🎯pnd | phd16_QM筆記 | QM-第6章 | 參考論文pdf | QM-Final Report |

Here’s a comprehensive summary of the paper, "The Predictability of Aggregate Consumption Growth in OECD Countries: A Panel Data Analysis":

1. Theory :

• The paper examines the predictability of aggregate private consumption growth within the framework of modern consumption theories.
• It incorporates elements such as habit formation, intertemporal substitution, current income consumption (rule-of-thumb consumers), and non-separabilities between private consumption, hours worked, and government consumption.
• Theoretical derivations are based on optimizing intertemporal utility models that challenge the random walk hypothesis implied by the Permanent Income Hypothesis (PIH).

  
  

理論基礎：


• 文章探討 OECD 國家私人消費增長的可預測性，結合了永久收入假說及消費理論中的新興要素，如習慣形成、跨期替代、當期收入消費以及私人消費與工時或政府消費之間的非分離性。
• 假設消費者在跨期優化時考慮習慣、利率變動、政府消費與工時等因素，同時考慮部分消費者為“規則型消費者”，即完全依賴當期可支配收入。

PIH恆常所得假說Permanent Income Hypothesis 由米爾頓·弗里德曼提出，認為消費者的消費決策主要基於其預期的長期平均收入（即恆常所得），而不是當期收入的波動。

隨機漫步假說Random Walk Hypothesis: 在這裡的應用是指，根據PIH，消費者的消費行為應該是不可預測的，因為它只會對新的、不可預測的信息做出反應。具體來說，如果消費者的預期是正確的，那麼每期的消費變動應該是隨機的，這就像隨機漫步一樣。

舉個例子，如果一個消費者的收入突然增加，但這個增加是暫時的，根據PIH，他們不會大幅增加消費，因為他們知道這不是恆常所得的一部分。相反，如果他們預期收入會長期增加，他們才會相應地增加消費。

使用“隨機漫步”這個名詞意味著消費變動是不可預測的，這與股票價格的隨機漫步假說類似，表示未來的變動僅取決於新的、不可預測的信息，而不是過去的數據。

恆常所得假說
Permanent Income Hypothesis (PIH)
Incomplete market 2006




---

2. Dependent Variable :

• The growth rate of private consumption in 15 OECD countries over the period 1972–2007.

  
  

  ---

依變數：OECD 國家的私人消費增長率。




---

3. Independent Variables :

• Lagged consumption growth (habit formation).
• Real interest rate (intertemporal substitution effects).
• Growth rate of government consumption (non-separability with private consumption).
• Growth rate of hours worked (non-separability with private consumption).
• Growth rate of current disposable income (rule-of-thumb or liquidity-constrained consumers).

  
  

解釋變數：包括消費的滯後項、實際利率、政府消費增長、工時增長和當期可支配收入增長。。




---

4. Methodology :

• Panel Data Analysis:
• The study uses a panel of 15 OECD countries covering 1972–2007.
• Estimation Methods:
• Employs a Generalized Method of Moments (GMM) estimator in combination with the Common Correlated Effects Mean Group (CCEMG) estimator.
• Addresses parameter heterogeneity, endogeneity, and cross-sectional dependence in the data.
• Monte Carlo Simulations:
• Used to validate the small-sample properties of the estimation approach.
• Corrects for biases due to omitted variables by accounting for cross-sectional dependence via latent common factors.

  
  

研究方法：

• 使用涵蓋 15 個 OECD 國家（1972–2007 年）的面板數據。
• 採用通用相關效應均值組估計（CCEMG），並結合廣義矩估計法（GMM）以解決參數異質性、內生性及誤差跨橫截面相關等問題。
• 通過蒙地卡羅方法模擬驗證方法的樣本特性。(也稱統計類比方法，與它對應的是確定性演算法)



---

5. Hypotheses :

• Aggregate consumption growth is predictable, contrary to the random walk hypothesis.
• Predictability arises due to factors like habit formation, intertemporal substitution, income effects, and non-separabilities.

  
  

假設：
• 假設消費增長受到多種因素的影響，如習慣形成、跨期替代、政府消費的非分離性、工時的非分離性及當期收入消費。



---

6. Findings :

• Significant Predictor:
• Only current income growth has predictive power for aggregate consumption growth, aligning with models emphasizing liquidity constraints or rule-of-thumb consumers.
• Insignificant Factors:
• Habit formation, intertemporal substitution, and non-separabilities between private consumption, hours worked, and government consumption were found to have no significant impact.
• Cross-sectional Dependence:
• Significant evidence of cross-sectional dependence suggests the influence of unobserved global factors (e.g., financial liberalization, synchronized business cycles).
• Methodological adjustments for endogeneity and cross-sectional dependence lead to more reliable estimates compared to studies that do not account for these complexities.

  
  

發現：

• 實證結果顯示，只有當期收入增長對消費增長具有顯著的預測能力，其他因素（如習慣形成、跨期替代、政府與私人消費的非分離性）對消費增長的影響並不顯著。
• 誤差中顯著的跨橫截面相關性表明未觀測到的共同因子影響消費增長的可預測性。



---

7. Contributions :

• Theoretical Contribution:
• Introduces a comprehensive model that integrates multiple dimensions of consumption predictability into a unified framework.
• Methodological Contribution:
• Applies advanced econometric methods (CCEMG and GMM) to handle heterogeneity, endogeneity, and cross-sectional dependence effectively.
• First study to estimate such a general consumption equation in a multi-country panel setting.


貢獻：

• 理論貢獻：提出了一個綜合性模型，涵蓋消費理論的最新進展。
• 方法論貢獻：改進估計方法以處理宏觀數據中的誤差相關性，特別是未觀測因素引起的偏誤。



---

Empirical Contribution:

• Provides robust international evidence on consumption growth predictability, filling gaps in earlier literature that focused on single-country analyses or failed to address cross-sectional dependence.

  
  

• Conclusion:

  
  The study challenges the Permanent Income Hypothesis by highlighting the role of income growth in predicting consumption growth, while refuting the importance of other commonly proposed factors like habit formation and intertemporal substitution. Its methodological advancements enhance the reliability of consumption studies in macroeconomic research.
  

...

SUMMARY

We examine aggregate consumption growth predictability. We derive a dynamic consumption equation which encompasses relevant predictability factors: habit formation, intertemporal substitution, current income consumption and non-separabilities between private consumption and both hours worked and government consumption. We estimate this equation for a panel of 15 OECD countries over the period 1972–2007, taking into account parameter heterogeneity, endogeneity and error cross-sectional dependence using a GMM version of the common correlated effects mean group estimator. Small-sample properties are demonstrated using Monte Carlo simulations. The estimation results support income growth as the only variable with significant predictive power for aggregate consumption growth. Copyright © 2013 John Wiley & Sons, Ltd.

1 INTRODUCTION

The permanent income hypothesis implies that aggregate private consumption follows a random walk (Hall, 1978). Empirical studies show that this random walk hypothesis is not supported by the data since aggregate consumption growth is predictable, at least to some extent. More sophisticated theoretical models capture this fact by introducing various forms of predictability in aggregate consumption growth. Relevant forms are caused by liquidity constraints (Campbell and Mankiw, 1989, 1990, 1991), habit formation (Campbell, 1998; Carroll et al., 2011), intertemporal substitution effects in response to real interest rate changes (Campbell and Mankiw, 1989) and non-separabilities in the utility function between private consumption and government consumption (Evans and Karras, 1998) and between private consumption and hours worked (Basu and Kimball, 2002). Empirically, an often reported finding is the positive impact of aggregate disposable income growth on private consumption growth (i.e. the ‘excess sensitivity’ puzzle), which can be rationalized from models incorporating consumers who base consumption on current income due to liquidity constraints (see Jappelli and Pagano, 1989; Campbell and Mankiw, 1990) or myopia (see Flavin, 1985). These current income consumers are often referred to as ‘rule-of-thumb’ consumers. Recent evidence in favour of current income consumption is provided by Kiley (2010). Other studies, such as Basu and Kimball (2002) and Carroll et al. (2011), argue that predictability stemming from the impact of current disposable income on consumption growth is less relevant once other forms of predictability are taken into account. As Gali et al. (2007) show that different predictability mechanisms have different macroeconomic implications, it is important to correctly identify the relevant forms of predictability. One drawback of all these studies is that they typically focus only on a subset of possible forms of predictability. Moreover, the empirical analysis is usually restricted to a single country (mainly the USA). Studies that present international evidence such as Campbell and Mankiw (1991) and Carroll et al. (2011) use a country-by-country approach. As a result, the additional information in the cross-sectional dimension of the data is not fully exploited. Evans and Karras (1998) and Lopez et al. (2000) use panel data methods but they do not tackle all the complications that arise when estimating aggregate consumption growth equations with macroeconomic data. In particular, they disregard cross-sectional dependence that may stem from the presence of unobserved variables that are common to all countries in the panel.

This paper examines the predictability of aggregate private consumption growth in a panel of OECD countries over the period 1972–2007. The contribution of the paper to the literature is both theoretical and methodological. Theoretically we present a model with consumers who optimize intertemporally. They form habits since their utility also depends on past consumption. They further substitute consumption intertemporally when confronted with real interest rate changes. Finally, their utility is affected by government consumption and also by the number of hours that they work. Following Campbell and Mankiw (1990) we also allow for rule-of-thumb consumers or current income consumers who consume their entire disposable income in each period. This model provides an expression for aggregate consumption growth that can be estimated using macroeconomic data. The five predictability factors incorporated in the model (habits, intertemporal substitution, non-separabilities in utility between consumption and government consumption and between consumption and hours worked, and current income consumption) lead to the dependence of aggregate private consumption growth on its own lag, on the real interest rate, on aggregate government consumption growth, on the growth rate in aggregate hours worked and on aggregate disposable income growth. These predictability factors constitute deviations from perfect consumption smoothing as implied by Hall’s (1978) random walk hypothesis. Our specification for aggregate consumption growth encompasses many of the recent developments in consumption theory. And while our specification nests a number of specifications that have been estimated in the literature previously, to the best of our knowledge no study has yet estimated a specification as general as ours.

Methodologically we estimate the dynamic consumption equation derived in our theoretical model for a panel of 15 OECD countries over the period 1972–2007, making full use of the panel structure of the data. First, we estimate country-specific coefficients which are then combined using the mean group (MG) estimator to obtain estimates for the average effects. This avoids obtaining biased and inconsistent parameter estimates when falsely assuming that the regression slope parameters are identical across countries (see, for example, Pesaran and Smith, 1995). Differences across countries in aggregate consumption growth predictability can, for instance, be due to cross-country differences in financial systems, government policies and demographics. The cross-country estimates from Campbell and Mankiw (1991) and Evans and Karras (1998) indeed show considerable disparity in the predictability estimates obtained from regressions of aggregate consumption growth on current income and government expenditures. Second, we exploit the cross-sectional dependence in the data. Recently, the panel literature has emphasized unobserved, time-varying heterogeneity that may stem from omitted common variables that have differential impacts on individual units (see, for example, Coakley et al., 2002; Phillips and Sul, 2003). These latent common variables induce error cross-sectional dependence and may lead to inconsistent estimates if they are correlated with the explanatory variables. Especially when studying macroeconomic data, such unobserved global variables or shocks are likely to be the rule rather than the exception (see, for example, Coakley et al., 2006; Westerlund, 2008). In the context of aggregate private consumption, common factors may, for instance, be induced by financial liberalization and business cycle synchronization. Rather than treating the resulting cross-sectional correlation as a nuisance, we exploit it to correct for a potential omitted variables bias stemming from unobserved common factors. To this end, we use the common correlated effects (CCE) methodology suggested by Pesaran (2006). The basic idea behind CCE estimation is to capture the unobserved common factors by including cross-sectional averages of the dependent and the explanatory variables as additional regressors in the model. We use the mean group (CCEMG) variant to allow for possible parameter heterogeneity. Next, we suggest a generalized method of moments (GMM) version of the CCEMG estimator to account for endogeneity of the explanatory variables. A Monte Carlo simulation shows that in a dynamic panel data model with both endogeneity and error cross-sectional dependence this CCEMG-GMM performs reasonably well, especially when compared to alternative estimators, for the modest sample size T = 35, N = 15 that is available for our empirical analysis.

The estimation results support rule-of-thumb or current income consumption as the only significant form of predictability. We do not find a significant impact of hours worked on consumption growth. Neither do we find support for habit formation, intertemporal substitution effects and non-separabilities between private consumption and government consumption. Taking into account endogeneity and cross-sectional dependence proves to be important as it has a marked effect on the coefficient estimates. The finding of significant cross-sectional dependence in particular suggests that one or more unobserved common factors affect the predictability of aggregate consumption growth. This suggests that the conclusions obtained by existing studies that use only a time series approach or that use a panel approach without allowing for cross-sectional dependence may be less reliable.

The paper is structured as follows. In Section 2 we derive a dynamic equation for aggregate private consumption growth from a model that encompasses most of the relevant predictability factors discussed in the consumption literature. We discuss specification issues that arise when implementing this equation empirically. In Section 3 we review the different estimators that can be used. Section 4 presents the estimation results for a panel of OECD countries. In Section 5, we investigate the small-sample properties of the considered estimators using a Monte Carlo experiment. Section 6 concludes.

2 THEORY

In this section we first derive a dynamic equation for aggregate private consumption growth from a model that encompasses most of the relevant predictability factors discussed in the literature. Then we discuss a number of specification issues that need to be taken into account before this equation can be estimated.

2.1 The Model

Consider an economy with intertemporally optimizing permanent income consumers. The contemporaneous utility function u of each consumer is of the constant relative risk aversion (CRRA) type and is given by

where Ct is the real per capita consumption level, Ht is the per capita number of hours worked and Gt is real per capita government consumption. The parameter θ is the elasticity of intertemporal substitution for which θ > 0. Under the CRRA utility this parameter is the inverse of the coefficient of relative risk aversion (1/θ). To correctly interpret the other parameters in the utility function (β, γ and π) we also assume that θ < 1 (i.e. the elasticity of intertemporal substitution is smaller than 1 and the coefficient of relative risk aversion is larger than 1). This restriction is supported by the estimation results reported below. The parameter β is the habit parameter for which β ≥ 0 (Campbell, 1998). The parameters γ and π capture respectively the impact of hours worked (Campbell and Mankiw, 1990) and government consumption (Evans and Karras, 1998) on the marginal utility of private consumption. When γ > 0 (< 0) hours worked and private consumption are complements (substitutes). When π > 0 (< 0) government consumption and private consumption are complements (substitutes). When γ = 0 and π = 0 hours worked and government consumption have no impact on the marginal utility of private consumption. Note that γ < 0 and π > 0 do not imply that hours worked increase and government consumption decrease total utility of consumption since a function ϕ(Ht,Gt) could be added to the utility function (with ϕH < 0 and ϕG > 0) without changing the first-order condition.

The first-order condition with respect to consumption Ct is given by

where 0 < δ < 1 is the rate of time preference, Et − 1 the expectations operator conditional on period t-1 information and Rt the time-varying but risk-free real interest rate for which Et − 1(Rt) = Rt. Substituting equation 1 into the first-order condition gives

where such that . We assume that the distribution of ΔlnCt, ΔlnHt, and ΔlnGt is jointly normal conditional on period t-1 information. As a result the distribution of lnXt is also normal conditional on period t-1 information. From the log-normal property1 we then have

where the conditional variance Vt − 1(lnXt) is assumed to be constant, i.e. , implying that the conditional variances of ΔlnCt, ΔlnHt, and ΔlnGt and the conditional covariances between ΔlnCt, ΔlnHt and ΔlnGt are all constant. We then substitute equation 4 into equation 3 and take logs of the resulting equality to obtain

where we have used the approximations ln(1 + δ) ≈ δ and ln(1 + Rt) ≈ Rt. We then substitute the expression for lnXt derived below equation 3 into equation 5 and rearrange terms to obtain

or

Suppose now that some consumers in the economy are not optimizing permanent income consumers but are instead rule-of-thumb consumers who consume their entire disposable labour income in each period due to, for instance, myopia (see Flavin, 1985) or liquidity constraints (see Jappelli and Pagano, 1989; Campbell and Mankiw, 1990). In that case the growth rate of real per capita consumption in the economy can be approximated by

where Yt is real per capita disposable labour income (see Campbell and Mankiw, 1991; Kiley, 2010) and where λ approximates the fraction of rule-of-thumb current income consumers (with 0 ≤ λ ≤ 1). Note that when λ = 0 equation 8 collapses to equation 7. The estimable form of equation 8 can be written as

Our consumption equation (equation 9) encompasses most of the relevant predictability factors discussed in the literature. The ‘stickiness’ parameter a1 ≥ 0 reflects habit formation. Its sign is determined by the structural parameter capturing habits, i.e. β ≥ 0. A non-zero value for a2 captures the non-separability between private consumption and hours worked. Its sign is determined by the structural parameter γ. When γ > 0 (< 0) and therefore a2 > 0 (< 0) aggregate hours worked and aggregate private consumption are complements (substitutes). A non-zero value for a3 captures the non-separability between private consumption and government consumption. Its sign is determined by the structural parameter π. When π > 0 (< 0) and therefore a3 > 0 (< 0) government consumption and aggregate private consumption are complements (substitutes). The parameter a4 > 0 reflects intertemporal substitution effects in consumption caused by interest rate changes. It is determined by the structural parameter θ (where 0 < θ < 1), i.e. the intertemporal elasticity of substitution. The parameter a5 (0 ≤ a5 ≤ 1) reflects the impact of current income on consumption (liquidity constraints, myopia). It equals the structural parameter λ (where 0 ≤ λ ≤ 1). It is important to mention that the structural parameters β, γ, π, θ and λ are uniquely identified from the parameters a1, a2, a3, a4 and a5. Note further that some of the coefficients in equation 9 could be given other interpretations. A positive coefficient a1 on lagged aggregate consumption growth could also be the result of the presence of consumers who are inattentive to macro developments (see Reis, 2006; Carroll et al., 2011). Further, a positive coefficient a5 on current aggregate labour income growth could also be the result of consumers who are imperfectly informed about the aggregate economy (see Goodfriend, 1992; Pischke, 1995).2

To the best of our knowledge no study has yet estimated a specification as general as ours. Equation 9 nests, however, a number of specifications that have been estimated in the literature previously. Campbell and Mankiw (1990) conduct regressions on a version of equation 9 with restrictions a1 = 0 (with ΔlnYt always included and either ΔlnHt, ΔlnGt, or Rt added as an additional regressor). Evans and Karras (1998) estimate a version of equation 9 with restrictions a1 = a2 = a4 = 0 (with ΔlnYt and ΔlnGt included). Basu and Kimball (2002) estimate a version of equation 9 with restrictions a1 = a3 = 0 (with ΔlnHt, ΔlnYt, and Rt included). Kiley (2010) estimates a version of equation 9 with the restriction a3 = 0 (with ΔlnHt, ΔlnYt, ΔlnCt − 1, and Rt included). Carroll et al. (2011) estimate a version of equation 9 with restrictions a2 = a3 = a4 = 0 (with ΔlnCt − 1 and ΔlnYt included).

2.2 Discussion

2.2.1 Endogeneity

According to the theoretical model the error term in equation 9 depends, by construction, on shocks to the real interest rate and on shocks to the growth rates in aggregate consumption, hours worked and government consumption. Hence the error term μt is expected to be contemporaneously correlated with the regressors ΔlnHt, ΔlnGt and Rt. Additionally, the error term μt is expected to be correlated with the regressor ΔlnYt because shocks to consumption are basically shocks to permanent income. The latter are correlated with current income growth ΔlnYt. As such, to estimate the parameters of equation 9 consistently, an instrumental variables approach is necessary. Details will be given in the following sections.

2.2.2 Autocorrelation

The error term μt in equation 9 is assumed to be unpredictable based on lagged information. Three features that are not incorporated in the model could lead to a violation of this assumption and to the occurrence of autocorrelation of the moving average (MA) form in the error term μt. First, Campbell and Mankiw (1990) note that transitory consumption and measurement error can lead to an MA structure in the error term.3 Second, Working (1960) shows that an MA component could be present in consumption growth if consumption decisions are more frequent than observed data. Third, if durable consumption components are present in Ct this could induce negative autocorrelation in ΔlnCt since durable consumption growth tends to be slightly negatively autocorrelated (see Mankiw, 1982). This negative autocorrelation could be reflected in less positive values for a1 or in negative MA coefficients in the error term.

2.2.3 Cross-Sectional Dependence

When estimating the equation for aggregate consumption growth equation 9 using a panel of OECD countries, it can be expected that the error term μt is not independent across countries. Common unobserved shocks or factors can affect all countries simultaneously and induce error cross-sectional dependence. A twofold interpretation can be given to common unobserved factors found in regressions for aggregate consumption growth. First, financial liberalization most likely affects all OECD countries simultaneously over the sample period and could increase the importance of the common factor through increased risk-sharing opportunities between countries. In that case we would expect that countries’ aggregate consumption growth rates move more closely with a common (‘world’) consumption growth rate (i.e. the idiosyncratic country-specific component of consumption growth becomes less important). Over the period 1973–1988, Obstfeld (1994) documents a general rise in the correlations of domestic consumption growth with world consumption growth for G7 countries. Second, increased business cycle synchronization could increase the importance of a common factor in aggregate consumption growth (see, for example, Kose et al., 2008, who provide evidence for aggregate consumption).

2.2.4 Long-Run Considerations

While the model presented above provides an expression for aggregate consumption growth which relates variables in the short run, it also implies a long-run relationship. In particular the solution of the optimization problem given by the Euler equation (equation 2) is a stochastic representation of the permanent income hypothesis (see Hall, 1978). Campbell (1987) shows that the permanent income hypothesis implies that consumption and disposable income are cointegrated. If all consumers are rule-of-thumb consumers instead of permanent income consumers then aggregate consumption equals disposable income in every period and consumption and disposable income are also cointegrated. Hence both models predict a long-run cointegration relationship between income and consumption. However, since no error correction term enters directly into our derived equation for consumption growth—equation 9—deviations from the equilibrium relationship between consumption and income are not subsequently corrected by changes in aggregate consumption. Then, by necessity, it is income that must adjust to the lagged difference between income and consumption to maintain the long-run equilibrium relationship between both variables (see Deaton, 1992, pp. 124–125). To deal with this cointegration relationship in our estimation of equation 9 we follow the empirical approach outlined for US data by Campbell and Mankiw (1990) and applied subsequently in a large number of papers (see, for example, McKiernan, 1996). The approach consists in imposing structure on the process followed by aggregate income growth when estimating equation 9. This can be done by adding an appropriate lag of ln(Yt) − ln(Ct) as an error-correction term in the instrument list for income growth (see Campbell and Mankiw, 1990, pp. 267–268). More details on the instruments used in our estimations will be given in the following sections.

3 ECONOMETRIC METHODOLOGY

In this section we outline our econometric methodology to estimate the model for aggregate consumption growth outlined in Section 2.1 using a panel dataset for 15 OECD countries over the period 1972–2007.

3.1 Model and Assumptions

Equation 9 is written in the form of a first-order autoregressive panel data model:

where yit = ΔlnCit and xit = (ΔlnHit, ΔlnGit, Rit, ΔlnYit)′. The individual effect αi captures unobserved time-invariant heterogeneity, while the heterogeneity in the parameters ρi and βi across countries may, for example, reflect differences across countries in financial market institutions and development, government policies and demographics. Following the recent panel literature, we allow for a multi-factor structure in μit in which ft is an m × 1 vector of unobserved common variables. This error structure is quite general as it allows for an unknown (but fixed) number of unobserved common components with heterogeneous factor loadings (heterogeneous cross-sectional dependence). As such, it nests common time effects or time dummies (homogeneous cross-sectional dependence) as a special case. As noted in Section 2.2.2, there are various reasons that could lead to the occurrence of MA type autocorrelation in the error term of equation 9. Therefore, we allow μit in the empirical model in equation 10 to have an MA(q) component where ϕ(L) = 1 + ϕ1L + … + ϕqLq is a lag polynomial of order q. We further make the following assumptions.

Assumption 1.(Error condition)

E(εit) = 0 for all i and t; E(εitεjs) = 0 for either i ≠ j, or t ≠ s, or both; E(εitαj) = 0 for all i, j and t.

Assumption 2.(Explanatory variables) E(xi,t − sεit) = 0 for all i, t and s > 0.

Assumption 3.(Random slope coefficients) ρi = ρ + ψ1i,  βi = β + ψ2i,  ψi = (ψ1i,ψ′2i)′ ∼ i. i. d. (0,Ω), where Ω is a 5 × 5 symmetric non-negative definite matrix and the random deviations ψi are distributed independently of εit and xit.

Assumption 4.(Cross-sectional dependence)

The unobserved factors ft can follow general covariance stationary processes; E(ftεis) = 0 for all i, t and s.

Assumption 1(a) and (b) states that εit is a mean zero error process which is mutually uncorrelated over time and over cross-sections. Assumption 1(c) states that the individual effects are exogenous. With respect to the explanatory variables, Assumption 2 allows the variables in xit to be endogenous but implies that appropriately lagged, i.e. depending on the order of the MA component in μit, values of xit are available as instruments. Note that we do not restrict xit to be uncorrelated with αi. Assumption 4 states that the unobserved factors in ft are exogenous but it is quite general as it allows ft to exhibit rich dynamics4 and to be correlated with xit and αi. As Assumption 1 states that εit is uncorrelated over cross-sections, any dependence across countries is restricted to the common factors.5

3.2 Estimation Methodology

3.2.1 Averaging over Country-by-Country Coefficient Estimates

Pesaran and Smith (1995) show that in a dynamic heterogeneous panel data model as in equation 10, pooled estimators such as the fixed effects estimator in general provide inconsistent (for large N and T) estimates for the average effects and . To overcome this problem, they suggest averaging over country-by-country coefficient estimates, i.e. and . This yields consistent estimates for the average effects and for both N, T → ∞ provided that and are consistent for T → ∞. In the remainder of this section we will outline four alternative estimators for the country-specific coefficients ρi and βi. This will result in four alternative estimators for the average effects. Following Pesaran (2006), the asymptotic covariance matrix Σ for each of these average estimators is consistently estimated nonparametrically by

3.2.2 Naive Estimators

Direct estimation of ρi and βi in the model in equations 10–11 is infeasible as the factors ft in the error term μit are unobserved. As a benchmark in the empirical analysis and in the Monte Carlo simulation below, we therefore start with two naive estimators that ignore ft. The first one estimates ρi and βi using ordinary least squares (OLS) on equation 10 ignoring the error structure in equation 11. The average over the N country-specific OLS estimates is referred to as the mean group (MG) estimator. Abstracting from endogeneity of xit, a possible MA(q) component in μit and cross-sectional dependence induced by the common factors ft, country-by-country OLS estimation of the autoregressive model in equation 10 yields biased but consistent (as T → ∞) estimates for ρi and βi. In this case, the MG estimator is consistent for both N, T → ∞.

Under Assumption 2, the MG estimator is inconsistent as the variables in xit are allowed to be endogenous, while the MA(q) component in μit implies that the predetermined yi,t − 1 is also correlated with μit. Therefore, our second estimator for ρi and βi is a GMM estimator using an appropriate number of periods lagged values of yi,t − 1 and xit as instruments. The appropriate lag depth depends on the order q of the MA component in μit, i.e. the first available lags are yi,t − 1 − q and xi,t − 1 − q. Adding deeper lags improves the efficiency of the GMM estimator. However, in order to avoid problems related to using too many instruments, we only use the first two available lags. This results in the following instrument set: (yi,t − 1 − q,yi,t − 2 − q,xi,t − 1 − q,xi,t − 2 − q,zit); where zit is a set of additional instruments which will be defined in the next section. The country-by-country GMM estimates are then averaged over the N countries to obtain the MG-GMM estimator.

3.2.3 Common Correlated Effects Estimators

The most obvious implication of ignoring error cross-sectional dependence is that it increases the variation of standard panel data estimators. Phillips and Sul (2003), for instance, show that if there is high cross-sectional correlation there may not be much to gain from using the cross-sectional dimension of the panel dataset. However, cross-sectional dependence can also introduce a bias and even result in inconsistent estimates. For a static panel data model, the Monte Carlo simulations in Pesaran (2006) reveal that the MG estimator ignoring the error component structure proposed in equation 11 is seriously biased and suffers from large size distortions. Essentially, as Assumption 4 allows the unobserved factors to be correlated with the explanatory variables, this is an omitted variables bias which does not disappear as T → ∞, N → ∞ or both. Thus the naive estimators presented above are biased and even inconsistent in this case. Second, Phillips and Sul (2007) show that in a dynamic panel data model cross-sectional dependence introduces additional small-sample bias.

Pesaran (2006) shows that the cross-sectional averages of yit, yi,t − 1 and xit are suitable proxies for ft. For a model with a single factor,6 this can be seen by inserting equation 11 in equation 10 and taking cross-sectional averages to obtain

where and similarly for the other variables. Solving equation 13 for ft yields

such that from using Assumption 1, which implies that for each t, we have

This is the main result in Pesaran (2006) that the cross-sectional averages can be used as observable proxies of ft. Although the construction of as a consistent estimator of ft requires knowledge of the unknown underlying parameters, the individual coefficients (ρi,βi) and their means can be consistently estimated from an augmented form which is obtained by inserting equation 14 in equation 10:

with , and . As implies that for N → ∞, equation 16 is a standard heterogeneous dynamic panel data model with cross-sectional independent error terms as N → ∞. Country-by-country least squares estimation of equation 16 is the CCE estimator suggested by Pesaran (2006). The CCEMG estimator is then the simple average of the individual CCE estimators. Given the dynamic nature of the model, the individual CCE estimator is biased for finite T, but conditional on xit being predetermined or exogenous and ϕ(L) = 1 this bias disappears as T → ∞. This implies that consistency of the CCEMG estimator requires both N and T → ∞.

Endogeneity of xit and/or an MA(q) component in μit imply that the CCEMG estimator is inconsistent even for both N and T → ∞. Therefore, we use GMM in the country-by-country estimation of equation 16. As N → ∞, such that , the cross-sectional averages and are exogenous, while appropriate instruments for yi,t − 1 and xit are as before (yi,t − 1 − q,yi,t − 2 − q,xi,t − 1 − q,xi,t − 2 − q,zit). Also, letting T → ∞, this will yield consistent country-by-country CCE-GMM estimates. These CCE-GMM estimates are then averaged over the N countries to obtain the CCEMG-GMM estimator.

4 EMPIRICAL RESULTS

The model in equations 10–11 is estimated using aggregate yearly data for 15 OECD countries over the period 1972–2007. The selection of the countries and the sample period is determined by data availability and the aim to have as many time periods as possible for a reasonably large set of countries. The data are described in Section A of the online Appendix (supporting information).7

As motivated in the previous section, the GMM estimators are constructed using yi,t − 1 − q, yi,t − 2 − q, xi,t − 1 − q, xi,t − 2 − q and zit as instruments for the endogenous variables yi,t − 1 and xit, where zit is a set of additional instruments. We report results for three alternative instrument sets, which differ according to the assumed MA(q) component in the error term μit. Instrument set (a) assumes q = 0, (b) assumes q = 1 and (c) assumes q = 2. We include the appropriately lagged error correction term lnYi,t − 1 − q − lnCi,t − 1 − q as an additional instrument in zit. As noted in Section 2.2.4, this serves to take into account the cointegration relationship that—according to our theoretical framework—exists between consumption and income. We have further experimented with other additional instruments in zit like the lagged inflation rate (see, for example, Kiley, 2010) but these had little or no impact on the results, which we therefore do not report. We use a two-step procedure with a consistent estimate for the optimal weighting matrix constructed from a White type of estimator, allowing for heteroscedasticity when using instrument set (a) and from a Newey–West type of estimator allowing for both heteroscedasticity and MA(1) or MA(2) errors when using instrument sets (b) and (c) respectively.

Table 1 reports the average effects of the unrestricted parameters a1 to a5 in equation 9 as well as the estimates of the structural parameters β, θ, λ, γ, and π, since they are uniquely identified from the parameters a1 to a5 as indicated by the parameter restrictions reported below equation 9.8 The estimation results for the individual countries can be found in Table B-1 of the online Appendix. In order to save space, individual country results for the GMM estimators are only reported for instrument set (b).

Table 1. Panel data estimation results. Dependent variable: ΔlnCit; sample period: 1972–2007, 15 countries

Before looking at the specific coefficient estimates, we perform some diagnostic tests. The panel test results are reported at the bottom of Table 1. The cross-sectional independence test CD is from Pesaran (2004). Cross-sectional independence is rejected when applying the test to the residuals of the regressions estimated with the MG and MG-GMM estimators but not for the regressions estimated with the CCEMG-GMM estimator. This result suggests that cross-sectional dependence is an issue and that, to err on the side of caution, more weight should be given to the results obtained from the CCE type estimators.

For the GMM estimators, the Hansen (1982) overidentifying restrictions J test is first calculated for each country individually. The results are reported in Table B-1 of the online Appendix. Next, the panel version F(J) is obtainedby combining the country-specific p-values using the Fisher (1925) combined probability test. From the panel results in Table 1 we see that the used moment conditions are rejected by the data only when instrument set (a) is used. For instrument sets (b) and (c) the moment conditions are not rejected. This suggests that the order of the MA component in the residuals is at most q = 1.

The difference-in-Hansen test, denoted ΔJ, tests whether the regressors (ΔHit, ΔGit, Rit, ΔYit) are actually exogenous by adding their contemporaneous values to the set of instruments and testing whether the resulting increase in the J statistic is significant. Individual country p-values are again combined using the Fisher test to obtain panel results. These show that exogeneity of the regressors can be rejected when using instrument set (a) but not when using instrument sets (b) and (c).

The results of the J and ΔJ tests may, however, not be very informative due to weak instruments. When comparing the average Cragg–Donald test statistic for weak instruments (Cragg and Donald, 1993) reported in Table 1 for the GMM estimators with the appropriate critical values from Stock and Yogo (2004), we cannot reject the null hypothesis that the instruments are weak. We find this result for all GMM estimators and for all instrument sets considered.

As a more direct test of the order of the MA component, we therefore also use the estimated error terms for each country and for each estimator to (i) estimate an MA(2) model and (ii) perform a Cumby and Huizinga (1992) (CH) autocorrelation test. The CH test is particularly suited as it allows the model to have MA errors and to be estimated by a variety of GMM estimators, including those used in this paper. The country-specific results are reported in Table B-1 of the online Appendix and summarized in Table 1 by reporting mean group estimates for the MA(1) and MA(2) parameters and combining the p-values of the CH autocorrelation test using a Fisher test. The results confirm, in particular for the CCEMG-GMM estimator, the above conclusion that the order of the MA component in the residuals is q = 1.

Given the finding of significant cross-sectional dependence, an MA(1) component in the error terms and possible endogeneity of the regressors, the CCEMG-GMM estimator with instrument set (b) is our preferred estimator.

When looking at the point estimates reported in Table 1, we note that the coefficient on lagged aggregate consumption growth is either insignificant or its significance is very low. In some cases it is estimated with a negative sign. The structural estimates for β are in line with this since the estimates for β are generally found to be insignificant. Carroll et al. (2011) find significant and positive values for this parameter in quarterly data. The lower significance of our estimates may be due to data frequency, i.e. habit formation may be an important predictability mechanism at the quarterly frequency but is probably less relevant in annual data.

We further find that the impact of the growth rate in hours worked and the estimates for γ are positive and often significant. For the CCEMG-GMM estimator with instrument set (b) the impact of hours worked on consumption is significant (but only at the 10% level), while the parameter γ is not—even when the estimates are insignificant their magnitude is rather high. So it seems that the results of Basu and Kimball (2002), who argue in favour of complementarity between consumption and labour in the USA, cannot be refuted completely.

The impact of government consumption growth on private consumption growth is never significant and the magnitude of the estimated impact is low. As a result, the estimates for π reported in the table are never significant. We conclude that there is no evidence to support the existence of non-separabilities between private consumption and government consumption. This stands in contrast to results reported, for instance, by Evans and Karras (1998) for a large sample of countries.

When looking at potential intertemporal substitution effects, i.e. the impact of the real interest rate on aggregate consumption growth, our results are in line with the literature in the sense that the evidence to support intertemporal substitution is not very strong (see, for example, Campbell and Mankiw, 1990). This result is confirmed by the estimates for the elasticity of intertemporal substitution θ reported in the table, which are insignificant even though they tend to have economically sensible values. There is one exception though. When the estimation is conducted with the MG-GMM estimator using instrument set (b) we find a positive and strongly significant impact of the real interest rate on aggregate consumption growth.

We finally find that the impact of aggregate disposable income growth on aggregate consumption growth—which equals the structural parameter λ —is positive and strongly significant across all estimators and instrument sets. Our parameter estimates are in line with studies by Campbell and Mankiw (1990) and Kiley (2010), who also find that current disposable income growth has a positive and significant impact on aggregate consumption growth. Contrary to Basu and Kimball (2002) and Carroll et al. (2011), we do not find that rule-of-thumb or current income consumption is less important once other forms of predictability are taken into account. Given the significance of the rule-of-thumb result and its robustness, it makes sense to check the individual country estimates of this parameter, which are reported in Table B-1 of the online Appendix. From that table we note that, across all estimators, for the majority of the countries considered a positive and significant impact of aggregate disposable income growth on aggregate consumption growth is found. With respect to the well-documented case of the USA we find point estimates for λ that lie between 0.5 and 0.66 —a result which is in accordance with the range of values for this parameter reported in the literature (see, for example, Campbell and Mankiw, 1990).

To summarize, the results that we obtain with our newly introduced CCEMG-GMM estimator—which we consider to be the most appropriate for the question at hand—suggest that aggregate consumption growth in a panel of OECD countries over the period 1972–2007 depends significantly (at the 1% level) only on the growth rate in aggregate disposable labour income. The impact of the growth rate in hours worked (non-separability between consumption and hours worked) is positive and significant at the 10% level but the coinciding structural model parameter is not significant. The coefficient estimates on lagged aggregate consumption growth (habit formation) and the interest rate (intertemporal substitution) are insignificant at the conventional significance levels, while their signs and magnitudes are economically meaningful. There is no evidence in favour of non-separabilities between private consumption and government consumption.

Our estimations may be hampered by two important complications. First, as noted by Stock and Yogo (2004), the weak instruments problem reported above can lead to biased estimates and to unreliable inference. Second, all of the estimators used are consistent for N, T → ∞. Since our sample size (T = 35 and N = 15) is relatively small—especially in the N dimension—small-sample biases might make our estimation results less reliable. To investigate the potential biasedness and reliability of inference of the estimators—in particular the CCEMG-GMM estimator—under both weak instruments and a relatively small sample size we conduct a Monte Carlo simulation in the next section.

5 MONTE CARLO SIMULATION

In this section we use a Monte Carlo experiment to examine the small-sample properties of the MG, MG-GMM, CCEMG and CCEMG-GMM estimators. To make sure that the Monte Carlo results are relevant for putting our empirical results in Section 4 into perspective, the data-generating process (DGP) and population parameters are chosen such that the properties of the simulated data match with those of the observed data as much as possible. Although we are mainly interested in the setting T = 35 and N = 15, we also present results for a range of alternative sample sizes to illustrate the more general properties of the estimators.

5.1 Experimental Design

The DGP is assumed to be

The DGP for ΔlnCit is a restricted version of the model in equations 10–11. First, for the sake of simplicity of the MC simulation, we restrict the set of explanatory variables to include only lagged consumption growth ΔlnCi,t − 1 and income growth ΔlnYit. ΔlnCi,t − 1 is included to maintain the dynamic panel structure, while ΔlnYit is included as this appears to be the only variable which is (robustly) significant in the empirical analysis. Second, we consider a single common factor (i.e. we restrict m = 1). This is without loss of generality as the CCE-type estimators are robust to multiple common factors. Third, as the empirical results suggest that the order of the MA component in the errors is at most 1, we restrict ξit to be an MA process of order 1. Finally, we also set the individual effects αi = 0. As all regressions include a country-specific constant, such that the individual effects are cancelled out exactly, this is without any loss of generality.

The DGP for the explanatory variable ΔlnYit is fairly general as it allows for correlation with the unobserved common factor in ΔlnCit (i.e. when γ2i ≠ 0), endogeneity (i.e. when φ ≠ 0) and, as explained in Section 2.2.4, error correction to the long-run relationship between lnCit and lnYit (i.e. when δi > 0). It is important to note that the addition of ΔlnCi,t − 1 to the DGP of ΔlnYit can also be given a theoretical justification. As noted by Campbell and Mankiw (1990), the permanent income hypothesis implies that current consumption summarizes consumers’ information about the future process for income. Then, assuming that consumers have better information about future income than that which is contained in the history of income growth, lagged values of consumption growth will help to predict income growth.

The heterogeneous slope coefficients are drawn as In order to obtain realistic parameter values, we calibrate the DGP outlined above to our observed sample of OECD data. More specifically, the parameter values are chosen such that the moments (standard deviations, cross-correlations, autocorrelations, cross-sectional dependence) of the simulated data match with those of the observed data as much as possible. We do this by first estimating the restricted consumption equation 17 and the income equation 18 to get an idea about the parameter values and their heterogeneity over countries. The value for the MA1 parameter ϕi is inspired by country-by-country auxiliary estimations of an MA(1) process on the estimated residuals of the consumption equation 17 (also see the country-specific estimation results in Table B-1 of the online Appendix). As the CCE-type of estimators do not provide direct estimates for the common factors, we further calibrate the parameters (τ1,τ2) governing the AR process of the common factors, the factor loadings (γ1i,γ2i,γ3i) and the error variances to the observed data. To shed light on the impact of cross-sectional dependence and endogeneity on the considered estimators, we conduct the following four experiments with parameter values given by All experiments (common parameter values): ρ = 0.20, β = 0.40, θ1 = 0.30, θ2 = 0.20, δ = 0.10, τ1 = 0.25, τ2 = 0.25, σρ = 0.10, σβ = 0.10, , , σδ = 0.03, and . Experiment 1 (no cross-sectional dependence, no endogeneity): γji = 0 for j = 1, 2, 3, φ = 0.4, σε = 0.0175 and σν = 0.0210. Experiment 2 (no cross-sectional dependence, endogeneity): γji = 0 for j = 1, 2, 3, φ = 0.4, σε = 0.0155 and σν = 0.0210. Experiment 3 (cross-sectional dependence, no endogeneity): γji ∼ i. i. d. U(0.25,1.75) for j = 1, 2, 3, φ = 0, σε = 0.0135 and σν = 0.0175. Experiment 4 (cross-sectional dependence, endogeneity): γji ∼ i. i. d. U(0.25,1.75) for j = 1, 2, 3, φ = 0.25, σε = 0.0125 and σν = 0.0175. For each of these four experiments, we consider two versions: MA(0) errors: ϕi = 0. MA(1) errors: ϕ = − 0.25 and σϕ = 0.10.

For each cross-section i, we generate data for ΔlnCit and ΔlnYit using the above DGP over the period t = − 49, …, 1, …, T with initial values ΔlnCi,− 49 = 0 and ΔlnYi,− 49 = 0. Data for lnCit and lnYit are obtained by simple accumulation of ΔlnCit and ΔlnYit with initial values lnCi,− 49 = 0 and lnYi,− 49 = 0. The actual sample is then obtained by discarding the first 50 observations.

Each experiment is replicated 5000 times for the (T,N) pairs with T = 20, 35, 50 and N = 15, 50. In each experiment, we compute the MG, MG-GMM, CCEMG and CCEMG-GMM estimators. The GMM estimators are two-step estimators using the instrument sets (a) and (b) defined above. In order to save space, instrument set (c) is not used as the order of the MA process in the simulated data is at most 1.

To shed some light on the relevance of the Monte Carlo design, Table 2 compares some of the moments of the observed data with those of the simulated data for the sample size T = 35, N = 15 in each of the four experiments. The moments of the simulated data are averages over the 5000 iterations. The results show that the moments of the data simulated using Experiment 4, which is the most general, are very much in line with the observed data. Experiments 1 and 2, which do not model cross-sectional dependence, fail to match the observed correlation in both ΔlnCit and ΔlnYit across countries. Experiments 1 and 3, which do not model endogeneity, fail to match the observed contemporaneous correlation between ΔlnCit and ΔlnYit within countries, although this is to a lesser degree the case in Experiment 3, as part of this contemporaneous correlation is captured by the common factor f1t, which shows up in both ΔlnCit and ΔlnYit. At the bottom of Table 2 we also report Cragg–Donald statistics for the various GMM estimators when applied to the observed and to the simulated data. This is very important for the relevance of our Monte Carlo simulation as instrument strength is an important determinant of the size of the bias of the considered GMM estimators and of the reliability of inference based on these estimators. The results show that the instrument strength in the simulated data is highly similar to that in the observed data.

5.2 Results

Tables B-2–B-5 in the online Appendix report results for the four MC experiments. The estimators are compared in terms of mean bias (bias), mean of the estimated standard errors (stde), standard deviation (stdv), root mean squared errors (rmse) and size at the nominal 5% level of t-tests for the null hypotheses that ρ = 0.20 and β = 0.40 respectively.

Before looking more closely at the results for each of the four experiments, some important, more general results can be noted. First, the performance of the CCEMG-type estimators in the experiments with no cross-sectional dependence is not much worse compared to their standard MG counterparts both in terms of bias and dispersion. This shows that there is no high cost involved in unnecessarily adding cross-sectional averages to account for possible cross-sectional dependence. Second, the bias of the CCEMG-type estimators is highly similar for N = 15 and N = 50. This suggests that our relatively low cross-sectional dimension (N = 15) is not really a source of concern. Third, the mean of the estimated standard errors is in most cases fairly close to the actual standard deviation of the estimates. This shows that the nonparametric estimator defined in equation (12) is reasonably accurate. At least, the weak instruments problem for our GMM estimators does not result in a severe underestimation of the standard errors, which is a well-known problem when using parametric estimates. The size distortions observed for some of the estimators, especially for larger values for T and N, are therefore driven mainly by the bias of the estimators. Fourth, despite a weak instruments problem, the bias and dispersion of the GMM estimators are not unacceptably high, especially when compared to the population values ρ = 0.20 and β = 0.40.

We now turn to some more specific results. Experiment 1(i) is for a heterogeneous dynamic panel data model with no cross-sectional dependence, no endogeneity and no MA component in the errors. As expected, the MG estimator outperforms the other estimators. This is especially the case for estimating β, for which the MG estimator shows almost no bias. Given the accurate estimation of the standard errors, the size of the MG estimator for β is close to its nominal level of 5%. In line with the results for homogeneous dynamic panel data models, the MG estimator for ρ is downward biased but this bias decreases in T and is fairly small for T = 35 (see, for example, Judson and Owen, 1999). Note that, given this bias, the MG is severely oversized, especially for larger values of N. Despite the weak instruments problem, the GMM estimators for ρ have a slightly smaller bias and when using instrument set (a) their dispersion, as measured by the stdv, is not much higher compared to the MG estimator. As such, they tend to outperform the MG estimator for ρ also in terms of rmse and size. Only when using instrument set (b), the stdv more or less doubles. The weak instruments problem shows up more clearly when estimating β, though. The GMM estimators are now relatively more biased and have a higher dispersion, irrespective of whether instrument set (a) or (b) is used. However, neither the bias nor the stdv is unacceptably high. Experiment 1(ii) adds an MA(1) component to the errors of the consumption equation. As this implies that lagged consumption growth ΔlnCi,t − 1 is endogenous, only the GMM estimators using instrument set (b) are consistent in this case. This shows up very clearly in the estimation results, which show a relatively high bias for ρ that does not disappear for higher values for T and N for all estimators but the GMM estimators using instrument set (b). Surprisingly, all estimators are more or less unbiased for β.

Experiment 2 adds endogeneity of ΔlnYit to the above experiment. The main implication of this is that the MG and CCEMG estimators become inconsistent in both Experiment 2(i) and 2(ii). In the simulation results, this shows up as a much bigger bias and a considerable size problem for these estimators for all sample sizes. With respect to the GMM estimators, both instrument sets are valid in 2(i) but only instrument set (b) is valid in 2(ii). The simulation results now show a relatively higher bias for both ρ and β when using instrument set (a) in experiment 2(ii).

Experiment 3 adds cross-sectional dependence, with the unobserved factor in ΔlnCit being correlated with ΔlnYit, but again assumes ΔlnYit to be exogenous with respect to εit. Both the MG and the MG-GMM estimators should now suffer from an omitted variables bias. In Experiment 3(i), the CCEMG estimator should now be the preferred estimator. In the simulation results, this shows up especially when estimating β for which the CCEMG estimator has no bias, the smallest stdv and no big size problem. When estimating ρ, the MG estimator tends to have a smaller bias but a bigger stdv, resulting in slightly smaller rmse. In Experiment 3(ii), the CCEMG-GMM(b) should be the preferred estimator. When estimating ρ, the CCEMG-GMM(b) estimator indeed has the smallest rmse for larger values of T, while for smaller values of T the MG-GMM(b) estimator has a slightly smaller rmse. As ΔlnYit is exogenous in this experiment, the CCEMG estimator still clearly has the smallest rmse when estimating β, though.

Experiment 4 includes both endogeneity and cross-section dependence. The CCEMG-GMM estimator is now the only consistent estimator. The simulation results show that the CCEMG-GMM(a) and the CCEMG-GMM(b) estimators for β indeed clearly outperform the other estimators in terms of bias and size for all sample sizes in Experiments 4(i) and 4(ii) respectively. Interestingly, for the sample size (T = 35, N = 15) that is available to us in the empirical analysis, the bias is negligibly small and the real sizes of 7.3% and 12.1% respectively are sufficiently close to the nominal level of 5%. Note that the CCEMG estimator has a much smaller dispersion, resulting in a smaller rmse compared to the CCEMG-GMM(a) estimator for smaller values of T, but its relatively larger bias, which does not decrease for larger values of T or N, results in very poor size properties. When estimating ρ, the MG-GMM estimators tend to show up as the preferred estimators in terms of bias and size but the CCEMG-GMM estimators are not lagging behind too much. Note that for the sample size (T = 35, N = 15) there is a moderate downward bias for ρ which results in a size problem. This problem decreases for larger values of T.

To summarize, in a heterogeneous dynamic panel data model with both endogeneity and error cross-sectional dependence the CCEMG-GMM is the preferred estimator, both in terms of bias and size. Especially when compared to the alternative estimators, it performs relatively well for the modest sample size T = 35, N = 15 that is available for the empirical analysis presented in Section 4. However, it should be noted that weak instruments may still imply a small to moderate bias and size distortions. These conclusions should be taken into account when reading the empirical results presented in Section 4.

6 CONCLUSIONS

This paper examines the sources of predictability in aggregate private consumption growth. We first derive a dynamic consumption equation which nests most of the relevant predictability factors discussed in the literature: rule-of-thumb or current income consumption, habit formation, intertemporal substitution effects and non-separabilities between private consumption and both hours worked and government consumption. Next, we estimate this dynamic consumption equation for a panel of 15 OECD countries over the period 1972–2007. We follow recent developments in panel data econometrics by allowing for unobserved common factors which have heterogeneous impacts on the countries in the panel. We develop a CCEMG-GMM estimator by combining the CCEMG estimator advanced by Pesaran (2006) to account for error cross-sectional dependence and the GMM estimator to account for endogeneity of the regressors. The moment conditions imposed by this CCEMG-GMM estimator are valid as N, T → ∞ jointly. A Monte Carlo experiment shows that the CCEMG-GMM estimator performs reasonably well, taking into account both that the instruments used in the estimations are not strong and that the sample size is relatively small. In our dynamic panel data setting with both endogeneity and error cross-sectional dependence, it is preferred over standard MG, MG-GMM and CCEMG estimators both in terms of bias of the estimated coefficients and in terms of inference.

Taking into account endogeneity and cross-sectional dependence proves to be important as it has a marked effect on our estimation results. These suggest that the growth rate in aggregate private consumption depends positively on the growth rate in current disposable income, which is found to be the only variable with significant predictive power for aggregate consumption growth. The estimates of the impact of lagged aggregate consumption growth (habit formation), the interest rate (intertemporal substitution), and the growth rate in hours worked (non-separability between consumption and hours worked) on aggregate consumption growth are insignificant at the conventional significance levels but their signs and magnitudes are economically meaningful. There is no evidence in favour of non-separabilities between private consumption and government consumption.

ACKNOWLEDGEMENTS

We thank Freddy Heylen, Raf Wouters and two anonymous referees for helpful suggestions and constructive comments on earlier versions of this paper. Gerdie Everaert acknowledges financial support from the Interuniversity Attraction Poles Program—Belgian Science Policy, contract no. P5/21.

The provided documents and dataset do not directly specify the exact government departments, methodologies, or publication details for the variables (CSUMPTN, GOV, HOURS, INC, R) used in the study. However, based on standard practices for such research involving OECD countries:

Data Sources:
These variables are likely sourced from official national statistics agencies or central banks of the respective countries. For example:
Japan: Statistics Bureau of Japan or Bank of Japan.
Canada: Statistics Canada or Bank of Canada.

CSUMPTN (Private Consumption): Derived from national accounts data, typically as part of GDP breakdowns.

GOV (Government Consumption): Reported in national accounts, covering government expenditure data.

HOURS (Hours Worked): Derived from labor force surveys or employment reports.

INC (Disposable Income): Calculated as total income minus taxes and social contributions, available in income and expenditure surveys.

R (Real Interest Rate): Often based on central bank reports or derived from nominal interest rates adjusted for inflation.

Timeframes:
The study covers data from 1972–2007, so the data would have been gathered during this period. Publication of such data often follows standard reporting periods (quarterly or annually).

Publication Venues:Typically published in:OECD databases (e.g., OECD.Stat).

Country-specific statistical yearbooks or online databases.Central bank reports and economic reviews.

If you have access to the study’s appendix or supplementary material, it might provide a detailed description of the data sources. Otherwise, the most precise way to confirm would be to consult OECD or national data repositories directly. Let me know if you want assistance in locating this information!