This chapter describes each menu option in the PcGets menu structure. From the menu bar at the top of the screen, pull-down menus give access to all the actions in PcGets. Examples of most options were given in the tutorial chapters. Often, after selecting a menu option, a dialog will take over to request the information needed to perform the requested task. Only a few dialogs are discussed in this chapter, but remember that all dialogs are featured in the tutorial chapters. Moreover, on-line help is available in each dialog by pressing F1 or the help button.
The File menu provides overall control of the program. The following commands are available:
The General-to-specific package is always active. No further functionality is offered in this version of PcGets, although conventional least-squares and instrumental-variables estimation without search are provided.
The Model menu allows you to formulate and estimate models, recall a previously-estimated model, and check the progress made in the modelling process. It also provides access to all the Options. The following commands are available:
The Test menu is used to evaluate the model, both graphically and through diagnostic testing.
It also offers facilities to print out the model, generate batch files, and store results.
Press the F1 key for help, or click on the Help menu and select Help Topics to see the help index.
The following commands are available:
Press F1 or click on the Help button in any dialog for context-specific help.
This menu item toggles the Always on Top status. When switched on, PcGets will always be above other windows, even if it does not have the focus. When PcGets is shrunk to a small window, this could be useful to allow easy access. When off, PcGets will behave as a normal window.
Note that the setting is persistent between runs.
Exits the application.
The general-to-specific (Gets) package is always active: no other options are offered in this version of PcGets. However, conventional OLS and IVE (i.e., without search) are available under Estimate, and recursive variants of both under Test.
Single equation modelling is described in detail in Chapter 12.
The Formulate command is used for dynamic model formulation: formulate (or reformulate) a model for estimation by selecting variables and lag lengths in the Formulate a model dialog. When you press OK, you will be taken automatically to the Estimation dialog. For instrumental variables estimation, mark variables in the model as endogenous or instruments. The remaining variables will be assumed exogenous.
A model in PcGets is formulated by:
PcGets is designed for single dynamic equations specified as autoregressive-distributed lag models:
|
|
In (eq:14.1), the lag polynomials are defined by:
| Bi(L)=åj=minibi,jLj with 0<mi<ni, i=1,...,k. |
`Solving' (eq:14.1) yields:
| yt=åi=1kHi(L)xi+ut, where Hi(L)= |
| . |
Zero is a legitimate order for a lag polynomial: thus, static or dynamic models are equally easily specified.
When a model has been formulated, it can be estimated and then analyzed. PcGets facilitates a general-to-specific modelling strategy.
The Model Formulation dialog can be used to formulate a new model, or reformulate an existing model.
Mark all the variables you wish to include in the new model or add to the existing model, in this Multiple-Selection List box, using the spacebar or the mouse.
After you have pressed Enter (or double-clicked if you are using a mouse), you can select a lag length for each variable (see lagged variables).
The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below.
This Multiple-Selection List box shows the current model.
The variable at the top of the list will by default become the endogenous (Y) variable.
To select a dependent variable which is listed further down:
If you have marked variables in the model, you can delete them, or assign a status to them.
Deletes the selection of variables from the current model.
Deletes the whole model, so that you can restart from scratch.
Clears the status of all selected model variables. Cleared variables behave as Z variables. You can also double click on a model variable to clear its status.
Label the currently selected variables as the regressand (this is not possible for lagged variables). The regressand is marked by Y in the model list. Only one variable can have this status.
Marks each selected model variable as an unmodelled variable. This is the default for an unmarked variable, so an X variable or unmarked variable are treated identically.
Forces the selected model variables to be included in the specific model for
GETS/GETSIVE.
In the case of OLS/IVE, F variables and X variables are treated in the
same way.
Label each selected model variable as an endogenous variable. Endogenous variables are preceded by E in the model list. Estimation requires at least as many additional instruments as endogenous regressors in the model.
Label each selected model variable as an additional instruments. This button is only relevant if you wish to use Instrumental Variables estimation. Additional instruments are marked by A in the model list.
If there are still database variables marked, this button will be called
Add. Press it to add the variables (or press Deselect All to
change it to OK). You will be prompted for a lag length if that is set to query. You can also double click on
a database variable to add it to the model.
Press OK to move to the Model Settings or Estimation.
Equivalent to pressing Esc, this will abort the model formulation.
Choose one of
Up to 100 estimated models are remembered during a session. The Recall command lets you recall any of these through the Recall dialog box. This requires that the variables involved are still in the same position in the database.
Use this dialog to recall a previously estimated model.
You will have to re-estimate it prior to accessing items on the Test
Menu.
Move the cursor to the model you wish to recall, and press OK.
Use these buttons to move between the estimated models.
Because of its central role in PcGets, the whole of Chapter 15 is devoted to the dialog following from this menu option.
The following information is needed to estimate an equation:
PcGets will elicit information on all these aspects. Models may be revised interactively after formulation and after estimation.
The Estimate command provides model estimation.
Select an estimation method, sample period, and number of forecasts for the formulated model.
Enter the sample period you wish to use for estimation (including
initialization and forecasts), e.g. 1960 1 to 1980 4. The maximum sample is
shown one line up.
The default is the sample of the previous estimation (of course, only if
possible). PcGets automatically excludes observations with missing values.
Enter the number of observations you wish to withhold for forecasting.
Allows setting the estimation options described in Chapter 13.
Pressing OK starts estimation, unless there still is something missing or invalid in the dialog.
Ordinary least squares is the standard textbook method. OLS is valid if the data model is congruent. The requirements for congruency are:
PcGets provides tests of most of the aspects of model congruency
GETS (general-to-specific) offers automatic model selection when the precise formulation of an econometric relationship is not known a priori.
Starting from a general model which is congruent with the data evidence, statistically-insignificant variables are eliminated, with diagnostic tests checking the validity of reductions, to ensure a congruent final selection.
A structural representation is parsimonious with parameters but has regressors which are correlated with the error term. IVE requires that the reduced form is a congruent data model. The Instrumental variables are the reduced form regressors. Instrumental variables includes two-stage least squares (2SLS) as a special case.
PcGets needs to know the status of the variables in the model:
GETSIVE operates like GETS but using Instrumental variables methods instead of Ordinary least squares.
Individual equation estimation is allowed by least squares (OLS) and instrumental variables (IVE).
Once a model has been specified, a sample period selected, and an estimation method chosen, the output appears.
The first column of these results records the names of the variables and the second, the estimated regression coefficients values. The following three columns give further information about each of the magnitudes described below in 3 to 5.
These are the square roots of the diagonal of the variance-covariance matrix.
These statistics are conventionally calculated to determine whether individual coefficients are significantly different from zero (called the null hypothesis, H0). When H0 is true (and the model is otherwise correctly specified), a Student t-distribution is used, since the sample size is often small and we only have an estimate of the parameter's standard error. However, as the sample size increases, t tends to a standard normal distribution under H0. Large values of t reject H0; but, in many situations, H0 may be of little interest to test. This depends on appropriate prior transformations of the variables, preferably to near orthogonality, such that the null is relevant for some of the transformations.
Gives the probability values of the t-statistics.
Beneath the columnar presentation, an array of summary statistics is provided as follows:
This is exactly what it states, with s^2= RSS/(T-k).
This is the standard deviation of the difference between the actual and fitted values in the regression. For a given dependent variable, s^ can be standardized as a percentage of the mean of the original level of the dependent variable y (except when its mean is zero) for comparisons across specifications. Since many economics magnitudes are inherently positive, that standardization is often feasible. If y is in logs, 100s^ is the percentage standard error.
This is a measure of the goodness of fit of the present regression.
This adjusts the squared multiple correlation coefficient R2 for the loss in the degrees of freedom.
Under normality, OLS maximizes the log-likelihood function which is given by:
| ln L=-T/2-T/2 ln (2p)-1/2RSS. |
This is the information criterion proposed by Akaike (1973); note that it tends to overselect asymptotically.
| AIC=-2 ln L/T+2k/T. |
This is the information criterion proposed by Hannan and Quinn (1979), which provides a consistent selection.
| HQ=-2 ln L/T+2k ln ( ln (T))/T. |
This is the Bayesian information criterion proposed by Schwarz (1978), which also provides a consistent selection.
| SC=-2 ln L/T+k ln T/T. |
These three measures differ in the `penalty' they impose for more parameters, where smaller values of all three are preferable, ceteris paribus. From these, other model selection criteria may be calculated. These, and related scalar measures, are often used to choose between alternative models in a class. The Progress output reports s^, RSS and SC.
In case of over-identifying instruments under IVE/GETSIVE, the Sargan test statistic (§12.6.3) is reported.
PcGets controls the validity of model reductions by diagnostic testing for:
Test statistics and marginal rejection probabilities are reported for the following diagnostics:
The test statistic proposed by Chow (1960) checks the parameter constancy of the model using:
|
The marginal rejection probability reported assumes an F distribution, which is the exact distribution for fixed regressors, but is only approximately (or asymptotically) so in dynamic models.
| T2åj=1srj2/(T-j) |
Note: The residual correlogram, autoregression and the Durbin--Watson test are not valid for models with lagged dependent variables, or only weakly (as opposed to strongly) exogenous variables, whereas the LM test for error autocorrelation (4. above) is valid.
The Normality test checks whether the residuals are normally distributed as:
| ut~IN[ 0,s2] with E[ ut3] =0, and E[ ut4] =3s2. |
A c2 test is reported (with 2 degrees of freedom), and the output includes all moments up to the fourth. The null hypothesis is normality, which will be rejected at (e.g.) the 5%level if a test statistic of more than 5.99 is observed. The reported test statistic has a small-sample correction.
Yields a Lagrange-multiplier (LM) test for serial correlation:
| ut=åi=srriut-i+et where 0<s<r<12, |
with et~IID(0,s2). The F-test is performed by an auxiliary regression of the residuals on the original variables and lagged residuals (missing lagged residuals at the start of the sample are replaced by zero, so no observations are lost). The null hypothesis is no autocorrelation, which would be rejected if the test statistic is too high. This LM test is valid for models with lagged dependent variables, whereas neither the DW nor the residual correlogram provide a valid test in that case.
Checks whether the residuals have an ARCH structure:
| E[ ut2½ut-1,...,ut-r] =åi=1raiut-i2, |
with 0<s<r<12 and e~IID(0,t2). An F-statistic is reported. The null hypothesis is no ARCH, which would be rejected if the test statistic is too high. This test is computed by regressing the squared residuals on a constant and lagged squared residuals (now some observations are lost at the beginning of the sample).
Tests if the {ut} have constant variance against the alternative that ut2 depends on the time-t original and squared regressors. The null hypothesis is no heteroscedasticity, which would be rejected if the test statistic is too high. The reported F-statistic is derived by an auxiliary regression of the squared residuals on a constant, the original regressors, and the original regressors squared.
The Progress command reports on the progress to date made in the general to specific modelling strategy the Progress dialog box is used to change the default model nesting sequence.
Use this dialog to review the progress made to date in the model reduction, when using different GUMs in the general to specific Modelling Strategy.
Already marked are the models that are sequentially nested in an older (i.e., lower in the list) model. However, PcGets might miss a model that could be nested through transformed variables. You can add such models to the nesting chain by marking them in this Multiple-Selection List box.
Moving from top to bottom through the marked models RSS must decrease, the sample period be constant, and the number of explanatory variables go up. Models that do not satisfy this requirement will be deleted.
Will exit the dialog and try to locate the output of the highlighted model in the GiveWin results window.
Generates batch code with the variables of the GUM, the estimation method and sample and writes it to the Results window.
The progress report consists of:
The detailed discussion and explanation of all the available options is provided in Chapter 15.
The mathematical formulations of all the statistics are described in Chapter 13.
The Graphic analysis command offers various options for graphing actual and fitted values, forecasts and residuals. Use the Zoom button to graph for periods other than the default.
Actual and fitted values
Shows the fitted and actual values of the dependent variable over time, over the whole sample period, including the forecast period.
As above, but now a cross-plot (scatter plot) of actual and fitted values.
Shows the scaled residuals against time over the sample period. The residuals are scaled by the residual standard deviation.
Shows the squared residuals against time over the sample period, scaled by the residual variance.
Residual analysis
Shows the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of the residuals.
Estimated density and histogram of the residuals. The normal density with the same mean and variance is drawn for reference.
Shows the spectral density of the residuals, using the lag length of the ACF as the truncation point.
Plots the ordered residuals in a QQ plot against the normal distribution.
To zoom a graph, adjust the area inside GiveWin.
Histograms are a way of looking at the sample distributions of statistics. Then, on the basis of the original data, density functions may be interpolated to give a clearer picture of the implied distributional shape: similarly, cumulative distribution functions may be constructed (and compared on-screen to a cumulative Normal density).
Non-parametric density estimation:
Given observations
| ( x1...xT) |
from some unknown probability density function f(×), about which little may be known a priori. To estimate that density without imposing too many assumptions about its properties, a non-parametric approach is used in PcGets based on a kernel estimator of f(×). The Kernel estimator, f, of the density f is defined by:
| f(X)= 1/Nh K{ 1/h åt=1T ( X-xt)}, |
where K{×} is the kernel function, and h is a `window width' or smoothing parameter, that corresponds to the width of the histogram bars. The kernel K used is the Normal or Gaussian kernel. Research suggests that the density estimate is little affected by the choice of kernel, but is largely governed by the choice of window width, h.
Since evaluating f(X) directly can be expensive in computer time, a method based on a fast Fourier transform is used in PcGets.
The window width in estimating the density, h=CsTP, is set to minimize the integrated mean-square error for normal densities: P=-0.2 and C=1.06.
The correlogram or autocorrelation function (ACF) of a variable, or of the residuals of an estimated model, plots the series of correlation coefficients rj between xt and xt-j.
The length s of the ACF is chosen by the user, leading to a figure which shows (r1,r2,...,rs) plotted against (1,2...,s).
A related statistic is the Portmanteau (also called Box--Pierce, or Q-statistic):
| Tåj=1srj2. |
The partial autocorrelation coefficients correct the autocorrelation for the effects of previous lags. So the first partial autocorrelation coefficient equals the first normal autocorrelation coefficient.
A stationary series can be decomposed into cyclical components with different frequencies and amplitudes. The spectral density provides a graphical representation of this. It is symmetric around 0, and only graphed for [0,p] (the horizontal axis in the PcGets graphs is scaled by p, and given as [0,1]).
The spectral density consists of a weighted sum of the autocorrelations, using the Parzen window as the weighting function. The truncation parameter m can be set (the larger m, the less smooth the spectrum becomes, but the lower the bias).
A white-noise series has a flat spectrum.
The Recursive analysis command provides graphs of the recursive output as generated by a recursive estimation. Multiple graphs allow for condensation of the information. For T observations and M observations for initialization, that output is:
The Recursive analysis command generates recursive estimates of the model and graphs the output.
Estimation starts with the initial number of observations at the end of the sample and increases the sample size towards the start gradually by one.
Estimation starts with the initial number of observations at the beginning of the sample and increases the sample size gradually by one.
As in the forward recursion, the estimation starts with the initial number of observations at the beginning of the sample. While the estimation window moves forward, the sample size is kept constant.
Enter the number of observations you wish to use for initializing the recursive estimation.
If the model contains many variables it is recommended to split up the output into three graphics windows:
Recursive least squares is OLS where coefficients are estimated sequentially, and is a powerful tool for investigating parameter constancy.
Recursive instrumental variables operates like Recursive least squares but using Instrumental variables methods.
The sample starts from a minimal number of observations N = K variables and statistics are recalculated adding observations one at a time.
The recursive output is analyzed graphically.
The graphical output of the recursive analysis option. For T observations and M initial values consists of:
Beta coefficients (±2SE) for all variables of the model; the graphs are centered on b^i with the approximate 95%confidence interval at each observation shown on either side.
The `t-statistic' = b^i/SEi for the ith coefficient.
Showing RSS at each t, based on the OLS/IVE residuals:
| RSS=ås=1tvs2 where vs=ys-xs'b^t. |
Plots ût=yt-xt'b^ with twice the equation standard error at each t on either side of zero. This will reveal any model deficiencies.
Graphs Chow test statistics for a break at each t.
Shows the marginal rejection probability of the Chow test statistic.
The Chow statistics are only shown in case of forward OLS recursions, owing to endogenous regressors in the case of IVE.
After estimation, the dynamic properties of models like (1) as defined by the Model Formulation can be analyzed.
PcGets produces the following output for dynamic models:
Produces a table of lag coefficients for every variable.
| B0(L)yt=c+B1(L)x1,t+B2(L)x2,t+...+Bik(L)xk,t+et, ~t=1,...,T, |
where
| Bi(L)=bi,0+bi,1L+bi,2L2+...+bi,nLn. |
If the roots of B0(L) lie outside the unit circle, we can rewrite (1) as (forgetting about c and et):
| yt=åi=1kHi(L)xi,t, where Hi(L)=Bi(L)/B0(L). |
If E[x] has remained at a constant level x for long enough, y will reach its long-run solution:
| E[ y] =åi=1kHi(1)E[ xi] , where Hi(1)=Bi(1)/B0(1). |
(reported with asymptotic standard errors).
A Wald test is reported which tests the joint significance of all the variables (excluding the constant) in
the long-run solution.
Reports the eigenvalues of the autoregressive dynamics (inverses of the roots of the autoregressive lag polynomial), and the roots of the distributed lags (if a given variable enters the model more than once).
The second-moment matrix MXX=T-1X'X, and the matrix of correlations:
| rij= |
| , |
are reported, together with the eigenvalues of the former, which should all be real and positive. §13.5 describes the theory.
Generates `dynamic' or `static' forecasts optionally with standard-error bars, bands or fans (±2 forecast standard errors). These are, however, conditional on the future values of all unmodelled variables: PcGive provides system procedures that endogenize all variables.
By default, this displays the last feasible forecast. If there are unmodelled variables in the model, forecasting is only possible while data is available. By default, the final date of the database is reported. If there are exogenous variables in the model, forecasting is only possible while data is available.
Forecast horizon
Generate the sequence of 1-step ahead forecasts using realized lagged endogenous variables.
Generate the sequence of 1,2,3,...H-step ahead forecasts.
Reporting
Write the information (the forecast, its standard error, the actual value, the forecast error and its t-value) to the Results window.
Produce graphical representation of the results.
Calculates forecasts for the cumuland of the modelled variable. For example, when a variable is modelled in differences, it produces forecasts of the level of the variable:
| yt+h=yt+åi=1hDyt+i. |
The output is normalized to zero at the time when the forecasts are made. Forecast-error standard errors will be computed analytically.
By default all observations are included from the pre-forecasting sample.
Forecast standard errors
For OLS/GETS, comprehensive h-step ahead forecasts are produced. For IVE/GETSIVE, the results have to be interpreted with caution since there are endogenous regressor variables. If required, PcGets will compute analytical standard errors of dynamic forecasts.
This tests if some variables of the GUM which have been deleted by Gets should be added to the model.
If the GUM is:
| y=Xb+Zg+v, |
and the estimated model is:
| y=Xb+u, |
then the omitted variables test, tests for gi=0 in
| y=Xb+Zigi+w. |
The Lagrange-multiplier F-statistic for single and joint tests is reported, and the null hypothesis is rejected when its value is significant at the preset level.
The Linear restrictions command allows you to impose restrictions on explanatory variables and test whether they are jointly significant. The general form of the test for linear restrictions is:
| ( Rb=r) |
Restrictions are given in the form of a matrix R, and a vector r. These are entered as one matrix ( R:r) in the Matrix Editor.
For example, the two restrictions a1=1 and a3=-a2 in
| mpt=a0+a1mpt-1+a2yt+a3yt-1 |
can be expressed as:
| R=( |
| ) ; r=( |
| ) . |
The null-hypothesis H0: Rb=r is rejected if we observe a significant test statistic. The matrix entered into PcGets is:
| ( R:r) =( |
| ) . |
Tests for linear restrictions are specified in the form of a matrix R, and a vector r. These are entered as one matrix (R:r) in the dialog.
For example, if the model is mp on Constant, mp_1, y, y_1, and we wish to test that the coefficients on y and y_1 add up to one, and that on mp_1 equals zero. Then the (R:r) matrix can be written as
| ( R:r) =( |
| ) . |
The first four columns are the columns of R, specifying two restrictions. The last column is r, which specifies what the restrictions should add up to.
The dimensions of the matrix must be specified in the rows and columns fields. It is the user's responsibility to specify the correct values: PcGets will not try to work it out (because elements of a row may be spread over several lines).
The number of rows in the matrix.
The number of columns in the matrix.
This window is a basic text editor in which you can edit a matrix file. Here you can enter the (R:r) matrix as in the above example.
This could be useful for creating an initial matrix. Select variables in the model box (this is a this multiple-selection list box), and press this button to specify the (R:r) matrix which corresponds to the restriction that each selected variable has a coefficient of zero (so generates one row for each selected variable)
Enables you to load an existing matrix file into the editor. Any existing matrix in the editor will be lost.
Enables you to save the contents of the editor in an matrix file, so that it can be used again.
The dialog box allows you to select explanatory variables and test whether they are jointly significant, so is a more general form of the test for linear restrictions.
Mark all the variables you wish to include in the test in this Multiple-Selection List box.
PcGets tests whether the selected variables can be deleted from the model.
The resulting output can be pasted to a LaTeX document: The GUM, the model selected by GETS/GETSIVE and the long-run equilibrium are reported in equation form. The tabulated estimation statistics and diagnostic tests complete the output.
Reports the recent settings of PcGets.
Generates batch code to re-estimate the GUM.
Generates batch code to re-estimate the Specific model.
Generates batch code to analyze the (specific) model with PcGive.
Generates batch code to prepare for system estimation by PcGive.
The Store in database command allows you to store residuals or fitted values, or recursive outcomes etc., in the current database.
Allows you to save in the database
Following the successful generation of forecasts, you can also save:
Following the detection of outliers during GETS/GETSIVE:
GiveWin will prompt for a variable name.
The Help topics command describes all the available topics in the Help system.
The Help Index lists the subjects covered in the Help database. You can go directly to a subject from the index by selecting the subject's name.
The About command displays a message that tells you what the application is, namely PcGets.
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