584—Chapter 19. Specification and Diagnostic Tests

Specification and Stability Tests

EViews provides a number of test statistic views that examine whether the parameters of your model are stable across various subsamples of your data.

One recommended empirical technique is to split the T observations in your data set of observations into T1 observations to be used for estimation, and T2 = T − T1 observations to be used for testing and evaluation. Using all available

sample observations for estimation promotes a search for a specification that best fits that specific data set, but does not allow for testing predictions of the model against data that have not been used in estimating the model. Nor does it allow one to test for parameter constancy, stability and robustness of the estimated relationship. In time series work, you will usually take the first T1 observations for estimation and the last T2 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 sub-set for testing.

There are no hard and fast rules for determining the relative sizes of T1 and T2 . 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.

EViews provides built-in procedures which facilitate variations on this type of analysis.

Chow's Breakpoint Test

The idea of the breakpoint Chow test is to fit the equation separately for each subsample and to see whether there are significant differences in the estimated equations. A significant difference indicates a structural change in the relationship. For example, you can use this test to examine whether the demand function for energy was the same before and after the oil shock. The test may be used with least squares and two-stage least squares regressions.

To carry out the test, we partition the data into two or more subsamples. Each subsample must contain more observations than the number of coefficients in the equation so that the equation can be estimated. The Chow breakpoint test compares the sum of squared residuals obtained by fitting a single equation to the entire sample with the sum of squared residuals obtained when separate equations are fit to each subsample of the data.

EViews reports two test statistics for the Chow breakpoint test. The F-statistic is based on the comparison of the restricted and unrestricted sum of squared residuals and in the simplest case involving a single breakpoint, is computed as:

586—Chapter 19. Specification and Diagnostic Tests

where ˜ ′ ˜ is the residual sum of squares when the equation is fitted to all sample u u T

observations, u′u is the residual sum of squares when the equation is fitted to T1 observations, and k 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 χ2 distribution with degrees of freedom equal to the number of forecast points T2 under the null hypothesis of no structural change.

To apply Chow’s forecast test, push View/Stability Tests/Chow Forecast Test… on the equation toolbar and specify the date or observation number for the beginning of the forecasting sample. The date should be within the current sample of observations.

As an example, suppose we estimate a consumption function using quarterly data from 1947:1 to 1994:4 and specify 1973:1 as the first observation in the forecast period. The test reestimates the equation for the period 1947:1 to 1972:4, and uses the result to compute the prediction errors for the remaining quarters, and reports the following results:

Chow Forecast Test: Forecast from 1973:1 to 1994:4

Neither of the forecast test statistics reject the null hypothesis of no structural change in the consumption function before and after 1973:1.

If we test the same hypothesis using the Chow breakpoint test, the result is:

Chow Breakpoint Test: 1973:1

Note that both of the breakpoint test statistics decisively reject the hypothesis from above. This example illustrates the possibility that the two Chow tests may yield conflicting results.

Ramsey's RESET Test

RESET stands for Regression Specification Error Test and was proposed by Ramsey (1969). The classical normal linear regression model is specified as:

588—Chapter 19. Specification and Diagnostic Tests

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 Z :

indicate the powers to which these predictions are raised. The first power is not included since it is perfectly collinear with the X matrix.

Output from the test reports the test regression and the F-statistic and log likelihood ratio for testing the hypothesis that the coefficients on the powers of fitted values are all zero. A study by Ramsey and Alexander (1984) showed that the RESET test could detect specification error in an equation which was known a priori to be misspecified but which nonetheless gave satisfactory values for all the more traditional test criteria—goodness of fit, test for first order serial correlation, high t-ratios.

To apply the test, select View/Stability Tests/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

ber 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 applicable only to an equation estimated by least squares.

Recursive Least Squares

In recursive least squares the equation is estimated repeatedly, using ever larger subsets of the sample data. If there are k coefficients to be estimated in the b vector, then the first k observations are used to form the first estimate of b . The next observation is then added to the data set and k + 1 observations are used to compute the second estimate of b . This process is repeated until all the T sample points have been used, yielding T − k + 1 estimates of the b vector. At each step the last estimate of b 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 Xt − 1 denote the ( t − 1) × k matrix of the regressors from period 1 to period t − 1 , and yt − 1 the corresponding vector of observations on the dependent vari-

Specification and Stability Tests—589

able. These data up to period t − 1 give an estimated coefficient vector, denoted by bt − 1 .

These residuals can be computed for t = k + 1, …, T . If the maintained model is valid, the recursive residuals will be independently and normally distributed with zero mean and constant variance σ2 .

To calculate the recursive residuals, press

View/Stability Tests/Recursive Estimates (OLS only)… on the equation toolbar. There are six options available for the recursive estimates view. The recursive estimates view is only available for equations estimated by ordinary least squares without AR and MA terms. The Save Results as Series option allows you to save the recursive residuals and recursive coefficients as

named series in the workfile; see “Save Results as Series” on page 592.

Recursive Residuals

This option shows a plot of the recursive residuals about the zero line. Plus and minus two standard errors are also shown at each point. Residuals outside the standard error bands suggest instability in the parameters of the equation.

CUSUM Test

The CUSUM test (Brown, Durbin, and Evans, 1975) is based on the cumulative sum of the recursive residuals. This option plots the cumulative sum together with the 5% critical lines. The test finds parameter instability if the cumulative sum goes outside the area between the two critical lines.

The CUSUM test is based on the statistic:

r = k + 1

590—Chapter 19. Specification and Diagnostic Tests

for t = k + 1, …, T , where w is the recursive residual defined above, and s is the standard error of the regression fitted to all T sample points. If the β vector remains constant from period to period, E( Wt) = 0 , but if β changes, Wt 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 t . The 5% significance lines are found by connecting the points:

[ k, ±−0.948 ( T − k)1 ⁄ 2] and [ T, ±3 × 0.948( T − k)1 ⁄ 2] . (19.31)

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

300

250

200

150

100

50

0

-50

The test clearly indicates instability in the equation during the sample period.

CUSUM of Squares Test

The CUSUM of squares test (Brown, Durbin, and Evans, 1975) is based on the test statistic:

which goes from zero at t = k to unity at t = T . The significance of the departure of S 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 St against t 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.

The cumulative sum of squares is generally within the 5% significance lines, suggesting that the residual variance is somewhat stable.

One-Step Forecast Test

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 t 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 One-Step Forecast Test option produces a plot of the recursive residuals and standard errors and the sample points whose probability value is at or below 15 percent. The plot can help you spot the periods when your equation is least successful. For example, the one-step ahead forecast test might look like this:

This test uses the recursive calculations to carry out a sequence of Chow Forecast tests. In contrast to the single Chow Forecast test described earlier, this test does not require the specification of a forecast period— it automatically computes all feasible cases, starting with the smallest possible sample size for estimating the forecasting equation and then adding one observation at a time. The plot from this test shows the recursive residuals at

592—Chapter 19. Specification and Diagnostic Tests

the top and significant probabilities (based on the F-statistic) in the lower portion of the diagram.

Recursive Coefficient Estimates

This view enables you to trace the evolution of estimates for any coefficient as more and more of the sample data are used in the estimation. The view will provide a plot of selected coefficients in the equation for all feasible recursive estimations. Also shown are the two standard error bands around the estimated coefficients.

If the coefficient displays significant variation as more data is added to the estimating equation, it is a strong indication of instability. Coefficient plots will sometimes show dramatic jumps as the postulated equation tries to digest a structural break.

To view the recursive coefficient estimates, click the Recursive Coefficients option and list the coefficients you want to plot in the Coefficient Display List field of the dialog box. The recursive estimates of the marginal propensity to consume (coefficient C(2)), from the sample consumption function are provided below:

named series. EViews will name the coeffi-

cients using the next available name of the form, R_C1, R_C2, …, and the corresponding standard errors as R_C1SE, R_C2SE, and so on.

If you check the Save Results as Series box with any of the other options, EViews saves the recursive residuals and the recursive standard errors as named series in the workfile. EViews will name the residual and standard errors as R_RES and R_RESSE, respectively.

Note that you can use the recursive residuals to reconstruct the CUSUM and CUSUM of squares series.