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(texreg) library(splitstackshape) #transform wide data (with stacked variables) to long data library(plm) #linear models for panel data ``` --- #Dealing with the real data - [Prepare the data](https://rpubs.com/fancycmn/1243424) --- #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 ;"> `$$Sat_i= \beta_{1}*partner_i + e_i$$` $$\text{OLS Exogeneity assumption}:E(e_i|x_i)= 0 $$ $$\text{But it is often violated due to endogeneity}:E(e_i|x_i)≠0 $$ $$\text{E.g if we don't control the personality in this case, the error term would be correlated with partnership status} $$ --- #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. In this approach, - both dependent and independent variables are demeaneded within each person - the de-meaned dependent variables becomes the new DV - the de-meaned independent variables becomes the new IV. - All time-constant factors (both observed and unobserved) drop out. - This is often called, the fixed effect estimator --- #Does partnership make you happier? Fixed effect estimator ```r twowaves_simple1 <- twowaves_long %>% select(id, check, wave, relstat, partner, sat)%>% group_by(id) %>% mutate( partner1=as.numeric(partner), pt_mean=mean(partner1), #generate the mean for partner1 by id pt_demean=partner1-pt_mean, #the process of de-meaning for partner sat_mean=mean(sat), #generate the mean for sat by id sat_demean=sat-sat_mean #the process of de-meaning for sat ) ``` --- #Does partnership make you happier? Fixed effect estimator <img src="https://github.com/fancycmn/2024-Session10/blob/main/Figure1.JPG?raw=true" width="100%" style="display: block; margin: 10px;"> --- #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, - the the difference between the independent variable and the lagged independent variable becomes the new independent variable. --- #Does partnership make you happier? First difference estimator ```r twowaves_simple2 <- twowaves_long %>% select(id, check, wave, relstat, partner, sat)%>% group_by(id) %>% mutate( partner1=as.numeric(partner), partner1_lag=dplyr::lag(partner1,1), #one time lag of partner1 pt_dif=partner1-partner1_lag, #the new IV sat_lag=dplyr::lag(sat,1), #one time lag for sat sat_dif=sat-sat_lag #the new DV ) ``` --- #Does partnership make you happier? First difference estimator <img src="https://github.com/fancycmn/2024-Session10/blob/main/Figure2.JPG?raw=true" width="100%" style="display: block; margin: 10px;"> --- #Define panel data: `pdata.frame()` a function of "plm package" ```r #define panel data twowaves_long <- pdata.frame(twowaves_long, index=c("id", "wave")) sixwaves_long <- pdata.frame(sixwaves_long, index=c("id", "wave")) ``` --- #If we use OLS (pooled OLS)? - Pooled OLS - Combining Data: Pooled OLS combines the data from all time periods into a single dataset, effectively ignoring the time dimension. - Estimation: A standard OLS regression is then performed on the pooled dataset, treating all observations as if they come from a single cross-section. - Assumption of Independence: Pooled OLS assumes that observations across time are independent of each other. (often not true) However, in longitudinal data, observations within the same entity (individual, company, etc.) may be correlated over time. Ignoring this correlation can lead to inefficient and biased estimates. --- #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 ~ partner , data=twowaves_long, model="pooling") summary(pols) ``` ``` ## Pooling Model ## ## Call: ## plm(formula = sat ~ partner, data = twowaves_long, model = "pooling") ## ## Unbalanced Panel: n = 2591, T = 1-2, N = 4562 ## ## Residuals: ## Min. 1st Qu. Median 3rd Qu. Max. ## -7.50134 -0.50134 0.49866 1.49866 2.49866 ## ## Coefficients: ## Estimate Std. Error t-value Pr(>|t|) ## (Intercept) 7.501339 0.027667 271.129 < 2e-16 *** ## partnerYes 0.458926 0.087799 5.227 1.8e-07 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Total Sum of Squares: 14428 ## Residual Sum of Squares: 14343 ## R-Squared: 0.0059559 ## Adj. R-Squared: 0.0057379 ## F-statistic: 27.3215 on 1 and 4560 DF, p-value: 1.7998e-07 ``` --- #Fixed effect estimator (Within-transformation): using `plm()` .pull-left[ ```r fixed <- plm(sat ~ partner, data=twowaves_long, model="within") summary(fixed) ``` ``` ## Oneway (individual) effect Within Model ## ## Call: ## plm(formula = sat ~ partner, data = twowaves_long, model = "within") ## ## Unbalanced Panel: n = 2591, T = 1-2, N = 4562 ## ## Residuals: ## Min. 1st Qu. Median 3rd Qu. Max. ## -5.0 -0.5 0.0 0.5 5.0 ## ## Coefficients: ## Estimate Std. Error t-value Pr(>|t|) ## partnerYes 0.569536 0.084075 6.7742 1.647e-11 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Total Sum of Squares: 3227.5 ## Residual Sum of Squares: 3154 ## R-Squared: 0.022764 ## Adj. R-Squared: -1.2625 ## F-statistic: 45.8893 on 1 and 1970 DF, p-value: 1.6466e-11 ``` ] .pull-right[ - Interpretation - A within-person change from single to partnered is associated with 0.57 scale points increase in life satisfaction (always correct interpretation). - When having a partner, life satisfaction is 0.57 points higher than when not (always correct interpretation). - A careful interpretation*: finding a partner increases life satisfaction by 0.57 scale (correct conditioning on when exogeneity assumption holds). ] --- #Dummy regression model: using `plm()` ```r dumreg <- plm(sat ~ partner + factor(id), data=twowaves_long, model="pooling") summary(dumreg) ``` ``` ## Pooling Model ## ## Call: ## plm(formula = sat ~ partner + factor(id), data = twowaves_long, ## model = "pooling") ## ## Unbalanced Panel: n = 2591, T = 1-2, N = 4562 ## ## Residuals: ## Min. 1st Qu. Median 3rd Qu. Max. ## -5.0000e+00 -5.0000e-01 -9.7826e-16 5.0000e-01 5.0000e+00 ## ## Coefficients: ## Estimate Std. Error t-value Pr(>|t|) ## (Intercept) 6.0000e+00 1.2653e+00 4.7419 2.269e-06 *** ## partnerYes 5.6954e-01 8.4075e-02 6.7742 1.647e-11 *** ## factor(id)423000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## factor(id)828000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)945000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)1133000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)1245000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)2078000 1.0000e+00 1.5497e+00 0.6453 0.5188148 ## factor(id)2605000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)2674000 1.0000e+00 1.5497e+00 0.6453 0.5188148 ## factor(id)3491000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)3682000 -2.0000e+00 1.7894e+00 -1.1177 0.2638425 ## factor(id)4372000 -1.0000e+00 1.7894e+00 -0.5588 0.5763364 ## factor(id)4504000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)4732000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)4849000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## factor(id)4889000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)4897000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)4962000 3.0000e+00 1.7894e+00 1.6765 0.0937968 . ## factor(id)5678000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)6243000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)6340000 2.7152e+00 1.5503e+00 1.7515 0.0800212 . ## factor(id)6480000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)6553000 1.0000e+00 1.5497e+00 0.6453 0.5188148 ## factor(id)6994000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)7343000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)7455000 -2.0000e+00 1.7894e+00 -1.1177 0.2638425 ## factor(id)7543000 -1.0000e+00 1.7894e+00 -0.5588 0.5763364 ## factor(id)7885000 2.7152e+00 1.5503e+00 1.7515 0.0800212 . ## factor(id)8003000 2.2152e+00 1.5503e+00 1.4289 0.1531800 ## factor(id)8041000 4.0000e+00 1.7894e+00 2.2353 0.0255061 * ## factor(id)8207000 2.7152e+00 1.5503e+00 1.7515 0.0800212 . ## factor(id)8277000 1.7152e+00 1.5503e+00 1.1064 0.2686824 ## factor(id)8878000 1.0000e+00 1.5497e+00 0.6453 0.5188148 ## factor(id)8881000 3.0000e+00 1.7894e+00 1.6765 0.0937968 . ## factor(id)9139000 1.7152e+00 1.5503e+00 1.1064 0.2686824 ## factor(id)9474000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)9766000 -5.0000e-01 1.5497e+00 -0.3226 0.7469985 ## factor(id)10086000 4.0000e+00 1.7894e+00 2.2353 0.0255061 * ## factor(id)10175000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## factor(id)10789000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## factor(id)11067000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)11854000 4.0000e+00 1.5497e+00 2.5812 0.0099187 ** ## factor(id)12027000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)12482000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)12715000 4.0000e+00 1.7894e+00 2.2353 0.0255061 * ## factor(id)12814000 -5.0000e-01 1.5497e+00 -0.3226 0.7469985 ## factor(id)13754000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)13841000 -2.5000e+00 1.5497e+00 -1.6132 0.1068561 ## factor(id)14047000 1.0000e+00 1.7894e+00 0.5588 0.5763364 ## factor(id)14256000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)14362000 -3.5000e+00 1.5497e+00 -2.2585 0.0240226 * ## factor(id)14587000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)14620000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)15196000 1.7152e+00 1.5503e+00 1.1064 0.2686824 ## factor(id)15291000 1.0000e+00 1.7894e+00 0.5588 0.5763364 ## factor(id)15401000 3.5000e+00 1.5497e+00 2.2585 0.0240226 * ## factor(id)15430000 4.0000e+00 1.5497e+00 2.5812 0.0099187 ** ## factor(id)15479000 3.2152e+00 1.5503e+00 2.0740 0.0382097 * ## factor(id)15627000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)15975000 2.2152e+00 1.5503e+00 1.4289 0.1531800 ## factor(id)16402000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)16671000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## factor(id)16774000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)17048000 1.0000e+00 1.5497e+00 0.6453 0.5188148 ## factor(id)18129000 -1.2848e+00 1.5503e+00 -0.8287 0.4073505 ## factor(id)18628000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)18814000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)19485000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)20137000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)20180000 -1.2848e+00 1.5503e+00 -0.8287 0.4073505 ## factor(id)20226000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)20618000 5.0000e-01 1.5497e+00 0.3226 0.7469985 ## factor(id)20663000 1.0000e+00 1.7894e+00 0.5588 0.5763364 ## factor(id)20822000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)21371000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)21518000 -2.4793e-11 1.5497e+00 0.0000 1.0000000 ## factor(id)21830000 -2.8477e-01 1.5503e+00 -0.1837 0.8542753 ## factor(id)22028000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## factor(id)22046000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)22406000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)23590000 -2.7848e+00 1.5503e+00 -1.7963 0.0725967 . ## factor(id)23601000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)23966000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## factor(id)24106000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)24345000 -5.0000e-01 1.5497e+00 -0.3226 0.7469985 ## factor(id)24950000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)25694000 -2.4632e-11 1.5497e+00 0.0000 1.0000000 ## factor(id)26219000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)26468000 1.2152e+00 1.5503e+00 0.7839 0.4332001 ## factor(id)26574000 1.7152e+00 1.5503e+00 1.1064 0.2686824 ## factor(id)26926000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)27023000 2.5000e+00 1.5497e+00 1.6132 0.1068561 ## factor(id)27056000 -1.0000e+00 1.7894e+00 -0.5588 0.5763364 ## factor(id)27152000 1.5000e+00 1.5497e+00 0.9679 0.3331960 ## factor(id)27590000 -1.0000e+00 1.7894e+00 -0.5588 0.5763364 ## factor(id)27695000 2.0000e+00 1.7894e+00 1.1177 0.2638425 ## factor(id)28816000 3.0000e+00 1.7894e+00 1.6765 0.0937968 . ## factor(id)28952000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)29206000 3.0000e+00 1.5497e+00 1.9359 0.0530267 . ## factor(id)29537000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## factor(id)29568000 2.0000e+00 1.5497e+00 1.2906 0.1970012 ## 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## Total Sum of Squares: 14428 ## Residual Sum of Squares: 3154 ## R-Squared: 0.7814 ## Adj. R-Squared: 0.4939 ## F-statistic: 2.71786 on 2591 and 1970 DF, p-value: < 2.22e-16 ``` the result from `lm()` and the result from `plm()` are the same. --- #First difference model: using `plm()` ```r twowaves_long$partner <- fct_relevel(twowaves_long$partner, "Yes","No") #R takes the reference category as 1 firstdif <- plm(sat ~ 0 + partner, data=twowaves_long, model="fd") #0 + partner, 0 is to remove the intercept summary(firstdif) ``` ``` ## Oneway (individual) effect First-Difference Model ## ## Call: ## plm(formula = sat ~ 0 + partner, data = twowaves_long, model = "fd") ## ## Unbalanced Panel: n = 2591, T = 1-2, N = 4562 ## Observations used in estimation: 1971 ## ## Residuals: ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## -10.0000 -1.0000 0.0000 0.0918 1.0000 9.4305 ## ## Coefficients: (1 dropped because of singularities) ## Estimate Std. Error t-value Pr(>|t|) ## partnerYes 0.569536 0.084075 6.7742 1.647e-11 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Total Sum of Squares: 6357.2 ## Residual Sum of Squares: 6308.1 ## R-Squared: 0.011128 ## Adj. R-Squared: 0.011128 ## F-statistic: 45.8893 on 1 and 1970 DF, p-value: 1.6466e-11 ``` --- #Compare the results ```r texreg::screenreg(list(pols, fixed, dumreg, firstdif), custom.model.names=c("Pooled regression", "Fixed effect(within)", "Dummy variable regression", "First difference"), include.ci = FALSE, omit.coef = "factor", center=TRUE) ``` ``` ## ## ================================================================================================= ## Pooled regression Fixed effect(within) Dummy variable regression First difference ## ------------------------------------------------------------------------------------------------- ## (Intercept) 7.50 *** 6.00 *** ## (0.03) (1.27) ## partnerYes 0.46 *** 0.57 *** 0.57 *** 0.57 *** ## (0.09) (0.08) (0.08) (0.08) ## ------------------------------------------------------------------------------------------------- ## R^2 0.01 0.02 0.78 0.01 ## Adj. R^2 0.01 -1.26 0.49 0.01 ## Num. obs. 4562 4562 4562 1971 ## ================================================================================================= ## *** p < 0.001; ** p < 0.01; * p < 0.05 ``` Note: When it is two waves, the first difference and within-transformation estimators are numerically equivalent. --- #When we have 6 waves, the results look like ```r pols_6wave <- plm(sat ~ partner , data=sixwaves_long, model="pooling") fixed_6wave <- plm(sat ~ partner, data=sixwaves_long, model="within") dumreg_6wave <- plm(sat ~ partner + factor(id), data=sixwaves_long, model="pooling") sixwaves_long$partner <- fct_relevel(sixwaves_long$partner,"Yes","No")#R takes the reference category as 1 firstdif_6wave <- plm(sat ~ 0 + partner, data=sixwaves_long, model="fd") ``` --- #When we have 6 waves, the results look like ```r texreg::screenreg(list(pols_6wave, fixed_6wave, dumreg_6wave, firstdif_6wave), custom.model.names=c("Pooled regression", "Fixed effect(within)", "Dummy variable regression", "First difference"), include.ci = FALSE, omit.coef = "factor", center=TRUE) ``` ``` ## ## ================================================================================================= ## Pooled regression Fixed effect(within) Dummy variable regression First difference ## ------------------------------------------------------------------------------------------------- ## (Intercept) 7.41 *** 6.50 *** ## (0.02) (0.89) ## partnerYes 0.43 *** 0.31 *** 0.31 *** 0.38 *** ## (0.04) (0.04) (0.04) (0.04) ## ------------------------------------------------------------------------------------------------- ## R^2 0.01 0.01 0.59 0.01 ## Adj. R^2 0.01 -0.32 0.46 0.01 ## Num. obs. 10370 10370 10370 7779 ## ================================================================================================= ## *** p < 0.001; ** p < 0.01; * p < 0.05 ``` When there are more than 2 waves, the result from Fixed effect are the same with the result from Dummy variable regression, but different from the result from first difference estimator. --- # when to fixed effect is better than first difference - **More than Two Time Periods**: fixed effect is generally preferred for panel data with three or more periods. With multiple time points, the FE estimator can more effectively capture within-individual variation, offering more stable estimates of time-varying effects. - **Within-Individual Variability**: If you’re interested in understanding changes within individuals (or entities) over time, and how within-entity differences correlate with the dependent variable, fixed effect is appropriate - **Persistent Unobserved Heterogeneity**: When there is strong time-invariant unobserved heterogeneity (e.g., personality traits, intrinsic productivity), and these unobserved factors correlate with your independent variables, fixed effect is a natural choice as it removes this bias. --- #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/1243446)