12.3 A Fixed Effects Model


z1 = 22470 panlag (z, a{, T})
yields the lagged (or trimmed) variables of the dataset z
{output, siga, sige} = 22473 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:

$\displaystyle \Delta s_{it} = \beta_0+ \beta_1 (r_l - r^*_l)_{it-1} + \beta_2 (r_s -r_s^*)_{it}+ \alpha_i + u_{it}$ (12.4)

With the country specific intercept $ \alpha_i$ it is possible to capture unobservable country specific heterogeneity. In our example we assume that the individual effects $ \alpha_i$ 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 $ u_{it}$ is uncorrelated across $ i$ and $ t$ with $ E(u_{it})=0$, $ E(u^2_{it})=\sigma^2$. 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 22476 panlag to compute the lagged variables. Specifically the 22479 panlag performs the transformation $ (1-a)z_t-a z_{t-1}$. To obtain the lag of the variable $ z_t$ we therefore set $ a=1$. In general the quantlet is called by
  z1=panlag(z,a {,T})
Using $ a=1$ 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 22482 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 22485 panfix quantlet is used. Generally, 22488 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 $ m=2$, all regressors are allowed to be correlated with the individual effect $ \alpha_i$. 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 22491 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 $ F$-test clearly rejects the null hypotheses of no effects and hence individual effects $ \alpha_i$ are significant. However, comparing estimates of $ \sigma^2_\alpha=6.43$ and $ \sigma^2_u=31.24$ it turns out that the remaining error dominates the individual effect.

Nevertheless, the coefficient estimates seem to be reasonable. While $ \beta_2$ is not significant on conventional significance levels, $ \beta_1$ is significantly different from zero on the 10% level. Furthermore the sign of $ \beta_1$ 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 22494 panfix output we conclude that there is a significant first order autocorrelation of the residuals. It follows that the standard deviations and $ t$-statistics may be biased if they are computed in the usual way. In the 22497 panfix quantlet the standard deviations and $ t$-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.