12.4 A Dynamic Panel Data Model


z1 = 22695 panlag (z, a,{, T})
yields the lagged (or trimmed) variables of the dataset z
{output, beta} = 22698 pandyn (z, p, IVmeth{, T})
computes 1-st stage GMM estimate of a dynamic linear model with p lags of the dependent variables

A dynamic panel data model is given by

$\displaystyle y_{it} = \gamma_1 y_{i,t-1} + \cdots + \gamma_p y_{i,t-p} + x_{it}^T \beta + \alpha_i + \varepsilon_{it}$ (12.5)

For such a model the within-group estimator is not applicable. Therefore, Arrelano and Bond (1991) suggest to estimate the model using a GMM estimation procedure. The idea is to estimate the differenced model

$\displaystyle \Delta y_{it} = \gamma_1 \Delta y_{i,t-1} + \cdots + \gamma_p \Delta y_{i,t-p} + x_{it}^T \beta + \Delta \varepsilon_{it}$ (12.6)

where $ \Delta$ is the difference operator such that $ \Delta y_{it}
= y_{it} - y_{i,t-1}$ by using the instruments:

$\displaystyle y_{i,t-2},y_{i,t-3},\ldots,y_{i1},\Delta x_{it}$ (12.7)

In general, it is possible to construct different GMM estimators using the instruments from (12.7). Arrelano and Bond (1991) suggest several instrumental variable matrices which are also implemented in XploRe . (For details on the different methods see Xplore Learning Guide, Section 12.5.) In our example we have a panel with $ T > N$. In this case we only use $ y_{i,t-2}$, $ \Delta x_{i,t}$ and $ \Delta x_{i,t-1}$ as instruments for lagged differences:

$\displaystyle \textrm{Method 1:} \quad \left( \begin{array}{ccccc} y_{ip} &
y_{...
... & \Delta x_{i,p+3} & \Delta x_{i,p+4} & . &
\Delta x_{iT}
\end{array}\right).
$

To continue our UIP example we rewrite our model in dynamic form as

$\displaystyle \Delta s_{it} = \gamma_1 \Delta s_{i,t-1} + \beta_1 (r_l - r^*_l)_{i,t-1} + \beta_2 (r_s -r_s^*)_{it}+ \alpha_i + u_{it}.$ (12.8)

We include the one-period lag of the exchange index growth rate as an additional regressor. To estimate equation (12.8) we use the XploRe quantlet 22710 pandyn with the general syntax
  {output,beta} = pandyn(z,p,IVmeth {,T})
where z is the data set, p the number of lagged dependent variables, IVmeth the method for constructing the instrument matrix and T the number of period covered in the data set. Note that the T is only needed if the panel is balanced. For the UIP data we just type
  pandyn(z1,1,1)
to get the following output table:
  [ 1,] "====================================================="
  [ 2,] "GMM-Estimation of the Dynamic Fixed-Effect Model:    "
  [ 3,] "y(i,t) = y(i,t-1)'gamma + x(i,t)'beta + a(i) + e(i,t)"
  [ 4,] "====================================================="
  [ 5,] "PARAMETERS        Estimate        SE          t-value"
  [ 6,] "====================================================="
  [ 7,] "beta[ 1 ]=          0.5454       0.2085         2.617"
  [ 8,] "beta[ 2 ]=         -0.1357       0.1842        -0.737"
  [ 9,] "Lag [ 1 ]=          0.2896       0.0846         3.424"
  [10,] "====================================================="
  [11,] "N*T=    362      N=   16      DF=     2              "
  [12,] "R-square (Levels):               0.1513              "
  [13,] "Hansen's J-statistic (p-val):    0.2848              "
  [14,] "Instruments according to method: 1                   "
  [15,] "====================================================="
As in the static model, $ \beta_2$ turns out to be insignificant on conventional levels and in line with results from the static fixed effects model $ \beta_1$ is now significantly different from zero on the 5% level. Moreover, the coefficient of the lagged dependent variable $ \Delta s_{i,t-1}$ is significant and about the same magnitude as in the test for autocorrelation in the static model. Now that we account for first order autocorrelation the parameters are estimated more precisely. These estimates support the evidence in favor of UIP theory.