chowtest

Chow test for structural change

Syntax

h = chowtest(X,y,bp)

[h,pValue,stat,cValue] = chowtest(X,y,bp)

StatTbl = chowtest(Tbl,bp)

___ = chowtest(___,Name=Value)

Description

Chow tests assess the stability of coefficients β in a multiple linear regression model of the form y = Xβ + ε. chowtest splits the data at specified break points. Coefficients are estimated in initial subsamples, then tested for compatibility with data in complementary subsamples.

h = chowtest(X,y,bp) returns a vector of test decisions h from conducting Chow tests on the multiple linear regression model y = Xβ + ε at the break points in bp. y is a vector of response data and X is a matrix of predictor data. Each element of bp results in a separate test.

[h,pValue,stat,cValue] = chowtest(X,y,bp) additionally returns vectors of p-values pValue, test statistics stat, and critical values cValue for the tests.

StatTbl = chowtest(Tbl,bp) returns the table StatTbl containing variables for the test results, statistics, and settings from conducting Chow tests on the variables of the table or timetable Tbl. Each row of StatTbl contains the results of the corresponding test.

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.

___ = chowtest(___,Name=Value) uses additional options specified by one or more name-value arguments, using any input-argument combination in the previous syntaxes. chowtest returns the output-argument combination for the corresponding input arguments.

In addition to bp, some options control the number of tests to conduct. For example, chowtest(Tbl,ResponseVariable="GDP",Test=["breakpoint" "forecast"],Intercept=false) conducts two tests for the presence of a structural break in the coefficients of the regression model of GDP on all other variables of the table Tbl without an intercept term. The first test assesses coefficient equality constraints directly, and the second test assesses forecast performance.

Examples

Conduct Chow Test for Structural Change

Conduct Chow tests to assess whether there are structural changes in the equation for food demand around World War II. 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 the war 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 World War II.

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 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")

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

Data relationships appear to be affected by the war.

Conduct two break point Chow tests at 5% level of significance. For the first test, set the break point at 1941. Set the break point of the other test at 1948.

bp = find(preWarIdx,1,"last");

X = [LP LI];
y = LQ;
h1941 = chowtest(X,y,bp)     

h1941 = logical
   1

h1948 = chowtest(X,y,bp + 1)

h1948 = logical
   0

h1941 = 1 indicates that there is enough evidence to reject the null hypothesis that the coefficients are stable when the break points occur before the war. However, h1948 = 0 indicates that there is not enough evidence to reject coefficient stability if the break point is after the war. This result suggests that the data at 1948 is influential.

Alternatively, you can supply a vector of break points to conduct three Chow tests.

h = chowtest(X,y,[bp bp+1]);

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

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

Sample size: 30
Breakpoint: 16

Test type: breakpoint
Coefficients tested: All

Statistic: 1.2942
Critical value: 3.0088

P value: 0.2992
Significance level: 0.0500

Decision: Fail to reject coefficient stability

By default, chowtest displays a summary of the test results for each test when you conduct more than one test.

Return Test p-Values and 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
X = log(Data(:,1:2));
y = log(Data(:,3));

Conduct two break point Chow tests at 5% level of significance. For the first test, set the break point at 1941. Set the break point of the other test at 1948. Return the test decision, $\mathit{p}$-Value, test statistic, and test critical value.

bp = find(dates <= 1941,1,"last");
[h,pValue,stat,cValue] = chowtest(X,y,bp)

h = logical
   1

pValue = 0.0049

stat = 5.5400

cValue = 3.0088

pValue < 0.01, which suggests that the evidence to reject the null hypothesis that all coefficients in the regression models, determined by the break point at 1941, are equal.

Conduct Chow Test for Structural Change on Table Variables

Conduct Chow tests to assess whether there are structural changes in the equation for food demand around World War II, 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 the war 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 two break point Chow tests at 5% level of significance. For the first test, set the break point at the end of 1941. Set the break point of the other test at the end of 1948.

bp1941 = find(LogTT.Time >= datetime(1941,12,31),1);
bp1948 = find(LogTT.Time >= datetime(1948,12,31),1);
bp = [bp1941 bp1948];

StatTbl = chowtest(LogTT,bp)

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

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

Sample size: 30
Breakpoint: 16

Test type: breakpoint
Coefficients tested: All

Statistic: 1.2942
Critical value: 3.0088

P value: 0.2992
Significance level: 0.0500

Decision: Fail to reject coefficient stability

StatTbl=2×8 table
                h       pValue       stat     cValue    Break Point    Alpha    Intercept         Test     
              _____    _________    ______    ______    ___________    _____    _________    ______________

    Test 1    true     0.0049125      5.54    3.0088        15         0.05       true       {'breakpoint'}
    Test 2    false      0.29918    1.2942    3.0088        16         0.05       true       {'breakpoint'}

StatTbl contains the decision statistics and options for each test (row).

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. You can choose a different set of predictor variables by using the PredictorVariables name-value argument.

Test Model of Real U.S. GNP for Structural Change

Apply the Chow test to assess the stability of an explanatory model of US real gross national product (RGNP) using the end of World War II as a break point.

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_.

Select the index corresponding to the end of World War II, September 2, 1945.

bp = find(LogTT.Time > datetime(1945,9,2),1);

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 break point test to assess whether all regression coefficients are stable. Use the end of WWII as a break point. Print a test summary to the command line.

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

RESULTS SUMMARY

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

Sample size: 56
Breakpoint: 31

Test type: breakpoint
Coefficients tested: All

Statistic: 4.0978
Critical value: 2.5652

P value: 0.0062
Significance level: 0.0500

Decision: Reject coefficient stability

StatTbl=1×8 table
                h       pValue       stat     cValue    Break Point    Alpha    Intercept         Test     
              _____    _________    ______    ______    ___________    _____    _________    ______________

    Test 1    true     0.0061633    4.0978    2.5652        31         0.05       true       {'breakpoint'}

StatTbl contains decision statistics and test options for the test. StatTbl.h = 1 and StatTbl.pValue < 0.01 indicate string evidence to reject the null hypothesis that the regression coefficients before and after WWII are equivalent.

Assess Stability of Subsets of Regression Coefficients

Conduct a Chow test to assess the stability of a subset of regression coefficients. This example expands on Conduct Chow Test for Structural Change.

Load the US food consumption data set. Convert the table to a timetable, and remove rows containing missing values.

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

Apply the log transformation to each series.

LogTT = varfun(@log,DataTable);

Identify the indices before World War II.

preWarIdx = dates <= datetime(1941,12,31);

Consider two regression models: one is log consumption onto log food price, and the other is log consumption onto log income. Plot scatter plots and regression lines for both models.

figure
plot(LogTT.log_P(preWarIdx),LogTT.log_Q(preWarIdx),"bo", ...
    LogTT.log_P(~preWarIdx),LogTT.log_Q(~preWarIdx),"r*");
axis tight
grid on
lsline
xlabel("Log Price")
ylabel("Log Consumption")
legend("Pre-war observations","Post-war observations")

figure
plot(LogTT.log_I(preWarIdx),LogTT.log_Q(preWarIdx),"bo", ...
    LogTT.log_I(~preWarIdx),LogTT.log_Q(~preWarIdx),"r*");
axis tight
grid on
lsline
xlabel("Log Income")
ylabel("Log Consumption")
legend("Pre-war observations","Post-war observations", ...
    Location="northwest")

A clear break in food price elasticity exists between subsamples before and after the war. However, income elasticity does not appear to have such a break.

Conduct two Chow tests to determine whether there is statistical evidence to reject model continuity for both regression models. Because there are more observations in the complementary subsample than coefficients, conduct a break point test. Consider the elasticities in the test only. That is, specify false for the intercept (first coefficient) and true for elasticity (second coefficient).

bp = find(preWarIdx,1,"last"); % Index for 1941
chowtest(LogTT,bp,Coeffs=[false true],Display="summary", ...
    ResponseVariable="log_Q",PredictorVariables="log_P");

RESULTS SUMMARY

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

Sample size: 30
Breakpoint: 15

Test type: breakpoint
Coefficients tested: 0  1

Statistic: 7.3947
Critical value: 4.2252

P value: 0.0115
Significance level: 0.0500

Decision: Reject coefficient stability

chowtest(LogTT,bp,Coeffs=[false true],Display="summary", ...
    ResponseVariable="log_Q",PredictorVariables="log_I");

RESULTS SUMMARY

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

Sample size: 30
Breakpoint: 15

Test type: breakpoint
Coefficients tested: 0  1

Statistic: 0.1289
Critical value: 4.2252

P value: 0.7225
Significance level: 0.0500

Decision: Fail to reject coefficient stability

The first test rejects the null hypothesis that price elasticities are equivalent across subsamples at 5% level of significance. The second test fails to reject the null hypothesis that income elasticities are equivalent across subsamples.

Consider a regression model of log consumption onto the logs of price and income. Conduct two break point tests: one that compares price elasticity across subsamples only, and another that compares income elasticity only.

Coeffs = [false true false;
          false false true];
chowtest(LogTT,bp,Coeffs=Coeffs,Display="summary", ...
    ResponseVariable="log_Q",PredictorVariables=["log_P" "log_I"]);

RESULTS SUMMARY

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

Sample size: 30
Breakpoint: 15

Test type: breakpoint
Coefficients tested: 0  1  0

Statistic: 0.0001
Critical value: 4.2597

P value: 0.9920
Significance level: 0.0500

Decision: Fail to reject coefficient stability

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

Sample size: 30
Breakpoint: 15

Test type: breakpoint
Coefficients tested: 0  0  1

Statistic: 2.8151
Critical value: 4.2597

P value: 0.1064
Significance level: 0.0500

Decision: Fail to reject coefficient stability

For both tests, there is not enough evidence to reject model stability at 5% level.

Model Structural Change

Simulate data for a linear model including a structural break in the intercept and one of the predictor coefficients. Then, choose specific coefficients to test for equality across a break point using the Chow test. Adjust parameters to assess the sensitivity of the Chow test.

Specify four predictors, 50 observations, and a break point at period 44 for the simulated linear model.

numPreds = 4;
numObs = 50;
bp = 44;
rng(1); % For reproducibility

Form the predictor data by specifying means for the predictors, and then adding random, standard Gaussian noise to each of the means.

mu = [0 1 2 3];
X = repmat(mu,numObs,1) + randn(numObs,numPreds);

Include a column of ones to the predictor data.

X = [ones(numObs,1) X];

Specify the true values of the regression coefficients and that the intercept and the coefficient of the second predictor jump by 10%.

beta1 = [1 2 3 4 5]';                                % Initial subsample coefficients
beta2 = beta1 + [beta1(1)*0.1 0 beta1(3)*0.1 0 0 ]'; % Complementary subsample coefficients
X1 = X(1:bp,:);                                      % Initial subsample predictors
X2 = X(bp+1:end,:);                                  % Complementary subsample predictors

Specify a 2-by-5 logical matrix that indicates to test first the intercept and second regression coefficient, and then test all other coefficients.

test1 = [true false true false false]; 
Coeffs = [test1; ~test1]

Coeffs = 2x5 logical array

   1   0   1   0   0
   0   1   0   1   1

The null hypothesis for the first test (Coeffs(1,:)) is equality of the intercepts and the coefficients of the second predictor across subsamples. The null hypothesis for the second test (Coeffs(2,:)) is equality of the first, third, and fourth predictors across subsamples.

Simulate data for the linear model

$$\text{y}=\left[\begin{array}{cc}X1& 0\\ 0& X2\end{array}\right]\left[\begin{array}{c}\text{beta1}\\ \text{beta2}\end{array}\right]+\text{innov}.$$

Create innov as a vector of random Gaussian variates with mean zero and standard deviation 0.2.

sigma = 0.2;
innov = sigma*randn(numObs,1);
y = [X1 zeros(bp,size(X2,2)); ...
    zeros(numObs-bp,size(X1,2)) X2]*[beta1; beta2]+innov;

Conduct the two break point tests indicated in Coeffs. Because there is an intercept in the predictor matrix X already, specify to suppress its inclusion in the linear model that chowtest fits.

chowtest(X,y,bp,Intercept=false,Coeffs=Coeffs, ...
    Display="summary");

RESULTS SUMMARY

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

Sample size: 50
Breakpoint: 44

Test type: breakpoint
Coefficients tested: 1  0  1  0  0

Statistic: 5.7102
Critical value: 3.2317

P value: 0.0066
Significance level: 0.0500

Decision: Reject coefficient stability

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

Sample size: 50
Breakpoint: 44

Test type: breakpoint
Coefficients tested: 0  1  0  1  1

Statistic: 0.2497
Critical value: 2.8387

P value: 0.8611
Significance level: 0.0500

Decision: Fail to reject coefficient stability

At the default significance level:

• The Chow test correctly rejects the null hypothesis that no structural breaks exist at period bp for the intercept and the second coefficient.

• The Chow test correctly failed to reject the null hypothesis for the other coefficients.

Compare the break point test results to the results of the forecast test.

chowtest(X,y,bp,Intercept=false,Coeffs=Coeffs, ...
    Test="forecast",Display="summary");

RESULTS SUMMARY

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

Sample size: 50
Breakpoint: 44

Test type: forecast
Coefficients tested: 1  0  1  0  0

Statistic: 3.7637
Critical value: 2.8451

P value: 0.0182
Significance level: 0.0500

Decision: Reject coefficient stability

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

Sample size: 50
Breakpoint: 44

Test type: forecast
Coefficients tested: 0  1  0  1  1

Statistic: 0.2135
Critical value: 2.6123

P value: 0.9293
Significance level: 0.0500

Decision: Fail to reject coefficient stability

In this case, the inferences from the tests are equivalent to those for the break point test.

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.

bp — Break points

positive integer | vector of positive integers

Break points for the tests, specified as a positive integer or a vector of positive integers.

Each break point is an index of a specific observation (row) in the data. The element bp(j) specifies to split the data into the initial and complementary samples indexed by 1:bp(j) and (bp(j) + 1):numObs, respectively.

Data Types: double

Note

• NaNs in X, y, or Tbl indicate missing values, and chowtest removes observations containing at least one NaN. That is, to remove NaNs in X or y, chowtest merges the variables [X y], and then it uses list-wise deletion to remove any row that contains at least one NaN. chowtest 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.

• If bp is a scalar, then the number of tests, numTests, is the common dimension of values in name-value argument. In this case, chowtest uses the same bp in each test. Otherwise, the length of bp determines numTests, and chowtest runs separate tests for each value in bp.

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: chowtest(Tbl,DataVariable="GDP",Test=["breakpoint" "forecast"],Intercept=true) conducts two tests for the presence of a structural break in the coefficients, including an intercept, of the regression model of GDP on all other variables of the table Tbl. The first test assesses coefficient equality constraints directly, and the second test assesses forecast performance.

Intercept — Flag to include intercept

true (default) | false | logical vector

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

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

chowtest 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 — Chow test type

"breakpoint" (default) | "forecast" | character vector | string vector | cell vector of character vectors

Chow test type, specified as a test name, or a string vector or cell vector of test names. This table contains the supported test names.

Test Name	Description
"breakpoint" (default)	chowtest directly assesses coefficient equality constraints using an F statistic. Both subsamples must have more than numCoeffs observations.
"forecast"	chowtest assesses forecast performance using a modified F statistic. The initial subsample must have more than numCoeffs observations.

For details on the value of numCoeffs, see the Intercept and Coeffs name-value arguments.

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

Example: Test=["breakpoint" "forecast"] conducts two Chow tests. The first test directly assesses coefficient equality constraints, and the second test assess forecast performance.

Data Types: char | string | cell

Coeffs — Flags indicating which elements of β to test for equality

logical vector | logical array

Flags indicating which elements of β to test for equality, specified as a logical vector or array. Vector values must be of length numCoeffs. Array values must be of size numTests-by-numCoeffs.

If the value of Intercept contains mixed logical values:

• numCoeffs is numPreds + 1

• chowtest ignores values in the first column of Coeffs for models without an intercept.

For example, suppose the regression model has three predictors (numPreds is 3) in a linear model, and you want to conduct two Chow tests (numTests is 2). Each test includes all regression parameters in the linear model. Also, you want chowtest to fit an intercept in the linear model for the first test only. Therefore, Intercept must be the logical array [1 0]. Because there is at least one model for which chowtest fits an intercept, Coeffs must be a 2-by-4 logical array (numTests is 2 and numCoeffs is numPreds + 1). The elements of Coeffs(:,1) correspond to whether to test the intercept irrespective of its presence in the model. Therefore, one way to specify Coeffs is true(2,4). For the second test, chowtest does not fit an intercept, and so it ignores the value true in Coeffs(2,1). Because chowtest ignores Coeffs(2,1), Coeffs = [true(1,4); false true(1,3)] yields the same result.

The default is true(numTests,numCoeffs), which tests all of β for all tests.

Example: Coeffs=[false true true; true true true] conducts two Chow tests on a model containing three coefficients. The first test assesses the second and third coefficients, and the second test assesses all coefficients.

Alpha — Nominal significance level

0.05 (default) | numeric scalar | numeric vector

Nominal significance level for the hypothesis test, specified as a numeric scalar in the interval (0,1) or a numeric vector of such values.

chowtest 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"	chowtest does not display results in the command window.	numTests = 1
"summary"	For each test, chowtest displays results in the command window.	numTests > 1

The value of Display applies to all tests.

Example: Display="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.

chowtest 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.

chowtest uses the same specified predictors for all tests.

By default, chowtest 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 chowtest 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. Vector values and Coeffs arrays must share a common dimension, equal to numTests.

• If you specify X and y, and bp, Intercept, Test, or Alpha are row vectors, chowtest returns row vectors.

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. chowtest returns h when you supply the inputs X and y.

The Chow test has the following hypotheses are:

• H₀: Regression coefficients β, selected by the Coeffs name-value argument, are identical across subsamples.

• H₁: At least one regression coefficient in β, selected by the Coeffs name-value argument, exhibits significant change across subsamples.

Elements of h have the following values and meanings.

• Values of 1 indicate rejection of the null hypothesis that the regression coefficients β, selected by Coeffs, are identical across subsamples model, in favor of the alternative hypothesis.

• Values of 0 indicate failure to reject the null hypothesis.

pValue — Test statistic p-values

numeric scalar | numeric vector

Test statistic p-values, returned as a numeric scalar or vector with length equal to the number of tests numTests. chowtest returns pValue when you supply the inputs X and y.

stat — Test statistics

numeric scalar | numeric vector

Test statistics, returned as a numeric scalar or vector with length equal to the number of tests numTests. chowtest returns stat when you supply the inputs X and y. For details, see Chow Tests.

cValue — Critical values

numeric scalar | numeric vector

Critical values, returned as a numeric scalar or vector with length equal to the number of tests numTests. chowtest returns cValue when you supply the inputs X and y.

Alpha determines the critical values.

StatTbl — Test summary

table

Test summary, returned as a table with variables for the outputs h, pValue, stat, and cValue, and with a row for each test. chowtest returns StatTbl when you supply the input Tbl.

StatTbl contains variables for the break point Break Point and test settings specified by Alpha, Intercept, and Test.

More About

Chow Tests

Chow tests assess the stability of the coefficients β in a multiple linear regression model of the form y = Xβ + ε. chowtest supports the two variations of the Chow test introduced in [1]: the break point and forecast tests.

The break point test is a standard F test from the analysis of covariance. The forecast test makes use of the standard theory of prediction intervals. Chow’s contribution is to place both tests within the general linear hypothesis framework, and then to develop appropriate test statistics for testing subsets of coefficients (see Coeffs). For test-statistic formulae, see [1].

Tips

• Chow tests assume continuity of the innovations variance across structural changes. Heteroscedasticity can distort the size and power of the test. Therefore, verify the innovations-variance-continuity assumption holds before using the test results for inference.

• If both subsamples contain more than numCoeffs observations, you can conduct a forecast test instead of a break point test (see Test). However, the forecast test might have lower power relative to the break point test [1]. Nevertheless, Wilson (1978) suggests conducting the forecast test in the presence of unknown specification errors [4].

• You can apply the forecast test to cases where both subsamples have size greater than numCoeffs, where you would typically apply a breakpoint test. In such cases, the forecast test might have significantly reduced power relative to a break point test [1]. Nevertheless, Wilson (1978) suggests use of the forecast test in the presence of unknown specification errors [4].

• The forecast test is based on the unbiased predictions, with zero mean error, that result from stable coefficients. However, zero mean forecast error does not, in general, guarantee coefficient stability. Therefore, forecast tests are most effective in checking for structural breaks, rather than model continuity [3].

• To obtain diagnostic statistics for each subsample, such as regression coefficient estimates, their standard errors, error sums of squares, and so on, pass the appropriate data to fitlm. For details on working with LinearModel model objects, see Multiple Linear Regression.

References

[1] Chow, G. C. "Tests of Equality Between Sets of Coefficients in Two Linear Regressions." Econometrica. Vol. 28, 1960, pp. 591–605.

[2] Fisher, F. M. "Tests of Equality Between Sets of Coefficients in Two Linear Regressions: An Expository Note." Econometrica. Vol. 38, 1970, pp. 361–66.

[3] Rea, J. D. "Indeterminacy of the Chow Test When the Number of Observations is Insufficient." Econometrica. Vol. 46, 1978, p. 229.

[4] Wilson, A. L. "When is the Chow Test UMP?" The American Statistician. Vol. 32, 1978, pp. 66–68.

Version History

Introduced in R2015b

R2022a: chowtest returns a results table when you supply a table of data

If you supply a table of time series data Tbl, chowtest returns a table containing variables for the test rejection decisions h, p-values pValue, tests statistics stat, and critical values cValue, with rows corresponding to separate tests.

Before R2022a, chowtest returned the numeric outputs in separate positions of the output when you supplied a table of input data.

Starting in R2022a, if you supply a table of input data, update your code to return all outputs in the first output position.

StatTbl = chowtest(Tbl,bp,Name=Value)

If you request more outputs, chowtest issues an error.

Also, access results by using table indexing. For more details, see Access Data in Tables.