Double ML: Causal Inference based on ML

.title[
# Double ML: Causal Inference based on ML
]
.subtitle[
## Part I: Introduction to Causal Machine Learning
]
.author[
### uai2022, August 1, 2022, Eindhoven
]
.date[
### Philipp Bach, Martin Spindler (UHH & Economic AI)
Collaborators: Victor Chernozhukov (MIT), Malte Kurz (TUM)
]

---

## Motivation for Causal Machine Learning

---

## Motivation for Causal ML

---

## Predictive vs. Causal ML

### Predictive ML

#### How can we build a good prediction rule, `$f(X)$`, that uses features `$X$` to predict `$Y$`?

Example: **Customer Churn**

> "*How well can we predict whether customers churn?*"

]

### Causal ML

#### What is the causal effect of a treatment `$D$` on an outcome `$Y$`?

> "*Why do customer churn?*"

> "*How can we retain customers?*"

]

---

## Motivation for Causal ML

#### Typical (research) questions in industry, business and economics:

* What is the **effect** of the **new website (feature)** on our **sales**?

]

* How much **additional revenue** did our **latest newsletter** **generate**?

]

#### Typical research question in academic studies:

* What is the causal effect of **active labor market policies** on **labor market participation**?

]

* How does a **new drug** change patients' **mortality** due to a particular disease?

]

### General Question

#### What is the causal effect of a treatment `$D$` on an outcome `$Y$`?

---

## Application: Randomized Experiments

#### Challenges in practice

1. No (pure) A/B-testing / experiments possible -> observational data 
    
2. A/B test suffers from low power

3. Heterogenous treatment effects

]

* **General:** What is the **effect** of a **certain variable** `$D$` on a relevant **outcome variable** `$Y$`?

* **Randomized experiments** are a direct way to estimate such effects (assuming they are conducted properly)

#### Solution with DoubleML

1. **Observational study**: Include control variables `$X$` which may also impact the variables `$Y$` or `$D$`

2. Include covariates `$X$` that help to predict the outcome `$Y$` using ML methods.

3. Detection of complex treatment effect patterns

]

---

## Example: Price Elasticity of Demand

#### **Price Elasticity of Demand: How does the price impact sales?**

]

- Absolute change in price (EUR 100) and the resulting absolute change in sales (10 million units) can be difficult to interpret

- **Price elasticity of demand**: Percentage change in quantity demanded `$Q$` when there is a one percent increase in price `$P$`

]

`$$E_d = \frac{\Delta Q / Q}{\Delta P / P} = \frac{-10 / 200}{100 / 1000} = \frac{-0.05}{0.1} = - 0.5$$`

**Econometric model** for estimating the **price elasticity** `$\theta_0$`:

`$$\log(Q) = \alpha + \theta_0 \log(P) + X' \beta + \varepsilon,$$`

where the vector of controls `$X$` can be very **high-dimensional**

---

## Motivation for Causal Machine Learning

- Machine Learning methods usually tailored for **prediction**

- In science and industry both prediction (stock market, demand, ...) and learning of **causal relationship** is of interest

- Here: **Focus on causal inference** with machine Learning methods

- Examples for causal inference: 
  
  - Effect of a new website, app design or the latest newsletter
  
  - Price elasticity of demand

- General: What is the **effect** of a certain **treatment** on a relevant **outcome** variable?

---

## Motivation for Causal Machine Learning

#### Challenge I: Identification of causal parameter

**Typical problem** - Potential endogeneity of the treatment assignment

* Potential sources

- Optimizing behavior of the individuals with regard to the outcome
  - Simultaneity (price elasticity of demand)
  - Unobserved heterogeneity
  - Omitted variables
  - The treatment assignment is observed rather than randomly assigned
  - Example: Covid vaccination

* Possible Solutions

- **Selection on observable characteristics (controls)**
  - Instrumental Variable (IV) estimation
  - ...

---

## Motivation for Causal Machine Learning

#### Challenge II: "Big data"

* **High-dimensional setting** with `$p$` (# of variables) even larger than `$n$` (# of observations), and / or

* a highly **non-linear functional form** for the relationship between variables of interest

#### Solution

* Use ML methods for estimation that ...

+ ... perform regularization, e.g., variable selection

+ ... are able to model non-linearities

---

## Motivation for Causal Machine Learning

#### Challenge I & II: Controls are needed for two reasons

####  **Identification:**

We need them to ensure `$D$` is as good as randomly assigned (exogenous) *conditional on* `$X$`. Along these lines, we can think of the confounding equation:

`$$D = m_0(X) + V, E[V|X] = 0.$$`

`$\rightarrow$` Variation in `$V$` is quasi-experimental

]

#### **Efficiency:**

Some controls may explain part of the variation in the outcome `$Y$`

ML methods may deliver more accurate results than OLS or `$t$`-tests

$$$$

`$\rightarrow$` The effect can be estimated more accurately

]

---

## What is Double/Debiased Machine Learning (DML)?

---

## What is Double/Debiased Machine Learning (DML)?

- **Double/debiased machine learning (DML)** surveyed  by Chernozhukov et al. (2018)

- General framework for causal inference and estimation of treatment effects based on machine learning tools using big data

- Combines the strength of **machine learning** and **econometrics**

- Our object-oriented implementation **DoubleML** (in R and Python) provides a general interface for the growing number of models and methods for DML

- **Documentation** & **user guide**: https://docs.doubleml.org

- Install the latest release via pip or conda, see [installation guide](https://docs.doubleml.org/stable/intro/install.html)

```python
pip install -U DoubleML
```

```python
conda install -c conda-forge doubleml
```

- Install development version from GitHub **https://github.com/DoubleML/doubleml-for-py**

---

## What is Double/Debiased Machine Learning (DML)?

- **Exploiting the strengths of two disciplines:**

]

- **Result / output** from the DML framework:

- Estimate of the causal effect (with valid confidence intervals `$\rightarrow$` statistical tests for effects)
  
  - Good statistical properties ( `$\sqrt{N}$` rate of convergence; unbiased; approximately Gaussian)
  
  - Multiple treatment effects, heterogeneous treatment effects, ...

---

## A Motivating Example: Basics of Double Machine Learning

---

## Partially Linear Regression

#### Partially linear regression (PLR) model

`$$\begin{align*}
&Y = D \theta_0 + g_0(X) + \zeta, & &\mathbb{E}[\zeta | D,X] = 0, \\
&D = m_0(X) + V, & &\mathbb{E}[V | X] = 0,
\end{align*}$$`
with 
* Outcome variable `$Y$`
* Policy or treatment variable of interest `$D$`
* High-dimensional vector of confounding covariates `$X = (X_1, \ldots, X_p)$`
* Stochastic errors `$\zeta$` and `$V$`

]

---

## Partially Linear Regression

#### Partially linear regression (PLR) model

#### Problem of simple "plug-in" approaches: Regularization bias

* If we use an ML model to estimate `$\hat{g}$` and simply plug in the predictions `$\hat{g}$`, the final estimate on `$\theta_0$` will not be unbiased and neither be asymptotically normal

---

## Partially Linear Regression

#### Illustration of naive approach: App

Example based on Chernozhukov et al. (2018) and

[https://docs.doubleml.org/stable/guide/basics.html](https://docs.doubleml.org/stable/guide/basics.html)

App available via GitHub: https://github.com/DoubleML/BasicsDML

]

---

## Frisch-Waugh-Lovell Theorem

### Solution to regularization bias: Orthogonalization

* Remember the Frisch-Waugh-Lovell (FWL) Theorem in a **linear regression model**

`$$Y = D \theta_0 + X'\beta + \varepsilon$$`
* `$\theta_0$` can be consistently estimated by **partialling out** `$X$`, i.e,

1. OLS regression of `$Y$` on `$X$`: `$\tilde{\beta} = (X'X)^{-1} X'Y$` `$\rightarrow$` Residuals `$\hat{\varepsilon}$`
    
    2. OLS regression of `$D$` on `$X$`: `$\tilde{\gamma} = (X'X)^{-1} X'D$` `$\rightarrow$` Residuals `$\hat{\zeta}$`
    
    3.  Final OLS regression of `$\hat{\varepsilon}$` on `$\hat{\zeta}$`

* Orthogonalization: The idea of the FWL Theorem can be generalized to using ML estimators instead of OLS

---

## Partially Linear Regression

#### Illustration of naive approach: App

Example based on Chernozhukov et al. (2018) and

[https://docs.doubleml.org/stable/guide/basics.html](https://docs.doubleml.org/stable/guide/basics.html)

App available via GitHub: https://github.com/DoubleML/BasicsDML

]

---

## The Key Ingredients of Double Machine Learning

---

## The Key Ingredients of DML

> #### 1. Neyman Orthogonality

> The inference is based on a score function `$\psi(W; \theta, \eta)$` that satisfies

> `$$E[\psi(W; \theta_0, \eta_0)] = 0,$$`
> where `$W:=(Y,D,X,Z)$` and with `$\theta_0$` being the unique solution that obeys the **Neyman orthogonality condition**

> `$$\left.\partial_\eta \mathbb{E}[\psi(W; \theta_0, \eta] \right|_{\eta=\eta_0} = 0.$$`

- `$\partial_{\eta}$` denotes the pathwise (Gateaux) derivative operator

- **Neyman orthogonality** ensures that the **moment condition** identifying `$\theta_0$` is **insensitive to small pertubations** of the nuisance function `$\eta$` around `$\eta_0$`

- Using a Neyman-orthogonal score **eliminates the first order biases** arising from the replacement of `$\eta_0$` with a ML estimator `$\hat{\eta}_0$`

---

## The Key Ingredients of DML

> #### 1. Neyman Orthogonality

> The inference is based on a score function `$\psi(W; \theta, \eta)$` that satisfies

> `$$E[\psi(W; \theta_0, \eta_0)] = 0,$$`
> where `$W:=(Y,D,X,Z)$` and with `$\theta_0$` being the unique solution that obeys the **Neyman orthogonality condition**

> `$$\left.\partial_\eta \mathbb{E}[\psi(W; \theta_0, \eta] \right|_{\eta=\eta_0} = 0.$$`

- For many models the Neyman orthogonal **score functions** are **linear** in `$\theta$`
`$$\psi(W;\theta, \eta) = \psi_a(W; \eta) \theta + \psi_b(W; \eta)$$`

- The estimator `$\tilde{\theta}_{0}$` then takes the form 
`$$\tilde{\theta}_0 = - \left(\mathbb{E}_N[\psi_a(W; \eta)]\right)^{-1}\mathbb{E}_N[\psi_b(W; \eta)]$$`
---

## The Key Ingredients of DML

> #### 1. Neyman Orthogonality

> The inference is based on a score function `$\psi(W; \theta, \eta)$` that satisfies

> `$$E[\psi(W; \theta_0, \eta_0)] = 0,$$`
> where `$W:=(Y,D,X,Z)$` and with `$\theta_0$` being the unique solution that obeys the **Neyman orthogonality condition**

> `$$\left.\partial_\eta \mathbb{E}[\psi(W; \theta_0, \eta] \right|_{\eta=\eta_0} = 0.$$`

* PLR example: Orthogonality by including the first-stage regression, i.e., the regression relationship of the treatment variable `$D$` and the regressors `$X$`.

* Orthogonal score function `$\psi(\cdot)= (Y-E[Y|X]-\theta (D - E[D|X]))(D-E[D|X])$`.

---

## Neyman Orthogonality

The two strategies rely on very different moment conditions for identifying and estimating `$\theta_0$`:
`$$\mathbb{E} [\psi(W, \theta_0, \eta_0)] = 0$$`

#### Naive approach
`\begin{align}
\psi (W, \theta_0, \eta) = & (Y - D\theta_0 - g_0(X))D
\end{align}`

Regression adjustment score

`\begin{align}
\eta &= g(X), \\
\eta_0 &= g_0(X),
\end{align}`

]

#### FWL partialling out

`\begin{align}
\psi (W, \theta_0, \eta_0) = &  ((Y- E[Y|X])-(D-E[D|X])\theta_0)\\
& (D-E[D|X])
\end{align}`

Neyman-orthogonal score (Frisch-Waugh-Lovell)

`\begin{align}
\eta &= (g(X), m(X)), \\
\eta_0 &= ( g_0(X), m_0(X)) = ( \mathbb{E} [Y \mid X], \mathbb{E}[D \mid X])
\end{align}`

]

Both estimators solve  the empirical analog of the moment conditions:
$$
\frac{1}{n} \sum_{i=1}^n  \psi (W_i, \theta, \hat \eta_0)  = 0,
$$
where instead of unknown nuisance functions we plug-in their ML-based (hold-out) estimators

---

## The Key Ingredients of DML

> #### 2. High-Quality Machine Learning Estimators
> The nuisance parameters are estimated with high-quality (fast-enough converging) machine learning methods.

- Different structural assumptions on `$\eta_0$` lead to the use of different machine-learning tools for estimating `$\eta_0$` (Chernozhukov et al., 
2018, Chapter 3)

> #### 3. Sample Splitting
> To avoid the biases arising from overfitting, a form of **sample splitting** is used at the stage of producing the estimator of the main parameter `$\theta_0$`.

- Cross-fitting performs well empirically (efficiency gain by switching roles)

---

## Key Ingredients of DML

#### Illustration of the cross-fitting algorithm

]

---

## Partially Linear Regression

#### Illustration of DML approach: App

Example based on Chernozhukov et al. (2018) and

[https://docs.doubleml.org/stable/guide/basics.html](https://docs.doubleml.org/stable/guide/basics.html)

App available via GitHub: https://github.com/DoubleML/BasicsDML

]

---

## References

---

## References

#### Double Machine Learning Approach

* Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W. and Robins, J. (2018), Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21: C1-C68, doi:10.1111/ectj.12097.

* Chernozhukov, V., Hansen, C., Kallus, N., Spindler, M., and Syrgkanis, V. (forthcoming), Applied Causal Inference Powered by ML and AI.

#### DoubleML Package for Python and R

* Bach, P., Chernozhukov, V., Kurz, M. S., and Spindler, M. (2021), DoubleML - An Object-Oriented Implementation of Double Machine Learning in R, [arXiv:2103.09603](https://arxiv.org/abs/2103.09603).

* Bach, P., Chernozhukov, V., Kurz, M. S., and Spindler, M. (2022), DoubleML - An Object-Oriented Implementation of Double Machine Learning in Python, Journal of Machine Learning Research, 23(53): 1-6, https://www.jmlr.org/papers/v23/21-0862.html.

---

## Appendix

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## Examples: Covid Vaccination

#### Does the COVID-19 vaccine increase mortality?

.footnote[Source: [Tagesschau.de](https://www.tagesschau.de/faktenfinder/impfquote-sterblichkeitsrate-101.html)]

---

## Examples: Covid Vaccination

#### Does the COVID-19 vaccine increase mortality?

]