To check that the GUM is a congruent specification (formally defined in Ch. 9), PcGets computes a battery of mis-specification tests, selected by the user. These tests seek to ascertain that the residuals approximate homoscedastic, normal, white noise, so are not autocorrelated, heteroscedastic (on several measures), or non-normal. There are also checks for parameter constancy. Empirical congruence is shown by satisfactory performance on all these checks.
It is important that congruence is tested and confirmed prior to model selection, as only then will conventional inference procedures be reliable. However, the larger the number of tests that are computed, and the looser the significance levels, the higher the probability that one will reject by chance (see Ch. 10 for a detailed discussion). Conversely, the fewer the tests and the tighter their significance levels, the lower their power to detect mis-specification. The user must balance such considerations, or select one of the two pre-programmed strategies.
Less obviously, but of equal importance, the nominal significance levels of the tests must be close to their actual null rejection frequencies for the models being analyzed. For some model types and sample sizes, some tests are `badly behaved', in that they reject far more often under the null than their nominal significance levels: examples noted below include the portmanteau statistic, and the general heteroscedasticity test (see §13.7.7).
It must be stressed that the tests are only applied once for mis-specification in the GUM. Thereafter, they act as diagnostics for invalid model simplifications; thus, their repeated application does not alter their status. However, re-specifying the GUM after a mis-specification test rejects does alter its characteristics, as it now becomes a design criterion -- although, of course, it has by then served its purpose of revealing model mis-specification.
The remaining sections explain the evaluation tests computed in PcGets, concluding with a brief examination of their behaviour in finite samples.
Graphic analysis focuses on graphical inspection of individual equations. Let yt, y^t denote respectively the actual (that is, observed) values and the fitted values of the selected equation, with residuals e^t=yt-y^t, t=1,...,T. When H observations are retained for forecasting, then y^T+1,...,y^T+H are the 1-step forecasts.
Eight different types of graph are available:
(yt,y^t) over t. This is a graph showing the fitted (y^t) and actual values (yt) of the dependent variable over time, including the forecast period;
y^t against yt, also including the forecast period;
( e^t/s^e) over t, where s^e2=(T-k)-1RSS is the full-sample equation error variance.
The static or dynamic forecasts can be plotted in a graph over time: y^T+h is shown with error bars, bands or fans of ±2SE( eT+h) centered on y^T+h (that is, an approximate 95% confidence interval for the forecast); yT+h is shown in-sample.
This plots the correlogram using e^t as the xt variable in (eq:13.15).
This plots the estimated spectral density (see the GiveWin book) using e^t as the xt variable.
By default the histogram of the standardized residuals e^t/Ö(T-1RSS), t=1,...,T, and the estimated density fe^(.)^ are graphed using the settings described in the GiveWin book.
The estimated QQ plot of the residuals is shown with a standard normal for comparison.
The residuals can be saved to the database for further inspection.
Recursive methods estimate the model at each t for t=M-1,...,T. For forward recursion, the process is initialized by estimation over 1,...,M-1, which is followed by recursive updating over M,...,T: essentially the order is reversed for backward recursion. The output generated by the recursive procedures is most easily studied graphically, viewing multiple graphs together on screen. Recursive estimation throws the spotlight on parameter constancy.
Let b^t denote the k parameters estimated from a sample of size t, and yj-xj'b^t the residuals at time j evaluated at the parameter estimates based on the sample 1,...,t.
We now consider the generated output:
The graph shows b^it±2SE[b^i,t] for each selected coefficient i ( i=1,...,k) over t=M,...,T.
b^it/SE[b^i,t] for each selected coefficient i ( i=1,...,k) over t=M,...,T.
The residual sum of squares at each t is RSSt=åj=1t(yj-xj'b^t)2 for t=M,...,T.
The standardized innovations (or standardized recursive residuals) for RLS are:
nt=(yt-xt'b^t-1)/(wt)1/2 where wt=1+xt'( Xt-1'Xt-1) -1xt for t=M,...,T.
As pointed out in §12.62, s2wt is the 1-step forecast error variance of (12.4), and b^M-1 are the coefficient estimates from the initializing OLS estimation.
The 1-step residuals yt-xt'b^t are shown bordered by 0±2s^e,t over M,...,T. Points outside the 2 standard-error region are either outliers or are associated with coefficient changes.
1-step forecast tests are F( 1,t-k-1) under the null of constant parameters, for t=M,...,T. A typical statistic is calculated as:
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Normality of yt is needed for this statistic to be distributed as an F.
Break-point F-tests are F( T-t+1,t-k-1) for t=M,...,T. These are, therefore, sequences of Chow tests and are called N¯ because the number of forecasts goes from N=T-M+1 to 1. When the forecast period exceeds the estimation period, this test is not necessarily optimal relative to the covariance test based on fitting the model separately to the split samples. A typical statistic is calculated as:
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This test is closely related to the CUSUMSQ statistic in Brown, Durbin and Evans (1975).
Forecast F-tests are F( t-M+1,M-k-1) for t=M,...,T, and are called N as the forecast horizon increases from M to T. This tests the model over 1 to M-1 against an alternative which allows any form of change over M to T. A typical statistic is calculated as:
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The statistics in (eq:13.1)--(eq:13.3) are variants of Chow (1960) tests: they are scaled by 1-off critical values from the F-distribution at any selected probability level as an adjustment for changing degrees of freedom, so that the significant critical values become a straight line at unity. Note that the first and last values of (eq:13.1) respectively equal the first value of (eq:13.3) and the last value of (eq:13.2).
Test statistics from (eq:13.1)--(eq:13.3) are not calculated if instrumental variables are present.
The general class of models estimable in PcGets has the form:
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where b0( L) and the bi( L) are polynomials in the lag operator L, and q is the number of regressors: k remains the number of estimated coefficients. For simplicity, we take all polynomials to be of equal length m:
| bi( L) =åj=0mbijLj, i=0,...,q, |
with b00=1. Letting a(L)=-åj=1majLj-1, we can write (eq:13.4) as:
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Finally, we use a=(a1,...,am)' and bi=(bi0,...,bim), i=1,...,q.
After estimation or selection, PcGets can analyze the polynomials involved, and it computes such functions as their roots and sums.
When working with dynamic models, concepts such as equilibrium solutions, steady-state growth paths, mean lags of responses etc., are generally of interest. In the simple model:
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where all the variables are stationary, a static equilibrium is defined by:
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in which case, E[ yt] =y* will also be constant if |a1|<1, and yt will converge to:
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For non-stationary, but cointegrated data, the first expression (eq:13.8) can be re-interpreted as E[ yt-kx1,t*] =0.
PcGets computes estimates of k, called static long-run parameters, and associated standard errors. If b0( 1) ¹0, the general long-run solution of (eq:13.4) is given by:
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The expression yt-Skixi,t is called the equilibrium-correction mechanism (EqCM) and can be stored in the data set.
The standard errors of k^=(k^1...k^q)' are calculated from:
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PcGets calculates J analytically (see Bårdsen, 1989).
Consider the simple autoregressive model:
| yt=a1yt-1+a2yt-2+et, |
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The eigenvalues of the dynamics li (inverses of the roots) are such that:
| ( 1-a1L-a2L2) =( 1-l1L) ( 1-l2L) , |
so a1=l1+l2 and a2=-l1l2. More generally:
| åi=0naiLi=Õi=1n( 1-liL) . |
PcGets reports the li.
A collinearity analysis is provided for both the general and specific models. The second-moment matrix MXX=T-1X'X , and the matrix of correlations:
| ri,j= |
| , |
are reported, together with the eigenvalues of the former, which should all be real and positive. A comparison of the eigenvalues before and after selection can be useful, as can large ratios of the biggest to the smallest eigenvalues. Eigenvalues of R close to zero indicate the presence of collinearity.
If dynamic forecasts are requested, a tabular presentation provides the forecasts and their standard errors; if the forecasting is in-sample, the actual outcome and the forecast error with the t-value thereof are also reported. As noted above, forecasts and their standard-error based bounds can be graphed.
The test of parameter constancy has the form of a Chow (1960) test:
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where H0 is parameter constancy. In (eq:13.12), RSS T is the full-sample residual sum of squares, RSST1 is for the relevant sub-sample, and there are k regressors. For fixed regressors, the Chow (1960) test is exactly distributed as an F, but is only approximately (or asymptotically) so in dynamic models.
The (Chow) test of parameter constancy out-of-sample has the form:
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where H0 is parameter constancy.
Irrespective of the estimator selected, a wide range of model evaluation criteria are automatically computed. Tests are available for residual autocorrelation, conditional heteroscedasticity, non-normality, unconditional heteroscedasticity, functional-form mis-specification and omitted variables. The diagnostic tests of this section concern the past (checking that the errors are a homoscedastic, normal, innovation process relative to the information available), whereas the forecast statistics discussed above concern the future (and encompassing tests concern information specific to rival models).
Many test statistics in PcGets have either a c2 distribution or an F distribution. Diagnostic tests are usually reported as:
value prob
normality test 7.5674 0.0227
AR 1-4 test 1.7694 0.1428
where the two tests shown are a c2(2) and an F(4,T-4) (with four degrees of freedom in the numerator, and T-4 in the denominator). The observed values are 7.5674 and 1.7694, and the probabilities of getting values as large (or larger) under their null distributions are 0.0227 and 0.1428. Test outcomes that exceed the user-selected significance levels are shown by two stars.
Probability values for F-tests are calculated using an algorithm based on Majunder and Bhattacharjee (1973a) and
Cran, Martin and Thomas (1977).
Those for the c2 are based on Shea (1988). The significance points of the F-distribution derive from Majunder and Bhattacharjee (1973b).
Many of the diagnostic tests are calculated using an auxiliary regression (see Pagan, 1984), so take the form of TR2 for the auxiliary regression: they are asymptotically distributed as c2( s) under their nulls, and have the usual additive property for independent c2s. Following Kiviet (1986) and Harvey (1990), F-approximations of the form:
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are calculated, as these seem to be better behaved in small samples. The precise statistics calculated are now described.
The correlogram of a variable xt is the series {rj} where rj is the correlation coefficient between xt and xt-j for j=1,...,s:
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Here, xi=åt=j+1Txt-i/(T-j) is the sample mean of xt-i (t=j+1,...,T), where i=0,1,...,s, so that rj corresponds to a correlation coefficient (note the difference from the definition in many time-series textbooks, where the denominator is defined in terms of å1T(xt-x)2: however, this difference tends to be small, and vanishes asymptotically).
The residual correlogram is defined as in (eq:13.15), but using the residuals from the fitted model, rather than the original data, i.e., correlations {rj} are computed between the residuals ût and ût-j. The portmanteau statistic is defined by:
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as in Box and Pierce (1970), but with the degrees-of-freedom correction suggested by Ljung and Box (1978). It was designed as a goodness-of-fit test for stationary, autoregressive moving-average models. Under the assumptions of the test, LB(s) is asymptotically distributed as c2(s-n) after fitting an AR(n) model. A value LB(s)³2s for large s suggests that the residuals are not white noise. However, Monte Carlo simulations of its behaviour in fitted conditional econometric equations suggest it should not form part of the test battery in model selection: see e.g., Teräsvirta (1976) and §13.7.7. An appropriate test for autocorrelated residuals is provided in §13.7.2 following.
This is the Lagrange-multiplier (LM) test for rth-order residual autocorrelation, distributed as c2( r) in large samples, under the null hypothesis that there is no autocorrelation (that is, that the errors are white noise). In standard usage, r~= 1/2 s for s in §13.7.1 above, so this provides a type of portmanteau test (see Godfrey, 1978). However, any orders from 1 up to 12 can be selected to test against:
| ut=åi=praiut-i+et where 0£p£r. |
The F-form suggested in (e.g.) Kiviet (1986) and Harvey (1990) is the recommended diagnostic test. The LM test is calculated by regressing the residuals on all the regressors of the original model and the lagged residuals for lags p to r (missing residuals are set to zero). The LM test c2(r-p+1) is TR2 from this regression (or the F-equivalent), and the error autocorrelation coefficients are estimated by the coefficients of the lagged residuals. For an excellent exposition, see Pagan (1984).
This is the ARCH (autoregressive conditional heteroscedasticity test: see Engle, 1982a) which in its present form tests the hypothesisg=0 in the model:
| E[ ut2½ut-1,...,ut-r] =c0+åi=1rgiut-i2 |
where g=( g1,...,gr) ' . Then TR2 from the regression of ût2 on a constant and ût-12 to u^t-r2 (called the ARCH test) is asymptotically distributed as c2( r) on H0: g=0: as usual, the F-form is reported. Both first-order and higher-order lag forms are easily calculated (see Engle, Hendry and Trumbull, 1985, for small-sample evidence on the former).
This is a test for whether the skewness and kurtosis of the residuals correspond to that of a normal distribution: see Doornik and Hansen (1994). For a variable xt, let m and sx2 denote its mean and variance, and write mi=E[ xt-m] i, so that sx2=m2 . The skewness and kurtosis are defined respectively as:
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A normal variate has Öb1=0 and b2=3.
Sample estimates of these four parameters are given by
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Bowman and Shenton (1975) consider a c2( 2) test based on Öb1 and b2 but deem it unsuitable unless the sample size is very large: the statistics Öb1 and b2 are not independently distributed, and the sample kurtosis approaches normality very slowly (see Shenton and Bowman, 1977, and D'Agostino, 1970). Instead, let z1 and z2 denote transformations of skewness and kurtosis, designed to make these statistics closer to the standard normal. The resulting test is (see Doornik and Hansen, 1994):
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Applied to residuals, this test gives appropriate rejection frequencies under the null of normality, using c2(2) critical values: see §13.7.7.
This test (called the hetero test in PcGets' output) is based on White (1980), and involves an auxiliary regression of {e^t2} on the original regressors {xi,t} and all their squares {xi,t2}. The null is unconditional homoscedasticity, and the alternative is that the variance of the {et} process depends on xt and on the xi,t2. Variables that are redundant when squared are automatically removed. The output comprises the F-test based on TR2.
White (1980) also suggested this test (called the hetero-x test in PcGets' output) which is only calculated where there is a sufficiently-large sample relative to the number of regressors. It is based on an auxiliary regression of the squared residuals {ût2} on all squares and cross-products of the original regressors (that is, on r=½k( k+1) variables). Then, if T>>k( k+1) , the test is calculated; redundant variables are automatically removed. The usual F-approximation to the c2 based on TR2 is reported. This is a general test for heteroscedastic errors: H0 is that the errors are homoscedastic or, if heteroscedasticity is present, it is unrelated to the xs.
Unfortunately, Monte Carlo simulations of its behaviour in fitted conditional econometric equations suggest it should not form part of the test battery in model selection, even in relatively large samples. Godfrey and Orme (1994) also show that this test does not have power against omitted variables.
Figure Figure:13.1 displays quantile--quantile (QQ) plots of the empirical distributions of seven potential mis-specification tests when estimating the correct specification, the general model, and the finally-selected model in a Monte Carlo study for 1000 replications with T=100 (from Krolzig and Hendry, 2001).
Some strong deviations are evident from the theoretical distributions (shown by the diagonals): the portmanteau statistic (see Box and Pierce, 1970) rejects serial independence of the errors too often in the correct specification, never in the general, and too rarely in the final model. The hetero-x test (see White, 1980) often has degrees-of-freedom problems for the GUM, but anyway performs badly for both the true and final models. Such incorrect finite-sample sizes of these two diagnostic tests would induce excess rejections of congruent GUMs, resulting in an increased overall size: thus, the portmanteau and the hetero-x diagnostics should be excluded from any battery of test statistics. Overall, therefore, we recommend the five mis-specification tests shown in table 13.1.
| Test | Alternative | Statistic | Sources |
| Chow(tT) | Predictive failure over a subset | Chow (1960, p.594-595), | |
| of (1-t) T obs. | F((1-t) T,tT-k) | Hendry (1979) | |
| portmanteau(r) | r-th order residual autocorrelation | c2(r) | Box and Pierce (1970) |
| normality test | Skewness and excess kurtosis | c2(2) | Jarque and Bera (1980), |
| Doornik and Hansen (1994) | |||
| AR 1-p test | p-th order residual autocorrelation | F(p,T-k-p) | Godfrey (1978), |
| Harvey (1981, p.173) | |||
| ARCH 1-p test | p-th order autoregressive | Engle (1982b), | |
| conditional heteroskedasticity | F(p,T-k-p) | Engle, Hendry and Trumbull (1985) | |
| hetero test | Heteroscedasticity quadratic | White (1980), | |
| in regressors xi2 | F(q,T-k-q-1) | Nicholls and Pagan (1983) | |
| There are T observations and k regressors in the model under the null. | |||
| The value of q may differ across statistics, as may those of k and T across models. | |||
| By default, PcGets sets p=4, r=12, and compute two Chow tests at t1=[0.5T]/T and t2=[0.9T]/T. | |||
Figure:13.2 compares the empirical distributions of these test statistics in large and small samples (T=1000 and T=100) for the DGP and GUM. The first two rows of graphs demonstrate that the test distributions are unaffected by the presence of (strongly-exogenous) nuisance regressors (i.e., variables that enter the GUM but not the DGP). For small samples (the second two rows), the properties of the mis-specification tests are still satisfactory -- except for the hetero test -- and are close to the distributions of those tests under the null of the DGP.
For any given model, PcNaive could be used to simulate the rejection frequencies under the null of the tests to be applied, simulating a GUM similar to that under analysis.
Writing the model in matrix form as y=Xb+e, the null hypothesis of p linear restrictions can be expressed as H0: Rb=r, with R a (p×k) matrix and r a p×1 vector. This test is well explained in most econometrics textbooks, and uses the unrestricted estimates (that is, it is a Wald test).
This subset form of the linear restrictions tests is for: H0: bi=...=bj=0: any choice of coefficients can be made, so a wide range of specification hypothesis can be tested.
This tests if any variables of the GUM which has been dropped by GETS or GETSIVE should be added to the selected model. The model itself remains unchanged. If the model is written in matrix form as
| y=Xb+Wg+e, |
then H0: g=0 is being tested. The test exploits the fact that on H0:
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Since ( X'X) -1 is already calculated, the F-statistic is easily computed by partitioned inversion. Computations for IVE are only slightly more involved.
Encompassing tests are used to select between final-selected models from path searches (see Cox, 1961, Pesaran, 1974, Ericsson, 1983, Mizon, 1984, Mizon and Richard, 1986, and Hendry and Richard, 1989). The F-test is for each model parsimoniously encompassing their union. This is the only encompassing test which is invariant to the choice of common regressors in models. Thus, the F -test yields the same numerical outcome for the first model parsimoniously encompassing either the union of the two models under consideration, or the orthogonal complement to the first model relative to the union.
Let the first model (M1) under consideration have k1+k2 regressors ( x1,t,x2,t) and the second (M2) have k2+k3 ( x2,t,x3,t) , so the x2,t are in common, and their union (MU) comprises the k=k1+k2+k3 non-redundant set:
| x1,t,x2,t,x3,t. |
Let RSS 1, RSS2 and RSSU denote the residual sums of squares from M1, M2 and MU respectively; and RSSG from the GUM. Then, the encompassing test of M1E M2 is equivalent to testing the parsimonious encompassing M1EpMU. For a sample of size T, PcGets uses the F-test:
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A similar construction is used for the other encompassing tests.
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