G. Growth and Institutions

Learning goals

Rich countries have more capital and more schooling than poor ones. But Chapter 4 and Appendix D showed that most of the gap comes from total factor productivity (TFP): rich countries also use their capital and their workers much better. Why? A leading answer is institutions, the rules that decide whether people can keep what they build, sign contracts that courts enforce, and invest without fear that a ruler will take it away. Countries with good institutions are clearly richer. The hard question is whether the institutions cause the prosperity, or only go together with it. In a famous paper, Acemoglu, Johnson and Robinson (2001), “AJR” from now on, proposed a way to answer it, with a clever use of history and a tool from econometrics called instrumental variables (IV). This appendix rebuilds their argument and runs their regressions on their data. By the end of it you will be able to:

derive how institutions enter the production and Solow models, and why they lead to a simple regression of log income on a measure of institutions;
explain three reasons why that regression, run by ordinary least squares (OLS), can give the wrong answer: omitted variables, reverse causality and measurement error;
state the conditions an instrument must meet (relevance and exclusion), and derive the IV estimate as the ratio of two slopes;
see in a simulation when OLS fails and when IV works, and why a weak instrument is a problem;
estimate OLS and two-stage least squares (2SLS) with the Python libraries pyfixest and linearmodels, and read the first-stage F statistic and the Wu–Hausman test;
interpret AJR’s estimate in economic terms, as a local effect, and judge its weak points.

This appendix builds on my tutorial Do Institutions Cause Prosperity? An IV Tutorial in Python, which goes further into the robustness checks of AJR (Section G.11 lists them).

A map of the appendix

Section	What you do	Builds on
G.1	The question: do institutions cause prosperity?	Chapter 4, Appendix D
G.2	Set up the libraries and the regression tools	Appendix A
G.3	Derive the model: institutions, income, and why a regression can mislead	Chapters 4 and 5, Appendix C
G.4	Simulation: when OLS fails and IV works	—
G.5	Meet AJR’s data	Chapter 3
G.6	Run the naive OLS regressions	Appendix F
G.7	The instrument: settler mortality, the first stage and the reduced form	—
G.8	The 2SLS estimate and its tests	—
G.9	OLS against IV: what the gap means	—
G.10	Discussion and limits	—
G.11	Go further	—

The regressions of Sections G.6 to G.9 replicate the main columns of Tables 2 and 4 of AJR. The robustness tables 5 to 8 of the paper are left for Section G.11 and the tutorial.

G.1 Why institutions?

Economists distinguish two kinds of causes of income differences. The proximate causes are the ones the production model of Chapter 4 measures: capital per worker, schooling, and TFP. But they only push the question back. Why do some countries invest so little, school their children so little, and use what they have so badly? The fundamental causes are the answers to that question. Three are discussed most: geography (climate, disease, soils, access to the sea), culture (values and beliefs), and institutions.

The economic historian Douglass North called institutions “the rules of the game in a society”. Good economic institutions protect property, enforce contracts, and give a broad part of the population the chance to invest and innovate. Bad ones let a small elite take what others produce. Two examples make the case vivid. North and South Korea share their history, culture and geography, but not their institutions since 1945; the satellite photograph of the Korean peninsula at night, discussed in Chapter 4, shows the result. And Botswana, which kept secure property rights and stable rules after independence, grew faster than almost any country in the world for three decades, while many of its neighbors stagnated.

Examples are not proof. Across all countries, a better measure of institutions goes with higher income (Fig. 2 shows it). But the correlation alone cannot tell us which way the arrow points. Rich countries can afford good courts. And something else, such as a healthy climate, may give a country both good institutions and high income. AJR’s idea was to find a piece of history that changed the institutions of some countries and not others, for reasons that have nothing to do with their income today. Section G.3 shows why such a piece of history, an instrument, solves the problem, and Sections G.7 and G.8 put AJR’s choice to work.

G.2 Setup

The setup cells install two libraries, load the others, and set the look of the charts. The two new libraries run regressions with the standard errors and tests that economists report:

pyfixest writes a regression as a formula, including IV regressions;
linearmodels runs IV regressions and reports the tests of instruments that this appendix needs.

They do not come with Colab, so the first cell installs them. It takes a few seconds.

# Install the two regression libraries of this appendix (Colab does not have them)
%pip install -q --disable-pip-version-check pyfixest linearmodels

Note: you may need to restart the kernel to use updated packages.

# Libraries used in this appendix
import numpy as np                        # numbers and arrays
import pandas as pd                       # data tables
import plotly.express as px               # quick charts from data tables
import plotly.graph_objects as go         # charts built piece by piece
import plotly.io as pio
import pyfixest as pf                     # regressions as formulas, including IV
from linearmodels.iv import IV2SLS        # IV regressions with tests of the instrument
from ipywidgets import interact, FloatSlider, Dropdown, ToggleButtons

# One light blue/green look for every chart
BLUE, TEAL, GREEN, GRAY, RUST = "#16528c", "#008070", "#7fb23a", "#8a96a3", "#c4651a"
pio.templates.default = "plotly_white"
# the same size in Colab and on the website
pio.templates["plotly_white"].layout.update(width=860, height=480)
px.defaults.color_discrete_sequence = [BLUE, TEAL, GREEN, RUST, "#8e5ea2", "#5aa0d8"]
WIDE = {"description_width": "initial"}   # show the full widget labels

# Colab only: let Plotly charts appear inside the interactive widgets
try:
    import google.colab
    google.colab.output.enable_custom_widget_manager()
except ImportError:
    pass                                  # not running in Colab: nothing to do

# The data file of this appendix lives in the book's GitHub repository
DATA_URL = "https://raw.githubusercontent.com/quarcs-lab/macro102-site/main/data/appG/"

Show the helper code

# @title Helper for the website versions (run this cell; you do not need to read it)
def button_figure(make_figure, name, values, start):
    """Precompute make_figure(name=value) for each value; one button per value.

    The website cannot run Python, so each interactive figure also gets this
    precomputed version.
    """
    figures = [make_figure(**{name: v}) for v in values]
    first = values.index(start)
    fig = go.Figure(layout=figures[first].layout)
    owner = []  # which precomputed figure each trace comes from
    for i, f in enumerate(figures):
        for trace in f.data:
            trace.visible = i == first
            fig.add_trace(trace)
            owner.append(i)
    buttons = []
    for i, f in enumerate(figures):
        layout = {
            "title": f.layout.title.to_plotly_json(),
            "yaxis": f.layout.yaxis.to_plotly_json(),
            "annotations": [a.to_plotly_json() for a in f.layout.annotations],
            "shapes": [s.to_plotly_json() for s in f.layout.shapes],
        }
        visible = [o == i for o in owner]
        buttons.append(
            dict(
                label=str(values[i]),
                method="update",
                args=[{"visible": visible}, layout],
            )
        )
    fig.update_layout(
        updatemenus=[
            dict(
                type="buttons",
                direction="right",
                buttons=buttons,
                active=first,
                x=0,
                y=1.02,
                xanchor="left",
                yanchor="bottom",
            )
        ],
        margin=dict(t=130),
    )
    return fig

G.2.1 Tools for regressions

We run the same few kinds of regression many times. Four short functions do the work. ols_fit runs OLS with pyfixest. Like AJR, it uses robust standard errors (called HC1), which stay valid when the spread of the errors differs from country to country. coef_table puts several OLS regressions side by side.

def ols_fit(formula, data):
    """OLS with robust (HC1) standard errors; rows with a missing value are dropped."""
    return pf.feols(formula, data=data, vcov="HC1")


def coef_table(models, names, term="avexpr"):
    """One row per regression: the coefficient on term, its standard error and N."""
    rows = []
    for model, name in zip(models, names):
        rows.append(
            {
                "regression": name,
                "coefficient": model.coef()[term],
                "std. error": model.se()[term],
                "countries": len(model.resid()),
            }
        )
    return pd.DataFrame(rows).set_index("regression").round(3)

Python note. pf.feols("logpgp95 ~ avexpr", data=d) reads like the regression equation: the variable on the left of ~ is explained by the variables on the right, and a constant is added automatically. The fitted model answers questions through its methods: .coef() gives the coefficients, .se() their standard errors and .resid() the residuals, one per country used. Each is a pandas Series indexed by the variable names, so model.coef()["avexpr"] is the coefficient on avexpr.

iv_fit runs the IV regression of AJR with linearmodels: log income on institutions, with institutions instrumented by log settler mortality, plus any control variables. add_line draws a fitted line on a chart, and first_stage_F reads off the test of the instrument that Section G.8 explains.

def iv_fit(data, controls=()):
    """2SLS of logpgp95 on avexpr, instrumented by logem4 (linearmodels)."""
    controls = list(controls)
    d = data.dropna(subset=["logpgp95", "avexpr", "logem4"] + controls)
    exog = d[controls].assign(const=1.0)  # the controls and a constant
    return IV2SLS(d["logpgp95"], exog, d[["avexpr"]], d[["logem4"]]).fit(
        cov_type="robust"
    )


def add_line(fig, x, y, name="OLS line"):
    """Add a dashed black line through the points (x, y) to a figure."""
    fig.add_trace(
        go.Scatter(x=x, y=y, mode="lines", name=name, line=dict(color="black", dash="dash"))
    )


def first_stage_F(result):
    """The robust F statistic of the first stage: how strongly Z moves institutions."""
    return float(result.first_stage.diagnostics.loc["avexpr", "f.stat"])

Python note. IV2SLS(y, exog, endog, instruments) takes four pieces: the outcome, the exogenous variables (the constant and the controls), the endogenous variable that needs an instrument, and the instruments. linearmodels does not add a constant by itself, so .assign(const=1.0) adds a column of ones. cov_type="robust" asks for robust standard errors.

G.3 Model overview

Picture Islandia again, but this time not alone. It sits in an archipelago of many island countries that make the same goods with the same kind of boats and tools. On some islands a boat builder can be sure that the boat she builds stays hers. On others a chief can seize it, and the courts do nothing. Islanders on the secure islands build more boats and try new ways to fish; on the insecure ones they do neither. This section turns that story into the models of Chapters 4 and 5, derives the regression that AJR estimate, and shows why running it the simple way can mislead.

The big idea

The question. Do better institutions cause higher income, and by how much?

The answer in one sentence. In the growth models, institutions raise income through productivity and investment, which gives a straight line between log income and institutions. A simple regression gets its slope wrong whenever institutions are tied to other causes of income or measured with error, but an instrument, a force that moved institutions and nothing else, recovers it as a ratio of two slopes.

What goes in and what comes out. We choose how strongly institutions raise productivity and investment. The model gives back the effect of institutions on long-run income, the slope that OLS finds, and the slope that IV finds.

Road map.

Section G.3.1, institutions in the growth model: the assumptions, the solution, and what the model predicts.
Section G.3.2, why a simple regression can mislead: omitted variables, reverse causality and measurement error.
Section G.3.3, an instrument: the first stage, the reduced form, and the IV estimate.
Section G.3.4, the model in code: the key results as Python functions, checked with numbers.

Notation. A bar over a letter marks a parameter. Subscript \(i\) is an island (or, in the data, a country).

Symbol	Meaning	Type
\(y_i\)	output per worker in the steady state	endogenous
\(\bar{A}_i\), \(\bar{s}_i\)	productivity (TFP) and the investment rate	depend on institutions
\(R_i\)	institutions: how secure property is, on a 0 to 10 scale	data
\(\bar{\phi}\), \(\bar{\psi}\)	the effects of institutions on \(\ln \bar{A}\) and \(\ln \bar{s}\)	parameters
\(\beta\)	the total effect of institutions on log income	parameter we want
\(u_i\)	everything else that moves log income	error term
\(Z_i\)	the instrument (in the data: log settler mortality)	data
\(\pi\), \(\rho\)	the slopes of the first stage and of the reduced form	parameters
\(\widehat{b}\)	a hat marks an estimate computed from data	estimate

G.3.1 Institutions in the growth model

Assumptions

1. Production. Each island makes output with capital and workers, with the production function of Chapter 4. Per worker it reads

\[y_i = \bar{A}_i k_i^{1/3}.\]

2. Capital accumulation. Each island invests a share \(\bar{s}_i\) of output, and capital wears out at the rate \(\bar{d}\), the same everywhere. As in Chapter 5, capital per worker settles where investment just covers depreciation, and steady-state output per worker is

\[y_i^* = \bar{A}_i^{3/2}\left(\frac{\bar{s}_i}{\bar{d}}\right)^{1/2}.\]

3. Institutions raise productivity and investment. An island with more secure property uses its tools and workers better and invests more. We write both effects in logs, with \(R_i\) the security of property:

\[\ln \bar{A}_i = a_0 + \bar{\phi}R_i + \varepsilon_i^A, \qquad \ln \bar{s}_i = s_0 + \bar{\psi}R_i + \varepsilon_i^s.\]

The terms \(\varepsilon_i^A\) and \(\varepsilon_i^s\) hold everything else that makes an island productive or thrifty: its soil, its harbors, its climate, its luck.

Institutions matter twice: they decide how well an island uses its tools, and how many tools it builds.

Solving the model

Step 1: take logs. The log of a product is the sum of the logs, and the log of a power is the exponent times the log (Appendix C):

\[\ln y_i^* = \tfrac{3}{2}\ln \bar{A}_i + \tfrac{1}{2}\ln \bar{s}_i - \tfrac{1}{2}\ln \bar{d}.\]

Step 2: put in the two rules of assumption 3. Collect the terms that are the same on every island into a constant \(\mu\), the terms with \(R_i\), and the rest:

\[\ln y_i^* = \underbrace{\tfrac{3}{2}a_0 + \tfrac{1}{2}s_0 - \tfrac{1}{2}\ln \bar{d}}_{\mu} + \underbrace{\left(\tfrac{3}{2}\bar{\phi} + \tfrac{1}{2}\bar{\psi}\right)}_{\beta}R_i + \underbrace{\tfrac{3}{2}\varepsilon_i^A + \tfrac{1}{2}\varepsilon_i^s}_{u_i}.\]

Key result: institutions and income

\[\ln y_i = \mu + \beta R_i + u_i \tag{1}\]

\[\beta = \tfrac{3}{2}\bar{\phi} + \tfrac{1}{2}\bar{\psi} \tag{2}\]

Equation (1) is the regression that AJR estimate, with log GDP per person for \(\ln y_i\). Equation (2) says what its slope means: the total long-run effect of institutions, through productivity (weighted by \(3/2\)) and through investment (weighted by \(1/2\)).

Islandia in numbers. In this appendix’s baseline, \(\bar{\phi} = \bar{\psi} = 0.5\), so \(\beta = 0.75 + 0.25 = 1\). One more point of secure property raises log income by 1, which multiplies income by \(e^1 \approx 2.7\).

In words

Productivity gets the larger weight for the same reason as in Chapter 5: better productivity raises output directly, and the extra output is partly invested, which raises capital and output again. A higher investment rate works only through capital, and diminishing returns damp it.

What the model predicts

Comparing two islands. Write equation (1) for two islands and subtract. The constant cancels:

\[\ln y_1 - \ln y_2 = \beta\,(R_1 - R_2) + (u_1 - u_2).\]

With \(\beta = 1\), an island whose institutions score 2 points higher is, other things equal, \(e^2 \approx 7.4\) times as rich. Institutions that differ by a few points can explain income gaps of the size we see between rich and poor countries.

A line through the cloud. If we plot \(\ln y_i\) against \(R_i\) for many islands, equation (1) says the points scatter around a straight line with slope \(\beta\). The scatter comes from \(u_i\): the islands differ in many ways besides their institutions.

In the model, log income is a straight line in institutions, and the slope is the effect we are after.

Check your understanding

1. Suppose institutions raise productivity (\(\bar{\phi} = 0.5\)) but do not change the investment rate (\(\bar{\psi} = 0\)). What is \(\beta\)?

Answer

Equation (2) gives \(\beta = \tfrac{3}{2}\times 0.5 + 0 = 0.75\). Institutions still matter a lot, through productivity alone.

2. Two islands differ by 1 point in \(R\), and \(\beta = 1\). Is the richer island exactly \(e \approx 2.7\) times as rich?

Answer

Only on average. The gap is \(\beta(R_1-R_2) + (u_1-u_2)\), and the second term can be positive or negative: the islands also differ in soil, harbors and luck.

G.3.2 Why a simple regression can mislead

The OLS slope

The obvious way to estimate \(\beta\) is to fit a line through the cloud of countries by OLS. The OLS slope is the covariance of log income with institutions divided by the variance of institutions. Put equation (1) into the covariance, and use \(\text{Cov}(R, R) = \text{Var}(R)\):

\[\widehat{b}_{OLS} = \frac{\text{Cov}(\ln y, R)}{\text{Var}(R)} = \frac{\text{Cov}(\mu + \beta R + u,\, R)}{\text{Var}(R)} = \beta + \frac{\text{Cov}(R, u)}{\text{Var}(R)}. \tag{3}\]

OLS finds \(\beta\) only when institutions are unrelated to everything else in \(u\). A regressor that is related to the error term is called endogenous. There are three reasons why institutions may be endogenous.

Three reasons for a bias

Omitted variables. Suppose a healthy climate makes an island productive (it is in \(u\)) and also lets its people build good institutions. Then \(\text{Cov}(R, u) > 0\), and OLS gives the climate’s credit to the institutions: the slope is too steep.

Reverse causality. Suppose income also shapes institutions, because rich islands can pay for honest judges. Then a lucky year, a high \(u_i\), raises income and through it \(R_i\). Again \(\text{Cov}(R, u) > 0\), and OLS is too steep.

Measurement error. We never see the true security of property, only an index \(\tilde{R}_i = R_i + m_i\), built from the judgment of experts, with an error \(m_i\) that is unrelated to everything else. Regress log income on the index instead of the truth. The numerator becomes \(\text{Cov}(\ln y, \tilde{R}) = \beta\,\text{Var}(R)\), the denominator \(\text{Var}(\tilde{R}) = \text{Var}(R) + \text{Var}(m)\), so

\[\widehat{b}_{OLS} = \beta\,\frac{\text{Var}(R)}{\text{Var}(R) + \text{Var}(m)}. \tag{4}\]

The fraction is below one: noise in the index flattens the line. This is called attenuation bias.

In words

The first two biases make OLS too large, the third makes it too small. Which one wins is an empirical question, and the answer in Section G.9 is a surprise: OLS turns out to be too small.

OLS measures how institutions and income move together, which is the causal effect only when nothing else moves with them.

Check your understanding

1. The noise in the index has half the variance of the true institutions, and \(\beta = 1\). Where does OLS end up?

Answer

Equation (4) gives \(1 \times 1/(1 + 0.5) = 2/3 \approx 0.67\): a third of the effect is lost.

2. Why does collecting more countries not fix any of the three biases?

Answer

Equations (3) and (4) do not depend on the number of countries. More data make the OLS slope more precise, but precisely wrong: it settles on \(\beta\) plus the bias, not on \(\beta\).

G.3.3 An instrument

Assumptions

Long ago, sailors explored the archipelago. On some islands deadly fevers killed most newcomers, so the sailors did not stay. They set up a few trading posts with harsh rules, built to ship goods away, and left. On the healthy islands they settled with their families and wrote rules that protected their own property. The rules outlived the sailors. Let \(Z_i\) be how deadly the fevers were. Two assumptions make \(Z_i\) an instrument:

Relevance. The fevers changed institutions. In the first stage, \[R_i = \pi_0 + \pi Z_i + v_i, \qquad \pi \neq 0. \tag{5}\]
Exclusion. The old fevers have no other path to income today: they are not in \(u_i\), so \(\text{Cov}(Z, u) = 0\). In particular, the fevers that killed sailors long ago do not make today’s islanders less productive, except through the rules they left behind.

Relevance can be checked in the data. Exclusion cannot: it is an argument about history, and Section G.10 weighs it.

Solving the model

Step 1: the reduced form. Put the first stage (5) into equation (1):

\[\ln y_i = \mu + \beta(\pi_0 + \pi Z_i + v_i) + u_i = (\mu + \beta\pi_0) + \beta\pi\,Z_i + (\beta v_i + u_i).\]

So log income is a straight line in \(Z_i\) too, the reduced form, with slope \(\rho = \beta\pi\).

Step 2: divide. The first-stage slope \(\pi\) says how much the fevers moved institutions; the reduced-form slope \(\rho\) says how much they moved income. If the fevers reach income only through institutions, the second is the first times \(\beta\), so \(\beta\) is their ratio. Each slope is a covariance with \(Z\) divided by \(\text{Var}(Z)\), which cancels:

Key result: the IV estimate

\[\rho = \beta\pi \tag{6}\]

\[\widehat{b}_{IV} = \frac{\widehat{\rho}}{\widehat{\pi}} = \frac{\text{Cov}(\ln y, Z)}{\text{Cov}(\tilde{R}, Z)} \tag{7}\]

The IV estimate is the reduced-form slope divided by the first-stage slope. It uses only the part of the differences in institutions that the instrument created.

Step 3: why measurement error does no harm. In equation (7) the index \(\tilde{R} = R + m\) appears only in a covariance with \(Z\). The noise \(m\) is unrelated to the old fevers, so \(\text{Cov}(\tilde{R}, Z) = \text{Cov}(R, Z)\), and the IV estimate is not flattened. This is one reason IV and OLS can differ.

Step 4: two stages give the same number. The name two-stage least squares describes another way to compute (7). First regress \(\tilde{R}\) on \(Z\) and keep the fitted values \(\widehat{R}_i = \widehat{\pi}_0 + \widehat{\pi}Z_i\). Then regress \(\ln y\) on \(\widehat{R}\) by OLS. Since \(\text{Cov}(\ln y, \widehat{R}) = \widehat{\pi}\,\text{Cov}(\ln y, Z)\) and \(\text{Var}(\widehat{R}) = \widehat{\pi}^2\,\text{Var}(Z)\), the slope of the second stage is \(\widehat{\rho}/\widehat{\pi}\) again.

In words

An instrument is like a natural experiment. History “assigned” deadly fevers to some islands and not others, and the fevers changed their rules. Comparing the incomes of fever and healthy islands, and scaling by how much their rules differ, gives the effect of the rules, with the climate, the luck and the noise in the index all left out.

What the model predicts

A weak instrument amplifies small problems. Suppose exclusion fails a little, \(\text{Cov}(Z, u) \neq 0\). Put equation (1) into the numerator of (7):

\[\widehat{b}_{IV} = \beta + \frac{\text{Cov}(Z, u)}{\text{Cov}(R, Z)}. \tag{8}\]

The bias is a small number divided by \(\text{Cov}(R, Z)\). If the instrument moves institutions only a little, the denominator is small, and a small violation, or plain bad luck in a sample of 64 countries, becomes a large error. That is why Section G.8 tests how strong the instrument is.

A local effect. If institutions matter more in some countries than in others, the IV estimate averages the effect over the countries whose institutions the instrument actually moved: the former colonies whose rules were set by how deadly they were for settlers. It is a local average treatment effect (LATE), not a statement about every country in the world.

IV trades one strong assumption, exclusion, for the many weak ones that OLS needs, and it is only as good as that assumption and the strength of the instrument.

Check your understanding

1. The fevers lower institutions by 0.6 points per unit (\(\pi = -0.6\)) and lower log income by 0.57 per unit (\(\rho = -0.57\)). What is the IV estimate?

Answer

Equation (7): \(-0.57/-0.6 = 0.95\). Both slopes are negative, so the ratio is positive: better institutions raise income.

2. Suppose the old fevers still make islanders sick today, and sick workers produce less. Which assumption fails, and in which direction does the IV estimate move?

Answer

Exclusion fails: the fevers now also sit in \(u\), lowering income, so \(\text{Cov}(Z, u) < 0\). The first stage is negative too, \(\text{Cov}(R, Z) < 0\), so the ratio in (8) is positive: the IV estimate is too large. Part of the health effect is counted as an effect of institutions.

Recap

The model gives a line: \(\ln y_i = \mu + \beta R_i + u_i\), with \(\beta = \tfrac{3}{2}\bar{\phi} + \tfrac{1}{2}\bar{\psi}\), the effect of institutions through productivity and investment.
OLS can mislead: omitted variables and reverse causality push it up, measurement error pulls it down (equations 3 and 4; the simulation of Fig. 1).
An instrument must move institutions (relevance) and have no other path to income (exclusion).
The IV estimate is the reduced-form slope over the first-stage slope (equation 7; Figs. 3 and 4), and two-stage least squares computes the same number.
Weak instruments turn small violations into large errors (equation 8), and the IV estimate is a local effect.

G.3.4 The model in code

Now we turn the key results into Python and check them with numbers. First equation (2), for Islandia’s baseline.

PHI, PSI = 0.5, 0.5  # effects of institutions on ln A and on ln s


def total_effect(phi, psi):
    """Equation (2): the long-run effect of institutions on log income per worker."""
    return 1.5 * phi + 0.5 * psi


BETA = total_effect(PHI, PSI)
print(f"beta = {BETA:.2f}: one more point multiplies income by {np.exp(BETA):.2f}")
print(f"two islands 2 points apart: one is {np.exp(2 * BETA):.1f} times as rich")

beta = 1.00: one more point multiplies income by 2.72
two islands 2 points apart: one is 7.4 times as rich

Next, an archipelago we can build ourselves. simulate_islands draws n islands that follow equations (1) and (5): an instrument fever, an omitted factor other (say, a healthy climate) that raises both institutions and income, the true institutions R, an index R_obs measured with noise, and log income ln_y. Three arguments set the three problems: confound (how strongly the omitted factor shapes institutions), noise (the measurement error) and strength (how strongly the fevers moved institutions).

def simulate_islands(n, beta=1.0, confound=0.5, noise=1.0, strength=0.6, seed=0):
    """An archipelago of n islands that follows equations (1) and (5): a data table."""
    rng = np.random.default_rng(seed)
    fever = rng.normal(size=n)  # the instrument Z
    other = rng.normal(size=n)  # an omitted factor, part of u
    R = 5 - strength * fever + confound * other + rng.normal(scale=0.8, size=n)
    ln_y = 2 + beta * R + 0.5 * other + rng.normal(scale=0.5, size=n)  # equation (1)
    R_obs = R + rng.normal(scale=noise, size=n)  # the measured index
    return pd.DataFrame({"fever": fever, "R": R, "R_obs": R_obs, "ln_y": ln_y})


def ols_slope(y, x):
    """The OLS slope Cov(y, x) / Var(x), as in equation (3)."""
    return np.cov(x, y)[0, 1] / np.var(x, ddof=1)


def iv_slope(y, x, z):
    """Equation (7): the IV slope Cov(y, z) / Cov(x, z)."""
    return np.cov(z, y)[0, 1] / np.cov(z, x)[0, 1]


def ols_limit(beta=1.0, confound=0.5, noise=1.0, strength=0.6):
    """Where OLS ends up in a huge archipelago: equations (3) and (4).

    The numbers are those of simulate_islands.
    """
    var_R = strength**2 + confound**2 + 0.8**2  # variance of the true institutions
    return (beta * var_R + 0.5 * confound) / (var_R + noise**2)

Python note. np.random.default_rng(seed) makes a random number generator; the same seed always gives the same numbers, so your results match the book’s. rng.normal(size=n) draws n numbers from a standard normal distribution. np.cov(x, y) returns a 2 by 2 table of covariances, and [0, 1] picks the covariance of x with y.

Now the checks. In an archipelago of 200,000 islands, sampling noise is tiny, so the estimates should land on the formulas.

big = simulate_islands(200_000)
pi_hat = ols_slope(big["R_obs"], big["fever"])  # first stage, equation (5)
rho_hat = ols_slope(big["ln_y"], big["fever"])  # reduced form
print(
    f"OLS slope {ols_slope(big['ln_y'], big['R_obs']):.3f}   "
    f"formula {ols_limit():.3f}   true beta {BETA}"
)
print(f"IV slope  {iv_slope(big['ln_y'], big['R_obs'], big['fever']):.3f}")
print(f"ratio rho/pi = {rho_hat:.3f} / {pi_hat:.3f} = {rho_hat / pi_hat:.3f}   (eq. 7)")

# Two-stage least squares by hand: regress ln_y on the fitted values of the first stage
R_hat = big["R_obs"].mean() + pi_hat * (big["fever"] - big["fever"].mean())
print(f"second stage slope {ols_slope(big['ln_y'], R_hat):.3f}   (the same number)")

OLS slope 0.668   formula 0.667   true beta 1.0
IV slope  1.003
ratio rho/pi = -0.605 / -0.604 = 1.003   (eq. 7)
second stage slope 1.003   (the same number)

What to notice. OLS lands on about 0.67, as the formula predicts, a third below the true effect of 1: in the baseline the measurement error, which flattens the line, outweighs the omitted factor, which steepens it. The IV slope, the ratio of the two slopes and the second stage of two-stage least squares all give the same number, close to 1.

G.4 Simulation: when OLS fails and IV works

AJR have 64 countries, not 200,000. With so few, every estimate also has sampling noise: a different set of 64 countries would give a different number. A simulation shows both problems at once. We build 1,000 archipelagos of 64 islands each, run OLS and IV in each, and look at the spread of the 1,000 estimates.

Fig. 1: OLS and IV estimates in 1,000 simulated archipelagos

Goal. See when OLS misses the true effect, and when IV finds it.

def sampling_estimates(confound=0.5, noise=1.0, strength=0.6, reps=1000, n=64):
    """OLS and IV estimates from reps archipelagos of n islands (true effect: BETA)."""
    ols_est, iv_est = [], []
    for r in range(reps):
        d = simulate_islands(n, BETA, confound, noise, strength, seed=r)
        ols_est.append(ols_slope(d["ln_y"], d["R_obs"]))
        iv_est.append(iv_slope(d["ln_y"], d["R_obs"], d["fever"]))
    return np.array(ols_est), np.array(iv_est)

simulation_figure draws the two histograms, a dashed line at the true effect, and reports in the title where each histogram is centered (its median).

def simulation_figure(confound=0.5, noise=1.0, strength=0.6):
    """Fig. 1: histograms of the OLS and IV estimates from 1,000 archipelagos."""
    ols_est, iv_est = sampling_estimates(confound, noise, strength)
    off = np.mean((iv_est < -1) | (iv_est > 3)) * 100  # IV estimates outside the chart
    bins = dict(start=-1, end=3, size=0.05)  # bars 0.05 wide
    fig = go.Figure()
    fig.add_trace(
        go.Histogram(x=ols_est, xbins=bins, name="OLS", marker_color=RUST, opacity=0.6)
    )
    fig.add_trace(
        go.Histogram(
            x=iv_est, xbins=bins, name="IV (2SLS)", marker_color=BLUE, opacity=0.6
        )
    )
    fig.add_vline(
        x=BETA,
        line=dict(color="black", dash="dash"),
        annotation_text="true effect",
        annotation_position="top right",
    )
    title = f"OLS centers on {np.median(ols_est):.2f}, IV on {np.median(iv_est):.2f}"
    if off >= 1:
        title += f" ({off:.0f}% of the IV estimates are off the chart)"
    fig.update_layout(
        title=title,
        barmode="overlay",
        legend=dict(orientation="h", y=-0.2),
        xaxis=dict(
            title="Estimated effect of institutions on log income",
            range=[-1, 3],
            tickvals=[-0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3],
        ),
        yaxis=dict(title="Number of archipelagos", range=[0, 450]),
    )
    return fig


simulation_figure()

interact(
    simulation_figure,
    confound=FloatSlider(
        min=-1, max=1, step=0.25, value=0.5, description="omitted factor", style=WIDE
    ),
    noise=FloatSlider(
        min=0,
        max=1.5,
        step=0.25,
        value=1.0,
        description="measurement error",
        style=WIDE,
    ),
    strength=FloatSlider(
        min=0.1,
        max=1,
        step=0.05,
        value=0.6,
        description="instrument strength",
        style=WIDE,
    ),
);

Show the code of the website version

# @title Website version: one button per scenario
SCENARIOS = {
    "Omitted factor only": dict(confound=0.5, noise=0.0, strength=0.6),
    "Measurement error only": dict(confound=0.0, noise=1.0, strength=0.6),
    "Both (baseline)": dict(confound=0.5, noise=1.0, strength=0.6),
    "Weak instrument": dict(confound=0.5, noise=1.0, strength=0.15),
}


def scenario_figure(scenario="Both (baseline)"):
    """Fig. 1 for one of the four scenarios."""
    return simulation_figure(**SCENARIOS[scenario])


button_figure(scenario_figure, "scenario", list(SCENARIOS), start="Both (baseline)")

Source. Simulation with simulate_islands (Section G.3.4): 1,000 archipelagos of 64 islands, true effect of institutions \(\beta = 1\).

What to notice.

Omitted factor only: OLS is centered above the true effect. The healthy climate raises both institutions and income, and OLS gives the climate’s credit to the institutions. IV stays centered on 1.
Measurement error only: OLS is centered below the true effect, the attenuation of equation (4). IV again stays on 1.
In both cases the OLS histogram is narrow: OLS is precise, but precisely wrong. The IV histogram is wider: IV pays for its honesty with more sampling noise.
Weak instrument: with strength 0.15 the IV histogram spreads out widely, some estimates fall off the chart, and its center slides toward OLS. A weak instrument gives no reliable answer in a sample of 64.

Your turn. In Colab, set the omitted factor to \(-0.5\) (the factor now weakens institutions while it still raises income) and the measurement error to 0. Which way does OLS miss now? Check your answer with ols_limit.

Hint

ols_limit(confound=-0.5, noise=0) gives about 0.8: with a negative confound, \(\text{Cov}(R, u) < 0\) in equation (3), so OLS is too small even without any measurement error. The IV histogram does not move.

# Your turn: try it here

G.5 The data of Acemoglu, Johnson and Robinson

AJR built their data from many sources. The book keeps one file with the variables of their Tables 1, 2 and 4, one row per country:

Column	What it measures	Unit
`logpgp95`	log GDP per person in 1995, at purchasing power parity: \(\ln y\)	log of dollars
`avexpr`	average protection against expropriation risk, 1985–1995: \(\tilde{R}\)	0 (none) to 10 (full)
`extmort4`, `logem4`	settler mortality: annual deaths per 1,000 settlers, and its log: \(Z\)	deaths, log
`lat_abst`	distance from the equator (absolute latitude divided by 90)	0 to 1
`africa`, `asia`, `other`	continent dummies (`other`: Australia, Malta, New Zealand)	0 or 1
`base`	1 for the 64 former colonies of AJR’s base sample	0 or 1
`euro1900`, `cons00a`, `democ00a`, `cons1`	European settlers in 1900, and early institutions	percent, scores

The measure of institutions comes from Political Risk Services, a firm that rates for investors how likely a government is to seize private property. Settler mortality is AJR’s compilation of the death rates of soldiers, bishops and sailors stationed in the colonies, mostly in the 17th to 19th centuries.

ajr = pd.read_csv(DATA_URL + "ajr2001.csv")
ajr["region"] = ajr["continent"].replace({"Other": "Australia, Malta, NZ"})
base = ajr[ajr["base"] == 1]  # the 64 former colonies
print(len(ajr), "countries;", len(base), "former colonies in the base sample")
ajr.head()

163 countries; 64 former colonies in the base sample

	code	iso3	country	continent	base	logpgp95	avexpr	logem4	extmort4	lat_abst	africa	asia	euro1900	cons00a	democ00a	cons1	loghjypl	region
0	AFG	AFG	Afghanistan	Asia	0	NaN	NaN	4.54010	93.7	0.366667	0	1	0.0	1.0	1.0	1.0	NaN	Asia
1	AGO	AGO	Angola	Africa	1	7.77065	5.36364	5.63479	280.0	0.136667	1	0	8.0	1.0	0.0	3.0	-3.411250	Africa
2	ARE	ARE	United Arab Emirates	Asia	0	9.80422	7.18182	NaN	NaN	0.266667	0	1	0.0	NaN	NaN	NaN	NaN	Asia
3	ARG	ARG	Argentina	Americas	1	9.13346	6.38636	4.23266	68.9	0.377778	0	0	60.0	3.0	3.0	1.0	-0.872274	Americas
4	ARM	ARM	Armenia	Asia	0	7.68248	NaN	NaN	NaN	0.444444	0	1	0.0	NaN	NaN	NaN	NaN	Asia

Table 1 of AJR describes the data. Here are the three main variables, for all countries and for the base sample:

VARS = ["logpgp95", "avexpr", "logem4"]
print("All countries")
print(ajr[VARS].describe().T[["count", "mean", "std", "min", "max"]].round(2))
print("\nBase sample: 64 former colonies")
print(
    base[VARS + ["extmort4"]]
    .describe()
    .T[["count", "mean", "std", "min", "max"]]
    .round(2)
)

All countries
          count  mean   std   min    max
logpgp95  148.0  8.30  1.11  6.11  10.29
avexpr    121.0  7.07  1.80  1.64  10.00
logem4     87.0  4.60  1.30  0.94   7.99

Base sample: 64 former colonies
          count    mean     std   min      max
logpgp95   64.0    8.06    1.04  6.11    10.22
avexpr     64.0    6.52    1.47  3.50    10.00
logem4     64.0    4.66    1.26  2.15     7.99
extmort4   64.0  245.91  472.62  8.55  2940.00

What to notice.

Income in the base sample ranges from \(e^{6.11} \approx 450\) dollars a person (Ethiopia) to \(e^{10.22} \approx 27{,}000\) (the United States): a factor of 60.
The former colonies have weaker institutions on average (6.5 against 7.1 for all countries).
Look at your data. Settler mortality runs from 8.55 deaths per 1,000 a year (Australia and New Zealand) to 2,940 (Mali). More than 1,000 deaths per 1,000 settlers means that, on average, each soldier had to be replaced more than once a year. Numbers like these come from small garrisons in a few bad years, and Section G.10 returns to how reliable they are. Taking logs keeps them from dominating the regressions.

Fig. 2: Institutions and income across countries

Goal. See the correlation that starts the debate: countries with more secure property are richer.

The vertical axis shows GDP per person on a ratio scale, so equal distances are equal percentage gaps, and a straight line on the chart is a straight line in \(\ln y\), as in equation (1).

SAMPLES = ["All countries", "Base sample"]


def income_figure(sample="All countries"):
    """Fig. 2: GDP per person in 1995 against institutions, with the OLS line."""
    d = ajr.dropna(subset=["logpgp95", "avexpr"])
    if sample == "Base sample":
        d = d[d["base"] == 1]
    fit = ols_fit("logpgp95 ~ avexpr", d)
    a, b = fit.coef()["Intercept"], fit.coef()["avexpr"]
    d = d.assign(
        gdp=np.exp(d["logpgp95"]),
        group=np.where(d["base"] == 1, "Former colony (base sample)", "Other country"),
    )
    fig = px.scatter(
        d,
        x="avexpr",
        y="gdp",
        color="group",
        hover_name="country",
        log_y=True,
        color_discrete_map={"Former colony (base sample)": BLUE, "Other country": GRAY},
        range_x=[1, 10.5],
        range_y=[300, 50000],
        labels={
            "avexpr": "Protection against expropriation, 1985–1995 (0 to 10)",
            "gdp": "GDP per person, 1995 (PPP dollars)",
            "group": "",
        },
        title=f"{len(d)} countries: one more point goes with {b:.2f} more log income",
    )
    x = np.linspace(1.5, 10, 50)
    add_line(fig, x, np.exp(a + b * x))
    fig.update_yaxes(tickvals=[500, 1000, 2000, 5000, 10000, 20000, 50000])
    fig.update_layout(legend=dict(orientation="h", y=-0.2), height=520)
    return fig


income_figure()

interact(
    income_figure,
    sample=Dropdown(
        options=SAMPLES, value="All countries", description="sample", style=WIDE
    ),
);

Show the code of the website version

# @title Website version: one button per sample
button_figure(income_figure, "sample", SAMPLES, start="All countries")

Source. Data: Acemoglu, Johnson and Robinson (2001), replication files (Table 2), from the World Bank (GDP) and Political Risk Services (institutions). Prepared file: data/appG/ajr2001.csv (see the data notes). Chart: inspired by Figure 2 in Acemoglu, Johnson and Robinson (2001).

What to notice.

The cloud slopes up steeply in both samples. Countries at the top of the scale, with 9 or 10 points, are around 20 times as rich as those near 4.
The slope is about 0.5 in both samples: one more point of protection goes with about \(e^{0.5} - 1 \approx 65\) percent more income.
The slope is a correlation. Section G.3 gave three reasons why it may differ from the effect of institutions.

Your turn. AJR also use log output per worker, loghjypl, from Hall and Jones (1999), measured relative to the United States. Run the OLS regression of loghjypl on avexpr for the base sample. Is the slope similar?

Hint

ols_fit("loghjypl ~ avexpr", base).coef() gives a slope of about 0.45, a little below the 0.52 for GDP per person. Countries with good institutions are also richer per worker.

# Your turn: try it here

G.6 The naive OLS benchmark

Before we instrument anything, we run the regression of equation (1) by OLS, as in Table 2 of AJR. We try four versions: all countries with data, the base sample, and all countries with two sets of controls, variables added to the regression to hold other causes of income constant. The controls are distance from the equator and the continent dummies.

ols_models = [
    ols_fit("logpgp95 ~ avexpr", ajr),
    ols_fit("logpgp95 ~ avexpr", base),
    ols_fit("logpgp95 ~ avexpr + lat_abst", ajr),
    ols_fit("logpgp95 ~ avexpr + lat_abst + africa + asia + other", ajr),
]
coef_table(
    ols_models,
    ["All countries", "Base sample", "+ latitude", "+ latitude and continents"],
)

	coefficient	std. error	countries
regression
All countries	0.532	0.029	111
Base sample	0.522	0.050	64
+ latitude	0.463	0.052	111
+ latitude and continents	0.390	0.051	111

What to notice.

The coefficient is 0.52 to 0.53 in the first two columns, with a small standard error: OLS is precise.
Adding latitude and the continent dummies lowers it to 0.39. Part of the correlation came from geography: the rich countries are farther from the equator and outside Africa.
These are the numbers of AJR’s Table 2. OLS says one more point of institutions goes with 40 to 70 percent more income. Is that the effect of institutions? Section G.3 says: only if none of the three biases is at work.

G.7 The instrument: settler mortality

AJR’s instrument is the island story of Section G.3.3 with real history. Where European settlers faced high death rates, from malaria and yellow fever, they did not settle. Instead they set up extractive states, built to move gold, rubber or slaves to Europe, with few protections for anyone else. Where settlers could survive, as in North America, Australia and New Zealand, they settled in large numbers and demanded the rights they had at home. Those early institutions persisted after independence. AJR summarize the argument as a chain:

\[\text{settler mortality} \;\Rightarrow\; \text{settlements} \;\Rightarrow\; \text{early institutions} \;\Rightarrow\; \text{current institutions} \;\Rightarrow\; \text{current income}.\]

The first stage, equation (5), tests the first links of the chain: does log settler mortality predict the expropriation risk of 1985–1995? The reduced form tests the whole chain: does it predict income in 1995? The two charts below show both, for the 64 former colonies.

Fig. 3: The first stage: settler mortality and institutions

Goal. Check relevance: did settler mortality move institutions?

The horizontal axis is on a ratio scale, so the slope is the change in institutions for a 1 log point (about 2.7-fold) rise in mortality. The menu drops groups of countries, to see whether the slope depends on a few of them. The neo-Europes are the four settler colonies with the lowest mortality and the best institutions: the United States, Canada, Australia and New Zealand.

NEO_EUROPES = ["USA", "CAN", "AUS", "NZL"]
SUBSAMPLES = [
    "All 64",
    "Without the neo-Europes",
    "Without Africa",
    "Without the Americas",
    "Without Asia",
]
COLORS = {"Africa": BLUE, "Americas": TEAL, "Asia": GREEN, "Australia, Malta, NZ": RUST}
MORTALITY_TICKS = [5, 10, 20, 50, 100, 200, 500, 1000, 3000]
# countries labeled on the charts (the others show their name when you hover over them)
LABELED = ["AUS", "NZL", "USA", "SGP", "HKG", "MLT", "CHL", "IND", "PAK", "ETH", "TZA",
           "GAB", "GMB", "NGA", "MLI", "SLE", "NER", "ZAR", "HTI", "BGD", "ZAF"]


def subsample(name):
    """The rows of the base sample that one menu choice keeps."""
    if name == "Without the neo-Europes":
        return base[~base["code"].isin(NEO_EUROPES)]
    if name.startswith("Without"):
        continent = name.replace("Without the ", "").replace("Without ", "")
        return base[base["continent"] != continent]
    return base


def sample_name(name, d):
    """The menu choice with its number of countries, for the chart titles."""
    return "64 former colonies" if name == "All 64" else f"{name} ({len(d)})"

add_highlight marks one country with an arrow, if it is in the data shown.

def add_highlight(fig, d, y, highlight):
    """Mark the country named highlight with an arrow (nothing if it is not in d)."""
    if highlight not in list(d["country"]):
        return
    row = d[d["country"] == highlight].iloc[0]
    fig.add_annotation(
        x=np.log10(row["extmort4"]),
        y=row[y],
        text=highlight,
        showarrow=True,
        arrowhead=2,
        ax=40,
        ay=-40,
        font=dict(color=RUST, size=13),
    )

mortality_scatter draws both charts of this section: the countries, colored by region, with the codes of some of them, and the OLS line through them. The highlight argument marks one country with an arrow.

def mortality_scatter(d, y, y_label, title, highlight):
    """A scatter of y against settler mortality (ratio scale), with the OLS line.

    Colored by region; the countries in LABELED show their codes.
    """
    fit = ols_fit(f"{y} ~ logem4", d)
    d = d.assign(label=np.where(d["code"].isin(LABELED), d["iso3"], ""))
    fig = px.scatter(
        d,
        x="extmort4",
        y=y,
        color="region",
        text="label",
        hover_name="country",
        log_x=True,
        hover_data={"label": False},
        color_discrete_map=COLORS,
        range_x=[4, 4000],
        labels={
            "extmort4": "Settler mortality (deaths per 1,000 a year, ratio scale)",
            y: y_label,
            "region": "",
        },
        title=title,
    )
    fig.update_traces(textposition="top center", textfont_size=9, marker_size=8)
    m = np.linspace(np.log(6), np.log(3500), 50)
    add_line(fig, np.exp(m), fit.coef()["Intercept"] + fit.coef()["logem4"] * m)
    add_highlight(fig, d, y, highlight)
    fig.update_xaxes(tickvals=MORTALITY_TICKS)
    fig.update_layout(legend=dict(orientation="h", y=-0.2), height=540)
    return fig

Python note. On a ratio-scale axis (log_x=True), px.scatter takes the data and range_x in deaths per 1,000, as they are. But an annotation added afterwards is placed in powers of ten, so a point at 200 deaths sits at \(\log_{10}(200) \approx 2.3\). That is why the annotation, and only the annotation, uses np.log10.

def first_stage_figure(sample="All 64", highlight="(none)"):
    """Fig. 3: protection against expropriation against settler mortality."""
    d = subsample(sample)
    res = iv_fit(d)
    slope = res.first_stage.individual["avexpr"].params["logem4"]
    title = (
        f"{sample_name(sample, d)}: slope {slope:.2f}, F = {first_stage_F(res):.1f}, "
        f"IV estimate {res.params['avexpr']:.2f}"
    )
    fig = mortality_scatter(
        d, "avexpr", "Protection against expropriation, 1985–1995", title, highlight
    )
    fig.update_yaxes(range=[3, 10.5])
    return fig


first_stage_figure()

interact(
    first_stage_figure,
    sample=Dropdown(
        options=SUBSAMPLES, value="All 64", description="countries", style=WIDE
    ),
    highlight=Dropdown(
        options=["(none)"] + sorted(base["country"]),
        value="(none)",
        description="highlight",
        style=WIDE,
    ),
);

Show the code of the website version

# @title Website version: one button per group of countries
button_figure(first_stage_figure, "sample", SUBSAMPLES, start="All 64")

Source. Data: Acemoglu, Johnson and Robinson (2001), replication files (Table 4). Prepared file: data/appG/ajr2001.csv. Chart: inspired by Figure 3 in Acemoglu, Johnson and Robinson (2001).

What to notice.

Relevance holds. In the base sample, one more log point of settler mortality goes with 0.61 points less protection against expropriation, and mortality alone explains about a quarter of the differences in institutions.
The F statistic of 16.8 measures the strength of the instrument; Section G.8 explains how to read it.
At the top left are the neo-Europes, with low mortality and secure property; at the bottom right, Mali, Nigeria and Sierra Leone. Without the four neo-Europes the slope falls by a third and the F statistic to about 7: they carry much of the first stage. Without Africa the first stage is even stronger.

Your turn. Highlight Gambia in Colab. It had very high settler mortality but scores 8.3 on institutions. How far is it from the line? Then find the residual of the first stage for every country and list the five largest.

Hint

fit = ols_fit("avexpr ~ logem4", base) and base.assign(resid=fit.resid()).nlargest(5, "resid")[["country", "avexpr", "logem4", "resid"]]. Gambia sits about 3.4 points above the line, the largest residual of all: a reminder that the first stage is a tendency, not a law.

# Your turn: try it here

Fig. 4: The reduced form: settler mortality and income

Goal. See the whole chain at once: did settler mortality long ago predict income in 1995?

def reduced_form_figure(sample="All 64", highlight="(none)"):
    """Fig. 4: log GDP per person in 1995 against settler mortality, one subsample."""
    d = subsample(sample)
    rho = ols_fit("logpgp95 ~ logem4", d).coef()["logem4"]
    pi = ols_fit("avexpr ~ logem4", d).coef()["logem4"]
    title = (
        f"{sample_name(sample, d)}: slope {rho:.2f}; "
        f"divided by the first stage {pi:.2f} gives {rho / pi:.2f}"
    )
    fig = mortality_scatter(d, "logpgp95", "Log GDP per person, 1995", title, highlight)
    fig.update_yaxes(range=[6, 10.6])
    return fig


reduced_form_figure()

interact(
    reduced_form_figure,
    sample=Dropdown(
        options=SUBSAMPLES, value="All 64", description="countries", style=WIDE
    ),
    highlight=Dropdown(
        options=["(none)"] + sorted(base["country"]),
        value="(none)",
        description="highlight",
        style=WIDE,
    ),
);

Show the code of the website version

# @title Website version: one button per group of countries
button_figure(reduced_form_figure, "sample", SUBSAMPLES, start="All 64")

Source. Data: Acemoglu, Johnson and Robinson (2001), replication files (Table 4). Prepared file: data/appG/ajr2001.csv. Chart: inspired by Figure 1 in Acemoglu, Johnson and Robinson (2001).

What to notice.

The reduced form is steep: one more log point of mortality goes with 0.57 less log income. Across the whole range of mortality, almost 6 log points, the line predicts an income gap of about 3.3 log points, a factor of almost 30.
The ratio is the IV estimate. Dividing the reduced-form slope by the first-stage slope of Fig. 3 gives \(-0.57/-0.61 = 0.94\), equation (7). The title of the chart computes it for each group of countries: 0.94 for all 64, 1.28 without the neo-Europes, 0.58 without Africa.
The vertical axis is log income, so the line in the chart is the straight line of the reduced form.

Your turn. Check equation (7) with the regressions themselves: run the reduced form and the first stage for the base sample with ols_fit, and divide the two slopes. Then compare the result with iv_fit(base).params["avexpr"].

Hint

rho = ols_fit("logpgp95 ~ logem4", base).coef()["logem4"] gives \(-0.573\) and pi = ols_fit("avexpr ~ logem4", base).coef()["logem4"] gives \(-0.607\). Their ratio, 0.944, is exactly the 2SLS estimate: with one instrument and one endogenous variable, 2SLS is this division.

# Your turn: try it here

G.8 The 2SLS estimate

Now the headline result of AJR, column 1 of their Table 4. We compute it three ways: by hand in two stages, with pyfixest and with linearmodels. Then we test the instrument and the need for it.

G.8.1 Two stages by hand

The first stage regresses institutions on the instrument and keeps the fitted values. The second regresses income on the fitted values.

stage1 = ols_fit("avexpr ~ logem4", base)
by_hand = base.assign(
    avexpr_hat=stage1.predict()
)  # the part of institutions the instrument predicts
stage2 = ols_fit("logpgp95 ~ avexpr_hat", by_hand)
print(
    f"first stage:  avexpr = {stage1.coef()['Intercept']:.3f} "
    f"+ ({stage1.coef()['logem4']:.3f}) x logem4"
)
print(
    f"second stage: logpgp95 = {stage2.coef()['Intercept']:.3f} "
    f"+ {stage2.coef()['avexpr_hat']:.3f} x avexpr_hat"
    f"   (standard error {stage2.se()['avexpr_hat']:.3f})"
)

first stage:  avexpr = 9.341 + (-0.607) x logem4
second stage: logpgp95 = 1.910 + 0.944 x avexpr_hat   (standard error 0.120)

The coefficient, 0.944, is the right one. Its standard error is not: the second stage treats the fitted values as if they were data, and forgets that they were estimated in the first stage. The IV commands below compute the standard error correctly.

G.8.2 With pyfixest and linearmodels

pyfixest writes an IV regression as a formula in three parts: the outcome, the exogenous variables (1, just a constant here), and after a second | the endogenous variable with its instrument, avexpr ~ logem4.

iv_pf = pf.feols("logpgp95 ~ 1 | avexpr ~ logem4", data=base, vcov="HC1")
iv_pf.tidy().round(3)

	Estimate	Std. Error	t value	Pr(>\|t\|)	2.5%	97.5%
Coefficient
Intercept	1.910	1.193	1.601	0.114	-0.475	4.294
avexpr	0.944	0.179	5.278	0.000	0.587	1.302

iv_lm = iv_fit(base)
ci = iv_lm.conf_int().loc["avexpr"]
print(
    f"pyfixest:     beta = {iv_pf.coef()['avexpr']:.4f}   "
    f"(standard error {iv_pf.se()['avexpr']:.4f})"
)
print(
    f"linearmodels: beta = {iv_lm.params['avexpr']:.4f}   "
    f"(standard error {iv_lm.std_errors['avexpr']:.4f})"
)
print(f"95% confidence interval (linearmodels): {ci['lower']:.2f} to {ci['upper']:.2f}")
print(f"OLS on the same 64 countries: {ols_models[1].coef()['avexpr']:.3f}")

pyfixest:     beta = 0.9443   (standard error 0.1789)
linearmodels: beta = 0.9443   (standard error 0.1761)
95% confidence interval (linearmodels): 0.60 to 1.29
OLS on the same 64 countries: 0.522

What to notice.

The 2SLS estimate is 0.94, the number of AJR’s Table 4. Both libraries agree on it to every decimal.
Their standard errors differ slightly, 0.179 and 0.176, because they make slightly different small-sample corrections. Both give a confidence interval of about 0.6 to 1.3, which does not include the OLS estimate.
IV is about 80 percent larger than OLS (0.94 against 0.52). Section G.9 asks why.

G.8.3 Is the instrument strong enough?

Equation (8) showed that a weak instrument turns small problems into large errors. The standard check is the first-stage F statistic: the test that the instrument’s coefficient in the first stage is zero. A common rule of thumb (Staiger and Stock 1997) asks for an F above 10. Stock and Yogo (2005) give a stricter threshold, 16.38, for an instrument whose IV tests are at most a little distorted.

print(f"first-stage F (robust, linearmodels):          {first_stage_F(iv_lm):.2f}")
r2 = iv_lm.first_stage.diagnostics.loc["avexpr", "rsquared"]
print(f"first-stage R-squared:                          {r2:.2f}")
iv_pf.IV_Diag()  # pyfixest computes its IV tests and stores them in the model
print(f"effective F (Olea and Pflueger 2013, pyfixest): {iv_pf._eff_F:.2f}")

first-stage F (robust, linearmodels):          16.85
first-stage R-squared:                          0.27
effective F (Olea and Pflueger 2013, pyfixest): 16.32

The F statistic is about 16 to 17, depending on the small-sample correction: above the rule of thumb of 10, and right at the Stock–Yogo threshold. The instrument is strong enough, but not by a wide margin. With controls that are themselves related to mortality, the F statistic falls, as Fig. 5 shows.

G.8.4 Is IV needed? The Wu–Hausman test

If institutions were not endogenous, OLS and IV would estimate the same thing, and OLS, being more precise, would be better. The Wu–Hausman test asks whether the gap between them is too large to be chance.

wh = iv_lm.wu_hausman()
print(f"Wu-Hausman test: statistic {wh.stat:.2f}, p-value {wh.pval:.6f}")

Wu-Hausman test: statistic 24.22, p-value 0.000007

The p-value is far below 0.01: the gap between OLS and IV is not chance. The data reject that OLS and IV estimate the same thing, so institutions are endogenous, as Section G.3 warned, and IV is needed. (The test assumes that the instrument itself is valid: it compares OLS with IV, not IV with the truth.)

Your turn. pyfixest and linearmodels gave standard errors of 0.179 and 0.176. Refit the linearmodels regression with cov_type="unadjusted", the textbook standard errors that assume the same spread of errors in every country. Is the standard error larger or smaller?

Hint

Use IV2SLS(base["logpgp95"], base[[]].assign(const=1.0), base[["avexpr"]], base[["logem4"]]).fit(cov_type="unadjusted") and read .std_errors["avexpr"]. It is smaller, about 0.154: in these data the spread of the errors differs across countries, and the robust standard error allows for it. The point estimate is the same 0.944. Libraries agree on estimates and differ on corrections, so always say which standard errors you report.

# Your turn: try it here

G.9 OLS against IV: what the gap means

Is the IV estimate fragile? AJR add controls that might explain both mortality and income: distance from the equator and the continent dummies. Fig. 5 puts OLS and IV side by side for each set of controls, the main columns of their Table 4.

Fig. 5: OLS and IV estimates with and without controls

Goal. See whether the IV estimate survives controls, and how its precision changes.

CONTROLS = {
    "None": [],
    "Latitude": ["lat_abst"],
    "Continents": ["africa", "asia", "other"],
    "Latitude and continents": ["lat_abst", "africa", "asia", "other"],
}


def estimates_table():
    """OLS and IV estimates with 95% intervals, for each set of controls."""
    rows = []
    for name, controls in CONTROLS.items():
        ols_m = ols_fit(" + ".join(["logpgp95 ~ avexpr"] + controls), base)
        iv_m = iv_fit(base, controls)
        rows.append(
            {
                "controls": name,
                "OLS": ols_m.coef()["avexpr"],
                "OLS low": ols_m.confint().loc["avexpr", "2.5%"],
                "OLS high": ols_m.confint().loc["avexpr", "97.5%"],
                "IV": iv_m.params["avexpr"],
                "IV low": iv_m.conf_int().loc["avexpr", "lower"],
                "IV high": iv_m.conf_int().loc["avexpr", "upper"],
                "first-stage F": first_stage_F(iv_m),
            }
        )
    return pd.DataFrame(rows).set_index("controls")


estimates = estimates_table()
estimates.round(2)

	OLS	OLS low	OLS high	IV	IV low	IV high	first-stage F
controls
None	0.52	0.42	0.62	0.94	0.60	1.29	16.85
Latitude	0.47	0.34	0.59	1.00	0.52	1.47	9.99
Continents	0.42	0.31	0.53	0.98	0.33	1.63	5.52
Latitude and continents	0.40	0.27	0.53	1.11	0.12	2.09	3.28

def estimates_trace(e, method, color, controls):
    """The estimates of one method (OLS or IV) with their 95% intervals."""
    opacity = []  # the chosen row in full color, the others faded
    for c in e["controls"]:
        opacity.append(1.0 if c == controls else 0.25)
    return go.Scatter(
        x=e[method],
        y=e["controls"],
        mode="markers",
        name=method,
        marker=dict(color=color, size=12, opacity=opacity),
        error_x=dict(
            type="data",
            symmetric=False,
            color=color,
            array=e[method + " high"] - e[method],
            arrayminus=e[method] - e[method + " low"],
        ),
    )

estimates_figure draws the two sets of estimates and writes the chosen row’s numbers in the title.

def estimates_figure(controls="None"):
    """Fig. 5: OLS and IV estimates with 95% intervals; the chosen row highlighted."""
    e = estimates.reset_index()
    fig = go.Figure()
    for method, color in [("OLS", RUST), ("IV", BLUE)]:
        fig.add_trace(estimates_trace(e, method, color, controls))
    row = estimates.loc[controls]
    fig.update_layout(
        title=f"Controls: {controls.lower()}. OLS {row['OLS']:.2f}, IV {row['IV']:.2f} "
        f"(first-stage F = {row['first-stage F']:.1f})",
        xaxis=dict(
            title="Effect of one more point of institutions on log income (95% CI)",
            range=[-0.2, 2.6],
            zeroline=True,
        ),
        yaxis=dict(title="", autorange="reversed"),
        legend=dict(orientation="h", y=-0.2),
    )
    return fig


estimates_figure()

interact(
    estimates_figure,
    controls=ToggleButtons(
        options=list(CONTROLS), value="None", description="controls", style=WIDE
    ),
);

Show the code of the website version

# @title Website version: one button per set of controls
button_figure(estimates_figure, "controls", list(CONTROLS), start="None")

Source. Data: Acemoglu, Johnson and Robinson (2001), replication files (Table 4). Prepared file: data/appG/ajr2001.csv. Chart: inspired by Table 4 in Acemoglu, Johnson and Robinson (2001) and Figure 3 of the tutorial Do Institutions Cause Prosperity?.

What to notice.

IV is above OLS with every set of controls: about 1 against 0.4 to 0.5. The controls barely move the IV estimate.
But the IV intervals widen as controls are added, and the first-stage F statistic falls from about 17 to about 3. Latitude and the continents are related to mortality, so they absorb part of what the instrument does. With all controls, the instrument is weak, and the wide interval should be read, not the point estimate.
The OLS intervals stay narrow: OLS is precise in every row. Precision is not the same as being right.

Your turn. Add a fifth set of controls: the share of Europeans in the population in 1900, euro1900. Does the IV estimate change? What happens to the first-stage F, and why might it?

Hint

r = iv_fit(base, ["euro1900"]), then r.params["avexpr"] and first_stage_F(r). The estimate barely moves (0.99), but the F statistic drops to about 5, because the share of Europeans in 1900 is one of the links of AJR’s chain: settlers settled where mortality was low. Controlling for a link of the chain takes away the variation the instrument works with.

# Your turn: try it here

G.9.1 What the coefficient means

An estimate of 0.94 means that one more point of protection against expropriation raises log income by 0.94, so income by \(e^{0.94} - 1 \approx 156\) percent. AJR’s own example compares Nigeria and Chile:

nga = base.loc[base["code"] == "NGA", "avexpr"].iloc[0]
chl = base.loc[base["code"] == "CHL", "avexpr"].iloc[0]
beta_iv = iv_lm.params["avexpr"]
gap = beta_iv * (chl - nga)
print(
    f"institutions: Nigeria {nga:.2f}, Chile {chl:.2f}, a gap of {chl - nga:.2f} points"
)
print(f"IV:  log income gap {gap:.2f}, a factor of {np.exp(gap):.1f}")
ols_gap = ols_models[1].coef()["avexpr"] * (chl - nga)
print(f"OLS: log income gap {ols_gap:.2f}, a factor of {np.exp(ols_gap):.1f}")

institutions: Nigeria 5.55, Chile 7.82, a gap of 2.27 points
IV:  log income gap 2.15, a factor of 8.6
OLS: log income gap 1.19, a factor of 3.3

If Nigeria had Chile’s institutions, the IV estimate predicts that it would be about 8 to 9 times as rich, close to the actual gap between the two countries. OLS would predict a factor of about 3.

Why is IV larger than OLS? Section G.3 gave two biases that push OLS up (omitted variables and reverse causality) and one that pulls it down (measurement error). IV is larger, so the downward pull must be the stronger. AJR argue that the index of expropriation risk is a noisy measure of the institutions that matter, which are broader and older than an investor rating of 1985–1995. The simulation of Fig. 1 showed exactly this pattern in the scenario “Measurement error only”.

A second reason is the local nature of the IV estimate (Section G.3.3): it is the effect for the former colonies whose institutions were shaped by settler mortality. If institutions matter more for them than for the average country, IV is larger than an OLS slope that averages over all.

G.10 Discussion and limits

AJR’s paper changed how economists think about growth. It also started a long debate. Four questions are at its center.

Is exclusion credible? The key assumption is that settler mortality affects income today only through institutions. The most serious doubt is disease: the malaria that killed settlers still makes workers sick today, and it might lower income directly. AJR control for malaria and life expectancy today and find a smaller, but still large, effect of institutions (about 0.6 to 0.7, in their Table 7). But today’s health is also partly an outcome of institutions, so controlling for it may hide part of their effect. Others argue that settlers brought human capital, their skills and schooling, not only institutions (Glaeser and coauthors 2004). Then mortality would affect income through schooling too, and the IV estimate would mix the two effects.

Are the mortality data reliable? Albouy (2012) went back to AJR’s sources. He found that 36 of the 64 countries in the base sample are assigned mortality rates from other countries, often neighbors, and that rates from very different groups, soldiers in barracks, soldiers on campaign, laborers and bishops, are combined. When he corrects for these problems, the first stage weakens, and the IV estimates become fragile. AJR (2012) reply that limiting the influence of the outliers, such as capping mortality at 250, leaves their results intact. The debate is a lesson for every data project: an instrument is only as good as the data it is built from.

What does the coefficient mean for policy? The IV estimate is a local effect of institutions that were set long ago by history. It does not tell a government how to raise its expropriation score, or how fast income would respond. Reforms change institutions in ways that history did not, and they can fail.

What does it mean for the growth models? In the model of Section G.3, institutions work through productivity and investment. AJR’s estimate is large enough to account for a big share of the income differences across former colonies, and so for much of the TFP gap that Chapter 4 and Appendix D measure. Institutions are a candidate answer to the question the production model leaves open: why some countries use their capital and workers so much better than others. Hall and Jones (1999), with the author of our textbook, reached a similar conclusion with a different instrument: what they called social infrastructure explains much of the differences in output per worker.

G.11 Go further

The tutorial behind this appendix, Do Institutions Cause Prosperity? An IV Tutorial in Python, reproduces the rest of AJR’s paper with the same tools. It comes with slides, a web app and a Stata version. There you can:

Add colonial, legal and religious controls (AJR’s Table 5): the identity of the colonizer, French legal origin and religion. The IV estimate stays between about 0.9 and 1.3.
Add geography (Table 6): temperature, humidity, soils, natural resources, being landlocked and ethnic fractionalization. The estimate stays between about 0.7 and 1.4, but the first stage often becomes weak.
Add health (Table 7): malaria, life expectancy and infant mortality today. This is the one place where the estimate falls, to about 0.55 to 0.7, and where the first stage is weakest.
Test overidentification (Table 8): with more instruments than endogenous variables (for example settler mortality together with the share of Europeans in 1900), the Hansen J (or Sargan) test checks whether the instruments agree. They do, but agreeing witnesses can share the same mistake.
Stress-test exclusion yourself. Pick a variable you think could break exclusion, add it to iv_fit as a control, and ask whether it is a cause of mortality (a real threat) or a result of institutions (a “bad control”).
Instrument two variables at once. In Table 7, AJR instrument both institutions and malaria with four instruments. pyfixest does not allow two endogenous variables; linearmodels’ IV2SLS does, and reports the Hansen J test (.sargan). Why can the test be computed here, but not with one instrument?

The original data and programs are in Daron Acemoglu’s data archive. The documentation of pyfixest and linearmodels has many more IV examples.

G.11.1 Common errors and how to fix them

Error or symptom	What it usually means	What to do
`ModuleNotFoundError: No module named 'pyfixest'`	the install cell was not run in this session	run the first cell of Section G.2 again, then the imports
`KeyError: 'avexpr'` after `.coef()`	the formula names another variable, or misspells one	print `model.coef()` to see the names
a regression has fewer countries than expected	rows with a missing value in any variable are dropped	count them with `base[["logpgp95", "lat_abst"]].isna().sum()`
the second stage by hand gives a different standard error	it ignores that the fitted values are estimated	use `pf.feols(... \\| avexpr ~ logem4 ...)` or `IV2SLS` for the standard error
`ValueError` from `IV2SLS` about the constant	the exogenous table has no constant, or two	build it with `d[controls].assign(const=1.0)`, as `iv_fit` does
a huge or negative IV estimate	the first stage is weak in that sample	look at `first_stage_F` before the estimate

G.12 Concluding remarks

In the production and Solow models, institutions raise income through productivity and investment, which gives a straight line between log income and institutions, equation (1). Its slope, equation (2), is the effect we want.
OLS finds the slope only if institutions are unrelated to everything else that moves income. Omitted variables and reverse causality push it up, measurement error pulls it down (equations 3 and 4; Fig. 1).
An instrument that moves institutions and has no other path to income recovers the slope as the ratio of the reduced form to the first stage, equation (7). Two-stage least squares computes the same number. A weak instrument makes the answer unreliable (equation 8).
With settler mortality as the instrument, AJR’s 64 former colonies give an IV estimate of 0.94 (Figs. 3 and 4), 80 percent above OLS, with a first-stage F of about 17 and a Wu–Hausman test that rejects that OLS is enough. The estimate survives the controls of Fig. 5, but the instrument weakens with them.
The result rests on the exclusion restriction and on old, partly borrowed mortality data. It is a local effect of history-made institutions, and the debate about disease, human capital and data quality is not settled. But it is strong evidence that institutions are a fundamental cause of the TFP differences that Chapter 4 and Appendix D measure.

G.12.1 References

Acemoglu, D., Johnson, S., and Robinson, J. A. (2001). The Colonial Origins of Comparative Development: An Empirical Investigation. American Economic Review 91(5), 1369–1401. doi:10.1257/aer.91.5.1369. Data: replication files.
Acemoglu, D., Johnson, S., and Robinson, J. A. (2012). The Colonial Origins of Comparative Development: An Empirical Investigation: Reply. American Economic Review 102(6), 3077–3110. doi:10.1257/aer.102.6.3077.
Albouy, D. Y. (2012). The Colonial Origins of Comparative Development: An Empirical Investigation: Comment. American Economic Review 102(6), 3059–3076. doi:10.1257/aer.102.6.3059.
Glaeser, E. L., La Porta, R., Lopez-de-Silanes, F., and Shleifer, A. (2004). Do Institutions Cause Growth? Journal of Economic Growth 9(3), 271–303. doi:10.1023/B:JOEG.0000038933.16398.ed.
Hall, R. E., and Jones, C. I. (1999). Why Do Some Countries Produce So Much More Output per Worker than Others? Quarterly Journal of Economics 114(1), 83–116. doi:10.1162/003355399555954.
Imbens, G. W., and Angrist, J. D. (1994). Identification and Estimation of Local Average Treatment Effects. Econometrica 62(2), 467–475.
Jones, C. I. (2024). Macroeconomics, 6th ed. W. W. Norton.
Mendez, C. (2026). Do Institutions Cause Prosperity? An IV Tutorial in Python.
North, D. C. (1990). Institutions, Institutional Change and Economic Performance. Cambridge University Press.
Olea, J. L. M., and Pflueger, C. (2013). A Robust Test for Weak Instruments. Journal of Business and Economic Statistics 31(3), 358–369.
Staiger, D., and Stock, J. H. (1997). Instrumental Variables Regression with Weak Instruments. Econometrica 65(3), 557–586.
Stock, J. H., and Yogo, M. (2005). Testing for Weak Instruments in Linear IV Regression. In D. W. K. Andrews and J. H. Stock (eds.), Identification and Inference for Econometric Models, Cambridge University Press, 80–108.