PcGive dialogs

Models for cross-section data: Dialogs for Cross-section Regression
Models for discrete data: Dialogs for Binary Discrete Choice Models; Dialogs for Multinomial Discrete Choice Models; Dialogs for Count Models
Models for financial data: Dialogs for GARCH Models
Models for time-series data: Dialogs for Single-equation Dynamic Modelling; Dialogs for Multiple-equation Dynamic Modelling; Dialogs for ARFIMA Models
Models for panel data: Dialogs for Static Panel Methods; Dialogs for Dynamic Panel Methods
Monte Carlo
Other models: Dialogs for Non-linear Modelling; Dialogs for Descriptive Statistics

Dialogs for Cross-section Regression

Dialogs for model formulation and estimation:: Formulate; Estimate; Progress; Options
Dialogs for model evaluation:: Graphic Analysis; Further Output; Test; Exclusion Restrictions; Linear Restrictions; General Restrictions; Omitted Variables; Store in Database

Formulate - Cross-section Regression

Use this dialog for single equation cross-section model formulation: to formulate a new model, or reformulate an existing model.

Database

Mark all the variables you wish to include in the new model or add to the existing model, using the spacebar or the mouse. After you have pressed << (or double-clicked if you are using a mouse), the database variables are added to the model.

The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below.

Specials

The listbox below the database variables shows the so-called special variables, which are pre-defined. Here it is only:

Constant
A constant will be added automatically in a new model but can be deleted.

Selection

This 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:

mark the current dependent variable and right-click to clear its status;
mark the new variable, and right-click to change to Y: endogenous.

If you have marked variables in the model, you can delete them, or assign a status to them.

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

The status drop-down box lists all the available variable types for the current model class. Variables are added to the model using the selected status. To change the status of variables that have been selected into the model, highlight the variable(s), choose a new status and set using the Set button. The status can also be changed by right-clicking on highlighted variables, and using the context menu.

Use default status The default status is used for variables that are added to the model.
Clear status Removes the current status so that the default applies.
Y: endogenous Label the current model selection as endogenous variables.
The first Y variable will be the dependent variable. Additional endogenous variables will be required for instrumental variables estimation. Endogenous variables are preceded by Y in the model list.
Z: regressor Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so a Z variable or unmarked variable are treated in the same way.
A: instrument Label the current model selection as additional instruments.
This button is only relevant if you wish to do an Instrumental Variables estimation. Additional instruments are marked by A in the model list.
Select By Cross-section modelling automatically drops all observations with missing values. This can be refined by marking a variable as Select By.
When a model variable is marked as S variable, only observations for which that variable has non-zero values are included for estimation. When predicting, the default is to use all valid observations which were not used in estimation, but it is also possible to only predict for observations which have a value 2 for the select by variable.

Recall a previous model

Use this to recall a previously estimated model.

OK Press OK to move to model estimation.

Estimate - Cross-section Regression

The estimation method is automatically selected for the formulated model.

estimation sample: Cross-section modelling defaults to using the full sample, while automatically dropping all observations with missing values. It is possible here to specify a subsample for estimation. This can be further refined by adding a select by variable in the model formulation stage.
OK: Pressing OK starts the estimation.

Graphic Analysis - Cross-section Regression

The Graphic analysis command gives various options to graph actual and fitted values, residuals, etc. The list box on the right lists the selected equations for which the graphs are drawn.

Actual and fitted values: Show the fitted and actual values of the dependent variable over time, over the whole sample period, including the forecast period.
Cross-plot of actual and fitted: As above, but now a cross-plot of actual and fitted values.
Residuals (scaled): Show the scaled residuals against time, over the whole sample period, including the forecast period. The residuals are scaled by the residual standard deviation.
Residual density and histogram (kernel estimate): Show the density estimate and histogram of the residuals. The normal density with the same mean and variance is drawn for reference.

To zoom a graph, adjust the area inside OxMetrics.

Test - Cross-section Regression

This dialog box gives access to a selection of diagnostic testing procedures. Mark the tests you want to be executed, then press OK. Many tests report a χ² and an F form. In the summary, only the F-test is reported, which is expected to have better small-sample properties.

Normality: Shows the first four moments, together with a test for normality.
Heteroscedasticity test (squares): Tests for the residuals being heteroscedastic owing to omitting squares of the regressors. Redundant variables (like the square of the Constant) are automatically eliminated. The test will be skipped if there are not enough observations.
Heteroscedasticity test (squares and cross products): This is the White test for heteroscedasticity, which includes all squares (as in the previous heteroscedasticity test) and all cross-products of variables. Redundant variables (like the square of the Constant) are automatically eliminated. The test will be skipped if there are not enough observations (which can happen easily in large models).
Reset test (using squares): The Reset test adds squares of the fitted y (only for OLS).

Dialogs for Single-equation Dynamic Modelling

Dialogs for model formulation and estimation:: Formulate; Model Settings; Estimate; Options; Progress
Dialogs for model evaluation:: Graphic Analysis; Recursive graphics; Dynamic analysis; Forecast; Further Output; Test; Exclusion Restrictions; Linear Restrictions; General Restrictions; Omitted Variables; Store in Database

Formulate - Single-equation Dynamic Modelling

Use this dialog for single equation dynamic model formulation: to formulate a new model, or reformulate an existing model.

Database

The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below.

Special

The listbox below the database variables shows the so-called special variables, which are pre-defined. Here it is:

Constant
A constant will be added automatically in a new model but can be deleted.
Trend
The trend has value 1,2,3,... with value 1 occurring for the first observation in the database. This may be different from the first observation in the estimation sample, for example when using lags (of course, this only affects the value of the constant term).
Seasonal
Seasonal is only present if the database has a non-annual frequency s. Selecting this variable will lead to s - 1 seasonals being added when a Constant is present in the model (s otherwise). For example, for quarterly data, this adds:
Seasonal (1 in quarter 1, zero otherwise),
Seasonal_1 (1 in quarter 2, zero otherwise),
Seasonal_2 (1 in quarter 3, zero otherwise).
CSeasonal
This behaves as Seasonal, except that the variable has zero mean within a year. For quarterly data, for example, CSeasonal has value 0.75 in the first quarter, and -0.25 in the remaining quarters.

Lags

At the top you can choose how the lag length is set with which variables are added to the model:

None: no lags.
Lag: just using the lag specified below.
Lag 0 to: from lag 0 to the lag specified below.

Selection

This 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:

mark the current dependent variable and right-click to clear its status;
mark the new variable, and right-click to change to Y: endogenous.

If you have marked variables in the model, you can delete them, or assign a status to them.

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

Use default status The default status is used for variables that are added to the model.
Clear status Removes the current status so that the default applies.
Y: endogenous Label the current model selection as endogenous variables (this is not possible for lagged variables).
The first Y variable will be the dependent variable. Additional endogenous variables will be required for instrumental variables estimation. Endogenous variables are preceded by Y in the model list.
Z: regressor Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so a Z variable or unmarked variable are treated in the same way.
A: instrument/unrestricted Label the current model selection as additional instruments (Instrumental Variables estimation). Additional instruments are marked by A in the model list.
Otherwise, when using Autometrics, the variable is entered as fixed: it is always forced to enter the model, and not a candidate for removal.

Recall a previous model

Use this to recall a previously estimated model.

OK

Press OK to move to the Model Settings or Estimation.

Model Settings - Single-equation Dynamic Modelling

This dialog is for choosing a model type.

The Model Type

Ordinary Least Squares
Instrumental Variables
This requires more than one endogenous variables, and as least as many additional instruments in the model.
Autoregressive Least Squares
Specify the first and last order of the autoregression of the error term. E.g.: u_t = ρ₂u_t-2+ ρ₃u_t-3+ ρ₄u_t-4+ ε_t
would be from order 2 to 4.

The Autometrics related options are:

Automatic model selection: Mark this box to activate Autometrics.
Target size: Select a target p-value at which the reduction should be run. Use advanced settings for a p-value thst is not listed.
Outlier detection: Select the outlier method: none, large residuals or dummy saturation (adding an impulse dummy for each observation).
Pre-search lag reduction: By default, pre-search lag reduction is switched on.
Advanced Autometrics settings: Mark this box for an additional dialog with advanced options.

Autometrics Settings - Single-equation Dynamic Modelling

Autometrics Settings - Multiple-equation Dynamic Modelling

Search settings

Outlier detection: Select the outlier method: none, large residuals or dummy saturation (adding an impulse dummy for each observation).
Pre-search lag reduction: By default, pre-search lag reduction is switched on.
Pre-search variable reduction: This is switched off by default.
Search effort: Changing the search effort may result in a different terminal model.
Backtesting: The default is backtesting with respect to GUM 0 (in PcGets this was w.r.t the current GUM).
Tie-breaker: When there are multiple terminal candidate models, the tie-breaker decides which one to choose as the final model.
Print level: Controls theamount of Autometrics output.
Target size: Select a target p-value at which the reduction should be run. Select User to specify a p-value directly in the next field.
User determined p-value: Active if User is selected in the previous field.
Diagnostic test p-value: The default is to run the test at 1%, independently of the reduction p-value.
Standard errors: Gives the option to use robust standard errors (HACSE or HCSE) in the reduction.
GIVE: first do reduced form: By default, the system reduction is used on the unrestricted reduced form, after which the IV equation is reduced.

Block identification when there are too many parameters

When k/T fraction exceeds: Determines the ratio of parameters to sample size at which the block reduction kicks in.
Block method: Allows experimentation with other block methods, which may take longer than the default (except for "quick", which does the block search with diagnostic testing switched off).
Maximum block size (-1: unlimited): The block size is roughly 0.5T times the k/T fraction, so 0.4T by default. This is subject to the maximum size specified here.

Diagnostic test set: Allows customization of the test battery used by Autometrics.

Estimate - Single-equation Dynamic Modelling

Select an estimation method, sample period, and number of forecasts for the formulated model. For recursive estimation also select the number of initializations.

Estimation sample

Enter the sample period you wish to use for the estimation (including initialization and forecasts), e.g. 1960(1) to 1980(4). The maximum sample is given one line up.
The default is the sample of the previous estimation (of course only if possible). PcGive automatically excludes observations with missing values.

Less forecasts

Enter the number of observations you wish to withhold for static forecasting.

Estimation method depends on the model settings:

Ordinary least squares,
Two Stage Least Squares for instrumental variables,

2SLS is only available the model has more than one endogenous variable, and at least as many additional instruments as endogenous regressors.

For r-th order Autoregressive Least Squares (RALS) the estimation method is non-linear estimation. An additional checkbox allows for automatic maximization (the default), or maximization through the maximization control dialog, which provides more control over the iterative process.

Recursive estimation:

Select this option to use recursive estimation. Recursive estimation is available with OLS and IV estimation.

Initialization:

Enter the number of observations you wish to use for initializing the recursive estimation.

OK

Pressing OK starts the estimation, unless there still is something missing or wrong in the dialog.

Progress

The Progress dialog is used to review the progress made to date in the model reduction, when using the general-to-specific modelling strategy.

To offer a default sequence, PcGive decides that model A could be nested in model B if the following conditions hold:

model A must have a lower log-likelihood (i.e.~higher RSS),
model A must have fewer parameters,
model A and B must have the same sample period and database.

PcGive does not check if the same variables are involved, because transformations could hide this. As a consequence PcGive does not always get the correct nesting sequence, and it is the user's responsability to ensure nesting.

E.g. DCONS = α + βDINC is nested in: CONS = a + b₁ CONS₁ + b₂ INC + b₃ INC₁ through the restrictions b₁ = 1 and b₃ = -b₂.

There are two options on the dialog to select a nesting sequence:

Mark Specific to General: Marks more general models, finding a nesting sequence with strictly increasing log-likelihood.
Mark General to Specific: Marks all specific models that have a lower log-likelihood.

The default selection is found by first setting the most recent model as specific, and then setting the general model that was found as the general model.

Additional dialog items are:

<: To move a model up in the modelling sequence.
>: To move a model down in the modelling sequence.
Del: Tp permanently delete a model from the modelling sequence.
OK: Prints the progress report, consisting of:
1. number of observations, paramaters, and log-likelihood.
2. Information criteria: reported are the Schwarz Criterion (SC), the Hannan-Quinn (HQ) Criterion, and the Akaike criterion (AIC).
3. F or Chi-squared tests of each reduction.

Options (all)

Controls maximization settings, and what is automatically printed after estimation (in addition to the normal estimation report). Model options referes to settings which are changed infrequently, and are persistent between runs of PcGive.

Maximization Settings

Maximum number of iterations:
Note that it is possible that the maximum number of iterations is reached before convergence. The maximum number of iterations also equals the maximum number of switches in cointegration.

Write results every:
By default no iteration progress is displayed in the results window. It is possible to write intermediate information to the Results window for a more permanent record. A zero (the default) will write nothing, a 1 every iteration, a 2 every other iteration, etc.

Write in compact form:
Writes one line per printed iteration (see Write results every).

Convergence tolerance:
Change the convergence tolerance levels (the smaller, the longer the estimation will take to converge). See under numerical optimization for an explanation of convergence decisions.

Default:
Resets the default maximization settings.

Additional output to be printed after estimation

A range of items can be selected for automatic printing after each estimation. Note that these can always be obtained from the Test menu as well.

Graphic Analysis (single-equation/non-linear/multiple-equation modelling)

The Graphic analysis command gives various options to graph actual and fitted values, forecasts and residuals, etc. The list box on the right lists the selected equations for which the graphs are drawn.

Actual and fitted values: Show the fitted and actual values of the dependent variable over time, over the whole sample period, including the forecast period.
Cross-plot of actual and fitted: As above, but now a cross-plot of actual and fitted values.
Residuals (scaled): Show the scaled residuals against time, over the whole sample period, including the forecast period. The residuals are scaled by the residual standard deviation.
Forecasts and outcomes: Show the static forecasts and actual values of the dependent variable over time, over the forecast period only. This option is only available if observations were withheld for forecasting when the estimation sample was selected.
Residual density and histogram: Show the density estimate and histogram of the residuals. The normal density with the same mean and variance is drawn for reference. To omit any of these items see under Further graphs.
Residual correlogram (ACF): Show the ACF of the residuals, using the lag length supplied in the text entry field.
Length of correlogram and spectrum: The lag length must be < T.
Partial autocorrelation function (PACF): Show the PACF of the residuals, using the lag length supplied in the text entry field.

Further graphs

Forecasts Chow tests: Shows the forecast Chow tests.
Residuals (unscaled): Show the residuals against time, over the whole sample period, including the forecast period.
Residual spectrum: Show the Spectral density of the residuals, using the lag length as the truncation point.
Residual QQ plot against N(0,1): Plots the ordered residuals against in a QQ plot against the normal distribution.
Cross-plots matrix of residuals: Show the cross-plots of the residuals, by default over the whole sample period, including the forecast period. This option is only available for multivariate models.
Residual density
Residual histogram: Show the density estimate of the residuals with the normal density for reference. If residual histogram is also checked, the histogram will be drawn on top of the density.
Residual distribution (normal quantiles): Show the distribution of the residuals in a QQ plot against the normal distribution. This is based on the smoothed density estimate.

Options

Order by equation: Choose an ordering for multiple-equation models.

Cointegration graphics (multiple-equation dynamic modelling)

Cointegration relations: β₀' (y_t;z_r) or β₀' r_1t.
Actual and fitted: The graphs of the cointegrating relations are split into two components: the actuals y_t and the fitted values y_t - β₀' (y_t;z_r). All lines are graphed in deviation from mean.
Components of relations: Graphs all the components of in deviations from their means.
Use (Y:Z) or (Y_1:Z) with lagged DY and U removed: Chooses between y_t;z_r and r_1t.

To zoom a graph, adjust the area inside OxMetrics.

Recursive Graphics: Single-equation Dynamic Modelling

The Recursive graphics command graphs the recursive output as generated by a recursive estimation.

Coefficients: Mark all the variables in the model you wish to include in the beta-coefficient and/or t-value graphs in this list box.
Beta coefficient ±2*S.E.: Graph the beta coefficient ±2*SE of all variables selected in the variables list box.
Beta t-value: Graph the t-values of all variables selected in the variables list box.
Residual Sums of squares: Graph the Residual Sums of Squares.
1-step Residuals ±2*S.E.: Graph the 1-step residuals with with error bands of two residual standard errors around zero.
Standardized innovations: Graph the Standardized innovations.
1-step Chow tests: Graph the 1-step Chow tests scaled by their critical values.
Break-point Chow tests: Graph the N decreasing Chow tests scaled by their critical values.
Forecast Chow tests: Graph the N increasing Chow tests scaled by their critical values.
Chow test p-value: The critical value by which the all the Chow tests need to be scaled. Default is 1%, enter 0 for unscaled chow tests.
Write results instead of graphing: Write the information to the Results window.

To zoom a graph, adjust the area inside OxMetrics.

Dynamic Analysis (single-equation/non-linear modelling)

The formulation and econometrics of dynamic analysis are described in Volume I.

Static long run solution: Determines whether the solved form is computed.
Lag structure analysis: This option gives a table of lag coefficients for every variable, F-tests on the significance of each lag and each variable, as well as the PcGive unit root test.
Roots of lag polynomials: Prints the roots of the lag polynomials
Test for common factor: The common factor test (COMFAC test) evaluates error-autocorrelation claims by checking if the model's lag polynomials have factors in common.

Lag weights

Graph normalized weights: Plot the normalized lag weights.
Graph cumulative normalized weights: Plot cumulative normalized lag weights, either instead of the normalized lag weights, or in addition to them.
Write lag weights: Write the information to the Results window.

Forecast (single-equation/non-linear/multiple-equation modelling)

Shows the dynamic forecasts or static (one-step) forecasts optionally with standard error bars, bands or fans (± 2 forecast standard errors). Dynamic forecasting is not possible for systems with identities (use the model in that case, so that the identities are known to PcGive).

If there are unmodelled variables in the model, forecasting is only possible while data is available.

Equation

Mark all the equations you wish to do the graphing for in this list box. In single-equation modelling there is only one equation, so only one variable listed in the box.

Number of forecasts

By default, this displays the maximum number of dynamic forecasts. If there are unmodelled variables in the model, forecasting is only possible while data is available.

The forecast types

Dynamic forecasts Select this to graph the dynamic forecasts (the sequence of 1,2,3,... H-step forecasts).
h-step forecasts Enter the required number for h in the edit field. Up to h forecasts, the graphs will be identical to the dynamic forecasts.

Forecast standard errors

Do not compute;
Error variance only;
With parameter uncertainty
(i.e., taking both parameter and error variance uncertainty into account).

Options

Type of error bars:

Use error bars
Use error bands
Use error fans

Critical value to use for errors bars

The default is ±2SE corresponding to 95% bands. Use 1.6 for 90% bands.

Number of pre-forecast observations to graph

By default 1 + the data frequency observations are included from the pre-forecasting sample.

Write results instead of graphing

Write the information to the Results window.

Transformations

Derived

Allows the specification for additional (derived) equations to forecast. The derived equations are specified in algebra code. For example, when a variable (CONS say) is in logs, you could add log(CONS). When more than one variable is derived, they must involve assignment, and be terminated by a semicolon, for example: x = exp(CONS); y = x + 2;

Forecast error standard errors will be computed numerically.

Linear derived expressions can also be created using identities.

Further Output

Information criteria: if checked: report information criteria
Heteroscedastic-consistent standard errors (HCSE): When selected the HCSEs will be computed: HCSE, HACSE (heteroscedasticity and autocorrelation consistent standard errors), JHCSE (jackknife HCSE, but only for single equation OLS).
R^2 relative to difference and seasonals: if checked: report R^2 relative to difference and seasonals
Correlation matrix of regressors: Print out the regressor correlation matrix, means and standard deviations.
Covariance matrix of estimated parameters: Print out the covariance matrix of the estimated parameters for each model, and the covariance matrix of constrained parameters following general restrictions.
Reduced form estimates: These can only be printed after instrumental variables or simultaneous equations models.
Static (1-step) forecasts: These can only be printed if observations were withheld at the formulation stage. Forecasts can also be made from the test menu.
Print large residuals: Check this box t list all observations that have an (absolute) residual exceeding if the specified value times the equation standard error (in other words, standardized residual in excess of the specified value).

Write model results

Equation format: write the results in equation format.
LaTeX format: This resulting output can be pasted to a LaTeX document.
Non-linear model format: This resulting output can be useed as a starting point for non-linear modelling.
Significant digits for parameters
Significant digits for std.errors: These control the format of the output.
Batch code to map CVAR to I(0) model: This option is available after estimating a cointegrated VAR. It prints the batch code to map it to a model with the cointegrating vectors as identities. This batch code can then be run from OxMetrics.

Test (single-equation/non-linear/multiple-equation modelling)

This dialog box gives access to a selection of diagnostic testing procedures. Mark the tests you want to be executed, then press OK. Many tests report a Chi^2 and an F form. In the summary, only the F-test is reported, which is expected to have better small-sample properties.

In multiple-equation models, there is a choice to compute vector tests, single-equation tests, or both.

Residual correlogram and Portmanteau statistic with length: Prints the residual correlogram (both ACF and PACF), as well as the Portmanteau statistic. You can change the lag length.
Error autocorrelation from lag .. to ..: Offers the choice of testing for autocorrelation, with the option to change the default starting and ending lag. Information on the auxiliary regression is printed in addition to the Chi^2 and F-form of the test statistic.
Normality: Shows the first four moments, together with a test for normality.
Heteroscedasticity (squares): Tests for the residuals being heteroscedastic owing to omitting squares of the regressors. Redundant variables (like the square of the Constant) are automatically eliminated. The test will be skipped if there are not enough observations.
Heteroscedasticity (squares and cross products): This is the White test for heteroscedasticity, which includes all squares (as in the previous heteroscedasticity test) and all cross-products of variables. Redundant variables (like the square of the Constant) are automatically eliminated. The test will be skipped if there are not enough observations (which can happen easily in large models).
ARCH with order (no vector form): Tests for Autoregressive Conditional Heteroscedasticity, for a user defined order. Information on the auxiliary regression is printed in addition to the F-form of the test statistic.
Reset up to power (only for single-equation OLS): The Reset test adds squares of the fitted y.
Instability tests (only for single-equation OLS): Tests for variance, joint, and coefficient instability.
Encompassing tests (only for single-equation OLS/IVE): Computes encompassing tests for single equation models estimated by OLS or IVE. Model 2 is the current model, and model 1 the previously estimated model (for the current database). These models must have the same dependent variable, and not be nested.

Test Exclusion Restrictions

Allows you to select explanatory variables and test whether they are jointly significant. A more general form is the test for linear restrictions.

Selection: Mark all the variables you wish to include in the test in this list box.

PcGive tests whether the selected variables can be deleted from the model.

Test Linear Restrictions

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. (This is a more general than testing for exclusion restrictions, but not as general as the general restrictions test.)

For example, if the model is CONS on Constant, CONS_1, INC, INC_1, and we wish to test that the coefficients on INC and INC_1 add up to one, and that on CONS_1 equals zero. Then the R:r matrix can be written as

The first four rows are the columns of R, specifying two restrictions. The last row 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.

Matrix: This window is a matrix editor in which you can specify the values, very similar to an OxMetrics database.
Rows: The number of rows in the matrix.
Columns: The number of columns in the matrix.
Load: Enables you to load an existing matrix file into the editor. Any existing matrix in the editor will be lost.
Save: Enables you to save the contents of the editor in an matrix file, so that it can be used again.

General Restrictions

This is the most general form for testing restrictions: restrictions are given in the form of expressions involving the coefficients. The mathematics is explained in tests for general restrictions.

Restrictions have to be entered when testing for parameter restrictions and for imposing parameter constraints for estimation. The syntax is similar to that of algebra, but simpler.

Restrictions code may consist of the following components:
(1) Comment
(2) Constants
(3) Arithmetic operators
These are all identical to algebra. In addition there are:
(4) Parameter references
Parameters are referenced by an ampersand followed by the parameter number. Counting starts at 0, so, for example, &2 is the third parameter of the model. What this parameter is depends on your model.

Ensure that when you enter restrictions through the batch language, you use the right order for the coefficients. In case of IV estimation PcGive will reorder your model so that the endogenous variables come first.

Restrictions for testing are entered in the format: f(c)=0. The following restrictions test the significance of the long-run parameters in an unconstrained model:

(&1 + &2) / (1 - &0) = 0; &3 / (1 - &0) = 0;

Omitted Variables

This implements the omitted variables test, which tests if some variables should be added to the model. For example, if the estimated model is

y = Xβ + u, then the omitted variables test, tests for γ= 0 in y = Xβ + Zγ +v,

The Lagrange Multiplier F-test is reported, and the null hypothesis is rejected when its value is significant.

This test is not available for autoregressive least squares or non-linear models.

Any variable can be selected, as long as it matches the present sample.

Database: Select variables to test as omitted from the model in this list box. Variables already in the model and variables which would reduce the sample size are not allowed and are automatically deleted.
Lag length: Specify the maximum lag length to use (not for cross-section regression).

Store in database

Allows you to save any of the listed items in the OxMetrics database. Note that forecasts must be generated using Test/Forecast before they can be stored.

OxMetrics will prompt for a variable name.

Dialogs for Non-linear Modelling

Dialogs for model formulation and estimation:: Formulate; Model Settings; Estimate; Options; Progress
Dialogs for model evaluation:: Graphic Analysis; Recursive graphics; Further Output; Test; Exclusion Restrictions; Linear Restrictions; General Restrictions; Store in Database

Formulate - Non-linear Modelling

[A] Non-linear least squares

A non-linear model is formulated in Algebra code. The following extensions are used:

parameter references
Parameters are referenced by an ampersand followed by the parameter number.
the numbering does not have to be consecutive, so your model can use,
for example &1, &3 and &4.

The following two variables must be defined for NLS to work:

actual defines the actual values of the dependent variable (the y variable, e.g. CONS).
fitted defines the fitted values (the y-hat variable). This is the formula for the explained component.

Consider, for example, the following specification of the fitted part:

fitted = &0*lag(CONS,1) + &1*INC + &3*INFLAT + &4;

Starting values are entered in the format: &parameter=value;. For example:

&0 = 0; &1 = 1; &3 = -1; &4 = 1;

Together, these formulate the whole non-linear model, as in the following example:

  actual = CONS;
  fitted = &0 + &1*lag(CONS,1) + &2*INC - &1*&2*lag(INC,1);
  &0 = 0; &1 = 1; &2 = -1;

[B] Maximum likelihood

Maximum likelihood models are defined using the three variables:

actual
fitted
loglik

Both actual and fitted only define the variables being used in the graphic analysis and the residual based tests. The loglik variable defines the function to be maximized. Parameters and starting values are as for NLS.

More information is available under non-linear models.

After estimating a linear model, and before starting non-linear estimation, you can use Test/Further Output to write the linear model in the form of non-linear model code. This could be a good starting point for formulating a non-linear model.

The dialog fields are:

Model: Edit field for formulating the non-linear model as outlined above.
OK: Moves to the estimation dialog.
Load: Loads a file with a non-linear model (as an algebra file: .ALG) from disk.
Save As: Saves the contents of the edit window to disk as an algebra file (.ALG).
Recall: Recalls the most recently estimated non-linear model.
Database drop-down box: Allows changing database, if multiple databases have been loaded into OxMetrics. The selected database is used for the non-linear estimation.
Database: At the bottom of the dialog is a list of all the variables in the database. Double clicking on a variable will paste it to the editor.

Estimate - Non-linear Modelling

Determines the estimation sample for non-linear models.

Estimation sample: Enter the sample period you wish to use for the estimation (including initialization and forecasts), e.g. 1960(1) to 1980(4). The maximum sample is given one line up.
The default is the sample of the previous estimation (of course only if possible). PcGive automatically excludes observations with missing values.
Less forecasts: Enter the number of observations you wish to withhold for static forecasting.
Estimation method: and non-linear models the choices is the estimation method is non-linear estimation. An additional checkbox allows for automatic maximization (the default), or maximization through the maximization control dialog, which provides more control over the iterative process.
Recursive estimation:: Select this option to use recursive estimation. Recursive estimation is available with OLS and IV estimation.
Initialization:: Enter the number of observations you wish to use for initializing the recursive estimation.
OK: Pressing OK starts the estimation, unless there still is something missing or wrong in the dialog.

Maximization Control

This dialog controls the estimation of non-linear models by numerical optimization.

Optimization status

Initially, the first line says: No convergence (yet) Subsequently, the first line shows the current convergence status:

No convergence
Aborted: no convergence
Function evaluation failed: no convergence
Failed to improve in line search: weak convergence
Failed to improve in line search: no convergence
Strong convergence
Maximum number of iterations reached: no convergence

Only after convergence will it be possible to press OK, which results in writing the estimation results. Note that grid plotting, and resetting parameter values, will result in loss of the previous convergence status.

Parameters

Make a selection and double-click: you can then edit the parameter value.

Optimization method

Lists the choice of optimization method (which could only be one).

Estimate

Start or continue the Numerical Optimization from the current parameter values.

Reset

Reset the parameters to the values they had when this dialog started.

Grid

Sparks off the grid dialog enabling grid searches over a single parameter.

OK

If the optimization process has converged, you will be able to press this button (then PcGive will write the estimation results), otherwise it is deactivated.

Cancel

Cancels the iterative model estimation, no model results will be available.

Options

Allows setting the estimation options, in particular the maximum number of iterations, the amount of intermediate output, and the convergence tolerance. See under numerical optimization for an explanation of convergence decisions.

Grid (all)

There are two types of grids:

Max: maximize over remaining parameters;
Fixed: keep all remaining parameters fixed.

When the grid is over one parameter, the max grid does a complete log-likelihood maximization over the remaining parameters, with the first fixed at the grid values. The fixed grid only involves likelihood evaluations, keeping the other parameters fixed at its current value, while computing the first over the grid coordinates. Therefore, the max grid method can be much slower, especially for a bivariate grid. For example, a 20 by 20 grid would require 400 likelihood maximizations (i.e.,~400 FIML estimates).

As long as you press Next Grid, this dialog keeps on asking for the next graph. When you press OK, the grids are drawn. So you can have many grids on-screen. These can include mixes of different parameters or the same parameter on different scales and/or locations.

3D grid: Check this to do a three-dimensional grid. A second column of grid parameters will appear to select the other parameter.
Parameter: Select the parameter over which you wish to do the grid search.
Grid center: The value on which the grid should be centred, the default is the current parameter value.
Number of steps: The number of steps over which the grid should be computed, with a default of 20.
Step length: The step length default is 0.1 which, together with the other defaults, gives: 20 * 0.1 equal to 2, which spans the range [-1,1]. For example, when the grid id centred at 0, the grid points are -1, -0.9,.., 0.9, 1.
Write grid values: This will write the function values to the results window.
Maximize over remaining parameters: Check this box to do the full maximization.

Recursive graphics: Non-linear Modelling

The Recursive graphics command graphs the output as generated by a recursive estimation.

Residual Sums of squares: Graph the Residual Sums of Squares.
1-step Residuals ±2*S.E.: Graph the 1-step residuals with with error bands of two residual standard errors around zero.
Log-likelihood/T (full sample): Graph the log-likelihood.
1-step Chow tests: Graph the 1-step Chow tests scaled by their critical values.
Break-point Chow tests: Graph the N decreasing Chow tests scaled by their critical values.
Forecast Chow tests: Graph the N increasing Chow tests scaled by their critical values.
Chow test p-value: The critical value by which the all the Chow tests need to be scaled. Default is 1%, enter 0 for unscaled chow tests.
Write results instead of graphing: Write the information to the Results window.

To zoom a graph, adjust the area inside OxMetrics.

Dialogs for Multiple-equation Dynamic Modelling

Dialogs for model formulation and estimation:: Formulate; Model Settings; Equations (for simultaneous equations); Cointegrated VAR Settings (for cointegrated VAR); Restrictions for Cointegration (for cointegrated VAR with general restrictions); Restrictions for CFIML (for CFIML); Estimate; Options; Progress
Dialogs for model evaluation:: Graphic Analysis; Recursive graphics; Dynamic analysis; Forecast; Simulation and Impulse Responses; Further Output; Test; Exclusion Restrictions; Linear Restrictions; General Restrictions; Omitted Variables; Store in Database

Formulate - Multiple-equation Dynamic Modelling

Use this dialog for Dynamic System Formulation: to formulate a vector autoregression (perhaps for cointegration analysis) or unrestricted reduced form of a simultaneous equations model, or reformulate an existing system.

Database

Variables that are not lagged (and not special variables) will by default become the endogenous (Y) variables. To select a different dependent variable, see below.

Specials

Constant
A constant will be added automatically in a new model but can be deleted.
Trend
The trend has value 1,2,3,... with value 1 occurring for the first observation in the database. This may be different from the first observation in the estimation sample, for example when using lags (of course, this only affects the value of the constant term).
Seasonal
Seasonal is only present if the database has a non-annual frequency s. Selecting this variable will lead to s - 1 seasonals being added when a Constant is present in the model (s otherwise). For example, for quarterly data, this adds:
Seasonal (1 in quarter 1, zero otherwise),
Seasonal_1 (1 in quarter 2, zero otherwise),
Seasonal_2 (1 in quarter 3, zero otherwise).
CSeasonal
This behaves as Seasonal, except that the variable has zero mean within a year. For quarterly data, for example, CSeasonal has value 0.75 in the first quarter, and -0.25 in the remaining quarters.

Lags

At the top you can choose how the lag length is set with which variables are added to the model:

None: no lags.
Lag: just using the lag specified below.
Lag 0 to: from lag 0 to the lag specified below.

Selection

This list box shows the current model. By default, the unlagged variables are added as endogenous variables. If you have marked variables in the model, you can delete them, or assign a different status to them. Variables marked with an Y are the endogenous variables. Those with an I are identity endogenous, those with a U are unrestricted, (i.e. partialled out prior to estimation), unmarked variables or Z variables are 'exogenous' (unmodelled).

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

Use default status The default status is used for variables that are added to the model.
Clear status Removes the current status so that the default applies.
Y: endogenous Label the current model selection as endogenous variables (this is not possible for lagged variables). Endogenous variables are preceded by Y in the model list.
Z: regressor Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so a Z variable or unmarked variable are treated in the same way.
I: Identity Label the current system selection as identity endogenous variables. Identities are preceded by I in the model list.
U: Unrestricted Label the current system selection as unrestricted. Such variables are preceded by U in the system list.

Recall a previous model

Use this to recall a previously estimated model.

OK

Press OK to move to the Model Settings or Estimation.

Model Settings - Multiple-equation Dynamic Modelling

This dialog is for choosing a model type for Dynamic System analysis.

Model type

Unrestricted system
Cointegrated VAR
Simultaneous equations model
Constrained simultaneous equations model

Cointegrated VAR settings: Multiple-equation Dynamic Modelling

This dialog specifies the restrictions for the cointegrated VAR. The rank of the long-run matrix can be set to the desired value, and further restrictions on alpha or beta can be imposed.

Cointegrating rank: Specify the rank of the long-run matrix.
Additional long-run restrictions: If required, choose a method for imposing additional restrictions.
Recursive estimation: Re-estimation of the short run during recursive estimation mimicks the cointegration procedure as it would be applied to a shorter sample. However, partialling out the short run estimated at the full sample leads to faster recursive estimation.

Restrictions for Cointegration - Multiple-equation Dynamic Modelling

General restrictions on α and β' may be expressed directly as a function of the unrestricted elements of α, β'. It allows general (non-linear) within and cross equation restrictions on the cointegration vectors β, as well as on the feedback coefficients α, and including imposing links between these. An important aim is to uniquely identify the parameters of the long-run relationships.

Using rank two and three variables, the elements are referenced as, for alpha:

&0 &1 &2 &3 &4 &5

and for beta:

&6 &7 &8 &9 &10 &11

Three examples of general restrictions are:

ex. 1: &0 = 0; &1 = 0;
ex. 2: &7 = -&6; &10 = -&9;
ex. 3: &6 = 1; &7 = -1; &8 = -1;

Identification of the cointegrating space is checked prior to estimation; afterwards, the degrees of freedom for the test statistic are properly computed if any restrictions are imposed.

Note that:

normalization restrictions on each beta vector have to be imposed explicitly.
A specification can impose restrictions, yet not identify all cointegrating vectors.

The standard errors of α are printed; if the specification is identifying, those of β are also printed. A chi-squared test of the overidentifying restrictions is reported. It is important to realise that not all impositions of fixed parameter values, or of relations between parameters, entail testable restrictions. The simplest example is when there is only one cointegrating vector: normalizing the first element does not impose a restriction, although it does fix the scale of the vector.

More generally, deriving the degrees of freedom involved in the chi-squared test is not straightforward, especially when α restrictions are involved, see Boswijk, H. P., and Doornik, J. A. (2004). "Identifying, estimating and testing restricted cointegrated systems: An overview" Statistica Neerlandica, 58, 440--465.

The following situations may occur:

some cointegrating vectors are identified, but others are not;
although restrictions have been imposed, these are just rotations, not affecting the likelihood;
restrictions have been imposed, but no identification achieved.

Assume that the identifying restrictions are imposed on β. In the unrestricted case of rank p, there are np parameters in α and np-pp in β. Restrictions on β are only binding if they cannot be `absorbed' by the αs, and vice versa. This is easily seen for rank n: restricting β'= I_n results in α = P_o, whereas imposing α = I_n gives β' = P_o. Nevertheless, setting α = 0 imposes n² restrictions (which, of course, violate cointegration). Fixing a row of β' constrains only n-p parameters, as the first p may be absorbed.

Finally, some forms of constraint on α and β can induce a failure of identification of the other under the null, in which case the tests need not have chi-squared-distributions (see, for example, Toda, H.Y. and Phillips, P.C.B. (1993), "Vector Autoregressions and Causality", Econometrica, 61, 1367--1393).

The dialog fields are:

Restrictions: Edit field for formulating the restrictions as outlined above.
OK: Moves to the estimation dialog.
Load: Loads a file with a non-linear model (as an algebra file: .ALG) from disk.
Save As: Saves the contents of the edit window to disk as an algebra file (.ALG).
Recall: Recalls the most recently estimated non-linear model.
Parameters: At the bottom of the dialog is a list of all the parameters in the model.

Restrictions for CFIML - Multiple-equation Dynamic Modelling

Parameter constraints for CFIML are written in the format: θ^*=g(θ);. First consider an example which restricts parameter 0 as a function of three other parameters, creating a model which is non-linear in the parameters:

    &0 = -(&1 - &2) * &3;
    &4 = 0;

The dialog fields are:

Restrictions: Edit field for formulating the restrictions as outlined above.
OK: Moves to the estimation dialog.
Load: Loads a file with a non-linear model (as an algebra file: .ALG) from disk.
Save As: Saves the contents of the edit window to disk as an algebra file (.ALG).
Recall: Recalls the most recently estimated non-linear model.
Parameters: At the bottom of the dialog is a list of all the parameters in the model.

Equations - Multiple-equation Dynamic Modelling

Use this dialog to formulate the simultaneous equations for a dynamic system.

Select from: Shows the current unrestricted reduced form (URF) from which the variables for each equation can be selected.
Mark all the variables you wish to add to the current equation, using the spacebar or the mouse. After you have pressed << (or double-clicked if you are using a mouse), the variables are added to the equation.
Equations: At the top of the left-hand side is a drop-down box allowing you to select an equation. You can also change to the next (or previous) equation using the spin buttons.
The list box shows the specification of the current equation.
<<: Adds the currently selected URF variable to the equation.
>>: Deletes the currently selected variables from the equation.
Clear>>: Deletes all variables from the current equation.
<<Default All: Resets all equations to their default, the unrestricted reduced form.
OK: Press OK to move to the Estimation.

Estimate -Multiple-equation Dynamic Modelling

The Estimate command provides dynamic model estimation.

Select an estimation method, sample period, and number of forecasts for the formulated model. For recursive methods also select the number of initializations.

Estimation sample

Less forecasts

Enter the number of observations you wish to withhold for static forecasting.

Estimation method

The estimation method depends on thge model type. For an unrestricted system it is only OLS, for a cointegrated VAR it is reduced rank regression, while for simultaneous equations models it is one of:

Full Information Maximum Likelihood (FIML),
Three-stage Least Squares (3SLS),
Two-stage Least Squares (2SLS),
Equation by Equation OLS (1SLS).

Only the first option is available for constrained simultaneous equations estimation.

An additional checkbox allows for automatic maximization (the default) when numerical optimization is required. Switch automatic maximization off to access the maximization control dialog, which provides more control over the iterative process.

Recursive estimation, initialization:

Select this option and enter the number of observations you wish to use for initializing the recursive estimation.

OK

Pressing OK starts the estimation, unless there still is something missing or wrong in the dialog.

Recursive graphics - Multiple-equation Dynamic Modelling

The Recursive graphics command graphs the output as generated by a recursive estimation.

Equations: Mark all the equations in the model you wish to include in the output.
Residual Sums of squares: Graph the Residual Sums of Squares.
1-step Residuals ±2*S.E.: Graph the 1-step residuals with with error bands of two residual standard errors around zero.
Log-likelihood (non-linear modelling): Graph the log-likelihood.
Log-likelihood/T (full sample) (multiple-equation modelling): Graph the log-likelihood.
1-step Chow tests: Graph the 1-step Chow tests scaled by their critical values.
Break-point Chow tests: Graph the N decreasing Chow tests scaled by their critical values.
Forecast Chow tests: Graph the N increasing Chow tests scaled by their critical values.
Chow test p-value: The critical value by which the all the Chow tests need to be scaled. Default is 1%, enter 0 for unscaled chow tests.
Write results instead of graphing: Write the information to the Results window.

For cointegrated VARs the entries are:

Eigenvalues: Recursively estimated eigenvalues from cointegration tests
Beta coefficients: The coefficients in the cointegrating vectors.
Log-likelihood/T (full sample) (multiple-equation modelling): Graph the log-likelihood.
Test for restrictions: The recursive tests for restrictions on the cointegration space.
Test p-value (%) =: The critical value for the tests.
Write results instead of graphing: Write the information to the Results window.

To zoom a graph, adjust the area inside OxMetrics.

Dynamic analysis - Multiple-equation Dynamic Modelling

The formulation and econometrics of dynamic analysis are described in Volume II.

Static long run: Determines whether the solved form is computed.
Roots of companion matrix: Prints the roots of the companion matrix.
Plot roots of companion matrix: Plots the roots of the companion matrix.
I(1) cointegration analysis: Performs the I(1) cointegration tests.
I(2) cointegration analysis: Performs the I(2) cointegration tests.

Simulation and Impulse Responses - Multiple-equation Dynamic Modelling

Use this dialog to perform dynamic simulation and impulse response analysis. Dynamic simulation is performed in the same way as dynamic forecasting, but starts at a point within the estimation sample, rather than immediately afterwards.

Note that dynamic simulation is not a valid technique for model evaluation, as discussed in Y. Chong and D.F. Hendry (1986), "Econometric Evaluation of Linear Macro-Economic Models", Review of Economic Studies, 53, 671--690.

Impulse response analysis ignores non-modelled variables and sets the history to zero, apart from the initial values. These can be taken as unity for each endogenous variable in turn, the equation standard error, orthogonal (based on the Choleski decomposition of the residual covariance matrix), or set by the user. By default this leads to n*n graphs.

Using unit or standard error initial values will only have the initial value of the i-th endogenous variable non-zero in the i-th set of graphs. Orthogonal initial values will have initial values up to the i-th endogenous variable non-zero in the i-th set of graphs.

Dialogs for Descriptive Statistics

Variables for Descriptive Statistics

Various types of data descriptions are offered:
  Means, standard deviations, correlations
  Normality tests and descriptive statistics
  Unit root tests

The first step is to select variables for descriptive statistics:

Database: Mark all the variables you wish to include in the descriptive statistics, in this list box, using the spacebar or the mouse.
Lag length: Specify the default lag length to use.
Change Database: Allows changing database, if multiple databases have been loaded into OxMetrics.

Descriptive Statistics

Means, standard deviations and correlations: Writes the means, standard deviations and correlations of all selected variables.
Normality tests and descriptive statistics: Writes the normality test, together with some summary statistics of all selected variables.
Unit root tests: Writes the unit root tests.
Compute unit root tests Tick this option to do unit root tests.
Report summary table only
This will produce a table of ADF tests, dropping one lag at a time. Also reported are the t-value and significance of the highest lag, and the p-value of the F-test on the lags dropped up to that point.
Lag length for differences
Enter the lag length you wish to use in the augmented Dickey-Fuller test (0 gives the Dickey-Fuller test only).
Constant
Tick this if you wish to include a constant term in the test.
Trend (and constant)
Tick this if you wish to include a trend and constant term in the test.
Add seasonals and constant
Tick this if you wish to include seasonals in the test (a constant term is also added).
Select sample: Allows selecting a subsample. The default is the full sample.

Dialogs for ARFIMA Models

Dialogs for model formulation and estimation:: Estimate; Formulate; Model Settings; Options; Progress
Dialogs for model evaluation:: Graphic Analysis; Forecast; Test; Exclusion Restrictions; Linear Restrictions; Store in database

Formulate - ARFIMA Models

Use this dialog for to formulate a new model, or reformulate an existing model.

Database

Mark all the variables you wish to include in the new model or add to the existing model, in this list box, using the spacebar or the mouse. After you have pressed << (or double-clicked if you are using a mouse), the database variables are added to the model with the default lag length.

The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below.

Special

The listbox below the database variables shows the so-called special variables, which are pre-defined. Here it is:

Constant
A constant will be added automatically in a new model but can be deleted.
Trend
The trend has value 1,2,3,... with value 1 occurring for the first observation in the database. This may be different from the first observation in the estimation sample, for example when using lags (of course, this only affects the value of the constant term).
Seasonal
Seasonal is only present if the database has a non-annual frequency s. Selecting this variable will lead to s - 1 seasonals being added when a Constant is present in the model (s otherwise). For example, for quarterly data, this adds:
Seasonal (1 in quarter 1, zero otherwise),
Seasonal_1 (1 in quarter 2, zero otherwise),
Seasonal_2 (1 in quarter 3, zero otherwise).
CSeasonal
This behaves as Seasonal, except that the variable has zero mean within a year. For quarterly data, for example, CSeasonal has value 0.75 in the first quarter, and -0.25 in the remaining quarters.

Lags

At the top you can choose how the lag length is set with which variables are added to the model:

None: no lags.
Lag: just using the lag specified below.
Lag 0 to: from lag 0 to the lag specified below.

Selection

This 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:

mark the current dependent variable and right-click to clear its status;
mark the new variable, and right-click to change to Y: endogenous.

If you have marked variables in the model, you can delete them, or assign a status to them.

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

Use default status The default status is used for variables that are added to the model.
Clear status Removes the current status so that the default applies (X).
Y: endogenous Label the current model selection as endogenous variables (this is not possible for lagged variables). Endogenous variables are preceded by Y in the model list.
X: variable Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so an X variable or unmarked variable are treated in the same way. X variables enter the mean of Y before the AR and fractional lag polynomials are applied.
Z: variable Marks the selected model variables as a Z regressor. Z variables enter the mean of Y after the AR and fractional lag polynomials are applied. Forecasting with Z variables is currently not available.
W: weight Optionally, a variable can be marked for weighted maximum likelihood estimation.

Recall a previous model

Use this to recall a previously estimated model.

Model Settings - ARFIMA Models

This dialog is for choosing an ARFIMA specification.

AR order, MA order: Specify the orders p and q for the ARMA(p,q) process.
Fix AR lags, Fix MA lags: These edit field allow for ARMA parameters to be fixed at zero. For example, when both p and q are set to 4, and both fields are set to
1;2;3
then only the fourth AR and MA coefficients are estimated.
Fractional parameter d: By default d is estimated, but it is possible to fix d at zero or another user-specified value.
Treatment of mean: The default usage is to add a Constant as regressor, and set the treatment of the mean to None. It is also possible to estimate the model in deviations from the sample mean, or to fix the mean to a user-specfied value.

Estimate - ARFIMA Models

Select an estimation method and sample period for the formulated model.

Estimation sample

Less forecasts

Enter the number of observations you wish to withhold from the sample.

Estimation method

Maximum Likelihood
Non-linear Least Squares
Modified Profile Likelihood
Starting values only
NLS with stationarity imposed

PcGive provides three estimation methods, Exact Maximum Likelihood (EML), Modified Profile Likelihood (MPL) and nonlinear least squares (NLS). By definition, EML and MPL impose -1 < d < 0.5. MPL is preferred over EML if the model includes regressor variables and the sample is not very large. NLS allows for d > -0.5 and can be used to estimate models for non-stationary processes directly, without a priori differencing. NLS estimation is usually fast.
Starting values only reports the GPH estimates of d, from the (frequency domain) log periodogram regression, the AR starting values from solving the Yule-Walker equations, and the MA parameters derived from Tunnicliffe-Wilson's method.
NLS with stationarity imposed enforces that the oots are inside the unit circle.

Graphic Analysis - ARFIMA Models

Actual and fitted values: Show the fitted and actual values of the dependent variable over time, over the whole sample period, including the forecast period.
Cross-plot of actual and fitted: As above, but now a cross-plot of actual and fitted values.
Residuals (scaled): Show the scaled residuals against time, over the whole sample period, including the forecast period. The residuals are scaled by the residual standard deviation.
Residual density and histogram: Show the density estimate and histogram of the residuals. The normal density with the same mean and variance is drawn for reference. To omit any of these items see under Further graphs.
Residual correlogram (ACF): Show the ACF of the residuals, using the lag length supplied in the text entry field.
Length of correlogram and spectrum: The lag length must be < T.

Further graphs

Partial autocorrelation function (PACF): Show the PACF of the residuals, using the lag length supplied in the text entry field.
Residuals (unscaled): Show the residuals against time, over the whole sample period, including the forecast period.
Residual spectrum: Show the Spectral density of the residuals, using the lag length as the truncation point.

To zoom a graph adjust the area inside OxMetrics.

Forecast - ARFIMA Models

Shows the dynamic forecasts forecasts optionally with standard error bars, bands or fans (± 2 forecast standard errors).

Number of forecasts: By default, this displays the maximum number of dynamic forecasts. If there are unmodelled variables in the model, forecasting is only possible while data is available.
Naive forecasts only: `Naive' forecasts are derived from the autoregressive representation of the process, truncated at T+h. Naive forecasts are faster to compute.

Undo data transformations

Base level for re-integration: Specify the level from which to re-integrate the series. This is the last observation of the levels (log-levels if growth rates are used) in the estimation sample. For second differences, specify two values separated by a comma (the last two observations in the estimation sample).
Undo logarithm: Click this box to takes exponents of the forecasts.

Options

Type of error bars:

Use error bars
Use error bands
Use error fans

Critical value to use for errors bars

The default is ± 2SE corresponding to 95% bands. Use 1.6 for 90% bands.

Number of pre-forecast observations

By default 1 + the data frequency observations are included from the pre-forecasting sample.

Write results

Write the information to the Results window.

Test - ARFIMA Models

Residual correlogram and Portmanteau statistic with length: Prints the residual correlogram (both ACF and PACF), as well as the Portmanteau statistic. You can change the lag length.
Normality: Shows the first four moments, together with a test for normality.
ARCH with order (no vector form): Tests for Autoregressive Conditional Heteroscedasticity, for a user defined order. Information on the auxiliary regression is printed in addition to the F-form of the test statistic.

Dialogs for GARCH Models

Dialogs for model formulation and estimation:: Estimate; Formulate; Model Settings; Options; Progress
Dialogs for model evaluation:: Graphic Analysis; Forecast; Recursive graphics; Test; Exclusion Restrictions; Linear Restrictions; Store in Database

Formulate - GARCH Models

Use this dialog for to formulate a new model, or reformulate an existing model.

Database

The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below.

Special

The listbox below the database variables shows the so-called special variables, which are pre-defined. Here it is:

Constant
A constant will be added automatically in a new model but can be deleted.
Trend
The trend has value 1,2,3,... with value 1 occurring for the first observation in the database. This may be different from the first observation in the estimation sample, for example when using lags (of course, this only affects the value of the constant term).
Seasonal
Seasonal is only present if the database has a non-annual frequency s. Selecting this variable will lead to s - 1 seasonals being added when a Constant is present in the model (s otherwise). For example, for quarterly data, this adds:
Seasonal (1 in quarter 1, zero otherwise),
Seasonal_1 (1 in quarter 2, zero otherwise),
Seasonal_2 (1 in quarter 3, zero otherwise).
CSeasonal
This behaves as Seasonal, except that the variable has zero mean within a year. For quarterly data, for example, CSeasonal has value 0.75 in the first quarter, and -0.25 in the remaining quarters.

Lags

At the top you can choose how the lag length is set with which variables are added to the model:

None: no lags.
Lag: just using the lag specified below.
Lag 0 to: from lag 0 to the lag specified below.

Selection

This 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:

mark the current dependent variable and right-click to clear its status;
mark the new variable, and right-click to change to Y: endogenous.

If you have marked variables in the model, you can delete them, or assign a status to them.

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model. You can also double click on a model variable to delete it.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

Use default status The default status is used for variables that are added to the model.
Clear status Clears the status of all selected model variables. Cleared variables behave as Z variables.
Y: endogenous Label the current model selection as endogenous variables (this is not possible for lagged variables).
The first Y variable will be the dependent variable. Additional endogenous variables will be required for instrumental variables estimation. Endogenous variables are preceded by Y in the model list.
Z: variable Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so a Z variable or unmarked variable are treated in the same way.
H: X in h_t Label the current model selection as regressors for the variance equation h_t.

Recall a previous model

Use this to recall a previously estimated model.

Model Settings - GARCH Models

This dialog is for choosing a GARCH or EGARCH specification.

GARCH(p,q)

Specify the orders p and q for the GARCH(p,q) process. Set p to zero for an ARCH(q) process.
Set a check in the EGARCH box to estimate an EGARCH model.
Optionally, a non-normal error distribution can be estimated. This is a standardized Student-t distribution for GARCH, and a generalized error distribution (GED) for EGARCH. The Student-t approaches the standard normal as the degrees of freedom go to infinity. The GED(nu) coincides with the normal when nu=2.

GARCH variations

Optionally, a threshold or asymmetry effect can be added to the GARCH model.
Both GARCH and EGARCH can be estimated with the conditional variance (or its square root or logarithm) entering as a regressor in the mean.

GARCH parameter restrictions

An important aspect of GARCH modelling is the choice of parameter space. In PcGive the options are:

Unrestricted estimation
All parameters are unrestricted, except for the intercept in the variance equation, which is forced to be non-negative.
Impose conditional variance >= 0
This imposes the Nelson & Cao conditions to keep the conditional variance positive.
Impose stationarity and alpha+beta >= 0
This imposes coefficients of the ARMA representation of the conditional variance to be positive, and bounds sum of the alpha and beta coefficients to be less than or equal to one (IGARCH boundary).
Impose alpha,beta >= 0
This imposes all coefficients to be non-negative.

Startup of GARCH variance recursion

There are two options for the start-up of the variance recursion: using the sample mean of the variance, or estimate the missing variance terms as extra parameters. The former is the default.

Preferred covariance estimator

Second derivatives
This uses minus the inverse of the numerical second derivative of the log-likelihood function.
Information matrix
The information matrix is computed analytically, but only for standard GARCH models. This is the default.
Outer product of gradients

In addition, the robust standard errors are printed by default when the information matrix J is available. These are of the form inv(JG inv(J), where G is the outer product of the gradients.

Search for global maximum after initial estimation

Especially when q>1, it is possible that the likelihood has multiple optima. This final set of advanced options allows for a search from random starting values. Because each of these involves maximization of the likelihood, this option can be time consuming.

Estimate - GARCH Models

Select an estimation method, and sample period for the formulated model. Optionally select recursive estimation.

Estimation method

Maximum Likelihood

ML is the only available estimation methods.

Estimation sample

Enter the sample period you wish to use for the estimation (including initialization and forecasts), e.g. 1960 1 to 1980 4. The maximum sample is given one line up.
The default is the sample of the previous estimation (of course only if possible). PcGive automatically excludes observations with missing values.

Recursive estimation, initialization:

Select this option and enter the number of observations you wish to use for initializing the recursive estimation. Recursive estimation is available for all GARCH and EGARCH type models, and estimates the model for all sample sizes down to the size specified for initialization.

Less forecasts

Enter the number of observations you wish to withhold from the estimation sample.

Options

Allows setting the estimation options.

OK

Pressing OK starts the estimation, unless there still is something missing or wrong in the dialog.

Options

Maximization Settings

Write in compact form:
Writes one line per printed iteration (see Write results every).

Default:
Resets the default maximization settings.

Graphic Analysis - GARCH Models

Actual and fitted values: Show the fitted and actual values of the dependent variable over time, over the whole sample period, including the forecast period.
Cross-plot of actual and fitted: As above, but now a cross-plot of actual and fitted values.
Residuals (scaled): Show the scaled residuals against time, over the whole sample period, including the forecast period. The residuals are scaled by the residual standard deviation.
Conditional standard deviation: Show the square root of the estimated conditional variance: sqrt(h_t).

Further graphs

Residual density and histogram: Show the density estimate and histogram of the residuals. The normal density with the same mean and variance is drawn for reference. To omit any of these items see under Further graphs.
Residual correlogram (ACF): Show the ACF of the residuals, using the lag length supplied in the text entry field.
Length of correlogram and spectrum: The lag length must be < T.
Partial autocorrelation function (PACF): Show the PACF of the residuals, using the lag length supplied in the text entry field.
Partial autocorrelation function (PACF): Show the PACF of the residuals, using the lag length supplied in the text entry field.
Residuals (unscaled): Show the residuals against time, over the whole sample period, including the forecast period.
Residual spectrum: Show the Spectral density of the residuals, using the lag length as the truncation point.
Residual correlogram of squares (scaled): Shows the ACF of the squared scaled residuals u_t²/h_t, using thelag length supplied in the text entry field.

To zoom a graph adjust the area inside OxMetrics.

Recursive Graphics - GARCH Models

The Recursive graphics command graphs the recursive output as generated by a recursive estimation.

Coefficients: Mark all the variables in the model you wish to include in the beta-coefficient and/or t-value graphs in this list box.
Coefficients ±2*S.E.: Graph the estimated coefficients ±2*SE of all variables selected in the variables list box.
t-value: Graph the t-values of all variables selected in the variables list box.
alpha(1)+beta(1): Graph α(1)+β(1).
Log-likelihood (non-linear modelling): Graph the log-likelihood.
Write results instead of graphing: Write the information to the Results window.

To zoom a graph adjust the area inside OxMetrics.

Forecast - GARCH Models

Shows the dynamic forecasts forecasts optionally with standard error bars, bands or fans (± 2 forecast standard errors).

Number of forecasts: By default, this displays the maximum number of dynamic forecasts. If there are unmodelled variables in the model, forecasting is only possible while data is available.
Conditional variance graph: Allows for the conditional variance to be graphed separately.

Options

Type of error bars:

Use error bars
Use error bands
Use error fans

Critical value to use for errors bars

The default is ±2SE corresponding to 95% bands. Use 1.6 for 90% bands.

Number of pre-forecast observations

By default 1 + the data frequency observations are included from the pre-forecasting sample.

Write results

Write the information to the Results window.

Test - GARCH Models

Residual correlogram and Portmanteau statistic with length: Prints the residual correlogram (both ACF and PACF), as well as the Portmanteau statistic. You can change the lag length.
Normality: Shows the first four moments, together with a test for normality.
ARCH with order (no vector form): Tests for Autoregressive Conditional Heteroscedasticity, for a user defined order. Information on the auxiliary regression is printed in addition to the F-form of the test statistic.

Dialogs for Discrete Choice and Count Models

Dialogs for model formulation and estimation:: Formulate - Binary Discrete Choice; Formulate - Multinomial Discrete Choice; Formulate - Count Data; Model Settings - Discrete Choice; Model Settings - Count Data; Estimate; Options; Progress
Dialogs for model evaluation:: Graphic Analysis; Norm Observation; Outliers; Predictions; Test; Further Output; Exclusion Restrictions; Linear Restrictions; Store in Database

Formulate - Binary Discrete Choice

Use this dialog for to formulate a new model, or reformulate an existing model.

Database

The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below.

Special

The listbox below the database variables shows the so-called special variables, which are pre-defined. Here it is:

Constant
A constant will be added automatically in a new model but can be deleted.

Selection

This 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:

mark the current dependent variable and right-click to clear its status;
mark the new variable, and right-click to change to Y: endogenous.

If you have marked variables in the model, you can delete them, or assign a status to them.

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model. You can also double click on a model variable to delete it.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

Use default status The default status is used for variables that are added to the model.
Clear status Clears the status of all selected model variables. Cleared variables behave as X variables.
Y: endogenous Label the current model selection as endogenous variables.
Normally, there is one dependent variable which holds the binary response 0 or 1 (although 1 and 2 are also allowed). If two Y variables are marked, the data are assumed to be grouped.
X: variable Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so an X variable or unmarked variable are treated in the same way.
S: Select By By default, all valid observations are used for estimation. Select by can be used to estimate over a sub-sample. When a model variable is marked as S variable, only observations which have a non-zero value are included for estimation. When predicting, the default is to use all valid observations which were not used in estimation, but it is also possible to only predict for observations which have a value 2 for the select by variable.

Recall a previous model

Use this to recall a previously estimated model.

Formulate - Multinomial Discrete Choice

Use this dialog for to formulate a new model, or reformulate an existing model.

Database

The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below.

Special

The listbox below the database variables shows the so-called special variables, which are pre-defined. Here it is:

Constant
A constant will be added automatically in a new model but can be deleted.

Selection

This 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:

mark the current dependent variable and right-click to clear its status;
mark the new variable, and right-click to change to Y: endogenous.

If you have marked variables in the model, you can delete them, or assign a status to them.

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model. You can also double click on a model variable to delete it.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

Use default status The default status is used for variables that are added to the model.
Clear status Clears the status of all selected model variables. Cleared variables behave as X variables.
Y: endogenous Label the current model selection as endogenous variables.
Normally, there is one dependent variable which holds the multinomial response 0,1,...,S-1 (although 1,2,...,S are also allowed). If S Y variables are marked, the data are assumed to be grouped or to just hold 0/1 indicators.
X: variable Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so an X variable or unmarked variable are treated in the same way.
Z: variable Marks the selected model variables as an alternative-dependent regressor. Because Z variables are dependent on the alternative, they must be entered in groups of size S at a time (where S is the number of alternatives). A model with only alternative-dependent regressors is sometimes called a conditional logit model.
W: variable Optionally, a variable can be marked for weighted maximum likelihood estimation.
S: Select By By default, all valid observations are used for estimation. Select by can be used to estimate over a sub-sample. When a model variable is marked as S variable, only observations which have a non-zero value are included for estimation. When predicting, the default is to use all valid observations which were not used in estimation, but it is also possible to only predict for observations which have a value 2 for the select by variable.

Recall a previous model

Use this to recall a previously estimated model.

Formulate - Count Data

Use this dialog for to formulate a new model, or reformulate an existing model.

Database

The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below.

Special

The listbox below the database variables shows the so-called special variables, which are pre-defined. Here it is:

Constant
A constant will be added automatically in a new model but can be deleted.

Selection

This 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:

mark the current dependent variable and right-click to clear its status;
mark the new variable, and right-click to change to Y: endogenous.

If you have marked variables in the model, you can delete them, or assign a status to them.

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model. You can also double click on a model variable to delete it.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

Use default status The default status is used for variables that are added to the model.
Clear status Clears the status of all selected model variables. Cleared variables behave as X variables.
Y:endogenous Label the current model selection as the endogenous variable. Only one endogenous variable, holding the counts, is allowed.
X:variable Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so an X variable or unmarked variable are treated in the same way.
S: Select By By default, all valid observations are used for estimation. Select by can be used to estimate over a sub-sample. When a model variable is marked as S variable, only observations which have a non-zero value are included for estimation.

Recall a previous model

Use this to recall a previously estimated model.

Model Settings

This dialog is for choosing a discrete choice model.

The Model Type

Logit
Probit
Only binary probit can be estimated.

Model Settings - Count Data

This dialog is for choosing a count data model.

The Model Type

Poisson
Negative binomial

Truncated above at (0 is untruncated)

Optionally, a truncated Poisson or negative binomial model can be estimated.

Negative binomial type

A type I (k=1) or type II (k=0) negative binomial model can be estimated, or k set directly.

Estimate

Select an estimation method for the formulated model.

Estimation method

Newton's method
BFGS method

Because the multinomial logit and binary probit likelihoods are concave, Newton's method, which uses analytical second derivatives, is the preferred estimation method. Count data models are always estimated by BFGS.

Estimation sample

Cross-section modelling automatically drops all observations with missing values. This can be refined by specifying a select by variable in the model formulation stage.

Options

Allows setting the estimation options.

OK

Pressing OK starts the estimation, unless there still is something missing or wrong in the dialog.

Options

Maximization Settings

Write in compact form:
Writes one line per printed iteration (see Write results every).

Default:
Resets the default maximization settings.

Graphic Analysis

Histograms of probabilities for each state: Plots the histograms of probabilities for each state separately (S histograms).
Histograms of probabilities of observed state: Plots the histograms of probabilities of observed state, for each state separately (S histograms), and all states together.
Number of bars: Sets the number of bars for the above histograms.
Cumulative correct predictions for each state: Plots the cumulative correct predictions, for each state separately (S graphs), and all states together.
Cumulative response for each state (sorted by probability): Plots the cumulative response for each state (S graphs), sorted by probability (with the highest probabilities first).

Predictions

Writes the predictions for the observations that were excluded from estimation or for observations which have a value 2 for the select by variable.

Further output

Allows for the printing of

Summary statistics for explanatory variables
Table of actual and predicted
Derivatives of probabilities at regressor means
Derivatives of probabilities at sample frequencies

Outliers

Writes the observations which have small estimated probabilities for the observed state.

Norm observation

Writes or graphs the probabilites observations for a `norm' observation with specified values for the explanatory variables.

Dialogs for Static and Dynamic Panel Models

Dialogs for model formulation and estimation:: (static and dynamic panel methods); Model Settings (static panel methods); Model Settings (dynamic panel methods); Estimate Model (static panel methods); Estimate Model (dynamic panel methods); Options; Progress
Dialogs for model evaluation:: Graphic Analysis; Dynamic Analysis; Further Output; Exclusion Restrictions; Linear Restrictions; Store in Database; Test

Formulate(static and dynamic panel methods)

Use this dialog for to formulate a new model, or reformulate an existing model.

Database

The variable at the top of the list will by default become the endogenous (Y) variable. To select a different dependent variable, see below. A Constant and other dummy variables can be entered at the next stage.

A year variable must always be added to the model.

Model

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:

mark the current dependent variable and clear its status;
mark the new variable, and press the Y:Endogenous button.

If you have marked variables in the model, you can delete them, or assign a status to them.

Lags

At the top you can choose how the lag length is set with which variables are added to the model:

None: no lags.
Lag: just using the lag specified below.
Lag 0 to: from lag 0 to the lag specified below.

Selection

This 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:

mark the current dependent variable and right-click to clear its status;
mark the new variable, and right-click to change to Y: endogenous.

If you have marked variables in the model, you can delete them, or assign a status to them.

<<

Adds the currently selected database or special variables to the model.

>>

Deletes the currently selected variables from the model. You can also double click on a model variable to delete it.

Clear>>

Deletes the whole model, so that you can start from scratch.

Status

Use default status The default status is used for variables that are added to the model.
Clear status Clears the status of all selected model variables. Cleared variables behave as Z variables.
Y: endogenous Label the current model selection as endogenous variable (there can only be one endogenous variable, which cannot be a lagged variable).
If endogenous regressors are present for instrumental variables estimation, these should be marked X, but not appear as instrument.
X: variable Marks the selected model variables as a normal regressor. This is the default for an unmarked variable, so an X variable or unmarked variable are treated in the same way.
A non-endogenous regressors for instrumental variables estimation can be marked both as X, and appear as instrument (I or L).
I: Instrument Label the current model selection as instruments. These instruments will be transformed along with the regressors (e.g. if the model is in first differences, I instruments will also be used in first differences).
L: Level instrument Label the current model selection as instruments. These instruments will not be transformed (e.g. if the model is in first differences, Level instruments will be used in levels).
R: Year Label the current variable as the year variable. Such a variable must always be defined, and there can only be one year variable.
N: Index An optional variable can be defined which holds the individual indices. If added, that variable will be used to determine the panel structure.
G: Group If group or group interaction dummies are to be used, a group variable must be defined which assigns individuals to groups.

Recall a previous model

Use this to recall a previously estimated model.

Model Settings - Static Panel Methods

This dialog is for further static panel model specification.

Dummies

The following dummy variables can be added to the model:

Constant
Time
Group
Time and Group
Individual

Group and time/group interaction dummies can only be used if a Group variables has been added to the model.

Specification tests

Check this box to include Wald tests for the significance of dummy variables and other regressors.

AR tests up to order

Unlike other PcGive packages, for panel data the order of the AR test must be specified in advanced. By default it is set to two. Set to zero to avoid the computation of the test.

Use robust standard errors

By default standard errors which are robust to heteroscedasticity are reported, and all tests based on the robust variance. This can be switched off, but not that two-step dynamic panel standard errors are particularly unreliable.

Concentrate dummies (not exact with instruments)

When this option is checked, the dependent variable, and all regressors and instruments are used after partialling the dummy variables out. This can help to reduce the dimensionality of the parameter space.

Transform dummies (OLS on differences)

This option is only relevant when estimating with OLS on differences. When selected, the first differences of the dummy variables is used in the model.

Model Settings - Dynamic Panel Methods

This dialog is for further dynamic panel model specification.

Dummies

The following dummy variables can be added to the model:

Constant
Time
Group
Time and Group
Individual

Group and time/group interaction dummies can only be used if a Group variables has been added to the model.

Transformations Select the transformations to applied prior to model estimation:

None
Differences
Othogonal Deviations
Within group
estimation replaces the dependent variable and regressors by deviations from time means (i.e.,subtracting the means of each time series).
Between group
estimation replaces the dependent variable and regressors by the means of each individual (leaving N observations).

Specification tests

Check this box to include Wald tests for the significance of dummy variables and other regressors.

AR tests up to order

Unlike other PcGive packages, for panel data the order of the AR test must be specified in advanced. By default it is set to two. Set to zero to avoid the computation of the test.

Use robust standard errors

Concentrate dummies (not exact with instruments)

Transform dummies (OLS on differences)

This option is only relevant when estimating with OLS on differences. When selected, the first differences of the dummy variables is used in the model.

Print contents of GMM instruments

This option can help understanding the format of the GMM-type instruments that was used in the estimation.

Estimate - Static Panel Methods

Select an estimation method for the formulated model.

Estimation method

OLS (pooled regression)
OLS on differences
LSDV (fixed effects)
Within groups estimation
Between groups estimation
GLS (using within/between)
GLS (using OLS residuals)
Maximum likelihood estimation

Estimation sample

Panel modelling automatically drops all observations with missing values.

Graphic analysis

Actual and fitted values: Show the fitted and actual values of the dependent variable over time, over the whole sample period, including the forecast period.
Cross-plot of actual and fitted: As above, but now a cross-plot of actual and fitted values.
Residuals (scaled): Show the scaled residuals against time, over the whole sample period, including the forecast period. The residuals are scaled by the residual standard deviation.
Residual density and histogram: Show the density estimate and histogram of the residuals. The normal density with the same mean and variance is drawn for reference. To omit any of these items see under Further graphs.
Residual correlogram (ACF): Show the ACF of the residuals, using the lag length supplied in the text entry field.
Length of correlogram and spectrum: The lag length must be < T.

Further graphs

Partial autocorrelation function (PACF): Show the PACF of the residuals, using the lag length supplied in the text entry field.
Residuals (unscaled): Show the residuals against time, over the whole sample period, including the forecast period.
Residual spectrum: Show the Spectral density of the residuals, using the lag length as the truncation point.

To zoom a graph adjust the area inside OxMetrics.

Further output

Covariance matrix of estimated parameters: Print out the covariance matrix of the estimated parameters for each model, and the covariance matrix of constrained parameters following general restrictions.

Write model results

Equation format: write the results in equation format.
LaTeX format: This resulting output can be pasted to a LaTeX document.
Significant digits for parameters
Significant digits for std.errors: These control the format of the output.

Lags

When creating lags, PcGive appends the lag length as extra characters in a name, preceded by an underscore. E.g. CONS_1 is CONS one period lagged.

Lagging a variable leads to the loss of observations, but seasonals can be lagged up to the frequency without loss. PcGive handles variables in models through lag polynomials.

Sample periods are automatically adjusted when lags are created.

PcGive stores the lag information, and uses it to recognize lagged variables for Dynamic Analysis. Lags created this way are not physically created, and do not consume any memory. However, when you compute a lag using the calculator, a new variable will be created in the database, which will NOT be treated as a lagged version of that variable, but as any other variable.

Dynamic model formulation

A dynamic equation is specified as an autoregressive-distributed lag model:

B₀(L) y_t = c + B₁(L) x_1,t + B₂(L) x_2,t + ... + B_k(L) x_k,t + e_t, t = 1,...,T. (1)

In (1), the lag polynomials are defined by:

B_i (L) = Σⁿⁱ_j=mi b_i,j L^j with 0 ≤ mi ≤ ni, i = 1,...,k.

`Solving' (1) yields:

y_t = Σ^k_i=1 H_i (L) x_it, where H_i (L)=B_i (L) / B₀(L).

Zero is a legitimate order for a lag polynomial. Thus, static or dynamic models are equally easily specified.

A model in PcGive is formulated by:

Which variables are involved;
The orders of the lag polynomials;
The status of variables (only when it is not legitimate to treat all regressors as valid conditioning variables, and you wish to use Instrumental Variables).

The following information is needed to estimate an equation:

The model formulation;
The sample period;
Optionally, the number of static forecasts to be withheld for testing parameter constancy;
The method of estimation;
Optionally, the number of observations to be used to initialize the recursive estimation (when available).

The available single-equation estimators are (see Volume I):

Ordinary Least Squares
Instrumental Variables
This requires more than one endogenous variables, and as least as many additional instruments in the model.
Autoregressive Least Squares
Non-linear Least Squares

Single-equation estimation output is discussed in Volume I. Models may be revised interactively after formulation and after estimation. Afterwards, the estimated model can be analyzed.

PcGive facilitates a general-to-specific modelling strategy.

Ordinary Least Squares (OLS)

Ordinary Least Squares is the standard textbook method. OLS is valid if the data model is congruent.

Congruency

The requirements for congruency are:

Homoscedastic innovation errors;
Weakly exogenous regressors;
Constant parameters;
Theory consistency;
Data admissibility;
Encompassing rival models.

PcGive provides tests of most of the aspects of model congruency.

Instrumental Variables (IV)

A structural representation is parsimonious with parameters but has regressors which are correlated with the error term. IV requires that the reduced form is a congruent data model. The Instrumental variables are the reduced form regressors. Instrumental Variables include two stage least squares (2SLS) as a special case.

PcGive needs to know the status of the variables in the model:
1. At least one endogenous variable on the right-hand side;
2. At least as many instruments as endogenous rhs variables.

PcGive computes:
1. The estimate of all the reduced form equations;
2. The estimate of the structural form equation;
3. Tests of the over-identifying restrictions.

Autoregressive least squares (RALS)

Autoregressive least squares requires that the restricted dynamic model is data congruent, where the restrictions correspond to COMFAC constraints selected (since an autoregressive error is a more parsimonious representation). Various orders of autoregression can be selected, and a grid is estimable for single orders.

Multiple optima to the likelihood function commonly occur in the COMFAC class, thus case 5. is recommended. Direct fitting of 4. may not find the optimum. .

RALS numerical optimization

The log-likelihood function f(θ) for RALS is a sum of squares of non-linear terms.

Let the regression and the autoregressive error parameters be β and ρ. Then f(β, ρ) is non-linear but is linear in β given ρ and conversely.

The Gauss-Newton method exploits this fact. It is a reliable choice, but need not find global optima. Like Newton-Raphson, Gauss-Newton uses analytical first and second derivatives. Hendry (1976) reviews alternative methods.

The autoregressive error can be written as

u_t = Σ^r_i=s ρ_i u_t-i + ε_t with ε_t ~ IN(0, σ²).

Numerical optimization

Numerical optimization is used to maximize the likelihood log L(θ) as an unconstrained non-linear function of θ.

PcGive maximization algorithms are based on a Newton scheme:

θ_k+1 = θ_k + s_kQ_k^-1 q_k,

with

θ parameter value at iteration k
s step length, normally unity
Q symmetric positive-defnite matrix (at iteration k)
q first derivative of the loglikelihood (at iteration k) (the score vector)
Δθ_k = θ_k - θ_k-1 is the change in the parameters

PcGive and PcGive use the quasi-Newton method developed by Broyden, Fletcher, Goldfarb, Shanno (BFGS) to update K = Q^-1 directly, estimating the first derivatives numerically.

Owing to numerical problems, it is possible (especially close to the maximum) that the calculated θ does not yield a higher likelihood. Then an s in [0,1] yielding a higher function value is determined by a line search. Theoretically, since the direction is upward, such an s should exist; however, numerically it might be impossible to find one.

The convergence decision is based on two tests:

1. based on likelihood elasticities (dlogLik/dlog|θ|) (scale invariant):

\| q_k,j θ_k,j \| ≤ eps	for all j when θ_k,j not zero,
\| q_k,j \| ≤ eps	for all j when θ_k,j = 0.

2. based on the one-step-ahead relative change in the parameter values (assuming step length 1) (scale variant, but relative change is infinite if any θ = 0)

\| θ_k+1,j - θ_k,j \| ≤ 10 * eps * \| θ_k,j \|	for all j when θ_k,j not zero,
\| θ_k+1,j - θ_k,j \| ≤ 10 * eps	for all j when θ_k,j = 0.

The status of the iterative process is given by the following messages:

No convergence!
Aborted: no convergence!
Function evaluation failed: no convergence!
Maximum number of iterations reached: no convergence!
Failed to improve in line search: no convergence!
s has become too small.
Test 1 was passed, using eps2.
Failed to improve in line search: weak convergence.
s has become too small.
Test 1 was passed, using eps2.
Strong convergence
Both tests were passed, using eps1.

The chosen default values are: eps1 = 1E-4, eps2 = 5E-3.

You can:

Set the initial values of the parameters to zero or the previous values;
Set the maximum number of iterations;
Write iteration output;
Change the convergence tolerances eps1 and eps2;
Care must be exercised with this: the defaults are `fine-tuned': some selections merely show the vital role of sensible choices!
Choose the maximization algorithm;
Plot a grid of the log-likelihood.
The `fineness', number of points and centre can be user-selected. Up to 16 grids can be plotted simultaneously. A grid may reveal potential multiple optima.

Options 1., 5. and 6 are mainly for teaching optimization.

NOTE: estimation can only continue after convergence.

Modelling strategy

PcGive has two modes of operation: general-to-specific and unordered.

General-to-specific: 1. Begin with the dynamic model formulation;
2. Check its data coherence and cointegration;
3. Transform to a set of variables with low intercorrelations but interpretable parameters;
4. Delete unwanted regressors to obtain a parsimonious model;
5. Check the validity of the model by thorough testing.

PcGive monitors the progress of the sequential reduction from the general to the specific and will provide the associated F-tests, Schwarz and σ values.

Unordered Search: Nothing commends unordered searches:
1. No control is offered over the significance level of testing;
2. A `later' reject outcome invalidates all earlier ones;
3. Until a model adequately characterizes the data, standard tests are invalid

Dynamic analysis

After estimation, unrestricted general models like (1) in the Dynamic Model Formulation are analysed:

B₀(L) y_t = c + B₁(L) x_1,t + B₂(L) x_2,t + ... + B_k(L) x_k,t + e_t, t = 1,...,T. (1)

where

B_i (L) = b_i,0 + b_i,1 L + b_i,2 L² + ... + b_i,n Lⁿ.

Static long-run solution

If the roots of B(L) lie outside the unit circle we can rewrite (1) as (forgetting about c and e): y_t = Σ^k_i=1 H_i (L) x_i,t, where H_i (L)=B_i (L) / B₀(L). (2)

If E[x] has remained at a constant level x for long enough, y will reach its long-run solution:

E[y] = Σ^k_i=1 H_i (1) E[x_i], where H_i (1)=B_i (1) / B₀(1). (3)

(reported with asymptotic standard errors).

Static Forecasting

PcGive allows you to retain observations to compute forecast statistics. For OLS/RLS/RALS these are comprehensive 1-step ahead forecasts.

For IV/RIV, since there are endogenous regressor variables, the only interesting issue is that of parameter constancy, and the only output is the forecast Chi² test.

Dynamic forecasts can be made from single equation models as well as from simultaneous equations system. PcGive will compute analytical standard errors of dynamic forecasts, and can take parameter uncertainty into account.

Correlations

The correlation matrix of selected variables is a symmetric matrix, with the diagonal equal to one. Each cell records the simple correlations between the two relevant variables.

The mean:

m = T^-1 Σ^T_i=1 x_i,

and standard deviation:

s = (T-1)^-1 Σ^T_i=1 (x_i - m)²

of the variables are also given.

NOTE that the standard deviation here is based on 1/(T-1).

Data density and histogram

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 ... x_T)

from some unknown probability density function f(X), 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 PcGive based on a kernel estimator. 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.

Owing to the importance of the window width h in estimating the density, the non-parametric density estimation menu offers control over the choice of window width, h = CσT^P. By default, P = -0.2 and C = 1.06 in PcGive. For normal densities this choice will minimize the Integrated Mean Square Error.

For more information see: Silverman B.W. (1986). Density Estimation for Statistics and Data Analysis, London: Chapman and Hall.

Correlogram (ACF, PACF)

The correlogram or autocorrelation function (ACF) of a variable, or of the residuals of an estimated model, plots the series of correlation coefficients { r_j } between x_t and x_t-j.

The length s of the ACF is chosen by the user, leading to a figure which shows (r₁, r₂, ..., r_s) plotted against (1,2,..., s).

A related statistic is the Portmanteau (also called Box-Pierce or Q-statistic):

T Σ^s_j=1 r_j².

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.

Spectrum

A stationary series can be decomposed in cyclical components with different frequencies and amplitudes. The spectral density gives a graphical representation of this. It is symmetric around 0, and only graphed for [0,π] (the horizontal axis in the PcGive graphs is scaled by π, 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.

Diagnostic testing

Test types

Many tests report a Chi^2 and an F form. In the summary, only the F-test is reported, which is expected to have better small-sample properties.

F-tests are usually reported as

F(num,denom) = Value [Probability] /*/**

For example

F(1, 155) = 5.0088 [0.0266] *

where the test statistic has an F distribution with 1 degree of freedom in the numerator, and 155 in the denominator. The observed value is 5.0088, and the probability of getting a value of 5.0088 or larger under this distribution is .0266. This is less than 5% but more than 1%, hence the star.

Significant outcomes at a 1% level are shown by two stars: **.

Chi^2 tests are also reported with probabilities, as e.g.:

Normality Chi^2(2)= 2.1867 [0.3351]

The 5% Chi^2 critical values with 2 degrees of freedom is 5.99, so here normality is not rejected (alternatively, Prob(Chi^2 ³ 2.1867) = 0.3351, which is more than 5%).

Auxiliary regression tests

Many diagnostic tests are done through an auxiliary regression.
In this case two forms of the test are reported:
1. TR^2 which has a Chi^2(r) distribution for r restrictions;
2. (T-k-r)R^2/r(1-R^2), which has an F(r,T-k-r) distribution.
The F-form may be better behaved in small samples.

Autoregressive Conditional Heteroscedasticity (ARCH)

Checks whether the residuals have an ARCH structure: E[ u_t² | u_t-1 , ..., u_t-r ] = Σ^r_i=s α_i u_t-i²,

with [0 ≤ s ≤ r ≤ 12] and e ~ IID(0, τ²). An F-statistic and the αs are reported. The null hypothesis is no ARCH, which would be rejected if the test statistic is too high. This test is done by regressing the squared residuals on a constant and lagged squared residuals (now some observations are lost at the beginning of the sample).

Normality

The Normality test checks whether the variable at hand (either a database variable or the residuals), here called u, are normally distributed as: u_t ~ IN(0,1) with E[u_t³] = 0, and E[u_t⁴] = 3σ².

A Chi^2 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 the 5% level, if a test statistic of more than 5.99 is observed.

Full report includes:
mean:

m = T^-1 Σ^T_i=1 x_i;

moments:

m_j = T^-1 Σ^T_i=1 (x_i - m)^j;

(reported as m₂^1/2);
skewness:

m₃ / m₂^3/2;

excess kurtosis:

m₄ / m₂² - 3.

The reported test statistic has a small-sample correction. Also reported is the asymptotic form of the test (skewness² *T/6 + excess_kurtosis² *T/24), which requires large samples for the asymptotic Chi²(2) distribution to hold.

NOTE that the standard deviation here is based on 1/T.

Tests for linear restrictions

If we write the model as

y = Xβ + u, where y is (T x 1), β is (k x 1) and X is (T x k),

then linear restrictions can be expressed in vector form as:

Rβ = r, where R is a (p x k) matrix, and r a (p x 1) vector.

E.g. the two restrictions: α = 1 and β = -γ in

CONS = b + α CONS₁ + β INC + γ INC₁

can be expressed as:

| 0 1 0 0 | R = | |, r' = [0 1]. | 0 0 1 1 |

PcGive allows you to test general linear restrictions by specifying R and r, in the form of a (p x k+1) matrix [R : r]. Simple linear restrictions of the form α =... = δ = 0 can be done by selecting the relevant variables.

The null-hypothesis Ho: Rβ = r is rejected if we observe a significant test statistic.

Two tests of linear restrictions are routinely reported in PcGive:
1. Ho: b = 0, where the test-statistic is the t-ratio of b.
2. Ho: α = ... = δ = 0 (all coefficients apart from the constant are zero).
Shown as the F-statistic which follows R^² (and can be derived from it).

Tests for general restrictions

Given the estimated coefficients θ, and their variance-covariance matrix V[θ], we can test for (non-) linear restrictions of the form:

f(θ) = 0;

The null hypothesis Ho: f(θ) = 0 will be tested against Ha: f(θ) ≠ 0 through a Wald test:

w = f(θ) ' (JV[θ] J')^-1 f(θ)

where J is the Jacobian of the transformation:

J = ∂ f(θ)/∂q'.

The statistic w evaluated at θ has a Chi^2(r) distribution, where r is the number of restrictions (i.e. equations in f(θ)). The null hypothesis is rejected if we observe a significant test statistic.

E.g. the two restrictions implied by the long-run solution of:

CONS = b + α CONS₁ + β INC + γ INC₁ + δ INFLAT

are expressed as

(β + γ) / (1 - α) = 0;
δ / (1 - α) = 0;

which has to be fed into PcGive as (coefficient numbering starts at 0!):

(&1 + &2) / (1 - &0) = 0; &3 / (1 - &0) = 0;

Common-Factor Test

The COMFAC test evaluates error-autocorrelation claims by checking if the model's lag polynomials have factors in common. If so, the model's lags can be simplified with an autoregressive error; if not, the model cannot be re-expressed with an autoregressive error. Chi^2 tests of each possible common factor and of sequences are shown.

The COMFAC test option is only feasible for unrestricted dynamic models (which have a closed lag system), which are not estimated by Autoregressive Least Squares.

The algorithm was developed and written by Denis Sargan and Juri Sylvestrowicz.

We have recently discovered that the COMFAC test outcome may change if ordering of the variables in the model is changed (but only if there are at least several lag polynomials of the same length). This is due to testing different formulations of the restrictions in the Wald test (i.e. computing determinants of different submatrices).

Omitted variables

This tests if some variables should be added to the model, which can be any variables in the database matching the present sample.

If the estimated model is

y = Xβ + u,

then the omitted variables test, tests for γ = 0 in

y = Xβ + Zγ +v,

The Lagrange Multiplier F-test is reported, and the null hypothesis is rejected when its value is significant.

This test is not available for Autoregressive Least Squares or non-linear models.

Encompassing tests

Encompassing evaluates against rival models to see if they embody specific information excluded from the model under test.

Encompassing tests are only available for single equation models estimated by OLS or IV.

Four tests are calculated:

1. The Cox non-nested hypotheses test (Cox, 1961): This tests whether the adjusted likelihoods of two rival models are compatible. It is equivalent to checking variance encompassing. This test is invalid for IV estimation, and omitted in that case.
2. The Ericsson Instrumental Variables test (Ericsson, 1983): This is an IV equivalent to the Cox test.
3. The Sargan restricted/unrestricted reduced form test (Sargan, 1964): This checks if the restricted reduced form of a structural model encompasses the unrestricted reduced form including exogenous regressors from rival models.
4. The joint model F-test: checks if each model parsimoniously encompasses the linear nesting model.

Invariance

The F-test is invariant to variables in common between the rival models. The Cox and the Ericsson tests are not invariant: their values change with the choice of overlapping variables.

Consult e.g. Ericsson (1983) or Hendry and Richard (1987) for details.

Status of variables

PcGive checks for valid choices of variables:
1. Endogenous variables are matched;
2. Instruments in Model 1 are treated as exogenous in Model 2 even if you denote them as endogenous;
3. The models must be non-nested.

Output

The output is summarized in an encompassing table:
1. The type of test statistic;
2. The value of each outcome;
3. The degrees of freedom of each test;
4. The null that Model 1 is valid is on the left;
5. The null that Model 2 is valid is on the right.

If the left-side tests are insignificant, Model 1 encompasses Model 2.
If the left-side tests are significant, Model 1 fails to encompass Model 2.
Similarly for the rightside tests with models 1 and 2 interchanged.

Model 1 encompasses Model 2 implies Model 1 also parsimoniously encompasses the linear nesting model. If not, Model 2 contains specific data information not captured by Model 1.

The algorithm incorporated in PcGive was written by Neil Ericsson.

Identities

Identities are exact (linear) relations between variables, as in the components of GNP adding up to the total by definition. In PcGive, identities are created by marking identity endogenous variables as such during dynamic system formulation.

Identities are ignored during system estimation/analysis. They come in at the model formulation level, where the identity is specified just like other equations. However, there is no need to specify the coefficients of the identity equation, as PcGive automatically derives these by estimating the equation (which must have an R^2 of at least 0.99).

Unrestricted variables

Variables can be classified as unrestricted during dynamic system formulation. Such variables will be partialled out, prior to estimation, and their coefficients will be reconstructed afterwards. Although unrestricted variables do not affect the basic estimation, there are important differences:

Following estimation:: the R^2 measures and corresponding F-test are relative to the unrestricted variables.
In recursive estimation:: the coefficient of unrestricted variables are fixed at the full sample values.
In cointegration analysis:: unrestricted variables are partialled out together with the short-run dynamics, whereas restricted variables (other then lags of the endogenous variables) are restricted to lie in the cointegrating space.
In simultaneous equations estimation:: unrestricted variables are partialled out prior to estimation. FIML estimation of the smaller model could improve convergence properties of the non-linear estimation process.

Dynamic System and Model Formulation

The simultaneous equations modelling process in PcGive starts by focusing on the System, often called the unrestricted reduced form (URF), which can be written as:

(1) y_t = π₀ + π_i y_t-i + π_j z_t-j + v_t, v_t ~ IN(0,Ω) i = 1,...,m, j = m+1,...,m+r.

where y_t, z_t are respectively (n x 1) and (q x 1) vectors of observations at time t, t = 1,...,T, on the endogenous and non-modelled variables. A more compact way of writing the system is:

(2) y_t = Πw_t + v_t

where w contains z, lags of z and lags of y, and Π is (n x k).

A vector autoregression (VAR) arises when there are no z's (but there could be a constant, seasonals or trend). An example of a 2-equation system is:

CONS = β₀ + β₁ CONS₁ + β₂ INC₁ + β₃ CONS₂ + β₄ INC₂ + β₅ INFL,
INC = β₆ + β₇ CONS₁ + β₈ INC₁ + β₉ CONS₂ + β₁₀ INC₂ + β₁₁ INFL.

This system would be a VAR when β₅ = β₁₁ = 0.

Non-modelled variables can be classified as unrestricted. Variables defined by identities are also allowed.

To obtain a structural dynamic model, premultiply the system (2) by a non-singular matrix B, which yields:

(3) By_t = BΠw_t + Bv_t.

We shall write this as:

(4) By_t + Cw_t = u_t, t = 1,...,T; u_t ~ IN(0,σ),

or succinctly:

Ax_t = u_t

The restricted reduced form (RRF) corresponding to this model is (note that the Π of (5) is a restricted version of that in (3)):

(5) y_t = Πw_t + v_t, with Π = -inv(B)C.

Identification of the model, through within equation restrictions on A, is required for estimation. Some equations of the model could be identities. An example of a model with the previous system as unrestricted reduced form is:

CONS = β₀ + β₁ CONS₁ + β₂ INC + β₃ INFL,
INC = β₄ + β₅ INC₁.

The philosophy behind PcGive is first to develop a congruent system. If the system displays symptoms of mis-specification, there is little point in imposing further restrictions on it. From a congruent system a model is derived.

A system in PcGive is formulated by:

which variables y_t, z_t are involved;
the orders of the lag polynomials;
classification of the ys in endogenous variables and identity endogenous variables;
any non-modelled variable may be classified as unrestricted. Such variables will be partialled out, prior to estimation, and their coefficients will be reconstructed afterwards.

A model in PcGive is formulated by:

which variables enter each equation, including identities;
coefficients of identity equations need not be specified, as PcGive automatically derives these by estimating the equation (requires an R^2 of at least 0.99);
constraints, if the model is going to be estimated by CFIML or RCFIML.

When a model has been formulated, it can be estimated and evaluated, a detailed description of estimators and tests is in Volume II. PcGive facilitates a general-to-simple modelling strategy.

Cointegration Analysis

The vector autoregression can be written in equilibrium-correction form as:

Δy_t=( π₁+π₂-I_n) y_t-1-π₂Δy_t-1+Φq_t+v_t,

or, writing P₀=π₁+π₂-I, and δ₁=-π:

Δy_t=P₀y_t-1+δ₁Δy_t-1+Φq_t+v_t.

(eq:1.1)

Equation (eq:1.1) shows that the matrix P₀ determines how the level of the process y enters the system: for example, when P₀=0, the dynamic evolution does not depend on the levels of any of the variables. This indicates the importance of the rank of P₀ in the analysis. P₀=∑π_i-I_n is the matrix of long-run responses. The statistical hypothesis of cointegration is:

H(p): rank( P₀) ≤p.

Under this hypothesis, P₀ can be written as the product of two matrices:

P₀=αβ',

where α and β have dimension n×p, and vary freely. As suggested by Søren Johansen, such a restriction can be analyzed by maximum likelihood methods.

So, although v_t~IN_n[0,Ω], and hence is stationary, the n variables in y_t need not all be stationary. The rank p of P₀ determines how many linear combinations of variables are I(0). If p=n, all variables in y_t are I(0), whereas p=0 implies that Δy_t is I(0). For 0<p<n there are p cointegrating relations β'y_t which are I(0). At this stage, we are not discussing I(2)-ness, other than assuming it is not present.

The approach in PcGive to determining cointegration rank, and the associated cointegrating vectors, is based on the Johansen procedure.

Estimator Generating Equation

All model estimation methods in PcGive are derived from the Estimator Generating Equation (EGE).

We require the reduced form to be a congruent data model, for which the structural specification is a more parsimonious representation. The structural model is:

BY' + CW' = U',

or using A = (B : C):

AX' = U',

with the restricted reduced form (RRF)

Y'= ΠW' + V'

(so Π = -inv(B)C). Writing Q' = (Π' : I), we have that AQ = 0, and can write the restricted reduced form as:

X'= QW' + V'.

The structural model involves regressors which are correlated with the error term. Instruments (reduced form regressors) are used in place of structural form regressors to estimate the unknown coefficients in A, denoted θ. The general estimation formulation is based on the EGE.

The available estimation methods are described in Volume II. 1SLS applies OLS to each equation, imposing a diagonal errorr varance matrix. This estimator is not consistent for a simultaneous system, but is offered for systems that are large relative to the data available, where its MSE properties may be the best that can be achieved.

Dynamic Forecasting

PcGive allows you to retain observations to compute forecasts and forecast statistics. Both 1-step ahead (static, ex-post) and h-step ahead (dynamic, ex-ante) forecasts are available. The 1-step forecasts are computed automatically after system and model estimation if observations are reserved. Three 1-step test statistics are offered:

Using Ω: This test ignores parameter uncertainty and intercorrelations between forecast errors, thus taking only innovation uncertainty into account.
using V[e]: This test takes parameter uncertainty into account, but ignores intercorrelations between forecast errors.
using V[E] (only for the system): This statistic takes both parameter uncertainty and intercorrelations between forecast errors into account, making it a better calibrated test statistic.

Dynamic forecasts are available separately, up to the end of the database sample period (observations are required for all exogenous variables, but not for endogenous variables and their lags). Dynamic forecasts can be with or without 95% error bars, but only the innovation uncertainty is allowed for in the computed error variances. Two types of forecasts are available for graphing:

Dynamic forecasts
Select this to graph the dynamic forecasts (the sequence of 1, 2, 3,...,h-step forecasts).
h-step forecasts
Up to h forecasts, the graphs will be identical to the dynamic forecasts. Thereafter values of the endogenous variables which go more than h periods back will use actual values.

The database sample can be extended with ease if longer-horizon forecasts are desired.

Matrix File

Seceral formats are available to load and save matrices:

in7, the OxMetrics format.
xls, the Excel format.
csv, the comma-separated spreadsheet format,
mat, a text file with a matrix, preceded by the matrix dimensions.

An example of a matrix file is: +---------+ ¦ 2 3 ¦ <-- dimensions, a 2 by 3 matrix ¦//comment¦ <-- a line of comment ¦ 1 0 0 ¦ <-- first row of the matrix ¦ 0 1 .5 ¦ <-- second row of the matrix +---------+

Closed lag system

With a closed lag system is meant that there are no gaps in the lag polynomials. So a closed system is e.g.:

CONS = b + α CONS₁ + β INC + γ INC₁

however, without INC (i.e. β = 0), it wouldn't be closed. You could then replace INC lagged by INC1 = lag(INC, 1), and close the lag system (because PcGive will not know that INC1 is a lagged variable; PcGive only recognizes lags when they are created within the model formulation dialog).

Missing values

The data sample for analysis is automatically selected to not include any missing values within the sample. In cross-section regression, any observation with missing values is automatically omitted from the analysis, so in-sample observations with missing values are simply skipped.

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