Econometric Modelling Using

PcGive I 0 Volume I l l

Panel Data Models

7.1 Introduction In a panel data setting. we have time-series observations on multiple entities, for ex- ample companies or individuals. We denote the cross-section sample size by x, and, in an ideal setting, have t = 1,. . . . T time-saies observations covering the same calendar period. This is called a bolaneed panel. In practice, it often happens that some cross- sections start earlier, or finish later. The panel data procedures in PcGive are expressly designed to handle such unbalanced panels.

When T is large, .V small. and the panel balanced, it will be possible to use the sin~ultaneous-equations modelling facilities of PcGive. When T is small, and the model dynamic (i.e. includes a lagged dependent variable), the estimation bias can be substan- tial (see Nickell, 1981). Methods to cope with such dynamic panel data models are the primary focus of this part of PcGive, particularly the GMM-type estimators of Arellano and Bond (1991), Arellano and Bover (1995), and Blundell and Bond (1998), but also some of the Anderson and Hsiao (1982) estimators. In addition, PcGive makes several of the standard static panel data estimators available, such as between and within groups and feasible GLS. Table 7.1 lists the available estimators.

Table 7.1 Static and dynamic panel data estimators of PcGive.

static panel data models dynamic panel data models OLS in levels O t S in levels . Between estimator one-step 1V estimation (Anderson-Hsiao 1V) Within estimator one-step GMM estimation Feasible GLS one-step estimation with robust std. errors GLS (OLS residuals) two-step estimation Maximunl likelihood (ML)

chapter only provides n cursory overview of panel d a ~ a methods. In addition 10 1 1 1 ~ ~ r l t ~ t n c c d literature. rhere are several text hooks on panel data. notably Hsiao

63

(1986) and mow recenrly Baltngi (1995). Many general econonietric rexr books have a chapter on panel data ( b u ~ rarely treating dynnniic models). see. e.g., Johnston and DiNardo ( 1997. Ch. I?).

7.2 Econometric methods for static panel data models 7.2.1 Static panel-data estimation

The static single-equation panel model can be written as:

The At and 17, are respectively time and individual specific effects and x,t is a E" vector of explanatory variables. i V is the number of cross-section observations. The total number of observations is NT.

Stacking the data for an individual according to time yields:

or. more concisely:

y, =X,y+A+Lzr),+v, , i = 1 >..,.I\-,

where y,, XI. and A, are T x 1, and L, is a T column of ones. Using D, for the matrix with dummies:

y z = X , y + D , 6 + v z , i = 1 ...... 1-.

The next step is to stack all individuals. combining the data inro W = [X : Dl:

y = W p + v . (7.2)

Baltagi ( 1995) reviews the standard estimation lnerhods used in this setting: OLS estimation. LSDV(1east squares dumniy variables) estiniation uses individual duninlies in the OLS regression. Wthir~ estinlation replaces y and W by dei-iarions lion) tinie means (i.e..subrracting the nirans of' each time series). Bcrwccn estiniarion replaces y and W by rhc means of eacl~ individual tlea\,in,e .V observations).

Fcnsible GLS (generalized leas1 squaws) estiniation replaces y and W by devi- ations from weighted tinie means. TEe outconle depends on the choice of the weight, 8. ML (niaxiniuni likelihood) estimation obtains 8 by iterating the GLS procedure.

More detail is provided in 5 10.3.

7.3 Econometric methods for dynamic panel data models The general model that can be estimated w+th PcGive is a single equation with indi- vidual effects of the forni:

where q, and At are respectively individual and time specific effects, r,t is a vector of explanatory variables. 3 ( L ) is a vector of associated polynomials in the lag operator and q is the maximum lag length in the model. The number of time periods available on the ith individual, T,, is small and the number of individuals, A*. is large. Identific- ation of the model requires restrictions on the serial correlation properties of the error term vrt and/or on the properties of the explanatory variables rZt. It is assumed that if the error term was originally autoregressive. the model has been transformed so that the coefficients a 's and 3's satisfy some set of common factor restrictions. Thus only serially uncorrelated or moving-average errors are explicitly allowed. The v,t are as- sumed to be independently distributed across individuals with zero mean. but arbitrary forms of heteroscedasticity across units and time are possible. The rZt may or may not be correlated with the individual effects qi. and for each of theses cases they may be strictly exogenous, predetermined or endogenous variables with respect to v,t. A case of particular interest is where the levels rZt are correlated with 7, but where lx,t (and possibly l y , t ) are uncorrelated with 7,: this allows the use of (suitably lagged) AX,, (and possibly Ay,,) as instruments for equations in levels.

The (TI - q) equations for individual i can be written conveniently in the form:

where 6 is a paranierer vecror including rhe ah 's . the d's and the X's, and W, is a data matrix conraining the time series of the lugged dependenr variables, the X'S and the time duniriiies. Lastly. L, is a (T, - q) x 1 vector of ones. PcGive can be used to compute various linellr GMM es~inlators of 6 with the general forni:

where

and W: and y: denote some transformation of W , and y, (e.g. levels, first differences. orthogonal deviations, combinations of first differences (or orthogonal deviations) and levels, deviations from individual means). Z , is a matrix of instrumental variables which may or may not be entirely internal, and HI is a possibly individual-specific weighting matrix.

If the number of columns of Z, equals that of W:. A,v becomes irrelevant and 5 reduces to

In particular. if Z, = W : and the transformed W , and y, are deviations from individual means or orthogonal deviations', then 5 is the within-groups estimator. As another example, if the transformation denotes first differences. Z , = IT> OX: and HI = G:G*'

1 ' where the G: are some consistent estimates of the first-differenced residuals, then 6 is the generalized three-stage least squares estimator of Chamberlain (1984). These two estimators require the x,t to be strictly exogenous with respect to vZt for consistency. In addition, the within-groups estimator can only be consistent as N - oo for fixed T if W : does not contain lagged dependent variables and all the explanatory variables are strictly exogenous.

When estimating dynamic models. we shall therefore typically be concerned with transformations that allow the use of lagged endogenous (and predetermined) variables as instruments in the transformed equations. Efficient GMM estimators will typically exploit a different number of instruments in each time period. Estimators of this type are discussed in Arellano (1988), Arellano and Bond (1991). Arellano and Bover (1995) and Blundell and Bond (1998). PcGive can be used to compute a range of linear GMM estimators of this type.

Where there are no instruments available that are uncorrelated with the individual effects 7, . the transformation must eliminate this component of the error term. The first difference and orthogonal deviations transformations are two examples of trans- formations that eliminate 7 , from the transformed error term, withou~ at the same time

'orthogonal deviations. as proposed by Arellano (1988) and Arellano and Bover (1995). ex- press each observation as the deviation from the averagr offtrrtrrc, obser\?ations in the sample for the same indi\:idual. and weight each deviation to standardize [he variance, i.e.

, ( J - , ( , c ~ ) + .-. + x,7 S,, = x , , - T - i ) ( T Z 1 ) l f01 / = 1. .... T - 1 I f rho original rrrors are serially uncorrelared and honiosctd;~.~ric.. rl~c r~.;~nsSomied errors will also he se~ially uncorrelared and honioscedasiic.

7.3 E~~or~or~~~rr - ;~ - ~~t&od.r,for- dsnarnic panel data nrodcls 67

introducing all lagged values of the disturbances vZt into the transformed error term.' Hence these transformations allowrbe use of suitably lagged endogenous (and predeter- mined) variables as instruments. Fa example. if the panel is balanced, p = 1, there are no explanatory variables nor time sects, the vlt are serially uncorrelated, and the ini- tial conditions y,, are uncorrelated with vLt for t = 2, ..., T . then using first differences we have:

Equations Instruments available AY,:~ = aAylz + A v , ~ Y11 Ay14 = a A y , ~ + Avt4 311 3 yi2

A Y ~ T = ~ A Y ~ . T - I + A ~ I T 911, y12, -.., y1.T-2

In this case, y: = (Ay13, ..., A y 1 ~ ) ' , W: = (Ay,2, -.., AY~.T-I ) ' and

Notice that precisely the same instrument set would be used to estimate the model in orthogonal deviations. Where the panel is unbalanced, for individuals with incomplete data the rows of Zi corresponding to the missing equations are deleted, and missing values in the remaining rows are replaced by zeros.

In PcGive, we call one-step estimates those which use some known matrix as the choice for H I . For a first-difference procedure, the one-step estimator uses

while for a levels or orthogonal deviations procedure. the one-step estimator sets HI to an identity matrix. If the vIt are heteroscedastic. a two-step estinlator which uses

where G: are one-step residuals, is more efficient (cf. White, 1982). PcGive produces both one-step and two-step GMM estimators, with asymptotic variance matrices that are

liere re are many other transfomiations which share these properties. See Arellano and Bover ( 1995) for further discussion.

heterc>scedasticity-consiste~it In both cases. Users should note that. particularly wher1 the vli are heteroscedastic. simulations suggcst that the asymptotic standard errors for the two-step estimators can be a poor guide for hypothesis testing in typical sample sizes. In these cases. inference based on asymptotic standard elTors for the one-step estimators seenis to be nlore reliable (see Arellano and Bond. 1991. and Blundell and Bond. 1998 for further discussion).

In models with explanatory variables. Z, may consist of sub-matrices with the block-diagonal form illustrated above (exploiting all 01. part of the rnonient restrictions available). concatenared to straightforward one-column instrunients. A judicious choice of the Z , matrix should strike a compromise between prior knowledge (from economic theory and previous enipirical work). the characteristics of the sample and computer limitations (see Arellano and Bond, 199 1 for an extended discussion and illustration). For example. if a predeternlined regressor xIt correlated with the individual effect, is added to the model discussed above. i.e.

E(x,tt.,,) = O for s > t # 0 otherwise

E(xzt 11,) # 0

then the corresponding optimal Z, matrix is given by

Where the number of colunins in Zi is very large. computational considerations may require those columns containing the least informative instruments to be deleted. Even when computing speed is not an issue. it may be advisable not to use the whole his- tory of the series as instruments in the later cross-sections. For a given cross-sectional sample size (N), the use of too many instrunients may result in (small sample) over- fitting biases. When overlitling results from the number of time periods ( T ) becoming large relative to the number of individuals (:V). and there are no endogenous regressors present. these GMM estimators are biased towards within groups, which is not a seri- ous concern since the within-groups estililator is itself consistent for models with pre- determined variables as T becomes large (see Alvarez and Arellano. 1998). However. in models with endogenous regressors, using roo many instri~ments in the later cross- sections could result in seriously biased estimn~es. This possibility can be in\.estigated in practice by comparing rhr GMM and wirhin-croi~ps esti~llntcs.

The assumption of no serial correlation in the I , , , is essc~~tial for the consistency of~estici lato~.~ such as [hose consiclcrccl i r i tlw pl-~.\-ious cxainplcs. tr.hich instrur~lcnr rhe

7.3 Ecor~or~lc.rr-ic. ~ ~ r r r h o d s , f i ~ r l ~ ~ ~ a ~ r r i c pa~lel data 111odc.l~ 69

lagged dependent variable with fulther lags of the same variable. Thus. PcGive re- ports tests for the absence of first-order andsecond-order serial correlation in the first- differenced residuals. If the disturbances qt are not serially correlated, there should he evidence of significant negative first-order serial correlation in differenced residuals (i.e. i?,;,t - Cl i - l ) . and no evidence of second-order serial correlation in the differenced residuals. These tests are based on the standardized average residual autocovariances which are asymptotically N(0, 1) \.ariables under the null of no autocorrelation. The tests reported are based on estimates of the residuals in first differences. even when the estimator is obtained using orthogonal deviations.' More generally, the Sargan (1964) tests of overidentifying restrictions are also reported. That is, if Ah- has been chosen optimally for any given Z,, the statistic

is asymptotically distributed as a \' with asmany degrees of freedom as overidentifying restrictions. under the null hypothesis of the validity of the instruments. Note that only the Sargan test based on the two-step GMM estimator is heteroscedasticity-consistent. Again, Arellano and Bond ( 199 1 ) provide a complete discussion of these procedures.

Where there are instrunients available that are uncorrelated with the individual ef- fects 7 1 ~ . these variables can be used as instruments for the equations in levels. Typically this will imply a set of moment conditions relating to the equations in first differences (or orthogonal deviations) and a set of moment conditions relating to the equations in levels. which need to be combined to obtain the efficient GMM e~t imator .~ For ex- ample. if the simple AR( I) model considered earlier is mean-stationary, then the first differences will be uncorrelated with q,. and this implies that A y , ~ - l can be used as instruments in the levels equations (see Arellano and Bover. 1995 and Blundell and Bond. 1998 for further discussion). In addition to the instruments available for the first-differenced equations that were described earlier, we then have:

Equations Instruments available yt3 = ay,2 + 71, + U t 3 A Y , ~

' ~ l l h o t ~ ~ h the validity of orthogonality conditions is not affected. the transformation to ortho- gonal dc.\.intions can induce serial correlation in the transfornied error term if the c,, are serially uncon-clntcd hut lle~sroscedastic.

'111 \(wc~i:tl c.;~ses il n1;ty efficient to use onlv [lie equatio~is in levels; for example, in a model 14 i t l i 11o 1a:gc.d depcl~rlent ~uiahles and all le~~-osso~.s str-ictl!. exogenous and uncorrelated with l l l d l \ j11tl;ll ~.Ifects.

7 0 Cl~aj>rer 7 Purlel Data Models 7.3 Econorner~-ic n~etl~ods fwdy~ranric purlel data models 7 1

Notice that no instruments are available in this case for the first levels equation (i.e., Y I ~ = a911 + 7~ + ~ ~ 2 ) . and that using further lags of Ap,, as instruments here would be redundant, given the instruments that are being used for the equations in first differences. In a balanced panel. we could use only the last levels equation (i-e., YIT = ~ Y I . T - ~ + v, + u,T). where (Ap,2, Apt:+, ..., A p z , ~ - l ) would all be valid instru- ments: however this approach does not extend conveniently to unbalanced panels.

In this case, we use

and

where Z? is the matrix of instruments for the equations in first differences. as described above. Again Z, would be precisely the same if the transformed equations in y: and W: were in orthogonal deviations rather than first differences. In models with explan- atory va,riables, it may be that the levels of some variables are uncorrelated with gi, in which case suitably lagged levels of these variables can be used as instruments in the levels equations, and in this case there may be instruments available for the first levels equation.

For the system of equations in first differences and levels, the one-step estimator computed in PcGive uses the weighting matrix

where HP is the weighting matrix described above for the first differenced estimator, and I, is an identity matrix with dimension equal to the number of Ievels equations ob- served for individual z. For the system of equations in orthogonal deviations and levels, the one-step estimator computed in PcGive sets H, to an identity matrix with dimension equal to the total number of equations in the system for individual i. In both cases the corresponding two-step estimator uses Hz = GlG:'. We adopt these particular one-step weighting matrices because they are equivalent in the following sense: for a balanced panel where all the available linear moment restrictions are exploited (i.e., no columns of 2, are omitted for computational or small-sample reasons), the associated one-step GMM estimators are numerically identical, regardless of whether the first difference or orthogonal deviat~ons transfornlation is used to construct the systen~. Not~ce though that the one-step estimator is asyniptotically inefficient relative to the two-step estlni-

ator for both of these systems, even if the prIt are hon~oscedastic.' Again simulations have suggested that asymptotic inference based on the one-step versions may be more reliable than asyn~ptotic inference based on the two-step versions, even in moderately large samples (see Blundell and Bond, 1998).

The validity of these extra instruments in the levels equations can be tested using the Sargan statistic provided by PcGive. Since the set of instruments used for the equa- tions in first differences (or orthogonal deviations) is a strict subset of that used in the system of first-differenced (or orthogonal deviations) and levels equations, a more spe- cific test of these additional instruments is a Difference Sargan test which compares the Sargan statistic for the system estimator and the Sargan statistic for the comsponding first-differenced (or orthogonal deviations)estimator. Another possibility is to compare these estimates using a Hausman (1978) specification test, which can be computed here by including another set of regressors that take the value zero in the equations in fust differences (or orthogonal deviations), and reproduce the levels of the right-hand side variables for the equations in levels6 The test statistic is then a Wald test of the hypo- thesis that the coefficients on these additional regressors are jointly zero. Full details of these test procedures can be found in AreUano and Bond (1991) and Arellano (1995).

'with levels equations included in the system. the optimal weight mamx depends on d o w n aaameters (for example, the ratio of var(q,) to var(v,t)) even in the homoscedasric case.

%us in the AR(l) case described above we would have

8.2 Data or~a,?i;atio~l 73

Chapter 8 Tutorial on Static Panel Data

Modelling

Year I F-I C-I Firnt 1935 317.6 3078.5 2.8 1

etc. 8.1 Introduction

GiveWin does not have special facilities for panel data management, and this is the The example for static panel modelling is based on Gmnfeld (1958), which is also used general format for all data sets. Although it is not necessary, it is useful to have an in Baltagi (1995, Ch. 2). The estimated model is an investment equation with N = 10 additional variable which indicates the finn or individual by a number, see 59.2. Figure and T = 20: 8.1 shows the data.

where

lit current gross investment of firm i, Ci,t-l beginning-of-year capital stock of firm i, Fi,t-l value of outstanding shares at beginning of year.

Writing uit = qi + vit corresponds to (7.1):

where q, -is a firm-specific effect, and vit IID across individuals (not serially correlated, but it may be heteroscedastic), x,.t may or may not be correlated with tii.

8.2 Data organization

The data set is provided as grunf eld-xls. which is in tl~e PcGive folder. Load this in Givewin. The database is ordered by firm, and within firm by year:

Figure 8.1 Gmnfeld(l958) investment data.

Here we have added the Firni variable using the insample command: firm = insample( 1.1, 20,l) ? 1; firm = insample( 21,1, 40.1) ? 2; firm = insample( 41,1, 60,l) ? 3; firm = insample( 61,1, 80.1) ? 4; firm = insample( 81,1,100,1) ? 5; firm = insample(101,1,120,1) ? 6; firm = insample(l21,1,140.1) ? 7;

74 Chapter 8 Tuforial on Slorir Porl~l Dmrr h . l o ~ l ~ / / ; ~ ~ ~ 8.3 Sroric panddata csriinarioll 7 5

firm = insample(l41.1.160.1) ? 8; firm = insample (161.1.180,l) ? 9;

nlodel \~ i t h I as dependent variable and F-1 and C-1 as regressors. The intercept be

firm = insample(181,1.200.1) ? 10; added in next step. PcGive must know about the panel StmCtUre of the data. In lhis

OLS estimation of (8.1) can be done using the Econometric ~ ~ d ~ l l i ~ ~ case. the easiest way is to define the Year variable: add Ymr to the model. and n'ak it package: as such selecting it in the model. and clicking on R: Year The model then is:

EQ( 1) Modelling I by OLS (using grunfeld.xls) The estimation sample is: 1 to 200

Coefficient Std.Error t-value t-prob Part.R-2 Constant -42.7144 9.512 -4.49 0.000 0.0929 F- 1 0.115562 0.005836 19.8 0.000 0.6656 C- 1 0.230678 0.02548 9.05 0.000 0.2939

sigma 94.4084 RSS 1755850.48 R-2 0.812408 F(2.197) = 426.6 [O.OOO]**

log-likelihood -908.015 DW 0.358 no. of observations 200 no. of parameters

3 mean(1) 145.958 var(1) 46799.7

8.3 Static panel data estimation We now assume that you have loaded the grunfeld. x l s data set into GiveWin and have started PcGive. In PcGive. select Panel Data Models (DPD) from the Package menu:

Click on OK, to see the Model Settings dialog. Switch off robust standard errors for the duration of this chapter:

On the Model menu, select Static Panel Methods:

Optlnns Ak+O

Next. select Formulate finnl the Model lllellu (or click on llle ~c)olbar icoll: rrlcctinp Static Panel Methods lion1 tllr Model nwnu will go tl~clr directly ). and fom~l~lrtc the

r'anstant T ~ m e Group T ~ m e and Group

The default adds a constant term, so keep that. and press OK again:

76 Clznipter- 8 Turorial ort Static Partel Data Modellirtg

mimum likelihood estimation

Starting with pooled regression replicates the earlier results:

DPD( 1) Modelling I by OLS (using grunfe1d.xl.s)

Coefficient Std.Error t-value t-prob F-1 0.115562 0.005836 19.8 0.000 c-1 0.230678 0.02548 9.05 0.000 Constant -42 -7144 9.512 -4.49 0.000

sigma 94.4084 sigma-2 8912 -947 R-2 0 .a12408 RSS 1755850.4841 TSS 9359943.9289 no. of observations 200 no. of parameters 3

Transformation used: none

constant : yes time dummies: 0 number of individuals 10 (derived from year) longest time series 20 11935 - 19541 shortest time series 20 (balanced panel)

Wald (joint): Chi-2(2) = 853.2 [0.0001 ** Wald(dummy): Chi'2(1)= 20.17[0.000~ ** AR(1) test: N(0,1)= 28.49[0.000] ** AR(2) test: N(0,1)= 23.4710.0001 **

The estimated coefficients and equation standard error (listed as sigma) are identical. Based on the Year variable, PcGive has also listed some information on the panel, namely IV = 10, T = 20 and that the panel is balanced. The first Wald test is for the sigl~ificance on all variables except the dummy (which is the constant term), so is the \' equivalent to the overall F-test. The next Wald test repons the significance of the constant lerm, and is just the squared t-value here. The two remaining tests are for

?A

S.3 Static /~arleldata estinzatiorz 77

first and second order serial correlation, which are both significant when one variable is involved.

ISR1 1-j k,,,

78 Cl~apter 8 Tutorial or1 Static Purlel Data Mndclling I

same coefficient estimates. The R' is different for within-groups, because it is based on the dependent variable after removing the individual means.

Table 8.1 Static panel estimates for Grunfeld data.

P1 02 roo OLS (pooled regression)

OLS 0.115562 0.230678 -42.7144 (0.005836) (0.02.548) (9 512)

OLS on differences OLS-diff 0.0897625 0.291767 -1.81889

(0.008364) (0.05375) (3.566)

LSDV (fired effects) . LSDV 0.110124 0.310065 -70.2967

(0.01186) (0.01735) (19.71)

Within-groups esrin~ariorl Within 0.110124 0.310065

(0.01186) (0.01735)

Bemeen-groups estinlation Between 0.134646 0.0320315 -8.52711

(0.02875) (0.1909) (47.52)

GLS (using withirdbemeen) GLS(w/b) 0.109781 0.3081 13 -57.8344

(0.01049) (0.01718) (28.90)

GLS (using OLS residuals) GLS 0.109734 0.307647 -57.6546

(0.01029) (0.01724) (26.46)

Marirnunt likelihood estimation ML 0.109763 0.307942 -57.7672

(0.01042) (0.01720) (27.91)

Chapter 9

Thtorial on Dynamic Panel Data Modelling

9.1 Introduction The applications in this chapter use the data from Arellano and Bond (1991). consisting of an unbalanced panel of 140 UK companies. The variables in the database are:

IND industry indicator YEAR Year EMP employment WAGE real product wage CAP gross capital INDOUTPT industry output 11 nit = log(EMP) W wit = log(WAGE) k kEt = log(CAP) vs yot = log(iND0UTPT)

The data are provided as abdata. in7labdata.bn7. Arellano and Bond (1991) consider dynamic employment equations of the form

where x:., = (wEt, kZt, ya). Time dummies are included to capture agega t e demand shocks.

9.2 Data organization The layout of a panel data set was introduced in 58.2. This section elaborates.

Note that we generally use the word it~di~lidual to denote the cross-sectional unit (which could be individual, firm. country, etc.).

80 Cl~olncr 9 fitroriol orr q\'rrotlric Parrcl DOIO Modcllirrg i

Because PcGive needs to recognize the structure of the data. and to allow for unbal- anced panels, the data must be organized prior to analysis:

Each variable is in a colunrrr. the columns are ordered by irrdividual. and within individual by time. Each r o w v refers to the same time period. a column with individual irrdex is recommended (when omitted, the year variable is used to create such an index). There must be one column which denotes the year. This does not have to be the first column. An optional period column may be present when the data is not annual (i.e., the frequency is not one). An optional group column may be present to facilitate creating group dummies.

For example:

vear irzdex employment wages , ~ -

1973 1 40 12 ,? " ii 1 indiv'a~ 197 1 520 1973 510 1973 9 5 10 14

Once the data set has the appropriate format, PcGive can determine where individu- als start and finish by differencing the irrdex variable. When the irzdex variable is not specified, PcGive will use the w a r variable instead. This is done by differencing the year variable. and will exclude subsequent individuals where the year of the last obser- vation on an individual is immediately after the year of the first observation on the next individual (which cannot happen with balanced panels. and may be a rare situation in unbalanced panels).

9.3 One-step GMM estimation We start by replicating Table 4 column ( a l ) from Arellano and Bond (1991. p. 1-90). The lag structure of x,.t is one lag of ujZt and two lags on kLt and y,t. The instrument matrix has the form:

9.3 Orre-srep GMM esrinrariorz 8 1

\ Ax:,, dunimies 1 0 0 0 0 0 - - - nil . - - nip /

AX:, duninues dumniies

Z, =

The first two columns are normal instruments, the remainder are GMM-type instru- ments.

7t11 1Zz2 0 0 0 . - - 0 . a - 0 0 0 T Z , ~ ~z,:] - - - 0 -.. 0

. (9.2)

TO formulate model (9.1) with instruments (9.2), first select Panel Data Models (DPD) from the Package menu. and then Dynamic Panel Methods from the Model menu. Formulate the model as described above, not forgetting to add the Year variable. In the panel data package, instruments must be marked as such explicitly. So, select a11 the variables wf to y s 2 in the model, and click on I: lnstrument and X: Regressor. The dialog should look like:

The difference between the two typesof instluments is as follows: I: lnstrument will be differenced when the model is estimated in differences, whereas L: Level instrument would use them unchanged.

Click on OK, to niove to Functions d~alog:

Clta,~cr 9 Tutorial on Dsi~ainic- Partel Doto Mvdelliltg I

If no GMM-type instrumenti are required, just leave this empty and press OK. To i add the GMM-type instrumens according to (9.2) select n in the database, set Lag1 to ; i 2 and Lag2 to 99, then press Add to see:' 1

In the model settings add a constant and time dumnues to the model, select Differ- ences to estimate the model in first differences:

l ~ c r e . 99 jc used as an upper limit. so that not more lhan 99 lags u e uad . In h i s c m the ac,ual llnnel limit is lower. but in general when T gets iwgcr. [he large number of instruments

-can_become a problem.

then scroll down in this dialog, and switch Use robust standard errors on (this was switched off in the previous chapter). Press OK and accept the default of I-step estim- ation:

The output is: DPD( 1) Modelling n by 1-step (using abdata.in7)

---- 1-step estimation using DPD ---- . Coefficient Std.Error t-value t-prob

Dn(-1) 0.686226 0.1446 4.75 0.000 Dn (-2) -0.0853582 0.05602 -1.52 0.128 Du -0.607821 0.1782 -3.41 0.001 Du(-1) 0.392623 0.1680 2.34 0.020 Dk 0.356846 0.05902 6.05 0.000 Dk(-1)

-0.0580010 0.07318 -0.793 0.428 Dk(-2) -0.0199475 0.03271 -0.610 0.542 DYS 0.608506 0.1725 3.53 0.000 Dys(-1)

-0.711164 0.2317 -3.07 0.002

Dys (-2) 0.105797 0.1412 0.749 0.454

Constant 0.00955445 0.01029 0.929 0.353 T1980 0.00290613 0.01580 0.184 0.854 T1981 -0.0433441 0.01700 -2.55 0.011 TI982 -0.0248388 0.02027 -1.23 0.221 TI983 -0.00381603 0.02194 -0.174 0.862 T1984 0.00406269 0.02190 0.186 0.853

sigma 0.1239809 sigma-2 0.01537128 sigma levels 0.08766777 FLSS 9.145908871 TSS 12.599978399 no. of observations 611 no. of parameters 16 Using robust variance matrix!

Transformation used: first differences Transformed instruments: v v(-1) k k(-1) kc-2) ys ~s(-l) ys(-2) Level instruments: Dummies Gmrn

86 Cl?nl~fer 9 firtor-jal CHI Dvrmmic Par~el Data Modeiiirtg I 9.6 Conlbiried GMM rstirnatiori 87

Here. A n , . t - l is treated as endogenous. and S I Z , . ~ - : ~ added as additional inst ment. As before. the model includes time dummies and is estimated in it has no GMM-type instruments. The results from I-step estimation match Table 5(e) (up. cit.).

The second Anderson-Hsiao type estimator has i z , . , - ~ as additional instrument i stead of An, . t -3 . Therefore, mark this variable as L: Level instrument. This ens that the untransformed variable is used as additional instrument. Again, 1:step est tion reproduces Table 5(f) (op. cit.).

Arellano and Bond (1991) implemented the within-group estimator as OLS aft applying the orthogonal deviations transformation. In unbalanced panels this is slight different from estimation in deviations from the means of each individual (but asym totically identical). A comparison using PcGive confirms that the difference is small.

9.6 Combined GMM estimation

PcGive also allows estimation using the combined GMM-SYS estimator proposed Arellano and Bover (1995) and Blundell and Bond (1998). In this type of estimatio the level equations are stacked on top of the transformed equations.

Blundell and Bond (1998) consider a dynamic employment equation of the form

using the same data set as Arellano and Bond (1991): I

The GMM-type instrunients for the differenced equations are

diag(11i.l . . . 12i.t-2, w i . 1 . . . u ' , , ~ - ? , k i , l . . . k i , , - 2 ) .

which is forniularcd as:

The GMM-style instruments in the levels equation are the lagged differences:

These are added in the same dialog. Note that now the first argument is the lag length, and the second whether to use differences (1) or not (0):

The presence of GMM-level instruments will automatically result in combined estima- tion. At the Model Settings, add time dummies, and estimate in differences by I and 3- step estimation. The '-step results are:

---- 2-step estimation using DPD ---- Coefflclent Std.Error t-value t-prob

Dn(-1) 0.872881 0.04528 19.3 0.000 Dw -0.779745 0.1166 -6.69 0.000 Dw(-l) 0.526803 0.1621 3.25 0.001

88 Chal~ter. 9 firtorial or1 Dyr~arnic Par~cl Data Modelling i !

Dk 0.470077 0.07986 5.89 0.000 Dk(-1) -0.357608 0.08003 -4.47 0.000 Constant 0.948489 0.3776 2.51 0.012 TI978 0.00580177 0.01971 0.294 0.769 T1979 0.0188976 0.02277 0.830 0.407 TI980 0.00281961 0.02407 0.117 0.907 TI981 -0.0200226 0.02744 -0.730 0.466 I TI982 0.0152802 0.02331 0.656 0.512 TI983 0.0317310 0.02350 1.35 0.177 T 1984 0.0224205 0.03107 0.722 0.471

sigma levels 0.1276452 sigma-2 levels 0.01629331 RSS 14.305523988 TSS 1601.0425015 no. of observations 891 no. of parameters 13 Using robust variance matrix!

Transformation used: first differences CMM-SYS estimation combines transformed and level equations

Instruments for transformed equation: Level instruments: Gmm(n,2,99) Cmm(v,2,99) Cmm(k,2,99) Instruments for level equations: Level instruments: Dummies CmmLevel(n, 1.1) GmmLevel(v. 1,l) CmmLevel(k,l,l)

constant: yes time dummies: 7 number of individuals 140 (derived from year) longest time series 7 [I978 - 19841 shortest time series 5 (unbalanced panel)

Wald (joint): Chi-2(5) = 3378. C0.0001 ** Wald (dummy): Chia2(8) = 28.51 C0.0001 ** Wald (time): Chie2(7) = 16.00 C0.0251 * Sargan test: Chi^ 2(100) = 111.6 C0.2011 ARC11 test: N(0.1) = -5.807 C0.0001 ** AR(2) test: N(0.1) = -0.1471 C0.8831

These results do not exactly match the last column of Blundell and Bond (1998, Table 4) for several reasons:

( I ) We report 2-step results instead of 1-step results. (2) The '-step results in PcGive now use the estimated variance matrix incorporating

the small-sample correction of Windmeijer (2000). (3) In Blundell and Bond (1998). the differenced dummies were used as instruments

in the transformed equations, and the H, matrix was set to the identity matrix.

Chapter 10

Panel Data Implementation Details

This section gives a summary of the statistical output of Pa ive , giving the formulae which are used in the computations.

10.1 Transformations none

x:t = x i t , t = I , . . . , T i . first differences

x : ~ = Axit = - xi.*-1, t = 2 , . . . , Ti, xfl = Axil = 0.

time means: 1 Ts 3.-- - z x i . .

Ti s=l deviations from time means

orthogonal deviations

The orthogonal deviations are stored with one lag, so that the first observation is lost instead of the last (this brings it in line with first differencing):

GLS deviations: x t = x - 6 t = 1 , . . . , T, ,

whew the choice of 8, determines the method, as discussed in the next section.

90 Cltapter- 10 Pajlel Data 1111l~le111e11tatiou Details 10.3 D>vza~nic panel data estimation 9 1 ! 10.2 Static panel-data estimation I

E The starting point is (7.2):

y = w p + v . The total number of observations is 0 = N T . In an unbaIanced panel, different indi- viduals may have a different number of observations, TI. In that case the total number of observations equals 0 = 1 T,. Baltagi (1995) reviews the standard estimation methods used in this setting:

The OLS estimates are:

3 = (wlw)-I W'y, O [3] = a: (wlw)-l , -2 'Ju = GIG/ (n - p ) , -

v = y-wa.

Here W has k columns, containing all specified regressors including dummies, s o p = k. The LSDV(1east squares dummy variables) estimates use individual dummies. The within estimates replace y and X by deviations from time means. The betweer1 estimates replace y and W by the individual means. The feasible GLS estimates replace y and W by deviations from weighted time means. The outcome depends on the choice of 0, which in PcGive is set by specifying a: and a::

When OLS residuals u = Gats are used, the GLS estimator can be based on:

-2 1E1 it - u,, = O - N >

PcGive computes 8, from 0: and 0: with the latter derived from 8; as:

In a balanced panel: iV /0 = 1/T. This is not optimal in an unbalanced panel, but seems reasonable. The standard feasible GLS estimator uses the between and within estimates:

2 - -2 0 I . l - ~ t % ~ l , / , l l ~ ~

nf = ~ , a ' l hrnrr,r,r~ '

For convenience, 8, is specified using three variance components a:, 02, and a;:

0; 0: u; OLS based GLS -2 -2 -2 0, 0, u n betweedwithin based GLS s : . ~ , ~ ~ ~ ~ 0

The ~~~axinlurn likelihood estimates obtain 0 by iterating the GLS procedure. The concennated likelihood is:

where T = U;/O:, SO that:

To summarize the implementation in PcGive:

number of degrees of freedom transforms observations, n lost in estimation, p dummies

OLS (no transformation) 0 k no within 0 k + N no between N k Yes GLS, ML 0 k Yes

It is important to note that the within transformations are only applied to X, and not to the dummies D. In the within estimator, individual dummies are redundant after sub- tracting the individual means. In the between estimator, time dummies are irrelevant, and individual dummies create a perfect fit. In GLS and ML, the individual means are subtracted with weights.

-.

10.3 Dynamic panel data estimation The single equation dynamic panel model can be written as:

s=l

As before. At and rIi are time and individual specific effects and x, is a k* explanatory variables. It is assumed that allowance for lagged observatio

92 Ci~apter I0 Par~el Darn fr~r~~lemr1iraric7n Details

made, so observations on y ,,-,, . . . , y,.- I are available. The total number of observ tions available for estimation equals 0 = C TI.

Stacking the data for an individual according to time: and using D for the matri with dummies:

y i=Xiy+DiS+v1 , i = O ,..., N - 1 . In formulating the model, we distinguish the following types of variables: I

y dependent variables, X regressors, including lagged dependent variables, D dummy variables, I normal instruments, L 'level' instruments, G GMM-style instruments.

The G, are described under the Gmm function. From these variables, the dependen variable, regressor and instrument matrices are formed as follows:

dimension ql = y f TI x 1 W, = [Xf : D:] Ti x li Zi = [Gi : If : Dg : L,] Ti x 2

where the * denotes some transformation which can be chosen from 3 10.1. This als& I affects the degrees of freedom, as shown in Table 10.1. The estimators and tests are! described in Arellano and Bond (1991), and have the form:

When the option to concentrate out dummies is used, y*. X', I*, and L are re- placed by the residuals from regressing on the set of dummies DS. Subsequently, the dummies are omitted from further computations.

In one-step estimation, HI is the identity matrix, except when estimating in first differences:

1 -112 0 . - . 0 -112 1 -112

H I ., = IT, , except: H?.: = 0 -1/2 1 ( 10.4)

10.3 Dyr~atnic parici data estirnatiorr 93

Table 10.1 Transformation and the treatment of dummy variables. transfom~ation number of degrees of freedom transforms dummies

observations, n lost in estimation, p differences 0 - N li Ds = D (optional: D*) orthogonal deviations 0 - 1V k Ds = D (optional: D*) none 0 k no: DS = D within 0 k + N no: DS = D* between N k yes: DS = D* GLS 0 k yes: DS = D*

In two-step estimation Hi is based on the previous step residuals: Hi M AN 6

one-step Hl.i, see (10.4) M1 A1.h~ 51 . i A = s;,i -* two step = 0; ,3;[, M2 A ~ . N ~ 2 . i = V2,i

with a subscript added to distinguish between one and two step estimation. The output reports the residual and total sum of squares:

RSS = GIG, TSS = qc- ( 3 ~ ) ~ 10.

So the TSS is in deviation from the mean. The one-step estimated variance of the coefficients is:

If selected, robust standard errors after one-step estimation are computed as:

When there are no instruments, (10.5) simplifies to:

( w l w ) - I (2 w:G,.:wz 1 ( w l w ) - I , which is the robust variance estimator proposed in Arellano (1987)~' The two-step

'This robust estima~or 1s nor ldentd to the standard White (1980) estimator for OLS, which uses as the central matnx:

N T C w:G,G,w,. , = I

asyn~ptotic variance matrix is:

This variance estimator can be severely downward biased in small san~ples, and is there- fore only reported when robust standard errors is switched off. The preffered solution (and the default) is to use the robust variance for two-step estimation, 92,.[3]. which uses the small sample correction as derived by Windmeijer (2000).

The AR test for order m is computed as:

Using w, for the residuals lagged m periods (substituting zero for missing lags): I

w,t = ~ , . t - ~ f o r t = m ,.... T,, and wzt = O f o r t < m . The d, are defined as:

do = C , w:u,, i dl = C , w:H,wt, i d2 = -2 (C, w:Wi) M-' (C, W:Z,) AX (C, Z:Hzwt), (10.7) i d3 = (C , w;wl) c [z] ( C , w;wl) - I I

1

The components are used as the notation suggests, except for Hz in one-step estimation: j Hi -2 M AN ui VIP] one step: ul,uHl.i Mi A ~ , N GF., Vl[,Z]

one step, robust: Hz., MI A ~ , N G;.i clr[PI] two step: H2.t M2 A ~ . N Gz.i c2[3] two step, robust: H2.i M2 A2.x G;., c2r[s ]

After estimation in orthogonal deviations, the AR test (10.6) is based on the residuals from the differenced model as follows (the superscript A refers to the regressors and residuals from the model in differenced form):

H: is H:!: after one-step estimation, and u:'u: otherwise. !P, equals

after one-step estimation, and u , u ~ ' w ~ after one-step robust or two-step estimation. \

The Sargan test with 2 - k degrees of fieedom is after one-step estimation:

and after two-step estimation (see 7.4 andequation (10) in Arellano and Bond, 1991):

Three Wald tests are routinely reported to test the significance of groups of vari- ables:

Wald (joint): on W, all regressors except dummies, Wald (dummy): on D, all dummies (including constant), Wald (time): on time dummies (including the constant in the

differenced/deviations model).

10.4 Dynamic panel data, combined estimation In combined estimation, GMM-SYS, the levels equation is used jointly with the trans- formed equation, see Blundell and Bond (1998). The IV estimation of (10.3) still ap- plies, but the data are organized as follows:

dimension

2T, x 1

X: D: wz = [:] = [ X D , ] 2T, x k ] 2Tz x ( r + zt)

G: are the GMM-style instruments for the levels equation, described under the GmmLevel function.? The dummies in the transformed equation are always trans- formed. When using differences or deviations, the effective number of observations in the transformed equations is as before (0 - AT), whereas the levels equations have 0 observations. Internally, both the transformed and levels equation i has Ti equations, but, when using differencesldeviations, the first observeion in the transformedtquation is set to zero, resulting in T, - 1 effective observations.

When the option to concentrate out dummies is used. y*, X*. I*, and L re-

placed by the residuals fronl reg~ss ing on the set of dun~nlies D*. and y and X on D-

he code used to estimate Blundell and Bond (1998) had 2; = (G , O I: D: L,)-

Clrapter. 10 Pnrlel Dnfa larple~~~c~~far io~~ Details

: that there is one time dumn~y extra in combined estimation compared to normal nation when using differences or deviations. After estimation, there are two sets of residuals:

2 = q' - ~ ' 3 (transformed equation), G+ = q+ - W+ 3 (levels equation).

L reported residual variance is based on the levels residuals: 2; + = Ot'pt/ (n - p). re n = 0 = xj"=, Ti and p are as before. The reported RSS and TSS are also based the levels, using G+ and q+. In onestep estimation, H, is as in (10.4) in the transformed equations, and the

mtitv matrix in the level equations (1/?I when using differences).' Only robust - - ,

~ndard errors are reported after one-step estimation. In two-step estimation. Hi is based on the previous step residuals 6; = (8:' :

;I).

The AR test (10.6) is based on the residuals from the transformed equation:

\her orthogonal deviations, the * is replaced by -' as in (10.8).

10.5 Panel batch commands t ! estimate ("mefhodU=OLS, year1 =-I, periodl=O, yea)-2-1, period2=0);

Estimate a system. Note that for panel data all sample arguments are ignored: i only the estimation rllerhod is used. 1

The nlerhod argument is one of:

If 1-step" 1-step estimation, "2-step" 1 and 2-step estimation. "OLS" OLS. " LSDV " Least-squares dummy variables, "Within" within-groups estimation. "Between" between-groups estimation. "GLS (w/b) " GLS (using withinhetween). "GLS" GLS (OLS based). "ML" Maximum likelihood.

'rile code used roestlnlarr In Hlundell and Bond (1998) had HI., = 11; also lo1 e\tlrnatlon in differences.

option ("opriorl", argrrnte~ll); option orgunlent value

! 1 robust Use robust standard errors 0 for off. 1 on, concentrate Concentrate dummies 0 for off, 1 on,

I (not exact with instruments) t r ans f ormdummy Transform dummies 0 for off, 1 on,

(differences/deviations) printgmm Print contents of GMM instruments 0 for off, 1 on, maxi t maximum number of iterations default: 1000, p r i n t print every # iteration 0: do not print, compact compact or extended output 0 for off, 1 on, strong strong convergence tolerance default: 0.005, weak set weak convergence tolerance default: 0.0001,

Lists the dummy variabIes to be added The arguments are separated by a comma and can be: "constant", "time", "group", "timegroup", "individual".

settest(ispec, iar) ; Sets the test options. The first argument is 0 or 1, and determines whether spe- cification tests are printed. The second argument sets the order for the AR test; the default is 2.

Lists the desired transformation. The argument can be: "differences", "deviations", "within", "between".

Use this command to store residuals, etc. into the database, the default name is used. Note that if the variable already exists, it is overwritten without warning. The name must be one of:

res iduals residuals resdif f differenced residuals (after combined estimation) resleve 1s levels residuals (after combined estimation)

system{Y= ... ; X= ... : I= ... ; L= ...; R= ... ; N= ... ;G= ... ; ) e Specify the system, consisting of the following components:

Y endogenous variable: X non-modelled X variable; I Instrument;

L Level instrument. R Year variable. N Index variable. G Group variable.

98 Cl~avtcr- I0 Par~el Data In~~~lerr~cr~rarior~ Derails

The variables listed are separated by commas, their base names (that is, name excluding lag length) must be in the database. If the variable names are not a valid token, the name must be enclosed in double quotes.

testlinres { . . . ) Test for linear restrictions. The content is the matrix dimensions followed by the (R : r ) matrix.

Part V

Time Series Models (ARFIMA) with Marius Ooms