Dynamic models

In a dynamic specification, the lagged dependent variable enters as an explanatory variable on the right hand side of the equation. In this setting, the assumption of strict exogeneity of the explanatory variables is violated since the lagged dependent variable is correlated with the error term by construction. Previously discussed fixed effects estimators (within estimators) are downward biased and inconsistent for N → ∞ and fixed T.³¹

The literature proposes two different solutions: a correction for the bias or, alternatively, estimation by the Generalized Method of Moments (GMM).

Bias-corrected fixed effects estimator: For dynamic panel data models with serially uncorrelated errors and strongly exogenous regressors, Kiviet (1995) derives an approximation for the bias of the fixed effects estimator. He proposes a corrected fixed effects estimator that subtracts a consistent estimate of this bias from the standard fixed effects estimator. In Kiviet (1999), a more accurate bias approximation is derived. Bun and Kiviet (2003) reformulate the approximation with simpler formulae for each term. Bruno (2005) extends these results to unbalanced panels. Simulation studies, which are included in the articles on bias approximation cited in this paragraph, show that this correction is an efficient method and can even be more efficient in finite samples than instrumental variables estimators.

31 For N → ∞ and T → ∞ the estimator would be consistent. However, a Monte Carlo study, conducted by Judson and Owen (1999), reveals that even for a relatively large time dimension (T=30) the bias may be equal to as much as 20% of the true value of the coefficient of interest.

Instrumental variables estimators are proposed as an alternative solution. This class of estimators eliminates the country-specific effects by first differencing. It then applies instrumental variables to the transformed equation.

Anderson and Hsiao (1981) propose the lagged level yi,t-2 or the lagged difference (yi,t-2 –yi,t-3) as instruments for yi,t-1. The use of the lagged difference has two drawbacks: it reduces the effective number of observations available for estimation and generally produces higher variances. Therefore, in practice, the lagged level is usually the preferred instrument. A valuable instrument must be correlated with the variable that it replaces and uncorrelated with the error term. These conditions should be fulfilled in our case where the lagged level of the reserves-to-GDP ratio is instrumented by its level lagged by two periods. Since the series of reserves is highly persistent, it is correlated over time. Moreover, whereas its lagged level is correlated with the error term by construction, it is reasonable to assume that it is uncorrelated with the error term of the period ahead.

This estimation leads to consistent estimates of the coefficients. However, it is not necessarily efficient because it does not make use of all available moment conditions and hence uses not all available instruments.

Difference GMM estimator: Arellano and Bond (1991) note that the number of instruments can be extended by using all feasible lagged values of the dependent variable as instruments.

More precisely, beginning with yi,t-2 all lagged levels of y until yi,0 can be used as instruments.

Hence, the number of instruments varies with t since with each forward period an extra instrument is added.

Whereas the Anderson-Hsiao estimator imposes only one moment condition, the Arellano-Bond estimator is based on a set of (T(T-1)/2) moment conditions. All these moment conditions can be exploited in a Generalized Method of Moments (GMM) framework.³² This estimator is also known as the difference GMM estimator because it is based on the first-differenced equations.

Estimation is executed by a two-step procedure. In the first step, an arbitrary weighting matrix – usually the identity matrix – is applied and, in the second step, the optimal weighting matrix – based on the inverse of the covariance matrix of the sample moments – is used. Since the

32 2SLS cannot be applied since it demands that the dimension of the matrix of instruments is constant for all t.

two-step standard errors tend to be biased downward in small samples, the one-step standard errors are used for inference. Alternatively, a finite sample correction that provides more accurate estimates of the standard errors is proposed by Windmeijer (2005).

While the Arellano-Bond estimator has been widely used in dynamic empirical studies, various authors have extended this estimator by additional moment conditions with intent to further improve its efficiency.

First, Ahn and Schmidt (1995) show that the assumptions of Arellano and Bond imply further (T-2) moment restrictions, which are not used in their GMM framework.

Second, additional moment conditions can be exploited if further assumptions are imposed.

Ahn and Schmidt (1990) show that the additional assumption of homoskedastic errors allows previously nonlinear moment conditions to be expressed linearly and adds (T-1) new moment conditions. As shown by Ahn and Schmidt (1995) the efficiency gain is small. Moreover, if the assumption of homoskedasticity is not valid, it may introduce inconsistency. For these reasons, we shall not use this estimator.

When the individual series are highly persistent and the number of time periods is moderately small, the Arellano-Bond estimator has been found to have poor finite sample properties. In this case, lagged levels of the series provide weak instruments for first differences.

Particularly, the instruments become less informative as the value of δ increases towards unity³³ and as the relative variance of the fixed effects ci increases

[ (

σc² σu²

)

^→∞

]

. Simulation results reported in Blundell and Bond (1998) show that the first-differenced GMM estimator may be subject to a large downward finite-sample bias in these cases.

A further shortcoming of the Arellano-Bond estimator – just as of the fixed effects estimator – consists in the fact that it makes use only of the time variation within each cross section, whereas the cross-sectional variation is not used for estimation. However, we are not only interested in the time-series relationship between the level of reserves and its determinants, but also in their cross-country relationship.

This loss of information is avoided and the bias due to persistent series can be reduced by an alternative estimator, the so-called System GMM estimator.

33 Bond (2002) notes that in this case the series are close to being random walks, so that their first-differences are close to being innovations.

System GMM estimator: This estimator, which was proposed by Arellano and Bover (1995) and fully developed by Blundell and Bond (1998), combines, in a system, the regression in differences with the regression in levels. The regression in differences, the first part of the system, and its instruments are the same as those of the Arellano-Bond estimator. For the regression in levels, the second part of the system, lagged first-differences of the corresponding variables are used as instruments. For consistency of this estimator, an additional stationarity assumption on the initial value of the level of reserves is required. If one can assume that the same process has generated the series for a long time, the condition will hold automatically.

Hence, the system estimator basically extends the Arellano-Bond estimator by adding the regression in levels.

Tests for the validity of the moment conditions in GMM estimation: The GMM model is overidentified by construction since there are several instruments, namely all lagged levels of a variable, for each explanatory variable. In such a context testing whether the additional instruments are valid in the sense that they are uncorrelated with the error term is indicated.

This is implemented by the Sargan test of overidentifying restrictions, which evaluates the null hypothesis that the overidentifying restrictions are valid. The Sargan test statistic is not robust to heteroskedasticity. Arellano and Bond report that the one-step Sargan test over-rejects in the presence of heteroskedasticity. Therefore, the Hansen J statistic is preferred since it is robust to heteroskedasticity.

A further test of the consistency of the GMM estimator is given by the Arellano-Bond test for autocorrelation in the residuals. The consistency of the GMM estimator hinges on the assumption that the residuals are uncorrelated over time. This implies that the differenced residuals (the residuals of the regression in differences) are first-order serially correlated by construction unless the residual in the level equation follows a random walk. Second-order serial correlation, however, implies that the original residual is also serially correlated.

Therefore, we test the null hypothesis that second-order serial correlation is absent.

Mean group estimator: Alike in the case of static models, the mean group estimator allows for parameter heterogeneity specifying an individual regression for each country.

This estimator leads to consistent estimates of the coefficients if N and T tend to infinity. For small T, however, there is the small-sample downward bias (Hurwicz bias) in the coefficient of the lagged dependent variable. Since each of the country estimates are subject to this bias, it will not be reduced by averaging across groups. The magnitude of the bias depends on 1/T and is less of concern if T and N are of the same order of magnitude. Hence, if one estimates a dynamic model for heterogeneous data, one has the choice between two biased estimators: an instrumental variables estimator à la Arellano-Bond, which is biased because of the neglect of slope heterogeneity, and the mean group estimator, which is subject to the Hurwicz bias.

3.4 Stationarity

For our case of a data field, characterized by a relatively long time-period under consideration, the time-series properties of the variables deserve more attention. From the time-series literature it is well-known that regressions based on nonstationary data may lead to misleading inferences due to non-standard distributions of the estimators. This is the problem of spurious regression where variables are statistically significant but without economic meaning. An important result in the context of panel data is that nonstationarity is not so much a topic of concern. Phillips and Moon (1999, 2000) and Kao (1999) show that panels make it possible to obtain consistent estimators as the number of cross-section units goes towards infinity. A long-run average parameter can be estimated consistently even when each of the individual time-series regressions is spurious. The averaging over N attenuates the noise in the individual estimators and thus facilitates a consistent estimator of the mean effect.

This is especially true if one uses cross-sections, pooled or averaged estimates across countries.

Despite this result, we opt to consider the issue of stationarity and nonstationarity of our variables. Although testing for unit roots in panels is recent, a number of different tests were proposed. Examples are Im, Pesaran and Shin (2003), Levin, Lin and Chu (2002) and Hadri (2000). In comparison with unit root tests on single time-series, which lack power relative to the alternative hypothesis of a persistent but stationary process, these panel unit root tests increase the power of the test by combining information across units. However, these tests require a balanced panel and can therefore not be applied in our case of an unbalanced panel.

One exception is the Fisher test, which was developed by Maddala and Wu (1999). It combines the p-values of independent unit root tests on each time-series. It tests the null hypothesis that all series are nonstationary against the alternative that at least one

time-series of the panel is stationary. Maddala and Wu find in a Monte Carlo study that the Fisher test performs the best compared to the three other panel unit root tests mentioned above.

Our findings are the following (see Table 3): For our dependent variables reserves over GDP as well as reserves over base money the hypothesis of nonstationarity can be rejected at the 1% level. Originally, real reserves defined as reserves deflated by the GDP deflator was also considered as a possible dependent variable. Since the nonstationarity hypothesis cannot be rejected, its use in the regression analysis was disregarded. The explanatory variables seem to be stationary with two inconclusive cases: The hypothesis of a nonstationary process cannot be rejected at reasonable levels of significance for GDP per capita in two out of eight specifications and for total external debt divided by GDP in five of a total of eight test specifications.

To test for the robustness of these results, Table 4 reports the results of the Levin-Lin-Chu unit root test. Since this tests requires a balanced panel, it is performed on a balanced subsample of our entire sample. Previous results are confirmed: Real GDP per capita and total external debt divided by GDP cannot always reject the hypothesis of a unit root. Moreover, lagged short-term debt and reserves over GDP now also lead to ambiguous results. With respect to the importance of these results one has to remember that the alternative hypothesis states that all series are stationary. Therefore, variables that fail to reject the null hypothesis might be characterized by a small fraction of series that follow nonstationary processes. This explanation is especially plausible for the variable reserves over GDP since it is well know that some countries (i.e. China, India, Mexico) accumulated large amounts of reserves in the recent past.

Despite their possible nonstationarity we include these variables in the econometric analysis since the results of Phillips and Moon suggest that the problem of spurious regression can be avoided by using pooled data. Moreover, we will check for spurious results in the section on robustness.

In document The Accumulation of International Reserves by Central Banks: Exploring Causes and Consequences (Page 72-77)