Skip to content

Improve clarity of TWFE explanation #522

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
merged 3 commits into from
Aug 29, 2025
Merged

Improve clarity of TWFE explanation #522

Changes from all commits
Commits
Show all changes
3 commits
Select commit Hold shift + click to select a range
ba72cda
Fixes #516
mercury0100 Aug 26, 2025
caf7ae1
docs: clarify TWFE explanation in Banking dataset notebook
Aug 26, 2025
638deb2
fix "\" appearing in the rendered docs preview in the post_treatment …
Aug 29, 2025
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
38 changes: 16 additions & 22 deletions docs/source/notebooks/did_pymc_banks.ipynb
View file
Open in desktop
Original file line number Diff line number Diff line change
@@ -1,7 +1,6 @@
{
"cells": [
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand Down Expand Up @@ -38,7 +37,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand Down Expand Up @@ -72,7 +70,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand Down Expand Up @@ -108,7 +105,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand Down Expand Up @@ -313,7 +309,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand Down Expand Up @@ -665,7 +660,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand Down Expand Up @@ -829,29 +823,30 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"## Analysis 4 - Two way fixed effects\n",
"## Analysis 4 - Two-way fixed effects\n",
"\n",
"Finally, we can evaluate the difference in difference model in its two-way fixed effects (TWFE) formulation. The two-way fixed effects model is widely used in econometrics for causal inference in panel data settings. It accounts for both unit-specific effects (e.g., differences between districts) and time-specific effects (e.g., shocks or trends affecting all units simultaneously). \n",
"Finally, we can evaluate the difference-in-differences model in its two-way fixed effects (TWFE) formulation. The TWFE model is widely used in econometrics for causal inference in panel data. It accounts for both unit-specific effects (e.g., differences between districts) and time-specific effects (e.g., shocks or trends affecting all units simultaneously).\n",
"\n",
"The TWFE model is equivalent to the classic 2$\\times$2 DiD model (Model 1) - but only in the situation of two groups and two time periods. Outside of this special case the approach is not equivalent and can potentially have some problems {cite:p}`hill2020limitations,imai2021twfepanel`. Readers should proceed with caution when using the TWFE model outside of the 2$\\times$2 case - see for example {cite:p}`kropko2018two,collischon2020let`.\n",
"The TWFE model is equivalent to the classic 2$\\times$2 DiD model (Model 1) only in the special case of two groups and two time periods. Outside of this case the approaches diverge, and TWFE can have important limitations {cite:p}`hill2020limitations,imai2021twfepanel`. Readers should proceed with caution when applying TWFE in richer settings {cite:p}`kropko2018two,collischon2020let`.\n",
"\n",
"The TWFE approach is similar to the previous model in that the `district:post_treatment` interaction term still gives you a treatment indicator variable and the assiated coefficient $\\beta_{\\Delta}$ is the causal effect of the intervention. But it is different in that there is no _linear_ `year` term, instead we have a _categorical_ `year` variable. This means that the model can capture any temporal trends in the data. These can be thought of as capturing time based schocks that affect all units.\n",
"The TWFE approach is similar to the previous model in that the `district:post_treatment` interaction term still defines a treatment indicator variable, and its coefficient $\\Delta$ represents the causal effect of the intervention. The difference is that there is no _linear_ `year` term, instead we use a _categorical_ year variable. This allows the model to flexibly capture any time-specific shocks that affect all units.\n",
"\n",
"The equation for the expected values is:\n",
"The expected values can be written as:\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"\\mu_i & \\quad + \\alpha[district_i] \\qquad \\textcolor{salmon}{\\text{(unit fixed effect)}}\\\\\n",
" & \\quad + \\beta[year_i] \\qquad \\textcolor{salmon}{\\text{(time fixed effect)}}\\\\\n",
" & \\quad + \\Delta \\cdot district_i \\cdot post~treatment_i \\qquad \\textcolor{salmon}{\\text{(treatment indicator)}}\n",
"\\mu_i &= \\gamma && \\color{#FA8072}{\\text{(global intercept)}} \\\\\n",
"&+ \\alpha[\\text{district}_i] && \\color{#FA8072}{\\text{(unit fixed effect)}} \\\\\n",
"&+ \\beta[\\text{year}_i] && \\color{#FA8072}{\\text{(time fixed effect)}} \\\\\n",
"&+ \\Delta \\cdot \\text{district}_i \\cdot \\text{post_treatment}_i && \\color{#FA8072}{\\text{(treatment indicator)}}\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"Typically the TWFE model is presented in matrix form, so the equation above might look less familiar. It has been adapted for long format data. In particular, note that:\n",
"* $\\alpha$ is a scalar intercept term.\n",
"* $\\alpha[district_i]$ is a vector of fixed effects for each district. There are only 2 districts, so this is a vector of length 2. The $district_i$ indexes the element of $\\alpha$ that corresponds to the district of the $i^{th}$ observation.\n",
"* $\\beta[year_i]$ is a vector of fixed effects for each year. There are 6 years, so this is a vector of length 6. The $year_i$ indexes the element of $\\beta$ that corresponds to the year of the $i^{th}$ observation.\n",
"* $\\Delta$ is a scalar representing the treatment effect, which is the same as the coefficient of the `district:post_treatment` interaction term."
"Here: \n",
"* $\\gamma$ is a scalar global intercept. \n",
"* $\\alpha[\\text{district}_i]$ is a vector of unit (district) fixed effects. With 2 districts, this is a vector of length 2. The index $\\text{district}_i$ selects the effect for the $i^{\\text{th}}$ observation. \n",
"* $\\beta[\\text{year}_i]$ is a vector of time (year) fixed effects. With 6 years, this is a vector of length 6. The index $\\text{year}_i$ selects the effect for the $i^{\\text{th}}$ observation. \n",
"* $\\Delta$ is a scalar treatment effect, corresponding to the coefficient on the `district:post_treatment` interaction term.\n"
]
},
{
Expand Down Expand Up @@ -1049,7 +1044,6 @@
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
Expand All @@ -1062,7 +1056,7 @@
],
"metadata": {
"kernelspec": {
"display_name": "CausalPy",
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
Expand All @@ -1076,7 +1070,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.13.2"
"version": "3.9.18"
}
},
"nbformat": 4,
Expand Down