2.1. Disaggregated Model

Suppose that the long-run dynamics of a k-dimensional I(1) time series z_t = (y_t′,x_1,t′)′ were generated at time interval Δt by the process

where y_t is k₀-dimensional, x_1,t is k₁-dimensional, and u_t ≡ (u_1,t′,u_2,t′)′ is a stationary process that satisfies the invariance principle

for r ∈ [0,1] as n → ∞, where W denotes a k-dimensional standardized Brownian motion with variance matrix Ω ≡ lim_n→∞ n⁻¹E [(Σ₁ⁿu_j)(Σ₁ⁿu_j)′], the long-run variance of u_t. Let us partition Ω according to u_t:

and define the conditional long-run variance Ω_11.2 ≡ Ω₁₁ − Ω₁₂Ω₂₂⁻¹Ω₂₁. We assume that the matrix Ω₂₂ is positive definite in such a way that x_1,t is individually I(1) but not CI(1,1). Under these assumptions, the process z_t is a CI(1,1) process with k₀ cointegrating relations and k₁ common stochastic trends.

Representation (1)–(3) has been used to analyze optimal inference on cointegrating vectors (see Phillips, 1991b) and to design a number of optimal regression-type estimators. Phillips and Hansen (1990) and Park (1992) propose semiparametric methods, Saikkonen (1991) and Stock and Watson (1993) dynamic regression methods, and Phillips (1991c) a frequency domain method. A very important property of optimal estimators of cointegrating vectors is their asymptotic mixed Gaussian distribution, meaning that a test for g restrictions on β₁ is χ_g² distributed (Phillips, 1991b). However, regression-type methods are not very well suited to model building because they impose the location and number of unit roots on the model rather than testing them. This has two important consequences. First, it is not possible to test sequentially for the number of cointegration relations, because a triangular model with k₀′ cointegrating relations is not nested into a triangular model with k₀′′ (>k₀′) cointegrating relations (see Johansen, 1994). Second, regression-type estimators impose normalization restrictions on cointegrating vectors, and this may lead to inconsistent inference if such restrictions do not hold. In addition, these methods were originally designed to test for a single cointegrating relation (k₀ = 1), and this restricts their applicability to very small systems. However, even small systems can be governed by more than one cointegrating relation, and although it is possible to estimate more than one cointegrating relation with regression methods, the properties of the estimator and tests for the extra cointegrating relations are not clear (see Hargreaves, 1994). For all these reasons, system methods that test for the dimension of the cointegrating space and do not impose normalization restrictions, such as the full maximum likelihood analysis of cointegrated vector autoregression (VAR) (see Johansen, 1991), are commonly used for model building.

The aim of this paper is to study the effects of different aggregation options on inference rather than on model building, and the triangular model is more suitable for this purpose than other representations of the cointegrated system, such as the VAR model or the moving average (MA) model, because of its nonparameterization of the short-run dynamics.

The main focus of the paper is CI(1,1) relations, but we consider two particular cases of cointegration with I(2) variables. To do so, let us suppose that the long-run dynamics of a k-dimensional time series z_t ≡ (y_t′,x_1,t′,x_2,t′)′ are generated by the following model:

where x_2,t is k₂-dimensional (k₀ + k₁ + k₂ = k), u_t = (u_1,t′,u_2,t′,u_3,t′)′ satisfies (3), and Ω₂₂ and Ω₃₃ are positive definite. This is a very general model that allows for many different cointegrating relationships. Matrix [I_k1,−β₃] denotes the CI(2,1) cointegrating vectors of (x_1,t′,x_2,t′)′, matrix [I_k0,−β₂] contains CI(2,2) cointegrating vectors of (y_t′,x_2,t′)′ when some or all the elements of β₁ are zero, and matrix [I_k0,−β₁] contains the CI(1,1) cointegrating vectors of (y_t′,x_1,t′)′ when some or all the vectors of matrix β₂ are zero. For the nonrestricted case, [I_k0,−β₂] denotes the CI(2,1) cointegrating vectors of (y_t′,x_2,t′)′, and [I_k0,−β₁] denotes the CI(1,1) vectors of (y_t′ − x_2,t′ β₂,x_1,t′)′. In this general setting, Stock and Watson (1993) proposed dynamic optimal inference methods and Kitamura (1995) full and partial information maximum likelihood methods. Under the very restrictive assumption that Ω₁₂ = Ω₁₃ = Ω₂₃ = 0, the ordinary least squares (OLS) estimator is optimal in the sense that it has a mixed Gaussian distribution allowing for a standard χ² inference (see Haldrup, 1994).

2.2. Aggregated Model

Let us use Z_T^{(j₀,j₁,j₂)} (with j₀,j₁,j₂ = {0,1}) to denote the observable vector time series that is measured at a longer sampling interval Δ_mT = mΔt than z_t. The representation of this temporally aggregated process is given in the following lemma.

LEMMA 2.1. If z_t is generated by (3)–(6), the observable temporally aggregated time series Z_T^{(j₀,j₁,j₂)} can be represented as

where L^mZ_T^{(j₀,j₁,j₂)} = Z_T−1^{(j₀,j₁,j₂)} , Δ_m ≡ 1 − L^m, and U_T^{(j₀,j₁+1,j₂+2)} ≡ (U_1,T^(j₀)′,U_2,T^(j₁+1)′,U_3,T^(j₂+2)′)′ satisfies the multivariate invariance principle

where Ω₁₁⁽⁰⁾ = m⁻¹(Ω₁₁ + Λ(m)), Ω₁₁⁽¹⁾ = mΩ₁₁, Λ(m) = 4πΣ_j=1^m−1F₁₁(2πj/m), and F₁₁(ω) is the spectral density matrix of u_1,t.

Proof. See the Appendix.

From Lemma 2.1, the number of cointegrating relations is invariant with any of the different combinations of aggregation schemes. However, the cointegrating spaces are not generally invariant with temporal aggregation. More specifically, when the aggregation schemes applied to the regressand and the respective regressors are the same (j_l = j_h), the cointegrating relations are invariant. If not, when some of the variables are aggregated with systematic sampling and others with average sampling, the cointegrating space changes with m. The weights m^j_l−j_h capture this change in cointegrating relations, which is exclusively attributable to a change in the unit of measurement. Ericsson, Hendry, and Tran (1994) analyze the effects of linear filters, z_t^a = g(L)z_t, on cointegrating vectors. They show that when all the filtered time series are measured by the same units, g(1) = I, then the filtered series z_t^a and the original series z_t have the same cointegrating space. This condition does not hold for mixed sampling, because the flow variables are measured in relation to the sampling interval, i.e., that month's income but not the stocks, i.e., money supply.

The second main effect of temporal aggregation on the cointegrated system is a change in the short-run dynamics. The different effects of temporal aggregation on the error terms of the cointegrated system change the short-run dynamics and also their contribution to the long run, represented by the aggregated long-run variance Ω^{(j₀,j₁+1,j₃+2)} . For any possible combination of aggregation schemes, the short-run dynamics will always vary. For example, when the aggregation schemes are the same (j₀ = j₁ = j₃ = j), the long-run variance of U_2,T^(j+1) increases in relation to the long-run variance of U_1,T^(j), and the long-run variance of U_3,T^(j+2) increases in relation to the long-run variances of the other errors. Note that when the regressand is systematically sampled, Y_T⁽⁰⁾, the long-run variance of U_1,T⁽⁰⁾ is an average of the disaggregated long-run variance Ω₁₁ and the variance of those seasonal cycles that emerge as a result of the aliasing effect, Λ(m). For instance, a quarterly stock variable with an important semiannual seasonal cycle (F₁₁(π)) systematically sampled as a semiannual variable (m = 2) will experience a substantial increase in the long-run variance of the semiannual systematically sampled cointegrating error U_1,T⁽⁰⁾. As discussed in the next section, this type of aggregation will make the mixed normal distribution of the estimator of the cointegrating vectors more disperse.

Let us focus on the CI(1,1) model. The following corollary particularizes the previous results to this situation.

COROLLARY 2.2. If z_t is generated by (1)–(3), the temporally aggregated time series Z_T^(j₀,j₁) can be represented as

where U_T^{(j₀,j₁+1)} ≡ (U_1,T^(j₀),U_2,T^(j₁+1)) satisfies the multivariate invariance principle

where Ω₁₁^(j₀) is defined as before.

Proof. It follows from Lemma 2.1.

In this case, we can differentiate four different temporal aggregation situations. We use the same classification as Chambers (2003) and identify the pure systematic sampling situation as aggregation type I (Z_T⁽⁰⁾ = (Y_T⁽⁰⁾′,X_1,T⁽⁰⁾′)′), the mixed sampled process Z_T⁽⁰¹⁾ = (Y_T⁽⁰⁾′,X_1,T⁽¹⁾′)′ as aggregation type II, the other mixed sampled situation Z_T⁽¹⁰⁾ = (Y_T⁽¹⁾′,X_1,T⁽⁰⁾′)′ as aggregation type III, and finally the pure average sampled process (Z_T⁽¹¹⁾ = (Y_T⁽¹⁾′,X_1,T⁽¹⁾′)′) as aggregation type IV. Table 1 shows the aggregation effect of these different aggregation schemes for m = 3 on the long-run variances Ω_11.2^(j₀), Ω₂₂^(j₁+1), and on Ξⁱ(m) ≡ m^j₀−j₁(Ω_11.2 + (1 − j₀)Λ(m))^1/2 (i = I, II, III, IV), which, as shown in the next section, is the relative dispersion of the mixed normal distribution across m for the different aggregation schemes.

Temporally aggregated long-run variances of the CI(1,1) model for m = 3

Let us consider a particular case of the I(1) cointegrated model. Assume that k₀ = k₁ = 1, u_1,t = ρu_1,t−1 + ε_1,t with |ρ| < 1, ε_1,t is i.i.d.N(0,1), u_2,t = ε_2,t is i.i.d.N(0,σ²), and E(ε_1,tε_2,t−j) = 0 for all j. The systematically sampled model is given by

where U_2,T⁽¹⁾ = E_2,T⁽¹⁾ = s_m(L)ε_2,mT, and U_1,T⁽⁰⁾ = ρ^mU_1,T−1⁽⁰⁾ + E_1,T⁽⁰⁾, E_1,T⁽⁰⁾ = [(1 − ρ^mL^m)/(1 − ρL)]ε_1,mT is serially uncorrelated and not correlated with E_2,T⁽¹⁾ for all lags and leads. From Pesaran and Shin (1996) the impact of a unit shock to the variable Y_T⁽⁰⁾ on the cointegrating relation after one T-period is given by the autoregressive coefficient ρ^m. Then, given |ρ| < 1, the longer the sampling interval with respect to the long-run dynamics generating time interval, the faster the adjustment of the system to a disequilibrium. Let us consider the two limiting aggregation cases. In the maximum aggregation case, i.e., when m → ∞, all the adjustment takes place in one period (lim_m→∞ ρ^m = 0), whereas for the maximum disaggregation case, when m → 0, no adjustment takes place (lim_m→0 ρ^m = 1). Therefore, the finer the sampling interval the closer the cointegrated model is to a noncointegrated system.

This example illustrates that when the long-run dynamics are measured at a much longer sampling interval than their generating time interval, then the speed of adjustment of the cointegrating relation should be very fast. In practice, for many cointegrated relations the estimated speed of adjustment with monthly or quarterly time series is low, and this suggests that the long-run dynamics for many cointegrated systems are not generated at a much shorter time interval.

3.1. Temporally Aggregated Mixed Normal Distribution

Suppose that a k-dimensional time series z_t = {(y_t,x_1,t′)′}_t=1ⁿ, where y_t is a scalar time series⁴ and x_1,t is a k₁-dimensional time series, is generated by

where u_t = (u_1,t,u_2,t′)′ is a stationary process that satisfies (3). Let

denote an optimal regression-type estimator of β₁ using the time series Z_T^(j₀,j₁) where i = I for Z_T⁽⁰⁾, i = II for Z_T⁽⁰¹⁾, i = III for Z_T⁽¹⁰⁾, and i = IV for Z_T⁽¹⁾. The following theorem presents its limiting distribution.

THEOREM 3.1. The asymptotic distribution of the optimal regression-type estimator of β₁ in model (10) and (11) with temporally aggregated time series is

where the relative dispersion across the sampling interval Ξⁱ(m) is Ξ^I(m) = (Ω_11.2 + Λ(m))^1/2, Ξ^II(m) = m⁻¹(Ω_11.2 + Λ(m))^1/2, Ξ^III(m) = mΩ_11.2^1/2, and Ξ^IV(m) = Ω_11.2^1/2.

Proof. See the Appendix.

From Theorem 3.1, comparing these asymptotic distributions with the one for an optimal estimator of the disaggregated model, Ω_11.2^1/2 ∫dW₁B₂′(∫B₂B₂′)⁻¹, there is no loss of efficiency when the regressand and the regressor are average sampled (aggregation IV), whereas there is a loss of efficiency when the regressors are systematically sampled, i.e., aggregation schemes I and III. These results are in keeping with those of Chambers (2003), who considers a similar scenario where the disaggregated model is a continuous time cointegrated model.

However, in contrast with Chambers (2003), in case II three situations are possible depending on the relative size of the aliasing component Λ(m) in relation to the conditional long-run variance Ω_11.2. To be more specific, when Λ(m) = (m² − 1)Ω_11.2, which implies that 2Σ_j=1^m−1F₁₁(2πj/m) = (m² − 1)F_11.2(0), there is no loss of efficiency, whereas for 2Σ_j=1^m−1F₁₁(2πj/m) > (m² − 1)F_11.2(0), there is a loss of asymptotic efficiency, and when 2Σ_j=1^m−1F₁₁(2πj/m) < (m² − 1)F_11.2(0) there is a gain in asymptotic efficiency. In any case Ξ^II(m) < Ξ^I(m), and when Λ(m) < (m⁴ − 1)Ω_11.2, then Ξ^II(m) < Ξ^III(m).

To summarize, type II aggregation is always more efficient than type I aggregation, and type IV aggregation is always more efficient than types I and III aggregation. When Λ(m) < (m² − 1)Ω_11.2 type II is more efficient than IV aggregation and therefore is the most efficient option.

Continuing with the example of Section 2.2 and for m = 3, a situation of no loss of asymptotic efficiency with aggregation II occurs when ρ = −0.27, for −1 < ρ < −0.27 there is a loss of efficiency, whereas for −0.27 < ρ < −1 there is a gain of efficiency.

Let us compare the relative dispersions Ξⁱ(m) with those derived with the continuous time aggregation approach (see Chambers, 2003), which we denote as Ξ_Cⁱ. To do so, let us consider a bivariate triangular continuous time cointegrated model:

where (u₁(t),u₂(t))′ is a stationary continuous time process with spectral density function F_C(ω) with −∞ < ω < ∞. From Chambers (2003, p. 59) and given that for j ≠ 0 h(2πij) = (2πij)⁻¹, h(2πij) + h(−2πij) = 0 and |h(2πij)|² = (4π²j²)⁻¹, we obtain the expressions

There are important differences between Ξ_Cⁱ and Ξⁱ(m). First of all, in the continuous time setting, type II aggregation is the most inefficient option for any F_C(ω), a result that contrasts with the findings obtained in the discrete time approach where type II aggregation may be the most efficient way to aggregate the time series and in any case is less efficient than schemes I. Second, the cointegration coefficient β₁ appears at the limiting distributions of the discrete time optimal estimator derived from the continuous time cointegrated system, a result that is in contradiction with the optimal theory of estimation of cointegrating vectors (see Saikkonen, 1991; Phillips, 1991b). Finally, in the continuous time approach, the aliasing affects the distributions through the error process of the cointegrated relations (2πΣ_j≠0 F_C,11(2πj)), and through the error process of the common stochastic trends (2πβ₁²Σ_j≠0(4π²j²)⁻¹F_C,22(2πj)), a surprising result because for both types of aggregation the process u₂(t) is at least cumulated once ∫u₂(t − s) d_s, a transformation that eliminates the components F_C,22(2πj) in scheme IV of continuous time aggregation for the errors of the cointegrating relations. Thus, the continuous time aggregation fails to explain the discrete time aggregation results.

Theorem 3.1 has practical implications. If we consider the common situation where the practitioner may decide how to aggregate a stock time series, then we have two different situations depending on whether it is the regressand or the regressor that is temporally aggregated. When a practitioner has to aggregate a stock regressand and the regressor is a stock, the best aggregation option depends on the sign of Λ(m) − (m² − 1)Ω_11.2. If this magnitude is negative, the best option, in terms of estimation efficiency, is to use systematic sampling (aggregation I), whereas if it is positive, the optimal choice is average sampling (aggregation III). When the regressor is a flow, the best option depends on Λ(m) − (m² − 1)Ω_11.2, in the sense that when this parameter is positive the stock regressand should be aggregated by average sampling (aggregation IV), whereas when it is negative the regressand should be aggregated by systematic sampling (aggregation II). When a practitioner has to aggregate a stock regressor, the best option, in terms of the asymptotic dispersion of the estimator, is always to apply average sampling to the regressor because aggregations II and IV are superior in efficiency to aggregations I and III, respectively. Therefore, the picture that emerges from the discrete time aggregation analysis is more complex than the one obtained from the continuous time approach where the best option seems to be in any situation to apply average sampling. We provide finite-sample evidence favoring our theoretical findings in Section 4.

In practice, when deciding whether to aggregate the stock regressand by systematic or average sampling, we need to compare Λ(m) with (m² − 1)Ω_11.2. However, because the flow variables are observable at a longer sampling interval than the stock variables, it is necessary to get Ω₁₂Ω₂₂⁻¹Ω₂₁ from the observable Ω₁₂^(j₀,2)Ω₂₂⁽³⁾⁻¹Ω₂₁^(2,j₁). Then, from the relation

we can estimate Ω₁₂Ω₂₂⁻¹Ω₂₁ and Ω_11.2 and compare the estimations of Λ(m) and (m² − 1)Ω_11.2.

To summarize, when a stock regressand must be temporally aggregated because some of the regressors are not available at the finest sampling interval, then depending on the magnitudes Λ(m) and (m² − 1)Ω_11.2 systematic or average sampling are the best aggregation options in terms of the dispersion of the mixed normal distribution. When a practitioner has to choose an aggregation option for the stock regressor, the best choice is always to apply average sampling.

The main results found for case CI(1,1) can be extended to models with I(2) variables. We do not aim to cover all possible situations of cointegration in models with I(2) variables but to show the robustness of the aggregation theory for some relevant examples. First, let us assume the following disaggregated model:

where u_t = (u_1,t,u_2,t′,u_3,t′)′ satisfies the assumptions described in Section 2 and the long-run covariances are zero Ω₁₂ = Ω₁₃ = Ω₂₃ = 0. For this model the optimal estimator is the OLS estimator. Theorem 3.2 shows the asymptotic distribution of the optimal estimator of β₁ and β₂ for the aggregation schemes Z_T⁽⁰⁾ (aggregation I), Z_T⁽⁰¹¹⁾ (aggregation II), Z_T⁽¹⁰⁰⁾ (aggregation III), and Z_T⁽¹⁾ (aggregation IV).

THEOREM 3.2. The asymptotic distribution of the optimal regression-type estimator of β = (β₁′, β₂′)′ in model (12)–(14) with temporally aggregated time series is

Proof. See the Appendix.

From Theorem 3.2, temporal aggregation has the same effect on the separate distribution of the estimator of β₁ as on the separate distribution of the estimator of β₂, and this effect is exactly like that found for case CI(1,1).

Now consider the estimation of CI(2,1) relations, which we assume are generated by the following model:

where u_t = (u_2,t,u_3,t′)′ satisfies the general assumptions for the cointegrated model with I(2) variables described in Section 2. In this situation, the optimal regression-type estimators of β₃ are those estimators for the CI(1,1) model with differenced variables. The limiting distribution for the different aggregation schemes is given in the next corollary.

COROLLARY 3.3. The asymptotic distribution of the optimal regression-type estimator of β₃ in model (15) and (16) with temporally aggregated time series is

where Ξ^I(m) = Ω_22.3^1/2, Ξ^II(m) = m⁻¹Ω_22.3^1/2, Ξ^III(m) = mΩ_22.3^1/2, and Ξ^IV(m) = Ω_22.3^1/2.

In this case, as a result of the absence of the aliasing effect because the errors are average and double-average sampled, we find slightly different results. Now, for pure aggregation schemes I and IV the limiting distribution remains invariant despite the sampling interval, whereas in the case of mixed sampling the dispersion is reduced with m for aggregation II, and it increases with m for aggregation III.

Now, when a practitioner has to aggregate a stock regressand, the best option is always to apply systematic sampling. This is slightly different from case CI(1,1) where in some situations the best way to aggregate the regressand was average sampling when the regressor was a flow. When it is the stock regressor that needs to be aggregated, the practitioner has to apply average sampling in all cases, in keeping with case CI(1,1).

The limiting properties of the optimal estimator of β₁ are derived under the assumption that the long-run variance Ω must be consistently estimated, and there may be a loss of valuable information when this matrix is estimated with temporally aggregated time series. Consequently, the aggregation effects on the estimation of Ω may alter the preceding conclusions reached on the best aggregation option for the estimation of long-run relations in finite samples. However, the Monte Carlo evidence provided in Section 4 shows that the estimation of Ω does not make any significant difference to the order in which the types of aggregation are ranked.

3.2. Temporally Aggregated Observed Information

When the purpose of the analysis is to test a hypothesis on the cointegrating vectors, such as the PPP theory or permanent income hypothesis, the aggregation effect on the marginal distribution of the estimator provides us with half the story, because for nonstationary cointegrating regressions, the observed information J_n(β₁) ≡ −∂² log L(β₁)/∂β₁∂β₁ where log L(β₁) is the log likelihood function, and not the variance of

, is used as the normalizing matrix of the χ²-distributed Wald test (see Johansen, 1995):

This situation is different from a stationary regression where

, and the marginal distribution of the estimator is sufficient to discuss the aggregation effect. However, the observed information in a cointegrated model is different from the inverse of the variance of the estimator, and the aggregation effect must be considered on both the estimator and the information. This task is straightforward when we only consider the asymptotic properties, because the χ² distribution of the Wald test implies that the aggregation effect on the observed information should neutralize the aggregation effect on the estimator. Thus, an increase in the dispersion of the asymptotic distribution of the estimator due to aggregation should be counterbalanced by a reduction in the dispersion of the distribution of the information. For example, for Λ(m) = (m² − 1)Ω_11.2, the asymptotic dispersion of the observed information increases for cases I and III, whereas it is invariant for cases II and IV. Therefore, in asymptotic terms it is not relevant which type of aggregation we select to test restrictions on β₁.

However, this asymptotic result may be different when we consider the more realistic finite-sample framework. In such a setting, as shown by Johansen (2002), the distribution of the test is slightly different from the χ² distribution, depending not only on the number of observations but also on the parameters of the model. As shown in Section 2, different combinations of aggregation schemes have different effects on the short-run dynamics, and therefore in finite samples one aggregation scheme may be clearly superior to another for hypothesis testing. Moreover, the faster convergence of the observed information n⁻²J_nⁱ(β₁) than the estimator

may contribute to the fact that the aggregation effect on J_n(β₁) has a bigger influence on the finite-sample properties of the test than the aggregation effect on

. However, the Monte Carlo experiment in Section 4 shows that, for the selected model, all the aggregation options have similar effects on the size of the test.

4.1. Design of the Experiment

In this section we study the finite-sample aggregation effect on the FM-OLS.⁵

where u_1,t = ρu_1,t−1 + ε_1,t, u_2,t = ε_2,t, and

The length of the sampling interval was fixed as m = 3. The relevant magnitudes for the aggregation effect are the aliasing component, Λ(3), and the conditional long-run variance of the cointegrating errors, Ω_11.2. This very simple model allows us to control these magnitudes through parameter ρ. More specifically, we consider three sizes for the aliasing effect, so that for ρ = −0.5 the aliasing component is very big in comparison with 8Ω_11.2 (Λ(3) = 4.54 and 8Ω_11.2 = 2.64); for ρ = 0 the aliasing component is slightly smaller than 8Ω_11.2 (Λ(3) = 4 and 8Ω_11.2 = 6), and for ρ = 0.5 the aliasing component is much bigger than 8Ω_11.2 (Λ(3) = 2.28 and 8Ω_11.2 = 24).

The experiment is performed with two spans, n = 120 and n = 360. We only present the results for the shorter span, n = 120, to compare the different aggregation effects on the estimator and test (see Table 2) because a comparison of the aggregation schemes is not (qualitatively) affected by the size of the span. As a measure of the aggregation effect on the precision of the estimator, we provide the ratio of the mean square error (MSE) of the aggregated estimator in relation to the disaggregated estimator, denoted M(i) where i ∈ {I, II, III, IV} is an aggregation scheme. As a measure of the aggregation effect on the test, we compute the variation of the size of the test for H₀ : β = 1 based on the 5% level, denoted S(i) with i ∈ {I, II, III, IV}. That is, S(i) stands for the difference between the size of the test in the aggregated models and the size in the disaggregated model. Three thousand replications of the experiment are used to obtain Monte Carlo estimations of the MSE and 10,000 for the size of the test. We also compare the aggregation effect with the span effect. To do so, with the disaggregated model we compute the variation in the MSE (M(n)) and size (S(n)) for a small sample n = 120 and a big sample n = 360. To avoid the influence of initial conditions, a constant is estimated together with the cointegrating coefficient and the first 50 observations are discarded.⁶

Comparison of the different temporal aggregation effects

4.2. Results

Table 2 shows the variation in the MSE and the size of the test when different types of aggregation are applied to the DGP described previously for n = 120. When 8Ω_11.2 > Λ(3), the best aggregation scheme, in terms of the precision of the estimator, is aggregation type II, with a lower MSE than the disaggregated estimator. The second best aggregation option is aggregation type IV. With this data transformation, the MSE hardly varies in comparison with the disaggregated estimator. Much less precise estimations of the cointegrated vector are obtained when the regressor is systematically sampled, with aggregation III being the worst option and there being a tremendous relative increase in the dispersion of the estimator. The performance of the estimator, when both variables are systematically sampled (i = I), is largely dependent on the relative size of the aliasing component, with the result that when the aliasing component is smaller than the long-run conditional variance, the precision of the estimator is closer to the precision of the estimator with aggregation IV.

If we look at the aggregation effect on the test, we find a different story. In this case, all the aggregation schemes increase the distortion of the test by approximately 3 to 5 points. The main difference, when compared with the aggregation effect on the precision of the estimator, is that now the different types of aggregation all share a very similar effect. The few differences that can be observed reveal that the aggregation effect on the test cannot be explained by the aggregation effect on the estimator. The most significant example is the case ρ = 0.5, where the least distorted test is obtained with aggregation III, the worst aggregation option in terms of the MSE. However, we cannot infer from this that the worst aggregation for estimation purposes is the best one for testing, because if we look at case III for ρ = −0.5, where there is the biggest loss in estimator precision (22.9), it is not the one with the least size distortion.

It is interesting to compare the aggregation effect with the effect of the span on the estimator and the test. In Table 3 we present the variation in the MSE and size when the sample is reduced from 360 to 120 observations, either by reducing the span by 3 (M(n) and S(n)), or by temporal aggregation (M(i) and S(i), i ∈ {I, II, III, IV}). If we examine the effects on the precision of the estimators, the reduction of the span has similar repercussions to the worst aggregation schemes (I, III), a situation similar to that one expects to find with a stationary regression. However, the reduction of the span clearly has worse consequences than aggregation schemes II and IV, which shows that the aggregation effect on the estimation of cointegrating regressions is not as important as the aggregation effect on stationary regressions.

Comparison between the span effect and the temporal aggregation effect

Focusing on the test, reducing the span has a slightly bigger impact on size than aggregation. However, once again the few exceptions that can be observed show how the effects on the test are different than on the estimator. For example, when ρ = 0, the size distortion of aggregation II is practically the same (2.2) as the distortion caused by the reduction of the span (2.1). However, for this case, if we consider the effect on the precision of the estimator, the reduction of the span affects the quality of the estimation far more than aggregation with scheme II.

To summarize the Monte Carlo results, different aggregation schemes have very different effects on the precision of the estimator, thus confirming the asymptotic theory derived in the paper. More specifically, when the regressor is average sampled, the precision is not affected or even improved by temporal aggregation, whereas when the regressor is systematically sampled the estimator is clearly less precise, especially when the regressand is average sampled. The aggregation effects on the test are very different from the effects on the estimator. In this case, there are no significant differences among the different aggregation options. Finally, we found that reducing the span has a similar effect on precision to those aggregation schemes where the regressor is systematically sampled (I, III) and a much greater effect than those aggregations where the regressor is average sampled (II, IV). Also, a reduction in the span has a slightly greater effect on size than the temporal aggregation of the variables.