Figure 1: Posterior distribution of parameter $$\psi _{1}$$ and $$\alpha $$
Initial
parametrization $$\psi _1=2$$. The grey line represents the prior distribution and the black line is the posterior distribution.
Two panels.
Given an initial parametrization of psi_1 equal to 2, the figure plots the prior and posterior distribution for the
parameter psi_1 and alpha.
Left panel:
Prior and posterior distribution for the parameter psi_1 only. The X axis ranges from 0 to 3, and the Y axis from 0 to 5.
Prior distribution plotted as a curve, while posterior distribution has two peaks. This panel shows that the prior
distribution is plotted as a concave curve with one peak at 1 for a value of psi_1 around 0.9. After the estimation, the
posterior distribution is bimodal: one peak is at 2.8 for a value of psi_1 of 0.8, and the second peak is at 4.5 for a
value of psi_1 slightly above 1. The distribution is bimodal because the algorithm jumps across the region of determinacy
and indeterminacy.
Right panel:
Prior and posterior distribution for the parameter alpha only. The X axis ranges from -0.1 to 2.1, and the Y axis from 0
to 1.5. Prior distribution plotted as a straight horizontal line at 0.5 over the range from 0 to 2. The posterior
distribution has also ranges from 0 to 2, and, even if it is not a straight line, it closely approximates the prior
distribution. This panel shows that the algorithm traveled across the region of determinacy and indeterminacy, as the
parameter alpha takes values both above and below 0. Moreover, the posterior distribution approximates the prior
distribution as the parameter alpha is not identified.
Figure 2: Draws of the parameter $$\alpha $$
Sequence of draws for $$\alpha $$ given an initial
parametrization $$\psi _1=2$$.
The figure plots the sequence of draws of parameter alpha given an initial parametrization of psi_1 equal to 2. The X axis ranges from 0 to 1,000,000 that corresponds to the number of draws executed by the algorithm. The Y axis ranges from 0 to 2. Over the range from 0 to approximately 400,000, the parameter alpha takes values between 1 and 2. Starting from 400,000, the parameter alpha takes values between 0 and 1. The figure shows that after approximately 400,000 draws of alpha in the region of determinacy (namely, outside the unit circle), the algorithm jumps to the indeterminate region (namely, within the unit circle) and never visits the determinacy region again.
Figure 3: Proposal distribution for different values of $$\alpha $$.
The proposal distribution is chosen to facilitate crossing the determinacy threshold and is obtained with a
mixture of normals. The upper (lower) panel assumes that $$\alpha $$ is currently above (below) the threshold of the determinacy region.
The figure plots the proposal distribution for different values of alpha. Two panels. In each of the two panels, there are
three curved lines that represent alternative proposal distributions depending on three alternative values of alpha
drawn from the algorithrm.
Top panel:
Proposal distribution when the value of alpha drawn by the algorithm is greater than 1 (namely, determinacy).
The X axis ranges from 0.2 to 1.8. The Y axis ranges from 0 to 10.
Each proposal distribution is plotted as a curved line that however peaks at different values of the X axis depending on
the last value drawn for alpha. As alpha assumes values closer to 1 coming from values above 1, the proposal distribution
peaks at values below and further away from 1. The figure shows that the posterior distribution is chosen to facilitate
the crossing of the determinacy threshold.
Lower panel:
Proposal distribution when the value of alpha drawn by the algorithm is smaller than 1 (namely, indeterminacy).
The X axis ranges from 0.2 to 1.8. The Y axis ranges from 0 to 10.
Each proposal distribution is plotted as a curved line that however peaks at different values of the X axis depending on
the last value drawn for alpha. As alpha assumes values closer to 1 coming from values below 1, the proposal distribution
peaks at values above and further away from 1. The figure shows that the posterior distribution is chosen to facilitate
the crossing of the determinacy threshold.
Figure 4:
The figure reports the distribution for the number of draws necessary to cross the determinacy threshold for the first time when using a Metropolis-Hastings algorithm to estimate the model of Lubik and Schorfheide (2004). Two cases are
considered. In the first case (blue/dark colored bars), the algorithm is implemented by drawing values for the auxiliary parameter $$\alpha $$. The value of $$\alpha $$ is then used to obtain the corresponding value of
$$\psi _{\pi }$$. In the second case (yellow/light colored bars), the algorithm is implemented by drawing directly the parameters of the model. The distribution is truncated at 100,000 draws.
The figure reports the distribution of the number of draws necessary to cross the determinacy threshold for the
first time when using a Metropolis-Hastings algorithm to estimate the model of Lubik and Schorfheide (2004).
The X axis ranges from 0 to 100,000. The Y axis ranges from 0 to 800.
The figure plots the distribution for the number of draws necessary to cross the determinacy regions under two cases.
For each case, the figure plots a sequence of bars that counts the number of draws necessary to cross. The first case
corresponds to the algorithm proposed in this paper and that is implemented by drawing values for the auxiliary parameter
alpha and then using the value of alpha to obtain the corresponding value of psi 1. The bars for this case start at a
value of 200 for a the lowest number of draws considered (namely 50,000) and then decrease homogeneously until reaching
almost 0 for very large number of draws denoted on the X axis.
Instead, the second case corresponds to the traditional algorithm that makes draws for the original parameter space. The
bars for this case are very close to 0 for any value on the X axis. However, there is only one high bar when the number
of draws on the X axis reaches 100,000.
The figure shows that the modified algorithm greatly facilitates crossing the determinacy region.
Figure 5:
The figures plot the posterior means (solid lines) and 90-percent probability intervals (dashed lines) for the impulse responses of output, inflation and nominal interest rate to a shock of one standard deviation for each orthogonalized
disturbance using a Cholesky decomposition with the same order as in the plots.
The figure plots the posterior mean and 90-percent probability intervals for the impulse responses of output, inflation
and nominal interest rate to a shock of one standard deviation for each of the five disturbances orthogonalized using a
Cholesky decomposition with the same order as chosen to describe the figure below.
Fifteen panels organized in five rows and three columns for each row. The columns correspond to output, inflation and
nominal interest rate, respectively. The rows correspond to the shocks: the creation of a new bubble, a supply side
shock, a monetary policy shock, sunspot shock to inflation expectations and sunspot shock to expectations about future
devations of output from its trend.
For all the panels, the X axis ranges from 0 to 20 quarters, while the Y axis is expressed in percentages.
Each panel has three curved lines: the posterior mean is plotted between the two lines that define the 90-percent
probability intervals for the impulse responses.
FIRST ROW OF THREE PANELS: Shock due to the creation of a new bubble. Impact on output, inflation and nominal interest rate.
Left panel. Y axis: from -0.4 to 0.3. All the three lines are mildy hump-shaped and convex.
Top line starts at 0.3 and ends at 0.08. Posterior mean starts at 0 and ends at -0.1. Bottom line stays constant at -0.3.
Center panel. Y axis: from -0.1 to 0.4.
Top line starts at 0.2 and increases gently at 0.4. Posterior mean starts at 0.05 and ends at 0.15. Bottom line stays
constant at -0.1.
Right panel. Y axis: from -0.12 to 0.4.
Top line starts at 0.1 and ends at 0.4. Posterior mean starts at 0 and ends at 0.2. Bottom line stays constant at -0.1.
SECOND ROW OF THREE PANELS: Supply side shock. Impact on output, inflation and nominal interest rate.
Left panel. Y axis: from -0.9 to -0.2. All the three lines are hump-shaped and convex.
Top line starts at -0.4 and ends at -0.2. Posterior mean starts at -0.5, is hump-shaped and ends again at -0.5. Bottom line
starts at -0.6 and ends at -0.8.
Center panel. Y axis: from -0.5 to 0.2. All the three lines are decreasing and convex.
Top line starts at 0.2 and decreases at -0.18. Posterior mean starts at 0.15 and ends at -0.28. Bottom line starts at 0.1
and ends at -0.5.
Right panel. Y axis: from -0.5 to 0. All the three lines are decreasing and convex.
Top line starts at 0 and ends at -0.1. Posterior mean starts at 0.08 and ends at -0.3. Bottom line starts at -0.05 and
ends at -0.5.
THIRD ROW OF THREE PANELS: Monetary policy shock. Impact on output, inflation and nominal interest rate.
Left panel. Y axis: from -0.3 to 0.4. All the three lines are hump-shaped and concave.
Top line starts at 0.1 and ends at 0.2. Posterior mean starts at -0.1 and ends at 0.1. Bottom line starts at -0.3 and
ends at -0.03.
Center panel. Y axis: from -0.4 to 0. All the three lines are gently decreasing.
Top line starts at 0 and decreases at -0.1. Posterior mean starts at -0.12 and ends at -0.22. Bottom line starts at -0.2
and ends at -0.4.
Right panel. Y axis: from -0.4 to 0.1. All the three lines are decreasing and convex.
Top line starts at 0.1 and decreases at -0.1. Posterior mean starts at 0.03 and ends at -0.23. Bottom line starts at -0.02
and ends at -0.4.
FOURTH ROW OF THREE PANELS: Sunspot shock to inflation expectations. Impact on output, inflation and nominal interest rate.
Left panel. Y axis: from -0.15 to 0.03. All the three lines are hump-shaped and convex.
Top line starts at 0, stays below 0 until 15 quarters and then crossed above 0 to end at 0.03. Posterior mean starts at 0
and ends at -0.03. Bottom line starts at 0 and ends at -0.08.
Center panel. Y axis: from 0.02 to 0.25. All the three lines are gently increasing.
Top line starts at 0.15 and increases at 0.25. Posterior mean starts at 0.08 and ends at 0.14. Bottom line starts at 0.02
and stays constant at 0.03.
Right panel. Y axis: from 0 to 0.25. All the three lines are increasing and mildly concave.
Top line starts at 0.08 and increases to 0.25. Posterior mean starts at 0.05 and ends at 0.15. Bottom line starts at 0.01
and stays constant at 0.03.
FIFTH ROW OF THREE PANELS: Sunspot shock to expectations about future devations of output from its trend. Impact on output,
inflation and nominal interest rate.
Left panel. Y axis: from 0 to 0.4. All the three lines are decreasing.
Top line starts at 0.4 and ends at 0.2. Posterior mean starts at 0.25 and ends at 0.1. Bottom line starts at 0.08 and
ends at 0.
Center panel. Y axis: from -0.23 to 0. All the three lines are decreasing.
Top line starts at 0 and ends at 0.02. Posterior mean starts at 0 and ends at -0.14. Bottom line starts at 0
and stays constant at -0.23.
Right panel. Y axis: from -0.25 to 0. All the three lines are decreasing.
Top line starts at 0 and decreases at 0.03. Posterior mean starts at -0.02 and ends at -0.14. Bottom line starts at -0.03
and ends at -0.25.