A Panel-Data Sets with:

*time-index* : t=1,...,T=30

*individual-index* : i=1,...,N=60

This panel-data set has a polynomial factor structure (3 common
factors) and *endogenous* regressors.

1 |

A data frame containing :

- Y
dependent variable as N*T-vector

- X1
first regressor as N*T-vector

- X2
second regressor as N*T-vector

- CF.1
first (unobserved) common factor:

*CF.1(t)=1*- CF.2
second (unobserved) common factor:

*CF.2(t)=(t/T)*- CF.3
thrid (unobserved) common factor:

*CF.3(t)=(t/T)^2*Remark: The time-index t is running "faster" than the individual-index i such that e.g. Y_it is ordered as:

*Y_{11},Y_{12},…,Y_{1T},Y_{21},Y_{22},…*

The panel-data set DPG2 is simulated according to the simulation-study
in Kneip, Sickles & Song (2012):
*Y_{it}=β_{1}X_{it1}+β_{2}X_{it2}+v_i(t)+ε_{it},
i=1,…,n, t=1,…,T*
-Slope parameters:
*beta_{1}=beta_{2}=0.5*

-Time varying individual effects being second order polynomials:
*v_i(t)=theta_{i0}+theta_{i1}*frac{t}{T}+theta_{i2}*(frac{t}{T})^2*
Where theta_i1, theta_i1, and theta_i1 are iid as N(0,4)

The Regressors X_it=(X_it1,X_it2)' are simulated from a bivariate VAR model:
*X_{it}=R X_{i,t-1}+eta_{it} with
R=matrix(c(0.4,0.05,0.05,0.4),2,2) and eta_{it}~N(0,I_2)*

After this simulation, the N regressor-series
*(X_{1i1},X_{2i1})',…,(X_{1iT},X_{2iT})'*
are additionally shifted such that there are three different
mean-value-clusters. Such that every third of the N regressor-series
fluctuates around on of the following mean-values
*mu_1=(5,5)', mu_2=(7.5,7.5)', and mu_3=(10,10)'*

In this Panel-Data Set the regressor X_it2 is made endogenous by the
re-definition:
*X_{it2}:=X_{it2}+0.5*v_i(t)*

See Kneip, Sickles & Song (2012) for more details.

Dominik Liebl

Kneip, A., Sickles, R. C., Song, W., 2012 “A New Panel Data Treatment for Heterogneity in Time Trends”,

*Econometric Theory*

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | ```
data(DGP2)
## Dimensions
N <- 60
T <- 30
## Observed Variables
Y <- matrix(DGP2$Y, nrow=T,ncol=N)
X1 <- matrix(DGP2$X1, nrow=T,ncol=N)
X2 <- matrix(DGP2$X2, nrow=T,ncol=N)
## Unobserved common factors
CF.1 <- DGP2$CF.1[1:T]
CF.2 <- DGP2$CF.2[1:T]
CF.3 <- DGP2$CF.3[1:T]
## Take a look at the simulated data set DGP2:
par(mfrow=c(2,2))
matplot(Y, type="l", xlab="Time", ylab="", main="Depend Variable")
matplot(X1, type="l", xlab="Time", ylab="", main="First Regressor")
matplot(X2, type="l", xlab="Time", ylab="", main="Second Regressor")
## Usually unobserved common factors:
matplot(matrix(c(CF.1,
CF.2,
CF.3), nrow=T,ncol=3),
type="l", xlab="Time", ylab="", main="Unobserved Common Factors")
par(mfrow=c(1,1))
## Esimation
KSS.fit <- KSS(Y~-1+X1+X2)
(KSS.fit.sum <- summary(KSS.fit))
plot(KSS.fit.sum)
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