|
Size: 8401
Comment:
|
Size: 8385
Comment:
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 105: | Line 105: |
| The posterior mean of [[latex($\mathbf A$)]] is a convex combination of [[latex($A^*(\theta)$)]], the prior mean, and of the OLS estimate of [[latex($\mathbf A$)]]. When [[latex($\lambda$)]] goes to infinity the posterior mean shrinks towards the prior mean, ''ie'' the projection of the DSGE model onto the VAR([[latex($p$)]]). We do not have a closed form expression for the joint posterior density of [[latex($\theta$)]] and [[latex($\lambda$)]] (the second term on the right hand side of \equaref{posterior1}). So the posterior distribution of [[latex($(\theta,\lambda)$)]] is recovered from an MCMC algorithm, as described in Del Negro and Schorfheide, except that we do estimate [[latex($\lambda$)]] as the deep parameters [[latex($\theta$)]]. | The posterior mean of [[latex($\mathbf A$)]] is a convex combination of [[latex($A^*(\theta)$)]], the prior mean, and of the OLS estimate of [[latex($\mathbf A$)]]. When [[latex($\lambda$)]] goes to infinity the posterior mean shrinks towards the prior mean, ''ie'' the projection of the DSGE model onto the VAR([[latex($p$)]]). We do not have a closed form expression for the joint posterior density of [[latex($\theta$)]] and [[latex($\lambda$)]] (the second term on the right hand side of [Q3]). So the posterior distribution of [[latex($(\theta,\lambda)$)]] is recovered from an MCMC algorithm, as described in Del Negro and Schorfheide, except that we do estimate [[latex($\lambda$)]] as the deep parameters [[latex($\theta$)]]. |
Consider the order latex($p$) VAR representation for the latex($1\times m$) vector of observed variables latex($y_t$):
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_a0d779b6c7c14e45e5f34a3cc8b26d0e41b45de1_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_a0d779b6c7c14e45e5f34a3cc8b26d0e41b45de1_p.log.
where latex($u_t\sim \mathcal N\left( 0,\Sigma_u\right)$). Let latex($z_t$) be the latex($mp\times 1$) vector latex($\left[y_{t-1}',...,y_{t-p}'\right]'$) and define latex($\mathbf{A}=\left[\mathbf A_1',...,\mathbf A_p'\right]'$), the VAR representation can then be written in matrix form as:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_39839dadd822540e9e83e6cc8e65f045bf8531f9_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_39839dadd822540e9e83e6cc8e65f045bf8531f9_p.log.
where latex($Y = (y_1',\dots,y_T')'$), latex($Z = (z_1',\dots,z_T')'$) and latex($\mathcal U = (u_1',\dots,u_T')'$).
Dummy observations prior for the VAR can be constructed using the VAR likelihood function for latex($\mathcal T = [\lambda T]$) artificial data simulated with the DSGE latex($\left( Y^{\ast },Z^{\ast}\right)$), combined with diffuse priors. The prior is then given by:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_80e95b87a51ea9815bc69e9866fe1033510b0531_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_80e95b87a51ea9815bc69e9866fe1033510b0531_p.log.
implying that latex($\Sigma$) follows an inverted Wishart distribution and latex($\mathbf A$) conditional on latex($\Sigma$) is gaussian. Assuming that observables are covariance stationary, Del Negro and Schorfheide use the DSGE theoretical autocovariance matrices for a given latex($n\times 1$) vector of model parameters latex($\theta$), denoted latex($\Gamma_{YY}\left( \theta \right)$), latex($\Gamma_{ZY}\left( \theta \right)$), latex($\Gamma_{YZ}\left( \theta\right)$), latex($\Gamma_{ZZ}\left( \theta \right)$) instead of the (artificial) sample moments latex(${Y^*}'Y^*$), latex(${Z^*}'Y^*$), latex(${Y^*}'Z^*$), latex(${Z^*}'Z^*$). In addition, the latex($p$)-th order VAR approximation of the DSGE provides the first moment of the prior distributions through the population least-square regression:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_0f7ac474acdcd7c90d82c31233e2bb594046b163_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_0f7ac474acdcd7c90d82c31233e2bb594046b163_p.log.
Conditional on the deep parameters of the DSGE latex($\theta $) and latex($\lambda$), the priors for the VAR parameters are given by:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_5d2eafb18a285f7ebb7aa6002cdc207212c5b850_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_5d2eafb18a285f7ebb7aa6002cdc207212c5b850_p.log.
where latex($\Gamma_{ZZ}(\theta)$) is assumed to be non singular and latex($\lambda \geq \frac{mp+m}{T}$) for the priors to be proper (Note that it would not be possible to estimate the VAR model by OLS (or maximum likelihood) if we had latex($\mathcal T < m(p+1)$). In this case we would not have more observations than parameters to estimate). The a priori density of latex($\mathbf A$) is defined by latex($n+1$) parameters (latex($\theta$) and latex($\lambda$)), which is likely to be less than latex($mp$) (the VAR number of parameters). If we have a one-to-one relationship (no identification issues) between latex($(\theta,\lambda)$) and latex($\mathbf A$) it will be a good idea to estimate latex($(\theta,\lambda)$) instead of latex($\mathbf A$), ie to estimate fewer free parameters. To do so, Del Negro and Schorfheide complete the prior by specifying a prior distribution over the structural model's deep parameters: latex($p_0(\theta)$). We still have to set the weight of the structural prior, latex($\lambda$). Del Negro and Schorfheide choose the value of latex($\lambda$) that maximizes the marginal density. They estimate a limited number of DSGE-VAR models with different values of latex($\lambda$). For each model they also estimate the marginal density and select the model (ie the value of latex($\lambda$)) with highest marginal density. Dynare can also estimate directly latex($\lambda$) as another parameter, instead of doing a loop over the values of this parameter. So we define a prior on latex($\lambda$), which is assumed to be independent from latex($\theta$). Finally, the DSGE-VAR model has the following prior structure:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_f6336bea699a384dccf1f8459be6a16f1240957a_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_f6336bea699a384dccf1f8459be6a16f1240957a_p.log.
where latex($p_0\left(\mathbf A, \Sigma \mid \theta ,\lambda \right)$) is defined by [P1a,P1b] and [P2].
The posterior distribution, may be factorized in the following way:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_2fcdbfa94c5a50eba79b7f28dbc50383c90bd9ea_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_2fcdbfa94c5a50eba79b7f28dbc50383c90bd9ea_p.log.
where latex($\mathcal Y_T$) stands for the sample. A closed form expression for the first density function on the right hand side of [Q3] is available. Conditional on latex($\theta$) and latex($\lambda$), [P1a,P1b] and [P2] define a conjugate prior for the VAR model, so its posterior density belongs to the same family: the distribution of latex($\mathbf A$) conditional on latex($\Sigma$), latex($\theta$), latex($\lambda$) and the sample is matric-variate normal, and the distribution of latex($\Sigma$) conditional on latex($\theta$), latex($\lambda$) and the sample is inverted Wishart. More formally, we have:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_30ce7ace1fe210596ce844ed2b62b8e6ad103c64_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_30ce7ace1fe210596ce844ed2b62b8e6ad103c64_p.log.
where:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_6b5382e5a043b3a5b91531da7705b18bdd3fc401_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_6b5382e5a043b3a5b91531da7705b18bdd3fc401_p.log.
with:
latex error! exitcode was 1 (signal 0), transscript follows:
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2019/dev/Debian) (preloaded format=latex)
entering extended mode
(./latex_72ce5e113f8bfde21517cc5d6aea0c67e9144c16_p.tex
LaTeX2e <2018-12-01>
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2018/09/03 v1.4i Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsxtra.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/dcolumn.sty
(/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/graphics-def/dvips.def)))
(/usr/share/texlive/texmf-dist/tex/latex/tools/tabularx.sty)
(/usr/share/texlive/texmf-dist/tex/latex/tools/hhline.sty)
(/usr/share/texlive/texmf-dist/tex/latex/psfrag/psfrag.sty)
! LaTeX Error: File `multirow.sty' not found.
Type X to quit or <RETURN> to proceed,
or enter new name. (Default extension: sty)
Enter file name:
! Emergency stop.
<read *>
l.35 ^^M
No pages of output.
Transcript written on latex_72ce5e113f8bfde21517cc5d6aea0c67e9144c16_p.log.
The posterior mean of latex($\mathbf A$) is a convex combination of latex($A^*(\theta)$), the prior mean, and of the OLS estimate of latex($\mathbf A$). When latex($\lambda$) goes to infinity the posterior mean shrinks towards the prior mean, ie the projection of the DSGE model onto the VAR(latex($p$)). We do not have a closed form expression for the joint posterior density of latex($\theta$) and latex($\lambda$) (the second term on the right hand side of [Q3]). So the posterior distribution of latex($(\theta,\lambda)$) is recovered from an MCMC algorithm, as described in Del Negro and Schorfheide, except that we do estimate latex($\lambda$) as the deep parameters latex($\theta$).
Example 1
References