Determine whether an explanatory model of real gross national product (GNP) is stable by plotting recursive residuals.

Load the Nelosson-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.

span = (1915 <= dates) & (dates <= 1970);

Consider the multiple linear regression model

Collect the model variables into a tabular array. Position the response as the last variable.

Mdl = DataTable(span,[4,5,10,1]);

Plot the test statistics.

cusumtest(Mdl);

The cusum series crosses the upper critical line after the 45th recursive regression, which indicates model instability.

Conduct cusum tests to assess whether there are structural changes in the equation for food demand around World War II. Implement forward and backward recursive regressions to obtain the test statistics.

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.

Consider a model for consumption as determined by food prices and disposable income, and assess its stability through the economic shock through the war.

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);

Assume that 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 indices before World War II. Plot log consumption with respect to the logs of food price and income.

preWarIdx = (dates <= 1941);

figure
scatter3(LP(preWarIdx),LI(preWarIdx),LQ(preWarIdx),[],'ro');
hold on
scatter3(LP(~preWarIdx),LI(~preWarIdx),LQ(~preWarIdx),[],'b*');
legend({'Pre-war observations','Post-war observations'},...
    'Location','Best')
xlabel('Log price')
ylabel('Log income')
zlabel('Log consumption')
title('{\bf Food Consumption Data}')
% Get a better view
h = gca;
h.CameraPosition = [4.3 -12.2 5.3];

Conduct forward and backward cusum tests using a 5% level of significance for each test. Plot the cusums.

cusumtest([LP,LI],LQ,'Direction',{'forward','backward'},'Plot','on');

RESULTS SUMMARY

***************
Test 1

Test type: cusum
Test direction: forward
Intercept: yes
Number of iterations: 27

Decision: Fail to reject coefficient stability
Significance level: 0.0500

***************
Test 2

Test type: cusum
Test direction: backward
Intercept: yes
Number of iterations: 27

Decision: Fail to reject coefficient stability
Significance level: 0.0500

The plots and test results at the command line indicate that neither test rejects the null hypothesis that coefficients are stable.

Compare the results of the cusum tests with the results of a Chow test. Unlike cusum tests, Chow tests require a guess for the time point at which the structural break occurs. Specify that the break point is 1941.

bp = find(preWarIdx,1,'last');
chowtest([LP,LI],LQ,bp,'Display','summary');

RESULTS SUMMARY

***************
Test 1

Sample size: 30
Breakpoint: 15

Test type: breakpoint
Coefficients tested: All

Statistic: 5.5400
Critical value: 3.0088

P value: 0.0049
Significance level: 0.0500

Decision: Reject coefficient stability

The test results reject the null hypothesis that the coefficients are stable.

The Chow and cusum test results are not consistent. For details on cusum test limitations, see Limitations.

Check whether a cusum of squares test can detect a structural break in volatility in simulated data.

Simulate a series of data from this regression model

$$x_t$$ is a series of observations from three standard Gaussian predictor variables. $${\epsilon }_{1t}$$ and $${\epsilon }_{2t}$$ are series of Gaussian innovations both with mean 0 and standard deviation 0.1 and 0.2, respectively.

rng(1); % For reproducibility
T = 100;
X = randn(T,3);
sigma1 = 0.1;
sigma2 = 0.2;
e = [sigma1*randn(T/2,1); sigma2*randn(T/2,1)];
b = (1:3)';
y = X*b + e;

Conduct a cusum of squares test using a 5% level of significance. Plot the test statistics and critical region bands. Indicate that there is no model intercept. Request to return whether the test statistics cross into critical region at each iteration.

[~,H] = cusumtest(X,y,'Test','cusumsq','Plot','on',...
    'Direction',{'forward','backward'},'Display','off','Intercept',false);

Because the test statistics cross the critical lines at least once for both tests, the tests reject the null hypothesis of constant volatility at 5% level. The test statistics change direction around iteration 50, which is consistent with the simulated break in volatility in the data.

H is a 2-by-97 logical matrix containing the sequence of decisions for each iteration of each cusum of squares test. The first row corresponds to the forward cusum of squares test, and the second row corresponds to the backward cusum of squares test.

For the forward test, determine the iterations that result in the test statistics crossing the critical line.

bp = find(H(1,:) == 1)

bp = 1×35

    24    25    26    27    28    29    30    31    32    33    34    35    36    37    38    39    40    41    42    43    44    45    46    47    48    49    50    51    52    53    54    55    56    57    58