Using notation introduced above we write the modified UIP relation as a fixed effects model of the form:
In order to estimate equation (12.4) we need to construct the appropriate variables from variables saved in the original data set. First we use
z[,3]=log(z[,3])*100to compute the logarithm of the exchange rate index and multiply by 100 in order to interpret the result as annual changes in percent. Then we use the quantlet
z1=panlag(z,a {,T})Using
The command
dz=panlag(z,0)-panlag(z,1)stores the first differences of the variables in z in the new dataset dz if z is an unbalanced dataset with the indices for the cross section units and the time periods in the first two columns.
In our case we apply the
panlag
quantlet as follows:
y1=panlag(z,1) y0=panlag(z,0)and compute the relevant data set as:
z1=y0[,1:2]~(y0[,3]-y1[,3])~y1[,4]~y0[,5]To estimate the fixed effects model (12.4) the
{output, siga, sige} = panfix(z, m{, T})The string output yields the output table of an estimation assuming the first given m explanatory variables as time varying and correlated with the individual effects. The remaining variables are assumed to be uncorrelated with the individual effect. The common time period T is included in the list of input parameters if the data is a balanced panel.
For the UIP example we simply type
panfix(z1,2)Since we have set
[ 1,] "=====================================================" [ 2,] "Fixed-Effect Model: y(i,t)=x(i,t)'beta+ a(i) + e(i,t)" [ 3,] "=====================================================" [ 4,] "PARAMETERS Estimate robust SE t-value" [ 5,] "=====================================================" [ 6,] "beta[ 1 ]= 0.38256 0.2029 1.885" [ 7,] "beta[ 2 ]= 0.21321 0.2176 0.980" [ 8,] "CONSTANT = -1.1781 0.6295 -1.872" [ 9,] "=====================================================" [10,] "Var. of a(i): 6.4327 e(i,t): 31.241" [11,] "AR(1)-test p-val: 0.0001 Autocorr.: 0.3633" [12,] "F(no eff.) p-val: 0.0000 R-square: 0.2549" [13,] "LM(siga=0) p-val: 0.0000 Log-Like: -613.194" [14,] "====================================================="The
Nevertheless, the coefficient estimates seem to be reasonable.
While is not significant on conventional significance
levels,
is significantly different from zero on the 10%
level. Furthermore the sign of
is in line with
expectations. We thus interpret this result as evidence in favor
of the UIP.
A potential problem with the fixed effects estimates is the
autocorrelation of the errors. From the
panfix
output we
conclude that there is a significant first order autocorrelation
of the residuals. It follows that the standard deviations and
-statistics may be biased if they are computed in the usual
way. In the
panfix
quantlet the standard deviations and
-statistics are however computed in a robust fashion as suggested
by Arellano (1987) so that possible autocorrelation or
heteroskedasticity of the errors do not bias the inference.
Nevertheless, the estimates may be inefficient and in many
applications one is interested to estimate the dynamic
relationship explicitely. In the next section we therefore
estimate a dynamic panel data model.