One common approach is to split the observations in your data set of observations into observations to be used for estimation, and observations to be used for testing and evaluation. In time series work, you will usually take the first observations for estimation and the last for testing. With cross-section data, you may wish to order the data by some variable, such as household income, sales of a firm, or other indicator variables and use a subset for testing.

There are no hard and fast rules for determining the relative sizes of and . In some cases there may be obvious points at which a break in structure might have taken place—a war, a piece of legislation, a switch from fixed to floating exchange rates, or an oil shock. Where there is no reason a priori to expect a structural break, a commonly used rule-of-thumb is to use 85 to 90 percent of the observations for estimation and the remainder for testing.

where is the restricted sum of squared residuals, is the sum of squared residuals from subsample , is the total number of observations, and is the number of parameters in the equation. This formula can be generalized naturally to more than one breakpoint. The F-statistic has an exact finite sample F-distribution if the errors are independent and identically distributed normal random variables.

The log likelihood ratio statistic is based on the comparison of the restricted and unrestricted maximum of the (Gaussian) log likelihood function. The LR test statistic has an asymptotic distribution with degrees of freedom equal to under the null hypothesis of no structural change, where is the number of subsamples.

The Wald statistic is computed from a standard Wald test of the restriction that the coefficients on the equation parameters are the same in all subsamples. As with the log likelihood ratio statistic, the Wald statistic has an asymptotic distribution with degrees of freedom, where is the number of subsamples.

The Quandt-Andrews Breakpoint Test tests for one or more unknown structural breakpoints in the sample for a specified equation. The idea behind the Quandt-Andrews test is that a single Chow Breakpoint Test is performed at every observation between two dates, or observations, and . The test statistics from those Chow tests are then summarized into one test statistic for a test against the null hypothesis of no breakpoints between and .

The distribution of these test statistics is non-standard. Andrews (1993) developed their true distribution, and Hansen (1997) provided approximate asymptotic p-values. EViews reports the Hansen p-values. The distribution of these statistics becomes degenerate as approaches the beginning of the equation sample, or approaches the end of the equation sample. To compensate for this behavior, it is generally suggested that the ends of the equation sample not be included in the testing procedure. A standard level for this “trimming” is 15%, where we exclude the first and last 15% of the observations. EViews sets trimming at 15% by default, but also allows the user to choose other levels. Note EViews only allows symmetric trimming, i.e. the same number of observations are removed from the beginning of the estimation sample as from the end.

We consider a standard multiple linear regression model with periods and potential breaks (producing regimes). For the observations in regime we have the regression model

for the regimes . Note that the regressors are divided into two groups. The variables are those whose parameters do not vary across regimes, while the variables have coefficients that are regime specific.

While it is slightly more convenient to define breakdates to be the last date of a regime, we follow EViews’s convention in defining the breakdate to be the first date of the subsequent regime. We tie down the endpoints by setting and .

Bai and Perron (1998) describe global optimization procedures for identifying the multiple breaks which minimize the sums-of-squared residuals of the regression model Equation (26.29).

Briefly, for a specific set of breakpoints, say , we may minimize the sum-of-squared residuals:

using standard least squares regression to obtain estimates . The global -break optimizers are the set of breakpoints and corresponding coefficient estimates that minimize sum-of-squares across all possible sets of -break partitions.

Note that the number of comparison models increases rapidly in both and so that efficient algorithms for computing the optimizers are required. Practical algorithms for computing the global optimizers for multiple breakpoint models are outlined in Bai and Perron (2003a).

These global breakpoint estimates are then used as the basis for several breakpoint tests. EViews supports both the Bai and Perron (1998) tests of -breaks versus none test (along with the double maximum variants of this test in which is determined as part of the testing procedure), and information criterion methods (Yao, 1988 and Liu, Wi, and Zidek, 1997) for determining the number of breaks.

Bai and Perron (1998) describe a generalization of the Quandt-Andrews test (Andrews, 1993) in which we test for equality of the across multiple regimes. For a test of the null of no breaks against an alternative of breaks, we employ an F-statistic to evaluate the null hypothesis that . The general form of the statistic (Bai-Perron 2003a) is:

where is the optimal -break estimate of , , and is an estimate of the variance covariance matrix of which may be robust to serial correlation and heteroskedasticity, whose form depends on assumptions about the distribution of the data and the errors across segments. (We do not reproduce the formulae for the estimators of the variance matrices here as there are a large number of cases to consider; Bai-Perron (2003a) offer detailed descriptions of the various cases.)

A single test of no breaks against an alternative of breaks assumes that the alternative number of breakpoints is pre-specified. In cases where is not known, we may test the null of no structural change against an unknown number of breaks up to some upper-bound, . This type of testing is termed double maximum since it involves maximization both for a given and across various values of the test statistic for .

The equal-weighted version of the test, termed chooses the alternative that maximizes the statistic across the number of breakpoints. An alternative approach, denoted applies weights to the individual statistics so that the implied marginal -values are equal prior to taking the maximum.

Yao (1988) shows that under relatively strong conditions, the number of breaks that minimizes the Schwarz criterion is a consistent estimator of the true number of breaks in a breaking mean model.

Bai and Perron (1998) describe a modified Bai (1997) approach in which, at each test step, the breakpoints under the null are obtained by global minimization of the sum-of-squared residuals. We may therefore view this approach as an versus test procedure that combines the global and sequential testing approaches.

Each test begins with the set of global optimizing breakpoints and performs a single test of parameter constancy using the subsample break that most reduces the-sum-of-squared residuals. Note that in this case, we only test for constancy in a single subsample.

The two sequential tests are based on the Bai sequential methodology as described in “Sequential Testing Procedures” above. The methods differ in whether, for a given breakpoints, we test for an additional breakpoint in each of the segments (Sequential tests all subsets), or whether we test for the single added breakpoint that most reduces the sum-of-squares (Sequential L+1 breaks vs. L).

The Global L breaks vs. none choice implements the Bai-Perron tests of globally optimized breaks against the null of no structural breaks, along with the corresponding and tests ( “Global L Breaks vs. None”).

The L+1 breaks vs. global L choice implements the Bai-Perron vs. testing procedure outlined in “Global Plus Sequential Testing”.

By default, all of the variables in your specification will be included in this list. To treat some of these variables as non-varying ‘s, you may simply delete them from the list. Note that there must be at least one variable in the list.

In testing settings, you will be prompted to specify a test size, and to make assumptions about the distribution of the errors across segments which determine the form for the estimator in Equation (26.31).

One final note. EViews will, by default, estimate the robust specification assuming heterogeneous distributions for the . Bai and Perron (2003a) who, with one exception, do not impose the restriction that the distribution of the is the same across regimes. In cases where you are testing using robust variances, EViews will offer you a choice of whether to assume a common distribution for the data across regimes.

To illustrate the use of these tools in practice, we consider a simple model of the U.S. ex-post real interest rate from Garcia and Perron (1996) that is used as an example by Bai and Perron (2003a). The data, which consist of observations for the three-month treasury rate deflated by the CPI for the period 1961q1–1983q3, are provided in the series RATES in the workfile “realrate.WF1”. The regression model consists of a constant regressor, and allows for serial correlation that differs across regimes through the use of HAC covariance estimation. We allow up to 5 breaks in the model, and employ a trimming percentage of 15% . Since there are 103 observations in the sample, the trimming value implies that regimes are restricted to have at least 15 observations.

The default Method setting (Sequential L+1 breaks vs. L) instructs EViews to perform sequential testing of versus breaks using the methods outlined by Bai (1997) and Bai and Perron (1998).

To perform the Bai-Perron tests of globally optimized breaks against the null of no structural breaks, along with the corresponding and tests, simply call up the dialog and change the Method drop-down to Global L breaks vs. none:

The remaining lines show the individual test statistics (original, scaled, weighted) along with the critical values for the scaled statistics. In each case, the statistics far exceed the critical value so that we reject the null of no breaks. Note that the values corresponding to the and statistics are shaded for easy identification.

The Chow forecast test estimates two models—one using the full set of data , and the other using a long subperiod . Differences between the results for the two estimated models casts doubt on the stability of the estimated relation over the sample period. The Chow forecast test can be used with least squares and two-stage least squares regressions.

where is the residual sum of squares when the equation is fitted to all sample observations, is the residual sum of squares when the equation is fitted to observations, and is the number of estimated coefficients. This F-statistic follows an exact finite sample F-distribution if the errors are independent, and identically, normally distributed.

The log likelihood ratio statistic is based on the comparison of the restricted and unrestricted maximum of the (Gaussian) log likelihood function. Both the restricted and unrestricted log likelihood are obtained by estimating the regression using the whole sample. The restricted regression uses the original set of regressors, while the unrestricted regression adds a dummy variable for each forecast point. The LR test statistic has an asymptotic distribution with degrees of freedom equal to the number of forecast points under the null hypothesis of no structural change.

where the disturbance vector is presumed to follow the multivariate normal distribution . Specification error is an omnibus term which covers any departure from the assumptions of the maintained model. Serial correlation, heteroskedasticity, or non-normality of all violate the assumption that the disturbances are distributed . Tests for these specification errors have been described above. In contrast, RESET is a general test for the following types of specification errors:

Under such specification errors, LS estimators will be biased and inconsistent, and conventional inference procedures will be invalidated. Ramsey (1969) showed that any or all of these specification errors produce a non-zero mean vector for . Therefore, the null and alternative hypotheses of the RESET test are:

The test of specification error evaluates the restriction . The crucial question in constructing the test is to determine what variables should enter the matrix. Note that the matrix may, for example, be comprised of variables that are not in the original specification, so that the test of is simply the omitted variables test described above.

In testing for incorrect functional form, the nonlinear part of the regression model may be some function of the regressors included in . For example, if a linear relation,

the augmented model has and we are back to the omitted variable case. A more general example might be the specification of an additive relation,

A Taylor series approximation of the multiplicative relation would yield an expression involving powers and cross-products of the explanatory variables. Ramsey's suggestion is to include powers of the predicted values of the dependent variable (which are, of course, linear combinations of powers and cross-product terms of the explanatory variables) in :

where is the vector of fitted values from the regression of on . The superscripts indicate the powers to which these predictions are raised. The first power is not included since it is perfectly collinear with the matrix.

To apply the test, select View/Stability Diagnostics/Ramsey RESET Test… and specify the number of fitted terms to include in the test regression. The fitted terms are the powers of the fitted values from the original regression, starting with the square or second power. For example, if you specify 1, then the test will add in the regression, and if you specify 2, then the test will add and in the regression, and so on. If you specify a large number of fitted terms, EViews may report a near singular matrix error message since the powers of the fitted values are likely to be highly collinear. The Ramsey RESET test is only applicable to equations estimated using selected methods.

In recursive least squares the equation is estimated repeatedly, using ever larger subsets of the sample data. If there are coefficients to be estimated in the vector, then the first observations are used to form the first estimate of . The next observation is then added to the data set and observations are used to compute the second estimate of . This process is repeated until all the sample points have been used, yielding estimates of the vector. At each step the last estimate of can be used to predict the next value of the dependent variable. The one-step ahead forecast error resulting from this prediction, suitably scaled, is defined to be a recursive residual.

More formally, let denote the matrix of the regressors from period 1 to period , and the corresponding vector of observations on the dependent variable. These data up to period give an estimated coefficient vector, denoted by . This coefficient vector gives you a forecast of the dependent variable in period . The forecast is , where is the row vector of observations on the regressors in period . The forecast error is , and the forecast variance is given by:

The recursive residual is defined in EViews as:

These residuals can be computed for . If the maintained model is valid, the recursive residuals will be independently and normally distributed with zero mean and constant variance .

for , where is the recursive residual defined above, and s is the standard deviation of the recursive residuals . If the vector remains constant from period to period, , but if changes, will tend to diverge from the zero mean value line. The significance of any departure from the zero line is assessed by reference to a pair of 5% significance lines, the distance between which increases with . The 5% significance lines are found by connecting the points:

Movement of outside the critical lines is suggestive of coefficient instability. A sample CUSUM is given below:

The expected value of under the hypothesis of parameter constancy is:

which goes from zero at to unity at . The significance of the departure of from its expected value is assessed by reference to a pair of parallel straight lines around the expected value. See Brown, Durbin, and Evans (1975) or Johnston and DiNardo (1997, Table D.8) for a table of significance lines for the CUSUM of squares test.

The CUSUM of squares test provides a plot of against and the pair of 5 percent critical lines. As with the CUSUM test, movement outside the critical lines is suggestive of parameter or variance instability.

If you look back at the definition of the recursive residuals given above, you will see that each recursive residual is the error in a one-step ahead forecast. To test whether the value of the dependent variable at time might have come from the model fitted to all the data up to that point, each error can be compared with its standard deviation from the full sample.

Let be the k-th column of the data matrix (the k-th variable in a linear equation, or the k-th gradient in a non-linear), and be the remaining columns. Let be the residuals from a regression of the dependent variable, on , and let be the residuals from a regression of on . The leverage plot for the k-th coefficient is then a scatter plot of on .

It can easily be shown that in an auxiliary regression of on a constant and , the coefficient on will be identical to the k-th coefficient from the original regression. Thus the original regression can be represented as a series of these univariate auxiliary regressions.