2. A TWO-LEVEL HIERARCHICAL MODEL

We assume that information is available at two levels: the macro level and the micro level. The macro level provides aggregate information, whereas the micro level provides disaggregate information, thus giving more detail for the components of the macro level. The macro level is indexed by t = 1,…,T; the micro level, in addition, by j = 1,…,p.

For the two levels we may have different models and different data sources. At the macro level we assume that data are available on the variable of interest η_t together with [ell ] × 1 vectors of exogenous explanatory variables z_t. For simplicity, we assume a linear model

where {u_t} is unobservable independent and identically distributed (i.i.d.) random noise with mean zero and variance σ_u² and z_t−1 is revealed at the end of period t − 1. If

denotes the ordinary least squares (OLS) estimator of γ based on T observations, then the macro forecast is given by

.

The data at the more detailed micro level consist of the vector of variables of interest y_t := (y_t,1,…,y_t,p)′ together with p × k matrices of exogenous explanatory variables X_t. We assume again a linear model

where the vectors {v_t} are unobservable i.i.d. random noise with mean zero and variance matrix Σ_v and X_t−1 is revealed at the end of period t − 1. If

denotes the generalized least squares (GLS) estimator of β based on T observations, then the micro forecast is given by

.

The data for the macro model are compiled in a macro office, and γ is estimated by a macro statistician. The data for the micro model are compiled in a micro office, and β is estimated by a micro statistician. The two offices and the two statisticians may or may not talk to each other. Thus, some of the information used by one statistician may also be used by the other but not necessarily so. The two levels are related to each other, even though the macro and micro statisticians do not utilize this relationship. In the background lives the linear constraint

where a is a known p × 1 vector. For example, if a = ı_p, the vector of ones, then the macro observations are the sum of the micro observations, and if a = ı_p/p, then the macro observations are the average of the micro observations. As a result, the two levels may be (and probably will be) correlated, but we shall assume that this correlation exists only within the same time period. Hence, cov(u_t,v_t) may be nonzero but cov(u_s,v_t) = 0 for s ≠ t. We emphasize that we do not require the linear constraint η_t = a′y_t to hold for every t. This would be the case of perfect model aggregation, which is rare in practice. If one, nevertheless, encounters this case, then the micro model contains the best information for both (micro and macro) levels.

The macro statistician provides a macro forecast

with an estimated variance

, and the micro statistician provides a micro forecast

with an estimated variance matrix

. (We will see later that only the relative precisions matter.) Our task is to combine the two forecasts, making use of all available information. In particular, we wish to obtain optimal one-period-ahead forecasts

(micro) and

(macro), satisfying the micro-macro relationship

Note that we do not assume that η_t = a′y_t holds at t = T + 1; only that the forecast satisfies (3). Note also that our assumption on the availability of data rules out the possibility of estimating of (1) and (2) jointly. We emphasize that the linearity assumptions in (1)–(3) are made for simplicity only and do not essentially affect our analysis.

We have two forecasts of η_T+1, one based only on macro data and one based only on micro data:

We wish to improve the forecasts at both levels, and we propose the combined macro forecast as the linear combination

in the spirit of Bates and Granger (1969). Conditional on the choice of α, this provides a common-sense improvement of the macro forecast. We call (4) a linear combination rather than a weighted average, because α may not lie between zero and one if the correlation between

and

is high and positive. This is somewhat counterintuitive, and Appendix A therefore provides a justification and explanation.

Two questions now arise. First, how do we choose α? Second, how do we update the micro forecast, given the improvement of the macro forecast? We shall discuss the second question first.

2.1. Updating the Micro Forecast

We assume first that α is known and that the macro forecast is given by (4). Given the improvement at the macro level, we now update the micro forecast.

Letting

, this is achieved by choosing y to maximize

subject to

Under normality, the solution is the point with the highest probability in the micro model satisfying the macro restriction. In the two-dimensional case, Figure 1 illustrates the intuition. We have a point estimate (c₁,c₂) and the associated probability distribution, and we want the sum of the two estimates to be equal to c′. To include this additional information, we choose the ellipse (constant-probability contour) that is tangent to the line c₁′ + c₂′ = c′. The tangent point satisfies the restriction and is the most probable point given the probability distribution. In the special case where the forecasts c₁ and c₂ are not correlated, they will be adjusted proportionally to their variances. In the special case where the variance of c₁ is zero, all adjustment will be absorbed by c₂.

Update procedure in the two-dimensional case.

The normality assumption is of course a simplification. We use it here because of its central role as an approximating distribution and because it provides explicit and intuitively clear results. Our procedure can be generalized to other densities, but a closed-form solution will then typically not be available and numerical optimization has to be performed.

The solution to (5) is given by

which can be calculated if we replace Σ by

and if we know α. Note that the forecast (6) is biased, because the expectations of

and

are, in general, not equal. Thus we obtain a macro forecast

and a micro forecast

, satisfying the restriction

and taking into account information at both levels.

2.2. The Choice of α

We next address the question how to choose α. We shall employ the mean squared error criterion as our guiding principle, although this criterion has some drawbacks (Clements and Hendry, 1993) and some other criteria have recently been proposed by Diebold and Kilian (2001), Inoue and Kilian (2006), and Yang (2004).

The mean squared error of the forecast (6) is given by

where

and

Hence,

—considered as a function of α—is “small” if and only if (αq − c)(αq − c)′ is “small.” The latter matrix has rank one, and it possesses one positive eigenvalue (αq − c)′(αq − c), which is minimized for

Choosing α = α₁ thus minimizes the trace of

.

³

Instead of the trace we might also consider minimizing

. This leads to α₁* = a′c/a′q. One may verify that α₁ = α₁* if and only if (a′Σa)(a′Σπ) = (a′Σ²a)(a′π), which is the case, for example, when π = 0 or Σ = I or Σa = π.

Using the facts that

we obtain

and hence

where

and

Because information on micro and macro forecasts reaches us separately, we do not have a direct estimate of π. We can, however, estimate π from a history of forecasts

, if such a history is available. Quite often, such a history will not be available. Interestingly, our simulations (to be discussed later) indicate that the assumption π = 0 is a reasonable, often optimal, assumption to make.

If the two processes {u_t} and {v_t} are uncorrelated, we have π = 0, and hence

which is a number between zero and one. For small values of the components of π, α₁ also lies between zero and one. But if the correlation between

and

is positive and high, then α₁ does not necessarily lie between zero and one.

To compare the various combined forecasts we consider the trace of

, which we denote generically by τ². For α = α₁ we obtain from (7)

and for α = α₂

3. A DIFFERENT ROUTE

The method of the previous section can be summarized as follows. Given a macro forecast

and a micro forecast

, we minimize the quadratic form

subject to the restriction

This produces a forecast

. We then choose α such that the mean squared error of

is minimized.

We now describe a second possible method, based on the idea that we should choose y such that a′y is close to

and that y is close to

. Thus we minimize

where

We can also phrase this problem as the regression problem

with

This formulation implies that

, so that E(δ) = 0. After somewhat tedious but straightforward calculations we find the GLS solution as

with

Note that we introduce α₃ to connect the GLS solution (12) to the previous solutions. There is, however, no optimization over α in this case. In this sense one can think about the GLS solution as optimal for both micro and macro level.

The forecast (12) is unbiased, and its variance is given by

so that

which should be viewed as the analogue to τ₁² in the previous section. There is no theoretical inequality between τ₁² and τ₃². For example, in the special case when there is no inconsistency between the models (so that z_T′γ − a′X_Tβ = 0), and the variance structure is simply Σ = I_p and a = ı_p, we find

The difference between τ₁² and τ₃² then depends on the relationship between

and both τ₁² < τ₃² and τ₁² > τ₃² are possible.

Given the micro forecast, the macro forecast is then given by

In the special case where π = 0 we obtain

with

The variance of this forecast is

so that

Notice that τ₄² ≤ τ₂² as a result of the fact that in the derivation of τ₄² we have assumed that the inconsistency parameter satisfies E(δ) = 0.

The result (15) can also be phrased in Bayesian terms by considering macro data together with a micro prior:

Using Theorem 1 of Magnus, van Tongeren, and de Vos (2000), we then obtain the mean and the variance of the posterior distribution of

as (15) and (17), respectively.

Finally, if more information were available than there is, we could try to incorporate macro information in the micro model, motivated by Hendry and Hubrich (2005). For example, we might assume

where the vector h will typically be the unit vector ı and a′h > 0. The formulation (19) implies that a′y_t is a weighted average of a′X_t−1β and z_t−1′γ, apart from some noise a′ζ_t.

We emphasize that this model requires knowledge of the separate micro and macro data sets, which we do not have. Nevertheless it is of interest to compare our feasible forecasts with some forecasts that require more data, because this provides information on the sensitivity of the feasible forecasts.

Let

and

denote the OLS estimators in this model. The unrestricted estimators

and

are related to the restricted estimator

(under the restriction γ = 0) by

for some matrix Π₁. Hence the micro forecast is given by

where Π′ = (Π₁′ : I) and its mean squared error is

where

We see that

in this case can be larger or smaller than

.

4. MONTE CARLO EVIDENCE: THE BASE SCENARIO

To compare the proposed forecasts and to assess the possible improvement resulting from the use of macro information, extensive simulations were performed on which we now report. First we generate five regressors:

These regressors can be combined in various data sets. We consider five data sets with two regressors and five with three regressors, as follows:

The number of regressors is therefore k = 2 or k = 3. Given one of these 10 data sets, we generate the p × k matrices X_t−1 for t = 1,…,T. We consider T = 24 or T = 96 years and within each year p = 3 or p = 9 micro units.

4.1. The Data-Generation Process

For the base scenario, the micro data are generated by

where all k elements of β are equal to one. Letting σ_v² = 1 we take independent draws from the N(0,σ_v²) distribution, generate the p × 1 vectors v_t, and compute y_t. The macro data are exact aggregates of the micro data, so that

where the aggregation vector a = ı_p/p. The macro data are thus generated by

where all k elements of γ are also set equal to one, u_t = a′v_t almost surely, and σ_u² = σ_v²a′a = 1/p.

4.2. The Model

The base model is the same as the data-generation process, except that the micro and macro statisticians do not know the exact relationship between the two models and hence do not utilize the relationship u_t = a′v_t (almost surely).

The OLS estimator in the micro model (21) is given by

and the one-period-ahead forecast can be computed as

. All sums, unless otherwise indicated, are for t = 1,…,T. The variance of

is

The macro model is given by (22). The OLS estimator for γ is

and the one-period-ahead forecast can be computed as

. The residuals are defined as

and the variance estimates as

One may verify that

because cov(y_t,η_t) = cov(v_t,a′v_t) = σ_v²a. Because the micro and macro models are estimated separately, the fact that u_t and v_t are correlated is not known and not used by our macro and micro statisticians. As a result, cov(y_t,η_t) is not estimated by

but rather by

, which is not the same. The estimate for π is then given by

and, although π = Σa, it is not true that

. We note that, in practice, we cannot compute this estimate, because our information consists only of the two forecasts and their precisions. In the simulations we can, however, compute the estimate, and it allows us to study the sensitivity of the feasible forecasts. All Monte Carlo experiments were performed with 10,000 replications.

4.3. Forecast Comparisons

We now compare the various forecasts assuming the data-generation process and the models from the base scenario. As a target for comparison, we shall use the forecast (6),

considered as a function of α. The trace of its mean squared error is

also a function of α. We define τ₀² as the minimum value of τ²(α), and we call τ₀ the target root mean squared error (RMSE). We emphasize again that, given the data constraints on our statistician, only three of the six forecasts can actually be computed in practice, namely, the micro forecast and the forecasts whose RMSEs are labeled τ₂ and τ₄. The remaining three forecasts are presented for comparison and sensitivity purposes only.

Table 1a gives the results for the micro forecast and for the five competing forecasts. In each case we present

that is, the performance of the relevant RMSE relative to the target RMSE. Because π = Σa in the base model, we obtain

in both (6) and (12). Hence, the forecast based on micro data only (and ignoring macro information) is the optimal forecast in this case. The relative loss compared to itself (Table 1a(1)) is obviously zero.

Relative RMSE comparisons (in %) for various forecasts and various scenarios

We do not present the results for each data set; instead we provide only the minimum, maximum, and median of the relative RMSE among the different data sets. If we replace the optimal α (i.e., α₁) by its estimate

, we obtain τ₁; see Table 1a(2). For most data sets, the results are close to Table 1a(1) because

is close to α₁. But for some data sets the relative loss is very large. Setting π = 0 we obtain

and τ₂, and the relative loss is reported in Table 1a(3). This should not be as good as Tables 1a(1) and 1a(2) because in fact π ≠ 0, but in fact the performance is better, both in terms of “worst-case scenario” (max) and in terms of the median.

In Tables 1a(4) and 1a(5) we report the RMSE for the forecasts (12) and (15). The forecast (12) does not perform well for some data sets, whereas the forecast (15) does perform well. Table 1a(4) is very close to Table 1a(2), whereas Table 1a(5) is slightly worse than Table 1a(3) because α₂ < α₄ because of the bias term.

Finally, the model that incorporates macro information into the micro model leads to forecast (20). Here we estimate parameters that are in fact zero, and hence we lose efficiency. This is reflected in Table 1a(6).

5. DEVIATIONS FROM THE BASE SCENARIO

In real-life applications, the micro model and the macro model will deviate more than in the idealized situation of the previous section. We now investigate four scenarios that mimic situations as they might occur in practice.

5.1. Measurement Errors in the Micro Regressors

We assume that there are two regressors (k = 2, data sets 1–5) and that the data are generated as in the base case. However, the second regressor is badly measured at the micro level, so that we observe x_t−1,2* = x_t−1,2 + ζ_t−1 instead of x_t−1,2. The {ζ_t} are i.i.d. N(0,σ_ζ²), and they are independent of all other errors. The estimates and forecast at the micro level are thus based on X_t−1* := (x_t−1,1,x_t−1,2*). The effect of the measurement error will be that the OLS estimator

is inconsistent with a bias toward zero.

Measurement errors can either accumulate or dampen when aggregated. Here we assume that they dampen and in fact that measurement errors are absent in the macro model. Thus, the regressors in the macro model are z_t−1 = X_t−1a and not z_t−1 = X_t−1*a. In Figure 2 we plot τ(α) (solid line) for three values of σ_ζ, namely, 0.0, 0.1, and 0.5. The case σ_ζ = 0 is identical to the base scenario discussed in the previous section. Hence, the minimum τ₀ occurs at α = 0. The values of τ₁,…,τ₅ are also plotted, and we see (for this specific data set) that τ₁ performs best and τ₅ worst, in correspondence to Table 1.

Effect of measurement errors in micro regressors, T = 24, p = 3, data set = 3, for three values of σζ.

If σ_ζ > 0, then the minimum of τ(α) will not be at α = 0. In this specific example, τ₄ performs best. This is the simple “Bayesian” solution, ignoring information on π.

Tables 1b and 1c provide more detailed information. When σ_ζ = 0.1 (Table 1b), the optimum is not at zero, and the naive micro forecast is not optimal any more. This is reflected in Table 1b(1). Table 1b(2) (reporting on τ₁) is further away from the optimum, because of the misspecification, which implies that α₁ is no longer the optimum value for α. Table 1b(3) is close to the optimum, showing that the wrong assumption π = 0 is less damaging than the misspecification. Table 1b(4) is again similar to Table 1b(2), but, because the optimal α is not zero, there is no strict dominance in this case. Table 1b(5) (reporting on τ₄) is uniformly closer to the minimum than Table 1b(4), which shows again that information on π is not necessarily useful. The explanation for Table 1b(6) is similar to Table 1a(6) with the addition that we not only estimate zeros in this case but also have a measurement error bias.

In Table 1c we consider the case where σ_ζ = 0.5. The differences between the various forecasts are larger now than in the case when σ_ζ = 0.1. The naive micro forecast in Table 1c(1) is not a good forecast any more. The correct specification assumption remains more important than the assumption that π = 0, so that Tables 1c(2) and 1c(4) are dominated by (bigger than) Tables 1c(3) and 1c(5), respectively. The forecast corresponding to τ₄ is the clear winner in this case.⁴

Notice the negative number in Table 1c(5), implying that τ₄ < τ₀ in one of the data sets (number 4). This can happen, because the α-curve and therefore the theoretical optimum is calculated by averaging the simulation results, keeping α fixed. When α is estimated, then the average is not necessarily on the α-curve (this is the reason why on the pictures, when α is estimated, the results are shown as a straight line rather than a point). Therefore it is possible that τ₁,…,τ₄ are less than the simulated optimum. This is, however, a rare event.

5.2. Nonavailability of Some Micro Regressors

We now assume that there are three regressors (k = 3, data sets 6–10). However, the third regressor is only available at the macro level, not at the micro level. As a result we have X_t−1* := (x_t−1,1,x_t−1,2), whereas X_t−1 := (x_t−1,1,x_t−1,2,x_t−1,3). When one of the regressors is not available at the micro level, then the forecast performance depends on which regressor is not available and how much useful information is lost. Figure 3a illustrates that the minimum of the τ(α)-curve is not at α = 0 and gives the RMSEs for the five competing forecasts for one data set. Table 1d provides more detailed information and shows that the loss can be substantial. For example, in Table 1d(1) the naive micro forecast suffers badly from the omitted variable bias, in particular in one of the data sets (number 7). (These outliers are the main reason for using the median rather than the mean in reporting the simulation results.) Tables 1d(2) and 1d(3) are slightly closer to the target, but not as much as Tables 1d(4) and 1d(5), which shows that the bias term in α₁ is poorly estimated. In most cases, Table 1d(4) gives better result than Table 1d(5).

Effect of three types of misspecification, T = 24, p = 3, data set = 8.

The model that incorporates macro information into the micro model (forecast (20)) is interesting in this case. In the previous scenarios, this forecast was far from the optimum, but here it gives a better result than the minimum of the α-curve. This is not a surprise, because we are using a macro regressor as a proxy for the corresponding micro regressor (which is not available) and this proxy is a very good proxy.

5.4. Correlation at the Micro Level

Finally, it may happen (indeed, it is likely) that the micro units are correlated, so that v_t ∼ N(0,Σ_v), where Σ_v is not proportional to the identity matrix. Here we assume that

where J := ı_pı_p′/p is an idempotent matrix of rank one and the coefficients are θ₁ := 1 + (p − 1)ρ and θ₂ := 1 − ρ. The statistician, however, is unaware of the correlation. We take ρ = 0.5. The data-generation process has changed, but the models are still the same.

In this case the theoretical minimum is often obtained for α < 0; see Figure 3c. We know that this is theoretically possible, and truncation at zero would lead to poorer performance. The forecasts reported as τ₁ and τ₃ perform poorly; the other forecasts perform reasonably well.

6. EXTENSION TO TIME SERIES

Let us summarize our findings so far. Our purpose was to construct an improved micro forecast

in the situation where we are given a macro forecast

with estimated variance

and a micro forecast

with estimated variance matrix

and where, in addition, we wish the improved forecasts to satisfy a constraint

. Based on the preceding theory and simulations we propose the micro forecast

where

and

denotes the weight in the associated macro forecast

The proposed micro forecast has a minimum mean squared error interpretation and is also optimal in a least squares and a Bayesian framework.

In a time-series framework the aggregate level (e.g., a quarterly model) forecasts the average outcome for several periods (say, the average over three months in one quarter), whereas the more detailed level (a monthly model) forecasts each period (month) separately. In this sense one can think about the aggregated (quarterly) level as a “macro” model and about the detailed (monthly) level as a “micro” model. This provides the link between the current section and the previous sections.

Implicit in the theory developed so far is that the data become available once per period, so that the additional data required to forecast period T + 1 all become available at the same time, namely, at the end of period T. This is reasonable if we consider p micro units in one macro unit, such as p firms in one industry or p industries in one sector or p sectors in one economy. But it is not reasonable in a time-series framework where we consider p = 3 months in one quarter or p = 4 quarters in one year. In that case the relevant micro data will become available during the year and not just at the end of the year. An extension of the theory is thus required.

The extension is straightforward. Suppose that p₁ micro periods have passed so that p − p₁ micro periods are to be forecasted. The first p₁ elements of y_T+1 are then known so that

The estimated variance matrix of this vector has the form

where Σ₂ is a (p − p₁) × (p − p₁) matrix. At the macro level there is no new information, and so there is no need to redefine the original macro forecast

.

With these modifications of

and

we can compute the micro forecast

and the macro forecast

from (23) and (24), and the same optimality results will hold.

As we proceed through period T + 1, the original macro forecast

does not change, but the vector of micro forecasts

is updated together with its variance

. This will affect α. One would expect that the weight of the micro information increases when more information becomes available, so that α decreases. To demonstrate this intuition assume that Σ = diag(σ₁²,…,σ_p²) is diagonal and known and that a = ı_p/p where ı_p denotes the p × 1 vector of ones. Then α as a function of p₁ will be

which obviously gives α(p₁) > α(p₁ + 1).

7. IMPROVING THE EURO YIELD FORECAST BY COMBINING MONTHLY AND QUARTERLY MODELS

We demonstrate our theory by presenting new predictions of European interest rates, in particular zero rates; see Figure 4. As predictors we take various macro and financial series. A detailed description of the data set is provided in Appendix B.

Euro zero rates for maturities 1, 2, and 5 years.

An important feature of the European series is its relative shortness. The series starts with the introduction of the Euro in January 1999, which gives us seven full years of observations. In addition, some of the macro series are released with a half-year delay, so that the final data set consists of 78 monthly observations and 26 quarterly observations. This is an extremely short series when compared to the U.S. series, which is available from the early 1960s. There is some evidence of convergence of the yield curves of the Euro-zone countries (see, e.g., Batten and Fetherston, 2004), but this does not allow us to extend the European series significantly.

It is therefore important to use as much information as possible to get a good forecast. The annual observations cannot provide a reliable forecast, but there are enough quarterly observations to construct a reasonable model, especially because they are much less noisy than the monthly data.

The shortness of the series also explains why relatively few papers exist on the European market. In contrast, the U.S. market has been studied extensively; see Barrett, Gosnell, and Heuson (2004), Piazzesi and Swanson (2004), and Diebold, Rudebusch, and Aruoba (2006). An excellent survey and empirical comparisons can be found in Diebold and Li (2006).

We shall be concerned with the prediction of maturities of 1, 2, and 5 years, rather than with the term structure of interest rates. The main reason is that bonds with higher maturities are not exercised during the observed period, so that there is no evidence of how well they were priced. We predict the selected yields directly rather than using factor models, because each additional step (extracting factors in this case) consumes information and the information is very limited in our case because of the short history. Model selection and pretesting issues are ignored. The final models are similar to the models typically used for the U.S. yields. The final improvement of the forecast is uniform over different possible sets of monthly and quarterly models.

To compare different forecasts we look again at the RMSE. The first half of the sample is used for estimation, and we use the second half for out-of-sample forecasting. The forecasts are based on the linear models (1) and (2) with the regressors described in Appendix B. For both models we use a one-step-ahead linear forecast with an increasing window. The results for the separate monthly and quarterly forecasts are given in Table 2. One should not conclude from Table 2 that the monthly model is better than the quarterly model, because they are not nested. Moreover, they are structurally different and use a different number of observations and predictors. However, one can see that the monthly model forecasts short-term yields better than long-term yields, whereas the quarterly model is more accurate for the long-term yields. Notice that in this case micro and macro data are both available but that the micro data do not become available at once—they come in three installments.

RMSE of separate forecasts

The quarterly yield is just the average of the monthly yields, but for the forecasts this is not necessarily the case. The procedure from Section 6 allows us to bring the quarterly and monthly forecasts in accordance with each other by taking the restriction vector to be

. Combining monthly and quarterly forecasts we obtain the results presented in Table 3.

RMSE comparison of the monthly forecasts

The updating procedure improves the monthly forecast. One can argue that the improvement is not significant, but if it is translated into basis points (bp: 100 bp = 1%), which is a standard measure for the bond market, then the improvement is 40, 100, and 180 bp, respectively, for 1, 2, and 5 years to maturity, which is considerable.

This is for the monthly (“micro”) forecasts. For the quarterly (“macro”) forecasts we have three updated versions; see Table 4. The first update (I) is made at the beginning of the quarter, when the monthly forecasts are made with one-, two-, and three-step-ahead models. At the beginning of the second month, the first month has been observed, and one- and two-step-ahead forecasts are used for the second and third months. This gives the second update (II). Finally, the third update (III) is made at the beginning of the third month when the values for the first and second months are known; the third month is forecasted with a one-step-ahead model. As Table 4 reveals, with the new monthly information received we can improve the quarterly forecast considerably, and the more information revealed, the more accurate our forecast becomes. This simple illustration thus demonstrates the importance of combining information at different levels and also suggests that our proposed method works well in practice.

RMSE comparison of the quarterly forecasts