CSSS/POLS 512

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.title[
# CSSS/POLS 512
]
.subtitle[
## Lab 5: Panel Data — Dynamics, FE/RE, Mundlak, IRF
]
.author[
### Ramses Llobet
]
.date[
### Spring 2026
]

---

# Today's Plan

.pull-left[
**Part 1: Theory + Data**
- DAG: `reg`, `edt` (moderator), `oil`
- The Przeworski 1950–1990 panel
- Within-vs-between framing

**Part 2: Dynamics workflow**
- ACF/PACF on detrended series
- Per-country ADF/PP scan
- Panel unit-root **constellation** (CD, MW/IPS, Hadri, CIPS)
- Verdict: I(1)-like; choose Path A
]

.pull-right[
**Part 3: Panel models family**
- Pooled / FE / TWFE / RE / Mundlak (REWB)
- Static vs dynamic (AR(1), ARDL(1,1))
- corARMA(1) for RE / Mundlak
- Hausman, PCSE, Driscoll–Kraay

**Part 4: Counterfactual via IRF + LRE**
- FE-ARDL(1,1) closed form
- Permanent democratization
- Long-run effect by **education**
]

---
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# Part 1 — Theory and Data

---

# A Toy Theory of Politics in Panel Data

We study **regime → growth** with three covariate roles:

.pull-left[
1. **Within-rich treatment**: `reg` (1 = democracy; flips on coups, pacts, transitions)
2. **Slow-moving covariate AND moderator**: `edt` (cumulative education) — moderates the regime-growth slope
3. **Time-invariant control**: `oil` (major oil exporter, 1/0)

**Outcome**: `$\ln$`(GDP per worker)

**Hypothesis**: more-educated workforces benefit more (or less) from democratization.
]

.pull-right[
<img src="output/panel_dag.png" alt="" width="100%" style="display: block; margin: auto;" />
]

---

# Within vs Between — Which `$\beta$` Are You Estimating?

| Specification | What `$\hat\beta$` identifies | Identifying assumption |
|:---|:---|:---|
| **Pooled OLS** | mixture of within + between | `$u_{it}\perp X_{it}$`; `$\alpha_i$` common |
| **Fixed effects (within)** | within only (`$\beta_W$`) | `$\varepsilon_{it}\perp x_{it}$` given `$\alpha_i$`; allows `$\text{Cov}(\alpha_i, X)\neq 0$` |
| **Random effects** | matrix-weighted average | `$\alpha_i\perp X_i$` |
| **Mundlak / REWB** | `$\beta_W$` and `$\beta_B$` separately | RE on residuals; `$\bar x_i$` absorbs cross-level confounding |

**Decision logic**:

- *"Does democratization, **for a given country**, raise GDP?"* → `$\beta_W$`
- *"Are democracies, **on average across countries**, richer?"* → `$\beta_B$`
- *"Both — and do they differ?"* → Mundlak / REWB returns both, `$\beta_W = \beta_B$` is testable

.footnote[*Bell & Jones (2015), Kropko & Kubinec (2020).*]

---

# The Przeworski Democracy Panel

.pull-left[
**Dimensions**: `$N = 135$` countries, 1950–1990, **unbalanced** (`$T_i \in [10, 40]$`).

**Variables**: `ln_gdpw`, `reg`, `edt`, `oil`, `region`.

`pvar()` confirms `oil` is between-only; `reg` and `edt` vary both ways.

**Spaghetti plot →** trends, drift, country-specific slopes — coloured by region.
]

.pull-right[
<img src="Lab5_slides_files/figure-html/p1-spaghetti-region-1.svg" alt="" width="432" style="display: block; margin: auto;" />
]

---

# Unbalanced Panel: `$T_i$` Distribution

Most countries have `$T_i \approx 30$`–40; African and Asian post-colonial states cluster at the lower end. Short `$T$` matters for unit-root power and Nickell bias.

---
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# Part 2 — Dynamics Workflow

---

# Why Test for Stationarity Before Estimating?

A near-unit-root outcome creates two problems:

1. **Spurious regression** — coefficients can look significant even when there is no real relationship.
2. **AR(1) on level becomes a signature**: `$\hat\phi$` on `ln_gdpw_lag` lands near 1 → the long-run multiplier `$\hat\beta/(1-\hat\phi)$` blows up.

Workflow (per Lab 5):

1. **Visualize** all series — spaghetti by region.
2. **Detrend** country-by-country (OLS); inspect ACF/PACF on residuals.
3. **Per-series ADF + PP** → histogram of unit-level `$p$`-values.
4. **Panel unit-root tests**: read a *constellation*, not a single verdict.
5. **Pick a path**: levels with ARDL(1) (Path A) or first differences (Path B).

---

# Per-Series Unit-Root Scan

After removing each country's linear trend, run ADF and PP on the residuals; plot the distribution of `$p$`-values.

Most countries land `$p > 0.05$`: low power at `$T \le 30$` does not let us reject the unit-root null country-by-country.

---

# Panel Unit-Root Constellation

We read **multiple tests jointly** because no single one is decisive at this `$T$`.

| Step | Test | `$H_0$` | Reading |
|:----|:----|:----|:----|
| (a) | **Pesaran CD** | cross-sectional independence | If rejected → 1st-gen tests biased |
| (b) | **Maddala-Wu/Fisher**, **IPS** | unit root in *all* panels | 1st-gen, assume CSI |
| (c) | **Hadri** | stationarity in *all* panels | KPSS-flip; over-rejects under CSD (Hlouskova & Wagner 2006) |
| (d) | **Pesaran CIPS** | unit root, common-factor corrected | 2nd-gen, robust to CSD |

**Verdict for `ln_gdpw`**: CSD present (a). Fisher rejects but IPS does not (b) — Fisher is sensitive to a few low-$p$ countries. Hadri rejects but discount under CSD. **CIPS does not reject**. → **strong persistence consistent with I(1)-type behavior**.

---

# Two Paths — We Pick Path A

**One important exception** — pooled OLS with AR(1) on the level shows `$\hat\phi \approx 0.99$` (the I(1) signature). For the pooled model only, we use **Path B (FD + ARDL(1,1))**: outcome `$\Delta y$`, lagged `$\Delta y$`, and lagged regressors. The growth-rate equation has a well-behaved `$\hat\phi$`.

For FE / TWFE / RE / Mundlak, we stay on Path A.

---
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# Part 3 — Estimating Panel Models

---

# One Equation, Five Estimators

$$
y_{it} \;=\; \alpha + \beta_1\,\text{reg}_{it} + \beta_2\,\text{edt}_{it} + \beta_3\,(\text{reg}_{it}\!\cdot\!\text{edt}_{it}) + \beta_4\,\text{oil}_i + u_{it}
$$

| Subsection | Static | AR(1) (lagged DV) | ARDL(1,1) (+ lagged `reg`, `edt`) | corARMA(1) |
|:---|:---:|:---:|:---:|:---:|
| Pooled OLS | ✓ | ✓ | ✓ | — |
| FE (within) | ✓ | ✓ | ✓ | — |
| TWFE | ✓ | ✓ | ✓ | — |
| RE | ✓ | — | — | ✓ |
| Mundlak / REWB | ✓ | — | — | ✓ |

**Why no LDV in RE / Mundlak?** RE-LDV faces the Wooldridge (2005) initial-conditions problem; Mundlak-LDV breaks the within-between decomposition. corARMA absorbs persistence in the residuals.

---

# Fitting the Family

**Pooled OLS**: level AR(1) shows the I(1) signature (`$\hat\phi \approx 0.99$`) → switch outcome to first differences. Pick **FD + ARDL(1,1)**.

**FE / TWFE**: stay in levels with **ARDL(1,1)** (lagged DV + lagged `reg` + lagged `edt` + lagged interaction).

**RE**: **corARMA(1)** — AR(1) on residuals; no LDV (Wooldridge initial conditions).

**Mundlak / REWB**: **corARMA(1)** + within/between split on `reg`, `edt`, and `reg × edt` (precomputed product).

---

# Pooled OLS — Where the Headline Lives

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 7.277 </td>
   <td style="text-align:right;"> 0.028 </td>
   <td style="text-align:right;"> 264.597 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> reg </td>
   <td style="text-align:right;"> 0.633 </td>
   <td style="text-align:right;"> 0.059 </td>
   <td style="text-align:right;"> 10.664 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> edt </td>
   <td style="text-align:right;"> 0.233 </td>
   <td style="text-align:right;"> 0.006 </td>
   <td style="text-align:right;"> 36.984 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> oil </td>
   <td style="text-align:right;"> 0.752 </td>
   <td style="text-align:right;"> 0.037 </td>
   <td style="text-align:right;"> 20.483 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> reg:edt </td>
   <td style="text-align:right;"> -0.027 </td>
   <td style="text-align:right;"> 0.009 </td>
   <td style="text-align:right;"> -2.878 </td>
   <td style="text-align:right;"> 0.004 </td>
  </tr>
</tbody>
</table>

- `$\hat\beta$` on `reg` and `reg × edt` is **identified mostly off cross-country variation** — more-educated countries differ from less-educated countries in their average regime-growth slope.
- Pooled OLS in levels is also the spec where the I(1) signature appears: AR(1) on the level gives `$\hat\phi \approx 0.99$`. Differencing the outcome restores stationarity (FD + ARDL(1,1) chosen for §2.8).

---

# Fixed Effects — Does the Moderation Survive Within?

- `oil` is absorbed (time-invariant).
- The within `reg × edt` interaction tells us whether the moderation is a *within-country* phenomenon: *as a country's education rises over time, does the regime effect change?*
- If the FE interaction is small while pooled OLS's was large → between-country variation was doing the talking. → **Same data, different stories under different identifying assumptions.**

---

# Mundlak / REWB — The Diagnostic

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> Value </th>
   <th style="text-align:right;"> Std.Error </th>
   <th style="text-align:right;"> DF </th>
   <th style="text-align:right;"> t-value </th>
   <th style="text-align:right;"> p-value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> reg_within </td>
   <td style="text-align:right;"> 0.055 </td>
   <td style="text-align:right;"> 0.019 </td>
   <td style="text-align:right;"> 2784 </td>
   <td style="text-align:right;"> 2.842 </td>
   <td style="text-align:right;"> 0.005 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> reg_between </td>
   <td style="text-align:right;"> 0.907 </td>
   <td style="text-align:right;"> 0.334 </td>
   <td style="text-align:right;"> 108 </td>
   <td style="text-align:right;"> 2.712 </td>
   <td style="text-align:right;"> 0.008 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> edt_within </td>
   <td style="text-align:right;"> 0.051 </td>
   <td style="text-align:right;"> 0.006 </td>
   <td style="text-align:right;"> 2784 </td>
   <td style="text-align:right;"> 8.028 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> edt_between </td>
   <td style="text-align:right;"> 0.226 </td>
   <td style="text-align:right;"> 0.031 </td>
   <td style="text-align:right;"> 108 </td>
   <td style="text-align:right;"> 7.338 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> reg_x_edt_within </td>
   <td style="text-align:right;"> -0.012 </td>
   <td style="text-align:right;"> 0.004 </td>
   <td style="text-align:right;"> 2784 </td>
   <td style="text-align:right;"> -2.779 </td>
   <td style="text-align:right;"> 0.005 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> reg_x_edt_between </td>
   <td style="text-align:right;"> -0.037 </td>
   <td style="text-align:right;"> 0.049 </td>
   <td style="text-align:right;"> 108 </td>
   <td style="text-align:right;"> -0.753 </td>
   <td style="text-align:right;"> 0.453 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> oil </td>
   <td style="text-align:right;"> 0.652 </td>
   <td style="text-align:right;"> 0.175 </td>
   <td style="text-align:right;"> 108 </td>
   <td style="text-align:right;"> 3.717 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
</tbody>
</table>

- `reg_within` / `reg_between` — within-country regime effect vs cross-country slope.
- `reg_x_edt_within` / `reg_x_edt_between` — does the moderation by education operate *within* (rising education within a country changes the regime effect) or *between* (more-educated countries have a different regime-growth slope)?
- `oil` retains its main effect (RE flexibility, FE shortcoming).

**This is exactly the diagnostic Hausman gives, with covariate-by-covariate detail.**

---

# Hausman Test

$$
H = (\hat\beta_{FE} - \hat\beta_{RE})^\top [V_{FE} - V_{RE}]^{-1} (\hat\beta_{FE} - \hat\beta_{RE}) \;\sim\; \chi^2_k
$$

- `$H_0$`: `$\text{Cov}(\alpha_i, X_{it}) = 0$` — RE consistent and efficient.
- `$H_1$`: RE inconsistent; only FE consistent.
- The variance of the difference collapses to `$V_{FE} - V_{RE}$` because RE is *efficient* under `$H_0$` (Hausman 1978).

```
## [1] "chisq = 61.75,  df = 3,  p = 0.0000"
```

**Bell & Jones (2015) caveat**: do not use Hausman as a model-selection switch. Even when it rejects, REWB is often the better choice — it returns *both* within and between estimates so you can report the relevant comparison.

---

# Panel Standard Errors

| SE | Allows | When |
|:---|:---|:---|
| **Cluster-robust by unit** | Within-country correlation | Many clusters; residuals roughly independent across countries |
| **PCSE (Beck & Katz 1995)** | Contemporaneous *cross-sectional* dependence | TSCS panels — common shocks hit many countries the same year |
| **Driscoll–Kraay (1998)** | Cross-sectional **+** serial correlation | Moderate-to-large `$T$`; safest default |

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> coefficient </th>
   <th style="text-align:right;"> cluster </th>
   <th style="text-align:right;"> pcse </th>
   <th style="text-align:right;"> driscoll </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> reg </td>
   <td style="text-align:right;"> 0.0888 </td>
   <td style="text-align:right;"> 0.0821 </td>
   <td style="text-align:right;"> 0.0628 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> edt </td>
   <td style="text-align:right;"> 0.0175 </td>
   <td style="text-align:right;"> 0.0163 </td>
   <td style="text-align:right;"> 0.0231 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> reg:edt </td>
   <td style="text-align:right;"> 0.0186 </td>
   <td style="text-align:right;"> 0.0175 </td>
   <td style="text-align:right;"> 0.0081 </td>
  </tr>
</tbody>
</table>

PCSE is the TSCS default; Driscoll–Kraay is the safer choice when residual autocorrelation persists after the lagged DV.

---
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# Part 4 — Counterfactual via IRF and LRE

---

# The Forecast Question

> *If a country democratized **permanently**, what would the trajectory of `$\ln$` GDP per worker look like over the next 20 years — and how does it depend on the workforce's education level?*

We use the **FE-ARDL(1,1)** model: country fixed effects + lagged DV + contemporaneous and lagged `reg`, `edt`, and `reg × edt`.

The output is the **impulse response function (IRF)** — cumulative effect at horizon `$h$` — and its asymptote, the **long-run equilibrium effect (LRE)**.

This is the panel analog of Lab 4's IRF, with the moderation by education made explicit.

---

# Closed-Form IRF

For the dynamic FE-ARDL(1,1) model under permanent democratization at `$t = 0$`, with `edt` held at level `$e$`:

$$
\mathrm{IRF}(h \mid e)
\;=\;
A(e) \cdot \frac{1 - \phi^{h+1}}{1 - \phi}
\;+\;
B(e) \cdot \frac{1 - \phi^{h}}{1 - \phi}
$$

where

- `$A(e) = \beta_0 + \gamma_0\,e$` is the **impact** effect (year of democratization).
- `$B(e) = \beta_1 + \gamma_1\,e$` is the **delayed transmission** through `reg_lag` and `reg_lag × edt_lag`.

**Long-run equilibrium effect** (asymptote as `$h \to \infty$`, `$|\phi| < 1$`):

$$
\mathrm{LRE}(e) \;=\; \frac{A(e) + B(e)}{1 - \phi} \;=\; \frac{\beta_0 + \beta_1 + (\gamma_0 + \gamma_1)\,e}{1 - \phi}
$$

Parametric simulation (King–Tomz–Wittenberg): draw `$S = 1{,}000$` coefficient vectors from MVN, evaluate IRF at each draw and horizon, take quantile bands.

---

# IRF: Permanent Democratization at Two Education Levels

---

# Long-Run Equilibrium Effect (LRE)

<table>
<caption>LRE: long-run effect of permanent democratization, FE-ARDL(1,1).</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> scenario </th>
   <th style="text-align:right;"> pe </th>
   <th style="text-align:right;"> lo </th>
   <th style="text-align:right;"> hi </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Low education (edt = 2.21, P25) </td>
   <td style="text-align:right;"> -0.222 </td>
   <td style="text-align:right;"> -0.556 </td>
   <td style="text-align:right;"> 0.083 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> High education (edt = 7.08, P75) </td>
   <td style="text-align:right;"> -0.052 </td>
   <td style="text-align:right;"> -0.370 </td>
   <td style="text-align:right;"> 0.269 </td>
  </tr>
</tbody>
</table>

**How to read it.** A larger LRE for the high-`edt` scenario says democratization's long-run growth payoff rises with the workforce's education — a within-country moderation by human capital. Wide CIs reflect the Nickell-bias caveat: at `$\hat\phi$` close to 1, the LRE is sensitive to small changes in `$\hat\phi$`.

---
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# Wrap-Up

---

# What We Learned Today

1. **Within vs between is the substantive question**, not a methodological footnote. Mundlak / REWB returns both — Bell & Jones (2015): make it the default.

2. **Test before you estimate**. The panel unit-root *constellation* (CD + Fisher-MW + IPS + Hadri + CIPS) gave us a defensible I(1)-like verdict where no single test was decisive at `$T \le 30$`.

3. **Path A (ARDL) for FE / TWFE; Path B (FD) for pooled OLS**. The level-AR(1) `$\hat\phi \approx 0.99$` in pooled is the I(1) signature.

4. **No LDV in RE / Mundlak**. corARMA(1) is the natural dynamic version — it absorbs persistence in the residuals without breaking the within-between split (REWB) or running into Wooldridge initial-conditions (RE).

5. **The IRF + LRE generalizes Lab 4's forecast to panels** — closed-form propagation, parametric simulation, moderator made visible at the horizon scale.

**Next week**: dynamic-panel GMM (Arellano–Bond, Arellano–Bover) to address the Nickell bias.

---

# References

- Beck, N., & Katz, J. N. (1995). What to do (and not to do) with TSCS data. *APSR*.
- Beck, N., & Katz, J. N. (2011). Modeling dynamics in TSCS political-economy data. *Annu. Rev. Polit. Sci.*
- Bell, A., & Jones, K. (2015). Explaining fixed effects: Random effects modeling of TSCS and panel data. *PSRM*.
- Driscoll, J. C., & Kraay, A. C. (1998). Consistent covariance matrix estimation with spatially dependent panel data. *RES*.
- Hadri, K. (2000). Testing for stationarity in heterogeneous panel data. *EJ*.
- Hlouskova, J., & Wagner, M. (2006). The performance of panel unit root and stationarity tests. *Econ. Rev.*
- Im, K. S., Pesaran, M. H., & Shin, Y. (2003). Testing for unit roots in heterogeneous panels. *J. Econometrics*.
- King, G., Tomz, M., & Wittenberg, J. (2000). Making the most of statistical analyses. *AJPS*.
- Kropko, J., & Kubinec, R. (2020). Interpretation and identification of within-unit and cross-sectional variation. *PLOS ONE*.
- Maddala, G. S., & Wu, S. (1999). A comparative study of unit root tests with panel data. *OBES*.
- Mundlak, Y. (1978). On the pooling of time series and cross section data. *Econometrica*.
- Nickell, S. (1981). Biases in dynamic models with fixed effects. *Econometrica*.
- Pesaran, M. H. (2007). A simple panel unit root test in the presence of cross-section dependence. *J. Appl. Econ.*

---
class: inverse, center, middle

# Questions?

`rllobet@uw.edu`