PcGive Volume IV: Interactive Monte Carlo Experimentation in Econometrics using PcNaive

These reference chapters have been taken from Volume IV, and use the same chapter and section numbering as the printed version.

Table of contents

2 The Data Generation Processes and Models of PcNaive

  2.1 AR(1) DGP

  2.1.1 Data generation process

  2.1.2 Model

  2.1.3 Monte Carlo Output

  2.1.4 Live Graphics

  2.2 Static DGP

  2.2.1 Data generation process

  2.2.2 Model

  2.2.3 Monte Carlo Output

  2.2.4 Live Graphics

  2.3 PcNaive and Reduced Form DGP

  2.3.1 Data generation process for PcNaive DGP

  2.3.1.1 PcNaive DGP in equilibrium-correction form

  2.3.1.2 PcNaive DGP with break

  2.3.2 Data generation process for the Reduced Form DGP

  2.3.3 Models

  2.3.4 Monte Carlo Output

  2.3.5 Live Graphics

Chapter 2 The Data Generation Processes and Models of PcNaive

2.1 AR(1) DGP

2.1.1 Data generation process

The AR(1) data generation process is:

where α or μ may be set to zero.

2.1.2 Model

The model is:

which is estimated by OLS. Either the constant term or the lagged dependent variable can be omitted.

The sample size is specified as T₁, T₁+s, T₁+2s,...,T₂, where the step size s may be zero. The Monte Carlo sample size is automatically adjusted if the model includes a lagged dependent variable.

2.1.3 Monte Carlo Output

The following estimates and tests can be investigated

• Coefficients -- β̂₀, β̂₁ (provided they were estimated),
• Standard errors -- SE_β̂0, SE_β̂1 (provided they were estimated),
• Goodness-of-fit -- σ̂², R²,
• t - tests -- t_{β̂0 = 0}, t_{β̂1 = 0} (provided they were estimated),
• AR1 test, DW -- test for AR(1) errors and Durbin--Watson statistic

2.1.4 Live Graphics

• Generated data -- generated data: y_t, t=1,...,T,
• Recursive coefficients and tests -- plots of estimates and tests as a function of sample size,
• Histograms of coefficients and tests -- histograms of the estimates and test statistics,
• Set plot frequency -- determines how often graphs are created:
  the default frequency is M/4, or M when a recursive Monte Carlo is run; it could be that a higher frequency is desired when data are graphed.

2.2 Static DGP

2.2.1 Data generation process

The static data generation process is:

where α₁, α₂, or ρ may be set to zero. The regressors can be kept fixed, or recreated in each replication (stochastic regressors).

2.2.2 Model

The model for the static DGP is:

which is estimated by OLS, and where the constant, z_a,t and/or z_b,t may be omitted. The z_a,t and z_b,t are labelled Za and Zb in the output.

The sample size is specified as T₁, T₁+s, T₁+2s,...,T₂, where the step size s may be zero.

2.2.3 Monte Carlo Output

This is the same as for the AR(1) DGP, for the same tests, and the estimates β̂₀, β̂₁,β̂₂, provided they are included in the model.

2.2.4 Live Graphics

This is the same as for the AR(1) DGP, for the same tests, and the estimates β̂₀, β̂₁,β̂₂, provided they are included in the model.

2.3 PcNaive and Reduced Form DGP

2.3.1 Data generation process for PcNaive DGP

The PcNaive DGP is a data generation process designed for use in (multivariate) dynamic econometric Monte Carlo experiments:

The vectors y_t,u_t,e_t are n×1, so that the coefficient matrices A₀,A₁,B₀,B₁ are n×n, and a₃ is n×1. The z_t vector is q×1, making a₂ n×q, C₀ q×q, and c₁,c₂ q×1. The zs can be kept fixed between experiments, or regenerated for each experiment.

When B₀=B₁=0 there are no ARMA errors. When A₀=0 the DGP is in reduced form, when also A₂=0 the DGP is a VAR(2), and when in addition A₅=0 the DGP is a VAR(1).

A distribution for e_t and v_t can be specified. Writing ε_t for either e_t or v_t, then:

Initial values for y₀ can be specified.

2.3.1.1 PcNaive DGP in equilibrium-correction form

The DGP can also be formulated as a cointegrated VAR in equilibrium-correction form:

The rank of the cointegrating space must be specified.

Note how in (eq:2.4) the z_t component enters both the cointegrating space and unrestrictedly. This offers complete flexibility: the zeros in A₂ and β determine what actually happens. For example, setting A₂=0 would force all zs into the cointegration space, unless, of course, the corresponding elements of β are also zero, in which case the zs do not enter at all.

2.3.1.2 PcNaive DGP with break

A sample period can be specified over which any of the matrices in the y_t equation of (eq:2.3) or the Δy_t equation of (eq:2.4) can have different values.

The break period is specified as T₁^b,...T₂^b, meaning that the break starts at T₁^b, The first post-break observation is T₂^b+1. For example, when the break period is [20,30], the break is active over 11 periods.

2.3.2 Data generation process for the Reduced Form DGP

The Reduced Form DGP is the most general data generation process in PcNaive, and therefore the most complex.

The form of the DGP in mathematical formulation is a reduced-form model:

where w contains z, r lags of z and m lags of y:

Take y_t as an n×1 vector, z_t as q×1, and w_t as k×1.

The DGP database is constructed as follows:

The distributions for the error term offer the same choice as the PcNaive DGP.

2.3.3 Models

Models of the PcNaive and general DGP can be estimated by:

1. single-equation OLS,
2. single-equation (generalized) instrumental variables,
3. multiple-equation OLS (including vector autoregressions).

The implementation corresponds to PcGive (see Doornik and Hendry, 2006a). There are four types of variables:

1. [Y] endogenous variable: more than one results in a multivariate model, unless additional instruments have been specified,
2. [Z] regressor (corresponding to unmarked regressors in the model formulation dialog),
3. [U] unrestricted regressor,
4. [A] additional instruments for IV estimation.

The distinction between U and Z only matters for cointegration tests.

The estimated quantities are split into `estimates' and `tests'. For the former, the output includes means, standard errors, biases, etc. For the latter, the output consists of the first four moments, rejection frequencies and critical values.

In case of recursive Monte Carlo, the experiment is run for sample sizes T=T₁,...,T₂. The Monte Carlo sample size is automatically adjusted to allow for lagged variables in the model.

2.3.4 Monte Carlo Output

Let n denote the number of equations in the econometric model, k the number of regressors, and T the sample size.

• Asymptotic analysis

  Theoretical analysis of the DGP

• Coefficients

  The nk estimated coefficients.

• Standard errors

  The nk estimated standard errors of the coefficients.

• Goodness of fit

  If n=1: σ̂² and R².
  If n>1: Ω̂_ii for i=1,...,n (the residual variance for each equation).

• Cointegration eigenvalues

  The eigenvalues μ_i, i=1,...,n from the reduced-rank estimation.

• t-tests

  The nk estimated t-values for all coefficients.

• AR1 test

  F-test on first-order residual autocorrelation. This is the F form of the test by Breusch (1978) and Godfrey (1978). When n>1, it is the multivariate version, see Doornik (1996). For n=1, the Durbin--Watson is also computed.

• Normality test

  The Doornik and Hansen (1994) test for normality, which is approximately χ²(2n) distributed under the null hypothesis.

• Forecast Chow tests

  When the model is estimated up to T, this tests for constancy up to T₂+H. Since T₂+H≥T, this is called the forecast Chow tests.

  When n=1 the test is:

  (RSST2+H-RSST)(T-k) RSST(T2+H-T)

  ~F(T2+H-T, T-k).

  For n>1 the vector form is reported, see Doornik and Hendry (2006a).

  Note that for the last sample size (or when the Monte Carlo is not recursive) and the number of forecasts is zero (H=0): T=T₂ and the test is zero, with p-value one.

• Breakpoint Chow tests

  When the model is estimated up to T, this tests for insample constancy from T₁≤T. Since this is insample, it is called the break-point Chow tests.

  When n=1 the test is:

  (RSST-RSST1)(T1-k) RSST(T-T1)

  ~F(T, T1-k).

  For n>1 the vector form is reported, see Doornik and Hendry (2006a).

  Note that when the Monte Carlo is not recursive: T=T₁ and the test is zero, with p-value one.

• Co/integration tests
  • When n=1 this reports the ECM or ADF test, depending on the presence of other regressors. Let q+1 be the number of distinct variables (q is the number of regressors not counting any lagged regressors and not counting the constant term or trend). The model for estimation is written as:

    a(L)yt=b(L)xt+...+ut,

    which can be written in equilibrium-correction form as:

    Δyt= [a(1)-1] yt-1 + a*(L)Δyt-1+b(L)xt+...+ut,

    where a(1) is the sum of the coefficients on the lagged dependent variable. The test statistic is the t-value of a(1)-1. When q=1 (no other regressors) this is the ADF(s) test, where s is the number of lagged ys minus one (i.e. the number of lagged differences). When q>1 this is the PcGive unit-root test for cointegration, denoted ECM(q). The 5% critical values of this test are based on a meta response surface for the results from Ericsson and MacKinnon (1999).
    

    [Note: Meta-response surface for quantiles Q when q>1, d=0,1,2:

    Q(Ti) = θ̂∞ + θ̂1[Ti-(2q-1)-d]-1+ui,

    θ̂∞5% = -2.056 -0.535d(1-q/20) -0.343 q(1 - q/40) θ̂∞1% = θ̂∞5% -0.607 θ̂15% = 7.09 -0.483k + 3.64 * θ̂∞5% * (1 - q / 10) θ̂11% = 13.8 -0.447k + 6.06 * θ̂∞1% * (1 - q / 20)

    with d=0 for no deterministic terms, d=1 for a constant, and d=2 for a trend. For the ADF test, q=1, the values for θ_j are taken from the relevant tables in Ericsson and MacKinnon (1999). ]

    Note that the test is sensitive to the treatment of deterministic terms, which is noted, e.g. ADF(1;c) when a constant is included as a regressor, and ADF(1;ct) for a constant and trend. There are no p-values available for this test, and the returned pseudo p-value is 0.0 when the test rejects, and 1.0 when it accepts.

  • When n>1 the cointegration test is based on the maximum likelihood procedure of Johansen (1988). The test statistic is the trace test H(p ≤i) for i=0,...,n-1. There are five versions, based on the treatment of deterministic terms. In the notation of Doornik, Hendry and Nielsen (1998) and Doornik and Hendry (2006a):

    Hypothesis	Constant	Trend
    Hql(p)	unrestricted	unrestricted
    Hl(p)	unrestricted	restricted
    Hlc(p)	unrestricted	none
    Hc(p)	restricted	none
    Hz(p)	none	none

    The terms unrestricted/restricted refers to the classification in model formulation, which determines whether a variable is restricted to the cointegrating space or not. The asymptotic p-values are based on the approximations in Doornik (1998).

• Cointegration (cond. on 1)

  This is the trace test for cointegration, but the asymptotic distribution is modified to assume that it is conditional on one stationary exogenous regressor. The p-values are based on Boswijk and Doornik (1999).

• Cointegration (cond. on 2)

  As above, but conditional on two regressors.

• Cointegration (asymptotic)

  This uses the generated u_t to compute the critical values for the maximum likelihood test for cointegration based on the discrete equivalents of the Brownian motions (see e.g. Johansen, 1995, Ch. 15, Simulations and Tables). The reported tests are the trace and maximum eigenvalue statistics for the treatment of the constant and trend adopted in the estimating model. No p-values are available.

2.3.5 Live Graphics

• Histograms of estimates

  Histogram and non-parametrically estimated density of the selected estimates.

• Histograms of tests

  Histogram and non-parametrically estimated density of the selected test statistics.

• Recursive mean of estimates

  The mean of the estimates at the sample sizes of the recursive Monte Carlo, shown with ±2MCSD bands. If the standard errors are also simulated, the ±2ESE bands are also shown.

• Recursive bias of estimates

  The mean bias of the estimates at the sample sizes of the recursive Monte Carlo.

• Recursive MCSD of estimates

  The MCSD and RMSE of the estimates at the sample sizes of the recursive Monte Carlo.

• Recursive mean of tests

  The mean of the test statistics at the sample sizes of the recursive Monte Carlo.

• Recursive rejection frequencies

  The rejection frequencies at the right tail of the test statistics at the sample sizes of the recursive Monte Carlo, at user-specified p-values.

• Recursive critical values

  The critical values at the right tail of the test statistics at the sample sizes of the recursive Monte Carlo, at user-specified p-values.

• Generated data

  The generated y_t, z_t and custom transformations.

• Standardized data

  The generated y_t, z_t in deviation from its mean and divided by the standard deviation.

• Set plot frequency

  This determines how often graphs are created. The default of zero means that the plots are drawn when the experiments for a particular sample size have finished. In this case, data graphs are always for the final replication. Enter a value here that is less than the number of replications to see the plots more frequently.

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