Mixed Frequency Panel Vector Autoregressions and the Inequality vs. Growth Nexus

Transcription

1 Mixed Frequency Panel Vector Autoregressions and the Inequaly vs. Growth Nexus Michael Binder Goethe Universy Frankfurt Melanie Krause Goethe Universy Frankfurt Highly Preliminary and Incomplete. Please Do not Quote. May 15, Abstract Empirical findings as to how income inequaly affects output growth seem to yield ltle consensus to date. This paper argues that new insight can be gained from taking into account the different measurement frequencies of the variables of interest: While data on output growth on a broad cross-country basis is available at least annually, notable changes in inequaly as captured by the Gini coefficient only manifest themselves over a multiple-years horizon. This suggests the use of mixed frequency data sampling (MIDAS), as first introduced by Ghysels et al. (2002). We extend the MIDAS vector autoregression approach of Ghysels (2012) to the panel data setting, and estimate our MIDAS Panel VAR model wh an uncondional quasi-maximum likelihood estimator. Our modeling approach also accounts for country-specific characteristics, the potential endogeney of the inequaly variable and possible state-dependencies in the relation. We provide Monte Carlo simulation evidence of the performance of the QML estimator in a such a setting and then apply to inequaly data from the Standardized World Income Inequaly Database 4.0 based on Solt (2009). We find, inter alia, that condioning on the level of income per capa influences how inequaly affects output growth, wh the impact being more detrimental in wealthier countries. JEL Classification: C33, C51, D30, O57 Keywords: MIDAS, Output Growth, Inequaly, Panel Data Vector Autoregressions Correspondence address: Goethe Universy Frankfurt, Faculty of Economics and Business Administration, Chair for International Macroeconomics & Macroeconometrics, Grueneburgplatz 1, House of Finance, Frankfurt am Main, Germany; mbinder@wiwi.uni-frankfurt.de. Goethe Universy Frankfurt, Address see above. melanie.krause@wiwi.uni-frankfurt.de The authors are grateful for comments and suggestions to participants at the (EC) 2 -Conference on The Econometric Analysis of Mixed Data Sampling in Nicosia, the Spring Meeting of Young Economists in Vienna, the 9th Winter School of Inequaly and Social Welfare Theory in Alba di Canazei, and the Brown Bag Seminar at Goethe Universy Frankfurt. 1

2 1 Introduction Will a decrease in income inequaly increase or decrease output growth? This question is of obvious relevance for policy makers and has been widely debated for some decades now. Multiple channels as to how inequaly might influence growth have been noted in theoretical models: On the one hand, incentive and moral hazard aspects (see, for example, Mirrlees, 1971), the higher propensy to save of the rich (Bourguignon, 1981) and investment indivisibilies Aghion et al. (1999) can explain why higher inequaly might spur output growth. On the other hand, there are a number of factors linking higher inequaly to lower growth, including social instabily and unrest (Gupta, 1990) investment-hostile re-distributive pressures (Persson and Tabellini, 1994), and insufficient human capal acquision (Galor and Zeira, 1993). Some or all of these factors might be present but no consensus has been established yet as to what the overall net effect is. In fact, empirical studies addressing how inequaly influences growth have produced widely varying results; consider for example, Barro (2000), Forbes (2000) and the meta-analysis by de Dominicis et al. (2008). This is in part due to differences in country samples and choices of control variables but also due to differences in econometric methodology. In this paper we revis the relation between inequaly and output growth and address an often-overlooked point, namely the different measurement frequencies of inequaly and output growth: While cross-country data on output growth is available at least annually, changes in the Gini coefficient only tend to materialize over a multiple-year horizon. Typically, researchers bring both variables to the same low frequency by taking the average output growth rate over several years. We argue that this simple averaging involves a loss of information that can potentially distort the results. In order to make optimal use of all available data, we employ methods from the mixed frequency data sampling (MIDAS) lerature. MIDAS was introduced by Ghysels et al. (2002, 2005, 2006). The idea is to regress the low-frequency variable on a suable aggregation of the high-frequency variable. Typical applications can, for example, be found in financial macroeconomics, where comparatively low-frequency real variables like employment growth are regressed on high-frequency financial indicators such as stock prices. Our case here is different in that our dependent variable, output growth is of relatively high frequency compared to the regressor, inequaly. We can address this issue by resorting to a MIDAS-VAR setup, as introduced by Ghysels (2012), which also has the side benef of contour- 2

3 ing the potential endogeney involved in our regression. Because our cross-country data set is a panel, we extend Ghysels s MIDAS-VAR model to panels. To our knowledge, MIDAS Panel VARs (PVARs) have not yet appeared in the lerature. We estimate our model using an uncondional quasi maximum likelihood (QML) estimator, extending the QML estimator proposed by Binder et al. (2005) to incorporate mixed frequency sampling. This estimator overcomes the bias associated wh "short T, large N" dynamic fixed-effects models such as ours, through modeling of inial observations. Furthermore, our model can incorporate possible nonlinearies in the relation between inequaly and output growth: By introducing a condional pooled mean group component as suggested by Binder and Offermanns (2007) into the MIDAS-PVAR, we let the impact of inequaly on output growth depend on, for instance, a country s development level (proxied by income per capa). This allows for the possibily that the effects vary systematically across different macroeconomic environments and can shed light on the role of the choice of the country sample for previous empirical findings. This paper is organized as follows: In Section 2 we summarize the lerature on the inequaly vs. growth nexus wh a focus on key results obtained and the econometric methodology employed. Section 3 introduces MIDAS regression models and extends them to the nonlinear MIDAS-PVARs we will use. We present our QML estimator in Section 4, where we also conduct a Monte Carlo simulation to gain some insights into s performance. In Section 5 we discuss key features of our data set and present and discuss our estimation results. Section 6 concludes. 2 Overview Concerning the Inequaly vs. Growth Nexus in the Lerature How increases in income inequaly affect output growth has been an issue of intense debate in the lerature. Theoretical models have put forward a number of reasons for the effects to be posive or negative, and a body of distinguished empirical studies has produced contradictory results. In the theoretical lerature, three main channels have been suggested as to how an increase in inequaly might have a posive effect on growth: The classical argument of performance, effort and moral hazard, formalized by Mirrlees (1971), says that performance-based wages will provide an incentive for individuals to work more if effort is posively correlated wh performance. The second potential posive effect of inequaly on growth relates to the higher propensy to save of the rich ("Kaldor s 3

4 Hypothesis"). Bourguignon (1981) showed that a convex savings function leads to steady state output which depends posively on the inequaly of the inial income distribution. As a third channel, investment indivisibilies have been put forward as to why an inegalarian distribution can be beneficial, see Aghion et al. (1999): New and innovative industries often involve high sunk costs, which, in the absence of well-functioning markets for firms shares, can only be stemmed by individuals wh concentrated wealth. At the same time, a number of channels have been identified in the lerature through which higher inequaly might translate into lower rates of growth: Social instabily and polical unrest have been investigated, inter alia, by Gupta (1990) and Alesina and Perotti (1996). Their model suggests that higher inequaly increases socio-polical instabily, which in turn decreases investment and output growth. The polical dimension is also emphasized in the hold-up problem of Alesina and Rodrik (1994) and Benhabib and Rustichini (1996). Persson and Tabellini (1994), and more recently Farhi et al. (2012), argue that distributional pressure leads to distortionary taxation of investment and other growth-stimulating activies. While one can think of this effect being stronger in democracies, where the distribution of polical power is more egalarian than the distribution of economic power, Clarke (1995) and Deininger and Squire (1998) find empirical evidence that inequaly adversely affects growth both in democracies and in non-democracies. Another channel for higher inequaly to cause lower rates of growth involves credmarket imperfections that may preclude the poor from borrowing. While Aghion and Bolton (1997) focus on borrowing constraints for investment in physical capal, Galor and Zeira (1993) deal wh the human capal case: If cred-market imperfections do not allow the poor to borrow against future earnings, they may not be able to obtain an adequate education and may have to work as unskilled, decreasing the overall productivy of the economy. A related argument by de la Croix and Doepke (2003) emphasizes the higher fertily of the poor. In their theoretical model, poor families choose a higher level of fertily given that the cost of education is fixed and that the opportuny cost of time to raise multiple children increases wh income. Empirically, de la Croix and Doepke (2003) find that the Gini coefficient becomes insignificant in a growth regression when the fertily differential - the fertily difference between the most and least educated women - is included. Detrimental growth effects then are tied to inegalarian access to education rather than to income in- 4

5 equaly self. 1 Many or all of these channels through which inequaly can influence output growth might be there in practice, possibly counteracting in different strengths under different circumstances. The empirical evidence in cross-country studies is ambiguous. Findings vary not only wh the samples of countries (wh varying data qualy), but also wh different model specification and estimation methods, see e.g. Perotti (1996) for a cross-sectional analysis and Barro (2000) and Forbes (2000) for panel analyses. In a meta analysis of growth regressed on inequaly de Dominicis et al. (2008) note that cross-sectional studies tend to find negative coefficient estimate and panel studies a posive one. This need not necessarily be a contradiction as the short- and medium-term effects of changes in inequaly may differ from the longterm effects. In a panel data context the regression equation of output growth on inequaly is typically given by log(y ) = α + φ log(y i,t 1 ) + m β k x k + γgini + ɛ (1) where y stands for per capa income in country i in period t, Gini for the Gini coefficient and x k for the k-th out of m addional regressors (for instance investment, schooling, fertily, inflation, rule of law etc). Using this linear regression model Barro (2000) finds a posive coefficient on the Gini coefficient in rich countries and a negative coefficient in poor ones. His threestep Least Squares estimation results are qualatively supported by Lin et al. (2009) using a threshold regression. The differing effects of inequaly on growth in rich as compared to poor countries may point to a nonlineary in the relation. This is an issue elaborated upon by Banerjee and Duflo (2003), who argue that linear models such as (1) are misspecified because the growth rate should be an inverted U-shaped function of net changes in inequaly. 2 They estimate such a model wh the help of 1 This line of reasoning is related to the idea of inequaly of opportuny, which has received notable attention in recent years, see for example Ferreira and Gignoux (2011) and references therein. Inequaly is considered to consist of two components, one of which is assumed to be performance-based and hence under individuals control, while the other one is determined by circumstances beyond individuals control, such as their ethnic background or their parents level of education. One might purport the first component to influence growth posively and the second one to influence negatively. Marrero and Rodriguez (2010) find evidence supporting this hypothesis for a U.S. micro-level data set. It is beyond the scope of this paper to pursue possible inequaly of opportuny decomposions at the aggregate cross-country level. 2 It is worth mentioning that there is a vast strand of lerature on nonlinear effects operating k=1 5

6 a kernel regression and show that different estimation results in the lerature can be explained as a consequence of this nonlineary. A theoretical model motivating this nonlinear relation has been put forward by Galor and Moav (2004): The key argument is that in an economy s early, physical capal-driven stages of development, inequaly helps to channel resources towards individuals wh a higher propensy to save, thus spurring growth; but in developed, human capal-based economies, inequaly and inegalarian access to education in particular, are detrimental for output growth. This is a motivation for including a particular nonlineary in the effect of inequaly on output growth, more specifically wh a dependence on income per capa, in our modeling approach. A second feature will be to allow for unobservable countryspecific characteristics, which can reduce the omted variable bias (see Forbes, 2000; Herzer and Vollmer, 2012). The third issue we address is the potential endogeney of the inequaly variable (see Deininger and Squire, 1998), which we can contour in a bivariate setting. Another feature of our paper is the use of the Standardized World Income Inequaly data set (Solt, 2009, version 4.0, released in September 2013), a data set covering relatively many countries and time periods, while maintaining comparabily. But the most salient idea and our main motivation for revising the output growth vs. inequaly nexus is the following: In this strand of the lerature, the different measurement frequencies of output growth and inequaly have hardly received any attention. In fact, output growth is observed at least annually in cross-country data sets, while the Gini coefficient proxying for inequaly only exhibs notable changes over a multi-year horizon. For some countries the Gini coefficient is simply not observed annually, for others is, but s persistence is so high that remains virtually unchanged on a year-to-year basis. For many researchers a natural way of dealing wh a Gini coefficient taken at the multi-annual frequency and output growth observed annually is to bring the latter to the same, low frequency: They simply take multi-year averages of output growth. This paper argues that simple averaging entails a loss of information which might distort the estimation results. in the other direction, i.e. how output growth affects inequaly. The Kuznets hypothesis (see Kuznets, 1955) states that inequaly increases wh income in the early stages of a country s industrial development and decreases when a certain income threshold has been reached, due to the trickle-down-effect and the welfare state. The empirical validy of the Kuznets hypothesis is, however, controversial (see Deininger and Squire, 1998). 6

7 We will instead turn to econometric techniques specifically designed for making optimal use of variables observed at different frequencies: In the next section we will review MIDAS models and extend them to build our nonlinear MIDAS Panel VAR of output growth and inequaly. 3 Extending the MIDAS approach to Panel Vector Autoregressions Mixed Data Sampling Models (MIDAS) were introduced into the lerature by Ghysels et al. (2002) and elaborated upon by Ghysels et al. (2005, 2006) and Andreou et al. (2010), wh the aim of making optimal use of all available data when some variables are measured at a higher frequency than others. A typical motivation has been the area of financial macroeconomics, where one might regress a monthly-observed real economic variable such as output or employment growth, inter alia, on financial variables like stock prices which are observed at a much higher frequency. The key idea of the MIDAS approach is to use a suable parametric aggregation scheme for the high-frequency (HF) variable, specifically a scheme striking a good balance between flexibily and parsimony. A simple MIDAS model for the regression of a low-frequency (LF) variable y t on the HF variable x (m) t can be wrten as y t = β 0 + β 1 K k=1 B(k; θ) x (m) + ɛ t k t, (2) m where m denotes the number of HF observations per LF observation, and L 1 m represents a lag operator such that L 1 m x (m) t = x (m). The weighting function B(k; θ) t k m wh parameter vector θ keeps the model parsimonious, irrespective of the number of HF lags, K. Possible weighting functions include the Exponential Almon Lag and the Beta Lag specifications, which, depending on the parameter vector θ, allow flat, decreasing, increasing and hump-shaped weights to be obtained. It should be noted that the special case of a MIDAS model wh flat weights equals a simple averaging of observations. However, Andreou et al. (2010) find that the tradional OLS estimator applied to averaged data, i.e. using flat weights, can be inconsistent if the data generating process is different, while the MIDAS estimator using flexible weights is consistent and asymptotically efficient. When now turning to use the MIDAS framework for our re-examination of the 7

8 growth vs. inequaly nexus wh panel data, we need to address two important issues: First, the classical MIDAS setting regresses an LF variable on an aggregation of HF variables. In our application, output growth as the dependent variable wh s at least annual observations is the HF variable relative to the more slowly changing Gini coefficient. Regressing a HF variable on an LF variable is actually a reverse MIDAS setting. Single-equation reverse MIDAS regression models are not yet as developed, but in addion would in any case suffer from the endogeney problem for our application. So we will work wh a bivariate setting for the joint determination of output growth and inequaly in a MIDAS framework. Secondly, this MIDAS-VAR has then to be extended into the panel dimension, which, to our knowledge, has not yet appeared in the lerature. Let us consider MIDAS-VARs, as discussed by Ghysels (2012) and Götz and Hecq (2014), in more detail. For each LF time period t, the m HF observations x H,s t (wh 1 s m) are stacked, followed by the LF observation x L t. The resultant (m + 1)-dimensional vector is called x t : x H,1 t x t = x H,m t x L t The structural MIDAS-VAR wh one lag, 3 (3) A c x t = A 1 x t 1 + u t, (4) in more detail reveals the underlying MIDAS-VAR structure: 1 0 ρ Now 1L 1 ρ m,m 1 1 ρ Now ml ρ Now L1 ρ Now Lm 1 x H,1 t x H,m t x L t ρ 11 ρ 1m ρ 1L = 0 ρ mm ρ ml ρ L1 ρ Lm ρ LL x H,1 t 1 x H,m t 1 x L t 1 + (5) The matrix A c refers to the contemporaneous relations between the LF and HF variables during the LF period t, while the entries of A 1 determine the impact of 3 While we will restrict the analysis to a model wh one lag for the ease of exposion, the argument easily carries over to further lags. u H,1 t u H,m t u L t 8

9 the lagged LF and HF variables from period t 1 on those in period t: The entries of the m m upper-left submatrix of A c capture the autoregressive dynamics of the HF variable of different subperiods whin the same LF period. It has a lower-triangular form, representing the fact that only HF observations of past rather than future subperiods can have an impact in any given subperiod. The first m rows last entries, ρ Now sl (wh 1 s m), capture the contemporaneous impact of the LF variable on the HF variable in the subperiods, also called Nowcasting Causaly in a MIDAS framework (see Götz and Hecq, 2014). Similarly, the first m entries of the last row of A c, ρ Now Ls (wh 1 s m), refer to the contemporaneous impact of the HF variable on the LF variable. Looking at the matrix A 1, we see that the entries of the m m upper-right submatrix govern the autoregressive dynamics of the HF variable of the past LF period. A MIDAS-PVAR of lag order 1 can incorporate an AR-process up to lag order m for the HF variable, leading to the upper-triangular structure of that submatrix. The first m rows last entries, ρ sl (wh 1 s m), will be the main focus of our analysis because they govern the impact of the lagged LF variable on the HF variable in the subperiods ("Reverse MIDAS"). Conversely, the first m entries of the last row of A 1, ρ Ls (wh 1 s m), measuring the influence of the (aggregated) lagged HF variable on the LF variable is the "Classical MIDAS" case. Obviously, we will later on impose some restrictions on the parameters in A c and A c and on the variance-covariance matrix of the disturbance vector. But let us now proceed to extend this pure time-series MIDAS-VAR to the panel dimension, which is one of the contributions of this paper. A panel data estimation has many well-known advantages compared to time series, and, inter alia, results can be obtained wh fewer time periods available ("short T, large N"). This is relevant for our investigation of the relation between inequaly and output growth on a cross-country data set. In general, for each cross-sectional un i, 1 i N, the data vectors x for all LF periods 1 t T, as defined in (3), are stacked below each other. Then the cross-sectional vectors x i are stacked. For a MIDAS-PVAR wh the high frequency variable being observed m times during each LF period, N cross-sectional uns, T LF time periods, un-specific fixed effects c 0 i and a homogeneous time trend c1, the model is of dimension N T (m + 1) and looks as follows: 9

10 (I N T A c ) x 11 x 1T x i1 x it x N1 x NT = c 0 1 c 0 1 c 0 i c 0 i c 0 N c 0 N c 1 c 1 c 1 + t + (I c 1 N T A 1 ) c 1 c 1 x 10 x 1,T 1 x i0 x i,t 1 x N0 x N,T 1 + u 11 u 1T u i1 u it u N1 (6) The matrices I N A c and I N A 1 are block-diagonal. The exclusion of crosssectional dependence keeps the model parsimonious and estimable, subject to the condion that A c and A 1 are appropriately parametrized. A further point is the introduction of state-dependencies into this MIDAS-PVAR. As has been argued above, is desirable that our model should be able to capture a possible nonlineary in the effects of inequaly on output growth. This might for instance be caused by dependence on a condioning variable cond, which, following the lines of thinking of Barro (2000) and Galor and Moav (2004), could be income per capa proxying for a country s level of development. We will draw on the methodology developed by Binder and Offermanns (2007), who discuss statedependent effects using semi- and nonparametric specifications, and use a simple affine-linear polynomial for the condioning functional. So we let an entry of A 1, say a, depend on the condioning variable by u NT a = a 0 + a 1 cond. (7) This condioning functional a appears in all the entries of the matrix A 1 which involve the impact of inequaly on output growth in a subperiod. 4 Having constructed 4 In fact, the dependence on the condioning variable will make the matrix A 1 look different for each cross-sectional un. To simplify notation we will suppress the i-subscript in this context. Working wh a condioning variable that is constant over time (for instance, taking the income per capa levels at the beginning of the sample period) and standardized around zero ensures that we can derive our uncondional QML estimator in an analogous way to Binder et al. (2005) whout condioning. 10

11 a nonlinear 5 MIDAS-PVAR, we will now address the question how to estimate. 4 Monte Carlo Simulation: Estimating a MIDAS-PVAR model wh Uncondional QML 4.1 The Estimator In essence, (6) is a particular case of a "short T, large N" PVAR. This structure suggests the use of the uncondional PVAR Quasi Maximum Likelihood Estimator (PVAR-QML) proposed by Binder et al. (2005): Specifically modeling the inial observations, they can overcome the bias wh fixed-effect models of this type. In their paper, Binder et al. (2005) conclude that the PVAR-QML estimator outperforms various GMM estimators in fine samples, whether or not the error terms are distributed normally. The PVAR-QML estimator does not yet seem to have been used in the MIDAS context, so we adjust and extend accordingly: We can wre the structural MIDAS-PVAR (6) for each cross-sectional un i, 1 i N, and LF period t, 1 t T, as Premultiplying (8) by A 1 c wh c 0 i = A 1 c A c x = c 0 i + c 1 t + A 1 x i,t 1 + u. (8) c 0 i, c1 = A 1 c yields the reduced-form representation: x = c 0 i + c 1 t + A x i,t 1 + ɛ, (9) c 1, A = A 1 c A 1 and Ω ɛ = A 1 c Ω u A 1 c. In order to carry out the estimation, will be useful to rewre (9) in terms of deviations from the time trend: (I AL)(x µ i γ t) = ɛ (10) where I is the identy matrix of dimension m + 1, L denotes the lag operator and holds that µ i = (I A) 1 (c 0 i A(I A) 1 c 1 ) and γ = (I A) 1 c 1. 5 When we use the term nonlinearies in the following, refers to the state-dependencies discussed here. It is beyond the scope of this paper to deal wh the inclusion of other types of nonlinearies into a MIDAS-PVAR model. 11

12 We will consider (10) the model we observe. As in Binder et al. (2005), we assume that the following assumptions hold: The available observations are x i0, x i1,, x it wh T 2. The disturbances ɛ, t T are independently and identically distributed (i.i.d.) for all i and t wh E(ɛ ) = 0 and V ar(ɛ ) = Ω ɛ, where the latter is a posive define matrix. Furthermore, the inial deviations ξ i0 = x i0 µ i (11) are assumed to be i.i.d. across i, wh zero means and the constant nonsingular variance V ar(ξ i0 ) = Ψ ξ0. In this paper we will assume that all eigenvalues of A fall inside the un circle, hence the process is weakly (trend-)stationary. This is a special case of Binder et al. (2005) and means the process (10) can eher start from an infine or fine past. 6 Taking first differences of (10) eliminates the un-specific effects: or, equivalently, in a demeaned way wh x = x γ. (I AL)( x γ) = ɛ, t = 2, 3,, T. (13) (I AL) x = ɛ, t = 2, 3,, T. (14) For the inial first-differenced observation x i0 then holds that x i0 i.i.d. (0, Ψ), (15) whose variance Ψ can be expressed as a function of the variance of the inial observation: Ψ = (I A)Ψ ξ0 (I A) + Ω ɛ (16) Also, we define x i = ( x i1, x i2,, x it ) and η i = ( x i1, ɛ i2,, ɛ it ). The variance of η i is 6 If the process has been in operation for a very long time, one can wre (11) as ξ i0 = A j ɛ i, j. (12) j=0 This proves useful for the technical implementation of the process inialization in our Monte Carlo simulation. 12

13 Wh Ψ Ω ɛ 0 0 Ω ɛ 2Ω ɛ Ω ɛ 0 0 Ω ɛ 2Ω ɛ Ω ɛ 0 Σ η =. (17) 0 Ω ɛ 2Ω ɛ Ω ɛ 0 0 Ω ɛ 2Ω ɛ I 0 0 A I 0 R =, (18) 0 A I Σ x = R 1 Σ η R 1 and S N, x = 1 N N i=1 x i x i, we follow the same steps as Binder et al. (2005) and argue: The MIDAS-PVAR-QML estimator ρ = (vec(a), vech(ω ɛ ), vech(ψ) ) can be obtained by maximizing the following log-likelihood function based on the joint probabily distribution of ( x i1, x i2, x it ): (m + 1)NT ll = log(2π) N 2 2 log Σ η N 2 tr(σ 1 x S N, x). (19) Wh m as the number of HF observations per LF period, the dimensionaly has been adjusted for the MIDAS case. It follows from Binder et al. (2005) that this MIDAS-PVAR-QML estimator is consistent and asymptotically efficient. 4.2 Monte Carlo Simulation: Data-Generating Process Before taking this uncondional MIDAS-PVAR-QML estimator to the data on inequaly and output growth, we intend to gain some insights into s performance by conducting a Monte Carlo simulation. The programming is carried out in Mata in Stata. Concerning the data generating process for the simulation, we will specialize the MIDAS-PVAR (6) so that captures important characteristics of cross-country data of output growth and inequaly. For instance, while we allow for Granger causaly from inequaly to output growth and vice versa, there is no nowcasting causaly in eher direction. 7 This implies that both the last row and the last column of A c, 7 That inequaly and output growth only affect each other wh a lag both reflects the consensus in the lerature and findings from our data set: As we will show later, the parameter estimate of 13

14 except for the last entry of 1, only contain zeros. As regards the dimensionaly of the model, we settle on m = 4, as we do in the empirical application, thus the HF variable is observed four times in every LF period t. 8 Wh the country-specific fixed effects c 0 i and a deterministic time trend c 1 we have the following structural MIDAS-PVAR, 1 i N, 1 t T : = x H,1 ρ x H,2 ρ 2 ρ x H,3 ρ 3 ρ 2 ρ 1 0 x H, x L c 0 ih c 1 c 0 H ρ 4 ρ 3 ρ 2 ρ a ih c 1 c 0 H 0 ρ 4 ρ 3 ρ 2 ρ HL,2 a ih + c c 0 1 H t ρ ih c 1 4 ρ 3 ρ HL,3 a c 0 H ρ 4 ρ HL,4 a il c 1 L b 1 b 2 b 3 b 4 α x H,1 i,t 1 x H,2 i,t 1 x H,3 i,t 1 x H,4 i,t 1 x L i,t 1 + u H,1 u H,2 u H,3 u H,4 u L (20), wh a = a 0 + a 1 cond. The interpretation of the parameters involved in (20) are as follows: 1. Our main parameters of interest are a, ρ HL,2, ρ HL,3 and ρ HL,3 because they determine the impact of the lagged LF variable (inequaly in our application) on the HF variable (output growth) in the current period s four subperiods. While a captures this impact in the first subperiod, the scaling factors ρ HL,2, ρ HL,3 and ρ HL,4 allow for the effect to vary over subperiods. Through the condioning functional, a self involves the two parameters a 0 (independent of the condioning variable cond) and a 1, capturing the impact of cond on the effect of the LF variable on the HF variable. By setting a 1 equal to 0, one can shut down the influencing channel of the condioning variable and end up wh a linear model again. 2. b 1, b 2, b 3 and b 4 determine the impact of the lagged HF variable on the current period LF variable and can be viewed as unrestricted aggregation weights. 9 the covariance between inequaly and output growth is not significantly different from zero. 8 Addional simulations have shown that the estimator performs similarly wh a different m. 9 We leave the four parameters b 1, b 2, b 3 and b 4 unrestricted instead of specifying them in the 14

15 3. The HF variable follows an AR(4)-process wh autoregressive coefficient ρ. Hence, the HF variable in a given subperiod depends on s realization of the previous subperiod by ρ, in the subperiod before by ρ 2 and so on, up to four subperiods back, which refers to the same subperiod in the previous LF period. Actually, the HF lag order of four is the highest that can be modeled in a MIDAS-PVAR wh one lag and m = 4. Note that the coefficient restrictions in both A c and A 1 need to be imposed to ensure that all HF observations of all subperiods depend on their previous subperiods in the same way, whether or not these observations occur at the beginning or the end of an LF period. 4. The LF variable follows an AR(1)-process wh autoregressive coefficient α, thus depending on s observation in the previous LF period. 5. This being a structural PVAR, the variance-covariance matrix of the disturbance vectors is diagonal. Wh σ HH, the variance of the HF variable (assumed to be the same in each subperiod) and σ LL, the variance of the LF variable, we have: σ HH σ HH Ω u = E(u u ) = 0 0 σ HH 0 0. (21) σ HH σ LL In order to obtain the reduced-form model for our estimation, we multiply the structural form (20) by ρ A 1 c = 2ρ 2 ρ (22) 4ρ 3 2ρ 2 ρ classical MIDAS way as an exponential Almon Lag or a Beta Lag. Although these specifications would save one parameter to estimate, they would entail an addional complexy in estimation by introducing yet another nonlineary. Leaving the three parameter weights unrestricted can also be considered a U-MIDAS model as proposed by Foroni et al. (2012). 15

16 The resultant reduced-form MIDAS-PVAR is + x H,1 x H,2 x H,3 x H,4 x L = c 0 ih c 0 ih cih 0 c 0 ih c 0 il + c 1 H c 1 H c 1 H c 1 H c 1 L ρ 4 ρ 3 ρ 2 ρ a ρ 5 2ρ 4 2ρ 3 2ρ 2 ρ HL a 2ρ 6 3ρ 5 4ρ 4 4ρ 3 ρ HL a 4ρ 7 6ρ 6 7ρ 5 8ρ 4 ρ HL a b 1 b 2 b 3 b 4 α t (23) x H,1 i,t 1 x H,2 i,t 1 x H,3 i,t 1 x H,4 i,t 1 x L i,t 1 + ɛ H,1 ɛ H,2 i,t ɛ H,3 ɛ H,4 ɛ L, where, for ξ {c 0 ih ; c1 H }, ξ = (ρ + 1) ξ, ξ = (2ρ 2 + ρ + 1) ξ, ξ = (4ρ 3 + 2ρ 2 + ρ + 1) ξ, as well as ρ HL = ρ + ρ HL,2, ρ HL = 2ρ2 + ρ ρ HL,2 + ρ HL,3 and ρ HL = 4ρ3 + 2ρ 2 ρ HL,2 + ρ ρ HL,3 + ρ HL,4. The variance-covariance matrix of the disturbance vector of this reduced-form MIDAS-PVAR is = Ω ɛ = E(ɛ t ɛ t) = A 1 c Ω u A 1 c (24) σ HH ρ σ HH 2ρ 2 σ HH 4ρ 3 σ HH 0 ρ σ HH (ρ 2 + 1) σ HH (2ρ 3 + ρ) σ HH (4ρ 4 + 2ρ 2 ) σ HH 0 2ρ 2 σ HH (2ρ 3 + ρ) σ HH (4ρ 4 + ρ 2 + 1) σ HH (8ρ 5 + 2ρ 3 + ρ) σ HH 0. 4ρ 3 σ HH (4ρ 4 + 2ρ 2 ) σ HH (8ρ 5 + 2ρ 3 + ρ) σ HH (16ρ 6 + 4ρ 4 + ρ 2 + 1) σ HH σ LL Hence, the number of parameters to estimate in our Monte Carlo simulation is 15 for the nonlinear model and 14 for the linear model (wh a 1 set to zero). The true parameter values of the underlying data generating process are chosen so that the real parts of the eigenvalues of the autoregressive matrix A in (23) are less than one in absolute values, which ensures (trend-)stationary. At the same time we attempt to represent stylized features of inequaly and output growth data, in particular a high persistence of the former (α = 0.8). We work wh three different parameter combinations to evaluate the performance of the estimator in different 16

17 plausible environments. Wh 50 cross-sectional uns 10 and 5 LF time periods, we create an appropriate "Small T, large N" setting. Our time-constant condioning variable is drawn from a normal distribution across the cross-sectional uns. For our simulations, we contour inialization problems by letting the data generating process run for 50 periods, which we discard, before we obtain the 5 observations per cross-sectional un to be used for the estimation repetions are conducted. While the data generating process in our simulation is the MIDAS-PVAR from (23), the performance of the MIDAS-PVAR-QML estimator can best be appreciated in comparison to a classical PVAR-QML estimator that uses all the data only at the low frequency. Intertemporally averaging the HF data to form a bivariate LF-only PVAR gives ( ) ( ) ( x H,av c 0 i1 x L = c 0 + i2 c 1 1 c 1 2 ) t + ( ρ av b ) ( ) ( ) a x H,av i,t 1 ɛ H,av α x L + i,t 1 ɛ L, (25) wh a = a 0 + a 1 cond. For the LF-only model the variance-covariance matrix of the error terms is ( ) E(ɛ t ɛ σ 11 σ 21 t) = (26) σ 21 σ 22 and there are 9 (10) parameters to estimate in the (non-)linear specification Monte Carlo Simulation: Results Tables (1), (2) and (3) show the three different parameter combinations ("true value") and present performance indicators of the MIDAS-PVAR-QML and LF-only PVAR-QML estimators, namely the bias, root mean squared error and ratio of the estimated standard error to the standard deviation over the 2000 replications. Both the largest eigenvalue and the VAR Model R 2 by Pesaran et al. (2000) give insights 10 Addional simulation results suggest, not surprisingly, that the performance of the MIDAS- PVAR-QML estimator is further enhanced when N is increased to 200, but keeping N at a moderate magnude is more relevant to us in the light of our empirical data set. 11 The σ 21 parameter should be zero if the data generating process is correct and there is no contemporaneous impact of eher variable on the other. In the empirical application we will estimate σ 21 in order to verify this assumption. 17

18 into the persistence of the process. 12 While the data generating process involves the condioning variable, we estimate the model using both the nonlinear and linear variants of the estimator, where the latter restricts a 1 to zero. Looking at the parameter combinations of the first setting in Table (1), we note a negative impact of the lagged LF variable on the HF variable (a 0 = 0.4), which varies over the HF subperiods, as ρ HL,2, ρ HL,3 and ρ HL,4 indicate. Also, this impact depends negatively on a condioning variable (a 1 = 0.2). The performance indicators of the nonlinear MIDAS-PVAR-QML estimator in this setting are very convincing overall, showing only very small biases and ratios of the standard error to standard deviation which are close to 1. It is not surprising that due to s misspecification the linear MIDAS-PVAR-QML estimator exhibs slightly larger biases, in particular for ρ and α. However, still does fairly well (and for a few parameters s estimates are even closer to the true parameters than the nonlinear estimates). Turning to the LF-only estimator, we find evidence for our argument that working wh averages entails a loss of information that can distort the estimation result. Several parameters are estimated wh larger biases, the most sizable being a 0, which is even closer to -0.5 than to s true value of Even if the LF-only estimator manages to come slightly closer to the true a 1 -value in measuring the dependence on the condioning variable, s MIDAS counterpart is clearly preferable. One should also not forget that the latter can adequately estimate a time-varying impact of the LF variable on the HF variable in different subperiods, which the LF-only estimator, by construction, cannot. Table (2) shows the second parameter setting, which is characterized by different time trends for the HF and LF variables, a stronger HF persistence parameter ρ, 12 The VAR Model R 2 by Pesaran et al. (2000) is computed as R x 2 [Ω ɛ] ss = 1 [, (27) j=0 CjΩɛC j ]ss wh C 0 = I, C 1 = (I A) and C j = C ja, j = 2, 3, and [G] ss denoting the s-th diagonal entry of the matrix G. It is smaller than 1 for a (trend-)stationary process but increases wh persistence. 13 One should keep in mind that not all MIDAS parameters have counterparts in the LF-only process and vice versa: For instance, the LF-only ρ AV parameter is the autoregressive coefficient for the averaged HF value and cannot be compared to the intra-lf autocorrelation coefficient ρ of the HF variable in the MIDAS model. Moreover, b, which captures the lagged impact of the average HF variable on the LF variable, would be equivalent to the sum of b 1, b 2, b 3 and b 4 in the MIDAS model, and σ 11 = 4 ( ) 1 2 σ 4 HH. This has been taken into account in the column of the "true value". 18

19 Para- True Nonlinear MIDAS Linear MIDAS meter Value Bias RMSE SE/SD Bias RMSE SE/SD c 1 H c 1 L ρ a a ρ HL, ρ HL, ρ HL, b b b b α σ HH σ LL Para- True Nonlinear LF-only Linear LF-only meter Value Bias RMSE SE/SD Bias RMSE SE/SD c c ρ av a a b α σ σ σ Largest Eigenvalue of A in the DGP: Model R 2 (see (27)): Table 1: MC Setup 1: Parameter estimates of the MIDAS-PVAR-QML estimator (based on (23) and (24)) and the tradional LF-only PVAR-QML estimator (based on (25) and (26)), respectively in the nonlinear and linear (a 1 = 0) variants. The data generating process is of the nonlinear MIDAS-PVAR form wh N= 50 and T = repetions are conducted. The bias is calculated as the average parameter estimate minus the true value. RMSE stands for the root mean squared error and SE/SD measures the ratio of the estimated standard error to the standard deviation of the estimator over the 2000 replications. negative b-parameters and a high posive a 0 for the lagged impact of the LF on the HF variable, while the dependence on the condioning variable is still negative (a 1 ). We can see that the performance is rather similar to the first setting. But while the nonlinear MIDAS-PVAR-QML estimator continues to do well, s linear counterpart does hardly worse, which may be due to the smaller magnude of the a 1. Concerning the LF-only estimator, s bias in estimating a 0 is even more severe than in the first setting - leading to an estimate of around 0.97 rather than although 19

20 some other parameters are estimated rather well. We conclude that this distortion of LF-only estimator seems to increase wh the magnude of the a 0 -parameter, while the MIDAS estimator does not show such a problem. Para- True Nonlinear MIDAS Linear MIDAS meter Value Bias RMSE SE/SD Bias RMSE SE/SD c 1 H c 1 L ρ a a ρ HL, ρ HL, ρ HL, b b b b α σ HH σ LL Para- True Nonlinear LF-only Linear LF-only meter Value Bias RMSE SE/SD Bias RMSE SE/SD c c ρ av a a b α σ σ σ Largest Eigenvalue of A in the DGP: Model R 2 (see (27)): Table 2: MC Setup 2: Parameter estimates of the MIDAS-PVAR-QML estimator (based on (23) and (24)) and the tradional LF-only PVAR-QML estimator (based on (25) and (26)), respectively in the nonlinear and linear (a 1 = 0) variants. The data generating process is of the nonlinear MIDAS-PVAR form wh N= 50 and T = repetions are conducted. The bias is calculated as the average parameter estimate minus the true value. RMSE stands for the root mean squared error and SE/SD measures the ratio of the estimated standard error to the standard deviation of the estimator over the 2000 replications. Our conclusions from the first two settings are confirmed in the third one, where a 0 and a 1 are both of the same magnude, the impact of the lagged LF variable on the HF variable varies more over the subperiods and there is no feedback from the lagged HF variable on the LF variable, i.e. the b-parameters are zero. The bias of 20

21 the LF-only PVAR-QML estimator is smaller than in the second setting but is still notable. All the while, the MIDAS-PVAR-QML estimator performs convincingly. Para- True Nonlinear MIDAS Linear MIDAS meter Value Bias RMSE SE/SD Bias RMSE SE/SD c 1 H c 1 L ρ a a ρ HL, ρ HL, ρ HL, b b b b α σ HH σ LL Para- True Nonlinear LF-only Linear LF-only meter Value Bias RMSE SE/SD Bias RMSE SE/SD c c ρ av a a b α σ σ σ Largest Eigenvalue of A in the DGP: Model R 2 (see (27)): Table 3: MC Setup 3: Parameter estimates of the MIDAS-PVAR-QML estimator (based on (23) and (24)) and the tradional LF-only PVAR-QML estimator (based on (25) and (26)), respectively in the nonlinear and linear (a 1 = 0) variants. The data generating process is of the nonlinear MIDAS-PVAR form wh N= 50 and T = repetions are conducted. The bias is calculated as the average parameter estimate minus the true value. RMSE stands for the root mean squared error and SE/SD measures the ratio of the estimated standard error to the standard deviation of the estimator over the 2000 replications. After this Monte Carlo simulation, which has highlighted the benefs of working wh a MIDAS model rather than LF-only averages, we are now ready to take our MIDAS-PVAR modeling technique to the data and see whether we can gain new insights on the relation between inequaly and output growth. 21

22 5 Empirical Application: MIDAS-PVAR of Inequaly and Output Growth Data 5.1 The data set When constructing a cross-country panel data set on income inequaly and output growth, one comes across the well-known problem that the data has to fulfill the conflicting goals of comparabily and wide coverage across countries and years. High qualy data on inequaly is particularly hard to come by; even data sets compiled by the Uned Nations and the World Bank combine data income definions that vary over countries and years. 14 Relying on high-qualy, comparable inequaly data would reduce the sample significantly and favor developed countries. However, recently, a new data set has been created by Solt (2009): The Standardized World Income Inequaly Database (SWIID) combines and standardizes inequaly data from various sources (including Uned Nations Universy s World Income Inequaly Databases, The World Bank s PovcalNet, Eurostat, Luxembourg Income Study and national statistical offices). In the standardization procedure missing data are imputed based on other variables and sources, taking into account data uncertainty. 15 An addional advantage is that the SWIID database is so new - we will use version 4.0 which was released in September that is not used in other studies we are aware of. Hence, we will take gini, the Gini coefficient of income inequaly after taxes and transfers, from SWIID. In order to compute output growth, we rely on the Penn World Tables 8.0 (Feenstra et al., 2013), which is a standard choice in cross-country growth regressions. We divide the rgdp(na) (real gross domestic product using national accounts) variable by the population size, pop. By taking the natural logarhm and computing yearon-year first differences, we obtain the annual output growth rate, growth. In levels, the GDP variable, income, will also be used as the condioning variable in the nonlinear specification. To ensure exogeney of the condioning variable, we will take 14 Varying definions include income inequaly measured before or after taxes and transfers, income inequaly at the household or individual level, inequaly of the whole country or only the urban population and so forth. 15 In fact, SWIID presents 100 imputed values considered plausible for every country every year. We will work wh the arhmetic mean of these 100 imputations. An explorative study shows that our results would hardly change if we carried out the estimation wh each of the 100 values separately and averaged the estimation results. 22

23 income-values from the year 1989 (the beginning of our sample period), demean cross-sectionally and standardize so that measures income per capa relative to the worldwide sample. We proceed analogously wh the other condional variables that we use in addional specifications, namely humancap and democracy. The former is the human capal index from the Penn World Tables 8.0 see (Feenstra et al., 2013), representing an educational index, while democracy, the index of democratic rule as captured by the Poly IV index (Marshall et al., 2013), might be seen in the context of redistributive pressure for distortionary taxation. Concerning our main variables of interest, we confirm that in our combined data set growth tends to be available for every year, in contrast to gini, which even in the extensive SWIID database does not have values for all countries continuously throughout the years. And in those countries wh yearly gini observations, there is hardly any variation in adjacent years. This makes our choice of a MIDAS model wh growth observed yearly and gini observed on a multiple-year horizon appropriate. As regards the length of this horizon, we experiment wh 3-year, 4-year and 5-yearperiods in order to strike the right balance between sufficient time variation of the Gini variable and the number of countries included in a balanced panel. As our MIDAS-PVAR-QML estimator works wh a model in first differences, we have to take first differences of growth and gini over the LF periods, requiring the availabily of observations even further back. Settling on a 4-year LF horizon leaves us wh a balanced panel of 63 countries and five quadrennial time periods from 1989 to 2008, which we can estimate appropriately wh our "short T, large N" MIDAS-PVAR-QML estimator. 16 As Table (7) in the Appendix shows, our sample includes countries from every region of the world. OECD-countries (26 out of 63) and Non-OECD countries are equally well-represented. The average Gini coefficients and average yearly output growth rates across the 1989 to 2008 sample period reveal strong cross-sectional variation. Maurius is the country wh the most egalarian income distribution on average (0.1746), closely followed by the Scandinavian countries wh Gini coefficients in the low 0.20 s. Countries wh average Gini coefficients higher than 0.50 during 16 For reasons of comparabily of the alternative specifications we only include countries where we have observations of all growth, gini, income, humancap and democracy. In line wh other cross-country studies, we drop Venezuela for being an oil producer and both Rwanda and Sierra Leone due to civil wars in the years covered. Robustness checks show that including these countries into the sample (or dropping some others) would not qualatively affect the conclusions. 23