presentation.qmd

---
title: "The Problem With Historical Instrumental Variables"
subtitle: "Alternative title: Identification of Bi-Directional Two-Variable System With Time-Invariant Instrument"
author: "John T.H. Wong"
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## Illustrating the issue with a DAG

- Many historical IV papers use the following strategy.
  - e.g., AJR use settlers mortality ($z_i$) to instrument for constraints on the government's executive ($d_{it}$), and then estimate the latter's effect on output growth ($y_{it}$).

\begin{figure}
\centering
\begin{tikzpicture}[node distance=2cm]

  % Nodes
  \node (z)    {$z_i$};
  \node (dt)    [right=of z] {$d_{it}$};
  \node (yt)    [below=of dt] {$y_{it}$};

  % Arrows
  \draw[->, bend left=45] (z) to (dt);
  \draw[->] (dt) -- (yt);

\end{tikzpicture}
\end{figure}

\pause

- $z_i$ must affect $y_{it}$ only through $d_{it}$. \pause

- But note that $z_i$ is time-invariant, whereas $d_{it}$ is time-variant.

## Illustrating the issue with a DAG
\begin{figure}
\centering
\begin{tikzpicture}[node distance=2cm]

  % Nodes
  \node (z)    {$z_{i}$};
  \node (dtm1)  [right=of z] {$d_{i,t-1}$};
  \node (dt)    [right=of dtm1] {$d_{it}$};
  \node (yt)    [below=of dt] {$y_{it}$};

  % Arrows
  \draw[->] (z) -- (dtm1);
  \draw[->, bend left=45] (z) to (dt);
  \draw[->] (dtm1) -- (dt);
  \draw[->] (dtm1) -- (yt);
  \draw[->] (dt) -- (yt);

\end{tikzpicture}
\end{figure}

- If $z_i$ affects $d_{it}$, then it must also also affect $d_{i,t-1}$. 
  - This violates exclusion restriction. \pause
- We can add $d_{i, t-2}$, $d_{i, t-3}$, and so forth, to the graph.

## Theoretical model

### Second-stage equation

$$
y_{it} = \beta_0 d_{it} + \beta_1 d_{i, t-1} + \epsilon_{y,it}.
$$

\pause
- Note that this is a dynamic panel data model (DPDM).
- This is quite similar to an autoregressive distributed lag (ADL) setup. \pause

### First-stage equation

$$
d_{it} = \delta z_i + \alpha_0 d_{i,t-1} + \epsilon_{d, it}.
$$

\pause

### What happens when we omit $d_{t-1}$ in the first stage?
- Obtain the particular solution of the first-stage equation:

$$
d_{it} = (\delta  \sum_{j =0}^{\infty} a_1^j)z_{i}  + \underbrace{\sum_{j=0}^{\infty} a_1^j \epsilon_{i,t-j}}_{\text{Not iid!}}.
$$


## Let me prove it to you

- I simulated a panel with 50 units, each with 1000 observations (to show the misspecified model is inconsistent). \pause

```{r}
#| results: asis
#| tbl-cap: "Two-Stage Least Squares Results With Omitted Treatment Lag"

source("code.R")
table_omitted

```

## Monte Carlo Results
- These results are consistently biased across samples. (Each unit has 100 observations.)

```{r fig.height=4}
#| fig-cap: "Monte Carlo Results, Omitted Treatment Lag (500 iterations; 50 units; 100 observations per unit; ± 2 SD shaded; black dotted line indicates true mean)"

plot_biased
```

## Solution
- Including $Ld_{it}$ in both stages of the equation leads to a consistent estimator on all variables.

```{r}
#| results: asis
#| tbl-cap: "Two-stage least squares results with treatment lag"

table_lagd

```

## Monte Carlo results

```{r fig.height=4}
#| fig-cap: "Monte Carlo results, with treatment lag (500 iterations; 50 units; 100 observations per unit; ± 2 SD shaded)"

plot_labrecque
```

## Generalize to bi-directional Granger causation

- What if $y_{t-1}$ feeds into $y_{it}$ *and* $d_{it}$?

\begin{figure}
\centering
\begin{tikzpicture}[node distance=2cm]

  % Nodes
  \node (z)    {$z_{i}$};
  \node (dtm1)  [right=of z] {$d_{i,t-1}$};
  \node (dt)    [right=of dtm1] {$d_{it}$};
  \node (yt)    [below=of dt] {$y_{it}$};
  \node (ytm1)  [below=of dtm1] {$y_{i,t-1}$};

  % Arrows
  \draw[->] (z) -- (dtm1);
  \draw[->, bend left=45] (z) to (dt);
  \draw[->] (dtm1) -- (dt);
  \draw[->] (dtm1) -- (yt);
  \draw[->] (dt) -- (yt);
  \draw[->, dashed] (dtm1) -- (ytm1);
  \draw[->, dashed] (ytm1) -- (dt);
  \draw[->, dashed] (ytm1) -- (yt);
\end{tikzpicture}
\end{figure}


### For example, the Solow model implies this system
$$
\begin{aligned}
& (1) \ \Delta k = k_{t} - k_{t-1} = sy_{t-1} - \delta k_{t-1}
\\ & \implies k_{t} = sy_{t-1} +(1-\delta) k_{t-1}
\end{aligned}
$$

\pause

$$
\begin{aligned}
& (2) \ y_{t} = f(k_{t})
\end{aligned}
$$


## Estimation

We simulate then estimate the following equations:

$$
\begin{aligned}
y_{it}  = \beta d_{it} + \alpha_{11} y_{i,t-1} + \alpha_{12} d_{i,t-1}  + \epsilon_{y,it}
\\  d_{it} = \alpha_{11} y_{i,t-1} + \alpha_{12} d_{i,t-1} + \delta z_i + \epsilon_{it}.
\end{aligned}
$$

\pause

In VAR terms:

$$
\begin{aligned}
\begin{bmatrix} 1 & -\beta \\ \color{red}{0} & 1\end{bmatrix} \begin{bmatrix} y_{it} \\ d_{it} \end{bmatrix} = \begin{bmatrix} \alpha_{11} & \alpha_{12} \\ \alpha_{21} & \alpha_{22} \end{bmatrix} \begin{bmatrix} y_{i, t-1} \\ d_{i,t-1}\end{bmatrix} + \begin{bmatrix} \color{red}{0} \\ \delta \end{bmatrix} z_i + \begin{bmatrix}\epsilon_{y,it} \\ \epsilon_{d,it}\end{bmatrix}.
\end{aligned}
$$

\pause
- Our procedure is analogous to a Cholesky decomposition. \pause
- Note that lagged outcome enters the first-stage equation.

\pause

### What most historical IV papers are doing, in VAR terms
$$
\begin{aligned}
\begin{bmatrix} 1 & -\beta \\ \color{red}{0} & 1\end{bmatrix} \begin{bmatrix} y_{it} \\ d_{it} \end{bmatrix} = \begin{bmatrix} \color{red}{0} & \color{red}{0} \\ \color{red}{0} & \color{red}{0} \end{bmatrix} \begin{bmatrix} y_{i, t-1} \\ d_{i,t-1}\end{bmatrix} + \begin{bmatrix} \color{red}{0} \\ \delta \end{bmatrix} z_i + \begin{bmatrix}\epsilon_{y,it} \\ \epsilon_{d,it}\end{bmatrix}.
\end{aligned}
$$


## Results

```{r}
#| results: asis
#| tbl-cap: "Two-stage least squares results, comparison"

table_solow

```


## Monte Carlo results

```{r fig.height=4}
#| fig-cap: "Monte Carlo results, with treatment and outcome lag (500 iterations; 50 units; 100 observations per unit; ± 2 SD shaded)"

plot_unbiased
```

\pause

### Note that
1. We didn't need a second set of instruments.
2. Our instrument didn't need to be time-variant.
3. We didn't need to instrument for $d_{t-1}$.

# Discussion

## Is anyone talking about this?
- Most of the papers are in epidemiology (see Labrecque and Swanson 2018 in particular). 
  - This is because they use genetic variants to predict disease's effect (e.g., smoking) on health outcome (e.g., life expectancy), i.e., Mendelian randomization.
  - Also, their theoretical derivations didn't hold up in my Monte Carlos.\pause
- There is one development econ paper which talks about this (Casey and Klemp 2021), but their solution is questionable. 
  - They propose to estimate $d_{it}$ and $d_{i, t-Q}$, and instrument for the latter with $z_i$, and use the resulting parameter to adjust the second-stage equation parameter. \pause
  - The lag length is arbitrary. And even if it works, their method only recovers a "long-run" parameter, not the instantaneous parameter that is of policy interest. \pause
- There are actually a lot of time series tools that can help analyze and solve the problem. But the Anderson-Rubin causal inference people don't talk to the time series people or something?


## How important is this result?
1. This does not require a time-variant treatment (unlike proxy SVARs, aka SVARs with external instruments). \pause
    - In lieu, this setup reverts back to a Cholesky identification (but this is not an issue, and in fact taken as given in the causal inference setting). \pause
2. The development economics literature has been misspecifying a lot of IV papers. \pause
3. A contribution to the causal revolution-revolution, e.g., issues with TWFE DID estimator (Callaway & Sant'Anna 2020), geographical IVs (Mellon 2022).