These reference chapters have been taken from Volume IV, and use the same chapter and section numbering as the printed version.
The AR(1) data generation process is:


where α or μ may be set to zero.
The model is:
y_{t}=β_{0} +β_{1} y_{t1} + u _{t} with u _{t}~IN[ 0,σ^{2}] , 
which is estimated by OLS. Either the constant term or the lagged dependent variable can be omitted.
The sample size is specified as T_{1}, T_{1}+s, T_{1}+2s,...,T_{2}, where the step size s may be zero. The Monte Carlo sample size is automatically adjusted if the model includes a lagged dependent variable.
The following estimates and tests can be investigated
The static data generation process is:


where α_{1}, α_{2}, or ρ may be set to zero. The regressors can be kept fixed, or recreated in each replication (stochastic regressors).
The model for the static DGP is:
y_{t}=β_{0} + β_{1} z_{a,t} + β_{2} z_{b,t} + u _{t}, u _{t}~IN[0,σ^{2}], 
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 T_{1}, T_{1}+s, T_{1}+2s,...,T_{2}, where the step size s may be zero.
This is the same as for the AR(1) DGP, for the same tests, and the estimates β̂_{0}, β̂_{1},β̂_{2}, provided they are included in the model.
This is the same as for the AR(1) DGP, for the same tests, and the estimates β̂_{0}, β̂_{1},β̂_{2}, provided they are included in the model.
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 A_{0},A_{1},B_{0},B_{1} are n×n, and a_{3} is n×1. The z_{t} vector is q×1, making a_{2} n×q, C_{0} q×q, and c_{1},c_{2} q×1. The zs can be kept fixed between experiments, or regenerated for each experiment.
When B_{0}=B_{1}=0 there are no ARMA errors. When A_{0}=0 the DGP is in reduced form, when also A_{2}=0 the DGP is a VAR(2), and when in addition A_{5}=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:
distribution  parametrization 
none  0 (no distribution) 
normal (IN)  ε_{it} ~N(α_{i},β_{i}) = N(0,1)×√β_{i}+α_{i} 
multivariate normal (MVN)  ε_{t} ~N_{n}(α, β) 
MVN with correlations  ε_{t} ~N_{n}(α, β) with standard deviations on 
diagonal, correlations on lower diagonal  
log normal  ε_{it} ~Λ(α_{i}, β_{i}) = exp {N(0,1)}×√α_{i} +β_{i} 
Studentt  ε_{it} ~t(α_{i}) 
F  ε_{it} ~F(α_{i}, β_{i}) 
exponential  ε_{it} ~exp(α_{i}) 
MVN with ARCH  ε_{t} ~N_{n}(0, α + β ε_{t1}ε_{t1}' β') 
MVN with heteroscedasticity  e_{t} ~N_{n}(0, α + β y_{t1}y_{ti}' β') 

Initial values for y_{0} can be specified.
The DGP can also be formulated as a cointegrated VAR in equilibriumcorrection 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 A_{2} and β determine what actually happens. For example, setting A_{2}=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.
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 T_{1}^{b},...T_{2}^{b}, meaning that the break starts at T_{1}^{b}, The first postbreak observation is T_{2}^{b}+1. For example, when the break period is [20,30], the break is active over 11 periods.
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 reducedform model:


where w contains z, r lags of z and m lags of y:
w_{t}'=( y_{t1}',...,y_{tm}',z_{t}',...,z_{tr}^{'}) . 
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:
0 ...s1  initial values for lagged observations, s ≥ max (1,m,r) 
T_{1} = s ...s+d1  space to allow for discarded observations, 
T_{1}+d ...T_{2}^{*}  remainder of generated data. 
The distributions for the error term offer the same choice as the PcNaive DGP.
Models of the PcNaive and general DGP can be estimated by:
The implementation corresponds to PcGive (see Doornik and Hendry, 2013a). There are four types of variables:
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=T_{1},...,T_{2}. The Monte Carlo sample size is automatically adjusted to allow for lagged variables in the model.
Let n denote the number of equations in the econometric model, k the number of regressors, and T the sample size.
Theoretical analysis of the DGP
The nk estimated coefficients.
The nk estimated standard errors of the coefficients.
If n=1: σ̂^{2} and R^{2}.
If n>1: Ω̂_{ii}
for i=1,...,n (the residual variance for each equation).
The eigenvalues μ_{i}, i=1,...,n from the reducedrank estimation.
The nk estimated tvalues for all coefficients.
Ftest on firstorder 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 DurbinWatson is also computed.
The Doornik and Hansen (1994) test for normality, which is approximately χ^{2}(2n) distributed under the null hypothesis.
When the model is estimated up to T, this tests for constancy up to T_{2}+H. Since T_{2}+H≥T, this is called the forecast Chow tests.
When n=1 the test is:
 ~F(T_{2}+HT, Tk). 
For n>1 the vector form is reported, see Doornik and Hendry (2013a).
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=T_{2} and the test is zero, with pvalue one.
When the model is estimated up to T, this tests for insample constancy from T_{1}≤T. Since this is insample, it is called the breakpoint Chow tests.
When n=1 the test is:
 ~F(T, T_{1}k). 
For n>1 the vector form is reported, see Doornik and Hendry (2013a).
Note that when the Monte Carlo is not recursive: T=T_{1} and the test is zero, with pvalue one.
a(L)y_{t}=b(L)x_{t}+...+u_{t}, 
which can be written in equilibriumcorrection form as:
Δy_{t}= [a(1)1] y_{t1} + a^{*}(L)Δy_{t1}+b(L)x_{t}+...+u_{t}, 
where a(1) is the sum of the coefficients on the lagged dependent
variable. The test statistic is the tvalue 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 unitroot
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 (2002).
Q(T_{i}) = θ̂_{∞} + θ̂_{1}[T_{i}(2q1)d]^{1}+u_{i}, 

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 (2002). ]
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 pvalues available for this test, and the returned pseudo pvalue is 0.0 when the test rejects, and 1.0 when it accepts.
Hypothesis  Constant  Trend 
H_{ql}(p)  unrestricted  unrestricted 
H_{l}(p)  unrestricted  restricted 
H_{lc}(p)  unrestricted  none 
H_{c}(p)  restricted  none 
H_{z}(p)  none  none 
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 pvalues are based on Boswijk and Doornik (1999).
As above, but conditional on two regressors.
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 pvalues are available.
Histogram and nonparametrically estimated density of the selected estimates.
Histogram and nonparametrically estimated density of the selected test statistics.
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.
The mean bias of the estimates at the sample sizes of the recursive Monte Carlo.
The MCSD and RMSE of the estimates at the sample sizes of the recursive Monte Carlo.
The mean of the test statistics at the sample sizes of the recursive Monte Carlo.
The rejection frequencies at the right tail of the test statistics at the sample sizes of the recursive Monte Carlo, at userspecified pvalues.
The critical values at the right tail of the test statistics at the sample sizes of the recursive Monte Carlo, at userspecified pvalues.
The generated y_{t}, z_{t} and custom transformations.
The generated y_{t}, z_{t} in deviation from its mean and divided by the standard deviation.
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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