Lab Week 4: Statistical Matching + Fixed Effects

A replication of study analysis in Ahmadia (2015)

Author

EDS 241 / ESM 244

Published

January 29, 2026

Lab Outline

  1. Load in the packages and data
  2. Take a look at focal variables
  3. Checking covariate imbalance (pre‑matching).
  4. Conduct a matching analysis (Mahalanobis matching)
  5. Evaluate balance after matching
  6. Estimate regression models for each outcome using the matched sample
  7. Practice fixed effects estimation

Applied study & data source

This lab uses open-access replication data from:

Ahmadia, G. N., Glew, L., Provost, M., Gill, D., Hidayat, N. I., Mangubhai, S., Purwanto, P., Fox, H. E. (2015). Integrating impact evaluation in the design and implementation of monitoring marine protected areas. Philosophical Transactions of the Royal Society B.

Big idea: MPAs are not randomly placed. They may be placed in “better” locations which means that treated and control groups are unlikely to make an apples-to-apples comparison. AKA, we are up against the nearly universal causal inference problem with non-experimental data, treatment assignment selection bias.

Ahmadia et al. (2015) uses statistical matching to construct a more apples-to-apples comparison group. In this lab exercise we will approximately replicate the main matching analysis conducted in this study.

Note

NOTE: When matching, the authors imposed additional restrictions (caliper-style constraints) on some habitat variables. We do not fully replicate those constraints here.

Reading reference:

MatchIt Vignette: https://kosukeimai.github.io/MatchIt/articles/MatchIt.html

0. Load packages + study data

library(tidyverse)   # data wrangling + plotting
library(here)        # portable file paths (project-root relative)
library(janitor)     # clean variable names
library(gtsummary)   # clean summary tables
library(gt)          # optional: render gtsummary nicely
library(MatchIt)     # matching estimation
library(cobalt)      # balance checks + love plots
library(jtools)      # clean regression output summaries

Read in the cleaned survey data

data_clean <- read_csv(here("week4", "Ahmadia_2015_DataClean.csv"), show_col_types = FALSE)

1. Create variable description table for key variables in the matching analysis

Variable Descriptions (Treatment, Outcomes, Matching Covariates) - Ahmadia (2015)
Label Description
treated_mpa (Treatment) Site is inside a Marine Protected Area (1 = inside MPA , 0 = outside MPA).
biomass_fisheries (Outcome) Biomass of key fisheries families (Serranidae, Lutjanidae, Haemulidae) measured in kg per hectare (kg/ha).
biomass_ecological (Outcome) Biomass of herbivorous fish families (Acanthuridae, Scaridae, Siganidae) measured in kg per hectare (kg/ha).
dist_deep_water (Covariate) Covariate: Distance to deep water (50 m depth contour), in meters (m).
ssta_freq (Covariate) Frequency of sea-surface temperature anomalies (SSTA)
reef_exposure (Covariate) Reef wave exposure (Exposed, Semi-exposed, Sheltered).
reef_slope (Covariate) Reef slope (Flat, Slope, Wall)
reef_type (Covariate) Reef type (Patch, Fringing, Barrier, Atoll).
dist_mangroves (Covariate) Distance to nearest mangrove habitat, in meters (m).
dist_fishing_settlement (Covariate) Distance to nearest fishing settlement, in meters (m).
dist_market (Covariate) Distance to primary market, in meters (m).
pollution_risk (Covariate) Watershed pollution risk. (1 = Low; 2 = Medium; 3 = High).
monsoon_direction (Covariate) Monsoon wind exposure direction. (Northwest (NW), Southeast (SE)).

2. Check imbalance before matching (covariate balance table)

Goal: Compare covariate distributions between treated_mpa and control sites before matching.

data_clean %>%
  select(
    treated_mpa, dist_to_deep_water, ssta_freq, reef_exposure,
    reef_slope, reef_type,
    dist_mangroves, dist_fish_settl, dist_market,
    pollution_risk, monsoon_direction) %>%
  tbl_summary(
    by = treated_mpa,
    statistic = list(
      all_continuous() ~ "{mean} ({sd})",
      all_categorical() ~ "{n} ({p}%)"
    )
  ) %>%
  modify_header(label ~ "**Covariate**") %>%
  modify_spanning_header(c("stat_1", "stat_2") ~ "**Group**")
Covariate
Group
Control (non‑MPA)
N = 531		 MPA site
N = 1081
dist_to_deep_water 656 (905) 627 (913)
ssta_freq 25 (8) 25 (5)
reef_exposure

    Exposed 35 (66%) 92 (85%)
    Semi-exposed 18 (34%) 16 (15%)
reef_slope

    Flat 7 (13%) 10 (9.3%)
    Slope 45 (85%) 90 (83%)
    Wall 1 (1.9%) 8 (7.4%)
reef_type

    Barrier 8 (15%) 10 (9.3%)
    Fringing 45 (85%) 93 (86%)
    Patch 0 (0%) 5 (4.6%)
dist_mangroves 9,679 (14,229) 4,618 (5,165)
dist_fish_settl 25,646 (28,199) 32,505 (31,537)
dist_market 124,152 (49,763) 139,918 (66,406)
pollution_risk

    High 0 (0%) 1 (0.9%)
    Low 33 (62%) 84 (78%)
    Medium 20 (38%) 23 (21%)
monsoon_direction

    NW 19 (36%) 62 (57%)
    SE 34 (64%) 46 (43%)
1 Mean (SD); n (%)

✅ Your turn (write answers in the lab)

Q1. Based on the balance table, name two covariates that look most different between MPA and non-MPA sites.

Response: _________________________

Q2. Pick one of those imbalanced covariates and explain why it might confound the estimate of the effect of MPAs on fish biomass.

Response: _________________________

Matching plan (what are we trying to estimate?)

In {MatchIt}, most distance-based matching procedures (like nearest-neighbor matching) target is to estimate:

ATT: Average Treatment effect on the Treated

Why? When using matching methods we typically keep treated units and then select control units that most closely match them.

✅ Your turn

Q4. What does ATT mean in the context of this MPA evaluation setting?

Response: _________________________

3. Mahalanobis Matching

Matching criteria used in Ahmadia (2015)

  • method = "nearest-neighbor": Nearest neighbor matching
  • distance = "mahalanobis": Mahalanobis distance
  • ratio = 2: Two controls matched to each treated unit (2:1 ratio; control/treated)
  • replace = TRUE: Controls can be reused (matched to multiple treated units)

NOTE: This is the main matching method implemented in Ahmadia (2015) with the exception of the caliper-style constraints.

# 1) Set a seed for reproducible matching
set.seed(2412026)

# 2) Fit a nearest-neighbor Mahalanobis matching model
match_model <- matchit(
  treated_mpa ~ dist_to_deep_water + ssta_freq + reef_exposure +
    reef_slope + reef_type + dist_mangroves + dist_fish_settl +
    dist_market + pollution_risk + monsoon_direction,
  data = data_clean,
  method = "nearest",         # Nearest neighbor matching
  distance = "mahalanobis",   # Mahalanobis distance 
  ratio = 2,                  # 2:1 control/treated ratio)
  replace = TRUE )            # With replacement 

match_model
A `matchit` object
 - method: 2:1 nearest neighbor matching with replacement
 - distance: Mahalanobis - number of obs.: 161 (original), 145 (matched)
 - target estimand: ATT
 - covariates: dist_to_deep_water, ssta_freq, reef_exposure, reef_slope, reef_type, dist_mangroves, dist_fish_settl, dist_market, pollution_risk, monsoon_direction
# Extract matched dataset
matched_data <- match.data(match_model)
# Inspect matching summary
summary(match_model)

Call:
matchit(formula = treated_mpa ~ dist_to_deep_water + ssta_freq + 
    reef_exposure + reef_slope + reef_type + dist_mangroves + 
    dist_fish_settl + dist_market + pollution_risk + monsoon_direction, 
    data = data_clean, method = "nearest", distance = "mahalanobis", 
    replace = TRUE, ratio = 2)

Summary of Balance for All Data:
                          Means Treated Means Control Std. Mean Diff.
dist_to_deep_water             627.0960      655.7936         -0.0314
ssta_freq                       25.4444       24.5094          0.1830
reef_exposureExposed             0.8519        0.6604          0.5390
reef_exposureSemi-exposed        0.1481        0.3396         -0.5390
reef_slopeFlat                   0.0926        0.1321         -0.1362
reef_slopeSlope                  0.8333        0.8491         -0.0422
reef_slopeWall                   0.0741        0.0189          0.2108
reef_typeBarrier                 0.0926        0.1509         -0.2013
reef_typeFringing                0.8611        0.8491          0.0349
reef_typePatch                   0.0463        0.0000          0.2203
dist_mangroves                4618.4108     9679.3287         -0.9799
dist_fish_settl              32505.0449    25646.2642          0.2175
dist_market                 139918.0030   124152.3646          0.2374
pollution_riskHigh               0.0093        0.0000          0.0967
pollution_riskLow                0.7778        0.6226          0.3732
pollution_riskMedium             0.2130        0.3774         -0.4016
monsoon_directionNW              0.5741        0.3585          0.4360
monsoon_directionSE              0.4259        0.6415         -0.4360
                          Var. Ratio eCDF Mean eCDF Max
dist_to_deep_water            1.0168    0.0271   0.0924
ssta_freq                     0.4116    0.0749   0.1651
reef_exposureExposed               .    0.1915   0.1915
reef_exposureSemi-exposed          .    0.1915   0.1915
reef_slopeFlat                     .    0.0395   0.0395
reef_slopeSlope                    .    0.0157   0.0157
reef_slopeWall                     .    0.0552   0.0552
reef_typeBarrier                   .    0.0584   0.0584
reef_typeFringing                  .    0.0121   0.0121
reef_typePatch                     .    0.0463   0.0463
dist_mangroves                0.1317    0.0755   0.1707
dist_fish_settl               1.2507    0.0780   0.1988
dist_market                   1.7808    0.1061   0.3515
pollution_riskHigh                 .    0.0093   0.0093
pollution_riskLow                  .    0.1551   0.1551
pollution_riskMedium               .    0.1644   0.1644
monsoon_directionNW                .    0.2156   0.2156
monsoon_directionSE                .    0.2156   0.2156

Summary of Balance for Matched Data:
                          Means Treated Means Control Std. Mean Diff.
dist_to_deep_water             627.0960      344.4758          0.3097
ssta_freq                       25.4444       25.5370         -0.0181
reef_exposureExposed             0.8519        0.8380          0.0391
reef_exposureSemi-exposed        0.1481        0.1620         -0.0391
reef_slopeFlat                   0.0926        0.0556          0.1278
reef_slopeSlope                  0.8333        0.9120         -0.2112
reef_slopeWall                   0.0741        0.0324          0.1591
reef_typeBarrier                 0.0926        0.0880          0.0160
reef_typeFringing                0.8611        0.9120         -0.1473
reef_typePatch                   0.0463        0.0000          0.2203
dist_mangroves                4618.4108     3903.1137          0.1385
dist_fish_settl              32505.0449    23907.6888          0.2726
dist_market                 139918.0030   138886.2537          0.0155
pollution_riskHigh               0.0093        0.0000          0.0967
pollution_riskLow                0.7778        0.7963         -0.0445
pollution_riskMedium             0.2130        0.2037          0.0226
monsoon_directionNW              0.5741        0.5093          0.1311
monsoon_directionSE              0.4259        0.4907         -0.1311
                          Var. Ratio eCDF Mean eCDF Max Std. Pair Dist.
dist_to_deep_water            3.0661    0.1022   0.2176          0.6162
ssta_freq                     1.4678    0.0244   0.0926          0.7032
reef_exposureExposed               .    0.0139   0.0139          0.1434
reef_exposureSemi-exposed          .    0.0139   0.0139          0.1434
reef_slopeFlat                     .    0.0370   0.0370          0.1278
reef_slopeSlope                    .    0.0787   0.0787          0.2112
reef_slopeWall                     .    0.0417   0.0417          0.1591
reef_typeBarrier                   .    0.0046   0.0046          0.0160
reef_typeFringing                  .    0.0509   0.0509          0.1473
reef_typePatch                     .    0.0463   0.0463          0.2203
dist_mangroves                1.6708    0.0714   0.1759          0.7967
dist_fish_settl               1.0892    0.1147   0.3056          0.6229
dist_market                   2.0937    0.1334   0.3889          0.5956
pollution_riskHigh                 .    0.0093   0.0093          0.0967
pollution_riskLow                  .    0.0185   0.0185          0.0445
pollution_riskMedium               .    0.0093   0.0093          0.0226
monsoon_directionNW                .    0.0648   0.0648          0.2247
monsoon_directionSE                .    0.0648   0.0648          0.2247

Sample Sizes:
              Control Treated
All             53.       108
Matched (ESS)   17.69     108
Matched         37.       108
Unmatched       16.         0
Discarded        0.         0

✅ Your turn

Q5. Report how many treated units were matched and how many control sites were included in the matched data.

Response: _________________________

Q6. Find the covariate with the largest standardized mean difference (SMD) after matching. Which covariate is it?

Response: _________________________

Visualize balance on the covariates using plot() with type = "density"

plot(match_model, type = "density", interactive = FALSE,
     which.xs = ~dist_to_deep_water + ssta_freq + reef_exposure)

4. Evaluate balance after matching (love plot + balance table)

# Create nicer variable labels for plots/tables
nice_names <- data.frame(
  old = c(
    "dist_to_deep_water", "ssta_freq", "reef_exposure", "reef_slope",
    "reef_type", "dist_mangroves", "dist_fish_settl", "dist_market",
    "pollution_risk", "monsoon_direction"),
  new = c(
    "Distance to deep water", "SSTA frequency", "Reef exposure", "Reef slope",
    "Reef type", "Distance to mangroves (m)", "Distance to fishing settlement (m)",
    "Distance to market (m)", "Watershed pollution risk", "Monsoon direction (SE/NW)"))

Create a Love plot to visualize standardized mean differences (before & after matching)

love.plot(
  match_model,
  stats = "mean.diffs",
  thresholds = c(m = 0.1),
  var.names = nice_names)

5. Estimate differences in outcomes using the matched sample

  • When matching with replacement, some control sites were matched multiple times and receive larger weights.
  • So we estimate outcome differences using the weights provided by match.data().

Regression (Outcome 1): Key fisheries families biomass

reg_ecological <- lm(
  biomass_ecological ~ treated_mpa,
  data = matched_data,
  weights = weights)

summ(reg_ecological, model.fit = FALSE)
Observations 145
Dependent variable biomass_ecological
Type OLS linear regression
Est. S.E. t val. p
(Intercept) 290.77 106.02 2.74 0.01
treated_mpaMPA site 79.50 122.84 0.65 0.52
Standard errors: OLS

Regression (Outcome 2): Key fisheries families biomass

reg_fisheries <- lm(
  biomass_fisheries ~ treated_mpa,
  data = matched_data,
  weights = weights)

summ(reg_fisheries, model.fit = FALSE)
Observations 145
Dependent variable biomass_fisheries
Type OLS linear regression
Est. S.E. t val. p
(Intercept) 64.23 52.47 1.22 0.22
treated_mpaMPA site 107.78 60.79 1.77 0.08
Standard errors: OLS

View biomass differences by treatment group as simple weighted mean comparisons:

matched_data %>%
  group_by(treated_mpa) %>%
  summarize(
    n_obs = n(),
    avg_biomass_ecological = weighted.mean(biomass_ecological, w = weights),
    avg_biomass_fisheries  = weighted.mean(biomass_fisheries,  w = weights),
    .groups = "drop") %>%
  gt() %>%
  tab_header(title = "Weighted mean outcomes in matched sample")
Weighted mean outcomes in matched sample
treated_mpa n_obs avg_biomass_ecological avg_biomass_fisheries
Control (non‑MPA) 37 290.7664 64.2348
MPA site 108 370.2682 172.0131

✅ Your turn

Q7. Interpret one coefficient from the regressions above (choose ecological or fisheries). Write the interpretation of coefficient in plain language.

Response: _________________________

Q8. What is the biggest remaining threat to causal interpretation of regression coefficients after matching?

Response: _________________________

6. Fixed Effects Estimation - An Applied Example

Replication of fixed effects estimator from applied study:

Dudney, J., Willing, C. E., Das, A. J., Latimer, A. M., Nesmith, J. C., & Battles, J. J. (2021). Nonlinear shifts in infectious rust disease due to climate change. Nature communications, 12(1), 5102.

In this section we will look at a simple fixed effects (FE) example using panel-style data, where the same units (plots) are observed over multiple years (1995, 2016).

The key issue: observations from the same plot are likely to share stable characteristics (soil type, microclimate, slope, management history) that we don’t measure well—and those unobserved factors can bias a naïve regression.

Read in the data & change fixed effext to factor variables

By Coding plot and year as factors lets us include them as indicator (dummy) variables in an FE regression.

data_fe <- read_csv(here("week4", "Dudney2021_study_data.csv"), show_col_types = FALSE) %>% 
  mutate(year=factor(year), plot=factor(plot))

Estimate a “no fixed effects” baseline model

mod1_nofe <- lm(perinc ~ vpd + I(vpd^2) + dbh + density,
          data = data_fe)

summ(mod1_nofe, model.fit = FALSE, digits = 3)
Observations 294
Dependent variable perinc
Type OLS linear regression
Est. S.E. t val. p
(Intercept) -0.615 0.122 -5.027 0.000
vpd 0.126 0.022 5.712 0.000
I(vpd^2) -0.005 0.001 -5.119 0.000
dbh -0.001 0.001 -2.499 0.013
density -0.000 0.000 -3.287 0.001
Standard errors: OLS

Estimate the fixed effects model: Add in the fixed effects for plot & year

  • plot fixed effects absorb all time-invariant differences across plots (e.g., baseline soil quality)
  • year fixed effects absorb common shocks shared by all plots in a given year (e.g., a region-wide drought year)
fe_lm <- lm(perinc ~ vpd + I(vpd^2) + dbh + density + plot + year,
          data = data_fe)

summ(fe_lm, model.fit = FALSE, digits = 3)
Observations 294
Dependent variable perinc
Type OLS linear regression
Est. S.E. t val. p
(Intercept) -1.199 0.770 -1.556 0.122
vpd 0.227 0.093 2.449 0.016
I(vpd^2) -0.010 0.002 -4.209 0.000
dbh -0.001 0.001 -0.973 0.332
density -0.001 0.000 -2.275 0.024
plot2 0.092 0.177 0.516 0.606
plot3 0.140 0.180 0.778 0.438
plot4 0.118 0.186 0.634 0.527
plot5 0.181 0.185 0.978 0.330
plot6 0.136 0.126 1.079 0.283
plot7 0.007 0.101 0.066 0.948
plot8 0.198 0.262 0.757 0.450
plot9 0.070 0.149 0.472 0.638
plot10 0.156 0.117 1.336 0.184
plot11 0.147 0.169 0.871 0.385
plot13 0.160 0.167 0.957 0.340
plot14 0.344 0.122 2.816 0.006
plot15 0.107 0.198 0.542 0.589
plot16 0.207 0.222 0.933 0.352
plot17 0.161 0.121 1.329 0.186
plot18 0.125 0.187 0.668 0.505
plot19 0.107 0.194 0.554 0.580
plot20 0.138 0.204 0.677 0.500
plot21 0.144 0.179 0.805 0.422
plot22 0.197 0.158 1.249 0.214
plot23 0.090 0.141 0.638 0.525
plot24 0.112 0.158 0.704 0.483
plot25 0.142 0.230 0.618 0.537
plot26 0.186 0.223 0.831 0.408
plot28 0.170 0.190 0.894 0.373
plot29 0.078 0.155 0.504 0.615
plot30 0.266 0.230 1.157 0.249
plot31 0.131 0.160 0.815 0.416
plot32 0.103 0.157 0.656 0.513
plot34 0.118 0.194 0.610 0.543
plot35 0.265 0.261 1.017 0.311
plot36 0.025 0.121 0.207 0.836
plot37 0.131 0.165 0.799 0.426
plot38 0.747 0.088 8.439 0.000
plot39 -0.012 0.091 -0.134 0.894
plot40 0.082 0.090 0.915 0.362
plot41 0.231 0.121 1.907 0.059
plot42 0.423 0.263 1.607 0.110
plot43 0.366 0.142 2.584 0.011
plot44 0.296 0.202 1.462 0.146
plot45 0.038 0.116 0.333 0.740
plot46 0.248 0.143 1.733 0.085
plot47 0.543 0.143 3.795 0.000
plot48 0.107 0.133 0.801 0.425
plot49 0.059 0.092 0.643 0.521
plot51 0.577 0.114 5.039 0.000
plot52 0.178 0.256 0.695 0.488
plot53 0.152 0.252 0.604 0.547
plot54 0.323 0.113 2.857 0.005
plot55 0.204 0.192 1.060 0.291
plot56 0.203 0.267 0.761 0.448
plot57 0.172 0.180 0.957 0.340
plot58 0.020 0.114 0.179 0.858
plot59 0.221 0.219 1.010 0.314
plot60 0.105 0.214 0.489 0.625
plot61 0.136 0.119 1.135 0.258
plot62 0.141 0.224 0.629 0.531
plot63 0.234 0.188 1.242 0.216
plot64 0.201 0.299 0.674 0.502
plot65 0.083 0.186 0.449 0.654
plot66 0.120 0.130 0.924 0.357
plot67 0.084 0.124 0.683 0.496
plot68 0.074 0.090 0.822 0.412
plot69 0.201 0.161 1.248 0.214
plot70 0.276 0.113 2.449 0.016
plot71 0.126 0.201 0.630 0.530
plot72 0.221 0.174 1.266 0.208
plot73 0.044 0.123 0.356 0.722
plot74 0.069 0.130 0.529 0.598
plot75 0.043 0.123 0.347 0.729
plot76 0.155 0.252 0.618 0.538
plot77 0.207 0.228 0.905 0.367
plot78 0.295 0.205 1.439 0.152
plot79 0.145 0.234 0.619 0.537
plot80 0.266 0.252 1.057 0.292
plot82 0.239 0.273 0.876 0.383
plot83 0.237 0.261 0.908 0.365
plot84 0.260 0.270 0.965 0.336
plot85 0.298 0.297 1.001 0.319
plot86 0.120 0.209 0.576 0.565
plot87 0.088 0.124 0.710 0.479
plot88 0.296 0.093 3.199 0.002
plot89 0.175 0.125 1.400 0.164
plot90 0.018 0.101 0.175 0.861
plot91 0.017 0.102 0.166 0.868
plot92 0.078 0.150 0.522 0.602
plot93 0.170 0.154 1.100 0.273
plot94 -0.001 0.095 -0.012 0.990
plot95 0.234 0.141 1.666 0.098
plot96 0.083 0.096 0.858 0.392
plot97 0.216 0.150 1.440 0.152
plot98 0.180 0.162 1.108 0.270
plot99 0.125 0.159 0.785 0.434
plot100 0.019 0.117 0.165 0.869
plot101 0.088 0.119 0.745 0.457
plot102 0.010 0.114 0.084 0.933
plot103 0.171 0.174 0.980 0.329
plot104 0.157 0.173 0.909 0.365
plot105 0.117 0.171 0.687 0.493
plot106 0.102 0.090 1.140 0.256
plot107 0.038 0.112 0.334 0.739
plot108 0.086 0.120 0.716 0.475
plot110 0.092 0.153 0.601 0.549
plot111 0.074 0.096 0.767 0.444
plot112 0.201 0.105 1.912 0.058
plot113 0.090 0.128 0.704 0.482
plot114 0.270 0.089 3.038 0.003
plot115 0.008 0.109 0.075 0.941
plot116 0.111 0.121 0.918 0.360
plot117 0.057 0.108 0.530 0.597
plot118 0.169 0.167 1.014 0.312
plot119 0.074 0.167 0.444 0.658
plot120 0.139 0.182 0.767 0.444
plot121 0.191 0.092 2.073 0.040
plot122 0.033 0.132 0.252 0.802
plot123 0.072 0.113 0.634 0.527
plot124 0.080 0.142 0.565 0.573
plot125 0.381 0.224 1.695 0.092
plot126 0.106 0.126 0.845 0.400
plot127 0.172 0.138 1.245 0.215
plot128 0.505 0.104 4.851 0.000
plot129 0.104 0.171 0.607 0.545
plot130 0.292 0.196 1.490 0.138
plot131 0.195 0.175 1.118 0.266
plot132 0.254 0.094 2.705 0.008
plot133 0.258 0.145 1.783 0.077
plot134 0.386 0.269 1.434 0.154
plot135 0.103 0.132 0.777 0.438
plot136 0.053 0.095 0.553 0.581
plot137 0.136 0.221 0.615 0.539
plot138 0.244 0.198 1.232 0.220
plot139 0.095 0.154 0.616 0.539
plot140 0.295 0.094 3.143 0.002
plot141 0.158 0.150 1.058 0.292
plot142 0.139 0.184 0.756 0.451
plot143 0.105 0.117 0.896 0.372
plot145 -0.020 0.090 -0.218 0.828
plot146 0.342 0.268 1.276 0.204
plot147 0.367 0.275 1.335 0.184
plot148 0.214 0.129 1.664 0.098
plot149 0.231 0.131 1.757 0.081
plot150 0.245 0.199 1.233 0.220
plot151 0.119 0.196 0.607 0.545
plot152 0.340 0.099 3.426 0.001
plot153 0.186 0.128 1.448 0.150
plot154 0.065 0.103 0.636 0.526
year2016 -0.050 0.072 -0.693 0.490
Standard errors: OLS

✅ Your turn

Q9. Compare the coefficient on vpd in the no-FE model vs the FE model. Did the estimate get larger, smaller, or change sign? What does that suggest about confounding from unobserved plot differences?

Response: _________________________

Q10. Why include year fixed effects? Give one concrete example of a “year shock” that could bias estimates if not controlled (choose a different example than provided above).

Response: _________________________

Q11. Why include plot fixed effects? Give one concrete example of a “time-invariant plot difference” that could bias estimates if not controlled (choose a different example than provided above).

Response: _________________________