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
[2]

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:

yt=μ+ αyt-1 + εt with εt~IN[ 0,1],
(eq:2.1)

where α or μ may be set to zero.

2.1.2 Model

The model is:

yt01 yt-1 + u t with u t~IN[ 0,σ2] ,

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

2.1.4 Live Graphics

2.2 Static DGP

2.2.1 Data generation process

The static data generation process is:

yt=α1 za,t + α2 zb,t + εt, εt~IN[0,1],
(
za,t
zb,t
) ~IN[ (
0
0
), (
1 ρ
ρ1
) ],
(eq:2.2)

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:

yt0 + β1 za,t + β2 zb,t + u t,  u t~IN[0,σ2],

which is estimated by OLS, and where the constant, za,t and/or zb,t may be omitted. The za,t and zb,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:

yt = A0yt+A1yt-1+A2zt+a3+A5yt-2 + ut,
ut = B0ut-1+et+B1et-1,
zt = C0zt-1+c1+c2t+vt.
(eq:2.3)

The vectors yt,ut,et are n×1, so that the coefficient matrices A0,A1,B0,B1 are n×n, and a3 is n×1. The zt 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 et and vt can be specified. Writing εt for either et or vt, then:

distribution parametrization
none 0 (no distribution)
normal (IN) εit ~Nii) = N(0,1)×√βii
multivariate normal (MVN) εt ~Nn(α, β)
MVN with correlationsεt ~Nn(α, β) with standard deviations on
diagonal, correlations on lower diagonal
log normal εit ~Λ(αi, βi) = exp {N(0,1)}×√αii
Student-t εit ~ti)
F εit ~Fi, βi)
exponential εit ~expi)
MVN with ARCH εt ~Nn(0, α + β εt-1εt-1' β')
MVN with heteroscedasticity et ~Nn(0, α + β yt-1yt-i' β')
The parameters α and β are captured in matrices as follows:

αi α βi β
et n×1 vector m0n×n matrix M0 n×1 vector s0n×n matrix S0
vt q×1 vector m1q×q matrix M1 q×1 vector s1q×q matrix S1

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:

Δyt = αβ' (
yt-1
zt
) +A2zt+ a3+A5*Δyt-1 + ut.
(eq:2.4)

The rank of the cointegrating space must be specified.

Note how in (eq:2.4) the zt 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 yt equation of (eq:2.3) or the Δyt equation of (eq:2.4) can have different values.

The break period is specified as T1b,...T2b, meaning that the break starts at T1b, The first post-break observation is T2b+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:

yt = Πwt+ut,   t=T1,..., T2,
zt = C0zt-1+vt,   t=T1,..., T2.
(eq:2.5)

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

wt'=( yt-1',...,yt-m',zt',...,zt-r') .

Take yt as an n×1 vector, zt as q×1, and wt as k×1.

The DGP database is constructed as follows:

0 ...s-1 initial values for lagged observations, s ≥ max (1,m,r)
T1 = s ...s+d-1 space to allow for discarded observations,
T1+d ...T2* remainder of generated data.

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, 2013a). 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.

2.3.5 Live Graphics

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