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Posterior Sampling

 

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Posterior sampler: Lets you choose which MCMC or SMC algorithm to use, the random walk Metropolis with a normal proposal density, the slice sampler, the random walk Metropolis with a Student-t proposal density, the fixed blocking RWM with a normal proposal density, the fixed blocking RWM with a Student-t proposal density, the random blocking RWM with a normal proposal density, the random blocking RWM with a Student-t proposal density, Sequential Monte Carlo with likelihood tempering, or Sequential Monte Carlo with data tempering.

MCMC Algorithms

Number of posterior draws per chain: Lets you select the total number of draws from the random-walk Metropolis algorithm or slice sampler for the DSGE model and for the Gibbs sampler for the VAR model with a steady state prior. Integer values between 1,000 and 2,000,000 draws as well as values from user input are supported. The selected value is also used by the gmhmaxlik optimization procedure.
Number of posterior sample batches to save per chain: Select how many times per MCMC chain that data will be saved to disk. Integer values between 1 and 10,000 are supported.
Number of parallel chains: Select how many parallel MCMC chains that YADA should run. Integer values between 1 and 50 are supported.
Number of posterior draws discarded as burn-in period: Select how many draws from the beginning of the MCMC chain that are discarded from the sample of posterior draws. Integer values between 0 and 1,000,000 as well as values from user input are supported. The selection is also used by the gmhmaxlik optimization procedure.
Method for estimating the inverse Hessian: Choose which general method that YADA should use for setting up an estimator of the inverse Hessian to be used for the proposal density of the random walk Metropolis algorithm. The methods are: (1) Output from the optimization routine; (2) Fitting of quadratic function to evaluated log posterior around the posterior mode; (3) Finite difference method; and (4) using "My estimate". For the second routine, the diagonal of the inverse Hessian is changed, while the correlation structure of (1) remains intact. For the 4th routine, you can use your own estimate of the inverse Hessian. YADA only requires that this estimate is stored in a Matlab mat-file under the variable name ParameterCovarianceMatrix. Such a matrix can, for example, be estimated using previous posterior draws. On the View menu the function Parameter Covariance Matrix can perform this task.
Maximum absolute correlation for inverse Hessian estimator: The user can change the maximum absolute correlation for the inverse Hessian estimator. The default is No restriction, bu values between 0 and 0.95 are supported. If you select a value such as 0.95, then this is only imposed on the inverse Hessian if some correlation is greater than 0.95 in absolute terms.
Scale factor for initializing the posterior sampler: You can choose a constant for influencing how the random walk Metropolis algorithm is initialized for a single chain. Values between 0 and 4 are supported. If this constant, c0, is set to 0, then the posterior mode is used for initialization. The initial value is otherwise a draw from a normal distribution with mean equal to the posterior mode, and covariance matrix given by the square of the constant times the selected inverse Hessian.
Scale factor for the posterior sampler: The constant c is squared and multiplied by the inverse Hessian estimator for the covariance matrix of the proposal density for the random walk Metropolis algorithm. For the slice sampler the constant c is multiplied by the selected standard deviation when setting up the hyperrectangle. This controls allows you to select a suitable value for c. Values between 0.05 and 4 are supported. The selection is also used by the gmhmaxlik optimization procedure.
Weight on randomization for initial values (multiple chains): This constant is used for initializing the parameters under multiple chains. Values between 0 and 1 are supported. A value of 0 means that the chain is started at the posterior mode. For other values, the initial value is drawn from a normal distribution with mean equal to the posterior mode and covariance matrix equal to you selected inverse Hessian estimator times 4 times the weight on randomization.
Percentage use of posterior draws for impulse responses, etc.: For most other distributional calculations, such as the posterior distribution of impulse responses, variance decompositions, etc., YADA computes the number of draws to use from this percentage control. For example, if you have selected 550,000 posterior draws with 50,000 burn-in draws, then selecting 1% of the draws for posterior calculations means that 5,000 draws from the post burn-in sample will be used.
Maximum number of posterior draws to use for prediction: When estimating the predictive density YADA uses simulation where the number of paths per parameter value is determined on the Forecasting frame of the Miscellaneous tab. The current control allows you to select how many draws of the parameters from the posterior distribution that will be used. Integer values between 100 and the minimum of 1,000,000 and the number of posterior draws minus the number of burn-in draws are supported. The selection of parameters is by default given by equal distance.  This means that if you have selected to use 100 parameter draws and the total number of posterior draws after the burn-in period is 10,000, the selected draws will be 100 draws apart with the used draws being 1, 101, 201, ... , 9901. An alternative selections scheme supported by YADA is to randomize among the candidate draws. Such a selection is performed if the "Randomize draws from posterior distribution" option on the Tools frame of the Miscellaneous tab is check marked. The value for this setting is also used as the default value for the number of draws  for the prior sampling function.

SMC Algorithms

Number of posterior draws (particles) per chain: Lets you select the total number of draws from the random-walk Metropolis algorithm or slice sampler for the DSGE model and for the Gibbs sampler for the VAR model with a steady state prior. Integer values between 1,000 and 2,000,000 draws as well as values from user input are supported. The selected value is also used by the gmhmaxlik optimization procedure.
Number of tempering stages per chain: Lets you select how many stages the SMC algorithm reweighs the likelihood function between 0 and 1, i.e., the length of the tempering schedule. Value from 100 to 10,000 are supported. This option is disabled when using data tempering rather than likelihood tempering.
Number of parallel chains: Select how many parallel MCMC chains that YADA should run. Integer values between 1 and 50 are supported.
Value of the tempering schedule (bending) parameter (lambda): For a given number of tempering stages, this parameter determines in an exponential fashion the bend of the tempering schedule. Low value quickly give a larger weight to the likelihood for a fixed number of tempering stages while a higher value gives lower weight to the likelihood. This option is disabled when using data tempering rather than likelihood tempering.
Number of parameter blocks (mutation step): Determines the number of fixed parameter blocks for the Metropolis-Hastings algorithm during the mutation step. Default value is 1 and integer values up to 100 are supported. In case the number of parameters to estimate directly is less than the number of selected parameter blocks, YADA uses the number of parameters.
Number of Metropolis-Hastings steps (mutation step): Determines the number of Metropolis-Hastings steps the algorithm runs during the mutation step. Default value is 1 and integer values up to 100 are supported.
Mixing weight (alpha) for proposal density (mutation step): Determines the weight for mixing three normal distributions during the Metropolis-Hastings step during the mutation step. Default is 1, which gives all weight to one of the normals and therefore gives a normal proposal, but values from 0.60 and greater are also possible..
Initial value of scale factor for proposal density (mutation step): The scale factor is computed adaptively to make it possible for the acceptance rate of the mutation step to come close to the targeted acceptance rate. The user selects an initial value for the adaptive determination. Values between 0.05 and 10 are supported.
Target acceptance rate (mutation step): The targeted value of the acceptance rate for the Metropolis-Hastings step during the mutations. Values between 0.2 and 0.5 are supported.
Resampling algorithm (selection step): Lets you select which resampling algorithm to use when the effective sample size is below the resampling threshold. YADA supports multinomial, stratified, systematic, and residual resampling.
Resampling threshold of the effective sample size (selection step): Lets you select the share of the number of posterior draws which acts as a threshold for the effective sample size. Below this threshold, resampling of the particles (draws) is undertaken during the selection step. The default is 50 percent, while values between 10 and 90 percent are supported. Note that this options is renamed "Recursion threshold of the effective sample size (correction step)" under data tempering, since resampling is performed during each recursion of the algorithm and where the effective sample size is used as a threshold to determine how many new data points to include for the current recursion.

Importance Sampling based on the MitISEM Algorithm

Number of posterior draws per chain: Lets you select the total number of draws from the random-walk Metropolis algorithm or slice sampler for the DSGE model and for the Gibbs sampler for the VAR model with a steady state prior. Integer values between 1,000 and 2,000,000 draws as well as values from user input are supported. The selected value is also used by the gmhmaxlik optimization procedure.
Number of parallel chains: Select how many parallel MCMC chains that YADA should run. Integer values between 1 and 50 are supported.
Tolerance level for coefficient of variation criterion of MitISEM algorithm: Lets you choose the tolerance level for the relative change of the coefficient of variation of the IS weights (standard deviation divided by the mean). Values between 0.0001 and 0.2 are supported with 0.1 being the default value.
Method for estimating the inverse Hessian: Choose which general method that YADA should use for setting up an estimator of the inverse Hessian to be used for the proposal density of the random walk Metropolis algorithm. The methods are: (1) Output from the optimization routine; (2) Fitting of quadratic function to evaluated log posterior around the posterior mode; (3) Finite difference method; and (4) using "My estimate". For the second routine, the diagonal of the inverse Hessian is changed, while the correlation structure of (1) remains intact. For the 4th routine, you can use your own estimate of the inverse Hessian. YADA only requires that this estimate is stored in a Matlab mat-file under the variable name ParameterCovarianceMatrix. Such a matrix can, for example, be estimated using previous posterior draws. On the View menu the function Parameter Covariance Matrix can perform this task.
Maximum absolute correlation for inverse Hessian estimator: The user can change the maximum absolute correlation for the inverse Hessian estimator. The default is No restriction, bu values between 0 and 0.95 are supported. If you select a value such as 0.95, then this is only imposed on the inverse Hessian if some correlation is greater than 0.95 in absolute terms.
Maximum number of Student-t mixture components: The use can determine the highest number of mixture components to allow for the candidate density. Values between 2 and 20 are supported, with 5 being the default.
Initial number of degrees of freedom for Student-t density: Determines the number of degrees of freedom for the Student-t candidate density used during initialization and adaption of the MitISEM algorithm, i.e., prior to estimating all the parameters of the Student-t density at the end of the adaption step. Values between 1 and 20 are allowed with 1 being the default.
Share of draws & weights to initialize added Student-t component: When initializing the location and scale parameters of an added Student-t mixture component, a share of the parameter draws and IS weights based on the largest weights. Multiple share values are allowed for with the one being selected which obtains the lowest coefficient of variation. Values between 0.01 and 0.2 are supported with a maximum of five different share values, with 0.1 being the default.
Initial weight on added Student-t component: When adding a Student-t density to the mixture of such densities, the initial value for the weight given to the new component must be set. Values between 0.01 and 0.2 and supported with 0.1 being the default.
Initial degrees of freedom for added Student-t component: The final parameter to determine when adding a new Student-t density to the mixture is the initial value for its degrees of freedom. Values between 1 and 20 are supported, with 1 being the default.

 

Additional Information

A more detailed description about the random walk Metropolis algorithms is found in Section 8.1, the slice sampler is discussed in Section 8.2, sequential Monte Carlo with likelihood and data tempering is presented in Section 8.4, while importance sampling based on the MitISEM algorithm is considered in Section 8.5 of the YADA Manual.

 

 


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