ࡱ; u!F"  &$#%45'()*+,-./0123s7689:?;<=>@ABDCEHGdIJKMLNOPQRSTUVXWYZ[]\^_`abcefgihkjmlnopqrstuvwxyz{|~}Root Entry F<&#@CompObjbWordDocumentObjectPoolE4&#E4&# 4@  !#$%&'()*,/2456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUWZ[\]^_`adfghijklmnopqrtwz|}~ FMicrosoft Word 6.0 Document MSWordDocWord.Document.6; ࡱ; L 4p@ࡱ;  4J 9 .1  &@ & MathTypePTimes New Roman-ܥe3 enTP\\\\\pv(\1 4֖֖֖֖֖֖>@@@)iTDc֖,?֖֖֖֖֖֖֖֖֖֖֖֖>Ƌ8^8T^֖>֖h֖6. Bias when slope parameters are heterogeneous, in the cases of stationarity and non-stationarity Suppose we have heterogeneity in groups, manifesting in different slope coefficients across groups. We will look at both static and dynamic models. Static model Suppose we are interested in the average effect of the regressors on the dependent variable.  EMBED Equation.2  (33) where the heterogeneity in slopes across individuals takes the form  EMBED Equation.2  Other assumptions:  EMBED Equation.2  Estimators that might be used include: 1. Swamy GLS random coefficient estimator / Mean group estimator  EMBED Equation.2  (34) which takes weighted means of the estimate for each individual. If the weights are 1/N, the estimator is the mean group estimator discussed in Pesaran and Smith. [To use this estimator the individual parameters we really need a long time series: it is infeasible unless T is large. We henceforth consider cases where T is large. Panels with large T and large N are known as data fields.] 2. Cross-section estimator This is the between regression, involving averages over time of all of the variables:  EMBED Equation.2  (35) where  EMBED Equation.2  (and similarly for  EMBED Equation.2  and  EMBED Equation.2 ) (Note: in the panel context we can average over individuals or time; I use the notation ~ (tilda) for averages over time, and - (upper bar) for averages over individuals.) 3. Aggregate time series estimator  EMBED Equation.2  (36) where  EMBED Equation.2  (and similarly for  EMBED Equation.2  and  EMBED Equation.2 ) 4. Pooled regression which assumes common coefficients across individuals:  EMBED Equation.2  (37) In this case, because the individual slope heterogeneity is omitted, it gets pushed into the error term:  EMBED Equation.2  Nevertheless, the pooled estimator is consistent as long as all the regressors  EMBED Equation.2  are strictly exogenous and all parameters are random and distributed independently from the regressors, since the error term  EMBED Equation.2  (i.e. each of its components  EMBED Equation.2  and  EMBED Equation.2 ) is independent of the regressors  EMBED Equation.2 . The error  EMBED Equation.2  will be heteroskedastic (and serially correlated if  EMBED Equation.2  is serially correlated), so the estimates will be inefficient, but this does not effect unbiasedness and consistency. [The error terms in estimators 2. and 3. have the same properties. Estimator 1. might be efficient.] [If the coefficients are correlated with  EMBED Equation.2 , only 1. is unbiased. We henceforth assume no such correlation.] In a static model where regressors are strictly exogenous and the coefficients differ randomly and are distributed independently of the regressors (across groups), all four estimators give unbiased and consistent estimates of the coefficient means. Dynamic model In contrast to the static, model, not all estimators are consistent in the dynamic case, even when T is large. [We have previously considered biases which occur in dynamic panel models when T is small.]  EMBED Equation.2  (38) We are now (also) interested in the average long-run effect of the exogenous regressors on the dependent variable. Without heterogeneity in slope parameters, the long-run effect is defined as:  EMBED Equation.2  (39) With heterogeneity, there are two ways the long-run effect can be defined: we can either average the  EMBED Equation.2  and the  EMBED Equation.2  over individuals, then take the ratio as in (39) above, or we can take the ratio for each individual, and then average these individual long-run coefficients: Either  EMBED Equation.2  (40) or  EMBED Equation.2  (41) where  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  Random parameters can also be defined in corresponding ways: Either  EMBED Equation.2  or  EMBED Equation.2  where  EMBED Equation.2  = the mean lag. i.e. the parameter heterogeneity (here characterised as random variation across individuals) can be characterised as coming from either the short-run or the long-run coefficients. 1. Swamy GLS random coefficient estimator / Mean group estimator This class of estimators is consistent for large T when N is large. No matter whether we have  EMBED Equation.2  or  EMBED Equation.2 , by estimating separate time-series regressions for each individual and averaging (either unweighted, or using the GLS weights of Swamy), you can get consistent estimates of short- and long-run parameters. 2. Cross-section estimator Averaging (38) over time gives:  EMBED Equation.2  (42) This is the between regression for the dynamic model. It gives inconsistent estimates of b and l even when N, T EMBED Equation.2  because, when parameters are heterogeneous, the error is correlated with the lagged dependent variable:  EMBED Equation.2  [Correlation between coefficients and regressors will also lead to bias in the cross-section estimator.] But it can be shown that the cross-section estimator gives a consistent estimate of the mean of the long-run coefficients. (42) can be rearranged as  EMBED Equation.2  (43) where  EMBED Equation.2  and likewise for all the other tilda variables and  EMBED Equation.2  captures the difference between  EMBED Equation.2  and  EMBED Equation.2 :  EMBED Equation.2  is in  EMBED Equation.2  but not  EMBED Equation.2 , and (theoretically, at least)  EMBED Equation.2  is in  EMBED Equation.2  but not  EMBED Equation.2 . Rearranging (43) we get:  EMBED Equation.2  (44) The coefficients in (44) are in fact the correct long-run coefficients:  EMBED Equation.2  (45) The cross-section estimator, however, imposes homogeneity on the slope coefficients:  EMBED Equation.2  (46) where  EMBED Equation.2  The cross-section estimates of the average long-run coefficients are given by  EMBED Equation.2  (47) Although the growth terms  EMBED Equation.2  are omitted from (47), the cross-section estimates are nevertheless consistent, since these growth terms are in fact uncorrelated with the levels terms, so their omission does not affect consistency. PS note the implication that cross-section estimates will be robust to dynamic misspecification of the underlying micro-model, whereas aggregate or pooled estimates are not (note that if  EMBED Equation.2  had appeared in the original model, the omission of  EMBED Equation.2  from the estimator would similarly not affect consistency). PS show that the consistency of the cross-section estimator holds whether we assume randomness takes either form  EMBED Equation.2  or  EMBED Equation.2 . 3. Aggregate time series estimator  EMBED Equation.2  (48) where  EMBED Equation.2  The aggregate (macro) disturbances  EMBED Equation.2  are correlated with the regressors, so OLS on (48) will be biased and inconsistent even when N, T EMBED Equation.2 . Instrumental variables wont work, since anything that is correlated with the variables that we would wish to instrument ( EMBED Equation.2  and  EMBED Equation.2 ) will be correlated with the errors  EMBED Equation.2  through  EMBED Equation.2  or  EMBED Equation.2  (or, the other way round, anything that is uncorrelated with the macro disturbances will also be uncorrelated with the regressors). 4. Pooled estimators The pooled estimation of a dynamic model involves the following regression:  EMBED Equation.2  (49) where  EMBED Equation.2 . The error  EMBED Equation.2  will be serially correlated, since  EMBED Equation.2  is serially correlated. Serial correlation in the presence of a lagged dependent variable leads to inconsistent estimators (cf. the static case, where serial correlation did not matter much). [Note another problem with the pooled regression: if the regressors are I(1) and the (micro) relationship for each i cointegrates, we would hope the (aggregate) relationship cointegrated for all i. But the pooled regression will not cointegrate in these circumstances, since the error term contains terms in the I(1) regressors (and cointegration requires a stationary error).] Instrumental variables estimation (e.g. attempting to instrument the lagged dependent variable) is impossible: the regressors all appear in the error term  EMBED Equation.2 , so anything that is correlated with the regressors (which correlation is a necessary condition for a useful instrument) will also be correlated with the error (which will lead to biased and inconsistent estimates). In summary, average effects in a dynamic model with heterogeneous slopes cannot be estimated from a pooled regression even when N and T are large. Estimates are inconsistent even if  EMBED Equation.2 , so that the only heterogeneity comes from the coefficients of the exogenous regressors. In the case that  EMBED Equation.2 , the degree of inconsistency will depend on the serial correlation in the  EMBED Equation.2 . Suppose  EMBED Equation.2  Then it can be shown that there are massive inconsistencies in the pooled estimates: as  EMBED Equation.2 ,  EMBED Equation.2  (i.e. the estimates of the effect of the exogenous variables are biased towards zero)  EMBED Equation.2  (i.e. the coefficient on the lagged dependent variable is biased towards unity) even if the true  EMBED Equation.2  (Robertson and Symons result) even if the true  EMBED Equation.2  (Pesaran and Smiths generalisation) Clearly, the estimate of the long run coefficients will also be inconsistent. Both numerator  EMBED Equation.2  and denominator (1- EMBED Equation.2 ) tend to zero, so the long-run estimates will be computationally unstable. To be more specific:  EMBED Equation.2  (50)  EMBED Equation.2  (51) where  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2 , where  EMBED Equation.2   EMBED Equation.2 , where  EMBED Equation.2  If  EMBED Equation.2 . Then we can see the direction of bias by examining the right-hand terms in (50) and (51):  EMBED Equation.2  has large sample bias > 0 (i.e.  EMBED Equation.2 ) when  EMBED Equation.2  (RH term of (50) > 0) < 0 (i.e.  EMBED Equation.2 ) when  EMBED Equation.2  (RH term of (50) < 0)  EMBED Equation.2  has large sample bias < 0 (i.e.  EMBED Equation.2 ) always (RH term of (51) > 0) And for the long-run effect:  EMBED Equation.2  (52) where  EMBED Equation.2   EMBED Equation.2  has large sample bias > 0 (i.e.  EMBED Equation.2 ) when  EMBED Equation.2  (denominator of (52) < 1) < 0 (i.e.  EMBED Equation.2 ) when  EMBED Equation.2  (denominator of (52) > 1) = 0 (i.e.  EMBED Equation.2  unbiased)  EMBED Equation.2  [Note that the unbiasedness of the long-run coefficient when  EMBED Equation.2  does not hold when there are lagged exogenous regressors in the equation.] The larger the degree of individual heterogeneity, i.e. the larger is  EMBED Equation.2 , the larger are all the biases. Biases remain if lagged exogenous variables are included in the regression:  EMBED Equation.2  (52) Then as  EMBED Equation.2 ,  EMBED Equation.2   EMBED Equation.2  (i.e. the current and lagged exogenous regressors will tend to appear with equal and opposite coefficients)  EMBED Equation.2  even if the true  EMBED Equation.2  Application: Labour demand across UK industries (Pesaran and Smith, 1995) 1956-84 (29 years) - large T 38 UK industries - large N Model:  EMBED Equation.2  where  EMBED Equation.2  = log of industry i employment (in man-hours)  EMBED Equation.2  = log of industry output  EMBED Equation.2  = log of industry real product wage  EMBED Equation.2  = aggregate output All the above are I(1):  EMBED Equation.2 . PS are mainly interested in the long-run effects of the real wage and output (either total or own) on employment. 1. Swamy GLS random coefficient estimator / Mean group estimator PS estimate 2 versions of the labour demand functions, one static and one dynamic. The dynamic version takes the form of an autoregressive distributed lag:  EMBED Equation.2  - dynamic version (53) The long-run coefficients are defined as:  EMBED Equation.2  - w.r.t. the real wage  EMBED Equation.2  - w.r.t. total output  EMBED Equation.2  - w.r.t. own output PS compute the means of the long-run coefficients as in (40) and (41) above, i.e. as the long-run calculated from the mean of the individual industry short-run coefficients (Table 1, panel A(ii), which gives both Swamy GLS weighted and unweighted estimates), or the mean of the individual industry long-run coefficients (Table 1, panel A(i)). With the mean group estimator, it is also possible to obtain long-run coefficient estimates as the mean of the coefficients in individual industry cointegrating regressions, which take the form of the static regressions  EMBED Equation.2  - static version (54) (see PS Table 1, panel B). Recall, these estimators are all supposedly consistent. But in practice, there are noticeable differences between them. (a) To weight or not? The weighting doesnt seem to matter all that much. It affects the mean of the individual long-runs most (pan_917792479FE4&#E4&#Ole PIC LMETA  2 `Xy 2 `x 2 `=ik 2 `\N 2 `Gtk 2 `NT Times New Roman- 2 it>> 2 Bi> 2 {it>> 2 it>>Symbol- 2 `2= 2 dF^ 2 `+ 2 `Ub 2 `e Times New Roman-2  8888888888Times New Roman-2 `} `````````` 2 ` = 2 `,` 2 `@.` 2 `.` 2 `*.` 2 `,` 2 `,` 2 `' ``` 2 `= 2 `,` 2 `V.` 2 `.` 2 `@.` 2 `,` 2 `1 2 `31 & "System- ࡱCompObj ZObjInfo Equation Native _917792646 FE4&#E4&# FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ; ࡱ; _HG G y it ="b i x it +e it                     i=1,...,N,   t=1,...,Tࡱ; el A(i)), which is explained in practical terms by a large empirical variation in the individual industry estimates of the long-run coefficients, and a large variation in the accuracy of these estimates. (b) To use cointegrating equations or not? There is some doubt whether all the individual industry relations cointegrated. The mean ADF(1) statistic was -3.37. 2. Aggregate time-series estimator This involved estimation of (53) at aggregate level, with terms in own output dropped (see PS Table 2, panel A - note that own-output elasticity is not identified at the aggregate level), and the potentially-cointegrating equation, (54) at aggregate level, again dropping the term and coefficients relating to own output (Table 2, panel B). The aggregate estimates differ markedly from the mean of the individual industry estimates. Why? There is positive serial correlation in the regressors, and hence in the residuals of the aggregate equation. From the discussion above, we would expect this to bias the coefficient on the lagged dependent variable upwards, which would also bias the long-run estimate upwards. But the sum of the lagged dependent variable coefficients of 0.65 using the aggregate estimator is the same as that using the Swamy estimator (the dynamics differ a lot, though - for example, the pattern among the lagged dependent variables differs: 0.49+0.16 for the aggregate estimator vs. 0.79-0.14 for the Swamy). The estimates of the long-run coefficients from the aggregate cointegrating regression are close to the mean of those from the individual industry cointegrating regressions. But again the ADF(1) statistic of -3.23 casts doubt on the stationarity of the residuals from the cointegrating regression. 3. Cross-section estimator The cross-section estimator gives the following results:  EMBED Equation.2  where () give OLS unadjusted s.e.s and [] give Whites heteroskedasticity-consistent s.e.s. The coefficient on  EMBED Equation.2  is an estimate of the mean long-run own-output elasticity  EMBED Equation.2 . The cross-section estimates seem reasonable point estimates of the long-run coefficients, although the estimate for the real wage elasticity is not well determined. PS investigate the stability of the coefficients to changes in the number of time periods averaged to form the cross-sections, by estimating recursive cross-sections, adding one year from 1958 onwards. They find remarkable stability in the coefficients. But they emphasise that the cross-section estimator will only be consistent when coefficients and regressors are independent. Furthermore, the inability to compute the aggregate-output elasticity will be a disadvantage where macro effects on individual groups are important, e.g. because there are interaction effects between groups, e.g. as is possible in wage setting. 4. Pooled estimators Both static (54) and dynamic (53) versions are estimated using pooled regressions. The 3 different standard assumptions are made about the intercept:  EMBED Equation.2  constant  EMBED Equation.2  individual-specific random effects ( EMBED Equation.2  are the random components)  EMBED Equation.2  individual-specific fixed effects Estimates are highly specification-sensitive; results are erratic (see PS Table 3). PS Table 4 gives the fixed-effects estimates. They accord with the discussion above: the sum of the coefficients on the lagged dependent variable is roughly unity (1.16-0.21=0.95); the coefficients on the current exogenous variables are close to the (consistent) Swamy estimates; the sum of the coefficients on the exogenous variables is approximately zero (they have equal and opposite signs, the coefficient on the lagged value being biased towards minus the true coefficient on the current value). Following from this, the long-run coefficient will be the ratio of 2 numbers tending to zero (0/(1-1)), which is in practice clearly computationally unstable. The estimate of long-run output elasticity from the fixed effects estimator is 0.04004/(1-0.95406)=0.87 (s.e.=0.13). This is not too far from the Swamy GLS estimate of 0.61 (s.e.=0.11), but the latter is the ratio of 0.22/(1-0.64) - clearly the short-run estimates from the fixed effects estimator are biased as expected. PS note that the fixed effects estimator is biased anyway if T is not large (this is the bias we have discussed in earlier lectures). Here T=38, which should be large enough that any such bias is small. Just in case fixed-T biases remain, PS also use the Anderson-Hsiao estimator, differencing to remove the individual fixed effect and then instrumenting using the current and first two lags of the differenced exogenous variables and the 3rd and 4th lags of the differenced dependent variable. The coefficients on the current values of the exogenous variables are similar to those from the Swamy and pooled estimators (i.e. seem unbiased); the bias in the lagged exogenous variables does not seem so extreme as in the pooled case; but the coefficient on the lagged dependent variable seems seriously underestimated. What improvement there is, PS put down to the differencing having removed some of the serial correlation induced by incorrect pooling of the heterogeneous slope coefficients. PS also report OLS estimation of the differenced equation (i.e. no instrumentation), which gives very similar results to AH (i.e. instrumenting makes almost no difference) (compare Table 3, panel C(ii) (OLS) with C(iii) (AH)). Conclusions from Pesaran-Smith Aggregating or pooling dynamic heterogeneous panels can give very misleading results. The homogeneity assumption and the adoption of pooling techniques, are so prevalent in applied work that these are very important results. There is some support for the reasonably-commonly-held idea that cross-section estimates will give a good representation of long-run coefficients, immune to dynamic misspecification. The lesson PS wish us to draw is that, when we have long enough time series, we should ideally choose the mean group estimator, estimating individual regressions, then averaging resulting parameters. Homogeneity assumptions can then be tested (NB the size of the test should be adjusted to take account of the large number of observations). 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CompObjNativeceFZObjInfo0fFEquation Native _918297953iFS[&#S[&#5L@ ࡱ; *_ABlB 2y i =-m i D T y i +"q i 2x i +11-l i "e i ࡱ; ࡱ; L4t @ࡱ; OleObj18h0881414&18p8 8(24(24PICnfoNative hkLMETAon Native FHCompObj8jlFZ4fN  .1   & & MathTypePTimes New Roman- 2 R~ 2 > ~ 2 ~Times New Roman- 2 `Xy 2 `y 2 `> x 2 ` u Times New Roman- 2 i> 2 T| 2 i> 2  i> 2 i>Symbol- 2 `= 2 `/- 2 `[+ 2 dU ^ 2 d ^ 2 ` + 2 `mD 2 `m q & "System-fZM ࡱ;  FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ; ObjInfo3mFEquation Native _917798006tiveY)pFS[&#S[&#<Ole97620 F@ ࡱ; *΀_A&BX(B 2y i =-mD T y i +"q2x i +2u iࡱ; ࡱ; L ࡱ; >U 7 .PICObj7 orFLMETAfo CompObjNativeqs ZObjInfo0tF @    !"#$%&'()+./0356789:;<?BDEFGHIJKMPSUVWXYZ[]`cefghijlqstuvwxy~1  @&` & MathType- &Times New Roman\- 2 H~ 2 1 ~ 2 f~Times New Roman- 2 `6u 2 `y 2 `. x Times New Roman\- 2 i> 2 i> 2 HT| 2 i> 2  i> 2  i> 2 i> 2 i>Symbol- 2 `= 2 `- 2 `[+ 2 ` + 2 - 2 `x 2 ` x 2 l 2 `re Times New Roman\- 2 1p 2 L 2pTimes New Roman- 2 o1 2  1Symbol- 2 `ED & "System-\- 2 ࡱ;  FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ; 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ࡱ; .@o;l+:,: w 22 =var(b i )ࡱ; _918236588ZF&#&#ZOlenfo pPICion Native Y\qLMETA9658 Fsh@ ࡱ; L*{$hࡱ; *{I  .1  @&&`& & MathTypePSymbolA- 2 ;Y1 Times New RomanĔ- 2 Y1p 2 2p 2 2p 2 # 2p 2 2p 2 O2p 2 2p 2 :22pp 2 2pTimes New Roman- 2 H 1 2  1 2 1 2 $1 2 %0Symbol- 2 Q= 2 ( - 2 - 2 + 2 d- 2 + 2 - 2 $>Times New Roman- 2 ( 2 /k 2 V)( 2  )( 2 a) 2 ( 2 }) 2 ( 2 !)Symbol- 2 s 2 t 2 6 r 2 lr 2 |l 2 r 2 w 2  r 2 1"bTimes New Roman- 2 # ``` & "System-ࡱ FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ; ࡱ; *_A&Bp(B Y 1 =(s 2 /t 2 )(1-r 2 )(1-lr) 2 +(1-l 2 r 2 )w 22 +(1-r 2 )bCompObj9tiveSt86886868[]FZObjInfoNative^FEquation Native F_918310581aF&#&#   >0 2 ( ࡱ; ࡱ; L "Xࡱ;  "  .1   &g & MathTypeSymbol- 2 @5s 2 @s Times New RomaOle36810 FPICObj `cLMETAcument HCompObjlbdZ@ n- 2 G2p 2 2p 2 ba1pSymbol- 2 @T= 2 Y Symbol- 2 b={ Times New Roman- 2 i> 2 bi> 2 NTimes New Romanw - 2 @NTimes New Roman- 2 @/k & "System- ࡱ;  FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ; ࡱ; *`!'C'C s 2 =s i2 /N i=1N  &ࡱ; ObjInfo7eFEquation Native |_918310579tivem_hF&#&#n<Ole38396 F@ PICObj7 gjFLMETAfo CompObjNativeikZObjInfo6lF@ L; {<hࡱ; ; {< 8 .1  @`&  & MathTypePSymbol- 2 5s 2  e Times New Roman- 2 @i> 2 it>> Times New Romanw - 2 G2pSymbol- 2 T=Times New Roman- 2 var( 2 ) & "System-w - 2 @Nࡱ FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ; ࡱ; Equation Native F|_918310487toF&#&#aOlenfoNative PICon Native nqFL@ *`!'C'C s i2 =var(e it )ࡱ; L "xXࡱ;  "   .1  &g & MathTypeSymbol- 2 @=META8390 FHCompObjprZObjInfoNativesEquation Native F|@ t 2 @t Times New Roman- 2 2p 2 2p 2 b(1pSymbol- 2 @= 2 Ya Symbol- 2 b={ Times New Roman- 2 i> 2 bKi> 2 NTimes New Romanw - 2 @CNTimes New Roman- 2 @/k & "System-ࡱ;  FMicrosoft Equation 2.0 DS Equation Equation.2ࡱ; ࡱ; *`!'Cx 'C t 2 =t i2 /N i=1N _918310555vF&#&#ZOlenfo PICion Native uxLMETA9658 F@ ࡱ; L{hࡱ; {  U .1  @ & & MathTypePSymbol- 2 =t Times New Roman- 2 i> 2 it>>Times New Romanw - 2 u Times New Roman- 2 2pSymbol- 2 =Times New Roman- 2 Yvar( 2 a) & "System-Times New Romanࡱ;  FMicrosoft Equation 2.0 DS EqCompObj7wyFZObjInfoNativezEquation Native F|_918236586=}F&#&#@ uation Equation.2ࡱ; 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