Tutorial 11: Mariage market and assortative mating

Roadmap

We are going to:

1. Discover the difference-in-differences (DD) estimation
2. Solve matching matrices
3. Empirically estimate assortative mating using French data

Difference-in-differences

Main intuition

Graphical intuition

• Assume we observe \(n\) individuals over \(t\) period (balanced panel data). Some individuals are treated (\(T_i=1\)) and some are not after a given period.

Graphical intuition

Graphical intuition

Counterfactual change

Graphical intuition

Time difference

Graphical intuition

Between group difference

Graphical intuition

Difference in differences

Key assumptions

• Parallel trend assumption: to obtain the counterfactual change (not possible to test for specifically)
• No spillover effects: only treated are actually treated
• Randomness in treatment allocation (for all EPP)

Model

The model writes:

\[ y_i = \beta_0 + \beta_1 \texttt{Treated}_i + \beta_2 \texttt{After}_i + \beta_3 \texttt{Treated}_i \times \texttt{After}_i + \epsilon_i \]

Assortative matching: Abramitzky et al. (2011)

Empirical evidence with difference-in-differences

• Marrying Up: The Role of Sex Ratio in Assortative Matching, Abramitzky, Delavande, Vasconcelos (AEJ:Ap, 2011)
• Use the reduction in the number of male after World War I in France, in a difference-in-differences setting
• Sex ratio goes from 1,087 men per 1,000 women in 1911 to 992 men per 1,000 women in 1921, with large variation
• No change in the skill/class composition as soldiers were uniformly selected in the population

Assortative matching: Abramitzky et al. (2011)

Empirical model

The specification writes:

\[ Y_{idt} = \delta_d + \lambda PW_t + \alpha M_d \times PW_t + \gamma Z_{idt} + \varepsilon_{idt} \]

• \(Y\) is a measure of matching at the individual \(i\), département \(d\), time \(t\) level
• \(PW\) is a dummy for the marriage taking place after WWI
• \(M\) is a mortality measure
• \(Z\) includes local covariates

Assortative matching: Abramitzky et al. (2011)

Key results

Assortative matching in practice

Output matrix

We consider the heterosexual marriage market. People’s quality are summarized by a single index. Society is composed of 6 individuals with the following indexes: three men, \(M \in \{1, 3, 5\}\), and three women, \(W \in \{2, 4, 6\}\). Couples produce a joint output \(Z(M,W)\). We consider the following input-output matrices (with males in columns and females in rows, and the output in each inner cell).

Assortative matching in practice

Output matrix

1. Verify that the output functions are increasing in both arguments.
2. Recall the condition for which a perfect assortative matching occurs.
3. Assume that output is shared following a competitive biding process. For each case, guess who will end up marrying whom and explain why. Is the matching positive? negative? none of them?
4. Assume that output is shared according to some predetermined sharing rule (couples just want to maximize their output). For each case, guess who will end up marrying whom and explain why.

Assortative matching in practice

R implementation

We investigate if larger cities provide better marriage markets due to higher density. Load the data data_couples and investigate their structure.

We run the following model: \[ \texttt{PartnerEducation}_i = \beta_0 + \beta_1 \texttt{Education}_i \times \texttt{CitySize}_i + \beta_2 \texttt{Age}_i + \beta_3 \texttt{Active} + e_i \]

1. What is your prediction for the sign of each \(\beta\)?
2. Run the model using the survey weights as weights
3. Add year FE. Interpret.
4. Plot the coefficients by city size. Interpret.
5. [Bonus] Rerun the model but instead of partner’s education use the partner’s labour force status