class: center, middle, inverse, title-slide .title[ # Advanced quantitative data analysis ] .subtitle[ ## Fixed effect I ] .author[ ### Mengni Chen ] .institute[ ### Department of Sociology, University of Copenhagen ] --- <style type="text/css"> .remark-slide-content { font-size: 20px; padding: 20px 80px 20px 80px; } .remark-code, .remark-inline-code { background: #f0f0f0; } .remark-code { font-size: 14px; } </style> #Let's get ready ```r #install.packages("plm") library(tidyverse) # Add the tidyverse package to my current library. library(haven) # Handle labelled data. library(broom) library(splitstackshape) #transform wide data (with stacked variables) to long data library(plm) #linear models for panel data ``` --- #Does partnership make you happier? .pull-left[ `$$\Delta_i=Y_{i,t0}^{T} - Y_{i,t0}^{C}$$` where T refers to treatment: have a partner; C refers to control: being single **The fundamental problem of causal inference: we cannot observe both!** ] .pull-right[ <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic1.PNG?raw=true" width="40%" style="display: block; margin: 10px 100px;"> <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic2.PNG?raw=true" width="20%" style="display: block; margin: 10px 180px;"> ] --- #Does partnership make you happier? .pull-left[ **A cross-sectional analysis** `$$\hat{\Delta_i}=Y_{i,t0}^{T} - Y_{j,t0}^{C}$$` - A cross-sectional analysis is between estimation - Compare different persons T(who are partnered) and C(who are single) - Assumption: T and C differ only in their partnership status ] .pull-right[ <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic1.PNG?raw=true" width="40%" style="display: block; margin: 10px 100px;"> <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic2.PNG?raw=true" width="20%" style="display: block; margin: 10px 180px;"> ] --- #Does partnership make you happier? <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic15.PNG?raw=true" width="60%" style="display: block; margin: 10px ;"> `$$Sat_i= \beta_{1}*partner_i + e_i$$` $$\text{OLS Exogeneity assumption}:E(e_i|x_i)= 0 $$ $$\text{But it is often violated due to endogeneity}:E(e_i|x_i)≠0 $$ $$\text{E.g if we don't control the personality in the current case, the error term would be correlated with partnership status} $$ --- #Does partnership make you happier? .pull-left[ **A panel analysis** `$$\hat{\Delta_i}=Y_{i,t1}^{T} - Y_{i,t0}^{C}$$` where T refers to treatment: have a partner; C refers to control: being single - A panel analysis is within estimation - Compare the same person **P** at t0 and t1 - Assumption: **P** changed only in their partnership status ] .pull-right[ <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic4.PNG?raw=true" width="30%" style="display: block; margin: 10px 100px;"> <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic5.PNG?raw=true" width="20%" style="display: block; margin: 10px 180px;"> ] --- #Does partnership make you happier? <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic15.PNG?raw=true" width="70%" style="display: block; margin: 10px ;"> `$$\text{With cross-sectional data}:Sat_i= \beta_{1}*partner_i + e_i$$` `$$\text{With panel data}:Sat_{i,t}= \beta_{1}*partner_{i,t} + u_{i} + \epsilon_{i,t}$$` $$u_{i}:\text{person-specific unobserved component, time-constant} $$ `$$\epsilon_{i,t}:\text{person-and-time-specific component, time-varying}$$` `$$\text{Exogeneity assumption}:E(\epsilon_{i,t}|x_{i,t})= 0$$` --- #Does partnership make you happier? - What are person-specific time-constant variables? - Some can be observed: e.g. gender, which we often specify in the cross-sectional model - Some can be unobserved: e.g. cognitive ability, genetic disposition, personality - But define which is time-constant component should be put in the specific context of your study - Years of education could be time-constant when you are studying older adults, but time-varying when you are studying young adults - How to deal with time-constant unobserved variable? - Fixed effect estimator (within transformation) - Dummy variable regression estimator - First difference estimator --- #Does partnership make you happier? Fixed effect estimator `$$Sat_{i,t}= \beta_{1}*partner_{i,t} + u_{i} + \epsilon_{i,t}$$` Now, take an average of each variable: `$$\bar{Sat_{i,t}}= \beta_{1}*\bar{partner_{i,t}} + u_{i} + \bar{\epsilon_{i,t}}$$` Then, let take difference between the two: `$$Sat_{i,t}-\bar{Sat_{i,t}}= \beta_{1}*(partner_{i,t}-\bar{partner_{i,t}}) + (\epsilon_{i,t} -\bar{\epsilon_{i,t}})$$` `$$\text{Note: } u_{i} \text{ are cancelled as they are constant.}$$` This process is called within transformation, i.e. to de-mean the dependent and independent variables within each person, and then estimate the model. All time-constant factors (both observed and unobserved) drop out. That is, persons are controlled or fixed. -> The fixed effect model --- #Does partnership make you happier? Dummy regression etimator `$$Sat_{i,t}= \beta_{1}*partner_{i,t} + u_{i} + \epsilon_{i,t}$$` Now, include a set of dummy variables for each person (with one excluded category). That is, if you have `n` persons, you should create `(n-1)` dummies `$$Sat_{i,t}= \beta_{1}*partner_{i,t} + \gamma_{2}D_{2} +\gamma_{3}D_{3}+ ... +\gamma_{n}D_{n} + \epsilon_{i,t}$$` This means we are looking at the relationship between partnership and life satisfaction one individual at a time. --- #Does partnership make you happier? first difference estimator `$$Sat_{i,t}= \beta_{1}*partner_{i,t} + u_{i} + \epsilon_{i,t}$$` Now, subtract from each variable its value one period earlier, and then estimate the model `$$Sat_{i,t-1}= \beta_{1}*partner_{i,t-1} + u_{i} + \epsilon_{i,t-1}$$` `$$Sat_{i,t}-Sat_{i,t-1}= \beta_{1}*(partner_{i,t}-partner_{i,t-1}) + (\epsilon_{i,t}- \epsilon_{i,t-1})$$` In the first difference model, the difference between the dependent variable and the lagged dependent variable becomes the new dependent variable, and the the difference between the independent variable and the lagged independent variable becomes the new independent variable. --- #Dealing with the real data - [Prepare the data](https://rpubs.com/fancycmn/960815) - Define panel data ```r #define panel data panel_data <- pdata.frame(long_data, index=c("id", "wave")) ``` --- #If we use OLS (pooled OLS), the result will be ```r #you can use: ols <- lm(sat ~ ptner, data=panel_data) #but now let us use plm() pols <- plm(sat ~ ptner , data=panel_data, model="pooling") summary(pols) ``` ``` ## Pooling Model ## ## Call: ## plm(formula = sat ~ ptner, data = panel_data, model = "pooling") ## ## Unbalanced Panel: n = 2591, T = 1-6, N = 10370 ## ## Residuals: ## Min. 1st Qu. Median 3rd Qu. Max. ## -7.84224 -0.84224 0.15776 1.15776 2.58852 ## ## Coefficients: ## Estimate Std. Error t-value Pr(>|t|) ## (Intercept) 7.411483 0.019444 381.177 < 2.2e-16 *** ## ptnerYes 0.430762 0.038558 11.172 < 2.2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Total Sum of Squares: 30676 ## Residual Sum of Squares: 30311 ## R-Squared: 0.011895 ## Adj. R-Squared: 0.011799 ## F-statistic: 124.81 on 1 and 10368 DF, p-value: < 2.22e-16 ``` --- #Fixed effect estimator (Within-transformation): using `plm()` .pull-left[ ```r fixed <- plm(sat ~ ptner, data=panel_data, model="within") summary(fixed) ``` ``` ## Oneway (individual) effect Within Model ## ## Call: ## plm(formula = sat ~ ptner, data = panel_data, model = "within") ## ## Unbalanced Panel: n = 2591, T = 1-6, N = 10370 ## ## Residuals: ## Min. 1st Qu. Median 3rd Qu. Max. ## -7.83333 -0.50000 0.00000 0.56371 5.66667 ## ## Coefficients: ## Estimate Std. Error t-value Pr(>|t|) ## ptnerYes 0.308867 0.037296 8.2815 < 2.2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Total Sum of Squares: 12535 ## Residual Sum of Squares: 12426 ## R-Squared: 0.0087404 ## Adj. R-Squared: -0.32147 ## F-statistic: 68.5825 on 1 and 7778 DF, p-value: < 2.22e-16 ``` ] .pull-right[ - Interpretation - A within-person change from single to partnered is associated with 0.309 scale points increase in life satisfaction (always correct interpretation). - When having a partner, life satisfaction is 0.309 points higher than when not (always correct interpretation). - A careful interpretaion*: finding a partner increases life satisfaction by 0.309 scale (correct conditioning on when exogeneity assumption holds). ] --- #Dummy regression model: using `plm()` ```r # or use code: dumreg1 <- lm(sat ~ ptner + factor(id), data=panel_data) dumreg2 <- plm(sat ~ ptner + factor(id), data=panel_data, model="pooling") summary(dumreg2) ``` ``` ## Pooling Model ## ## Call: ## plm(formula = sat ~ ptner + factor(id), data = panel_data, model = "pooling") ## ## Unbalanced Panel: n = 2591, T = 1-6, N = 10370 ## ## Residuals: ## Min. 1st Qu. Median 3rd Qu. Max. ## -7.8333e+00 -5.0000e-01 6.9385e-14 5.6371e-01 5.6667e+00 ## ## Coefficients: ## Estimate Std. 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factor(id)746541000 3.5000e+00 1.5480e+00 2.2610 0.0237894 * ## factor(id)746596000 3.5000e+00 1.0946e+00 3.1975 0.0013919 ** ## factor(id)746610000 -1.3848e+00 1.0320e+00 -1.3418 0.1796895 ## factor(id)746813000 1.5000e+00 1.5480e+00 0.9690 0.3325844 ## factor(id)747395000 -4.8777e-01 1.0322e+00 -0.4726 0.6365405 ## factor(id)747479000 2.9608e+00 1.1541e+00 2.5654 0.0103230 * ## factor(id)747635000 5.0000e-01 1.0320e+00 0.4845 0.6280511 ## factor(id)747865000 -8.3333e-01 1.0320e+00 -0.8075 0.4194119 ## factor(id)748446000 7.9409e-01 1.0323e+00 0.7692 0.4417775 ## factor(id)749700000 5.0000e-01 1.5480e+00 0.3230 0.7467085 ## factor(id)749731000 2.2382e+00 1.0575e+00 2.1165 0.0343356 * ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Total Sum of Squares: 30676 ## Residual Sum of Squares: 12426 ## R-Squared: 0.59493 ## Adj. R-Squared: 0.45999 ## F-statistic: 4.40896 on 2591 and 7778 DF, p-value: < 2.22e-16 ``` the result from `lm()` and the result from `plm()` are the same. --- #First difference model: using `plm()` ```r firstdif <- plm(sat ~ ptner, data=panel_data, model="fd") summary(firstdif) ``` ``` ## Oneway (individual) effect First-Difference Model ## ## Call: ## plm(formula = sat ~ ptner, data = panel_data, model = "fd") ## ## Unbalanced Panel: n = 2591, T = 1-6, N = 10370 ## Observations used in estimation: 7779 ## ## Residuals: ## Min. 1st Qu. Median 3rd Qu. Max. ## -9.975889 -0.975889 0.024111 1.024111 10.024111 ## ## Coefficients: ## Estimate Std. Error t-value Pr(>|t|) ## (Intercept) -0.024111 0.019398 -1.243 0.2139 ## ptnerYes 0.389912 0.041313 9.438 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Total Sum of Squares: 21873 ## Residual Sum of Squares: 21625 ## R-Squared: 0.011324 ## Adj. R-Squared: 0.011197 ## F-statistic: 89.0756 on 1 and 7777 DF, p-value: < 2.22e-16 ``` --- #Compare the results ```r texreg::htmlreg(list(pols, fixed, dumreg2, firstdif), custom.model.names=c("OLS regression", "Fixed effect(within transformation)", "Dummy variable regression", "First difference regression"), include.ci = FALSE, omit.coef = "factor", center=TRUE, file = "compare.html") ``` ``` ## The table was written to the file 'compare.html'. ``` <img src="https://github.com/fancycmn/slide9/blob/main/S9_pic11.PNG?raw=true" width="55%" style="display: block; margin: 10px ;"> Note: When it is two waves, the first difference and within-transformation estimators are numerically equivalent. When there are more than 2 waves, they are not. If the error term are homoskedastic and have not serial correlation, within-transformation estimators are more efficient. --- #Another case of fixed effect analysis https://www.youtube.com/watch?v=1SchyQ77VFg&t=231s --- #Take home 1. Understand the fixed effect 2. Know three approaches to deal with fixed effect - Within transformation (fixed effect estimator) - Dummy variable regression - First difference 3. Important codes: - `pdata.frame()` to define a panel data - `plm()`, where `model="pooling", "within", "fd"` --- class: center, middle #[Exercise](https://rpubs.com/fancycmn/966430)