Getting Started in Fixed/Random Effects Models using R

1 Getting Started in Fixed/Random Effects Models using R/RStudio(v. )Oscar 2010 ~otorres/IntroPanel data (also known as longitudinal or cross-sectional time-series data) is a dataset in which the behavior of entities are observed across entities could be states, companies, individuals, countries, data looks like thiscountryyearYX1X2 a brief introduction onthe theory behind panel data analysis please see the following document: contents of this document rely heavily on the document: Panel Data Econometricsin R: theplmpackage notes from the ICPSR s Summer Program in Quantitative Methods of Social Research(summer 2010)Exploring panel data3library(foreign)Panel <- (" ")coplot(y ~ year|country, type="l", data=Panel) # Linescoplot(y ~ year|country, type="b", data=Panel) # Points and lines# Bars at top indicates corresponding graph ( countries)from left to right starting on the bottom row (Muenchen/Hilbe.)

2 355)Exploring panel data4library(foreign)Panel <- (" ")library(car)scatterplot(y~year|country, boxplots=FALSE, smooth=TRUE, , data=Panel)FIXED- Effects MODEL(Covariance Model, Within Estimator, Individual Dummy Variable Model, Least Squares Dummy Variable Model)Fixed Effects : Heterogeneity across countries (or entities)library(foreign)Panel <- (" ")library(gplots)plotmeans(y ~ country, main="Heterogeineityacross countries", data=Panel)# plotmeansdraw a 95% confidence intervalaround the meansdetach("package:gplots")# Remove package gplots from the workspace6 Heterogeneity: unobserved variables that do not change over time Fixed Effects : Heterogeneity across years library(foreign)Panel <- (" ")library(gplots)plotmeans(y ~ year, main="Heterogeineityacross years", data=Panel)# plotmeansdraw a 95% confidence intervalaround the meansdetach("package:gplots")# Remove package gplots from the workspace7 Heterogeneity: unobserved variables that do not change over time OLS regression8> library(foreign)> Panel < (" ")> ols<- lm(y ~ x1, data=Panel)> summary(ols)Call:lm(formula = y ~ x1, data = Panel)Residuals:Min 1Q Median 3Q Max +09 +09 +08 +09 +09 Coefficients.

3 Estimate Std. Error t value Pr(>|t|) (Intercept) +09 +08 *x1 +08 +08 ---Signif. codes: 0 ** ** * . 1 Residual standard error: +09 on 68 degrees of freedomMultiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 68 DF, p-value: > yhat <-ols$fitted> plot(Panel$x1, Panel$y, pch=19, xlab="x1", ylab="y")> abline(lm(Panel$y~Panel$x1),lwd=3, col="red")Regular OLS regression does not consider heterogeneity across groups or timeFixed Effects using Least squares dummy variable model9> library(foreign)>Panel < (" ") > <-lm(y ~ x1 + factor(country) -1, data=Panel)> summary( )Call:lm(formula = y ~ x1 + factor(country) -1, data = Panel)Residuals.

4 Min 1Q Median 3Q Max +09 +08 +08 +09 +09 Coefficients:Estimate Std. Error t value Pr(>|t|) x1 +09 +09 * factor(country)A +08 +08 factor(country)B +09 +09 factor(country)C +09 +09 factor(country)D +09 +08 **factor(country)E +08 +09 factor(country)F +09 +09 . factor(country)G +08 +09 ---Signif. codes: 0 ** ** * . 1 Residual standard error: +09 on 62 degrees of freedomMultiple R-squared: , Adjusted R-squared: F- statistic: on 8 and 62 DF, p-value: For the theory behind fixed Effects , please see squares dummy variable model10> yhat<- $fitted> library(car)> scatterplot(yhat~Panel$x1|Panel$country, boxplots=FALSE, xlab="x1", ylab="yhat",smooth=FALSE)> abline(lm(Panel$y~Panel$x1),lwd=3, col="red")OLS regressionComparing OLS vsLSDV modelEach component of the factor variable (country) is absorbing the Effects particular to each country.

5 Predictor x1 was not significant in the OLS model, once controlling for differences across countries, x1became significant in the OLS_DUM ( LSDV model).11> library(apsrtable)> apsrtable(ols, , c("OLS", "OLS_DUM")) # Displays a table in Latex form> cat(apsrtable(ols, , c("OLS", "OLS_DUM"), Sweave=F), file=" ")# Exports the table to a text file (in Latex code).\begin{table}[!ht]\caption{}\label {} \begin{tabular}{ l D{.}{.}{2}D{.}{.}{2} } \hline& \multicolumn{ 1 }{ c }{ OLS } & \multicolumn{ 1 }{ c }{ OLS_DUM } \\\hline% & OLS & OLS_DUM \\(Intercept) & ^* & \\& ( ) & \\x1 & & ^*\\& ( ) & ( ) \\factor(country)A & & \\& & ( ) \\factor(country)B & & \\& & ( ) \\factor(country)C & & \\& & ( ) \\factor(country)D & & ^*\\& & ( ) \\factor(country)E & & \\& & ( )

6 \\factor(country)F & & \\& & ( ) \\factor(country)G & & \\& & ( ) \\$N$ & 70 & 70 \\$R^2$ & & \\adj. $R^2$ & & \\Resid. sd& & \\ \ hline\multicolumn{3}{l}{\footnotesize{St andard errors in parentheses}}\\\multicolumn{3}{l}{\footn otesize{$^*$ indicates significance at $p< $}} \end{tabular} \end{table}The coefficient of x1indicates how much Ychanges overtime, controlling by differences in countries, when Xincreases by one unit.

7 Notice x1is significant in the LSDV modelThe coefficient of x1indicates how much Ychanges when Xincreases by one unit. Notice x1is not significant in the OLS modelFixed Effects : nentity-specific intercepts ( using plm)> library(plm)> fixed <-plm(y ~ x1, data=Panel, index=c("country", "year"), model="within")> summary(fixed)Oneway(individual) effect Within ModelCall:plm(formula = y ~ x1, data = Panel, model = "within", index = c("country", "year"))Balanced Panel: n=7, T=10, N=70 Residuals :Min. 1st Qu. Median Mean 3rd Qu. Max. +09 +08 +08 +09 +09 Coefficients :Estimate Std. Error t-value Pr(>|t|) x1 2475617827 1106675594 *---Signif.

8 Codes: 0 ** ** * . 1 Total Sum of Squares: +20 Residual Sum of Squares: +20R- Squared : Adj. R-Squared : F- statistic: on 1 and 62 DF, p-value: > fixef(fixed) # Display the fixed Effects (constants for each country)A B C D E F 880542404 -1057858363 -1722810755 3162826897 -602622000 2010731793 G -984717493> pFtest(fixed, ols) # Testing for fixed Effects , null: OLS better than fixedF test for individual effectsdata: y ~ x1 F = , df1 = 6, df2 = 62, p-value = hypothesis : significant effectsFixed Effects optionOutcome variablePredictor variable(s)Panel settingn = # of groups/panels, T = # years, N = total # of observationsPr(>|t|)= Two-tail p- values test the hypothesis that each coefficient is different from 0.

9 To reject this, the p-value has to be lower than (95%, you could choose also an alpha of ), if this is the case then you can say that the variable has a significant influence on your dependent variable (y)If this number is < then your model is ok. This is a test (F) to see whether all the coefficients in the model are different than the p-value is < then the fixed Effects model is a better choiceThe coeffof x1 indicates how much Ychanges overtime, on average per country, when Xincreases by one MODEL(Random Intercept, Partial Pooling Model)Random Effects ( using plm)> random <-plm(y ~ x1, data=Panel, index=c("country", "year"), model="random")> summary(random)Oneway(individual) effect Random Effect Model (Swamy-Arora's transformation)Call:plm(formula = y ~ x1, data = Panel, model = "random", index = c("country", "year"))Balanced Panel.

10 N=7, T=10, N=70 shareidiosyncratic +18 +09 +18 +09 : Residuals :Min. 1st Qu. Median Mean 3rd Qu. Max. +09 +09 +08 +09 +09 Coefficients :Estimate Std. Error t-value Pr(>|t|)(Intercept) 1037014284 790626206 1247001782 902145601 Sum of Squares: +20 Residual Sum of Squares: +20R- Squared : Adj. R-Squared : F- statistic: on 1 and 68 DF, p-value: # Setting as panel data (an alternative way to run the above <- (Panel, index = c("country", "year"))# Random Effects using panel setting (same output as above) <- plm(y ~ x1, data = , model="random")summary( )Random Effects optionOutcome variablePredictor variable(s)Panel settingn = # of groups/panels, T = # years, N = total # of observationsPr(>|t|)= Two-tail p- values test the hypothesis that each coefficient is different from 0.)