12.3 A Fixed Effects Model
- z1 =
panlag(z, a{, T})
- yields the lagged (or trimmed) variables of the dataset z
- {output, siga, sige} =
panfix(z, m{, T})
- estimates a fixed effects (or mixed) model
|
Using notation introduced above we write the modified UIP relation
as a fixed effects model of the form:
 |
(12.4) |
With the country specific intercept
it is possible to
capture unobservable country specific heterogeneity. In our
example we assume that the individual effects
is
deterministic. A comprehensive overview of this model type is
given by Hsiao (1986). In a fixed effects model
regressors may be correlated with the individual effects, while
the error term
is uncorrelated across
and
with
,
. However, the panfix
quantlet does not require such restrictive assumptions.
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])*100
to 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
panlag
to compute the
lagged variables. Specifically the
panlag
performs the
transformation
. To obtain the lag of the
variable
we therefore set
. In general the quantlet is
called by
z1=panlag(z,a {,T})
Using
returns a dataset z1 with the lagged variables
of z. As usual, the optional parameter T is used to
indicate the common number of time periods in a balanced dataset.
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
panfix
quantlet is used. Generally,
panfix
is called as:
{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
, all regressors are allowed to be
correlated with the individual effect
. In this case an
within-group estimator is applied. The standard errors of this
estimator are estimated in a robust fashion, that is, the standard
errors are valid for quite general forms of autocorrelation and
heteroskedasticity. The results from
panfix
are given as:
[ 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
-test clearly rejects the null hypotheses of no effects and
hence individual effects
are significant. However,
comparing estimates of
and
it turns out that the remaining error dominates
the individual effect.
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.