Unpooling Models for Time-Series Cross-Sectional Data

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.title[
# Unpooling Models for Time-Series Cross-Sectional Data
]
.subtitle[
## APSA 2025
]
.author[
### William Poirier
]
.author[
### Matthew J. Lebo
]
.author[
### David A. Armstrong II
]
.date[
### 2025-09-12
]

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## Plan of Presentation

1. Problems with PCSTS and Unpooling-PCSTS

2. Potential Solutions

3. An Example

4. Next Steps

---

## Pooled Cross-Sectional Time Series

- Data measured over multiple individual units, `$i$`, at mutliple points in time, `$t$`. (From `$y_t$` to `$y_{it}$`)

---

## Pooled Cross-Sectional Time Series

- Data measured over multiple individual units, `$i$`, at mutliple points in time, `$t$`. (From `$y_t$` to `$y_{it}$`)

.pull-left[
### Advantages:

1. Overcoming missing data issues in individual series;
2. Gain statisitcal power;
3. A single panel unit root test per variable;
4. Hypothesis falsified via a single coefficient.

]
.pull-right[
### Disadvantages:
]

---
## Pooled Cross-Sectional Time Series

- Data measured over multiple individual units, `$i$`, at mutliple points in time, `$t$`. (From `$y_t$` to `$y_{it}$`)

.pull-left[
### Advantages:

1. Overcoming missing data issues in individual series;
2. Gain statisitcal power;
3. A single panel unit root test per variable;
4. Hypothesis falsified via a single coefficient.

]
.pull-right[
### Disadvantages:

1. Assumes that series diagnostics apply across cases:
  - Sationarity;
  - Co-integration/Fractional integration;
  - Equation balance.
2. Assumes dynamic structure homogeneity;
3. Assumes cross-sectional independence;
4. Assumes effect homogenity.

]

.purpleb[Pooled models cannot be used to produce evidence that these assumptions may be reasonable.]

---

## Unpooling PCSTS

- Estimate a model for each individual unit `$i$`, at mutliple points in time `$t$`.

---

## Unpooling PCSTS

- Estimate a model for each individual unit `$i$`, at mutliple points in time `$t$`.

.pull-left[
### Advantages:

1. Removes the need for homogeneity assumptions;
2. Removes the need for cross-sectional independence;
3. Directly assesses heterogeneity in effects.

]
.pull-right[
### Disadvantages:
]

---

## Unpooling PCSTS

- Estimate a model for each individual unit `$i$`, at mutliple points in time `$t$`.

.pull-left[
### Advantages:

1. Removes the need for homogeneity assumptions;
2. Removes the need for cross-sectional independence;
3. Directly assesses heterogeneity in effects.

]
.pull-right[
### Disadvantages:

1. Multiply the number of diagnostics by `$i$`;
2. Reduces statistical power;
3. Reduces degrees of freedom;
4. Diagnostic tools can become useless.

**Hence, the general to specific modelling process that is the hallmark of conventional single-series analysis is not feasible as `$N$` gets large.**
]

.purpleb[We need a way to check for case heterogeneity in specification and effects, while not being bound by the uncertainty of underpowerred tests.]

---

## Potential Solutions — Bounds Method

.panelset[
  .panel[.panel-name[What?]
  Webb, Lebo, and Linn (2020):
  - ADL: `$y_t=\alpha_0+\alpha_1 y_{t-1}+\beta_0 x_t+\beta_1x_{t-1}+\varepsilon_t$`
  - LRM: `$\lambda = \frac{\beta_0+\beta_1}{1-\alpha_1}$`, gives the total effect of a unit change in x on y.
  - Use Bewley or Delta method to estimate `$\lambda$` and its SE.
  - Distribution of the LRM t-statistic depends on the univariate characteristics of variables in the model, deterministic features of DGP for y, number of IVs, and sample size.
  - Webb, Linn, and Lebo (2019) provide us with bounds, such that:
    - Below the lower bound, we would **fail to reject the null** whatever the true univariate characteristics of the series.
    - Above the upper bound, we would **reject the null** whatever the true univariate characteristics of the series.
    ] 
  .panel[.panel-name[Problem Solved?]
  - Solves part of the problem, no need to bother with Dickey-Fuller tests. ✅
  
  - But, still need to check for model specification (appropriate lag-lengths). ❌
  
  - Might be overtly conservative (less type 1 errors, a lot of type II errors). ❌
  ]
  .panel[.panel-name[Quasi-Bayesian t-statistic]
  - King et al. (2000):
    - generate sampling variance estimates for quantities that:
      1. are not normally distributed;
      2. but are derived from approximately normally distributed parameters.
  - Idea is to use this to generate the worst case scenario bounds as in Webb, Linn, and Lebo (2019).
  - Use draws from posterior distribution of numerator and denominator of LRM to calculate draws from the posterior distribution of the LRM.
  - Simulations show QBT to be superior in long (T>75) series to the original bounds and other techniques (Delta method; Hinkley distribution; and Fieller procedure) in terms of coverage rate of the 95% CI.
  - Recommend the original bounds method for shorter series.
  ]
  .panel[.panel-name[Problem Solved Strike 2?]
  - Solves part of the problem, no need to bother with Dickey-Fuller tests. ✅
  
  - But, **still need to check for model specification (appropriate lag-lengths).** ❌
  
  - Might be overtly conservative (less type 1 errors, a lot of type II errors). ✅
 ]
]

---

## Potential Solutions — Go Bayesian
.panelset[
 .panel[.panel-name[What?]
- 3 ideas:
 1. Using posterior distribution to estimate uncertainty of LRMs;
 2. Nieman and Peterson (unpublished manuscript): "semi-informed" priors on the lag of the DV (-1 to 1);
 3. Using regularizing priors via horseshoe distribution to allow the model to fit the number of lags.
- Horseshoe distribution:
<img src="APSA_uppcsts_2025_files/figure-html/unnamed-chunk-2-1.png" width="80%" style="display: block; margin: auto;" />
 ]
 .panel[.panel-name[Model 1/2]
- Model in ADL form: `$y_t = \alpha_0 + \alpha_1 y_{t-1} + \beta_1 x_t + \sum_{m=1}^{\# Lags} \theta_{m} x_{t-m} + \epsilon_t$`
- Priors, where `$\boldsymbol \gamma=[\alpha_0,\boldsymbol\beta,\boldsymbol\theta]$`:

`$$\begin{aligned}
\alpha_1 &\sim U(-1,1)\\
\gamma_{m} &\sim N(0, \lambda_{\gamma,m} \tau_\theta) \\
\lambda_{\gamma,m}^2 &\sim t_3^+(0, 1) \\
\tau_\gamma^2 &\sim t_3^+(0, 1)
\end{aligned}$$`

]
  .panel[.panel-name[Model 2/2]
- Model in ADL form: `$y_t = \alpha_0 + \alpha_1 y_{t-1} + \beta_1 x_t + \sum_{m=1}^{\# Lags} \theta_{m} x_{t-m} + \epsilon_t$`
- Priors, where `$\boldsymbol \gamma=[\alpha_0,\boldsymbol\beta,\boldsymbol\theta]$`:

`$$\begin{aligned}
\color{red}{\alpha_1} &\color{red}{\sim U(-1,1)}\\
\color{red}{\gamma_{m}} &\color{red}{\sim N(0, \lambda_{\gamma,m} \tau_\theta)} \\
\lambda_{\gamma,m}^2 &\sim t_3^+(0, 1) \\
\tau_\gamma^2 &\sim t_3^+(0, 1)
\end{aligned}$$`

]
  .panel[.panel-name[Problem Solved?]
  - Simulations show that estimator is unbiased for:
    1. Stationary X and Y;
    2. Co-integrating X and Y;
    3. Non-stationary independent X and Y.
  
  - No need to bother with Dickey-Fuller tests. ✅
  - No need to check for model specification (appropriate lag-lengths). ✅
  ]
]

---

## An Example: Elkjær & Iversen (2023) — PCSTS
.pull-left[
.purpleb[The Democratic State and Redistribution]
- View that gvt. lost their ability to redistribute income through taxes and transfers.
- RQ: What is driving changes in transfer rates over time?
- Three-class theory of redistribution (low-, middle-, or high-income).
- DV: relative transfer rate of top 1%, middle and bottom 20%.
- IVs: Top- and bottom-end inequality; Partisanship; Capital market and Trade openness; Public good consumption.
- Data from the World Inequality Database.
- 17 countries from ~1980 to 2016.
]
.pull-right[
<img src="APSA_uppcsts_2025_files/figure-html/ei22_p-1.png" width="100%" style="display: block; margin: auto;" />
.small[Replication of right panel of Figure 2 (Elkjær & Iversen, 2023, p.398)]

]

---

## An Example: Unpooling Figure 2

---

## An Example: Replicating Table 1

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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c17714ed{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c17714f6{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c17a02ec'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c17713d4"></th><th colspan="2"class="cl-c17713d5">Δ Top 1% transfer rate</th><th colspan="2"class="cl-c17713df">Δ Middle 20% transfer rate</th><th colspan="2"class="cl-c17713e9">Δ Bottom 20% transfer rate</th></tr><tr style="overflow-wrap:break-word;"><th class="cl-c17713f3"> </th><th class="cl-c17713f4">Excl. in-kind transfer</th><th class="cl-c17713fc">Incl. in-kind transfer</th><th class="cl-c1771406">Excl. in-kind transfer </th><th class="cl-c1771407">Incl. in-kind transfer </th><th class="cl-c1771408">Excl. in-kind transfer </th><th class="cl-c1771409">Incl. in-kind transfer </th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c1771410">Relative transfer ratet-1</td><td class="cl-c1771411">-0.31*</td><td class="cl-c177141a">-0.29*</td><td class="cl-c177141b">-0.35*</td><td class="cl-c177141c">-0.25*</td><td class="cl-c1771424">-0.49*</td><td class="cl-c1771425">-0.32*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(0.04)</td><td class="cl-c177142f">(0.04)</td><td class="cl-c1771438">(0.04)</td><td class="cl-c1771439">(0.04)</td><td class="cl-c177143a">(0.05)</td><td class="cl-c1771442">(0.04)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771443">Δ Top-end inequality</td><td class="cl-c1771444">2.67</td><td class="cl-c177144c">-0.48</td><td class="cl-c177144d">4.10*</td><td class="cl-c177144e">10.18*</td><td class="cl-c177144f">5.51</td><td class="cl-c1771456">8.67*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(5.40)</td><td class="cl-c177142f">(7.39)</td><td class="cl-c1771438">(0.76)</td><td class="cl-c1771439">(0.69)</td><td class="cl-c177143a">(3.62)</td><td class="cl-c1771442">(1.49)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771443">Top-end inequalityt-1</td><td class="cl-c1771444">0.74</td><td class="cl-c177144c">-2.57</td><td class="cl-c177144d">1.97*</td><td class="cl-c177144e">3.40*</td><td class="cl-c177144f">0.69</td><td class="cl-c1771456">3.49*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(2.25)</td><td class="cl-c177142f">(3.30)</td><td class="cl-c1771438">(0.37)</td><td class="cl-c1771439">(0.54)</td><td class="cl-c177143a">(1.72)</td><td class="cl-c1771442">(0.71)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771443">Δ Bottom-end inequality</td><td class="cl-c1771444">0.44</td><td class="cl-c177144c">0.73</td><td class="cl-c177144d">-0.25*</td><td class="cl-c177144e">-0.22*</td><td class="cl-c177144f">3.81*</td><td class="cl-c1771456">4.79*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(0.52)</td><td class="cl-c177142f">(0.67)</td><td class="cl-c1771438">(0.10)</td><td class="cl-c1771439">(0.09)</td><td class="cl-c177143a">(0.75)</td><td class="cl-c1771442">(0.24)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771443">Bottom-end inequalityt-1</td><td class="cl-c1771444">-0.08</td><td class="cl-c177144c">-0.14</td><td class="cl-c177144d">0.06</td><td class="cl-c177144e">0.07</td><td class="cl-c177144f">1.00</td><td class="cl-c1771456">1.56*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(0.43)</td><td class="cl-c177142f">(0.57)</td><td class="cl-c1771438">(0.07)</td><td class="cl-c1771439">(0.07)</td><td class="cl-c177143a">(0.56)</td><td class="cl-c1771442">(0.28)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771457">Δ Partisanship (left)</td><td class="cl-c1771460">-0.03</td><td class="cl-c1771461">-0.15</td><td class="cl-c1771462">0.05</td><td class="cl-c177146a">0.04</td><td class="cl-c177146b">0.09</td><td class="cl-c177146c">-0.07</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(0.82)</td><td class="cl-c177142f">(1.05)</td><td class="cl-c1771438">(0.11)</td><td class="cl-c1771439">(0.09)</td><td class="cl-c177143a">(0.59)</td><td class="cl-c1771442">(0.19)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771457">Partisanship (left)t-1</td><td class="cl-c1771460">-0.06</td><td class="cl-c1771461">-0.15</td><td class="cl-c1771462">-0.00</td><td class="cl-c177146a">-0.02</td><td class="cl-c177146b">0.23*</td><td class="cl-c177146c">0.06*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(0.10)</td><td class="cl-c177142f">(0.12)</td><td class="cl-c1771438">(0.02)</td><td class="cl-c1771439">(0.01)</td><td class="cl-c177143a">(0.06)</td><td class="cl-c1771442">(0.02)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771474">Δ Capital market openness</td><td class="cl-c1771475">5.24</td><td class="cl-c177147e">5.36</td><td class="cl-c177147f">0.24</td><td class="cl-c1771480">0.21</td><td class="cl-c1771481">-9.92*</td><td class="cl-c1771488">-0.18</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(4.33)</td><td class="cl-c177142f">(5.44)</td><td class="cl-c1771438">(0.76)</td><td class="cl-c1771439">(0.63)</td><td class="cl-c177143a">(4.93)</td><td class="cl-c1771442">(1.61)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771474">Capital market opennesst-1</td><td class="cl-c1771475">-3.71*</td><td class="cl-c177147e">-4.87*</td><td class="cl-c177147f">0.44</td><td class="cl-c1771480">0.40</td><td class="cl-c1771481">0.30</td><td class="cl-c1771488">0.23</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(1.65)</td><td class="cl-c177142f">(2.21)</td><td class="cl-c1771438">(0.26)</td><td class="cl-c1771439">(0.22)</td><td class="cl-c177143a">(1.45)</td><td class="cl-c1771442">(0.45)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771489">Δ Trade openness</td><td class="cl-c1771492">-7.42</td><td class="cl-c1771493">-6.28</td><td class="cl-c1771494">1.78</td><td class="cl-c177149c">1.21</td><td class="cl-c177149d">-3.02</td><td class="cl-c177149e">-1.90</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(5.67)</td><td class="cl-c177142f">(7.32)</td><td class="cl-c1771438">(0.91)</td><td class="cl-c1771439">(0.79)</td><td class="cl-c177143a">(3.62)</td><td class="cl-c1771442">(1.46)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771489">Trade opennesst-1</td><td class="cl-c1771492">1.85</td><td class="cl-c1771493">1.75</td><td class="cl-c1771494">0.52</td><td class="cl-c177149c">0.51</td><td class="cl-c177149d">0.55</td><td class="cl-c177149e">-0.31</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e">(2.51)</td><td class="cl-c177142f">(3.25)</td><td class="cl-c1771438">(0.36)</td><td class="cl-c1771439">(0.31)</td><td class="cl-c177143a">(1.29)</td><td class="cl-c1771442">(0.57)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c17714a6">Δ Public goods consumption</td><td class="cl-c17714a7"></td><td class="cl-c17714a8">-70.29</td><td class="cl-c17714a9"></td><td class="cl-c17714b0">16.19*</td><td class="cl-c17714ba"></td><td class="cl-c17714bb">66.16*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e"></td><td class="cl-c177142f">(52.37)</td><td class="cl-c1771438"></td><td class="cl-c1771439">(4.68)</td><td class="cl-c177143a"></td><td class="cl-c1771442">(11.01)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c17714a6">Public goods consumptiont-1</td><td class="cl-c17714a7"></td><td class="cl-c17714a8">-21.91</td><td class="cl-c17714a9"></td><td class="cl-c17714b0">3.71</td><td class="cl-c17714ba"></td><td class="cl-c17714bb">29.03*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1771426"></td><td class="cl-c177142e"></td><td class="cl-c177142f">(27.20)</td><td class="cl-c1771438"></td><td class="cl-c1771439">(2.75)</td><td class="cl-c177143a"></td><td class="cl-c1771442">(6.62)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c17714bc">R2</td><td class="cl-c17714c4">0.159</td><td class="cl-c17714c5">0.155</td><td class="cl-c17714ce">0.250</td><td class="cl-c17714cf">0.404</td><td class="cl-c17714d8">0.331</td><td class="cl-c17714d9">0.676</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c17714da">N</td><td class="cl-c17714db">576</td><td class="cl-c17714e2">576</td><td class="cl-c17714e3">576</td><td class="cl-c17714ec">576</td><td class="cl-c17714ed">576</td><td class="cl-c17714f6">576</td></tr></tbody></table></div>

---

## An Example: Replicating Table 1
<div class="tabwid"><style>.cl-c1f58b74{}.cl-c1f1bab2{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-c1f1babc{font-family:'Helvetica';font-size:6.6pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;position: relative;top:3.3pt;}.cl-c1f1babd{font-family:'Helvetica';font-size:6.6pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;position: relative;bottom:3.3pt;}.cl-c1f319d4{margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:0;padding-top:0;padding-left:0;padding-right:0;line-height: 1;background-color:transparent;}.cl-c1f319d5{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:0;padding-top:0;padding-left:0;padding-right:0;line-height: 1;background-color:transparent;}.cl-c1f32ac8{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32ad2{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32ad3{width:1.433in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32ad4{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32adc{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32add{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32ae6{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32ae7{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32af0{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32af1{width:1.433in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32afa{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32afb{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32afc{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32afd{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b04{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b05{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b0e{width:1.433in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b0f{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b18{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b19{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b1a{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b1b{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b22{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b23{width:1.433in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b2c{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b2d{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b2e{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b36{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b37{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32b38{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32c28{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32c29{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32c30{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c1f32c31{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c1f58b74'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c1f32ac8"></th><th colspan="2"class="cl-c1f32ad2">Δ Top 1% transfer rate</th><th colspan="2"class="cl-c1f32ad4">Δ Middle 20% transfer rate</th><th colspan="2"class="cl-c1f32add">Δ Bottom 20% transfer rate</th></tr><tr style="overflow-wrap:break-word;"><th class="cl-c1f32ae7"> </th><th class="cl-c1f32af0">Excl. in-kind transfer</th><th class="cl-c1f32af1">Incl. in-kind transfer</th><th class="cl-c1f32afa">Excl. in-kind transfer </th><th class="cl-c1f32afb">Incl. in-kind transfer </th><th class="cl-c1f32afc">Excl. in-kind transfer </th><th class="cl-c1f32afd">Incl. in-kind transfer </th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b04">Relative transfer ratet-1</td><td class="cl-c1f32b05">-0.31*</td><td class="cl-c1f32b0e">-0.29*</td><td class="cl-c1f32b0f">-0.35*</td><td class="cl-c1f32b18">-0.25*</td><td class="cl-c1f32b19">-0.49*</td><td class="cl-c1f32b1a">-0.32*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(0.04)</td><td class="cl-c1f32b23">(0.04)</td><td class="cl-c1f32b2c">(0.04)</td><td class="cl-c1f32b2d">(0.04)</td><td class="cl-c1f32b2e">(0.05)</td><td class="cl-c1f32b36">(0.04)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b37">Δ Top-end inequality</td><td class="cl-c1f32b38">2.67</td><td class="cl-c1f32b40">-0.48</td><td class="cl-c1f32b41">4.10*</td><td class="cl-c1f32b4a">10.18*</td><td class="cl-c1f32b4b">5.51</td><td class="cl-c1f32b4c">8.67*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(5.40)</td><td class="cl-c1f32b23">(7.39)</td><td class="cl-c1f32b2c">(0.76)</td><td class="cl-c1f32b2d">(0.69)</td><td class="cl-c1f32b2e">(3.62)</td><td class="cl-c1f32b36">(1.49)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b54">Top-end inequalityt-1</td><td class="cl-c1f32b55">0.74</td><td class="cl-c1f32b5e">-2.57</td><td class="cl-c1f32b5f">1.97*</td><td class="cl-c1f32b60">3.40*</td><td class="cl-c1f32b68">0.69</td><td class="cl-c1f32b69">3.49*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b72"></td><td class="cl-c1f32b73">(2.25)</td><td class="cl-c1f32b74">(3.30)</td><td class="cl-c1f32b7c">(0.37)</td><td class="cl-c1f32b7d">(0.54)</td><td class="cl-c1f32b7e">(1.72)</td><td class="cl-c1f32b86">(0.71)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b37">Δ Bottom-end inequality</td><td class="cl-c1f32b38">0.44</td><td class="cl-c1f32b40">0.73</td><td class="cl-c1f32b41">-0.25*</td><td class="cl-c1f32b4a">-0.22*</td><td class="cl-c1f32b4b">3.81*</td><td class="cl-c1f32b4c">4.79*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(0.52)</td><td class="cl-c1f32b23">(0.67)</td><td class="cl-c1f32b2c">(0.10)</td><td class="cl-c1f32b2d">(0.09)</td><td class="cl-c1f32b2e">(0.75)</td><td class="cl-c1f32b36">(0.24)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b37">Bottom-end inequalityt-1</td><td class="cl-c1f32b38">-0.08</td><td class="cl-c1f32b40">-0.14</td><td class="cl-c1f32b41">0.06</td><td class="cl-c1f32b4a">0.07</td><td class="cl-c1f32b4b">1.00</td><td class="cl-c1f32b4c">1.56*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(0.43)</td><td class="cl-c1f32b23">(0.57)</td><td class="cl-c1f32b2c">(0.07)</td><td class="cl-c1f32b2d">(0.07)</td><td class="cl-c1f32b2e">(0.56)</td><td class="cl-c1f32b36">(0.28)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b90">Δ Partisanship (left)</td><td class="cl-c1f32b91">-0.03</td><td class="cl-c1f32b92">-0.15</td><td class="cl-c1f32b9a">0.05</td><td class="cl-c1f32b9b">0.04</td><td class="cl-c1f32ba4">0.09</td><td class="cl-c1f32ba5">-0.07</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(0.82)</td><td class="cl-c1f32b23">(1.05)</td><td class="cl-c1f32b2c">(0.11)</td><td class="cl-c1f32b2d">(0.09)</td><td class="cl-c1f32b2e">(0.59)</td><td class="cl-c1f32b36">(0.19)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b90">Partisanship (left)t-1</td><td class="cl-c1f32b91">-0.06</td><td class="cl-c1f32b92">-0.15</td><td class="cl-c1f32b9a">-0.00</td><td class="cl-c1f32b9b">-0.02</td><td class="cl-c1f32ba4">0.23*</td><td class="cl-c1f32ba5">0.06*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(0.10)</td><td class="cl-c1f32b23">(0.12)</td><td class="cl-c1f32b2c">(0.02)</td><td class="cl-c1f32b2d">(0.01)</td><td class="cl-c1f32b2e">(0.06)</td><td class="cl-c1f32b36">(0.02)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32ba6">Δ Capital market openness</td><td class="cl-c1f32ba7">5.24</td><td class="cl-c1f32bae">5.36</td><td class="cl-c1f32baf">0.24</td><td class="cl-c1f32bb8">0.21</td><td class="cl-c1f32bc2">-9.92*</td><td class="cl-c1f32bcc">-0.18</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(4.33)</td><td class="cl-c1f32b23">(5.44)</td><td class="cl-c1f32b2c">(0.76)</td><td class="cl-c1f32b2d">(0.63)</td><td class="cl-c1f32b2e">(4.93)</td><td class="cl-c1f32b36">(1.61)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32ba6">Capital market opennesst-1</td><td class="cl-c1f32ba7">-3.71*</td><td class="cl-c1f32bae">-4.87*</td><td class="cl-c1f32baf">0.44</td><td class="cl-c1f32bb8">0.40</td><td class="cl-c1f32bc2">0.30</td><td class="cl-c1f32bcc">0.23</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(1.65)</td><td class="cl-c1f32b23">(2.21)</td><td class="cl-c1f32b2c">(0.26)</td><td class="cl-c1f32b2d">(0.22)</td><td class="cl-c1f32b2e">(1.45)</td><td class="cl-c1f32b36">(0.45)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32bcd">Δ Trade openness</td><td class="cl-c1f32bce">-7.42</td><td class="cl-c1f32bd6">-6.28</td><td class="cl-c1f32bd7">1.78</td><td class="cl-c1f32bd8">1.21</td><td class="cl-c1f32be0">-3.02</td><td class="cl-c1f32be1">-1.90</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(5.67)</td><td class="cl-c1f32b23">(7.32)</td><td class="cl-c1f32b2c">(0.91)</td><td class="cl-c1f32b2d">(0.79)</td><td class="cl-c1f32b2e">(3.62)</td><td class="cl-c1f32b36">(1.46)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32bcd">Trade opennesst-1</td><td class="cl-c1f32bce">1.85</td><td class="cl-c1f32bd6">1.75</td><td class="cl-c1f32bd7">0.52</td><td class="cl-c1f32bd8">0.51</td><td class="cl-c1f32be0">0.55</td><td class="cl-c1f32be1">-0.31</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22">(2.51)</td><td class="cl-c1f32b23">(3.25)</td><td class="cl-c1f32b2c">(0.36)</td><td class="cl-c1f32b2d">(0.31)</td><td class="cl-c1f32b2e">(1.29)</td><td class="cl-c1f32b36">(0.57)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32bea">Δ Public goods consumption</td><td class="cl-c1f32beb"></td><td class="cl-c1f32bec">-70.29</td><td class="cl-c1f32bf4"></td><td class="cl-c1f32bf5">16.19*</td><td class="cl-c1f32bf6"></td><td class="cl-c1f32bfe">66.16*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22"></td><td class="cl-c1f32b23">(52.37)</td><td class="cl-c1f32b2c"></td><td class="cl-c1f32b2d">(4.68)</td><td class="cl-c1f32b2e"></td><td class="cl-c1f32b36">(11.01)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32bea">Public goods consumptiont-1</td><td class="cl-c1f32beb"></td><td class="cl-c1f32bec">-21.91</td><td class="cl-c1f32bf4"></td><td class="cl-c1f32bf5">3.71</td><td class="cl-c1f32bf6"></td><td class="cl-c1f32bfe">29.03*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32b1b"></td><td class="cl-c1f32b22"></td><td class="cl-c1f32b23">(27.20)</td><td class="cl-c1f32b2c"></td><td class="cl-c1f32b2d">(2.75)</td><td class="cl-c1f32b2e"></td><td class="cl-c1f32b36">(6.62)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32bff">R2</td><td class="cl-c1f32c08">0.159</td><td class="cl-c1f32c09">0.155</td><td class="cl-c1f32c0a">0.250</td><td class="cl-c1f32c12">0.404</td><td class="cl-c1f32c13">0.331</td><td class="cl-c1f32c14">0.676</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c1f32c1c">N</td><td class="cl-c1f32c26">576</td><td class="cl-c1f32c27">576</td><td class="cl-c1f32c28">576</td><td class="cl-c1f32c29">576</td><td class="cl-c1f32c30">576</td><td class="cl-c1f32c31">576</td></tr></tbody></table></div>

---

## An Example: Replicating Table 1

<div class="tabwid"><style>.cl-c26a7f38{}.cl-c26690e4{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-c26690ee{font-family:'Helvetica';font-size:6.6pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;position: relative;top:3.3pt;}.cl-c26690f8{font-family:'Helvetica';font-size:6.6pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;position: relative;bottom:3.3pt;}.cl-c267f52e{margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:0;padding-top:0;padding-left:0;padding-right:0;line-height: 1;background-color:transparent;}.cl-c267f538{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:0;padding-top:0;padding-left:0;padding-right:0;line-height: 1;background-color:transparent;}.cl-c2680956{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680960{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680961{width:1.433in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680962{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c268096a{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c268096b{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680974{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680975{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680976{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c268097e{width:1.433in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c268097f{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680980{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680988{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680989{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c268098a{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680a8d{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680a8e{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680a96{width:1.433in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680a97{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680aa0{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680aa1{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680aa2{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680aaa{width:2.036in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680aab{width:1.484in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680aac{width:1.433in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680ab4{width:1.526in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680ab5{width:1.476in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680abe{width:1.569in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2680abf{width:1.518in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c26a7f38'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c2680956"></th><th colspan="2"class="cl-c2680960">Δ Top 1% transfer rate</th><th colspan="2"class="cl-c2680962">Δ Middle 20% transfer rate</th><th colspan="2"class="cl-c268096b">Δ Bottom 20% transfer rate</th></tr><tr style="overflow-wrap:break-word;"><th class="cl-c2680975"> </th><th class="cl-c2680976">Excl. in-kind transfer</th><th class="cl-c268097e">Incl. in-kind transfer</th><th class="cl-c268097f">Excl. in-kind transfer </th><th class="cl-c2680980">Incl. in-kind transfer </th><th class="cl-c2680988">Excl. in-kind transfer </th><th class="cl-c2680989">Incl. in-kind transfer </th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c268098a">Relative transfer ratet-1</td><td class="cl-c2680992">-0.31*</td><td class="cl-c2680993">-0.29*</td><td class="cl-c2680994">-0.35*</td><td class="cl-c268099c">-0.25*</td><td class="cl-c268099d">-0.49*</td><td class="cl-c26809a6">-0.32*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(0.04)</td><td class="cl-c26809b1">(0.04)</td><td class="cl-c26809ba">(0.04)</td><td class="cl-c26809bb">(0.04)</td><td class="cl-c26809c4">(0.05)</td><td class="cl-c26809c5">(0.04)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809ce">Δ Top-end inequality</td><td class="cl-c26809cf">2.67</td><td class="cl-c26809d0">-0.48</td><td class="cl-c26809d8">4.10*</td><td class="cl-c26809d9">10.18*</td><td class="cl-c26809e2">5.51</td><td class="cl-c26809e3">8.67*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(5.40)</td><td class="cl-c26809b1">(7.39)</td><td class="cl-c26809ba">(0.76)</td><td class="cl-c26809e4">(0.69)</td><td class="cl-c26809c4">(3.62)</td><td class="cl-c26809c5">(1.49)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809ec">Top-end inequalityt-1</td><td class="cl-c26809ed">0.74</td><td class="cl-c26809f6">-2.57</td><td class="cl-c26809f7">1.97*</td><td class="cl-c2680a00">3.40*</td><td class="cl-c2680a01">0.69</td><td class="cl-c2680a02">3.49*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a0a"></td><td class="cl-c2680a0b">(2.25)</td><td class="cl-c2680a0c">(3.30)</td><td class="cl-c2680a14">(0.37)</td><td class="cl-c2680a15">(0.54)</td><td class="cl-c2680a1e">(1.72)</td><td class="cl-c2680a1f">(0.71)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809ce">Δ Bottom-end inequality</td><td class="cl-c26809cf">0.44</td><td class="cl-c26809d0">0.73</td><td class="cl-c26809d8">-0.25*</td><td class="cl-c26809d9">-0.22*</td><td class="cl-c26809e2">3.81*</td><td class="cl-c26809e3">4.79*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(0.52)</td><td class="cl-c26809b1">(0.67)</td><td class="cl-c26809ba">(0.10)</td><td class="cl-c26809bb">(0.09)</td><td class="cl-c26809c4">(0.75)</td><td class="cl-c26809c5">(0.24)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809ce">Bottom-end inequalityt-1</td><td class="cl-c26809cf">-0.08</td><td class="cl-c26809d0">-0.14</td><td class="cl-c26809d8">0.06</td><td class="cl-c26809d9">0.07</td><td class="cl-c26809e2">1.00</td><td class="cl-c26809e3">1.56*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(0.43)</td><td class="cl-c26809b1">(0.57)</td><td class="cl-c26809ba">(0.07)</td><td class="cl-c26809bb">(0.07)</td><td class="cl-c26809c4">(0.56)</td><td class="cl-c26809c5">(0.28)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a20">Δ Partisanship (left)</td><td class="cl-c2680a28">-0.03</td><td class="cl-c2680a29">-0.15</td><td class="cl-c2680a2a">0.05</td><td class="cl-c2680a32">0.04</td><td class="cl-c2680a33">0.09</td><td class="cl-c2680a3c">-0.07</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(0.82)</td><td class="cl-c26809b1">(1.05)</td><td class="cl-c26809ba">(0.11)</td><td class="cl-c26809bb">(0.09)</td><td class="cl-c26809c4">(0.59)</td><td class="cl-c26809c5">(0.19)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a20">Partisanship (left)t-1</td><td class="cl-c2680a28">-0.06</td><td class="cl-c2680a29">-0.15</td><td class="cl-c2680a2a">-0.00</td><td class="cl-c2680a32">-0.02</td><td class="cl-c2680a33">0.23*</td><td class="cl-c2680a3c">0.06*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(0.10)</td><td class="cl-c26809b1">(0.12)</td><td class="cl-c26809ba">(0.02)</td><td class="cl-c26809bb">(0.01)</td><td class="cl-c26809c4">(0.06)</td><td class="cl-c26809c5">(0.02)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a3d">Δ Capital market openness</td><td class="cl-c2680a46">5.24</td><td class="cl-c2680a47">5.36</td><td class="cl-c2680a48">0.24</td><td class="cl-c2680a50">0.21</td><td class="cl-c2680a51">-9.92*</td><td class="cl-c2680a5a">-0.18</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(4.33)</td><td class="cl-c26809b1">(5.44)</td><td class="cl-c26809ba">(0.76)</td><td class="cl-c26809bb">(0.63)</td><td class="cl-c26809c4">(4.93)</td><td class="cl-c26809c5">(1.61)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a3d">Capital market opennesst-1</td><td class="cl-c2680a46">-3.71*</td><td class="cl-c2680a47">-4.87*</td><td class="cl-c2680a48">0.44</td><td class="cl-c2680a50">0.40</td><td class="cl-c2680a51">0.30</td><td class="cl-c2680a5a">0.23</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(1.65)</td><td class="cl-c26809b1">(2.21)</td><td class="cl-c26809ba">(0.26)</td><td class="cl-c26809bb">(0.22)</td><td class="cl-c26809c4">(1.45)</td><td class="cl-c26809c5">(0.45)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a5b">Δ Trade openness</td><td class="cl-c2680a5c">-7.42</td><td class="cl-c2680a64">-6.28</td><td class="cl-c2680a65">1.78</td><td class="cl-c2680a6e">1.21</td><td class="cl-c2680a6f">-3.02</td><td class="cl-c2680a70">-1.90</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(5.67)</td><td class="cl-c26809b1">(7.32)</td><td class="cl-c26809ba">(0.91)</td><td class="cl-c26809bb">(0.79)</td><td class="cl-c26809c4">(3.62)</td><td class="cl-c26809c5">(1.46)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a5b">Trade opennesst-1</td><td class="cl-c2680a5c">1.85</td><td class="cl-c2680a64">1.75</td><td class="cl-c2680a65">0.52</td><td class="cl-c2680a6e">0.51</td><td class="cl-c2680a6f">0.55</td><td class="cl-c2680a70">-0.31</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0">(2.51)</td><td class="cl-c26809b1">(3.25)</td><td class="cl-c26809ba">(0.36)</td><td class="cl-c26809bb">(0.31)</td><td class="cl-c26809c4">(1.29)</td><td class="cl-c26809c5">(0.57)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a78">Δ Public goods consumption</td><td class="cl-c2680a79"></td><td class="cl-c2680a82">-70.29</td><td class="cl-c2680a83"></td><td class="cl-c2680a84">16.19*</td><td class="cl-c2680a85"></td><td class="cl-c2680a8c">66.16*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0"></td><td class="cl-c26809b1">(52.37)</td><td class="cl-c26809ba"></td><td class="cl-c26809bb">(4.68)</td><td class="cl-c26809c4"></td><td class="cl-c26809c5">(11.01)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a78">Public goods consumptiont-1</td><td class="cl-c2680a79"></td><td class="cl-c2680a82">-21.91</td><td class="cl-c2680a83"></td><td class="cl-c2680a84">3.71</td><td class="cl-c2680a85"></td><td class="cl-c2680a8c">29.03*</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c26809a7"></td><td class="cl-c26809b0"></td><td class="cl-c26809b1">(27.20)</td><td class="cl-c26809ba"></td><td class="cl-c26809bb">(2.75)</td><td class="cl-c26809c4"></td><td class="cl-c26809c5">(6.62)</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680a8d">R2</td><td class="cl-c2680a8e">0.159</td><td class="cl-c2680a96">0.155</td><td class="cl-c2680a97">0.250</td><td class="cl-c2680aa0">0.404</td><td class="cl-c2680aa1">0.331</td><td class="cl-c2680aa2">0.676</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2680aaa">N</td><td class="cl-c2680aab">576</td><td class="cl-c2680aac">576</td><td class="cl-c2680ab4">576</td><td class="cl-c2680ab5">576</td><td class="cl-c2680abe">576</td><td class="cl-c2680abf">576</td></tr></tbody></table></div>

---

## An Example: Table 1 — Comparing Models

<div class="tabwid"><style>.cl-c276ba64{}.cl-c27359b4{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-c27359be{font-family:'Helvetica';font-size:6.6pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;position: relative;top:3.3pt;}.cl-c274a1fc{margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:0;padding-top:0;padding-left:0;padding-right:0;line-height: 1;background-color:transparent;}.cl-c274a206{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:0;padding-top:0;padding-left:0;padding-right:0;line-height: 1;background-color:transparent;}.cl-c274b386{width:1.973in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b387{width:1.051in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b390{width:0.966in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b391{width:1.008in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b392{width:1.973in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b39a{width:1.051in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b39b{width:0.966in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b3a4{width:1.008in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b3a5{width:1.973in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b430{width:1.008in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b43a{width:1.973in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b43b{width:1.051in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b444{width:0.966in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c274b44e{width:1.008in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c276ba64'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c274b386"> </th><th class="cl-c274b387">ADL</th><th class="cl-c274b390">ADL RP (1)</th><th class="cl-c274b391">ADL RP (1:5)</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c274b392">Relative transfer ratet-1</td><td class="cl-c274b39a">0.75</td><td class="cl-c274b39b">0.74</td><td class="cl-c274b3a4">0.68</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[0.68, 0.83]</td><td class="cl-c274b3af">[0.68, 0.79]</td><td class="cl-c274b3b8">[0.61, 0.75]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3b9">Top-end inequality</td><td class="cl-c274b3c2">10.18</td><td class="cl-c274b3c3">8.93</td><td class="cl-c274b3c4">8.64</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3cc"></td><td class="cl-c274b3cd">[8.83, 11.53]</td><td class="cl-c274b3d6">[7.68, 10.19]</td><td class="cl-c274b3d7">[7.26, 10.04]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3b9">Top-end inequalityt-1</td><td class="cl-c274b3c2">-6.78</td><td class="cl-c274b3c3">-5.45</td><td class="cl-c274b3c4">-4.89</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3cc"></td><td class="cl-c274b3cd">[-8.40, -5.16]</td><td class="cl-c274b3d6">[-6.86, -4.02]</td><td class="cl-c274b3d7">[-6.53, -3.20]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3d8">Bottom-end inequality</td><td class="cl-c274b3e0">-0.22</td><td class="cl-c274b3e1">-0.20</td><td class="cl-c274b3ea">-0.11</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[-0.40, -0.03]</td><td class="cl-c274b3af">[-0.35, -0.03]</td><td class="cl-c274b3b8">[-0.29, 0.06]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3d8">Bottom-end inequalityt-1</td><td class="cl-c274b3e0">0.29</td><td class="cl-c274b3e1">0.27</td><td class="cl-c274b3ea">0.16</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[0.10, 0.47]</td><td class="cl-c274b3af">[0.11, 0.43]</td><td class="cl-c274b3b8">[-0.01, 0.39]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3eb">Partisanship (left)</td><td class="cl-c274b3f4">0.04</td><td class="cl-c274b3f5">0.01</td><td class="cl-c274b3fe">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[-0.13, 0.21]</td><td class="cl-c274b3af">[-0.13, 0.19]</td><td class="cl-c274b3b8">[-0.11, 0.26]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3eb">Partisanship (left)t-1</td><td class="cl-c274b3f4">-0.06</td><td class="cl-c274b3f5">-0.03</td><td class="cl-c274b3fe">-0.04</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[-0.22, 0.11]</td><td class="cl-c274b3af">[-0.21, 0.11]</td><td class="cl-c274b3b8">[-0.42, 0.09]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3ff">Capital market openness</td><td class="cl-c274b400">0.21</td><td class="cl-c274b408">0.40</td><td class="cl-c274b409">0.49</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[-1.02, 1.45]</td><td class="cl-c274b3af">[-0.52, 1.15]</td><td class="cl-c274b3b8">[-0.21, 1.15]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3ff">Capital market opennesst-1</td><td class="cl-c274b400">0.19</td><td class="cl-c274b408">0.02</td><td class="cl-c274b409">0.00</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[-1.03, 1.41]</td><td class="cl-c274b3af">[-0.71, 0.92]</td><td class="cl-c274b3b8">[-0.33, 0.33]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b412">Trade openness</td><td class="cl-c274b413">1.21</td><td class="cl-c274b41c">0.54</td><td class="cl-c274b41d">0.74</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[-0.34, 2.77]</td><td class="cl-c274b3af">[-0.30, 1.59]</td><td class="cl-c274b3b8">[-0.03, 1.50]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b412">Trade opennesst-1</td><td class="cl-c274b413">-0.71</td><td class="cl-c274b41c">-0.03</td><td class="cl-c274b41d">-0.00</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[-2.28, 0.86]</td><td class="cl-c274b3af">[-1.11, 0.72]</td><td class="cl-c274b3b8">[-0.36, 0.29]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b41e">Public goods consumption</td><td class="cl-c274b426">16.19</td><td class="cl-c274b427">3.54</td><td class="cl-c274b430">2.12</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b3a5"></td><td class="cl-c274b3ae">[6.99, 25.38]</td><td class="cl-c274b3af">[-0.20, 8.19]</td><td class="cl-c274b3b8">[-1.71, 6.19]</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b41e">Public goods consumptiont-1</td><td class="cl-c274b426">-12.48</td><td class="cl-c274b427">-0.07</td><td class="cl-c274b430">-0.00</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c274b43a"></td><td class="cl-c274b43b">[-21.57, -3.39]</td><td class="cl-c274b444">[-3.58, 1.10]</td><td class="cl-c274b44e">[-0.41, 0.34]</td></tr></tbody></table></div>

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## An Example: Unpooled LRM with QBT
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solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45ba{width:0.661in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45bb{width:1.19in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45c4{width:1.222in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45c5{width:0.617in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45c6{width:0.887in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45ce{width:0.963in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45cf{width:1.558in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45d8{width:0.661in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45d9{width:1.19in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45e2{width:1.222in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45e3{width:0.617in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45e4{width:0.887in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45ec{width:0.963in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45ed{width:1.558in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b45ee{width:0.661in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4600{width:0.887in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b460a{width:0.963in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b460b{width:1.558in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4628{width:1.558in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4629{width:0.661in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b462a{width:1.19in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4632{width:1.222in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4633{width:0.617in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4634{width:0.887in;background-color:rgba(242, 237, 247, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4646{width:1.19in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4650{width:1.222in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4651{width:0.617in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4682{width:1.222in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4683{width:0.617in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b468c{width:0.887in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b468d{width:0.963in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b468e{width:1.558in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4696{width:0.661in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4697{width:1.19in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b4698{width:1.222in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46a0{width:0.617in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46a1{width:0.887in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46aa{width:0.963in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46ab{width:1.558in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46b4{width:0.661in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46b5{width:1.19in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46be{width:1.222in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46bf{width:0.617in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46c8{width:0.887in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46c9{width:0.963in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46ca{width:1.558in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46cb{width:0.661in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46d2{width:1.19in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46d3{width:1.222in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46d4{width:0.617in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46dc{width:0.887in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46dd{width:0.963in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46e6{width:1.558in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46e7{width:0.661in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46f0{width:1.19in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46f1{width:1.222in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46f2{width:0.617in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46f3{width:0.887in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c28b46fa{width:0.963in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c28d5cec'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c28b45b0">Country</th><th class="cl-c28b45ba">LRM</th><th class="cl-c28b45ba">SE</th><th class="cl-c28b45bb">95% QBT-L</th><th class="cl-c28b45c4">95% QBT-H</th><th class="cl-c28b45c5">t</th><th class="cl-c28b45c6">t-Critical</th><th class="cl-c28b45ce">MC Error</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c28b45cf">Germany</td><td class="cl-c28b45d8">19.48</td><td class="cl-c28b45d8">3.31</td><td class="cl-c28b45d9">12.29</td><td class="cl-c28b45e2">28.14</td><td class="cl-c28b45e3">5.88</td><td class="cl-c28b45e4">3.38</td><td class="cl-c28b45ec">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b45ed">Italy</td><td class="cl-c28b45ee">14.07</td><td class="cl-c28b45ee">2.73</td><td class="cl-c28b45f6">8.55</td><td class="cl-c28b45f7">20.20</td><td class="cl-c28b45f8">5.15</td><td class="cl-c28b4600">3.29</td><td class="cl-c28b460a">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b45ed">Norway</td><td class="cl-c28b45ee">18.74</td><td class="cl-c28b45ee">3.98</td><td class="cl-c28b45f6">9.02</td><td class="cl-c28b45f7">32.51</td><td class="cl-c28b45f8">4.71</td><td class="cl-c28b4600">3.33</td><td class="cl-c28b460a">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b460b">Ireland</td><td class="cl-c28b460c">17.92</td><td class="cl-c28b460c">5.21</td><td class="cl-c28b460d">9.11</td><td class="cl-c28b4614">34.07</td><td class="cl-c28b4615">3.44</td><td class="cl-c28b461e">3.32</td><td class="cl-c28b461f">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b4628">Greece</td><td class="cl-c28b4629">28.55</td><td class="cl-c28b4629">8.30</td><td class="cl-c28b462a">14.60</td><td class="cl-c28b4632">75.76</td><td class="cl-c28b4633">3.44</td><td class="cl-c28b4634">3.31</td><td class="cl-c28b4635">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b463c">Sweden</td><td class="cl-c28b463d">8.36</td><td class="cl-c28b463d">3.74</td><td class="cl-c28b4646">1.08</td><td class="cl-c28b4650">19.75</td><td class="cl-c28b4651">2.24</td><td class="cl-c28b4652">3.32</td><td class="cl-c28b465a">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b465b">France</td><td class="cl-c28b465c">6.49</td><td class="cl-c28b465c">4.38</td><td class="cl-c28b4664">-6.30</td><td class="cl-c28b4665">31.12</td><td class="cl-c28b466e">1.48</td><td class="cl-c28b466f">3.33</td><td class="cl-c28b4670">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b4678">United Kingdom</td><td class="cl-c28b4679">5.09</td><td class="cl-c28b4679">4.96</td><td class="cl-c28b467a">-7.83</td><td class="cl-c28b4682">17.83</td><td class="cl-c28b4683">1.03</td><td class="cl-c28b468c">3.29</td><td class="cl-c28b468d">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b463c">United States</td><td class="cl-c28b463d">3.13</td><td class="cl-c28b463d">4.93</td><td class="cl-c28b4646">-21.05</td><td class="cl-c28b4650">14.25</td><td class="cl-c28b4651">0.63</td><td class="cl-c28b4652">3.46</td><td class="cl-c28b465a">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b468e">Netherlands</td><td class="cl-c28b4696">4.87</td><td class="cl-c28b4696">8.57</td><td class="cl-c28b4697">-24.72</td><td class="cl-c28b4698">20.67</td><td class="cl-c28b46a0">0.57</td><td class="cl-c28b46a1">3.40</td><td class="cl-c28b46aa">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b463c">Switzerland</td><td class="cl-c28b463d">0.89</td><td class="cl-c28b463d">4.51</td><td class="cl-c28b4646">-9.92</td><td class="cl-c28b4650">11.03</td><td class="cl-c28b4651">0.20</td><td class="cl-c28b4652">3.41</td><td class="cl-c28b465a">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b465b">Denmark</td><td class="cl-c28b465c">0.52</td><td class="cl-c28b465c">3.09</td><td class="cl-c28b4664">-5.70</td><td class="cl-c28b4665">8.01</td><td class="cl-c28b466e">0.17</td><td class="cl-c28b466f">3.36</td><td class="cl-c28b4670">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b46ab">Spain</td><td class="cl-c28b46b4">-1.45</td><td class="cl-c28b46b4">12.14</td><td class="cl-c28b46b5">-98.20</td><td class="cl-c28b46be">47.81</td><td class="cl-c28b46bf">-0.12</td><td class="cl-c28b46c8">3.37</td><td class="cl-c28b46c9">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b46ca">Portugal</td><td class="cl-c28b46cb">-3.18</td><td class="cl-c28b46cb">11.84</td><td class="cl-c28b46d2">-40.67</td><td class="cl-c28b46d3">20.26</td><td class="cl-c28b46d4">-0.27</td><td class="cl-c28b46dc">3.31</td><td class="cl-c28b46dd">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c28b46e6">Finland</td><td class="cl-c28b46e7">-6.46</td><td class="cl-c28b46e7">22.51</td><td class="cl-c28b46f0">-209.73</td><td class="cl-c28b46f1">180.72</td><td class="cl-c28b46f2">-0.29</td><td class="cl-c28b46f3">3.34</td><td class="cl-c28b46fa">0.02</td></tr></tbody></table></div>

---

## An Example: Unpooled LRM with QBT
<div class="tabwid"><style>.cl-c2a17f10{}.cl-c29e0010{font-family:'Helvetica';font-size:14pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-c29f4484{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2pt;padding-top:2pt;padding-left:2pt;padding-right:2pt;line-height: 1;background-color:transparent;}.cl-c29f448e{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2pt;padding-top:2pt;padding-left:2pt;padding-right:2pt;line-height: 1;background-color:transparent;}.cl-c29f5370{width:1.558in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54b2{width:0.887in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54ba{width:0.963in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54c4{width:1.558in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54c5{width:0.661in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54ce{width:1.19in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54cf{width:1.222in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54d0{width:0.617in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54d8{width:0.887in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c29f54d9{width:0.963in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c2a17f10'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c29f5370">Country</th><th class="cl-c29f537a">LRM</th><th class="cl-c29f537a">SE</th><th class="cl-c29f537b">95% QBT-L</th><th class="cl-c29f537c">95% QBT-H</th><th class="cl-c29f5384">t</th><th class="cl-c29f5385">t-Critical</th><th class="cl-c29f538e">MC Error</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c29f5398">Germany</td><td class="cl-c29f5399">19.48</td><td class="cl-c29f5399">3.31</td><td class="cl-c29f539a">12.29</td><td class="cl-c29f53a2">28.14</td><td class="cl-c29f53a3">5.88</td><td class="cl-c29f53a4">3.38</td><td class="cl-c29f53a5">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f53ac">Italy</td><td class="cl-c29f53ad">14.07</td><td class="cl-c29f53ad">2.73</td><td class="cl-c29f53ae">8.55</td><td class="cl-c29f53b6">20.20</td><td class="cl-c29f53b7">5.15</td><td class="cl-c29f53b8">3.29</td><td class="cl-c29f53c0">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f53ac">Norway</td><td class="cl-c29f53ad">18.74</td><td class="cl-c29f53ad">3.98</td><td class="cl-c29f53ae">9.02</td><td class="cl-c29f53b6">32.51</td><td class="cl-c29f53b7">4.71</td><td class="cl-c29f53b8">3.33</td><td class="cl-c29f53c0">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f53c1">Ireland</td><td class="cl-c29f53ca">17.92</td><td class="cl-c29f53ca">5.21</td><td class="cl-c29f53cb">9.11</td><td class="cl-c29f53cc">34.07</td><td class="cl-c29f53d4">3.44</td><td class="cl-c29f53d5">3.32</td><td class="cl-c29f53de">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f53e8">Greece</td><td class="cl-c29f53e9">28.55</td><td class="cl-c29f53e9">8.30</td><td class="cl-c29f53ea">14.60</td><td class="cl-c29f53f2">75.76</td><td class="cl-c29f53f3">3.44</td><td class="cl-c29f53f4">3.31</td><td class="cl-c29f53f5">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f53fc">Sweden</td><td class="cl-c29f53fd">8.36</td><td class="cl-c29f53fd">3.74</td><td class="cl-c29f53fe">1.08</td><td class="cl-c29f5406">19.75</td><td class="cl-c29f5407">2.24</td><td class="cl-c29f5408">3.32</td><td class="cl-c29f5410">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f5411">France</td><td class="cl-c29f5412">6.49</td><td class="cl-c29f5412">4.38</td><td class="cl-c29f541a">-6.30</td><td class="cl-c29f541b">31.12</td><td class="cl-c29f5424">1.48</td><td class="cl-c29f5425">3.33</td><td class="cl-c29f5426">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f542e">United Kingdom</td><td class="cl-c29f542f">5.09</td><td class="cl-c29f542f">4.96</td><td class="cl-c29f5430">-7.83</td><td class="cl-c29f5431">17.83</td><td class="cl-c29f5438">1.03</td><td class="cl-c29f5439">3.29</td><td class="cl-c29f543a">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f5442">United States</td><td class="cl-c29f5443">3.13</td><td class="cl-c29f5443">4.93</td><td class="cl-c29f544c">-21.05</td><td class="cl-c29f544d">14.25</td><td class="cl-c29f5456">0.63</td><td class="cl-c29f5457">3.46</td><td class="cl-c29f5460">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f5461">Netherlands</td><td class="cl-c29f5462">4.87</td><td class="cl-c29f5462">8.57</td><td class="cl-c29f546a">-24.72</td><td class="cl-c29f546b">20.67</td><td class="cl-c29f5474">0.57</td><td class="cl-c29f5475">3.40</td><td class="cl-c29f547e">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f5442">Switzerland</td><td class="cl-c29f5443">0.89</td><td class="cl-c29f5443">4.51</td><td class="cl-c29f544c">-9.92</td><td class="cl-c29f544d">11.03</td><td class="cl-c29f5456">0.20</td><td class="cl-c29f5457">3.41</td><td class="cl-c29f5460">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f5411">Denmark</td><td class="cl-c29f5412">0.52</td><td class="cl-c29f5412">3.09</td><td class="cl-c29f541a">-5.70</td><td class="cl-c29f541b">8.01</td><td class="cl-c29f5424">0.17</td><td class="cl-c29f5425">3.36</td><td class="cl-c29f5426">0.02</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f547f">Spain</td><td class="cl-c29f5488">-1.45</td><td class="cl-c29f5488">12.14</td><td class="cl-c29f5489">-98.20</td><td class="cl-c29f5492">47.81</td><td class="cl-c29f5493">-0.12</td><td class="cl-c29f5494">3.37</td><td class="cl-c29f549c">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f549d">Portugal</td><td class="cl-c29f54a6">-3.18</td><td class="cl-c29f54a6">11.84</td><td class="cl-c29f54a7">-40.67</td><td class="cl-c29f54b0">20.26</td><td class="cl-c29f54b1">-0.27</td><td class="cl-c29f54b2">3.31</td><td class="cl-c29f54ba">0.03</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c29f54c4">Finland</td><td class="cl-c29f54c5">-6.46</td><td class="cl-c29f54c5">22.51</td><td class="cl-c29f54ce">-209.73</td><td class="cl-c29f54cf">180.72</td><td class="cl-c29f54d0">-0.29</td><td class="cl-c29f54d8">3.34</td><td class="cl-c29f54d9">0.02</td></tr></tbody></table></div>

---

## An Example: Unpooled LRM with Bayesian Regularization

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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2b27cd5{width:0.714in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c2b4a22a'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c2b27be4">Country</th><th class="cl-c2b27be5">Posterior Mean</th><th class="cl-c2b27be6">Posterior Probability</th><th class="cl-c2b27bee">Credible Int.-L</th><th class="cl-c2b27bef">Credible Int.-H</th><th class="cl-c2b27bf0">HDI-L</th><th class="cl-c2b27bf8">HDI-H</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c2b27bf9">Norway</td><td class="cl-c2b27bfa">16.81</td><td class="cl-c2b27bfb">1.00</td><td class="cl-c2b27c02">12.55</td><td class="cl-c2b27c03">21.74</td><td class="cl-c2b27c04">12.27</td><td class="cl-c2b27c05">21.37</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c0c">Sweden</td><td class="cl-c2b27c0d">6.93</td><td class="cl-c2b27c0e">1.00</td><td class="cl-c2b27c16">2.80</td><td class="cl-c2b27c17">10.09</td><td class="cl-c2b27c18">3.30</td><td class="cl-c2b27c19">10.41</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c20">United States</td><td class="cl-c2b27c21">7.83</td><td class="cl-c2b27c22">0.94</td><td class="cl-c2b27c23">-3.34</td><td class="cl-c2b27c2a">17.30</td><td class="cl-c2b27c2b">-2.00</td><td class="cl-c2b27c2c">18.14</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c2d">United Kingdom</td><td class="cl-c2b27c2e">5.92</td><td class="cl-c2b27c34">0.88</td><td class="cl-c2b27c35">-3.36</td><td class="cl-c2b27c36">15.57</td><td class="cl-c2b27c37">-3.16</td><td class="cl-c2b27c3e">15.74</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c3f">Germany</td><td class="cl-c2b27c40">5.68</td><td class="cl-c2b27c48">0.79</td><td class="cl-c2b27c49">-2.83</td><td class="cl-c2b27c4a">21.55</td><td class="cl-c2b27c4b">-3.29</td><td class="cl-c2b27c52">20.98</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c53">Greece</td><td class="cl-c2b27c54">7.05</td><td class="cl-c2b27c55">0.75</td><td class="cl-c2b27c56">-6.33</td><td class="cl-c2b27c5c">31.77</td><td class="cl-c2b27c5d">-7.67</td><td class="cl-c2b27c5e">29.53</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c20">Switzerland</td><td class="cl-c2b27c21">1.19</td><td class="cl-c2b27c22">0.75</td><td class="cl-c2b27c23">-1.21</td><td class="cl-c2b27c2a">6.00</td><td class="cl-c2b27c2b">-1.44</td><td class="cl-c2b27c2c">5.68</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c5f">Ireland</td><td class="cl-c2b27c60">4.60</td><td class="cl-c2b27c66">0.68</td><td class="cl-c2b27c67">-7.90</td><td class="cl-c2b27c68">26.71</td><td class="cl-c2b27c69">-10.03</td><td class="cl-c2b27c70">22.99</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c71">Italy</td><td class="cl-c2b27c72">0.95</td><td class="cl-c2b27c73">0.66</td><td class="cl-c2b27c7a">-2.41</td><td class="cl-c2b27c7b">8.44</td><td class="cl-c2b27c7c">-2.90</td><td class="cl-c2b27c7d">7.71</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c84">Netherlands</td><td class="cl-c2b27c85">1.12</td><td class="cl-c2b27c86">0.63</td><td class="cl-c2b27c87">-2.41</td><td class="cl-c2b27c8e">9.73</td><td class="cl-c2b27c8f">-2.42</td><td class="cl-c2b27c90">9.72</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c91">Denmark</td><td class="cl-c2b27c98">0.17</td><td class="cl-c2b27c99">0.57</td><td class="cl-c2b27c9a">-1.10</td><td class="cl-c2b27ca2">2.28</td><td class="cl-c2b27ca3">-1.32</td><td class="cl-c2b27ca4">1.97</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27c91">France</td><td class="cl-c2b27c98">0.20</td><td class="cl-c2b27c99">0.55</td><td class="cl-c2b27c9a">-5.26</td><td class="cl-c2b27ca2">6.41</td><td class="cl-c2b27ca3">-5.19</td><td class="cl-c2b27ca4">6.46</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27ca5">Spain</td><td class="cl-c2b27cac">-0.63</td><td class="cl-c2b27cad">0.54</td><td class="cl-c2b27cae">-8.81</td><td class="cl-c2b27caf">8.28</td><td class="cl-c2b27cb0">-7.85</td><td class="cl-c2b27cb6">9.08</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27cb7">Portugal</td><td class="cl-c2b27cb8">-0.26</td><td class="cl-c2b27cb9">0.53</td><td class="cl-c2b27cba">-8.60</td><td class="cl-c2b27cc0">8.42</td><td class="cl-c2b27cc1">-8.43</td><td class="cl-c2b27cc2">8.56</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2b27cc3">Finland</td><td class="cl-c2b27cc4">0.18</td><td class="cl-c2b27cca">0.52</td><td class="cl-c2b27ccb">-6.04</td><td class="cl-c2b27ccc">5.87</td><td class="cl-c2b27cd4">-6.06</td><td class="cl-c2b27cd5">5.83</td></tr></tbody></table></div>

---
## An Example: Unpooled LRM with Bayesian Regularization

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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2c518e5{width:0.725in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2c518e6{width:0.714in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c2c739b2'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c2c51754">Country</th><th class="cl-c2c51755">Posterior Mean</th><th class="cl-c2c5175e">Posterior Probability</th><th class="cl-c2c5175f">Credible Int.-L</th><th class="cl-c2c51768">Credible Int.-H</th><th class="cl-c2c51769">HDI-L</th><th class="cl-c2c5176a">HDI-H</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c2c51772">Norway</td><td class="cl-c2c51773">16.81</td><td class="cl-c2c5177c">1.00</td><td class="cl-c2c5177d">12.55</td><td class="cl-c2c5177e">21.74</td><td class="cl-c2c51786">12.27</td><td class="cl-c2c51787">21.37</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c51788">Sweden</td><td class="cl-c2c51790">6.93</td><td class="cl-c2c51791">1.00</td><td class="cl-c2c51792">2.80</td><td class="cl-c2c5179a">10.09</td><td class="cl-c2c5179b">3.30</td><td class="cl-c2c517a4">10.41</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c517a5">United States</td><td class="cl-c2c517a6">7.83</td><td class="cl-c2c517a7">0.94</td><td class="cl-c2c517ae">-3.34</td><td class="cl-c2c517af">17.30</td><td class="cl-c2c517b0">-2.00</td><td class="cl-c2c517b1">18.14</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c517b8">United Kingdom</td><td class="cl-c2c517b9">5.92</td><td class="cl-c2c517ba">0.88</td><td class="cl-c2c517c2">-3.36</td><td class="cl-c2c517c3">15.57</td><td class="cl-c2c517cc">-3.16</td><td class="cl-c2c517cd">15.74</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c517ce">Germany</td><td class="cl-c2c517d6">5.68</td><td class="cl-c2c517d7">0.79</td><td class="cl-c2c517e0">-2.83</td><td class="cl-c2c517e1">21.55</td><td class="cl-c2c517e2">-3.29</td><td class="cl-c2c517ea">20.98</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c517eb">Greece</td><td class="cl-c2c517ec">7.05</td><td class="cl-c2c517f4">0.75</td><td class="cl-c2c517f5">-6.33</td><td class="cl-c2c517f6">31.77</td><td class="cl-c2c517fe">-7.67</td><td class="cl-c2c517ff">29.53</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c51808">Switzerland</td><td class="cl-c2c51809">1.19</td><td class="cl-c2c51812">0.75</td><td class="cl-c2c51813">-1.21</td><td class="cl-c2c5181c">6.00</td><td class="cl-c2c5181d">-1.44</td><td class="cl-c2c5181e">5.68</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c51826">Ireland</td><td class="cl-c2c51827">4.60</td><td class="cl-c2c51828">0.68</td><td class="cl-c2c51830">-7.90</td><td class="cl-c2c51831">26.71</td><td class="cl-c2c51832">-10.03</td><td class="cl-c2c5183a">22.99</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c5183b">Italy</td><td class="cl-c2c5183c">0.95</td><td class="cl-c2c51844">0.66</td><td class="cl-c2c51845">-2.41</td><td class="cl-c2c51846">8.44</td><td class="cl-c2c51847">-2.90</td><td class="cl-c2c5184e">7.71</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c5184f">Netherlands</td><td class="cl-c2c51858">1.12</td><td class="cl-c2c51862">0.63</td><td class="cl-c2c51863">-2.41</td><td class="cl-c2c51864">9.73</td><td class="cl-c2c5186c">-2.42</td><td class="cl-c2c5186d">9.72</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c51876">Denmark</td><td class="cl-c2c51877">0.17</td><td class="cl-c2c51880">0.57</td><td class="cl-c2c51881">-1.10</td><td class="cl-c2c51882">2.28</td><td class="cl-c2c5188a">-1.32</td><td class="cl-c2c5188b">1.97</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c51876">France</td><td class="cl-c2c51877">0.20</td><td class="cl-c2c51880">0.55</td><td class="cl-c2c51881">-5.26</td><td class="cl-c2c51882">6.41</td><td class="cl-c2c5188a">-5.19</td><td class="cl-c2c5188b">6.46</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c51894">Spain</td><td class="cl-c2c51895">-0.63</td><td class="cl-c2c5189e">0.54</td><td class="cl-c2c5189f">-8.81</td><td class="cl-c2c518a8">8.28</td><td class="cl-c2c518b2">-7.85</td><td class="cl-c2c518b3">9.08</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c518b4">Portugal</td><td class="cl-c2c518bc">-0.26</td><td class="cl-c2c518bd">0.53</td><td class="cl-c2c518c6">-8.60</td><td class="cl-c2c518c7">8.42</td><td class="cl-c2c518c8">-8.43</td><td class="cl-c2c518c9">8.56</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2c518d0">Finland</td><td class="cl-c2c518d1">0.18</td><td class="cl-c2c518d2">0.52</td><td class="cl-c2c518da">-6.04</td><td class="cl-c2c518e4">5.87</td><td class="cl-c2c518e5">-6.06</td><td class="cl-c2c518e6">5.83</td></tr></tbody></table></div>

---

## An Example: Group-level results

.purpleb[Once we unpool, we might be interested in the effect of group-level variables on the LRMs.]

2 possibilities:
  1. Use Rubin (1987)'s rules to estimate the effect of group-level variables net of measurement variability in the LRM;
  2. Use Monte Carlo integration to obtain the marginal posterior of the effects of group-level variables.

---
## An Example: Group-level results

.purpleb[Once we unpool, we might be interested in the effect of group-level variables on the LRMs.]

<div class="tabwid"><style>.cl-c2d4a02a{}.cl-c2d1c12a{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-c2d2e56e{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-c2d2e56f{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-c2d2f432{width:1.049in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 1pt solid rgba(102, 102, 102, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f433{width:1.228in;background-color:transparent;vertical-align: middle;border-bottom: 1pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 1pt solid rgba(102, 102, 102, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f43c{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 1pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f43d{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 1pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f466{width:0.786in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f46e{width:1.049in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 1pt solid rgba(102, 102, 102, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f46f{width:1.228in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 1pt solid rgba(102, 102, 102, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f470{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f478{width:0.752in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f482{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f483{width:0.947in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2d2f484{width:0.786in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c2d4a02a'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c2d2f432"> </th><th class="cl-c2d2f433">Parameter</th><th class="cl-c2d2f43c">Estimate</th><th class="cl-c2d2f43d">SE</th><th class="cl-c2d2f446">95% CI-L</th><th class="cl-c2d2f447">95% CI-H</th><th class="cl-c2d2f450">p-value</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c2d2f451">Frequentist</td><td class="cl-c2d2f45a">Parliamentary</td><td class="cl-c2d2f45b">4.81</td><td class="cl-c2d2f45c">320.56</td><td class="cl-c2d2f464">-623.58</td><td class="cl-c2d2f465">633.2</td><td class="cl-c2d2f466">0.99</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2d2f46e"></td><td class="cl-c2d2f46f">(Intercept)</td><td class="cl-c2d2f470">6.76</td><td class="cl-c2d2f478">291.63</td><td class="cl-c2d2f482">-564.97</td><td class="cl-c2d2f483">578.5</td><td class="cl-c2d2f484">0.98</td></tr></tbody></table></div><div class="tabwid"><style>.cl-c2e26b38{}.cl-c2de6218{font-family:'Helvetica';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-c2dfa556{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2dfb2ef{width:0.583in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2dfb2f0{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2dfb2f8{width:0.947in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-c2dfb2f9{width:0.915in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-c2e26b38'><thead><tr style="overflow-wrap:break-word;"><th class="cl-c2dfb29e"> </th><th class="cl-c2dfb29f">Parameter</th><th class="cl-c2dfb2a8">Mean</th><th class="cl-c2dfb2a9">SE</th><th class="cl-c2dfb2b2">95% CI-L</th><th class="cl-c2dfb2b3">95% CI-H</th><th class="cl-c2dfb2bc">Pr(ω &gt; 0)</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-c2dfb2c6">Bayesian</td><td class="cl-c2dfb2c7">Parliamentary</td><td class="cl-c2dfb2d0">2.10</td><td class="cl-c2dfb2d1">20.9</td><td class="cl-c2dfb2d2">-5.24</td><td class="cl-c2dfb2da">10.10</td><td class="cl-c2dfb2db">0.71</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-c2dfb2e4"></td><td class="cl-c2dfb2e5">(Intercept)</td><td class="cl-c2dfb2ee">3.02</td><td class="cl-c2dfb2ef">16.4</td><td class="cl-c2dfb2f0">-3.25</td><td class="cl-c2dfb2f8">8.62</td><td class="cl-c2dfb2f9">0.78</td></tr></tbody></table></div>

---

## So ... What Have We Learned?

1. Unpooling reveals heterogeneity that might be of theoretical importance.

2. Can improve on the type II error rate of the bounds method using QB t-statistics.

3. Regularizing priors seem to be working. Also, it performs better than other techniques with shorter series.

---

## Next Steps

1. Find an example in which the pooled relationship holds when we unpool.

2. Run (longer) simulations on a wider variety of specifications.

3. Find a way to ease the presentation of unpooled results.

4. Build the whole thing into a user friendly package.

---

class: inverse,middle,center

## Questions? Comments? Testimonies?

.pull-left[
### Contacts

wpoirier@uwo.ca

@olsisblue.bsky.social

WilliamPo1

Or visit my website! `$\longmapsto$`

]
.pull-right[
<img src="figs/website_rev.png" width="70%" style="display: block; margin: auto;" />
]

---

## Appendix: Primer on Time Series Analysis
.pull-left[
- From `$y_i$` to `$y_t$`.
- Creates autocorrelation, the best predictor of `$y_t$` is `$y_{t-1}$`.
- Care about stationarity of series.
- Sationarity and unit root tests have problems:
    - Can't use usual t-distributions;
    - Low power;
    - Heteroskedasticity and serial correlation (bias SE);
    - Highly dependent on model specification.
- Lots of work describing each variable.
- Care about equation balance.
- Two common, isomorphic, models:
    - ADL(1,1:1): `$y_t=\alpha_0+\alpha_1 y_{t-1}+\beta_0 x_t+\beta_1x_{t-1}+\varepsilon_t$`
    - GECM: `$\Delta y_t=\alpha_0+\alpha^*_1 y_{t-1}+\beta_0^*\Delta x_t +\beta_1^* \Delta x_{t-1}+\varepsilon_t$`
]

.pull-right[

<img src="maplemeth_files/figure-html/series_monthly.png" width="100%" style="display: block; margin: auto;" />
 .small[An example of stationary series describing monthly tone of in chamber exchanges from 1961 to 2024.] 
 
]

---

## Appendix: Intuition for LRM Uncertainty

- ADL: `$y_t=\alpha_0+\alpha_1 y_{t-1}+\beta_0 x_t+\beta_1x_{t-1}+\varepsilon_t$`
- LRM: `$\lambda = \frac{\beta_0+\beta_1}{1-\alpha_1}$`, gives the total effect of a unit change in x on y.
- If `$x$` and `$y$` are unit roots: 
  - Cointegrated `$\rightarrow \alpha_1\neq1 \rightarrow \sum\beta_q\neq0\rightarrow\lambda\neq0$`;
  - Not cointegrated `$\rightarrow \alpha_1=1 \rightarrow \lambda$` is undefined.
- If `$x$` and `$y$` are stationary:
  - Then `$\rightarrow \alpha_1\neq1$` by definition;
  - `$\lambda\neq0$` if there exist a relationship;
  - `$\lambda=0$` otherwise.
- Hence, the LRM will be unbiased whatever the univariate characteristics of the variables.
- Still, the estimation of the SE remains problematic, hence the bounds defined in Webb, Linn, and Lebo (2019).

---

## An Example: Why the Horseshoe? (USA)

---

## An Example: Why the Horseshoe? (Germany)

---

## Appendix: Partial Pooling

.purpleb[Two problems:]
  1. Regularization of the horshoe would shrink toward central value (average LRM) instead of 0. 
  2. Becomes computationally cumbersome to estimate due to the addition of random effects.
  
---
## Appendix: Reasonableness of Horseshoe Priors

.panelset[
 .panel[.panel-name[Stationary X and Y]
 `$$\begin{aligned}
 x_1 &=z_1=y_1=0\\
 x_t &=\rho_{x} x_{t-1} + e_x \sim N(0,1)\\
 z_t &=\rho_{z} z_{t-1} + e_z \sim N(0,1)\\
 y_t &= \alpha_0 + \alpha_1 y_{t-1} + \beta_1 x_t + \theta_1 x_{t-1} + e_y \sim N(0,1)\\
 \rho_x &=\rho_z = .5\\
 \alpha_1 &= .6\\
 \beta_1 &= .5 \\
 \theta_1 &= 1
 \end{aligned}$$`
 
 - `$LRM_x=3.75, LRM_z=0$`
 
 ]
 .panel[.panel-name[Results]
<img src="APSA_uppcsts_2025_files/figure-html/unnamed-chunk-14-1.png" width="100%" />
 ]
 .panel[.panel-name[Co-integrating X and Y]
 `$$\begin{aligned}
 x_1 &=z_1=0\\
 u_1 &= N(0,1)\\
 y_1 &=\beta_1 x_1 + u_1\\
 x_t &=\rho_{x} x_{t-1} + e_{x,t} \sim N(0,1) ~~\forall~~ t=2,...,T\\
 z_t &=\rho_{z} z_{t-1} + e_{z,t} \sim N(0,1) ~~\forall~~ t=2,...,T\\
 u_t &=\alpha_{1} u_{t-1} + e_{u,t} \sim N(0,1) ~~\forall~~ t=2,...,T\\
 y_t &= \beta_1 x_t + \theta_1 x_{t-1} + e_{y,t} \sim N(0,1) ~~\forall~~ t=2,...,T\\
 \rho_x &= 1 \\
 \rho_z &= .5\\
 \alpha_1 &= .6\\
 \beta_1 &= .3 \\
 \theta_1 &= .3
 \end{aligned}$$`
 
 - `$LRM_x=1.5, LRM_z=0$`
 ]
 .panel[.panel-name[Results]
<img src="APSA_uppcsts_2025_files/figure-html/unnamed-chunk-15-1.png" width="95%" style="display: block; margin: auto;" />
 ]
]

---

## Appendix: Test for the LRMs

.panelset[
  .panel[.panel-name[Simulation]
  `$$\begin{align}
    z_{1,j} &= x_1 = y_1 = 0\\
    z_{t,j} &= \rho_z z_{t-1,j} + e_{t,j}\sim N(0,1)\\
    x_{t} &= \rho_x x_{t-1} + e_{t}\sim N(0,1)\\
    y_{t} &= \rho_y y_{t01} + b_1x_t + b_2 x_{t-1} + e_{t} \sim N(0,1)
\end{align}$$`
We used all combinations of the parameters:

`$$\begin{align}
 \rho_x &= [0,.5, .75, .9, 1]\\
 \rho_y &= [0,.5, .75, .9, 1]\\
 \rho_z &= [0, .5]\\
 j &= [1,2,3,4]\\
 T &= [25, 50, 75, 100]\\
 b_1 &= [0,1]\\
 b_2 &= [0,.5]
\end{align}$$`
 ]
 .panel[.panel-name[Coverage Rate by Procedure]
<img src="figs/all_conds_by_model.png" width="50%" style="display: block; margin: auto;" />
 ]
 .panel[.panel-name[Coverage Rate by Parameter]
<img src="APSA_uppcsts_2025_files/figure-html/unnamed-chunk-17-1.png" width="100%" />
 
 ]
 .panel[.panel-name[Recursive Partitioning]
<img src="figs/rpart_cvg.png" width="50%" style="display: block; margin: auto;" />
 ]
]