...
disptbl = table(Recessions,NBERRec);
           Recessions                  NBERRec        
    ________________________    ______________________

    6.7842e+05    6.7897e+05    1857-06-15  1858-12-15
    6.7964e+05    6.7988e+05    1860-10-15  1861-06-15
    6.8128e+05    6.8226e+05    1865-04-15  1867-12-15
           Recessions                  NBERRec        
    ________________________    ______________________

    7.3092e+05    7.3117e+05    2001-03-15  2001-11-15
    7.3339e+05    7.3394e+05    2007-12-15  2009-06-15
    7.3784e+05     7.379e+05    2020-02-15  2020-04-15

Create Array of Visited States

Assume that the sampling period begins on NBERRec(1,1) = 1857-06-15 and ends on 2021-12-10. Create a datetime array of all consecutive quarters in the interval 1857-06-15 through 2021-12-10.

timeline = NBERRec(1,1):calquarters(1):datetime("2021-12-10");
T = numel(timeline); % Sample size

Identify which quarters occur during a recession.

isrecession = @(x)any(isbetween(x,NBERRec(:,1),NBERRec(:,2)));
idxrecession = arrayfun(isrecession,timeline);

Create a MATLAB® timetable containing a variable that identifies the economic state of each quarter in the sampling period. Display the first few observations and the last few observations in the timetable.

EconState = repmat("Expansion",T,1); % Preallocation
EconState(idxrecession) = "Recession";

Tbl = timetable(EconState,'RowTimes',timeline);
       Time        EconState 
    __________    ___________

    1857-06-15    "Recession"
    1857-09-15    "Recession"
    1857-12-15    "Recession"
       Time        EconState 
    __________    ___________

    2021-03-15    "Expansion"
    2021-06-15    "Expansion"
    2021-09-15    "Expansion"

Tbl is a data set of the observed, quarterly economic states, from which transitions can be counted.

Count State Transitions

Consider creating a 2-by-2 matrix containing the number of:

  • Transitions from a recession to an expansion

  • Transitions from an expansion to a recession

  • Self-transitions of recessions

  • Self-transitions of expansions

Row j and column j of the matrix correspond to the same economic state, and element (j,k) contains the number of transitions from state j to state k.

First, assign indices to members of the state space.

[idxstate,states] = grp2idx(Tbl.EconState);
states = 2x1 cell

numstates = numel(states);
P = zeros(numstates); % Preallocate transition matrix

The state indices idxstate correspond to states (rows and columns) of the transition matrix, and states defines the indices. P(1,1) is the number of self-transitions of recessions, P(1,2) is the number of transitions from a recession to an expansion, and so on. Therefore, to count the transition types in the data efficiently, for every transition t to t+1, tally the observed transition by indexing into the transition matrix using state indices at times t (row) and t+1 (column).

for t = 1:(T - 1)
    P(idxstate(t),idxstate(t+1)) = P(idxstate(t),idxstate(t+1)) + 1;

ans=2×2 table
                 Recession    Expansion
                 _________    _________

    Recession       171           34   
    Expansion        33          419   

P is the matrix of observed transitions. P(1,2) = 34 means that the US economy transitioned from a recession to an expansion 34 times between 1857-06-15 and 2021-12-10.

Create Discrete-Time Markov Chain

Create a discrete-time Markov chain model characterized by the transition matrix P.

mc = dtmc(P,'StateNames',states);
ans=2×2 table
                 Recession    Expansion
                 _________    _________

    Recession     0.83415      0.16585 
    Expansion    0.073009      0.92699 

mc is a dtmc object. dtmc normalizes P to create the right-stochastic transition matrix mc.P (each row sums to 1). The element mc.P(1,2) = 0.1659 means that the empirical probability of transitioning to an expansion at time t + 1, given that the US economy is in a recession at time t, is 0.1659.

Plot a directed graph of the Markov chain.


Figure contains an axes object. The axes object contains an object of type graphplot.

The Markov chain suggests that the current economic state tends to persist, from quarter to quarter.

See Also



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