Use this dialog for single equation dynamic model formulation: to formulate a new model, or reformulate an existing 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.
This dialog is for choosing the components of the model.
For every explanatory variable you can choose the type of regression coefficient by selecting the variable (using Tab or Arrow keys and pressing spacebar, or by clicking) and then selecting the type of coefficient from the list that will be shown on the right of the variable, by using Tab or Arrow keys and pressing Enter or by clicking. Press OK when finished. This will take you to the next option or, if you did not select any other option, to Estimation
When this dialog opens, it will show a list of interventions. By default the list will consist of one irregular intervention.
You can change the type of intervention by clicking irregular and selecting a type from the list that will be shown.
You also have the option to include interventions in Level and Slope.
One cell to the right you can select the period in which the intervention takes place. If you wish to include the intervention in the model,
mark the box under Select.
If you wish to fix any of the parameters of the components used in the model, simply mark the box under Fix of the component you wish to fix the parameter for. You can then set the fixed value by selecting the cell under Value which currently contains a default value and then by changing this into the desired value.
Select the sample period and an estimation method.
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, STAMP decides that model A could be nested in model B if the following conditions hold:
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:
Additional dialog items are:
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 STAMP.
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.
This dialog box gives access to a selection of diagnostic testing procedures. Mark the tests you want to be executed, then press OK.
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.
When creating lags, STAMP 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. STAMP handles variables in models through lag polynomials.
Sample periods are automatically adjusted when lags are created.
STAMP 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.
A dynamic equation is specified as an autoregressive-distributed lag model:
In (1), the lag polynomials are defined by:
`Solving' (1) yields:
Zero is a legitimate order for a lag polynomial. Thus, static or dynamic models are equally easily specified.
A model in STAMP is formulated by:
The following information is needed to estimate an equation:
The available single-equation estimators are (see Volume I):
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.
STAMP facilitates a general-to-specific modelling strategy.
Ordinary Least Squares is the standard textbook method. OLS is valid if the data model is congruent.
Congruency
The requirements for congruency are:
STAMP provides tests of most of the aspects of model congruency.
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.
STAMP 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.
STAMP 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 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. .
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
Numerical optimization is used to maximize the likelihood log L(θ) as an unconstrained non-linear function of θ.
STAMP maximization algorithms are based on a Newton scheme:
with
STAMP and STAMP 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.
| | qk,j θk,j | ≤ eps | for all j when θk,j not zero, |
| | qk,j | ≤ eps | for all j when θk,j = 0. |
| | θk+1,j - θk,j | ≤ 10 * eps * | θk,j | | for all j when θk,j not zero, |
| | θk+1,j - θk,j | ≤ 10 * eps | for all j when θk,j = 0. |
The status of the iterative process is given by the following messages:
The chosen default values are: eps1 = 1E-4, eps2 = 5E-3.
You can:
Options 1., 5. and 6 are mainly for teaching optimization.
NOTE: estimation can only continue after convergence.
STAMP has two modes of operation: general-to-specific and unordered.
STAMP monitors the progress of the sequential reduction from the general to the specific and will provide the associated F-tests, Schwarz and σ values.
After estimation, unrestricted general models like (1) in the Dynamic Model Formulation are analysed:
where
If E[x] has remained at a constant level x for long enough, y will reach its long-run solution:
(reported with asymptotic standard errors).
STAMP 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 Chi˛ test.
Dynamic forecasts can be made from single equation models as well as from simultaneous equations system. STAMP will compute analytical standard errors of dynamic forecasts, and can take parameter uncertainty into account.
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:
and standard deviation:
of the variables are also given.
NOTE that the standard deviation here is based on 1/(T-1).
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).
Given observations:
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 STAMP 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 = CσTP. By default, P = -0.2 and C = 1.06 in STAMP. 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.
The correlogram or autocorrelation function (ACF) of a variable, or of the residuals of an estimated model, plots the series of correlation coefficients { rj } between xt and xt-j.
The length s of the ACF is chosen by the user, leading to a figure which shows (r1, r2, ..., rs) plotted against (1,2,..., s).
A related statistic is the Portmanteau (also called Box-Pierce or Q-statistic):
The partial autocorrelation coefficients correct the autocorrelation for the effects of previous lags. So the first partial autocorrelation coefficient equals the first normal autocorrelation coefficient.
A stationary series can be decomposed 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 STAMP 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.
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
For example
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.:
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%).
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.
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).
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:
moments:
(reported as m21/2);
skewness:
excess kurtosis:
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.
If we write the model as
then linear restrictions can be expressed in vector form as:
E.g. the two restrictions: α = 1 and β = -γ in
can be expressed as:
STAMP 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 STAMP:
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).
Given the estimated coefficients θ, and their variance-covariance matrix V[θ], we can test for (non-) linear restrictions of the form:
The null hypothesis Ho: f(θ) = 0 will be tested against Ha: f(θ) ≠ 0 through a Wald test:
where J is the Jacobian of the transformation:
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:
are expressed as
which has to be fed into STAMP as (coefficient numbering starts at 0!):
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).
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
then the omitted variables test, tests for γ = 0 in
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 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:
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
STAMP 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 STAMP was written by Neil Ericsson.
Identities are exact (linear) relations between variables, as in the components of GNP adding up to the total by definition. In STAMP, 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 STAMP automatically derives these by estimating the equation (which must have an R^2 of at least 0.99).
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:
The simultaneous equations modelling process in STAMP starts by focusing on the System, often called the unrestricted reduced form (URF), which can be written as:
where yt, zt 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:
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:
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:
We shall write this as:
or succinctly:
The restricted reduced form (RRF) corresponding to this model is (note that the Π of (5) is a restricted version of that in (3)):
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:
The philosophy behind STAMP 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 STAMP is formulated by:
A model in STAMP is formulated by:
When a model has been formulated, it can be estimated and evaluated, a detailed description of estimators and tests is in Volume II. STAMP facilitates a general-to-simple modelling strategy.
The vector autoregression can be written in equilibrium-correction form as:
| Δyt=( π1+π2-In) yt-1-π2Δyt-1+Φqt+vt, |
or, writing P0=π1+π2-I, and δ1=-π:
|
|
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=∑πi-In 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 vt~INn[0,Ω], and hence is stationary, the n variables in yt 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 yt are I(0), whereas p=0 implies that Δyt is I(0). For 0<p<n there are p cointegrating relations β'yt which are I(0). At this stage, we are not discussing I(2)-ness, other than assuming it is not present.
The approach in STAMP to determining cointegration rank, and the associated cointegrating vectors, is based on the Johansen procedure.
All model estimation methods in STAMP 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:
or using A = (B : C):
with the restricted reduced form (RRF)
(so Π = -inv(B)C). Writing Q' = (Π' : I), we have that AQ = 0, and can write the restricted reduced form as:
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.
STAMP 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:
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:
The database sample can be extended with ease if longer-horizon forecasts are desired.
Seceral formats are available to load and save matrices:
With a closed lag system is meant that there are no gaps in the lag polynomials. So a closed system is e.g.:
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 STAMP will not know that INC1 is a lagged variable; STAMP only recognizes lags when they are created within the model formulation dialog).
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.