cusumtest

Cusum test for structural change

Syntax

h = cusumtest(X,y)

h = cusumtest(Tbl)

h = cusumtest(___,Name=Value)

[h,H,Stat,W,B] = cusumtest(___)

cusumtest(___)

cusumtest(ax,___)

[___,sumPlots] = cusumtest(___)

Description

h = cusumtest(X,y) returns test rejection decision from conducting a cusum test on the multiple linear regression model y = Xβ + ε, where y is a vector of response data and X is a matrix of predictor data.

h = cusumtest(Tbl) conducts a cusum test on the variables of the input table or timetable. The response variable in the regression is the last table variable, and all other variables are the predictor variables. To select a different response variable for the regression, use the ResponseVariable name-value argument. To select different predictor variables, use the PredictorNames name-value argument.

h = cusumtest(___,Name=Value) uses additional options specified by one or more name-value arguments. Some options control the number of tests to conduct. The following conditions apply when cusumtest conducts multiple tests:

• cusumtest treats each test as separate from all other tests.

• All outputs expand their singleton dimension to contain results from each test.

For example, cusumtest(Tbl,ResponseVariable="RGDP",Test=["cusum" cusumsq"]) conducts two cusum tests using GDP as the response variable in the regressions and all other variables in the table Tbl as predictors. The first test uses the cusum test statistic and the second test uses the cusum of squares test statistic.

[h,H,Stat,W,B] = cusumtest(___) also returns the following decision statistics from conducting a cusum test, using any input-argument combination in the previous syntaxes:

• h, the test decision

• H, the sequence of decisions for each iteration of the test

• Stat, the sequence of test statistics

• W, the sequence of recursive residuals

• B, the sequence of coefficient estimates

cusumtest(___) plots both the sequence of cusums and the critical lines resulting from the cusum tests.

cusumtest(ax,___) plots on the axes specified by ax instead of the current axes (gca). ax can precede any of the input argument combinations in the previous syntaxes.

[___,sumPlots] = cusumtest(___) additionally returns handles to plotted graphics objects. Use elements of sumPlots to modify properties of the plot after you create it.

Examples

Conduct Cusum Test for Structural Change

Conduct a cusum test to assess whether there is a structural break in the equation for food demand. Input the predictor series as a matrix and input the response series as a vector.

Load the US food consumption data set Data_Consumption.mat, which contains annual measurements from 1927 through 1962 with missing data due to World War II in the matrix Data.

load Data_Consumption

Suppose that you want to develop 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])
axis tight
grid on
xlabel("Year")
ylabel("Index")
legend(["Price" "Income" "Consumption"],Location="southeast")

Measurements are missing from 1942 through 1947, which correspond to WWII.

Stabilize each series by applying the log transformation.

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.

$${\text{LQ}}_t={\beta }_0+{\beta }_1{\text{LI}}_t+{\beta }_2\text{LP}+{\epsilon }_t.$$

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

Identify the indices before WWII. 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")

% Obtain better view
h = gca;
h.CameraPosition = [4.3 -12.2 5.3];

Data relationships appear to be affected by the war.

Conduct a cusum test to assess whether there is a significant structural change. Use default values.

X = [LP LI];
y = LQ;
h = cusumtest(X,y)

h = logical
   0

h = 0 indicates that there is not enough evidence to reject the null hypothesis that the coefficients are equal across subsamples.

Conduct Cusum Test for Structural Change on Table Variables

Conduct a cusum test to assess whether there is a structural change in the equation for food demand, where the time series are variables in a table.

Load the US food consumption data set Data_Consumption.mat, which contains annual measurements from 1927 through 1962 with missing data due to World War II in the table DataTable. Convert the table to a timetable, and remove rows containing missing values.

load Data_Consumption
dates = datetime(dates,12,31);
TT = table2timetable(DataTable,RowTimes=dates);
TT.Row = [];
TT = rmmissing(TT);

Apply the log transform to all variables in the table.

LogTT = varfun(@log,TT);
LogTT.Properties.VariableNames

ans = 1x3 cell
    {'log_P'}    {'log_I'}    {'log_Q'}

Conduct a cusum test to assess whether there is a structural change in the regression model of log food consumption log_Q on log price log_P and log income log_I.

h = cusumtest(LogTT)

h = logical
   0

By default, chowtest selects the last table variable as the response, and selects all other variables as predictors. You can select a different variable by using the ResponseVariable name-value argument, and you can choose a different set of predictor variables by using the PredictorVariables name-value argument.

Return Test Decision Statistics

Load the US food consumption data set Data_Consumption.mat. Consider a model for log food consumption as determined by log food prices and log disposable income.

load Data_Consumption
LogDT = varfun(@log,DataTable);
numObs = height(LogDT) - sum(any(ismissing(LogDT),2))

numObs = 
30

numPreds = width(LogDT) - 1

numPreds = 
2

Conduct a cusum test using default values. Return all test decision statistics.

[h,H,Stat,W] = cusumtest(LogDT)

h = logical
   0

H=1×28 table
               H1       H2       H3       H4       H5       H6       H7       H8       H9       H10      H11      H12      H13      H14      H15      H16      H17      H18      H19      H20      H21      H22      H23      H24      H25      H26      H27      H28 
              _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____    _____

    Test 1    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false    false

Stat=1×28 table
              Stat1      Stat2       Stat3       Stat4       Stat5       Stat6       Stat7      Stat8      Stat9     Stat10     Stat11     Stat12      Stat13     Stat14    Stat15    Stat16    Stat17    Stat18    Stat19    Stat20    Stat21    Stat22    Stat23    Stat24    Stat25    Stat26      Stat27      Stat28 
              _____    _________    ________    ________    ________    ________    _______    _______    _______    _______    ______    ________    ________    ______    ______    ______    ______    ______    ______    ______    ______    ______    ______    ______    ______    _______    ________    ________

    Test 1     NaN     -0.012438    0.064511    -0.50784    -0.55747    -0.42687    -2.7881    -3.0973    -3.7625    -3.5417    -1.913    -0.65794    -0.35743    2.3762    3.3104    3.7509    2.8851    3.7395    4.2295    4.641     4.2412    4.496     3.2467    2.0001    1.5324    0.81729    0.053352    -0.98812

W=1×28 table
              W1         W2             W3            W4            W5            W6           W7            W8            W9           W10         W11         W12         W13         W14          W15          W16          W17           W18          W19          W20          W21           W22          W23          W24          W25           W26           W27           W28   
              ___    ___________    __________    __________    ___________    _________    _________    __________    __________    _________    ________    ________    ________    ________    _________    _________    __________    _________    _________    _________    __________    _________    _________    _________    __________    __________    __________    _________

    Test 1    NaN    -0.00012823    0.00079327    -0.0059004    -0.00051169    0.0013464    -0.024342    -0.0031882    -0.0068572    0.0022762    0.016791    0.012938    0.003098    0.028181    0.0096305    0.0045417    -0.0089262    0.0088086    0.0050515    0.0042415    -0.0041209    0.0026269    -0.012879    -0.012851    -0.0048219    -0.0073721    -0.0078755    -0.010737

cusumtest returns the overall rejection decision h, and tables containing the sequence of rejection decision from the forward recursions of the cusum test H, the corresponding sequence of test statistics Stat, and the corresponding recursive residuals W. Each table contains numObs - numPreds + Intercept = 28 variables corresponding to results of each recursion in the cusum test. Stat1 = W1 = NaN indicates the presence of a model intercept.

Plot Recursive Residuals and Critical Lines

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

Load the Nelson-Plosser data set Data_NelsonPlosser.mat, which contains the table of data DataTable.

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.

Convert the table to a timetable. Focus the sample to measurements from the end of 1915 through the end of 1970.

dates = datetime(dates,12,31);
span = isbetween(dates,datetime(1915,12,31),datetime(1970,12,31),"closed");
TT = table2timetable(DataTable,RowTimes=dates);
TT.Dates = [];
TT = TT(span,:);

Consider a predictive model of the US RGNP GNPR given measurements of the industrial production index IPI, total employment E, and real wages WR.

Plot the series in the model.

prednames = ["IPI" "E" "WR"];

tiledlayout(2,2)
for j = ["GNPR" prednames]
    nexttile
    plot(TT.Time,TT{:,j})
    ylabel(j)
end

To address exponential growth, apply the log transform to the series.

LogTT = varfun(@log,TT);

LogTT is a timetable containing the transformed variables in TT, but with names prepended with log_.

Assume that an appropriate multiple regression model to describe real GNP is

$$\log ({\text{GNPR}}_t)={\beta }_0+{\beta }_1\log ({\text{IPI}}_t)+{\beta }_2\log ({\text{E}}_t)+{\beta }_3\log ({\text{WR}}_t).$$

Conduct a cusum test to assess whether all regression coefficients are stable. Print a test summary to the command line. Plot the test statistics.

lprednames = "log_" + prednames;
cusumtest(LogTT,ResponseVariable="log_GNPR", ...
    PredictorVariables=lprednames,Display="summary")

RESULTS SUMMARY

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

Test type: cusum
Test direction: forward
Intercept: yes
Number of iterations: 52

Decision: Fail to reject coefficient stability
Significance level: 0.0500

ans = logical
   0

The cusum series does not cross the critical lines, which indicates model stability.

Test Consumption Model for Structural Change

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 US food consumption data set Data_Consumption.mat, which contains annual measurements from 1927 through 1962 with missing data due to the World War II in the table DataTable. Convert the table to a timetable, and remove rows containing missing values.

load Data_Consumption
dates = datetime(dates,12,31);
TT = table2timetable(DataTable,RowTimes=dates);
TT.Row = [];
TT = rmmissing(TT);

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

Apply the log transform to all variables in the table.

LogTT = varfun(@log,TT);

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

[h,~,~,W] = cusumtest(LogTT,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(LogTT.Time >= datetime(1941,12,31),1);
chowtest(LogTT,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.

Test for Structural Break in Volatility

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

$$\{\begin{array}{c}{y}_t=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]x_t+{\epsilon }_{1t};t=1,...,50\\ y_t=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]x_t+{\epsilon }_{2t};t=51,...,100.\end{array}$$

$$x_t$$ 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

Input Arguments

X — Predictor data X

numeric matrix

Predictor data X for the multiple linear regression model, specified as a numObs-by-numPreds numeric matrix.

Each row represents one of the numObs observations and each column represents one of the numPreds predictor variables.

Data Types: double

y — Response data y

numeric vector

Response data y for the multiple linear regression model, specified as a numObs-by-1 numeric vector. Rows of y and X correspond.

Data Types: double

Tbl — Combined predictor and response data

table | timetable

Combined predictor and response data for the multiple linear regression model, specified as a table or timetable with numObs rows. Each row of Tbl is an observation.

The test regresses the response variable, which is the last variable in Tbl, on the predictor variables, which are all other variables in Tbl. To select a different response variable for the regression, use the ResponseVariable name-value argument. To select different predictor variables, use the PredictorNames name-value argument to select numPreds predictors.

ax — Axes on which to plot

vector of Axes objects

Axes on which to plot, specified as a vector of Axes objects with length numTests.

By default, cusumtest plots each test to a separate figure.

Note

NaNs in X, y, or Tbl indicate missing values, and cusumtest removes observations containing at least one NaN. That is, to remove NaNs in X or y, cusumtest merges the variables [X y], and then it uses list-wise deletion to remove any row that contains at least one NaN. cusumtest also removes any row of Tbl containing at least one NaN. Removing NaNs in the data reduces the sample size and can create irregular time series.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: cusumtest(Tbl,ResponseVariable="RGDP",Test=["cusum" cusumsq"]) conducts two cusum tests using GDP as the response variable in the regressions and all other variables in the table Tbl as predictors. The first test uses the cusum test statistic and the second test uses the cusum of squares test statistic.

Intercept — Flag to include intercept

true (default) | false | logical vector

Flag to include an intercept when cusumtest fits the regression model, specified as a value in this table or a length numTests vector of such values.

Value	Description
true	cusumtest includes an intercept when fitting the regression model. numCoeffs = numPreds + 1.
false	cusumtest does not include an intercept when fitting the regression model. numCoeffs = numPreds.

cusumtest conducts a separate test for each value in Intercept.

Example: Intercept=false excludes an intercept from the model for each test.

Data Types: logical

Test — Type of cusum test

"cusum" (default) | "cusumsq" | character vector | string vector of test names | cell vector of test names

Type of cusum test, specified as a test name, or a string vector or cell vector of test names of length numTests.

Test Name	Description
"cusum"	Cusum test statistic. See [1].
"cusumsq"	Cusum of squares test statistic. See [1].

cusumtest conducts a separate test for each test name in Test.

Example: Test=["cusum" "cusumsq"] conducts two cusum tests. The first test uses the cusum test statistic and the second test uses the cusum of squares test statistic.

Data Types: char | cell | string

Direction — Iteration direction

"forward" (default) | "backward" | character vector | string vector of direction names | cell vector of direction names

Iteration direction, specified as a direction name, or a string vector or cell vector of direction names of length numTests.

Direction Name	Description
"forward"	cusumtest computes recursive residuals beginning with the first numCoeffs + 1 observations. Then, cusumtest adds one at a time until it reaches numObs observations.
"backward"	cusumtest reverses the order of the observations, and then follows the same steps as in "forward".

cusumtest conducts a separate test for each value in Direction.

Example: Test=["cusum" "cusumsq"] conducts two cusum tests. The first test computes recursive residuals using the forward method and the second test computes recursive residuals using the backward method.

Data Types: char | cell | string

Alpha — Nominal significance levels

0.05 (default) | numeric scalar | numeric vector

Nominal significance levels for the tests, specified as a numeric scalar or numeric vector of length numTests.

• For cusum tests (Test="cusum"), all elements of Alpha must be in the interval (0,1).

• For cusum of squares tests (Test="cusumsq"), all elements of Alpha must be in the interval [0.01,0.20].

cusumtest conducts a separate test for each value in Alpha.

Example: Alpha=[0.01 0.05] uses a level of significance of 0.01 for the first test, and then uses a level of significance of 0.05 for the second test.

Data Types: double

Display — Flag for command window display of results

"off" | "summary"

Flag for a command window display of results, specified as a value in this table.

Value	Description	Default Value When
"off"	cusumtest does not display results in the command window.	numTests = 1
"summary"	For each test, cusumtest displays results in the command window.	numTests > 1

The value of Display applies to all tests.

Example: Display="off"

Data Types: char | string

Plot — Flag indicating whether to plot test results

"off" | "on"

Flag indicating whether to plot test results, specified as a value in this table.

Value	Description	Default Value When
"off"	cusumtest does not produce any plots.	cusumtest returns any output argument.
"on"	cusumtest produces individual plots for each test.	cusumtest does not return any output arguments.

Depending on the value of Test, the plots show the sequence of cusums or cusums of squares together with critical lines determined by the value of Alpha.

The value of Plot applies to all tests.

Example: Plot="off"

Data Types: char | string

ResponseVariable — Variable in Tbl to use for response

first variable in Tbl (default) | string vector | cell vector of character vectors | vector of integers | logical vector

Variable in Tbl to use for response, specified as a string vector or cell vector of character vectors containing variable names in Tbl.Properties.VariableNames, or an integer or logical vector representing the indices of names. The selected variables must be numeric.

cusumtest uses the same specified response variable for all tests.

Example: ResponseVariable="GDP"

Example: ResponseVariable=[true false false false] or ResponseVariable=1 selects the first table variable as the response.

Data Types: double | logical | char | cell | string

PredictorVariables — Variables in Tbl to use for the predictors

string vector | cell vector of character vectors | vector of integers | logical vector

Variables in Tbl to use for the predictors, specified as a string vector or cell vector of character vectors containing variable names in Tbl.Properties.VariableNames, or an integer or logical vector representing the indices of names. The selected variables must be numeric.

cusumtest uses the same specified predictors for all tests.

By default, cusumtest uses all variables in Tbl that are not specified by the ResponseVariable name-value argument.

Example: PredictorVariables=["UN" "CPI"]

Example: PredictorVariables=[false true true false] or DataVariables=[2 3] selects the second and third table variables.

Data Types: double | logical | char | cell | string

Note

• When cusumtest conducts multiple tests, the function applies all single settings (scalars or character vectors) to each test.

• All vector-valued specifications that control the number of tests must have equal length.

• If the value of any option is a row vector, so is output h. Array and table outputs retain their specified dimensions.

Output Arguments

h — Test rejection decisions

logical scalar | logical vector

Test rejection decisions, returned as a logical scalar or vector with length equal to the number of tests numTests. cusumtest returns h when you supply the inputs X and y.

Hypotheses are independent of the value of Test.

• H₀: Coefficients in β are equal in all sequential subsamples.

• H₁: Coefficients in β change during the period of the sample.

Elements of h have the following values and meanings.

• Values of 1 indicates rejection of H₀ in favor of H₁.

• Value of 0 indicates failure to reject H₀.

H — Sequence of test rejection decisions

logical matrix | table

Sequence of test rejection decisions for each iteration of the cusum tests, returned as a numTests-by-(numObs – numPreds) logical matrix or table of logical variables.

Rows correspond to separate cusum tests and columns or variables correspond to iterations. When H is a table, variable j has label Hj.

• For tests in which Direction is "forward", columns or variables correspond to times numPreds + 1,...,numObs.

• For tests in which Direction is "backward", columns or variables correspond to times numObs – (numPreds + 1),...,1.

Rows corresponding to tests in which Intercept is true contain one less iteration, and the value in the first column of H defaults to false.

For a particular test (row), if any test decision in the sequence is 1, then h is true; that is, h = any(H,2). Otherwise, h is false.

Stat — Sequence of test statistics

numeric matrix | table

Sequence of test statistics for each iteration of the cusum tests, returned as a numTests-by-(numObs – numPreds) numeric matrix or table of numeric variables.

Rows correspond to separate cusum tests and columns or variables correspond to iterations. When Stat is a table, variable j has label Statj.

Values in any row depend on the value of Test. Array indices corresponds to the indexing in H. When W is a table, variable j has label Wj.

Rows corresponding to tests in which Intercept is true contain one less iteration, and the value in the first column of Stat defaults to NaN.

W — Sequence of standardized recursive residuals

numeric matrix | table

Sequence of standardized recursive residuals, returned as a numTests-by-(numObs – numPreds) numeric matrix or table of numeric variables.

Values in any row depend on the value of Test. Array indices correspond to the indexing in H. When W is a table, variable j has label Wj.

Rows corresponding to tests in which Intercept is true contain one less iteration, and the value in the first column of W defaults to NaN.

B — Sequence of recursive regression coefficient estimates

numeric array

Sequence of recursive regression coefficient estimates, returned as a (numPreds + 1)-by-(numObs – numPreds)-by-numTests numeric array.

• B(i,j,k) corresponds to coefficient i at iteration j for test k.

  At iteration j of test k, cusumtest estimates the coefficients using

  B(:,j,k) = X(1:numPreds+j,inRegression)\y(1:numPreds+j);

  inRegression is a logical vector indicating the predictors in the regression at iteration j of test k.

• During forward iterations, initially constant predictors can cause multicollinearity. Therefore, cusumtest holds out constant predictors until their data changes. For iterations in which cusumtest excludes predictors from the regression, corresponding coefficient estimates default to NaN. Similarly, for backward regression, cusumtest holds out terminally constant predictors. For more details, see [1].

• Tests in which:

  • Intercept is true contain one less iteration, and all values in the first column of B default to NaN.

  • Intercept is false contain one less coefficient, and the value in the first row, which corresponds to the intercept, defaults to NaN.

sumPlots — Handles to plotted graphics objects

graphics array

Handles to plotted graphics objects, returned as a 3-by-numTests graphics array. sumPlots contains unique plot identifiers, which you can use to query or modify properties of the plot.

Limitations

Cusum tests have little power to detect structural changes in the following cases.

• Late in the sample period

• When multiple changes produce cancellations in the cusums

More About

Cusum Tests

Cusum tests provide useful diagnostics for various model misspecifications, including gradual structural change, multiple structural changes, missing predictors, and neglected nonlinearities. The tests, formulated in [1], are based on cumulative sums, or cusums, of residuals resulting from recursive regressions (standardized one-step-ahead forecast errors) computed iteratively from nested subsamples of the data. Under the null hypothesis of coefficient constancy, values of the sequence outside an expected range suggest structural change in the model over time.

Tips

• The cusum of squares test:

  • Is a “useful complement to the cusum test, particularly when the departure from constancy of the [recursive coefficients] is haphazard rather than systematic” [1]

  • Has greater power for cases in which multiple shifts are likely to cancel

  • Is often suggested for detecting structural breaks in volatility

• Alpha specifies the nominal significance levels for the tests. The actual size of a test depends on various assumptions and approximations that cusumtest uses to compute the critical lines. Plots of the recursive residuals are the best indicator of structural change. Brown, et al. suggest that the tests “should be regarded as yardsticks for the interpretation of data rather than leading to hard and fast decisions” [1].

• To produce basic diagnostic plots of the recursive coefficient estimates having the same scale for test n, enter

  plot(B(:,:,n)')

  recreg produces similar plots, optionally using robust standard error bands.

Algorithms

• cusumtest handles initially constant predictor data using the method suggested in [1]. If a predictor's data is constant for the first numCoeffs observations and this results in multicollinearity with an intercept or another predictor, then cusumtest drops the predictor from regressions and the computation of recursive residuals until its data changes. Similarly, cusumtest temporarily holds out terminally constant predictors from backward regressions. Initially constant predictors in backward regressions, or terminally constant predictors in forward regressions, are not held out by cusumtest and can lead to rank deficiency in terminal iterations.

• cusumtest computes critical lines for inference in essentially different ways for the two test statistics. For cusums, cusumtest solves the normal CDF equation in [1] dynamically for each value of Alpha. For the cusums of squares test, cusumtest interpolates parameter values from the table in [2], using the method suggested in [1]. Sample sizes with degrees of freedom less than 4 are below tabulated values, and cusumtest cannot compute critical lines. Sample sizes with degrees of freedom greater than 202 are above tabulated values, and cusumtest uses the critical value associated with the largest tabulated sample size.

References

[1] Brown, R. L., J. Durbin, and J. M. Evans. "Techniques for Testing the Constancy of Regression Relationships Over Time." Journal of the Royal Statistical Society, Series B. Vol. 37, 1975, pp. 149–192.

[2] Durbin, J. "Tests for Serial Correlation in Regression Analysis Based on the Periodogram of Least Squares Residuals." Biometrika. Vol. 56, 1969, pp. 1–15.

Version History

Introduced in R2016a

R2022a: cusumtest returns some outputs in tables when you supply a table of data

If you supply a table of time series data Tbl, cusumtest returns H, Stat, and W as tables containing variables for recursive decision statistics with rows corresponding to separate tests.

Before R2022a, cusumtest returned the matrix outputs when you supplied a table of input data.

Starting in R2022a, if you supply a table of input data, update your code by accessing results in H, Stat, and W using table indexing. For more details, see Access Data in Tables.