Since its automatic model-selection capabilities are what distinguish PcGets, this is the key tutorial. Exit PcGets and re-enter, and clear the existing Results window to start the chapter with a clean slate, so model numbering will coincide with the output presented below.
First, we formulate a general unrestricted model (GUM) in §5.1, then discuss model settings for selection in §5.2 before showing the sequential simplification capabilities of PcGets, both for levels in §5.3, then for I(0) transformations in §5.5. The two pre-programmed strategies are illustrated in §5.6, namely the Liberal strategy and the Conservative strategy, followed by a brief run through multi-path selection, before describing how to ensure that theoretically-important variables enter the selected model in §5.7. A detailed description of the many settings under `expert options' is provided in §5.8. The chapter concludes with some advice on model formulation and selection settings in §5.10. Throughout, we use the M1UK data set to illustrate the procedures.
Return to the Model/Formulate dialog to create a new, general model. In substantive research, the starting point should be based on previous empirical research evidence (to test in due course that earlier findings are parsimoniously encompassed), economic (or other relevant subject-matter) theory, the data frequency -- and common sense. This is one aspect that remains dependent on the user's expertise: PcGets cannot perform well if the starting point is unsatisfactory.
Here, we base the initial model on Hendry and Ericsson (1991b) (denoted HE below), so begin by formulating an equation with mp, y, R, Dp and Constant as its basic variables, with two lags each. Notice that the data are seasonally adjusted, so two lags should suffice even though five lags is often a better initial lag length for quarterly data (as in Davidson, Hendry, Srba and Yeo, 1978). Even so, there are 12 regressors in our illustration, shown on the next page.
Accept, and we come to the Model Settings dialog, noted above, but not yet discussed.
To distinguish conventional estimation from selection subject to valid reductions, we call the PcGets procedure testimation (denoted GETS in batch). The settings dialog offers the following options:
The theory behind these selection procedures is discussed in Chapter 10, but our experience is that they act together to filter out many irrelevant variables yet retain almost all relevant variables.
Three are available, namely:
Again, there are three levels of detail:
On this first run, set reporting to level i, but otherwise accept the default settings as shown in the Model Settings dialog.
This brings up the Estimate Model dialog:
Do not retain any forecasts, and select full sample Testimation (GETS). The output appears as usual in the GiveWin Results window.
GUM( 1) Modelling mp by GETS (using M1UK.in7), 1963 (4) - 1989 (2)
Coeff StdError t-value t-prob
Constant -1.09093 0.12920 -8.444 0.0000
mp_1 0.63594 0.10064 6.319 0.0000
mp_2 0.26933 0.09326 2.888 0.0048
y 0.00156 0.10301 0.015 0.9879
y_1 0.21036 0.12774 1.647 0.1031
y_2 -0.11041 0.10527 -1.049 0.2970
R -0.46797 0.11105 -4.214 0.0001
R_1 -0.25988 0.16828 -1.544 0.1260
R_2 0.03278 0.12034 0.272 0.7859
Dp -0.88250 0.18659 -4.730 0.0000
Dp_1 0.00640 0.22637 0.028 0.9775
Dp_2 0.21312 0.18576 1.147 0.2543
RSS 0.01573 sigma 0.01315 R^2 0.99588 Radj^2 0.99538
LogLik 452.51069 AIC -8.55361 HQ -8.42928 SC -8.24665
T 103 p 12 FpNull 0.00000 FpConst 0.00000
Significance levels set: 0 Test(s) to be excluded.
value prob alpha
Chow(1976:3) 0.5144 0.9873 0.0100
Chow(1986:4) 0.5211 0.8706 0.0100
normality test 2.7932 0.2474 0.0100
AR 1-4 test 2.0189 0.0988 0.0100
ARCH 1-4 test 0.5600 0.6923 0.0100
hetero test 1.1837 0.2914 0.0100
Specific model of mp, 1963 (4) - 1989 (2)
Coeff StdError t-value t-prob Split1 Split2 reliable
Constant -1.02838 0.10226 -10.057 0.0000 0.0000 0.0000 1.0000
mp_1 0.70878 0.07420 9.552 0.0000 0.0000 0.0000 1.0000
mp_2 0.20459 0.07123 2.872 0.0050 0.0012 0.0026 1.0000
y_1 0.09576 0.00916 10.459 0.0000 0.0000 0.0000 1.0000
R -0.62871 0.06881 -9.136 0.0000 0.0000 0.0000 1.0000
Dp -0.72536 0.12914 -5.617 0.0000 0.0000 0.0000 1.0000
RSS 0.01677 sigma 0.01315 R^2 0.99561 Radj^2 0.99538
LogLik 449.23498 AIC -8.60650 HQ -8.54434 SC -8.45303
T 103 p 6 FpNull 0.00000 FpGUM 0.43292
value prob
Chow(1976:3) 0.5968 0.9632
Chow(1986:4) 0.5055 0.8819
normality test 5.6595 0.0590
AR 1-4 test 2.0043 0.1003
ARCH 1-4 test 0.6739 0.6118
hetero test 0.9417 0.4999
The output starts with the estimated GUM and the congruence tests thereon: no mis-specification test rejects at the selected level (denoted alpha), so simplification is worthwhile, especially as seven coefficient estimates have small t-values. The intermediate stages are omitted (deliberately here) and the final model is:
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This is also congruent, and provides a parsimonious representation of the information in the GUM (shown by the insignificance of the FpGUM test). All the coefficients are 100%reliable (so are significant overall, and in both overlapping sub-samples), with s^=1.3%, a reduction on the estimated models reported in earlier tutorials. Select Dynamic analysis to compute the long-run solution:
Lag structure
Lag 0 Lag 1 Lag 2 Sum LongRun
mp 0 0.7088 0.2046 0.9134 1.0000
SE 0 0.0742 0.0712 0.0112
y 0 0.0958 0 0.0958 1.1055
SE 0 0.0092 0 0.0092 0.1291
R -0.6287 0 0 -0.6287 -7.2576
SE 0.0688 0 0 0.0688 0.7096
Dp -0.7254 0 0 -0.7254 -8.3734
SE 0.1291 0 0 0.1291 1.9205
Constant -1.0284 0 0 -1.0284 -11.8712
SE 0.1023 0 0 0.1023 1.4177
Roots of the autoregressive lag polynomial
real
0.9290
-0.2202
Note the coefficient values (for example, y is close to unity, R to -7 and Dp to -8) and their small standard errors (but y is not significantly different from unity). The linear restrictions test confirms that long-run homogeneity of mp with respect to y is not rejected at the 5%level.
Test for linear restrictions (Rb=r):
R matrix
Constant mp_1 mp_2 y_1 R Dp
0 1 1 1 0 0
r vector
1
LinRes Chi^2(1) = 0.774 [0.3790]
Return to the Formulate Model dialog, and mark Dp as endogenous, adding Dm_2 and marking it as an instrument as shown:
Do not retain any forecasts, and select full sample IVE-testimation (denoted GETSIVE in batch commands). The output appears as usual in the GiveWin Results window.
Specific model of mp, 1964 (1) - 1989 (2)
Coeff StdError t-value t-prob Split1 Split2 reliable
Constant -1.03034 0.10449 -9.861 0.0000 0.0000 0.0511 0.6000
mp_1 0.72552 0.07577 9.576 0.0000 0.0000 0.0000 1.0000
mp_2 0.18703 0.07301 2.562 0.0120 0.0001 0.7632 0.2000
y_1 0.09586 0.00934 10.260 0.0000 0.0000 0.0419 1.0000
R -0.64952 0.07423 -8.750 0.0000 0.0000 0.0000 1.0000
Dp -0.61002 0.17327 -3.521 0.0007 0.0003 0.8463 0.2000
RSS 0.01672 sigma 0.01320 R^2 0.99561 Radj^2 0.99539
LogLik 444.50931 AIC -8.59822 HQ -8.53570 SC -8.44381
T 102 p 6 FpNull 0.00000 FpGUM 0.53346
value prob
normality test 4.9792 0.0829
AR 1-4 test 2.5743 0.0428
ARCH 1-2 test 0.3602 0.6985
hetero test 1.1053 0.3678
Thus, the same model is selected in this instance.
Such information suggests a further simplification, following up the cointegration (I(0)) transformation discussed in §4.4.1: reparameterize the GUM to the more orthogonal representation of differences and the equilibrium correction. Since Dp enters the last of these, it should enter the GUM as D2pt (the change in inflation), so use Algebra to compute DDp (= diff(Dp,1);) and:
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which is adjusted for its sample mean.
The new GUM is:
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It is important to remove any I(1) variables at this stage, as PcGets uses conventional critical values at the chosen significance levels, not those based on Brownian motion calculations (see, e.g., Banerjee, Dolado, Galbraith and Hendry, 1993). As discussed in §10.2.1.1, either formulate the model in levels, or carefully specify its I(0) transforms.
Thus, we recommence from the GUM comprising: Dmp, DDp, Dy, DR, their first lags, and e1. This time, however, choose the option of condensed output (see §5.5.5 for an explanation): we will walk through these results in steps, commencing from the GUM.
GUM( 2) Modelling Dmp by GETS (using M1UK.in7), 1963 (4) - 1989 (2)
Coeff StdError t-value t-prob
Constant 0.00709 0.00185 3.830 0.0002
Dmp_1 -0.25230 0.09203 -2.742 0.0073
Dy -0.01944 0.10077 -0.193 0.8475
Dy_1 0.10255 0.10431 0.983 0.3281
DR -0.46371 0.10714 -4.328 0.0000
DR_1 -0.04406 0.11089 -0.397 0.6920
DDp -0.92965 0.17565 -5.293 0.0000
DDp_1 -0.17093 0.17753 -0.963 0.3381
e_1 -0.09157 0.00916 -9.995 0.0000
RSS 0.01614 sigma 0.01310 R^2 0.76608 Radj^2 0.74617
LogLik 451.19518 AIC -8.58631 HQ -8.49307 SC -8.35610
T 103 p 9 FpNull 0.00000 FpConst 0.00000
Significance levels set: 0 Test(s) to be excluded.
value prob alpha
Chow(1976:3) 0.5349 0.9836 0.0100
Chow(1986:4) 0.5488 0.8504 0.0100
normality test 3.9477 0.1389 0.0100
AR 1-4 test 1.4257 0.2319 0.0100
ARCH 1-4 test 0.4265 0.7891 0.0100
hetero test 0.9898 0.4764 0.0100
As before, the GUM is congruent, but non-parsimonious (in fact, DDp_1 `extends' the GUM, as it implicitly introduces a third lag in the level p).
The first attempted reductions are pre-searches:
Stage-0 (Step 1): F pre-search testing (lag-order pre-selection)
Check lag 1: F-prob =0.0000, Tests failed = 2; Invalid reduction.
Stage-0 (Step 2): F pre-search testing (top-down)
Remove 1 variable with t-prob > 0.8475: F-prob =0.8475,
Tests failed = 1; Invalid reduction.
Stage-0 (Step 3): F pre-search testing (top-down)
Remove 1 variable with t-prob > 0.8475: F-prob =0.8475,
Tests failed = 0;
Remove 2 variables with t-prob > 0.6920: F-prob =0.8982,
Tests failed = 0;
Remove 3 variables with t-prob > 0.3381: F-prob =0.6906,
Tests failed = 1; Invalid reduction.
Stage-0 (Step 4): F pre-search testing (bottom-up)
Found 5 variables with t-prob < 0.0500.
Include 5 variables with t-prob < 0.3281: F-prob =0.6850,
Tests failed = 0; Valid reduction found.
Stage-0 (Step 5): Test of the additional restrictions imposed by the
bottom-up reduction.
F test 1.0521 [0.3532] Not rejected.
At every stage, the diagnostic tests are re-computed, and a successful outcome is shown by Tests failed = 0: should Tests failed exceed zero, that step is not followed as it has clearly generated an invalid reduction.
The outcomes of the pre-search reductions here are, therefore:
achieves no simplifications;
: the first achieves no simplifications, but the second eliminates two variables;
finds that 5 variables can be eliminated with an F-test value of 1.05 [0.3532].
All of these pre-search reductions are conducted at loose significance levels, the settings for which can be ascertained by using Test/PcGets settings:
PcGets Settings
Testimation algorithm
F pre-search testing (top - down) TRUE
F pre-search testing (bottom - up) TRUE
F presearch testing (lag order preselection) TRUE
Sample split analysis TRUE
Sample-size adjusted significance levels TRUE
Outlier correction FALSE
Information criterion HQ
Significance levels
t - tests 0.0500
F - tests 0.0750
F - test of GUM 0.7500
Encompassing test 0.0750
Diagnostics (high) 0.0100
Diagnostics (low) 0.0050
F pre-search tests
F - tests (lag pre-selection) 0.7500
F - tests (step 1) 0.7500
F - tests (step 2) 0.5000
F - tests (bottom-up) 0.0750
Marginal t-prob (step 1) 0.0750
Marginal t-prob (step 2) 0.0500
Marginal t-prob (bottom-up) 0.0500
Two-step pre-search testing TRUE
Sample split analysis
Significance level 0.0750
Size of the subsample (fraction) 0.7500
Penalty for failed t-test in full sample 0.2000
Penalty for failed t-test in subsample 1 0.4000
Penalty for failed t-test in subsample 2 0.4000
Block search
Check groups with t-pvals > 0.90 TRUE
Check groups with t-pvals > 0.70 TRUE
Check groups with t-pvals > 0.50 TRUE
Check groups with t-pvals > 0.25 TRUE
Check groups with t-pvals > 0.10 TRUE
Check groups with t-pvals > 0.05 TRUE
Check groups with t-pvals > 0.01 TRUE
Check groups with t-pvals > 0.001 TRUE
Diagnostic tests
Chow test 1 TRUE
Chow test 2 TRUE
Portmanteau FALSE
Normality TRUE
AR test TRUE
ARCH test TRUE
Hetero test TRUE
Test options
Chow test breakpoint 1 0.50
Chow rest breakpoint 2 0.90
Portmanteau max lag 12
AR test min lag 1
AR test max lag 4
ARCH test min lag 1
ARCH test max lag 4
Lag-order pre-selection is at 25%, the two stages of top-down at 95%and 50%respectively, and the bottom-up at 5%. These choices are set because the GUM is not heavily over-parameterized, and any variable eliminated at this stage is removed permanently, so one should not risk omitting an effect that might matter but is not very significant in the GUM.
Following these findings, PcGets simplifies the starting point for the multi-path searches by eliminating the irrelevant variables and re-estimating the GUM:
Stage-1: General model of Dmp, 1963 (4) - 1989 (2)
Coeff StdError t-value t-prob
Constant 0.00768 0.00138 5.555 0.0000
Dmp_1 -0.24820 0.08904 -2.788 0.0064
DR -0.48426 0.10029 -4.828 0.0000
DDp -0.85920 0.16365 -5.250 0.0000
e_1 -0.09420 0.00835 -11.282 0.0000
RSS 0.01653 sigma 0.01299 R^2 0.76041 Radj^2 0.75063
LogLik 449.96096 AIC -8.64002 HQ -8.58822 SC -8.51212
T 103 p 5 FpNull 0.00000 FpGUM 0.68504
Stage-1: Multiple-path encompassing search
All variables are significant: General -> Specific.
In fact, as PcGets reports, all variables are now significant, so the general and specific models coincide, and the simplification is terminated.
Specific model of Dmp, 1963 (4) - 1989 (2)
Coeff StdError t-value t-prob Split1 Split2 reliable
Constant 0.00768 0.00138 5.555 0.0000 0.0000 0.0000 1.0000
Dmp_1 -0.24820 0.08904 -2.788 0.0064 0.0062 0.0009 1.0000
DR -0.48426 0.10029 -4.828 0.0000 0.0000 0.0000 1.0000
DDp -0.85920 0.16365 -5.250 0.0000 0.0000 0.0000 1.0000
e_1 -0.09420 0.00835 -11.282 0.0000 0.0000 0.0000 1.0000
RSS 0.01653 sigma 0.01299 R^2 0.76041 Radj^2 0.75063
LogLik 449.96096 AIC -8.64002 HQ -8.58822 SC -8.51212
T 103 p 5 FpNull 0.00000 FpGUM 0.68504
value prob
Chow(1976:3) 0.5219 0.9880
Chow(1986:4) 0.5753 0.8300
normality test 9.1419 0.0103
AR 1-4 test 1.3173 0.2692
ARCH 1-4 test 0.4795 0.7507
hetero test 0.5672 0.8021
The search has delivered a congruent and interpretable simplification: we will discuss each coefficient in turn.
The intercept of 0.00768 entails an `autonomous' growth rate for real money of about 3%per annum, corresponding to the growth of y -- but somewhat larger -- since from (eq:5.2), Dm-Dp=Dy assuming DR=D2p=0. The lagged-dependent variable suggests some `overshooting', consistent with some theories of money demand (such as Smith, 1986). The impact effects from changes in interest rates and inflation are much smaller than their long-run effects, suggesting relatively slow adjustment, but sensibly signed. Finally, the feedback coefficient is very significant, and suggests that about 10%of any disequilibrium is removed each quarter (see Akerlof, 1979).
To check that the reduction has lost no significant information, select Test/Omitted variables to see:
LM test for omitted variables
Single t-prob Joint t-prob
DDp_1 -1.055 0.2941 -0.963 0.3381
Dy_1 0.909 0.3659 0.983 0.3281
DR_1 -0.508 0.6129 -0.397 0.6920
Dy -0.405 0.6864 -0.193 0.8475
The first column lists the variable; the next two report the actual t value and its probability when testing that variable in isolation; and the last two columns show the t value and its probability for that variable when testing all variables for inclusion. Thus, as anticipated, no omitted variable matters.
The one diagnostic that is close to significance above is the normality test, which reflects the outliers noted earlier. However, PcGets can automatically detect and remove outliers, as we now show.
Mark Outlier correction in Model Settings to obtain:
Specific model of Dmp, 1963 (4) - 1989 (2)
Coeff StdError t-value t-prob Split1 Split2 reliable
Constant 0.00792 0.00121 6.561 0.0000 0.0000 0.0000 1.0000
Dmp_1 -0.28444 0.07903 -3.599 0.0005 0.0002 0.0005 1.0000
DR -0.44715 0.08770 -5.099 0.0000 0.0000 0.0000 1.0000
DDp -0.96733 0.14392 -6.721 0.0000 0.0000 0.0000 1.0000
e_1 -0.09873 0.00737 -13.395 0.0000 0.0000 0.0000 1.0000
I1969:2 -0.03630 0.01162 -3.123 0.0024 0.0004 0.0000 1.0000
I1971:1 0.03894 0.01138 3.422 0.0009 0.0001 0.0002 1.0000
I1974:4 0.04141 0.01156 3.582 0.0005 0.0001 0.0001 1.0000
RSS 0.01213 sigma 0.01130 R^2 0.82417 Radj^2 0.81122
LogLik 465.89645 AIC -8.89119 HQ -8.80831 SC -8.68655
T 103 p 8 FpNull 0.00000 FpGUM 0.95508
value prob
Chow(1976:3) 0.8017 0.7775
Chow(1986:4) 0.8082 0.6213
normality test 3.3252 0.1896
AR 1-4 test 1.7544 0.1450
ARCH 1-4 test 0.5337 0.7113
hetero test 0.4476 0.9293
Three are detected, namely 1969(2), 1971(1), and 1974(4) -- check the economic history to see if they correspond to important events (e.g., Competition and Credit Control regulations were introduced in 1971 -- but later that year). However, the other coefficient estimates are hardly altered, the main changes being an increase in that of Dp with a corresponding decrease in that of R. All the diagnostic tests are acceptable, and every coefficient is 100%reliable (which is pure luck for the indicator variables, as they could well have been zero in one of the sub-samples).
Figure Figure:5.1 shows the graphical analysis of this final model. The graphical diagnostics are all acceptable, the indicators having removed the previous non-normality, and the fitted-actual cross plot is close to a straight line at 45o. Notice how easy such re-selections are -- how long would this have taken by `hand'? Can you `outperform' PcGets by finding a congruent model that parsimoniously encompasses and dominates its selection?
Finally, the resulting model can be re-estimated recursively to check for parameter constancy, as Figure Figure:5.2 illustrates. All of the parameter estimates are constant, the RSS increases almost linearly, s^ is constant, and none of the break-point Chow tests rejects.
When condensed output is selected, the following symbols are used:
symbol reduction path information
. single reduction step
: a variable or group of variables has been removed;
* reduction failed, path returns to the previous
specification;
f reduction failed, path is not continued.
c reduction path converged to a previously found reduction;
t terminal specification found.
We have seen how this operates in some detail above; it aims to keep as many as possible of the GUM variables that matter in the DGP. Naturally, the risk is that it will thereby retain irrelevant variables more often.
This is the `opposite' extreme to the Liberal strategy, in that its user must be concerned to avoid retaining irrelevant variables, at the risk of omitting variables that really matter in the DGP. Try re-selecting using this strategy -- now no outliers are detected at the more stringent significance levels used, but the same model is selected as above.
Another interesting variant is to switch off all the pre-selections and use only the multi-path selection algorithm. For the Conservative strategy, this yields:
Stage-1: Multiple-path encompassing search
Path 1: Check variables with t-prob > 0.0010 .t
Path 2: Check variables with t-prob > 0.0100 .t
Path 3: Check variables with t-prob > 0.0500 .c
Path 4: Check variables with t-prob > 0.1000 .c
Path 5: Check variables with t-prob > 0.2500 .c
Path 6: Check variables with t-prob > 0.5000 ...c
Path 7: Check variables with t-prob > 0.7000 ..c
Path 8: Remove Dy .c
Path 9: Remove DR_1 ..c
Path 10: Remove DDp_1 ....c
Path 11: Remove Dy_1 ...c
Union model
Coeff StdError t-value t-prob
Constant 0.00768 0.00138 5.555 0.0000
Dmp_1 -0.24820 0.08904 -2.788 0.0064
DR -0.48426 0.10029 -4.828 0.0000
DDp -0.85920 0.16365 -5.250 0.0000
e_1 -0.09420 0.00835 -11.282 0.0000
RSS 0.01653 sigma 0.01299 R^2 0.76041 Radj^2 0.75063
LogLik 449.96096 AIC -8.64002 HQ -8.58822 SC -8.51212
T 103 p 5 FpNull 0.00000 FpGUM 0.68504
Encompassing tests
Model 1: F test 7.7704 [0.0064] Removed.
Model 2: F test -.---- [0.0000]
All variables are significant: General -> Specific.
Thus, in this relatively simple case, all roads lead to the same
destination.
How many hours would all this have taken with another program (even PcGive)?
Theoretical models often specify some variables as central, and an investigator may then be interested in finding the best model subject to ensuring that such variables enter it. This too is easily achieved in PcGets. Moreover, it is easy to compare the resulting selection with the best unconstrained model to check that the former is not significantly dominated.
On the Model/Formulate dialog, create a new general model for mp, y, R, Dp and Constant, with two lags each. Highlight the current-dated value of y and mark it F:fixed leaving all other variables at their default settings, accept, and testimate using the `conservative strategy'. The results are as follows.
Specific model of mp, 1963 (4) - 1989 (2)
Coeff StdError t-value t-prob Split1 Split2 reliable
Constant -1.03026 0.10329 -9.974 0.0000 0.0000 0.0000 1.0000
mp_1 0.69947 0.07513 9.311 0.0000 0.0000 0.0000 1.0000
mp_2 0.21357 0.07206 2.964 0.0038 0.0013 0.0019 1.0000
R -0.62529 0.06903 -9.059 0.0000 0.0000 0.0000 1.0000
Dp -0.71530 0.12959 -5.520 0.0000 0.0000 0.0000 1.0000
y 0.09582 0.00924 10.373 0.0000 0.0000 0.0000 1.0000
RSS 0.01692 sigma 0.01321 R^2 0.99557 Radj^2 0.99534
LogLik 448.78253 AIC -8.59772 HQ -8.53555 SC -8.44424
T 103 p 6 FpNull 0.00000 FpGUM 0.34649
value prob
Chow(1976:3) 0.5494 0.9809
Chow(1986:4) 0.5136 0.8763
normality test 7.2391 0.0268
AR 1-4 test 2.0991 0.0871
ARCH 1-4 test 0.7082 0.5884
hetero test 0.7902 0.6382
The corresponding model with no constraints by the same strategy is:
Coeff StdError t-value t-prob Split1 Split2 reliable
Constant -1.02838 0.10226 -10.057 0.0000 0.0000 0.0000 1.0000
mp_1 0.70878 0.07420 9.552 0.0000 0.0000 0.0000 1.0000
mp_2 0.20459 0.07123 2.872 0.0050 0.0012 0.0026 1.0000
y_1 0.09576 0.00916 10.459 0.0000 0.0000 0.0000 1.0000
R -0.62871 0.06881 -9.136 0.0000 0.0000 0.0000 1.0000
Dp -0.72536 0.12914 -5.617 0.0000 0.0000 0.0000 1.0000
RSS 0.01677 sigma 0.01315 R^2 0.99561 Radj^2 0.99538
LogLik 449.23498 AIC -8.60650 HQ -8.54434 SC -8.45303
T 103 p 6 FpNull 0.00000 FpGUM 0.43292
value prob
Chow(1976:3) 0.5968 0.9632
Chow(1986:4) 0.5055 0.8819
normality test 5.6595 0.0590
AR 1-4 test 2.0043 0.1003
ARCH 1-4 test 0.6739 0.6118
hetero test 0.9417 0.4999
The models differ only in using yt rather that yt-1 so an F-test on the RSS values is not possible; but the tiny difference in their s^ values suggests both are acceptable.
This is one of five possible outcomes from fixing variables, namely:
The final set of options we will consider are for Expert settings. In this mode, every significance level can be set:
Access Model Options to see:
We have considered the settings for all the pre-search tests, so will focus on the remaining options.
First, consider significance levels for selection statistics:
Significance levels can be set for t-tests, F-tests (in block reductions), F-test of the GUM, encompassing tests, and diagnostic tests (high and low). The last of these concerns when the mis-specification tests are significant at the pre-specified level in the GUM: in the automatic procedure, the required significance level is lowered, and search paths are terminated only when that lower level is violated.
To change any desired setting, point to the relevant item, press on the underlined text, and type the new number in the highlighted box.
Next, block searches concern testing groups with t-probabilities >a, so reduction paths start by removing the associated group of variables.
Thirdly, we have the selection of the information criterion. When there are several mutually-encompassing congruent reductions, PcGets selects between these using one of four information criteria defined in §11.12.3. Simply click on the desired choice.
The same screen capture shows the settings for the Sample split analysis.
The significance level for the t-tests in sub-samples can be set: this should never be more stringent than the full-sample setting.
Next, the size of the sub-sample (as a fraction of the full sample) can be set.
Also, penalties for t-tests failing to achieve significance in the full sample, sub-sample 1, and sub-sample 2 can be set as fractions adding to unity.
Finally, on this screen, the outlier-correction criterion (as a number of s^GUM units) can be chosen: the smaller the value, the more dummies will be added.
The next set of expert options concerns the choice of mis-specification tests, their characteristics and significance levels.
As each item is checked that test is included in the test battery, for the mid-sample Chow test; forecast-type Chow test; portmanteau statistic; Lagrange-multiplier (LM) test for residual autocorrelation (AR test); LM test for autoregressive conditional heteroscedasticity in the residuals (ARCH test); and LM test for heteroscedasticity (Hetero test).
allows any parameters of these tests to be changed, namely the break points of the two Chow tests (as fractions of the full sample); the number of lags for calculating the portmanteau statistic; and the minimum and maximum lags of the AR and ARCH tests.
Finally, the overall selection strategy can be reset to Liberal or Conservative.
For the present GUM, the expert strategy picks the same model as both others, and declines to add outlier dummies.
However, that need not be the case, as §6.4 illustrates.
Simulation and empirical studies differ in the key feature that the model nests the DGP in the former, but the relation between model and DGP is unknown in the latter. Nevertheless, considerable effort has been devoted to the two modelling examples of UK money demand and aggregate consumers' expenditure, so we report how PcGets performs on these. Although the `truth' of the results cannot be established, the question of interest is whether the selected models match, or even beat, those of earlier authors.
How well did PcGets do compared to the original authors Hendry and Ericsson (1991b) in modelling narrow money demand in the UK? Hendry and Ericsson (1991b) commenced from a GUM with 2 lags of m-p, y, Dp, and R, and selected (after simplifications and transformations: also see Hendry and Doornik, 1994):
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From the levels GUM above, two models resulted (one in I(1), the other in I(0), transformations), so we compare both with (eq:5.3). From the levels GUM in ( m-p) , Dp, y and R, PcGets selected:
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The selection in (eq:5.4) corresponds closely to the model in (eq:5.3) without imposing the unit long-run income restriction, leaving a further valid reduction, but omitting Dyt-1. Solving for the long run with income homogeneity imposed yielded e in (eq:5.2), with cointegration well determined. The outcome in (eq:5.2) was used in the second GUM leading to:
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No diagnostic tests were significant in any model. The three values of s^ are very close, as are the three representations found. In fact, if the union of (eq:5.3) and (eq:5.5) is formed, then no variable from (eq:5.3) is significant, whereas three from (eq:5.5) remain highly significant (D2pt, et-1 and DRt). Thus, PcGets is certainly at least as successful as the authors. In general, an `expert system' is required to understand the links between regressors to formulate appropriate terms (such as D( m-p-y) t-1). Nevertheless, PcGets does remarkably well against the `experts' on these empirical problems, although its performance clearly benefits from the `fore-knowledge' which the initial investigators took some time to acquire.
Again, this data set (or extensions thereof) has been thoroughly studied by many investigators including Davidson, Hendry, Srba and Yeo (1978) (denoted by the acronym DHSY), Bean (1977, 1978), Hendry and von Ungern-Sternberg (1981), Carruth and Henley (1990), Hendry, Muellbauer and Murphy (1990), Hendry (1994), and Muellbauer (1994). We re-consider the first of these. The equation used as a baseline by DHSY corresponds to the GUM:
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where ct denotes aggregate real consumers' expenditure on non-durable goods and services, yt is real personal disposable income, pt is the implicit deflator of ct (all in logs), D4pt=pt-pt-4, the St-i are seasonal dummies, and Dt is an indicator for budget effects in 1968 and the introduction of VAT in 1973, equal to +1 in 1968(1) and 1973(1), and -1 in 1968(2) and 1973(2) (see footnote 5 in Davidson, Hendry, Srba and Yeo, 1978). We want to see if PcGets recovers their model (8.45)**:
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Commencing from the GUM in levels (see §10.2.1.1 on modelling integrated variables), the selection results were similar for the three basic strategies. The `liberal' outcome for the sample period 1959(2)--1975(4), when F pre-selection tests were used, was:
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The `conservative' and `expert' strategies omitted D4pt-1; and all three were unaffected by the presence or absence of seasonals. When F pre-selection tests were not used, the `conservative' strategy selected the same model, whereas the `expert' now selected (eq:5.8), and the `liberal' retained three more variables.
(eq:5.8) is a level's version of essentially the same equation as DHSY (all coefficients were 100%reliable except D4pt-1 at 20%and yt-4 at 60%). The resulting long-run solution (cointegrating vector) is:
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although cointegration is not strong (the PcGive unit-roott-test, tur=-2.8). Re-arranging (eq:5.8) in differences and the cointegrating vector from (eq:5.9) yields:
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as against the re-estimated values:
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Enforcing the same lag length as DHSY used, therefore, PcGets selects essentially the same model as that chosen by Davidson et al., and all strategy choices produce much the same final selection.
Expressing the GUM in terms of 4 lags of D4yt-i, D4ct-i, D4pt-i, with (c-y)t-4 and D4Dt, increases the lag lengths of ct and yt to eight, albeit in a restricted parametrization, but is an admissible specification. Commencing from that GUM, the `expert' strategy retained one more variable than (eq:5.8) (namely D4ct-3 -- which the original authors noted was significant), with one selection switched (D4pt-3 replaced D4pt-1). Now, s^=0.0058, and SC =-9.97: no diagnostic test was significant, and all coefficients were 100%reliable. The `conservative' strategy dropped D4pt-1 but added a seasonal; and the `liberal' took the union of these (adding D4ct-3 and a seasonal -- which was 60%reliable -- while switching D4pt-3 for D4pt-1). These specifications are more general than any considered by DHSY, and encompass their model.
Locating a specific model with k variables embedded in a large GUM of n>k variables is analogous to finding some proverbial needles in a haystack. Indeed, categorizing the problem facing PcGets in a related way is useful, and structuring its significance levels appropriately could improve its performance. The smaller the haystack to be searched (i.e., the closer the specification of the GUM is to the final model), the easier the search will be. Thus, initial parsimony could pay off. Conversely, too small a GUM entails omitting important relevant variables, which would jeopardize the whole exercise -- one is searching the wrong haystack. The true state of nature will be unknown, but the user may have information about its likely form from past research.
| p×( n-k) =0.025×(40-6)=0.85. |
The opposite risk is of missing slightly more than one relevant variable, which can be calculated roughly as follows. First, P( | t100| >2.275|H0) =0.025 defines the significance level. Then, the power to reject the null when E[ t] =3 is (see Ch. 10):
| P( t>2.275½E[ t] =3) ~=P( t>-0.725½H0) ~=0.765. |
Finally, when there are six such t-values, the probability of keeping every variable is 0.7656~=0.2 and the average number omitted is 6×( 1-0.765) ~=1.4. As in case I, the smaller the haystack to be searched, the looser the significance levels can be set for a given sample size.
In any given situation, you can check the likely performance of PcGets by generating artificial data using PcNaive (see Doornik and Hendry, 2001b) to simulate the GUM in which the specific model is hidden, and see how well PcGets does in finding your DGP.
Akaike, A. (1973). "Information theory and an extension of the maximum likelihood principle" In Petrov, B. N., and Saki, F. L.(eds.), Second International Symposium of Information Theory. Budapest.
Akaike, H. (1985). "Prediction and entropy" In Atkinson, A. C., and Fienberg, S. E.(eds.), A Celebration of Statistics, pp. 1--24. New York: Springer-Verlag.
Akerlof, G. A. (1979). "Irving Fisher on his head: The consequences of constant target-threshold monitoring of money holdings" Quarterly Journal of Economics, 93, 169--188.
Amemiya, T. (1980). "Selection of regressors" International Economic Review, 21, 331--354.
Anderson, T. W. (1971). The Statistical Analysis of Time Series. New York: John Wiley & Sons.
Andrews, D. W. K. (1991). "Heteroskedasticity and autocorrelation consistent covariance matrix estimation" Econometrica, 59, 817--858.
Banerjee, A., Dolado, J. J., Galbraith, J. W., and Hendry, D. F. (1993). Co-integration, Error Correction and the Econometric Analysis of Non-Stationary Data. Oxford: Oxford University Press.
Bårdsen, G. (1989). "The estimation of long run coefficients from error correction models" Oxford Bulletin of Economics and Statistics, 50.
Bean, C. R. (1977). "More consumers' expenditure equations" Academic panel paper (77)35, H.M. Treasury, London.
Bean, C. R. (1978). "The determination of consumers' expenditure in the UK" Government economic service working paper 4, H.M. Treasury, London.
Bontemps, C., and Mizon, G. E. (2001). "Congruence and encompassing" In Stigum, B.(ed.), Studies in Economic Methodology. Cambridge, Mass.: MIT Press.
Boswijk, H. P. (1992). Cointegration, Identification and Exogeneity, Vol. 37 of Tinbergen Institute Research Series. Amsterdam: Thesis Publishers.
Bowman, K. O., and Shenton, L. R. (1975). "Omnibus test contours for departures from normality based on Öb1 and b2" Biometrika, 62, 243--250.
Box, G. E. P., and Jenkins, G. M. (1976). Time Series Analysis, Forecasting and Control. San Francisco: Holden-Day. First published, 1970.
Box, G. E. P., and Pierce, D. A. (1970). "Distribution of residual autocorrelations in autoregressive-integrated moving average time series models" Journal of the American Statistical Association, 65, 1509--1526.
Breusch, T. S. (1990). "Simplified extreme bounds" in Granger 1990, pp. 72--81.
Brown, R. L., Durbin, J., and Evans, J. M. (1975). "Techniques for testing the constancy of regression relationships over time (with discussion)" Journal of the Royal Statistical Society B, 37, 149--192.
Burns, A. F., and Mitchell, W. C. (1946). Measuring Business Cycles. New York: NBER.
Campos, J., and Ericsson, N. R. (1999). "Constructive data mining: Modeling consumers' expenditure in Venezuela" Econometrics Journal, 2, 226--240.
Carruth, A., and Henley, A. (1990). "Can existing consumption functions forecast consumer spending in the late 1980s?" Oxford Bulletin of Economics and Statistics, 52, 211--222.
Chatfield, C. (1995). "Model uncertainty, data mining and statistical inference" Journal of the Royal Statistical Society, A, 158, 419--466. With discussion.
Chow, G. C. (1960). "Tests of equality between sets of coefficients in two linear regressions" Econometrica, 28, 591--605.
Chow, G. C. (1981). "Selection of econometric models by the information criteria" In Charatsis, E. G.(ed.), Proceedings of the Econometric Society European Meeting 1979, Ch. 8. Amsterdam: North-Holland.
Clayton, M. K., Geisser, S., and Jennings, D. E. (1986). "A comparison of several model selection procedures" In Goel, P., and Zellner, A.(eds.), Bayesian Inference and Decision Techniques: Elsevier Science.
Clements, M. P., and Hendry, D. F. (1998). Forecasting Economic Time Series. Cambridge: Cambridge University Press.
Clements, M. P., and Hendry, D. F. (1999a). Forecasting Non-stationary Economic Time Series. Cambridge, Mass.: MIT Press.
Clements, M. P., and Hendry, D. F. (1999b). "Modelling methodology and forecast failure" Unpublished typescript, Economics Department, University of Oxford.
Coen, P. G., Gomme, E. D., and Kendall, M. G. (1969). "Lagged relationships in economic forecasting" Journal of the Royal Statistical Society A, 132, 133--163.
Cox, D. R. (1961). "Tests of separate families of hypotheses" In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 105--123 Berkeley: University of California Press.
Cox, D. R. (1962). "Further results on tests of separate families of hypotheses" Journal of the Royal Statistical Society B, 24, 406--424.
Cran, G. W., Martin, K. J., and Thomas, G. E. (1977). "A remark on algorithms. AS 63: The incomplete beta integral. AS 64: Inverse of the incomplete beta function ratio" Applied Statistics, 26, 111--112.
D'Agostino, R. B. (1970). "Transformation to normality of the null distribution of g1" Biometrika, 57, 679--681.
Davidson, J. E. H., and Hendry, D. F. (1981). "Interpreting econometric evidence: The behaviour of consumers' expenditure in the UK" European Economic Review, 16, 177--192. Reprinted in Hendry, D. F., op. cit., (1993) and (2000).
Davidson, J. E. H., Hendry, D. F., Srba, F., and Yeo, J. S. (1978). "Econometric modelling of the aggregate time-series relationship between consumers' expenditure and income in the United Kingdom" Economic Journal, 88, 661--692. Reprinted in Hendry, D. F., op. cit., (1993) and (2000).
Deaton, A. S. (1982). "Model selection procedures or, does the consumption function exist" In Chow, G. C., and Corsi, P.(eds.), Evaluating the Reliability of Macro-Economic Models, Ch. 5. New York: John Wiley.
Doan, T., Litterman, R., and Sims, C. A. (1984). "Forecasting and conditional projection using realistic prior distributions" Econometric Reviews, 3, 1--100.
Doornik, J. A. (1999). Object-Oriented Matrix Programming using Ox 3rd ed. London: Timberlake Consultants Press.
Doornik, J. A., and Hansen, H. (1994). "A practical test for univariate and multivariate normality" Discussion paper, Nuffield College.
Doornik, J. A., and Hendry, D. F. (1996). GiveWin: An Interactive Empirical Modelling Program. London: Timberlake Consultants Press.
Doornik, J. A., and Hendry, D. F. (2001a). Econometric Modelling using PcGive 10, Volume II. London: Timberlake Consultants Press.
Doornik, J. A., and Hendry, D. F. (2001b). Interactive Monte Carlo Experimentation in Econometrics using PcNaive. London: Timberlake Consultants Press.
Engle, R. F. (1982a). "Autoregressive conditional heteroscedasticity, with estimates of the variance of United Kingdom inflation" Econometrica, 50, 987--1007.
Engle, R. F. (1982b). "Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation". 50, 987--1008.
Engle, R. F., Hendry, D. F., and Richard, J.-F. (1983). "Exogeneity" Econometrica, 51, 277--304. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000; and in Ericsson, N. R. and Irons, J. S. (eds.) Testing Exogeneity, Oxford: Oxford University Press, 1994.
Engle, R. F., Hendry, D. F., and Trumbull, D. (1985). "Small sample properties of ARCH estimators and tests" Canadian Journal of Economics, 43, 66--93.
Engle, R. F., and White, H.(eds.)(1999). Cointegration, Causality and Forecasting. Oxford: Oxford University Press.
Ericsson, N. R. (1983). "Asymptotic properties of instrumental variables statistics for testing non-nested hypotheses" Review of Economic Studies, 50, 287--303.
Ericsson, N. R., Campos, J., and Tran, H.-A. (1990). "PC-GIVE and David Hendry's econometric methodology" Revista De Econometria, 10, 7--117.
Ericsson, N. R., Hendry, D. F., and Prestwich, K. M. (1998). "The demand for broad money in the United Kingdom, 1878--1993" Scandinavian Journal of Economics, 100, 289--324.
Ericsson, N. R., and Irons, J. S.(eds.)(1994). Testing Exogeneity. Oxford: Oxford University Press.
Faust, J., and Whiteman, C. H. (1997). "General-to-specific procedures for fitting a data-admissible, theory-inspired, congruent, parsimonious, encompassing, weakly-exogenous, identified, structural model of the DGP: A translation and critique" Carnegie--Rochester Conference Series on Public Policy, 47, 121--161.
Friedman, M., and Schwartz, A. J. (1982). Monetary Trends in the United States and the United Kingdom: Their Relation to Income, Prices, and Interest Rates, 1867--1975. Chicago: University of Chicago Press.
Frisch, R., and Waugh, F. V. (1933). "Partial time regression as compared with individual trends" Econometrica, 1, 221--223.
Gilbert, C. L. (1986). "Professor Hendry's econometric methodology" Oxford Bulletin of Economics and Statistics, 48, 283--307. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press.
Godfrey, L. G. (1978). "Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables" Econometrica, 46, 1303--1313.
Godfrey, L. G., and Orme, C. D. (1994). "The sensitivity of some general checks to omitted variables in the linear model" International Economic Review, 35, 489--506.
Godfrey, L. G., and Veale, M. R. (1999). "Alternative approaches to testing by variable addition" Mimeo, York University, UK.
Gourieroux, C., and Monfort, A. (1995). "Testing, encompassing, and simulating dynamic econometric models" Econometric Theory, 11, 195--228.
Granger, C. W. J. (1969). "Investigating causal relations by econometric models and cross-spectral methods" Econometrica, 37, 424--438.
Granger, C. W. J.(ed.)(1990). Modelling Economic Series. Oxford: Clarendon Press.
Haavelmo, T. (1944). "The probability approach in econometrics" Econometrica, 12, 1--118. Supplement.
Hannan, E. J., and Quinn, B. G. (1979). "The determination of the order of an autoregression" Journal of the Royal Statistical Society, B, 41, 190--195.
Harris, R. I. D. (1995). Using Cointegration Analysis in Econometric Modelling. London: Prentice Hall.
Harvey, A. C. (1981). The Econometric Analysis of Time Series. Deddington: Philip Allan.
Harvey, A. C. (1990). The Econometric Analysis of Time Series, 2nd ed. Hemel Hempstead: Philip Allan.
Hendry, D. F. (1976). "The structure of simultaneous equations estimators" Journal of Econometrics, 4, 51--88. Reprinted in Hendry, D. F., op. cit., (1993) and (2000).
Hendry, D. F. (1979). "Predictive failure and econometric modelling in macro-economics: The transactions demand for money" In Ormerod, P.(ed.), Economic Modelling, pp. 217--242. London: Heinemann. Reprinted in Hendry, D. F., op. cit., (1993) and (2000).
Hendry, D. F. (1980). "Econometrics: Alchemy or science?" Economica, 47, 387--406. Reprinted in Hendry, D. F., Econometrics: Alchemy or Science? Oxford: Blackwell Publishers, 1993, and Oxford University Press, 2000.
Hendry, D. F. (1985). "Monetary economic myth and econometric reality" Oxford Review of Economic Policy, 1, 72--84. Reprinted in Hendry, D. F., op. cit., (1993) and (2000).
Hendry, D. F. (1994). "HUS revisited" Oxford Review of Economic Policy, 10, 86--106.
Hendry, D. F. (1995a). Dynamic Econometrics. Oxford: Oxford University Press.
Hendry, D. F. (1995b). "Econometrics and business cycle empirics" Economic Journal, 105, 1622--1636.
Hendry, D. F. (1996). "On the constancy of time-series econometric equations" Economic and Social Review, 27, 401--422.
Hendry, D. F. (1997). "On congruent econometric relations: A comment" Carnegie--Rochester Conference Series on Public Policy, 47, 163--190.
Hendry, D. F. (1999). "An econometric analysis of US food expenditure, 1931--1989" in Magnus, and Morgan 1999, pp. 341--361.
Hendry, D. F. (2000a). Econometrics: Alchemy or Science? Oxford: Oxford University Press. New Edition.
Hendry, D. F. (2000b). "Epilogue: The success of general-to-specific model selection" In Econometrics: Alchemy or Science?, pp. 467--490. Oxford: Oxford University Press. New Edition.
Hendry, D. F., and Doornik, J. A. (1994). "Modelling linear dynamic econometric systems" Scottish Journal of Political Economy, 41, 1--33.
Hendry, D. F., and Doornik, J. A. (1999). "The impact of computational tools on time-series econometrics" In Coppock, T.(ed.), Information Technology and Scholarship, pp. 257--269. Oxford: Oxford University Press.
Hendry, D. F., and Doornik, J. A. (2001). Econometric Modelling using PcGive 10: Volume I. London: Timberlake Consultants Press.
Hendry, D. F., and Ericsson, N. R. (1991a). "An econometric analysis of UK money demand in `Monetary Trends in the United States and the United Kingdom' by Milton Friedman and Anna J. Schwartz" American Economic Review, 81, 8--38.
Hendry, D. F., and Ericsson, N. R. (1991b). "Modeling the demand for narrow money in the United Kingdom and the United States" European Economic Review, 35, 833--886.
Hendry, D. F., and Krolzig, H.-M. (1999a). "General-to-specific model selection using PcGets for Ox" Unpublished paper, Economics Department, Oxford University.
Hendry, D. F., and Krolzig, H.-M. (1999b). "Improving on `Data mining reconsidered' by K.D. Hoover and S.J. Perez" Econometrics Journal, 2, 202--219.
Hendry, D. F., and Krolzig, H.-M. (2000). "The econometrics of general-to-simple modelling" Mimeo, Economics Department, Oxford University.
Hendry, D. F., Leamer, E. E., and Poirier, D. J. (1990). "A conversation on econometric methodology" Econometric Theory, 6, 171--261.
Hendry, D. F., and Mizon, G. E. (1978). "Serial correlation as a convenient simplification, not a nuisance: A comment on a study of the demand for money by the Bank of England" Economic Journal, 88, 549--563. Reprinted in Hendry, D. F., op. cit., (1993) and (2000).
Hendry, D. F., and Mizon, G. E. (1990). "Procrustean econometrics: or stretching and squeezing data" in Granger 1990, pp. 121--136.
Hendry, D. F., and Mizon, G. E. (1993). "Evaluating dynamic econometric models by encompassing the VAR" In Phillips, P. C. B.(ed.), Models, Methods and Applications of Econometrics, pp. 272--300. Oxford: Basil Blackwell.
Hendry, D. F., and Mizon, G. E. (1999). "The pervasiveness of Granger causality in econometrics" in Engle, and White 1999.
Hendry, D. F., and Mizon, G. E. (2000). "Reformulating empirical macro-econometric modelling" Oxford Review of Economic Policy, 16, 138--159.
Hendry, D. F., and Morgan, M. S. (1995). The Foundations of Econometric Analysis. Cambridge: Cambridge University Press.
Hendry, D. F., Muellbauer, J. N. J., and Murphy, T. A. (1990). "The econometrics of DHSY" In Hey, J. D., and Winch, D.(eds.), A Century of Economics, pp. 298--334. Oxford: Basil Blackwell.
Hendry, D. F., and Neale, A. J. (1987). "Monte Carlo experimentation using PC-NAIVE" In Fomby, T., and Rhodes, G. F.(eds.), Advances in Econometrics, Vol. 6, pp. 91--125. Greenwich, Connecticut: Jai Press Inc.
Hendry, D. F., and Richard, J.-F. (1982). "On the formulation of empirical models in dynamic econometrics" Journal of Econometrics, 20, 3--33. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press and in Hendry D. F., op. cit., (1993) and (2000).
Hendry, D. F., and Richard, J.-F. (1989). "Recent developments in the theory of encompassing" In Cornet, B., and Tulkens, H.(eds.), Contributions to Operations Research and Economics. The XXth Anniversary of CORE, pp. 393--440. Cambridge, MA: MIT Press.
Hendry, D. F., and von Ungern-Sternberg, T. (1981). "Liquidity and inflation effects on consumers' expenditure" In Deaton, A. S.(ed.), Essays in the Theory and Measurement of Consumers' Behaviour, pp. 237--261. Cambridge: Cambridge University Press. Reprinted in Hendry, D. F., op. cit., (1993) and (2000).
Hendry, D. F., and Wallis, K. F.(eds.)(1984). Econometrics and Quantitative Economics. Oxford: Basil Blackwell.
Hoover, K. D., and Perez, S. J. (1999). "Data mining reconsidered: Encompassing and the general-to-specific approach to specification search" Econometrics Journal, 2, 167--191.
Jarque, C. M., and Bera, A. K. (1980). "Efficient tests for normality, homoscedasticity and serial independence of regression residuals" Economics Letters, 6, 255--259.
Johansen, S. (1988). "Statistical analysis of cointegration vectors" Journal of Economic Dynamics and Control, 12, 231--254. Reprinted in R.F. Engle and C.W.J. Granger (eds), Long-Run Economic Relationships, Oxford: Oxford University Press, 1991, 131--52.
Johansen, S. (1992). "Testing weak exogeneity and the order of cointegration in UK money demand" Journal of Policy Modeling, 14, 313--334.
Judge, G. G., and Bock, M. E. (1978). The Statistical Implications of Pre-Test and Stein-Rule Estimators in Econometrics. Amsterdam: North Holland Publishing Company.
Judge, G. G., Griffiths, W. E., Hill, R. C., Lütkepohl, H., and Lee, T.-C. (1985). The Theory and Practice of Econometrics, 2nd ed. New York: John Wiley.
Kent, J. T. (1986). "The underlying nature of nonnested hypothesis tests" Biometrika, 73, 333--343.
Keynes, J. M. (1939). "Professor Tinbergen's method" Economic Journal, 44, 558--568.
Keynes, J. M. (1940). "Comment" Economic Journal, 50, 154--156.
Kiviet, J. F. (1985). "Model selection test procedures in a single linear equation of a dynamic simultaneous system and their defects in small samples" Journal of Econometrics, 28, 327--362.
Kiviet, J. F. (1986). "On the rigor of some mis-specification tests for modelling dynamic relationships" Review of Economic Studies, 53, 241--261.
Koopmans, T. C. (1947). "Measurement without theory" Review of Economics and Statistics, 29, 161--179.
Koopmans, T. C., Rubin, H., and Leipnik, R. B. (1950). "Measuring the equation systems of dynamic economics" In Koopmans, T. C.(ed.), Statistical Inference in Dynamic Economic Models, No. 10 in Cowles Commission Monograph, Ch. 2. New York: John Wiley & Sons.
Krolzig, H.-M. (2001). "General-to-specific reductions of vector autoregressive processes" Economics discussion paper 2000-W34, Nuffield College, Oxford.
Krolzig, H.-M., and Hendry, D. F. (2001). "Computer automation of general-to-specific model selection procedures" Journal of Economic Dynamics and Control, 25, 831--866.
Leamer, E. E. (1978). Specification Searches. Ad-Hoc Inference with Non-Experimental Data. New York: John Wiley.
Leamer, E. E. (1983a). "Let's take the con out of econometrics" American Economic Review, 73, 31--43. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press.
Leamer, E. E. (1983b). "Model choice and specification analysis" In Griliches, Z., and Intriligator, M. D.(eds.), Handbook of Econometrics, Vol. 1, Ch. 5. Amsterdam: North-Holland.
Leamer, E. E. (1984). "Global sensitivity results for generalized least squares estimates" Journal of the American Statistical Association, 79, 867--870.
Leamer, E. E. (1990). "Sensitivity analyses would help" in Granger 1990, pp. 88--96.
Ljung, G. M., and Box, G. E. P. (1978). "On a measure of lack of fit in time series models" Biometrika, 65, 297--303.
Lovell, M. C. (1983). "Data mining" Review of Economics and Statistics, 65, 1--12.
Lütkepohl, H. (1991). Introduction to Multiple Time Series Analysis. Berlin: Springer.
Magnus, J. R., and Morgan, M. S.(eds.)(1999). Methodology and Tacit Knowledge: Two Experiments in Econometrics. Chichester: John Wiley and Sons.
Majunder, K. L., and Bhattacharjee, G. P. (1973a). "Algorithm AS 63. The incomplete beta integral" Applied Statistics, 22, 409--411.
Majunder, K. L., and Bhattacharjee, G. P. (1973b). "Algorithm AS 64. Inverse of the incomplete beta function ratio" Applied Statistics, 22, 411--414.
Mayo, D. (1981). "Testing statistical testing" In Pitt, J. C.(ed.), Philosophy in Economics, pp. 175--230: D. Reidel Publishing Co. Reprinted as pp. 45--73 in Caldwell B. J. (1993), The Philosophy and Methodology of Economics, Vol. 2, Aldershot: Edward Elgar.
McAleer, M., Pagan, A. R., and Volker, P. A. (1985). "What will take the con out of econometrics?" American Economic Review, 95, 293--301. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press.
Mizon, G. E. (1977a). "Inferential procedures in nonlinear models: An application in a UK industrial cross section study of factor substitution and returns to scale" Econometrica, 45, 1221--1242.
Mizon, G. E. (1977b). "Model selection procedures" In Artis, M. J., and Nobay, A. R.(eds.), Studies in Modern Economic Analysis, pp. P97--120. Oxford: Basil Blackwell.
Mizon, G. E. (1984). "The encompassing approach in econometrics" in Hendry, and Wallis 1984, pp. 135--172.
Mizon, G. E. (1995). "Progressive modelling of macroeconomic time series: the LSE methodology" In Hoover, K. D.(ed.), Macroeconometrics: Developments, Tensions and Prospects, pp. 107--169. Dordrecht: Kluwer Academic Press.
Mizon, G. E., and Richard, J.-F. (1986). "The encompassing principle and its application to non-nested hypothesis tests" Econometrica, 54, 657--678.
Moore, H. L. (1914). Economic Cycles -- Their Law and Cause. New York: MacMillan.
Muellbauer, J. N. J. (1994). "The assessment: Consumer expenditure" Oxford Review of Economic Policy, 10, 1--41.
Nicholls, D. F., and Pagan, A. R. (1983). "Heteroscedasticity in models with lagged dependent variables" Econometrica, 51, 1233--1242.
Pagan, A. R. (1984). "Model evaluation by variable addition" in Hendry, and Wallis 1984, pp. 103--135.
Pagan, A. R. (1987). "Three econometric methodologies: A critical appraisal" Journal of Economic Surveys, 1, 3--24. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press.
Paroulo, P. (1996). "On the determination of integration indices in I(2) systems" Journal of Econometrics, 72, 313--356.
Pesaran, M. H. (1974). "On the general problem of model selection" Review of Economic Studies, 41, 153--171.
Pesaran, M. H.(ed.)(1987). The Limits of Rational Expectations. Oxford: Basil Blackwell.
Pike, M. C., and Hill, I. D. (1966). "Logarithm of the gamma function" Communications of the ACM, 9, 684.
Rahbek, A., Kongsted, H. C., and Jørgensen, C. (1999). "Trend-stationarity in the I(2) cointegration model" Journal of Econometrics, 90, 265--289.
Sargan, J. D. (1964). "Wages and prices in the United Kingdom: A study in econometric methodology (with discussion)" In Hart, P. E., Mills, G., and Whitaker, J. K.(eds.), Econometric Analysis for National Economic Planning, Vol. 16 of Colston Papers, pp. 25--63. London: Butterworth Co. Reprinted as pp. 275--314 in Hendry D. F. and Wallis K. F. (eds.) (1984). Econometrics and Quantitative Economics. Oxford: Basil Blackwell, and as pp. 124--169 in Sargan J. D. (1988), Contributions to Econometrics, Vol. 1, Cambridge: Cambridge University Press.
Sargan, J. D. (1973). "Model building and data mining" Discussion paper, London School of Economics. Presented to the Association of University Teachers of Economics, Meeting, Manchester, April 1973.
Sargan, J. D. (1980). "Some tests of dynamic specification for a single equation" Econometrica, 48, 879--897. Reprinted as pp. 191--212 in Sargan J. D. (1988), Contributions to Econometrics, Vol. 1, Cambridge: Cambridge University Press.
Sargan, J. D. (1981). "The choice between sets of regressors" Mimeo, Economics Department, London School of Economics.
Savin, N. E. (1984). "Multiple hypothesis testing" In Griliches, Z., and Intriligator, M. D.(eds.), Handbook of Econometrics, Vol. 2--3, Ch. 14. Amsterdam: North-Holland.
Sawa, T. (1978). "Information criteria for discriminating among alternative regression models" Econometrica, 46, 1273--1292.
Schwarz, G. (1978). "Estimating the dimension of a model" Annals of Statistics, 6, 461--464.
Shea, B. L. (1988). "Algorithm AS 239: Chi-squared and incomplete gamma integral" Applied Statistics, 37, 466--473.
Shenton, L. R., and Bowman, K. O. (1977). "A bivariate model for the distribution of Öb1 and b2" Journal of the American Statistical Association, 72, 206--211.
Shibata, R. (1980). "Asymptotically efficient selection of the order of the model for estimating parameters of a linear process" Annals of Statistics, 8, 147--164.
Sims, C. A. (1980). "Macroeconomics and reality" Econometrica, 48, 1--48. Reprinted in Granger, C. W. J. (ed.) (1990), Modelling Economic Series. Oxford: Clarendon Press.
Sims, C. A., Stock, J. H., and Watson, M. W. (1990). "Inference in linear time series models with some unit roots" Econometrica, 58, 113--144.
Smith, G. W. (1986). "A dynamic Baumol-Tobin model of money demand" Review of Economic Studies, 53, 465--469.
Spanos, A. (1989). "On re-reading Haavelmo: A retrospective view of econometric modeling" Econometric Theory, 5, 405--429.
Sullivan, R., Timmermann, A., and White, H. (1998). "Dangers of data-driven inference: The case of calendar effects in stock returns" Mimeo, Economics Department, University of California at San Diego.
Summers, L. H. (1991). "The scientific illusion in empirical macroeconomics" Scandinavian Journal of Economics, 93, 129--148.
Teräsvirta, T. (1976). "Effect of feedback on the distribution of the portmanteau statistic" Manuscript, London School of Economics.
Theil, H. (1971). Principles of Econometrics. London: John Wiley.
Tinbergen, J. (1940a). Statistical Testing of Business-Cycle Theories. Geneva: League of Nations. Vol. I: A Method and its application to Investment Activity.
Tinbergen, J. (1940b). Statistical Testing of Business-Cycle Theories. Geneva: League of Nations. Vol. II: Business Cycles in the United States of America, 1919--1932.
Vuong, Q. H. (1989). "Likelihood ratio tests for model selection and nonnested hypotheses" Econometrica, 50, 1--25.
White, H. (1980). "A heteroskedastic-consistent covariance matrix estimator and a direct test for heteroskedasticity" Econometrica, 48, 817--838.
White, H. (1984). Asymptotic Theory for Econometricians. London: Academic Press.
White, H. (1990). "A consistent model selection" in Granger 1990, pp. 369--383.
Wooldridge, J. M. (1999). "Asymptotic properties of some specification tests in linear models with integrated processes" in Engle, and White 1999, pp. 366--384.
Wright, P. G. (1915). "Moore's economic cycles" Quarterly Journal of Economics, 29, 631--641.
Yancey, T. A., and Judge, G. G. (1976). "A Monte Carlo comparison of traditional and stein-rule estimators under squared error loss" Journal of Econometrics, 4, 285--294.