Analysis of Sign-Restricted Models • bsvarPost

This article presents posterior analyses for a PosteriorBSVARSIGN object estimated with bsvarSIGNs. The identifying scheme can combine sign and zero restrictions on impulse responses, sign restrictions on the structural matrix and structural shocks, and narrative restrictions on structural shocks and historical decompositions.

Three distinct posterior quantities are considered:

Question	Function	What the result means
Which admissible draw is representative of the posterior distribution?	`most_likely_admissible_irf()` or `most_likely_admissible_cdm()`	One posterior draw selected according to its admissibility weight
What is the posterior probability that each restriction is satisfied?	`restriction_audit()`	Posterior satisfaction probability for each fitted restriction
Does the stored sample contain sufficient effective information and admissibility support?	`acceptance_diagnostics()`	Effective sample size (ESS), search settings, restriction counts, and the distribution of admissibility weights

These quantities have different interpretations. A representative draw is one joint draw from the posterior distribution and does not imply high posterior probabilities for all restrictions. Similarly, high restriction probabilities do not imply a large effective sample size or a dispersed distribution of admissibility weights.

Load demonstration draws

library(bsvarPost)
post_sign <- readRDS(system.file("extdata", "sign_post.rds",
  package = "bsvarPost", mustWork = TRUE))
post_sign_alt <- readRDS(system.file("extdata", "sign_post_alt.rds",
  package = "bsvarPost", mustWork = TRUE))
metadata <- read.csv(system.file("extdata", "demo-metadata.csv",
  package = "bsvarPost", mustWork = TRUE))
metadata[3:4, c("file", "p", "seed", "draws", "bsvarSIGNs_version")]
#>                file p seed draws bsvarSIGNs_version
#> 3     sign_post.rds 4  123   200                  3
#> 4 sign_post_alt.rds 2  456   200                  3

Each model contains 200 retained draws for demonstration; this does not establish sampling adequacy. The baseline has four lags and the alternative two. All analysis chunks below execute using these installed assets. See ?acceptance_diagnostics or help(package = "bsvarPost") for offline help.

Posterior distribution

The example uses the optimism data from bsvarSIGNs. The zero restriction excludes the contemporaneous response of productivity to the optimism shock, whereas the sign restriction imposes a positive contemporaneous response of stock prices. The model is specified with specify_bsvarSIGN$new() and its posterior distribution is estimated with estimate().

The frozen input is multiplied by 100 before estimation. Seeds, versions, restrictions and sample ranges are recorded in demo-metadata.csv. The estimation recipe appears at the end; it is optional when using the bundled draws.

Representative admissible draws

most_likely_admissible_irf() orders posterior draws by the admissibility kernel of the sign-restricted model. When several draws attain the largest weight, it selects the draw closest to the posterior median impulse responses. The result contains the impulse responses associated with one joint posterior draw rather than pointwise marginal quantiles.

rep_irf <- most_likely_admissible_irf(post_sign, horizon = 12)
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.

rep_irf$draw_index
#> [1] 4
rep_irf$score
#> [1] 0.0131408

rep_path <- subset(summary(rep_irf),
  variable == "productivity" & shock == "productivity" &
    horizon %in% c(0, 4, 8, 12),
  select = c(variable, shock, horizon, median, draw_index, method, score))
rep_path
#> # A tibble: 4 × 7
#>   variable     shock        horizon  median draw_index method              score
#>   <chr>        <chr>          <dbl>   <dbl>      <int> <chr>               <dbl>
#> 1 productivity productivity       0  0               4 most_likely_admis… 0.0131
#> 2 productivity productivity       4 -0.138           4 most_likely_admis… 0.0131
#> 3 productivity productivity       8 -0.112           4 most_likely_admis… 0.0131
#> 4 productivity productivity      12 -0.0729          4 most_likely_admis… 0.0131

The plot() method displays the selected impulse responses together with their pointwise posterior summaries. The red line represents the selected posterior draw. It is neither a credible interval nor a comparison of model specifications.

representative_plot <- publish_bsvar_plot(rep_irf, preset = "paper")

This plot object contains all response panels retained in rep_irf. The selected table above is only the productivity-to-productivity panel.

most_likely_admissible_cdm() applies the same criterion to cumulative dynamic responses computed for each posterior draw.

rep_cdm <- most_likely_admissible_cdm(post_sign, horizon = 12)
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.
head(summary(rep_cdm))
#> # A tibble: 6 × 24
#>   model object_type variable shock horizon    mean  median    sd   lower   upper
#>   <chr> <chr>       <chr>    <chr>   <dbl>   <dbl>   <dbl> <dbl>   <dbl>   <dbl>
#> 1 mode… cdm         product… prod…       0  0       0         NA  0       0     
#> 2 mode… cdm         product… prod…       1 -0.0283 -0.0283    NA -0.0283 -0.0283
#> 3 mode… cdm         product… prod…       2 -0.0965 -0.0965    NA -0.0965 -0.0965
#> 4 mode… cdm         product… prod…       3 -0.233  -0.233     NA -0.233  -0.233 
#> 5 mode… cdm         product… prod…       4 -0.371  -0.371     NA -0.371  -0.371 
#> 6 mode… cdm         product… prod…       5 -0.511  -0.511     NA -0.511  -0.511 
#> # ℹ 14 more variables: analysis_type <chr>, interval_probability <dbl>,
#> #   interval_method <chr>, scale_by <chr>, scale_var <chr>, scale_factor <dbl>,
#> #   units <chr>, source_package <chr>, source_version <chr>, n_draws <int>,
#> #   draw_index <int>, selected_draw_id <chr>, method <chr>, score <dbl>
representative_cdm_plot <- publish_bsvar_plot(rep_cdm, preset = "paper")

The two functions answer different empirical questions. The first selects a draw using impulse responses, whereas the second uses cumulative dynamic responses. Therefore, they need not select the same posterior draw.

Posterior probabilities of restrictions

restriction_audit() evaluates the impulse-response, zero, structural, and narrative restrictions specified for a PosteriorBSVARSIGN object. For each restriction, the posterior probability is estimated by the proportion of stored posterior draws that satisfy it.

audit <- restriction_audit(post_sign, zero_tol = 1e-6)
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.
audit_focus <- audit[, c("restriction_type", "restriction", "relation",
                         "posterior_prob")]
audit_focus
#> # A tibble: 2 × 4
#>   restriction_type restriction                         relation posterior_prob
#>   <chr>            <chr>                               <chr>             <dbl>
#> 1 irf_zero         irf[productivity,productivity,0]==0 ==0                   1
#> 2 irf_sign         irf[stock_prices,productivity,0]>0  >0                    1

The argument zero_tol sets the numerical tolerance for zero restrictions. Its value should be chosen with respect to the scale of the responses and reported with the results.

plot_restriction_audit() plots the posterior probabilities of the selected restrictions. Formatted labels change their presentation but not their definitions.

plot_restriction_audit(audit, label_style = "pretty",
  restriction_types = c("irf_sign", "irf_zero"))

Restrictions imposed during estimation should generally have satisfaction probabilities near one, allowing for numerical precision and the chosen zero tolerance. A low value calls for investigation of the restriction definition, response scale and tolerance; it is not automatically evidence against the identifying assumption. A fitted restricted posterior cannot independently validate assumptions already imposed on it.

These probabilities are conditional on the estimated posterior distribution and, for zero restrictions, on the chosen tolerance. They are not sampler acceptance rates and do not describe the concentration of admissible posterior draws.

Posterior sample diagnostics

acceptance_diagnostics() summarises the stored posterior sample. It reports the number of draws, importance-weight effective sample size, max_tries, the numbers of restrictions, and the distribution of admissibility weights.

diag <- acceptance_diagnostics(post_sign, ess_threshold = 100,
  sparse_threshold = 0.10)
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.

subset(diag,
  metric %in% c(
    "posterior_draws", "effective_sample_size", "max_tries",
    "kernel_zero_share", "kernel_cv"),
  select = c(metric, value, flag, message))
#> # A tibble: 5 × 4
#>   metric                  value flag  message                                   
#>   <chr>                   <dbl> <lgl> <chr>                                     
#> 1 posterior_draws       200     FALSE ""                                        
#> 2 effective_sample_size  72.3   TRUE  "ESS below threshold."                    
#> 3 max_tries             Inf     FALSE ""                                        
#> 4 kernel_zero_share       0.11  TRUE  "Large share of near-zero admissibility w…
#> 5 kernel_cv               0.715 FALSE ""

effective_sample_size reports the upstream importance-weight ESS, $(\sum_s w_s)^2 / \sum_s w_s^2$ , computed before weighted resampling. It describes concentration of importance weights, not an autocorrelation-based MCMC convergence diagnostic. It cannot establish convergence or mixing. kernel_zero_share is the proportion of approximately zero admissibility weights, and kernel_cv measures their relative dispersion. The restriction counts and max_tries describe the search for admissible rotations.

The thresholds indicate a small effective sample or limited admissibility support. They are not general decision rules. Moreover, the stored posterior object does not contain the complete sequence of proposed and rejected draws.

The following plot uses exactly the loaded 200-draw baseline.

figure <- plot_acceptance_diagnostics(diag,
  metrics = c("effective_sample_size", "kernel_zero_share", "kernel_cv"),
  title = "Stored-sample diagnostics")

Importance-weight ESS and stored-sample weight concentration

Baseline p=4, 200 retained draws: upstream importance-weight ESS and reconstructed admissibility-weight concentration. Red markers flag the illustrative diagnostic thresholds. These metrics are neither an MCMC convergence check nor sampler acceptance rates.

Historical decompositions

Historical decompositions attribute the observed series to the estimated structural shocks. For a selected period, the posterior contributions can be reported in levels or as proportions of their total magnitude. The figures below focus on productivity and present both summaries over observations 1 to 4.

figure <- plot_hd_event(post_sign, start = 1, end = 4,
  variables = "productivity", probability = 0.90) +
  ggplot2::labs(y = "Productivity contribution (model units)",
    x = "Event start (observations 1–4)") +
  ggplot2::theme(legend.position = "none")

Posterior structural-shock contributions over observations one to four

Structural-shock contributions summed within each draw over observations 1–4. Productivity is in the transformed model-data units (optimism multiplied by 100). Points and intervals show posterior medians and 90% equal-tailed intervals.

figure <- plot_hd_event_share(post_sign, start = 1, end = 4, top_n = 4,
  variables = "productivity",
  probability = 0.90, share = "absolute")

Proportional structural-shock contributions over observations one to four

Absolute shares of event productivity contributions over observations 1–4: four contributors and Other. Shares are unitless and describe absolute magnitude rather than signed net contributions; the plot shows the absolute value of each median contribution divided by their sum.

Comparison of model specifications

The posterior probabilities and sample diagnostics can be compared across alternative sign-restricted specifications. The specifications should contain the same variables and comparable identifying restrictions. The argument names identify the specifications in the resulting tables.

The loaded post_sign_alt uses the same restrictions and input with two lags.

The restriction probabilities and posterior sample diagnostics are computed separately.

restriction_comparison <- compare_restrictions(baseline = post_sign,
  shorter_lag = post_sign_alt, zero_tol = 1e-6)
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.

diagnostic_comparison <- compare_acceptance_diagnostics(
  baseline = post_sign,
  shorter_lag = post_sign_alt,
  ess_threshold = 100,
  sparse_threshold = 0.10)
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.
#> Argument standardise is forcibly set to FALSE due to zero restrictions imposed on the diagonal element(s) of the on-impact impulse response matrix.

compare_restrictions() reports posterior restriction probabilities for each specification. compare_acceptance_diagnostics() reports the corresponding posterior sample diagnostics. These comparisons describe the estimated posterior distributions and do not rank the economic plausibility of the identifying assumptions.

plot_compare_restrictions(restriction_comparison,
  restriction_types = c("irf_sign", "irf_zero"))


plot_acceptance_diagnostics(
  diagnostic_comparison,
  metrics = c("effective_sample_size", "kernel_zero_share", "kernel_cv"))

Estimate your own sign models

This recipe matches the bundled settings and frozen input. The demonstration sampling length is not a research recommendation. Numerical results can change with R or upstream versions even with the same seeds.

optimism_data <- readRDS(system.file("extdata", "optimism.rds",
  package = "bsvarPost", mustWork = TRUE))
sign_irf <- matrix(c(0, 1, rep(NA_real_, 23)), 5, 5)
set.seed(123)
spec_sign <- bsvarSIGNs::specify_bsvarSIGN$new(optimism_data * 100, p = 4,
  sign_irf = sign_irf)
post_sign <- bsvars::estimate(spec_sign, S = 200, thin = 1,
  show_progress = FALSE)
set.seed(456)
spec_sign_alt <- bsvarSIGNs::specify_bsvarSIGN$new(optimism_data * 100, p = 2,
  sign_irf = sign_irf)
post_sign_alt <- bsvars::estimate(spec_sign_alt, S = 200, thin = 1,
  show_progress = FALSE)

Summary

For the analysis of sign-restricted models:

Report the effective sample size and the distribution of admissibility weights using acceptance_diagnostics().
Report posterior restriction probabilities using restriction_audit().
Use a most-likely-admissible impulse or cumulative response when one joint posterior draw is required.
Compare restriction probabilities and posterior sample diagnostics across economically relevant model specifications.

For posterior summaries, probability statements, response timing, plots, and tables, see Post-estimation Analysis with bsvarPost. The function reference documents filters, tolerances, and plotting arguments.