Apply recursive regressions using nested windows to look for instability in an explanatory model of real GNP for a period spanning World War II.

Load the Nelson-Plosser data set.

load Data_NelsonPlosser

The time series in the data set contain annual, macroeconomic measurements from 1860 to 1970. For more details, a list of variables, and descriptions, enter Description in the command line.

Several series have missing data. Focus the sample to measurements from 1915 to 1970. Identify the breakpoint index corresponding to 1945, the end of the war.

span = (1915 <= dates) & (dates <= 1970);
bp = find(dates(span) == 1945);

Consider the multiple linear regression model

Collect the model variables into a tabular array. Position the predictors in the first three columns, and the response in the last column. Compute the number of coefficients in the model.

Mdl = DataTable(span,[4,5,10,1]);
numCoeff = size(Mdl,2); % Three predictors and an intercept

Estimate the coefficients using recursive regressions, and return separate plots for the iterative estimates. Identify the iteration corresponding to the end of the war.

recreg(Mdl);

bpIter = bp - numCoeff

bpIter = 27

By default, recreg forms the subsamples using nested windows. The end of the war (1945) occurs at the 27th iteration.

All coefficients show some initial, transient instability during the "burn-in" period (see Tip). The plot of WR seems stable since the line is relatively flat. However, the plots of E, IPI, and the intercept (Const) show instability, particularly just after iteration 27.

Check coefficient estimates for instability in a model of food demand around World War II. Implement forward and backward recursive regressions in a rolling window.

Load the U.S. food consumption data set, which contains annual measurements from 1927 through 1962 with missing data due to the war.

load Data_Consumption

For more details on the data, enter Description at the command prompt.

Plot the series.

P = Data(:,1); % Food price index
I = Data(:,2); % Disposable income index
Q = Data(:,3); % Food consumption index 

figure;
plot(dates,[P I Q],'o-')
axis tight
grid on
xlabel('Year')
ylabel('Index')
title('{\bf Time Series Plot of All Series}')
legend({'Price','Income','Consumption'},'Location','SE')

Measurements are missing from 1942 through 1947, which correspond to World War II.

To examine elasticities, apply the log transformation to each series.

LP = log(P);
LI = log(I);
LQ = log(Q);

Consider a model in which log consumption is a linear function of the logs of food price and income. In other words,

$${\epsilon }_t$$ is a Gaussian random variable with mean 0 and standard deviation $${\sigma }^2$$.

Identify the breakpoint index at the end of the war, 1945. Ignore missing years with missing data.

numCoeff = 4; % Three predictors and an intercept
T = numel(dates(~isnan(P))); % Sample size
bpIdx = find(dates(~isnan(P)) >= 1945,1) - numCoeff

bpIdx = 12

The 12th iteration corresponds to the end of the war.

Plot forward recursive-regression coefficient estimates using a rolling window of size 1/4 the sample size. Indicate to plot the coefficients of LP and LI only in the same figure.

X = [LP LI];
y = LQ;
window = ceil(T*1/4);
recreg(X,y,'Window',window,'Plot','combined','PlotVars',[0 1 1],...
    'VarNames',{'Log-price' 'Log-income'});

Plot forward recursive-regression coefficient estimates using a rolling window of size 1/3 the sample size.

window = ceil(T*1/3);
recreg(X,y,'Window',window,'Plot','combined','PlotVars',[0 1 1],...
    'VarNames',{'Log-price' 'Log-income'});

Plot forward recursive-regression coefficient estimates using a rolling window of size 1/2 the sample size.

window = ceil(T*1/2);
recreg(X,y,'Window',window,'Plot','combined','PlotVars',[0 1 1],...
    'VarNames',{'Log-price' 'Log-income'});

As the window size increases, the lines show less volatility, but the coefficients do exhibit instability.

If a linear regression model violates classical linear model assumptions, then OLS coefficient standard errors are incorrect. However, recreg has options to estimate coefficients and standard errors that are robust to heteroscedastic or autocorrelated innovations.

Simulate a series from this piecewise regression model with AR(1) errors whose regression coefficient changes at time 51.

$${\epsilon }_t$$ is a series of Gaussian innovations with mean 0 and standard deviation 0.5. $$x_t$$ is Gaussian with mean 1 and standard deviation 0.25.

rng(1); % For reproducibility
T = 100;
muX = 1;
sigmaX = 0.25;
x = sigmaX*randn(T,1) + muX;
ar = 0.6;
sigma = 0.5;
c = 5;
b = [3 -1];
y = zeros(T,1);
Mdl1 = regARIMA('AR',ar,'Variance',sigma,'Intercept',c,'Beta',b(1));
y(1:T/2) = simulate(Mdl1,T/2,'X',x(1:T/2));
Mdl2 = regARIMA('AR',ar,'Variance',sigma,'Intercept',c,'Beta',b(2));
y((T/2 + 1):T) = simulate(Mdl2,T/2,'X',x((T/2 + 1):T));

Estimate recursive regression coefficients using OLS.

[CoeffOLS,SEOLS] = recreg(x,y,'Plot','separate');

After transient effects, 5 is within the confidence bounds of the intercept estimates. There is an insignificant, but persistent shock at iteration 50. The coefficient estimates show the structural change after iteration 60.

To account for autocorrelated innovations, estimate recursive regression coefficients using OLS, but with Newey-West robust standard errors. For estimating the HAC standard errors, use the quadratic-spectral weighting scheme.

hacOptions.Weights = 'QS';
[CoeffNW,SENW] = recreg(x,y,'Estimator','hac','Options',hacOptions,...
    'Plot','separate');

The HAC coefficient estimates are the same as the OLS estimates. The confidence bounds are slightly different because the standard error estimators are different.