Static versus dynamic specification

Apart from the determination of an appropriate set of explanatory variables, one has to make an assumption concerning the properties of the dependent variable. Either it is static or it follows a dynamic adjustment process.

From a theoretical perspective, models often relate to a stationary or static world in which the equilibrium value of a variable is exclusively determined by its determinants and in particular

independent of its own realization in the previous period. However, most empirical economic relationships are characterized by a dynamic adjustment process where the equilibrium value is not reached immediately. In these cases, the current value of a variable depends on its past value, the current value may deviate from its optimal or desired value and the analysis of the time path to its equilibrium value is important.

Nevertheless, empirical models are often based on static estimation when a dynamic one might be warranted. However, since the availability of data sets over long time periods has improved and estimation methods for dynamic models, which are more complicated and require stronger assumptions than those of static models, have been developed, the addiction to static models is no longer justified.

Moreover, the neglect of dynamics is not without cost. An incorrect specification of the process governing the dependent variable may also affect the estimated coefficients of its explanatory variables.

In terms of empirical modelling, a static model is represented by equation (1) whereas a dynamic one includes the lagged dependent variable as a right-hand-side variable:

it i it i,t

it δy x β c

y = ₋₁ + + +ε (2)

where δ is the coefficient of the lagged dependent variable.

This demonstrates that the neglect of dynamics is a special case of an omitted variable problem. The standard result of the literature on omitted variables states that the coefficients of the included explanatory variables are biased if the neglected variable is correlated with the included ones. In the case of neglected dynamics, the omitted variable contains all past values of the explanatory variables. Therefore, consistent estimation is only viable if the explanatory variables are uncorrelated over time.

An alternative illustration of this problem can be obtained from an examination of the relation between explanatory variables and the error term. Estimation of fixed effects models by pooled OLS relies on the assumption that the explanatory variables are exogenous, i.e.

uncorrelated with the error term. If the lagged dependent variable is neglected, it becomes part of the error term. By repeated substitution, the error term can be expressed as:

∑

It is noteworthy that this error term contains all past values of the explanatory variables xi. In consequence, if the explanatory variables are correlated over time, then the error term is as well correlated with the current explanatory variables in equation (2) and there is an endogeneity problem. This leads to a bias in the estimation.

I would like to illustrate these theoretical considerations with an example. The current level of external debt is likely to depend on its past value; it moves slowly over time and shows cyclical rather than erratic movements. External debt is serially correlated. In our application, external debt belongs to the set of control variables that determine the demand for reserves.

Hence, reserves and external debt are correlated in any given time period. If reserves follow a dynamic process that is, however, neglected in the estimation, lagged external debt becomes part of the current error term. By assumption, the error term – containing lagged external debt – is then correlated with one of the current explanatory variables, namely current external debt.

For a fixed time dimension T and a cross section dimension N that goes to infinity, the omission of dynamics may result in a substantial bias. The static fixed effects estimator is downward biased if the coefficients (β and δ) are positive. It is the larger, the slower the dependent variable adjusts and the lower the serial correlation of the explanatory variables.

Hence, it is important to determine whether the dependent variable of a panel data study is generated by a static or dynamic data generating process. A first technique to detect dynamics simply consists in the estimation of a dynamic relationship (equation (2)) and the verification if δ is significantly different from zero. Alternatively, one might estimate a static model like (1) and test for serial correlation of the error term. The presence of serially correlated errors might be an evidence for the presence of dynamics.

In our study of the demand for reserves, we have to determine whether the dependent variable – international reserves divided by GDP – follows a static or dynamic process. There are several arguments why a dynamic behaviour is more plausible.

As a first approach, one might consider the nature of this variable. The level of reserves is a stock variable that would be constant over time if the central bank were totally passive (under a freely floating exchange rate system). The initial stock of reserves is not zero, but there is an existing level which is inherited from the previous period. When a central bank determines the level of reserves, it always starts from this level and has to define the desired changes.

Therefore, the determination of the level of reserves is a natural candidate for a dynamic specification that includes the lagged level of reserves as one of its determinants.

Analytically, the inclusion of the lagged dependent variable can be motivated by a partial adjustment or habit-persistence model (see Appendix B for a theoretical illustration). A central bank might only partially adjust the level of reserves to its desired level when the adjustment is costly and when an optimising behaviour is assumed, trading off the costs of making the adjustment and the costs of not having the desired level of reserves. In addition, the lagged dependent variable might be interpreted as a measure of inertia or historical persistence. A central bank that evaluates the past level of reserves as adequate will be inclined to stick to this level, even if the determinants of reserve holdings call for a reduction.

Bordo and Eichengreen (2001) show that inertia is at least present for gold holdings of central banks.

Persistence of reserves might also be the result of a central bank considering the link between reserves and the confidence which is attributed to its paper money. Economic agents still associate the reliability of paper money, namely its price stability, with the level of reserves.

If a central bank reduced abruptly but appropriately the level of reserves – for example in response to the move to a less managed exchange rate system – economic agents might question the reliability of national money, expect higher inflation, substitute national paper money for other sources of wealth, thereby destabilize the national money market and finally self-fulfil their expectations. Therefore, if fundamentals call for a reduction of reserves, central banks might sell reserves only smoothly such that the public is not unsettled.

Finally, empirical tests also point to dynamics in the level of reserves. The lagged dependent variable is highly significant in our estimations independently of the included set of control variables (see Table 7). The Wooldridge test for serial correlation in linear panel data models (Wooldridge 2002, pp. 274-276) strongly rejects the null hypothesis of no serial correlation (p=0.00) for the specification of Table 5, column 2. This might be due to omitted dynamics.

Regardless of these considerations, the standard approach of the literature on the determinants of international reserves uses static models. There are only two exceptions (Bordo and Eichengreen 2001 and Dreher and Vaubel 2009).