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    1 \chapter{Dynamic panel models}
2 \label{chap:dpanel}
3
4 \newcommand{\by}{\boldsymbol{y}}
5 \newcommand{\bx}{\boldsymbol{x}}
6 \newcommand{\bv}{\boldsymbol{v}}
7 \newcommand{\bX}{\boldsymbol{X}}
8 \newcommand{\bW}{\boldsymbol{W}}
9 \newcommand{\bZ}{\boldsymbol{Z}}
10 \newcommand{\bA}{\boldsymbol{A}}
11 \newcommand{\biota}{\bm{\iota}}
12
13 \DefineVerbatimEnvironment%
14 {code}{Verbatim}
15 {fontsize=\small, xleftmargin=1em}
16
17 \newenvironment%
18 {altcode}%
19 {\vspace{1ex}\small\leftmargin 1em}{\vspace{1ex}}
20
21 The primary command for estimating dynamic panel models in gretl is
22 \texttt{dpanel}. The closely related \texttt{arbond} command predated
23 \texttt{dpanel}, and is still present, but whereas \texttt{arbond}
24 only supports the so-called difference'' estimator
25 \citep{arellano-bond91}, \texttt{dpanel} in addition offers the
26 system'' estimator \citep{blundell-bond98}, which has become the
27 method of choice in the applied literature.
28
29 \section{Introduction}
30 \subsection{Notation}
31 \label{sec:notation}
32
33 A dynamic linear panel data model can be represented as follows
34 (in notation based on \cite{arellano03}):
35 \begin{equation}
36   \label{eq:dpd-def}
37   y_{it} = \alpha y_{i,t-1} + \beta'x_{it} + \eta_{i} + v_{it}
38 \end{equation}
39
40 The main idea behind the difference estimator is to sweep out the
41 individual effect via differencing.  First-differencing eq.\
42 (\ref{eq:dpd-def}) yields
43 \begin{equation}
44   \label{eq:dpd-dif}
45   \Delta y_{it} = \alpha \Delta y_{i,t-1} + \beta'\Delta x_{it} +
46   \Delta v_{it} = \gamma' W_{it} + \Delta v_{it} ,
47 \end{equation}
48 in obvious notation. The error term of (\ref{eq:dpd-dif}) is, by
49 construction, autocorrelated and also correlated with the lagged
50 dependent variable, so an estimator that takes both issues into
51 account is needed. The endogeneity issue is solved by noting that all
52 values of $y_{i,t-k}$ with $k>1$ can be used as instruments for
53 $\Delta y_{i,t-1}$: unobserved values of $y_{i,t-k}$ (whether missing
54 or pre-sample) can safely be substituted with 0. In the language of
55 GMM, this amounts to using the relation
56 \begin{equation}
57   \label{eq:OC-dif}
58   E(\Delta v_{it} \cdot y_{i,t-k}) = 0, \quad k>1
59 \end{equation}
60 as an orthogonality condition.
61
62 Autocorrelation is dealt with by noting that if $v_{it}$ is white
63 noise, the covariance matrix of the vector whose typical element is
64 $\Delta v_{it}$ is proportional to a matrix $H$ that has 2 on the main
65 diagonal, $-1$ on the first subdiagonals and 0 elsewhere.  One-step
66 GMM estimation of equation (\ref{eq:dpd-dif}) amounts to computing
67 \begin{equation}
68 \label{eq:dif-gmm}
69   \hat{\gamma} = \left[
70     \left( \sum_{i=1}^N \bW_i'\bZ_i \right) \bA_N
71     \left( \sum_{i=1}^N \bZ_i'\bW_i \right) \right]^{-1}
72     \left( \sum_{i=1}^N \bW_i'\bZ_i \right) \bA_N
73     \left( \sum_{i=1}^N \bZ_i'\Delta \by_i \right)
74 \end{equation}
75 where
76 \begin{align*}
77   \Delta \by_i  & =
78      \left[ \begin{array}{ccc}
79          \Delta y_{i,3} & \cdots & \Delta y_{i,T}
80        \end{array} \right]' \\
81   \bW_i  & =
82      \left[ \begin{array}{ccc}
83          \Delta y_{i,2} & \cdots & \Delta y_{i,T-1} \\
84          \Delta x_{i,3} & \cdots & \Delta x_{i,T} \\
85        \end{array} \right]' \\
86   \bZ_i  & =
87      \left[ \begin{array}{ccccccc}
88          y_{i1} & 0 & 0 & \cdots & 0 & \Delta x_{i3}\\
89          0 & y_{i1} & y_{i2} & \cdots & 0 & \Delta x_{i4}\\
90          & & \vdots \\
91          0 & 0 & 0 & \cdots & y_{i, T-2} & \Delta x_{iT} \\
92        \end{array} \right]' \\
93   \intertext{and}
94   \bA_N & = \left( \sum_{i=1}^N \bZ_i' H \bZ_i \right)^{-1}
95 \end{align*}
96
97 Once the 1-step estimator is computed, the sample covariance matrix of
98 the estimated residuals can be used instead of $H$ to obtain 2-step
99 estimates, which are not only consistent but asymptotically
100 efficient. (In principle the process may be iterated, but nobody seems
101 to be interested.) Standard GMM theory applies, except for one thing:
102 \cite{Windmeijer05} has computed finite-sample corrections to the
103 asymptotic covariance matrix of the parameters, which are nowadays
104 almost universally used.
105
106 The difference estimator is consistent, but has been shown to have
107 poor properties in finite samples when $\alpha$ is near one. People
108 these days prefer the so-called system'' estimator, which
109 complements the differenced data (with lagged levels used as
110 instruments) with data in levels (using lagged differences as
111 instruments). The system estimator relies on an extra orthogonality
112 condition which has to do with the earliest value of the dependent
113 variable $y_{i,1}$. The interested reader is referred to \citet[pp.\
114 124--125]{blundell-bond98} for details, but here it suffices to say
115 that this condition is satisfied in mean-stationary models and brings
116 an improvement in efficiency that may be substantial in many cases.
117
118 The set of orthogonality conditions exploited in the system approach
119 is not very much larger than with the difference estimator since most
120 of the possible orthogonality conditions associated with the equations
121 in levels are redundant, given those already used for the equations in
122 differences.
123
124 The key equations of the system estimator can be written as
125
126 \begin{equation}
127 \label{eq:sys-gmm}
128   \tilde{\gamma} = \left[
129     \left( \sum_{i=1}^N \tilde{\bW}'\tilde{\bZ} \right) \bA_N
130     \left( \sum_{i=1}^N \tilde{\bZ}'\tilde{\bW} \right) \right]^{-1}
131     \left( \sum_{i=1}^N \tilde{\bW}'\tilde{\bZ} \right) \bA_N
132     \left( \sum_{i=1}^N \tilde{\bZ}'\Delta \tilde{\by}_i \right)
133 \end{equation}
134 where
135 \begin{align*}
136   \Delta \tilde{\by}_i  & =
137      \left[ \begin{array}{ccccccc}
138          \Delta y_{i3} & \cdots & \Delta y_{iT} & y_{i3} & \cdots & y_{iT}
139        \end{array} \right]' \\
140   \tilde{\bW}_i  & =
141      \left[ \begin{array}{cccccc}
142          \Delta y_{i2} & \cdots & \Delta y_{i,T-1} & y_{i2} & \cdots & y_{i,T-1} \\
143          \Delta x_{i3} & \cdots & \Delta x_{iT}  & x_{i3} & \cdots & x_{iT} \\
144        \end{array} \right]' \\
145   \tilde{\bZ}_i  & =
146      \left[ \begin{array}{ccccccccc}
147          y_{i1} & 0 & 0       & \cdots & 0  & 0  & \cdots & 0 & \Delta x_{i,3}\\
148          0 & y_{i1} & y_{i2} & \cdots & 0  & 0  & \cdots & 0 & \Delta x_{i,4}\\
149          & & \vdots \\
150          0 & 0 & 0 & \cdots & y_{i, T-2} & 0  & \cdots & 0  & \Delta x_{iT}\\
151          & & \vdots \\
152          0 & 0 & 0 & \cdots & 0 & \Delta y_{i2} & \cdots & 0  & x_{i3}\\
153          & & \vdots \\
154          0 & 0 & 0 & \cdots & 0 & 0 & \cdots & \Delta y_{i,T-1}  & x_{iT}\\
155        \end{array} \right]' \\
156   \intertext{and}
157   \bA_N & = \left( \sum_{i=1}^N \tilde{\bZ}' H^* \tilde{\bZ} \right)^{-1}
158 \end{align*}
159
160 In this case choosing a precise form for the matrix $H^*$ for the
161 first step is no trivial matter. Its north-west block should be as
162 similar as possible to the covariance matrix of the vector $\Delta 163 v_{it}$, so the same choice as the difference'' estimator is
164 appropriate. Ideally, the south-east block should be proportional to
165 the covariance matrix of the vector $\biota \eta_i + \bv$, that is
166 $\sigma^2_{v} I + \sigma^2_{\eta} \biota \biota'$; but since
167 $\sigma^2_{\eta}$ is unknown and any positive definite matrix renders
168 the estimator consistent, people just use $I$. The off-diagonal blocks
169 should, in principle, contain the covariances between $\Delta v_{is}$
170 and $v_{it}$, which would be an identity matrix if $v_{it}$ is white
171 noise. However, since the south-east block is typically given a
172 conventional value anyway, the benefit in making this choice is not
173 obvious. Some packages use $I$; others use a zero matrix.
174 Asymptotically, it should not matter, but on real datasets the
175 difference between the resulting estimates can be noticeable.
176
177 \subsection{Rank deficiency}
178 \label{sec:rankdef}
179
180 Both the difference estimator (\ref{eq:dif-gmm}) and the system
181 estimator (\ref{eq:sys-gmm}) depend for their existence on the
182 invertibility of $\bA_N$. This matrix may turn out to be singular for
183 several reasons. However, this does not mean that the estimator is not
184 computable: in some cases, adjustments are possible such that the
185 estimator does exist, but the user should be aware that in these cases
186 not all software packages use the same strategy and replication of
187 results may prove difficult or even impossible.
188
189 A first reason why $\bA_N$ may be singular could be the unavailability
190 of instruments, chiefly because of missing observations. This case is
191 easy to handle. If a particular row of $\tilde{\bZ}_i$ is zero for all
192 units, the corresponding orthogonality condition (or the corresponding
193 instrument if you prefer) is automatically dropped; of course, the
194 overidentification rank is adjusted for testing purposes.
195
196 Even if no instruments are zero, however, $\bA_N$ could be rank
197 deficient. A trivial case occurs if there are collinear instruments,
198 but a less trivial case may arise when $T$ (the total number of time
199 periods available) is not much smaller than $N$ (the number of units),
200 as, for example, in some macro datasets where the units are
201 countries. The total number of potentially usable orthogonality
202 conditions is $O(T^2)$, which may well exceed $N$ in some cases. Of
203 course $\bA_N$ is the sum of $N$ matrices which have, at most, rank $2T - 204 3$ and therefore it could well happen that the sum is singular.
205
206 In all these cases, what we consider the proper'' way to go is to
207 substitute the pseudo-inverse of $\bA_N$ (Moore--Penrose) for its regular
208 inverse. Again, our choice is shared by some software packages, but
209 not all, so replication may be hard.
210
211
212 \subsection{Treatment of missing values}
213
214 Textbooks seldom bother with missing values, but in some cases their
215 treatment may be far from obvious. This is especially true if missing
216 values are interspersed between valid observations. For example,
217 consider the plain difference estimator with one lag, so
218 $219 y_t = \alpha y_{t-1} + \eta + \epsilon_t 220$
221 where the $i$ index is omitted for clarity. Suppose you have an
222 individual with $t=1\ldots5$, for which $y_3$ is missing. It may seem
223 that the data for this individual are unusable, because
224 differencing $y_t$ would produce something like
225 $226 \begin{array}{c|ccccc} 227 t & 1 & 2 & 3 & 4 & 5 \\ 228 \hline 229 y_t & * & * & \circ & * & * \\ 230 \Delta y_t & \circ & * & \circ & \circ & * 231 \end{array} 232$
233 where $*$ = nonmissing and $\circ$ = missing. Estimation seems to be
234 unfeasible, since there are no periods in which $\Delta y_t$ and
235 $\Delta y_{t-1}$ are both observable.
236
237 However, we can use a $k$-difference operator and get
238 $239 \Delta_k y_t = \alpha \Delta_k y_{t-1} + \Delta_k \epsilon_t 240$
241 where $\Delta_k = 1 - L^k$ and past levels of $y_t$ are perfectly
242 valid instruments. In this example, we can choose $k=3$ and use $y_1$
243 as an instrument, so this unit is in fact perfectly usable.
244
245 Not all software packages seem to be aware of this possibility, so
246 replicating published results may prove tricky if your dataset
247 contains individuals with gaps between valid observations.
248
249 \section{Usage}
250
251 One of the concepts underlying the syntax of \texttt{dpanel} is that
252 you get default values for several choices you may want to make, so
253 that in a standard'' situation the command is very concise.  The
254 simplest case of the model (\ref{eq:dpd-def}) is a plain AR(1)
255 process:
256 \begin{equation}
257 \label{eq:dp1}
258   y_{i,t} = \alpha y_{i,t-1} + \eta_{i} + v_{it} .
259 \end{equation}
260 If you give the command
261 \begin{code}
262   dpanel 1 ; y
263 \end{code}
264 gretl assumes that you want to estimate (\ref{eq:dp1}) via the
265 difference estimator (\ref{eq:dif-gmm}), using as many orthogonality
266 conditions as possible.  The scalar \texttt{1} between \texttt{dpanel}
267 and the semicolon indicates that only one lag of \texttt{y} is
268 included as an explanatory variable; using \texttt{2} would give an
269 AR(2) model. The syntax that gretl uses for the non-seasonal AR and MA
270 lags in an ARMA model is also supported in this context.\footnote{This
271   represents an enhancement over the \texttt{arbond} command.} For
272 example, if you want the first and third lags of \texttt{y} (but not
273 the second) included as explanatory variables you can say
274 \begin{code}
275   dpanel {1 3} ; y
276 \end{code}
277 or you can use a pre-defined matrix for this purpose:
278 \begin{code}
279   matrix ylags = {1, 3}
280   dpanel ylags ; y
281 \end{code}
282 To use a single lag of \texttt{y} other than the first you need to
283 employ this mechanism:
284 \begin{code}
285   dpanel {3} ; y # only lag 3 is included
286   dpanel 3 ; y   # compare: lags 1, 2 and 3 are used
287 \end{code}
288
290 option, as in
291 \begin{code}
292   dpanel 1 ; y --system
293 \end{code}
294 The level orthogonality conditions and the corresponding instrument
295 are appended automatically (see eq.\ \ref{eq:sys-gmm}).
296
297 \subsection{Regressors}
298
299 If we want to introduce additional regressors, we list them after the
300 dependent variable in the same way as other gretl commands, such as
301 \texttt{ols}.
302
303 For the difference orthogonality relations, \texttt{dpanel} takes care
304 of transforming the regressors in parallel with the dependent
305 variable. Note that this differs from gretl's \texttt{arbond} command,
306 where only the dependent variable is differenced automatically; it
307 brings us more in line with other software.
308
309 One case of potential ambiguity is when an intercept is specified but
310 the difference-only estimator is selected, as in
311 \begin{code}
312   dpanel 1 ; y const
313 \end{code}
314 In this case the default \texttt{dpanel} behavior, which agrees with
315 Stata's \texttt{xtabond2}, is to drop the constant (since differencing
316 reduces it to nothing but zeros). However, for compatibility with the
317 DPD package for Ox, you can give the option \verb|--dpdstyle|, in
318 which case the constant is retained (equivalent to including a linear
319 trend in equation~\ref{eq:dpd-def}).  A similar point applies to the
320 period-specific dummy variables which can be added in \texttt{dpanel}
321 via the \verb|--time-dummies| option: in the differences-only case
322 these dummies are entered in differenced form by default, but when the
323 \verb|--dpdstyle| switch is applied they are entered in levels.
324
325 The standard gretl syntax applies if you want to use lagged
326 explanatory variables, so for example the command
327 \begin{code}
328   dpanel 1 ; y const x(0 to -1) --system
329 \end{code}
330 would result in estimation of the model
331 $332 y_{it} = \alpha y_{i,t-1} + 333 \beta_0 + \beta_1 x_{it} + \beta_2 x_{i,t-1} + 334 \eta_{i} + v_{it} . 335$
336
337
338 \subsection{Instruments}
339
340 The default rules for instruments are:
341 \begin{itemize}
342 \item lags of the dependent variable are instrumented using all
343   available orthogonality conditions; and
344 \item additional regressors are considered exogenous, so they are used
345   as their own instruments.
346 \end{itemize}
347
348 If a different policy is wanted, the instruments should be specified
349 in an additional list, separated from the regressors list by a
350 semicolon. The syntax closely mirrors that for the \texttt{tsls}
351 command, but in this context it is necessary to distinguish between
352 regular'' instruments and what are often called GMM-style''
353 instruments (that is, instruments that are handled in the same
354 block-diagonal manner as lags of the dependent variable, as described
355 above).
356
357 Regular'' instruments are transformed in the same way as
358 regressors, and the contemporaneous value of the transformed variable
359 is used to form an orthogonality condition. Since regressors are
360 treated as exogenous by default, it follows that these two commands
361 estimate the same model:
362
363 \begin{code}
364   dpanel 1 ; y z
365   dpanel 1 ; y z ; z
366 \end{code}
367 The instrument specification in the second case simply confirms what
368 is implicit in the first: that \texttt{z} is exogenous. Note, though,
369 that if you have some additional variable \texttt{z2} which you want
370 to add as a regular instrument, it then becomes necessary to
371 include \texttt{z} in the instrument list if it is to be treated
372 as exogenous:
373 \begin{code}
374   dpanel 1 ; y z ; z2   # z is now implicitly endogenous
375   dpanel 1 ; y z ; z z2 # z is treated as exogenous
376 \end{code}
377
378 The specification of GMM-style'' instruments is handled by the
379 special constructs \texttt{GMM()} and \texttt{GMMlevel()}.  The first
380 of these relates to instruments for the equations in differences, and
381 the second to the equations in levels. The syntax for \texttt{GMM()}
382 is
383
384 \begin{altcode}
385 \texttt{GMM(}\textsl{name}\texttt{,} \textsl{minlag}\texttt{,}
386 \textsl{maxlag}\texttt{)}
387 \end{altcode}
388
389 \noindent
390 where \textsl{name} is replaced by the name of a series (or the name
391 of a list of series), and \textsl{minlag} and \textsl{maxlag} are
392 replaced by the minimum and maximum lags to be used as
393 instruments. The same goes for \texttt{GMMlevel()}.
394
395 One common use of \texttt{GMM()} is to limit the number of lagged
396 levels of the dependent variable used as instruments for the equations
397 in differences. It's well known that although exploiting all possible
398 orthogonality conditions yields maximal asymptotic efficiency, in
399 finite samples it may be preferable to use a smaller subset (but see
400 also \cite{OkuiJoE2009}).  For example, the specification
401
402 \begin{code}
403   dpanel 1 ; y ; GMM(y, 2, 4)
404 \end{code}
405 ensures that no lags of $y_t$ earlier than $t-4$ will be used as
406 instruments.
407
408 A second use of \texttt{GMM()} is to exploit more fully the potential
409 block-diagonal orthogonality conditions offered by an exogenous
410 regressor, or a related variable that does not appear as a regressor.
411 For example, in
412
413 \begin{code}
414   dpanel 1 ; y x ; GMM(z, 2, 6)
415 \end{code}
416 the variable \texttt{x} is considered an endogenous regressor, and up to
417 5 lags of \texttt{z} are used as instruments.
418
419 Note that in the following script fragment
420 \begin{code}
421   dpanel 1 ; y z
422   dpanel 1 ; y z ; GMM(z,0,0)
423 \end{code}
424 the two estimation commands should not be expected to give the same
425 result, as the sets of orthogonality relationships are subtly
426 different.  In the latter case, you have $T-2$ separate orthogonality
427 relationships pertaining to $z_{it}$, none of which has any
428 implication for the other ones; in the former case, you only have one.
429 In terms of the $\bZ_i$ matrix, the first form adds a single row to
430 the bottom of the instruments matrix, while the second form adds a
431 diagonal block with $T-2$ columns; that is,
432 $433 \left[ \begin{array}{cccc} 434 z_{i3} & z_{i4} & \cdots & z_{it} 435 \end{array} \right] 436$
437 versus
438 $439 \left[ \begin{array}{cccc} 440 z_{i3} & 0 & \cdots & 0 \\ 441 0 & z_{i4} & \cdots & 0 \\ 442 & \ddots & \ddots & \\ 443 0 & 0 & \cdots & z_{it} 444 \end{array} \right] 445$
446
447 \section{Replication of DPD results}
448 \label{sec:DPD-replic}
449
450 In this section we show how to replicate the results of some of the
451 pioneering work with dynamic panel-data estimators by Arellano, Bond
452 and Blundell.  As the DPD manual \citep*{DPDmanual} explains, it is
453 difficult to replicate the original published results exactly, for two
454 main reasons: not all of the data used in those studies are publicly
455 available; and some of the choices made in the original software
456 implementation of the estimators have been superseded.  Here,
457 therefore, our focus is on replicating the results obtained using the
458 current DPD package and reported in the DPD manual.
459
460 The examples are based on the program files \texttt{abest1.ox},
461 \texttt{abest3.ox} and \texttt{bbest1.ox}. These are included in the
462 DPD package, along with the Arellano--Bond database files
463 \texttt{abdata.bn7} and \texttt{abdata.in7}.\footnote{See
465 Arellano--Bond data are also provided with gretl, in the file
466 \texttt{abdata.gdt}. In the following we do not show the output from
467 DPD or gretl; it is somewhat voluminous, and is easily generated by
468 the user. As of this writing the results from Ox/DPD and gretl are
469 identical in all relevant respects for all of the examples
470 shown.\footnote{To be specific, this is using Ox Console version 5.10,
471   version 1.24 of the DPD package, and gretl built from CVS as of
472   2010-10-23, all on Linux.}
473
474 A complete Ox/DPD program to generate the results of interest takes
475 this general form:
476
477 \begin{code}
478 #include <oxstd.h>
479 #import <packages/dpd/dpd>
480
481 main()
482 {
483     decl dpd = new DPD();
484
486     dpd.SetYear("YEAR");
487
488     // model-specific code here
489
490     delete dpd;
491 }
492 \end{code}
493 %
494 In the examples below we take this template for granted and show just
495 the model-specific code.
496
497 \subsection{Example 1}
498
499 The following Ox/DPD code---drawn from \texttt{abest1.ox}---replicates
500 column (b) of Table 4 in \cite{arellano-bond91}, an instance of the
501 differences-only or GMM-DIF estimator. The dependent variable is the
502 log of employment, \texttt{n}; the regressors include two lags of the
503 dependent variable, current and lagged values of the log real-product
504 wage, \texttt{w}, the current value of the log of gross capital,
505 \texttt{k}, and current and lagged values of the log of industry
506 output, \texttt{ys}. In addition the specification includes a constant
507 and five year dummies; unlike the stochastic regressors, these
508 deterministic terms are not differenced. In this specification the
509 regressors \texttt{w}, \texttt{k} and \texttt{ys} are treated as
510 exogenous and serve as their own instruments. In DPD syntax this
511 requires entering these variables twice, on the \verb|X_VAR| and
512 \verb|I_VAR| lines. The GMM-type (block-diagonal) instruments in this
513 example are the second and subsequent lags of the level of \texttt{n}.
514 Both 1-step and 2-step estimates are computed.
515
516 \begin{code}
517 dpd.SetOptions(FALSE); // don't use robust standard errors
518 dpd.Select(Y_VAR, {"n", 0, 2});
519 dpd.Select(X_VAR, {"w", 0, 1, "k", 0, 0, "ys", 0, 1});
520 dpd.Select(I_VAR, {"w", 0, 1, "k", 0, 0, "ys", 0, 1});
521
522 dpd.Gmm("n", 2, 99);
523 dpd.SetDummies(D_CONSTANT + D_TIME);
524
525 print("\n\n***** Arellano & Bond (1991), Table 4 (b)");
526 dpd.SetMethod(M_1STEP);
527 dpd.Estimate();
528 dpd.SetMethod(M_2STEP);
529 dpd.Estimate();
530 \end{code}
531
532 Here is gretl code to do the same job:
533
534 \begin{code}
535 open abdata.gdt
536 list X = w w(-1) k ys ys(-1)
537 dpanel 2 ; n X const --time-dummies --asy --dpdstyle
538 dpanel 2 ; n X const --time-dummies --asy --two-step --dpdstyle
539 \end{code}
540
541 Note that in gretl the switch to suppress robust standard errors is
542 \verb|--asymptotic|, here abbreviated to \verb|--asy|.\footnote{Option
543   flags in gretl can always be truncated, down to the minimal unique
544   abbreviation.} The \verb|--dpdstyle| flag specifies that the
545 constant and dummies should not be differenced, in the context of a
546 GMM-DIF model. With gretl's \texttt{dpanel} command it is not
547 necessary to specify the exogenous regressors as their own instruments
548 since this is the default; similarly, the use of the second and all
549 longer lags of the dependent variable as GMM-type instruments is the
550 default and need not be stated explicitly.
551
552 \subsection{Example 2}
553
554 The DPD file \texttt{abest3.ox} contains a variant of the above that
555 differs with regard to the choice of instruments: the variables
556 \texttt{w} and \texttt{k} are now treated as predetermined, and are
557 instrumented GMM-style using the second and third lags of their
558 levels. This approximates column (c) of Table 4 in
559 \cite{arellano-bond91}.  We have modified the code in
560 \texttt{abest3.ox} slightly to allow the use of robust
561 (Windmeijer-corrected) standard errors, which are the default in both
562 DPD and gretl with 2-step estimation:
563
564 \begin{code}
565 dpd.Select(Y_VAR, {"n", 0, 2});
566 dpd.Select(X_VAR, {"w", 0, 1, "k", 0, 0, "ys", 0, 1});
567 dpd.Select(I_VAR, {"ys", 0, 1});
568 dpd.SetDummies(D_CONSTANT + D_TIME);
569
570 dpd.Gmm("n", 2, 99);
571 dpd.Gmm("w", 2, 3);
572 dpd.Gmm("k", 2, 3);
573
574 print("\n***** Arellano & Bond (1991), Table 4 (c)\n");
575 print("        (but using different instruments!!)\n");
576 dpd.SetMethod(M_2STEP);
577 dpd.Estimate();
578 \end{code}
579
580 The gretl code is as follows:
581
582 \begin{code}
583 open abdata.gdt
584 list X = w w(-1) k ys ys(-1)
585 list Ivars = ys ys(-1)
586 dpanel 2 ; n X const ; GMM(w,2,3) GMM(k,2,3) Ivars --time --two-step --dpd
587 \end{code}
588 %
589 Note that since we are now calling for an instrument set other then
590 the default (following the second semicolon), it is necessary to
591 include the \texttt{Ivars} specification for the variable \texttt{ys}.
592 However, it is not necessary to specify \texttt{GMM(n,2,99)} since
593 this remains the default treatment of the dependent variable.
594
595 \subsection{Example 3}
596
597 Our third example replicates the DPD output from \texttt{bbest1.ox}:
598 this uses the same dataset as the previous examples but the model
599 specifications are based on \cite{blundell-bond98}, and involve
600 comparison of the GMM-DIF and GMM-SYS (system'') estimators. The
601 basic specification is slightly simplified in that the variable
602 \texttt{ys} is not used and only one lag of the dependent variable
603 appears as a regressor. The Ox/DPD code is:
604
605 \begin{code}
606 dpd.Select(Y_VAR, {"n", 0, 1});
607 dpd.Select(X_VAR, {"w", 0, 1, "k", 0, 1});
608 dpd.SetDummies(D_CONSTANT + D_TIME);
609
610 print("\n\n***** Blundell & Bond (1998), Table 4: 1976-86 GMM-DIF");
611 dpd.Gmm("n", 2, 99);
612 dpd.Gmm("w", 2, 99);
613 dpd.Gmm("k", 2, 99);
614 dpd.SetMethod(M_2STEP);
615 dpd.Estimate();
616
617 print("\n\n***** Blundell & Bond (1998), Table 4: 1976-86 GMM-SYS");
618 dpd.GmmLevel("n", 1, 1);
619 dpd.GmmLevel("w", 1, 1);
620 dpd.GmmLevel("k", 1, 1);
621 dpd.SetMethod(M_2STEP);
622 dpd.Estimate();
623 \end{code}
624
625 Here is the corresponding gretl code:
626
627 \begin{code}
628 open abdata.gdt
629 list X = w w(-1) k k(-1)
630 list Z = w k
631
632 # Blundell & Bond (1998), Table 4: 1976-86 GMM-DIF
633 dpanel 1 ; n X const ; GMM(Z,2,99) --time --two-step --dpd
634
635 # Blundell & Bond (1998), Table 4: 1976-86 GMM-SYS
636 dpanel 1 ; n X const ; GMM(Z,2,99) GMMlevel(Z,1,1) \
637  --time --two-step --dpd --system
638 \end{code}
639
640 Note the use of the \verb|--system| option flag to specify GMM-SYS,
641 including the default treatment of the dependent variable, which
642 corresponds to \texttt{GMMlevel(n,1,1)}. In this case we also want to
643 use lagged differences of the regressors \texttt{w} and \texttt{k} as
644 instruments for the levels equations so we need explicit
645 \texttt{GMMlevel} entries for those variables. If you want something
646 other than the default treatment for the dependent variable as an
647 instrument for the levels equations, you should give an explicit
648 \texttt{GMMlevel} specification for that variable---and in that case
649 the \verb|--system| flag is redundant (but harmless).
650
651 For the sake of completeness, note that if you specify at least one
652 \texttt{GMMlevel} term, \texttt{dpanel} will then include equations in
653 levels, but it will not automatically add a default \texttt{GMMlevel}
654 specification for the dependent variable unless the \verb|--system|
655 option is given.
656
657 \section{Cross-country growth example}
658 \label{sec:dpanel-growth}
659
660 The previous examples all used the Arellano--Bond dataset; for this
661 example we use the dataset \texttt{CEL.gdt}, which is also included in
662 the gretl distribution. As with the Arellano--Bond data, there are
663 numerous missing values.  Details of the provenance of the data can be
664 found by opening the dataset information window in the gretl GUI
665 (\textsf{Data} menu, \textsf{Dataset info} item). This is a subset of
666 the Barro--Lee 138-country panel dataset, an approximation to which is
667 used in \citet*{CEL96} and \citet*{Bond2001}.\footnote{We say an
668   approximation'' because we have not been able to replicate exactly
669   the OLS results reported in the papers cited, though it seems from
670   the description of the data in \cite{CEL96} that we ought to be able
671   to do so.  We note that \cite{Bond2001} used data provided by
672   Professor Caselli yet did not manage to reproduce the latter's
673   results.}  Both of these papers explore the dynamic panel-data
674 approach in relation to the issues of growth and convergence of per
675 capita income across countries.
676
677 The dependent variable is growth in real GDP per capita over
678 successive five-year periods; the regressors are the log of the
679 initial (five years prior) value of GDP per capita, the log-ratio of
680 investment to GDP, $s$, in the prior five years, and the log of annual
681 average population growth, $n$, over the prior five years plus 0.05 as
682 stand-in for the rate of technical progress, $g$, plus the rate of
683 depreciation, $\delta$ (with the last two terms assumed to be constant
684 across both countries and periods).  The original model is
685 \begin{equation}
686 \label{eq:CEL96}
687 \Delta_5 y_{it} = \beta y_{i,t-5} + \alpha s_{it} + \gamma (n_{it} +
688 g + \delta) + \nu_t + \eta_i + \epsilon_{it}
689 \end{equation}
690 which allows for a time-specific disturbance $\nu_t$. The Solow model
691 with Cobb--Douglas production function implies that $\gamma = 692 -\alpha$, but this assumption is not imposed in estimation. The
693 time-specific disturbance is eliminated by subtracting the period mean
694 from each of the series.
695
696 Equation (\ref{eq:CEL96}) can be transformed to an AR(1) dynamic
697 panel-data model by adding $y_{i,t-5}$ to both sides, which gives
698 \begin{equation}
699 \label{eq:CEL96a}
700 y_{it} = (1 + \beta) y_{i,t-5} + \alpha s_{it} + \gamma (n_{it} +
701 g + \delta) + \eta_i + \epsilon_{it}
702 \end{equation}
703 where all variables are now assumed to be time-demeaned.
704
705 In (rough) replication of \cite{Bond2001} we now proceed to estimate
706 the following two models: (a) equation (\ref{eq:CEL96a}) via GMM-DIF,
707 using as instruments the second and all longer lags of $y_{it}$,
708 $s_{it}$ and $n_{it} + g + \delta$; and (b) equation
709 (\ref{eq:CEL96a}) via GMM-SYS, using $\Delta y_{i,t-1}$, $\Delta 710 s_{i,t-1}$ and $\Delta (n_{i,t-1} + g + \delta)$ as additional
711 instruments in the levels equations. We report robust standard errors
712 throughout. (As a purely notational matter, we now use $t-1$'' to
713 refer to values five years prior to $t$, as in \cite{Bond2001}).
714
715 The gretl script to do this job is shown below. Note that the final
716 transformed versions of the variables (logs, with time-means
717 subtracted) are named \texttt{ly} ($y_{it}$), \texttt{linv} ($s_{it}$)
718 and \texttt{lngd} ($n_{it} + g + \delta$).
719 %
720 \begin{code}
721 open CEL.gdt
722
723 ngd = n + 0.05
724 ly = log(y)
725 linv = log(s)
726 lngd = log(ngd)
727
728 # take out time means
729 loop i=1..8 --quiet
730   smpl (time == i) --restrict --replace
731   ly -= mean(ly)
732   linv -= mean(linv)
733   lngd -= mean(lngd)
734 endloop
735
736 smpl --full
737 list X = linv lngd
738 # 1-step GMM-DIF
739 dpanel 1 ; ly X ; GMM(X,2,99)
740 # 2-step GMM-DIF
741 dpanel 1 ; ly X ; GMM(X,2,99) --two-step
742 # GMM-SYS
743 dpanel 1 ; ly X ; GMM(X,2,99) GMMlevel(X,1,1) --two-step --sys
744 \end{code}
745
746 For comparison we estimated the same two models using Ox/DPD and the
747 Stata command \texttt{xtabond2}. (In each case we constructed a
748 comma-separated values dataset containing the data as transformed in
749 the gretl script shown above, using a missing-value code appropriate
750 to the target program.) For reference, the commands used with
751 Stata are reproduced below:
752 %
753 \begin{code}
754 insheet using CEL.csv
755 tsset unit time
756 xtabond2 ly L.ly linv lngd, gmm(L.ly, lag(1 99)) gmm(linv, lag(2 99))
757   gmm(lngd, lag(2 99)) rob nolev
758 xtabond2 ly L.ly linv lngd, gmm(L.ly, lag(1 99)) gmm(linv, lag(2 99))
759   gmm(lngd, lag(2 99)) rob nolev twostep
760 xtabond2 ly L.ly linv lngd, gmm(L.ly, lag(1 99)) gmm(linv, lag(2 99))
761   gmm(lngd, lag(2 99)) rob nocons twostep
762 \end{code}
763
764 For the GMM-DIF model all three programs find 382 usable observations
765 and 30 instruments, and yield identical parameter estimates and
766 robust standard errors (up to the number of digits printed, or more);
767 see Table~\ref{tab:growth-DIF}.\footnote{The coefficient shown for
768   \texttt{ly(-1)} in the Tables is that reported directly by the
769   software; for comparability with the original model (eq.\
770   \ref{eq:CEL96}) it is necesary to subtract 1, which produces the
771   expected negative value indicating conditional convergence in per
772   capita income.}
773
774 \begin{table}[htbp]
775 \begin{center}
776 \begin{tabular}{lrrrr}
777 & \multicolumn{2}{c}{1-step} & \multicolumn{2}{c}{2-step} \\
778 & \multicolumn{1}{c}{coeff} & \multicolumn{1}{c}{std.\ error} &
779   \multicolumn{1}{c}{coeff} & \multicolumn{1}{c}{std.\ error} \\
780 \texttt{ly(-1)} & 0.577564 & 0.1292 & 0.610056 & 0.1562 \\
781 \texttt{linv} & 0.0565469 & 0.07082 & 0.100952 & 0.07772 \\
782 \texttt{lngd} & $-$0.143950 & 0.2753 & $-$0.310041 & 0.2980 \\
783 \end{tabular}
784 \caption{GMM-DIF: Barro--Lee data}
785 \label{tab:growth-DIF}
786 \end{center}
787 \end{table}
788
789 Results for GMM-SYS estimation are shown in
790 Table~\ref{tab:growth-SYS}. In this case we show two sets of gretl
791 results: those labeled gretl(1)'' were obtained using gretl's
792 \verb|--dpdstyle| option, while those labeled gretl(2)'' did not use
793 that option---the intent being to reproduce the $H$ matrices used by
794 Ox/DPD and \texttt{xtabond2} respectively.
795
796 \begin{table}[htbp]
797 \begin{center}
798 \begin{tabular}{lrrrr}
799 & \multicolumn{1}{c}{gretl(1)} &
800   \multicolumn{1}{c}{Ox/DPD} &
801   \multicolumn{1}{c}{gretl(2)} &
802   \multicolumn{1}{c}{xtabond2} \\
803 \texttt{ly(-1)} & 0.9237 (0.0385) &
804   0.9167 (0.0373) &
805     0.9073 (0.0370) &
806       0.9073 (0.0370) \\
807 \texttt{linv} & 0.1592 (0.0449) &
808   0.1636 (0.0441) &
809     0.1856 (0.0411) &
810       0.1856 (0.0411) \\
811 \texttt{lngd} & $-$0.2370 (0.1485) &
812   $-$0.2178 (0.1433) &
813     $-$0.2355 (0.1501) &
814       $-$0.2355 (0.1501)
815 \end{tabular}
816 \caption{2-step GMM-SYS: Barro--Lee data (standard errors in parentheses)}
817 \label{tab:growth-SYS}
818 \end{center}
819 \end{table}
820
821 In this case all three programs use 479 observations; gretl and
822 \texttt{xtabond2} use 41 instruments and produce the same estimates
823 (when using the same $H$ matrix) while Ox/DPD nominally uses
824 66.\footnote{This is a case of the issue described in
825   section~\ref{sec:rankdef}: the full $\bA_N$ matrix turns out to be
826   singular and special measures must be taken to produce estimates.}
827 It is noteworthy that with GMM-SYS plus messy'' missing
828 observations, the results depend on the precise array of instruments
829 used, which in turn depends on the details of the implementation of
830 the estimator.
831
832 \section{Auxiliary test statistics}
833
834 We have concentrated above on the parameter estimates and standard
835 errors. It may be worth adding a few words on the additional test
836 statistics that typically accompany both GMM-DIF and GMM-SYS
837 estimation. These include the Sargan test for overidentification, one
838 or more Wald tests for the joint significance of the regressors (and time
839 dummies, if applicable) and tests for first- and second-order
840 autocorrelation of the residuals from the equations in differences.
841
842 As in Ox/DPD, the Sargan test statistic reported by gretl is
843 $844 S = \left(\sum_{i=1}^N \hat{\bv}^{*\prime}_i \bZ_i\right) 845 \bA_N \left(\sum_{i=1}^N \bZ_i' \hat{\bv}^*_i\right) 846$
847 where the $\hat{\bv}^*_i$ are the transformed (e.g.\ differenced)
848 residuals for unit $i$.  Under the null hypothesis that the
849 instruments are valid, $S$ is asymptotically distributed as chi-square
850 with degrees of freedom equal to the number of overidentifying
851 restrictions.
852
853 In general we see a good level of agreement between gretl, DPD and
854 \texttt{xtabond2} with regard to these statistics, with a few
855 relatively minor exceptions. Specifically, \texttt{xtabond2} computes
856 both a Sargan test'' and a Hansen test'' for overidentification,
857 but what it calls the Hansen test is, apparently, what DPD calls the
858 Sargan test. (We have had difficulty determining from the
859 \texttt{xtabond2} documentation \citep{Roodman2006} exactly how its
860 Sargan test is computed.) In addition there are cases where the
861 degrees of freedom for the Sargan test differ between DPD and gretl;
862 this occurs when the $\bA_N$ matrix is singular
863 (section~\ref{sec:rankdef}). In concept the df equals the number of
864 instruments minus the number of parameters estimated; for the first of
865 these terms gretl uses the rank of $\bA_N$, while DPD appears to use
866 the full dimension of this matrix.
867
868 \section{Post-estimation available statistics}
869 \label{sec:dpanel-post}
870
871 After estimation, the \dollar{model} accessor will return a bundle
872 containing several items that may be of interest: most should be
873 self-explanatory, but here's a partial list:
874
875 \begin{center}
876 \begin{tabular}{rp{0.6\textwidth}}
877   \hline
878   \textbf{Key} & \textbf{Content} \\
879   \hline
880   \texttt{AR1}, \texttt{AR2} & 1st and 2nd order autocorrelation test
881                                statistics \\
882   \texttt{sargan}, \texttt{sargan\_df} & Sargan test for
883                                          overidentifying restrictions
884                                          and corresponding degrees of freedom \\
885   \texttt{wald}, \texttt{wald\_df} & Wald test for
886                                      overall significance
887                                      and corresponding degrees of
888                                      freedom \\
889   \texttt{GMMinst} & The matrix $\bZ$ of instruments (see equations
890                      (\ref{eq:dpd-dif}) and (\ref{eq:sys-gmm}) \\
891   \texttt{wgtmat} & The matrix $\bA$ of GMM weights (see equations
892                     (\ref{eq:dpd-dif}) and (\ref{eq:sys-gmm}) \\
893   \hline
894 \end{tabular}
895 \end{center}
896
897 Note, however, that \texttt{GMMinst} and \texttt{wgtmat} (which may be
898 quite large matrices) are not saved in the \dollar{model} bundle by
899 default; that requires use of the \option{keep-extra} option with the
900 \cmd{dpanel} command. The script in Table \ref{tab:dpanel-rep}
901 illustrates use of these matrices to replicate via hansl commands the
902 calculation of the GMM estimator.
903
904 \begin{table}[htbp]
905 \label{tab:dpanel-rep}
906   \begin{scode}
907 set verbose off
908 open abdata.gdt
909
910 # compose list of regressors
911 list X = w w(-1) k k(-1)
912 list Z = w k
913
914 dpanel 1 ; n X const ; GMM(Z,2,99) --two-step --dpd --keep-extra
915
916 ### --- re-do by hand ----------------------------
917
918 # fetch Z and A from model
919 A = $model.wgtmat 920 mZt =$model.GMMinst # note: transposed
921
922 # create data matrices
923 series valid = ok($uhat) 924 series ddep = diff(n) 925 series dldep = ddep(-1) 926 list dreg = diff(X) 927 928 smpl valid --dummy 929 930 matrix m_reg = {dldep} ~ {dreg} ~ 1 931 matrix m_dep = {ddep} 932 933 matrix uno = mZt * m_reg 934 matrix due = qform(uno', A) 935 matrix tre = (uno'A) * (mZt * m_dep) 936 matrix coef = due\tre 937 938 print coef 939 \end{scode} 940 \caption{replication of built-in command via hansl commands} 941 \end{table} 942 943 \section{Memo: \texttt{dpanel} options} 944 \label{sec:options} 945 946 \begin{center} 947 \begin{tabular}{lp{.7\textwidth}} 948 \textit{flag} & \textit{effect} \\ [6pt] 949 \verb|--asymptotic| & Suppresses the use of robust standard errors \\ 950 \verb|--two-step| & Calls for 2-step estimation (the default being 1-step) \\ 951 \verb|--system| & Calls for GMM-SYS, with default treatment of the 952 dependent variable, as in \texttt{GMMlevel(y,1,1)} \\ 953 \verb|--time-dummies| & Includes period-specific dummy variables \\ 954 \verb|--dpdstyle| & Compute the$H\$ matrix as in DPD; also suppresses
955                       differencing of automatic time dummies and omission of intercept
956                       in the GMM-DIF case\\
957   \verb|--verbose| & When \verb|--two-step| is selected, prints
958                      the 1-step estimates first \\
959   \verb|--vcv| & Calls for printing of the covariance matrix \\
960   \verb|--quiet| & Suppresses the printing of results \\
961   \verb|--keep-extra| & Save additional matrices in \dollar{model}
962                         bundle (see above) \\
963 \end{tabular}
964 \end{center}
965
966 The time dummies option supports the qualifier \texttt{noprint}, as
967 in
968
969 \verb|  --time-dummies=noprint|
970
971 This means that although the dummies are included in the specification
972 their coefficients, standard errors and so on are not printed.
973
974 %%% Local Variables:
975 %%% mode: latex
976 %%% TeX-master: "gretl-guide"
977 %%% End: