Palgrave Handbook of Econometrics: Applied Econometrics

666 Panel Methods to Test for Unit Roots and Cointegration

These results apply even when the break dates are unknown.^22

Analysis of Model 2
Here things are slightly more complicated, since differencing the processes does
not eliminate the dependence of the relevant test statistics on the break fractions.
For Model 2 we have, upon differencing:

yi,t=Ft′πi+δi+

∑mi

k= 1

γi,kDUi,k,t+ei∗,t, (13.15)

whereDUi,k,t=1 whenT>Tbi,kand zero elsewhere are the step dummies which
arise from differencing the trend breaks.
To follow Bai and Carrion-i-Silvestre’s notation, let:

di=(δi,γi,1,...,γi,mi)′

ai,t=(1,Di,1,t,...,Di,mi,t)′

ai=(ai,2,...,ai,T)′.

Then the first-differenced model can be rewritten (using the notation established
previously) as:
y ̃i=fπi+aidi+zi.

Since the break dates are assumed known the matrixaiis completely specified.
Conditional onδibeing known, the variablewi=y ̃i−aidihas a “complete” fac-
tor structure, in the sense of assumption (i), which follows (13.5)–(13.7) above.f
andcan therefore be estimated based onw=(w 1 ,w 2 ,...,wN). If, on the other
hand,fπiis known, regressingy ̃i−fπionaileads to consistent estimates ofdi. Bai
and Carrion-i-Silvestre therefore propose (conditional upon the break dates being
known) an iterative procedure for estimating the model as follows:

Step 1:Estimatediby least squares – that is,d ̃i=(a′iai)−^1 a′iy ̃i– ignoring the pres-

ence of the factors, which are assumed to have zero mean and will thus be

included in the regression errors.

Step 2:Givend ̃i, construct the seriesw ̃i=y ̃i−aid ̃iand estimate the factors and

factor loadings to givef ̃π ̃i.

Step 3:Regress w ̃i onai to obtain updated estimates ofdi and iterate until

convergence.

Step 4:Denoting the final estimates byfˆ,πˆanddˆi, computezˆi=y ̃i−fˆπˆi−aˆidˆi

and cumulate to obtaineˆi,t=

∑t

s= 2 zˆi,s. Compute theMSBstatistic for each

unit.

Then, asT→∞,

MSB(i,λi)⇒

m∑i+ 1

k= 1

(λi,k−λi,k− 1 )^2

∫ 1

0

Vi^2 ,k(b)db,