Debt, In ation and Growth

Transcription

1 Debt, In ation and Growth Robust Estimation of Long-Run E ects in Dynamic Panel Data Models Alexander Chudik a, Kamiar Mohaddes by, M. Hashem Pesaran c, and Mehdi Raissi d a Federal Reserve Bank of Dallas, USA b Faculty of Economics and Girton College, University of Cambridge, UK c Deartment of Economics, University of Southern California, USA and rinity College, Cambridge, UK d International Monetary Fund, Washington DC, USA ovember 203 Abstract his aer investigates the long-run e ects of ublic debt and in ation on economic growth. Our contribution is both theoretical and emirical. On the theoretical side, we develo a cross-sectionally augmented distributed lag (CS-DL) aroach to the estimation of long-run e ects in dynamic heterogeneous anel data models with crosssectionally deendent errors. he relative merits of the CS-DL aroach and other existing aroaches in the literature are discussed and illustrated with small samle evidence obtained by means of Monte Carlo simulations. On the emirical side, using data on a samle of 40 countries over the eriod, we nd signi cant negative long-run e ects of ublic debt and in ation on growth. Our results indicate that, if the debt to GDP ratio is raised and this increase turns out to be ermanent, then it will have negative e ects on economic growth in the long run. But if the increase is temorary, then there are no long-run growth e ects so long as debt to GDP is brought back to its normal level. We do not nd a universally alicable threshold e ect in the relationshi between ublic debt and growth. We only nd statistically signi cant threshold e ects in the case of countries with rising debt to GDP ratios. Keywords: Long-run relationshis, estimation and inference, large dynamic heterogeneous anels, cross-section deendence, debt, in ation and growth, debt overhang. JEL Classi cations: C23, E62, F34, H6. We are grateful to Luis Catão, Markus Eberhardt, homas Moutos, Kenneth Rogo, Alessandro Rebucci, Ron Smith, Martin Weale, Mark Wynne, and seminar articiants at the International Monetary Fund and articiants at the Conference on MEA Economies 203 for constructive comments and suggestions. he views exressed in this aer are those of the authors and do not necessarily reresent those of Federal Reserve Bank of Dallas, the Federal Reserve System, the International Monetary Fund or IMF olicy. Hashem Pesaran acknowledges nancial suort under ESRC Grant ES/I03626/. y Corresonding author. address: km48@cam.ac.uk.

2 Introduction he debt-growth nexus has received renewed interest among academics and olicy makers alike in the aftermath of the recent global nancial crisis and the subsequent euro area sovereign debt crisis which has triggered trillions of dollars in scal stimulus across the globe. his aer investigates whether a build-u of ublic debt slows down the economy in the long run. he conventional view is that ublic debt (arising from de cit nancing) can stimulate aggregate demand and outut in the short run, but crowds out caital and reduces outut in the long run. In addition, there are ossible non-linear e ects where the build-u of debt can harm economic growth esecially when the level of debt exceeds a certain threshold, as estimated, for examle, by Reinhart and Rogo (200) to be around 90% of the GDP. However, such results are obtained under strong homogeneity assumtions across countries, and without adequate attention to dynamics, feed-back e ects from debt to GDP, and error cross-sectional deendencies that exist across countries, due to unobserved common factors or sill-over e ects that tend to magnify at times of nancial crises. Due to the intrinsic cross-country heterogeneities, the thresholds are most-likely country seci c and estimation of a universal threshold based on ooling of observations across countries might not be informative to olicy makers interested in a articular economy and their use could be even misleading. Relaxing the homogeneity assumtion, whilst ossible in a number of dimensions (as seen below), is di cult when it comes to the estimation of country-seci c thresholds, because due to the non-linearity of the relationshis involved, identi cation and estimation of country-seci c thresholds require much larger time series data than are currently available. In this aer we model the growth rates, as oosed to levels of (log) GDP and debt to GDP, which allows us to make inferences about the long-term e ects of debt on growth, regardless of thresholds. Using recent develoments in the literature on dynamic heterogeneous anels, we rovide a fresh re-examination of debt-growth nexus while allowing for dynamic heterogeneities and cross-sectional error deendencies. Our focus will be on the long-run imacts of debt and in ation on GDP growth which will be shown to be robust to feedbacks from growth to debt and in ation. We use a relatively large anel of advanced and emerging market economies, and jointly model in ation, debt, and growth. We consider the role of in ation in our long-run analysis because, in some countries in the anel that do not have active government bond markets, de cit nancing is often achieved through money creation with high in ation. Like excessively high levels of debt, high levels of in ation, when ersistent, can also be detrimental for growth. By considering both in ation and debt we allow the regression analysis to accommodate both tyes of economies in the anel. he aer also makes a theoretical contribution to the econometric analysis of the long run. A new aroach to the estimation of the long-run coe cients in dynamic heterogeneous anels with cross-sectionally deendent errors is roosed. he aroach is based on a

3 distributed lag reresentation that does not feature lags of the deendent variable, and allows for a residual factor error structure and weak cross-section deendence of idiosyncratic errors. Similarly to Common Correlated E ects (CCE) estimators roosed by Pesaran (2006), we aroriately augment the individual regressions by cross-section averages to deal with the e ects of common factors. We derive the asymtotic distribution of the roosed crosssection augmented distributed lag (or CS-DL in short) mean grou and ooled estimators under the coe cient heterogeneity and large time ( ) and cross section () dimensions. We also investigate consequences of various deartures from our maintained assumtions by means of Monte Carlo exeriments, including unit root in factors and/or in regressors, homogeneity of coe cients or breaks in error rocesses. he small samle evidence suggests that the CS-DL estimators often outerform the traditional aroach based on estimating the full autoregressive distributed lag (ARDL) seci cation. However, the CS-DL aroach should be seen as comlementary and not as suerior to the ARDL aroach due to its two drawbacks: unlike the anel ARDL aroach it does not allow for feedback e ects from the deendent variable onto the regressors, and its small samle erformance deteriorates when the roots of the AR olynomial in the ARDL reresentation are close to the unit circle. he relative merits of di erent aroaches are carefully documented in the aer. Our emirical contribution is in estimating long-run e ects of debt and in ation on economic growth in a anel of 40 countries over the eriod Cross-country exerience shows that some economies have run into debt di culties and exerienced subdued growth at relatively low debt levels, while others have been able to sustain high levels of indebtedness for rolonged eriods and grow strongly without exeriencing debt distress. his suggests that the e ects of ublic debt on growth varies across countries, deending critically on country-seci c factors and institutions. account of cross-country heterogeneity. It is therefore imortant that we take he dynamics should also be modelled roerly, otherwise the estimates of the long-run e ects might be inconsistent. Last but not least, it is now widely agreed that conditioning on observed variables seci c to countries alone need not ensure error cross-section indeendence that underlies much of the anel data literature. It is, therefore, also imortant that we allow for the ossibility of cross-sectional error correlations, which could arise due to omitted common e ects, ossibly correlated with the regressors. eglecting such deendencies can lead to biased estimates and surious inference. We adot a cross-section augmented ARDL aroach (CS-ARDL), advanced in Chudik and Pesaran (203a), and a CS-DL aroach develoed in this aer. his estimation strategy takes into account all three key features of the anel (i.e. dynamics, heterogeneity and cross-sectional deendence) jointly, in contrast with the earlier literature surveyed in Section 5. We study whether there is a common threshold for government debt ratios above hese might include rosects for rimary scal surluses and growth; cost of borrowing including both the interest cost of debt already contracted and market ercetions of a country s ability to service future borrowings; regulatory requirements; nature of the investor base and the track record of meeting its debt obligations (whether it had debt distress/lost market access); and vulnerability to shocks (con dence e ects). 2

4 which long-term growth rates are adversely a ected (esecially if the country is on an uward debt trajectory). We articularly look into debt trajectory beyond certain debt threshold levels as to our knowledge no such systematic analysis has been carried out in the ast. We do not nd a universally alicable threshold e ect in the relationshi between debt and growth. We only nd a statistically signi cant threshold e ect in the case of countries with rising debt to GDP ratios. he debt trajectory seems much more imortant than the level of debt itself. Provided that debt is on a downward ath, a country with a high level of debt can grow just as fast as its eers. his "no-simle-debt-threshold-level" nding can be driven, among other ossible factors, by cross-country di erences in (i) overall net wealth (international investment osition) and the deth of nancial system; (ii) investor behavior (home bias); (iii) ability to generate rimary surluses and interest costs growth considerations; and (iv) con dence factors. Our results also show that, regardless of the threshold, there are signi cant and robust negative long-run e ects of debt on economic growth. By comarison, the evidence of a negative e ect of in ation on growth is less strong, although it is statistically signi cant in the case of most seci cations considered. Our results suggest that if the debt level is raised and this increase is ermanent, then it will have negative e ects on growth in the long run. On the other hand, if the debt rises (for instance to hel smooth out business cycle uctuations) and this increase is temorary, then there are no long-run negative e ects on outut growth. he key in debt nancing is the reassurance, backed by commitment and action, that the increase in government debt is temorary and will not be a ermanent dearture from the revailing norms. he remainder of the aer is organized as follows. We begin with the de nition of longrun coe cients and discuss their estimation in Section 2. he next section introduces the CS-DL aroach to the estimation of long-run relationshis. Section 4 investigates the small samle erformance of the CS-DL aroach and comares it with the erformance of the CS-ARDL aroach by means of Monte Carlo exeriments. Section 5 reviews the literature on long-run e ects of in ation and debt on economic growth. Section 6 resents emirical ndings on the long-run e ects of debt and in ation on economic growth in our anel of countries. he last section concludes. Mathematical derivations and other suorting material are relegated to the Aendix. A brief word on notation: All vectors are column vectors reresented by bold lower case letters and matrices are reresented by bold caital letters. kak = % (A 0 A) is the sectral norm of A, % (A) is the sectral radius of A. 2 a n = O(b n ) denotes the deterministic sequence fa n g is at most of order b n. Convergence in robability and convergence in distribution are denoted by! and!, d resectively. (; )! j denotes joint asymtotic in and ; with and!, in no articular order. We use K to denote a ositive xed constant that does not vary with or. x. 2 ote that if x is a vector, then kxk = % (x 0 x) = x 0 x corresonds to the Euclidean length of vector 3

5 2 Estimation of long-run or level relationshis in economics Estimating long-run or level relationshis is of great imortance in economics. he concet of the long-run in economics is associated with the steady-state solution of a structural model. Often the same long-run relations can also be obtained from arbitrage conditions within and across markets. As a result many long-run relationshis in economics are free of articular model assumtions; examles being urchasing ower arity, uncovered interest arity and the Fisher in ation arity. Other long-run relations, such as those between macroeconomic aggregates like consumtion and income, outut and investment, technological rogress and real wages, are less grounded in arbitrage and hence are more controversial, but still form a major art of what is generally agreed in emirical macro modelling. his is in contrast to the analysis of short-run e ects which are model seci c and subject to identi cation roblems. he estimation of long-run relations can be carried out with or without constraining the short-run dynamics (ossibly from a articular theory). In this section we focus on the estimation of long-run relations without restricting the short-run dynamics. In view of the emirical alication that we have in mind, we shall assume that there exists a single longrun relationshi between the deendent variable, y t, and a set of regressors. 3 For illustrative uroses, suose that there is one regressor x t and suose that z t = (y t ; x t ) 0 is jointly determined by the following vector autoregressive model of order, VAR(), z t = z t + e t, () where = ( ij ) is a 22 matrix of unknown arameters, and e t = (e yt ; e xt ) 0 is 2-dimensional vector of reduced form errors. Denoting the covariance of e yt and e xt by!v ar (e xt ), we can write e yt = E (e yt je xt ) + u t =!e xt + u t, (2) where by construction u t is uncorrelated with e xt, namely E (u t je xt ) = 0. Substituting (2) for e yt, the equation for the deendent variable y t in () is y t = y t + 2 x t +!e xt + u t. (3) Using the equation for the regressor x t in (), we obtain the following exression for e xt e xt = x t 2 y t 22 x t, 3 he roblem of estimation and inference in the case of multile long-run relations is further comlicated by the identi cation roblem and simultaneous determination of variables. he case of multile long-run relations is discussed for examle in Pesaran (997). 4

6 and substituting this exression for e xt back in (3) yields the following conditional model for y t, y t = 'y t + 0 x t + x t + u t, (4) where ' =! 2 ; 0 =!; = 2! 22 : (5) ote that u t is uncorrelated with the regressor x t and its lag by construction. (4) is ARDL(,) reresentation of y t conditional on x t, and the short-run coe cients ', 0, and can be directly estimated from (4) by least squares. Model (4) can also be written as the following error-correction model, y t = ( ') (y t x t ) + 0 x t + u t, or as the following level relationshi y t = x t + (L) x t + ~u t, where the level coe cient is de ned by the ratio = 0 + ', ~u t = ( 'L) u t is uncorrelated with regressor x t and its lags, and (L) = P `=0 `L`, with ` = P s=`+ s, for ` = 0; ; 2; :::, and (L) = P `=0 `L` = ( 'L) ( 0 + L). ote that if z t is I () then (; ) 0 is the cointegrating vector and the level relation is also cointegrating. he level coe cient can still be motivated as the long-run outcome of a counterfactual exercise even if z t is stationary. One ossible counterfactual is to consider the e ects of a ermanent shock to the x t rocess on y t in the long run. Let and similarly g yt = lim s! E y t+s y;t+s I t ; e x;t+h = x, for h = 0; ; 2; :::, g xt = lim s! E x t+s x;t+s I t ; e x;t+h = x, for h = 0; ; 2; :::, where yt and xt, resectively, are the deterministic comonents of y t and x t (in the current illustrative examle deterministic comonents are zero) and I t is the information set containing all information u to the eriod t. Using () and noting that E (e yt je xt ) =!e xt, 5

7 we obtain g yt = g y, g xt = g x, 4 g = g y g x! = (I 2 )!! x =!+ 2! ! ! x, and g y g x =! + 2! 22 (! 2 ) ; which uon using (5), yields, g y = g x, namely the long-run imact of a ermanent change in the mean of x on y is given by. ote that only in the secial case when the reduced form errors are uncorrelated (! = 0) then the short-run coe cient 0 in the ARDL model (4) is equal to 0 and the long-run coe cient reduces to 2 = ( ). But in general, when! 6= 0, the short-run coe cient 0 is non-zero and contemoraneous values of the regressor should not be excluded from (4). In the stationary case with regressors not strictly exogenous, deends also on the arameters of the x t rocess and the estimation of should therefore be based on (4). An alternative way to show that is equal to the ratio g y =g x is to consider the ARDL reresentation (4) for the future eriod t + s; given the information at time t. We rst note that y t+s = 'y t+s + 0 x t+s + x t+s + u t+s, and after taking the conditional exectation with resect to fi t ; e x;t+h = x, for h = 0; ; 2; :::g, taking limits as s!, and noting that in the stationary case g yt = g y and g xt = g x, we obtain g y = 'g y + 0 g x + g x, and hence g y g x = 0 + ' =, as desired. Regardless of whether the variables are I (0) or I (), or whether the regressors are exogenous or not, the level coe cient is well de ned and can be consistently estimated. he rates of convergence and the asymtotic distributions of the ARDL estimates of are established in Pesaran and Shin (999). See in articular their heorem wo aroaches to the estimation of long-run e ects Let y it be the deendent variable in country i, x it be the k vector of country-seci c regressors, and suose that the object of interest is the long-run coe cient vector of country i, denoted as i, or, in a multicounty context, the average long-run coe cients vector, = i. In modelling the relationshi between the deendent variable and the 4 ote that in the stationary case P `=0 ` = (I ). 6

8 regressors in a anel context, we need to allow for sloe heterogeneity, dynamics and crosssectional deendence. his is accomlished by assuming that the deendent variable is given by the following ARDL( yi ; xi ) seci cation, yi X X xi y it = ' i`y i;t ` + 0 i`x i;t ` + u it, (6) `= `=0 u it = 0 if t + " it, (7) for i = ; 2; :::; and t = ; 2; :::;, where f t is an m vector of unobserved common factors, and yi and xi are the lag orders chosen to be su ciently long so that u it is a serially uncorrelated rocess across all i. he vector of long-run coe cients is then given by i = P xi `=0 i` P yi `= ' i`. (8) here are two aroaches to estimating the long-run coe cients. One aroach, considered in the literature, is to estimate the individual short-run coe cients f' i`g and f i`g in the ARDL relation, (6), and then comute the estimates of long-run e ects n^i`o using formula (8) with the short-run coe cients relaced by their estimates f^' i`g and. We shall refer to this aroach as the "ARDL aroach to the estimation of long-run e ects". he advantage of this aroach is that the estimates of short-run coe cients are also obtained. But when the focus is on the long-run then, under certain conditions to be clari ed below, an alternative aroach roosed in this aer can be undertaken to estimate i directly. his is ossible by observing that the ARDL model, (6), can be written as y it = i x it + 0 i (L) x it + ~u it, (9) where ~u it = ' (L) P yi u it, ' i (L) = `= ' i`l`, i = i (), i (L) = ' i (L) i (L) = P `=0 i`l`, i (L) = P xi `=0 i`l`, and i (L) = P P `=0 s=`+ sl`. We shall refer to the estimation of i based on the distributed lag reresentation (9) as the "distributed lag (DL) aroach to the estimation of long-run e ects". Under the usual assumtions on the roots of ' i (L) falling strictly outside the unit circle, then the coe cients of i (L) are exonentially decaying; and it is ossible to show that, in the absence of feedback e ects from lagged values of y it onto the regressors x it, a consistent estimate of i can be obtained directly based on the least squares regression of y it on x it and fx it `g `=0 ; where the truncation lag order is chosen aroriately as an increasing function of the samle size. But, when the feedback e ects from the lagged values of the deendent variable to the regressors are resent, ~u it will be correlated with x it and the DL aroach would no longer be consistent. ote that strict exogeneity is, however, not necessarily required for the consistency of the DL aroach, since arbitrary correlations amongst the individual reduced form innovations 7

9 in e t are still allowed. After the individual estimates ^ i are obtained, either using ARDL or DL aroach, they can then be averaged across i to obtain a consistent estimate of the average long-run e ects, given by ^ = i ^ i. 2.2 Pros and cons of the two aroaches to the estimation of longrun e ects Consider rst the ARDL aroach, where the estimates of long-run e ects are comuted based on the estimates of the short-run coe cients in (6). In the case where the unobserved common factors are serially uncorrelated and are also uncorrelated with the regressors, the long-run coe cients can be estimated consistently from the Ordinary Least Squares (OLS) estimates of the short-run coe cients, irresective of whether the regressors are strictly exogenous or jointly determined with y it, in the sense that z it = (y it ; x 0 it) 0 follows a VAR model. he long-run estimates are also consistent irresective of whether the underlying variables are integrated of order one, I () for short, or integrated of order zero, I (0). hese robustness roerties are clearly imortant in emirical research. However, the ARDL aroach has also a number of drawbacks. he samling uncertainty could be large esecially when the seed of convergence towards the long-run relation is rather slow and the time dimension is not su ciently long. his is readily aarent from (8) since even a small change P yi to `= ^' i` could have large imact on the estimates of i when P yi `= ^' i` is close to unity. In this resect, a correct seci cation of lag orders could be quite imortant for the erformance of the ARDL estimates of i. Underestimating the lag orders leads to inconsistent estimates, whilst overestimating the lag orders could result in loss of e ciency and low ower when the ARDL long-run estimates are used for inference. In the more general case when the unobserved common factors are correlated with the regressors then LS estimation of ARDL model is no longer consistent and the e ects of unobserved common factors need to be taken into account. here are so far two ossible estimators develoed in the literature for this case: 5 a rincial-comonents based aroach by Song (203) who extends the interactive e ects estimator originally roosed Bai (2009) to dynamic heterogeneous anels, and the dynamic common correlated e ects mean grou estimator suggested by Chudik and Pesaran (203a). A recent overview of these methods is rovided in Chudik and Pesaran (203b). hese estimators have (so far) been roosed only for stationary anels, and are subject to the small bias of the ARDL aroach discussed above. Bias correction techniques can also be used, but overall they do not seem to be e ective when the seed of adjustment to the steady state is slow. 6 he main merits of the DL aroach that we develo below is that, once (9) is aro- 5 Related is also the quasi maximum likelihood estimator for dynamic anels by Moon and Weidner (200), but this estimators has been develoed only for homogeneous anels. 6 Chudik and Pesaran (203a) consider the alication of two bias correction rocedures to dynamic CCE tye estimators, but nd that they do not fully eliminate the bias. 8

10 riately augmented by cross-section averages, it is robust along a number of dimensions that are imortant in ractice and it tends to show better small samle erformance when the time dimension is not very large. his includes robustness to the ossibility of unit roots in regressors and/or factors, heterogeneity or homogeneity of short and/or long-run coe cients, arbitrary serial correlation in " it and f t (note that i is identi ed even when " it is serially correlated), number of unobserved common factors (subject to certain conditions), and weak cross-sectional deendence in the idiosyncratic errors, " it. hese are very imortant considerations in alied work. In addition, the CS-DL aroach does not require secifying the individual lag orders, yi and xi, and is robust to ossible breaks in " it. he main drawback of the CS-DL aroach, however, is that ~u it = ' (L) u it is correlated with x it when there are feedback e ects from lagged values of y it onto the regressors, x it. his correlation in turn introduces a bias that will not vanish as the samle size increase and therefore the CS-DL estimation of the long-run e ects is consistent only in the case when the feedback e ects (or reverse causality) are not resent. he second drawback is that the small samle erformance is very good only when the eigenvalues of ' (L) are not close to the unit circle. We will rovide small samle evidence on the two aroaches by means of Monte Carlo exeriments in Section 4. 3 Cross section augmented distributed lag (CS-DL) aroach to estimation of mean long-run coe cients 3. he ARDL anel data model Suose y it is generated according to the anel ARDL data model (6) with yi = and xi = 0, y it = ' i y i;t + 0 ix it + 0 if t + " it, (0) for i = ; 2; :::; and t = ; 2; :::;. o allow for correlation between the m unobserved factors, f t ; and the k observed regressors, x it, suose that the latter is generated according to the following canonical factor model x it = 0 if t + v it, () for i = ; 2; :::; and t = ; 2; :::;, where i is m k matrix of factor loadings, and v it are the idiosyncratic comonents of x it which are assumed to be distributed indeendently of the idiosyncratic errors, " it. he anel data model (0) and () is identical to the model considered by Pesaran (2006) with the excetion that the lagged deendent variable is included in (0). We have also omitted observed common e ects and deterministics (such as intercets and time trends) from (0) to simlify the exosition. Introducing these terms and additional lags of the deendent variable and regressors is relatively straightforward. 9

11 We are interested in the estimation of the mean long-run coe cients = E ( i ), where i, i = ; 2; :::; are the cross section seci c long-run coe cients de ned by (8), which for yi = and xi = 0 reduces to i = We ostulate the following assumtions. i ' i. (2) Assumtion (Individual Seci c Errors) Individual seci c errors " it and v jt 0 are indeendently distributed for all i; j; t and t 0. " it follows a linear stationary rocess with absolute summable autocovariances (uniformly in i), " it = X "i` i;t `, (3) `=0 for i = ; 2; :::;, where the vector of innovations t = ( t ; 2t ; :::; t ) 0 is satially correlated according to t = R& t, in which the elements of & t are indeendently and identically distributed (IID) with mean zero, unit variance and nite fourth-order cumulants and the matrix R has bounded row and column matrix norms, namely krk < K and krk < K. In articular, V ar (" it ) = X 2 "i` 2 i = 2 i K <, (4) `=0 for i = ; 2; :::;, where 2 i = V ar ( it). v it follows a linear stationary rocess with absolute summable autocovariances uniformly in i, v it = X S i` i;t `, (5) `=0 for i = ; 2; :::;, where it is k vector of IID random variables, with mean zero, variance matrix I k and nite fourth-order cumulants. In articular, X kv ar (v it )k = `=0 S i`s 0 i` = k ik K <, (6) for i = ; 2; :::;, where kak is the sectral norm of the matrix A. Assumtion 2 (Common E ects) he m vector of unobserved common factors, f t = (f t ; f 2t ; :::; f mt ), is covariance stationary with absolute summable autocovariances, distributed indeendently of & it 0 and v it 0 for all i; t and t 0. Fourth moments of f`t, for ` = ; 2; :::; m, are bounded. 0

12 Assumtion 3 (Factor Loadings) Factor loadings i, and i are indeendently and identically distributed across i, and of the common factors f t, for all i and t, with xed mean and and, resectively, and bounded second moments. In articular, i = + i, i IID 0 m ; vec ( i ) = vec ( ) + i, i IID 0 ; km, for i = ; 2; :::;,, for i = ; 2; :::;, where and are m m and k m k m symmetric nonnegative de nite matrices, kk < K, k k < K, k k < K, and k k < K. Assumtion 4 (Coe cients) he level coe cients i, de ned in (2), follow the random coe cient model i = + i, i IID 0 ;, for i = ; 2; :::;, (7) k where kk < K, k k < K, is k k symmetric nonnegative de nite matrix, and the random deviations i are indeendently distributed of j, j, & jt, v jt, and f t for all i,j, and t. he coe cients ' i are distributed with a suort strictly inside the unit circle. he olynomial ( ' i L) we obtain ' i L is invertible under Assumtion 4, and multilying (0) by y it = ( ' i L) 0 ix it + ( ' i L) 0 if t + ( ' i L) " it = i x it 0 i (L) x it + 0 i ~ f it + ~" it, for i = ; 2; :::;, (8) where x it = x it x i;t, i (L) = P `=0 '`+ i ( ' i ) i L`; ~ f it = ( ' i L) f t and ~" it = ( ' i L) " it. he distributed lag seci cation in (8) does not include lagged values of the deendent variable, and as a result the CCE estimation rocedure can be alied to (8) directly. he level regression of y it on x it is estimated by augmenting the individual regressions by di erences of unit seci c regressors x it and their lags, in addition to the augmentation by the cross section averages that take care of the e ects of unobserved common factors. Let w = (w ; w 2 ; :::; w ) 0 granularity conditions be an vector of weights that satis es the following kwk = O 2, (9) w i kwk = O 2 uniformly in i, (20)

13 and the normalization condition X w i =. (2) De ne the cross section averages z wt = (y wt ; x 0 wt) 0 = P w iz it, and consider augmenting the regressions of y it on x it and the current and lagged values of x it, with the following set of cross section averages, S t = z wt [ fx w;t `g `=0. Cross section averages aroximate the unobserved common factors arbitrarily well if # f = f t E (f t j S t )! 0, (22) uniformly in t, as and j!. Su cient conditions for result (22) to hold are given by Assumtions -4 and if the rank condition rank ( ) = m holds. Di erent sets of cross section averages could also be considered. For examle, if the set of cross section averages is de ned as S zt = fz wt `g z `=0, then the su cient condition for (22) to hold under Assumtion -4 would be the usual rank condition rank (C) = m, where C = (; ). Using covariates to enlarge the set of cross section averages could also be considered, as in Chudik and Pesaran (203a). heses rank conditions can be relaxed in the case i and i are indeendently distributed. 7 In this case the asymtotic variance of the CCE estimators does deend on the rank condition, nevertheless the CS-DL estimators are consistent and the roosed nonarametric estimators of the covariance matrix of the CS-DL estimators given below are also valid regardless of whether the rank condition holds. Let us also introduce the following notations, which will rove useful for setting u of the roosed estimators. Let y i = (y i;+ ; y i;+2 ; :::;y i; ) 0, X i = x i;+ ; x i;+2 ; :::; x i; 0, Z w = (z w;+ ; z w;+2 ; :::; z w; ) 0, X i = ( )k 0 x 0 i;+ x 0 i; x 0 i2 x 0 i;+2 x 0 i;+ x 0 i3... x 0 i x 0 i; x 0 i; + C A, X w = P w ix i, Q wi = Z w ; X w ; X i, and the de ne the rojection matrix M qi = I Q wi (Q 0 wiq wi ) + Q 0 wi, (23) for i = ; 2; :::;, where = ( ) is a chosen non-decreasing truncation lag function such that 0 <, and A + is the Moore-Penrose seudoinverse of the matrix A. We use the Moore-Penrose seudoinverse as oosed to standard inverse in (23) because the column vectors of Q wi could be asymtotically (as! ) linearly deendent. 7 Correlation of i and i could introduce a bias in the rank de cient case, as noted by Sara dis and Wansbeek (202). 2

14 he CS-DL mean grou estimator of the mean long-run coe cients is given by b MG = X b i, (24) where b i = (X 0 im qi X 0 i) X 0 im qi y i. (25) he CS-DL ooled estimator of the mean long-run coe cients is b P =! X w i X 0 im qi X i X w i X 0 im qi y i. (26) Estimators b MG and b P di er from the mean grou and ooled CCE estimator develoed in Pesaran (2006), which only allows for the inclusion of a xed number of regressors, whilst the CS-DL tye estimators include lags of x it and their cross section averages, where increases with, albeit at a slower rate. In addition to Assumtions -4 above, we shall also require the following assumtion to hold. Assumtion 5 below ensures that b MG and b P and their asymtotic distributions are well de ned. P Assumtion 5 (a) he matrix lim j ;;! w i i = exists and is nonsingular, and su i; < K, where i = lim X 0 im hi X i, and M hi is de ned in (A.3). i (b) Denote the t-th row of matrix X e i = M hi X i by ex 0 it = (ex it ; ex i2t ; ::::; ex ikt ). he individual elements of ex it have uniformly bounded fourth moments, namely there exists a ositive constant K < such that E (ex 4 ist) < K; for any t = ; 2; :::; ; i = ; 2; :::; and s = ; 2; :::; k. (c) here exists 0 such that for all 0 ; P w ix 0 im qi X i = exists. (d) here exists 0, 0 and 0 = ( 0 ) such that for all 0, 0 and ( ) ( 0 ), the k k matrices (X 0 im qi X i = ) exist for all i, uniformly. Our main ndings are summarized in the following theorems. heorem (Asymtotic distribution of b MG ) Suose y it, for i = ; 2; :::; and t = ; 2; :::; is given by the anel data model (0)-(), Assumtions -5 hold, and (; ; ( ))! j such that ( )! 0; for any constant 0 < < and ( ) 3 =! {, 0 < { <. hen, if rank ( ) = m we have bmg d! (0; ), (27) 3

15 where = V ar ( i ) and b MG is given by (24). If rank ( ) 6= m and i is indeendently distributed of i, we have bmg d! (0; MG ), (28) where MG = + lim ;! " X i Q if Q 0 if i #, (29) in which = V ar ( i ), i = lim! X 0 im hi X i and Q if = lim! X 0 im hi F. In both cases, the asymtotic variance of b MG can be consistently estimated nonarametrically by b MG = X 0 bi MG b bi MG b. (30) heorem 2 (Asymtotic distribution of b P ) Suose y it, for i = ; 2; :::; and t = ; 2; :::; are generated by the anel data model (0)-(), Assumtions -5 hold, and (; ; ( )) j! ; such that ( )! 0; for any constant 0 < < and ( ) 3 =! {, 0 < { <. hen, if i is indeendently distributed of where b P is given by (26), i, we have! =2 X wi 2 bp d! (0; P ), (3) P = R, = lim! X w i i, (32) R = R + R, R = lim! X ew i 2 i i, R = lim! X ew i 2 Q if Q 0 if, = V ar ( i ), = V ar ( i ), i = lim X 0 im hi X i, Q if = lim X 0 im hi F, and ew i = P =2. wi w2 i If rank ( ) = m; then i is no longer required to be indeendently distributed of i and (3) continues to hold with P = R. In both cases, P can be consistently estimated by ^ P de ned by equation (A.25) in the Aendix. heorems -2 establish asymtotic distribution of b MG and b P under sloe heterogeneity. hese theorems distinguish between cases where the rank condition that ensures (22) is satis ed or not. In the former case, unobserved common factors can be aroximated by cross section averages when is large and regardless of whether i is correlated with i, b MG and b P are consistent and asymtotically normal. In the latter case, where the unobserved common factors cannot be aroximated by cross section averages when is 4

16 large, then so long as i and i are indeendently distributed, both b MG and b P continue to be consistent and asymtotically normal, but the asymtotic variance deends also on unobserved common factors and their loadings. In both (full rank or rank de cient) cases, the asymtotic variance of the CS-DL estimators can be estimated consistently using the same non-arametric formulae as in the full rank case. here are several deartures from the assumtions of these theorems that might be of interest in alied work, such as the consequences of breaks in the error rocesses, " it, ossibility of unit roots in factors and/or regressor seci c comonents, and situations where some or all coe cients are homogeneous over the cross-section units. hese theoretical extensions are outside the scoe of the resent aer but we investigate the robustness of the roosed CS-DL estimator to such deartures by means of Monte Carlo simulations in the next section. 4 Monte Carlo exeriments his section investigates small samle roerties of the CS-DL estimators and comare them with the estimates obtained from the anel ARDL aroach using the dynamic CCEMG estimator of the short-run coe cients advanced in Chudik and Pesaran (203a), which we denote by CS-ARDL. First, we resent results from the baseline exeriments with heterogeneous sloes (long- and short-run coe cients), and then we document small samle erformance of the alternative estimators under various deviations from the baseline exeriments, including robustness of the estimators to the introduction of unit roots in the regressors or factors, ossible breaks in the idiosyncratic error rocesses, and the consequences of feedback e ects from lagged values of y it onto x it. Second, we investigate whether it is ossible to imrove on the estimation of short-run coe cients, rovided the model is correctly seci ed, by imosing CS-DL estimates of the long-run coe cients. We start with a brief summary of the estimation methods and a descrition of the data generating rocesses. hen we resent ndings on the estimation of mean long-run coe cient and on the extent to which estimates of the short-run coe cients can be imroved by using the CS-DL estimators of the long-run e ects. 4. Estimation methods he CS-DL estimators are based on the following auxiliary regressions: X y it = c yi + 0 ix it + i`x i;t ` + `=0 y X `=0! y;i`y t ` + x X `=0! 0 x;i`x t ` + e it, (33) where x t = P x it, y t = P y it, x is set equal to the integer art of =3, denoted as =3, = x and y is set to 0. We consider both CS-DL mean grou and 5

17 ooled estimators based on (33). he CS-ARDL estimator is based on the following regressions: y it = c yi + y X `= ' i`y i;t ` + x X `=0 0 i`x i;t ` + z X `=0 0 i`z t ` + e it, (34) where z t = (y t ; x 0 t) 0, z = =3 and two otions for the remaining lag orders are considered: ARDL(2,) seci cation, y = 2 and x =, and ARDL(,0) seci cation, y = and x = 0. he CS-ARDL estimates of individual mean level coe cient are then given by ^ CS ARDL;i = P x ^ `=0 P i` y `= ^' i`, (35) where the estimates of short run coe cients (^' i`,^ i`) are based on (34). he mean longrun e ects are estimated as P ^ CS ARDL;i and the inference is based on the usual non-arametric estimator of asymtotic variance of the mean grou estimator. 4.2 Data generating rocess he deendent variable and regressors are generated from the following ARDL(2,) anel data model with factor error structure, y it = c yi + ' i y i;t + ' i2 y i;t 2 + i0 x it + i x i;t + u it, u it = 0 if t + " it, (36) and x it = c xi + yi y i;t + 0 xif t + v it. (37) We generate y it ; x it for i = ; 2; :::;, and t = 99; :::; 0; ; 2; :::; with the starting values y i; 0 = y i; 00 = 0; and the rst 00 time observations (t = 99; 48; :::; 0) are discarded to reduce the e ects of the initial values on the outcomes. he xed e ects are generated as c iy IID (; ), and c xi = c yi + & cxi, where & cxi IID (0; ), thus allowing for deendence between x it and c yi. We consider three cases deending on the heterogeneity/homogeneity of the sloes: (heterogeneous sloes - baseline) ' i = ( + { 'i ) 'i, ' i2 = { 'i 'i, { 'i IIDU (0:2; 0:3), 'i IIDU (0; ' max ). he long-run coe cients are generated as i IID (; 0:2 2 ) and the regression coe cient are generated as i0 = { i i, i = ( { i ) i, where i = i = ( ' i ' i2 ) and { i IIDU (0; ). (homogeneous long-run, heterogenous short-run sloes) i = for all i and the remaining coe cients (' i ; ' i2 ; i0 ; i ) are generated as in the revious fully heterogeneous case. 6

18 (homogeneous long- and short-run sloes) ' i = :5' max =2, ' i2 = 0:5' max =2, i =, and i0 = i = 0:5= ( ' max =2). We also consider the case of ARDL(,0) anel model by setting { 'i = 0 and { i = for all i, which gives ' i2 = i = 0 for all i. We consider three values for ' max = 0:6, 0:8 or 0:9. he unobserved common factors in f t and the unit-seci c comonents, v it ; are generated as indeendent AR() rocesses: f t` = f`f t ;` + & ft`, & ft` IID 0; 2 &f`, (38) v it = xi v i;t + it, & xit IID 0; 2 i, (39) for i = ; 2; :::;, ` = ; 2; ::; m, and for t = 99; :::; 0; ; 2; :::; with the starting values f`; 00 = 0, and v i; 00 = 0. he rst 00 time observations (t = 99; 48; :::; 0) are discarded. We consider three ossibilities for the AR() coe cients f` and xi : (stationary baseline) xi IIDU [0:0:95], 2 i = 2 xi for all i; f` = 0:6, and 2 &f` = 2 f` for ` = ; 2; :::; m. (nonstationary factors) xi IIDU [0:0:95], 2 i = 2 xi for all i; and f` =, 2 &f` = 0:2 for ` = ; 2; :::; m. (nonstationary regressors and stationary factors) xi =, 2 i = 0: 2 for all i; and f` = 0:6, 2 &f` = 2 f`, for ` = ; 2; :::; m. We consider also two otions for the feedback coe cients yi : no feedback e ects, yi = 0 for all i, and with feedback e ects, yi IIDU (0; 0:2). Factor loadings are generated as i` IID `; 0:2 2 and xi` IID x`; 0:2 2, for ` = ; 2; ::; m; and i = ; 2; :::;. Also, without loss of generality, the means of factor loadings are calibrated so that V ar( 0 if t ) = V ar ( 0 xif t ) = in the stationary case. We set ` = b, and x` = `b x, for ` = ; 2; :::; m, where b = =m 0:2 2 ; and b x = 2= [m (m + )] 2= (m + ) 0:2 2. his ensures that the contribution of the unobserved factors to the variance of y it does not rise with m in the stationary case. We consider m = 2 or 3 unobserved common factors. Finally, the idiosyncratic errors, " it, are generated to be heteroskedastic, weakly crosssectionally deendent and serially correlated. Seci cally, " it = "i " i;t + it, (40) 7

19 where t = ( t ; 2t ; :::; t ) 0 are generated using the following satial autoregressive model (SAR), in which the elements of & t are drawn as IID 0; 2 2 i ( 2 "i), with 2 i obtained as indeendent draws from 2 (2) distribution, t = a S t + & t, (4) S = , C 2 2 A and the satial autoregressive arameter is set to a = 0:6. ote that f it g is cross-sectionally weakly deendent for ja j <. We consider "i = 0 for all i or "i IIDU (0; 0:8). We also consider the ossibility of breaks in " it by generating for each i random break oints b i 2 f; 2; :: g and " it = a "i" i;t + it, for t = ; 2; :::; b i " it = b "i" i;t + it, for t = b i + ; b i + 2; :::;, where a "i; b "i IIDU (0; 0:8), and t = ( t ; 2t ; :::; t ) 0 is generated using SAR model (4) with & it IID 0; 2 2 i ( a2 "i ). he above DGP is more general than the other DGPs used in MC exeriments in the literature and allows the factors and regressors to be correlated and ersistent. he above DGPs also include models with unit roots, breaks in the error rocesses, and allows for correlated xed e ects. o summarize, we consider the following cases:. (3 otions for heterogeneity of coe cients) heterogeneous baseline, homogeneous longrun with heterogeneous short-run, and both long-and short-run homogeneous, 2. (2 otions for lags) ARDL(2,) baseline, and ARDL(,0) model where { 'i = 0 and { i = for all i, which gives ' i2 = i = 0 for all i. 3. (3 otions for ' max ) ' max = 0:6 (baseline), 0:8, or 0:9 4. (3 otions for the ersistence of factors and regressors) stationary baseline, I() factors, or I() regressor seci c comonents v it, 5. (2 otions for the number of factors) full rank case baseline m = 2, or rank de cient case m = 3, 8

20 6. (3 otions for the ersistence of idiosyncratic errors) serially uncorrelated baseline "i = 0, "i IIDU (0; 0:8), or breaks in the error rocess. 7. (2 otions for feedback e ects) yi = 0 for all i (baseline), or yi IIDU (0; 0:2). Due to the large number of ossible cases (648 in total), we only consider baseline exeriments and various deartures from the baseline. We consider the following combinations of samle sizes: ; 2 f30; 50; 00; 50; 200g, and set the number of relications to R = 2; 000, in the case of all exeriments. 4.3 Monte Carlo ndings on the estimation of mean long-run coe cients he results for the baseline DGP are summarized in able. his table shows good erformance of the CS-DL estimators in the baseline exeriments. his table also shows roblems with the CS-ARDL aroach when is not large (<00) due to the small samle roblems arising when P y `= ^' i` is close to unity. Also, CS-ARDL estimates based on misseci ed lags orders are inconsistent, as to be exected. ext, we investigate robustness of the results to di erent assumtions regarding sloe heterogeneity. able 2 resents ndings for the exeriment that deart from the baseline DGP by assuming homogeneous long-run sloes, while allowing the short-run sloes to be heterogeneous. able 3 gives the results when both long- and short-run sloes are homogeneous. hese results show that the CS-DL estimators continue to have good size and ower roerties in all cases. Exeriments based on the ARDL(,0) seci cation (as the DGP) are summarized in able 4. CS-DL estimators continue to erform well, showing their robustness to the underlying ARDL seci cation. he e ects of increasing the value of ' max on the roerties of the various estimators are summarized in ables 5 (for ' max = 0:8) and 6 (for ' max = 0:9). Small samle erformance of the CS-DL estimators deteriorates as ' max moves closer to unity, as to be exected. ables 5-6 show that the erformance deteriorates substantially for values of ' max close to unity, due to the bias that results from the truncation of lags for the rst di erences of regressors. It can take a large lag order for the truncation bias to be negligible when the largest eigenvalue of the dynamic seci cation (given by the lags of the deendent variable) is close to one. We see quite a substantial bias when ' max = 0:9. herefore, it is imortant that the CS-DL aroach is used when the seed of convergence towards equilibrium is not too slow and/or is su ciently large so that biases arising from the aroximation of dynamics by distributed lag functions can be controlled. he robustness of the results to the number of unobserved factors (m) is investigated in able 7. his table rovides a summary in the case of m = 3 factors, which reresents the 9

21 rank de cient case. It is interesting to note that desite the failure of the rank condition, the CS-DL estimators continue to erform well (the results are almost unchanged as comared with those in able ), while the CS-ARDL estimates are a ected by two tyes of biases (the time series bias and the bias due to rank de ciency) that oerate in oosite directions. Consider now the robustness of the results to the resence of unit roots in the unobserved factors (able 8) or in the regressors (able 9). As can be seen the CS-DL estimators continue to erform well when factors contain unit roots. able 9, on the other hand, shows large RMSE and low ower for = 30 and 50, when the idiosyncratic errors have unit roots. But, interestingly enough, the reorted size is correct and biases are very small for all samle sizes. he results in able 0 consider the robustness of the CS-DL estimators to the roblem of serial correlation in the errors, whilst those in able consider the robustness of these estimators to the breaks in the error rocesses. As can be seen, and as redicted by the theory, the CS-DL estimators are robust to both of these deartures from the baseline scenario, whereas the CS-ARDL aroach is not. Recall, that CS-ARDL aroach requires that the lag orders are correctly seci ed, and does not allow for residual serial correlation and/or breaks in the error rocesses, whilst CS-DL does. Last but not least, the consequences of feedback e ects from y it to the regressors, x it, is documented in able 2. his table shows that the CS-ARDL aroach is consistent regardless of the feedback e ects, rovided that the lag orders are correctly seci ed, again as redicted by the theory. But a satisfactory erformance (in terms of bias and size of the test) for the CS-ARDL aroach requires to be su ciently large. On the other hand, in the resence of feedbacks, the CS-DL estimators are inconsistent and show ositive bias even for su ciently large. But the bias due to feedback e ects seem to be quite small; between and 0.06, and the CS-DL estimators tend to outerform the CS-ARDL estimators when < 00. Given the above MC results, and considering that outut growth is only moderately ersistent 8, and given that the time dimension is 45 years, the CS-DL estimates are likely to rovide a valuable comlement to the ARDL estimates in our emirical investigation below. 4.4 Monte Carlo ndings on the imrovement in estimation of short-run coe cients As a nal exercise, we consider if it is ossible to imrove on the estimation of short-run coe cients by imosing the CS-DL estimates of the long-run, before estimating the short-run coe cients. We consider the exeriment that dearts from the baseline model by assuming a homogeneous long-run coe cient, whilst all the short-run sloes are heterogeneous, 8 In our emirical alication the rst order autoregressive coe cient of outut growth ranges from 0:53 (Morocco) to 0:65 (Jaan), with mean and median of 0:274 and 0:273, resectively. 20

22 and use the ARDL(,0) as the data generating rocess. More seci cally, we imose the CS-DL ooled estimator of the long-run coe cient, ^ P, when estimating the short-run coef- cients using the CS-ARDL aroach. In articular, we estimate the following unit-seci c regressions, y it = c yi + i y i;t ^P x it + z X `=0 0 i`z t ` + " it, (42) for i = ; 2; :::;, and the resulting mean grou estimator of E (' i ) = + E ( i ) is denoted by ~' ;MG = X ~' i ; ~' i = i ~, where ~ i is the least square estimate of i based on (42). he results of these exeriments are summarized in able 3. Imosing the CS-DL ooled estimator of the long-run coe cient imroves the small samle roerties of the short-run estimates substantially, about 80-90% reduction of the di erence between the RMSE of the infeasible CS-ARDL estimator and the RMSE of the unconstrained estimator when = 30. We are now in a osition to aly the various estimation techniques discussed in this aer to our central emirical question of interest, namely the relationshi between in ation, debt to GDP and outut growth across a anel of develoed and emerging economies. But rst we rovide an overview of the literature so that our emirical results can be laced within the extant literature. 5 E ects of in ation and debt on economic growth: a literature review 5. Debt and growth Economic theory rovides mixed results on the relationshi between ublic debt and growth. Elmendorf and Mankiw (999) argue that ro igate debt-generating scal olicy (and high ublic debt) can have a negative imact on long-term growth by crowding out rivate investment, although it is argued that this e ect is quantitatively small. he negative growth e ect of ublic debt could be larger in the resence of olicy uncertainty or exectations of future con scation (ossibly through in ation and nancial reression). See, for examle, Cochrane (20a) and Cochrane (20b). Contrary to this view, DeLong and Summers (202) argue that hysteresis arising from recessions can lead to a situation in which exansionary scal olicies may have ositive e ect on long-run growth. Krugman (988) argues that nonlinearities and threshold e ects can arise from the resence of external debt overhang, but it is not clear whether such an argument is alicable to advanced economies where the majority of debt-holders are residents. onlinearities may also arise if there is a turning oint above 2