Journal of Econometrics 158 (2010) 246–261

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Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom

Regression models with mixed sampling frequencies Elena Andreou a,1 , Eric Ghysels b,c,∗,2 , Andros Kourtellos a,1 a

Department of Economics, University of Cyprus, Cyprus

b

Department of Economics, University of North Carolina, USA

c

Department of Finance, Kenan-Flagler Business School, USA

article

info

Article history: Available online 18 January 2010 JEL classification: C13 C32 C51 Keywords: High frequency data Temporal aggregation

abstract We study regression models that involve data sampled at different frequencies. We derive the asymptotic properties of the NLS estimators of such regression models and compare them with the LS estimators of a traditional model that involves aggregating or equally weighting data to estimate a model at the same sampling frequency. In addition we propose new tests to examine the null hypothesis of equal weights in aggregating time series in a regression model. We explore the above theoretical aspects and verify them via an extensive Monte Carlo simulation study and an empirical application. © 2010 Elsevier B.V. All rights reserved.

1. Introduction We study regression models that involve data sampled at different frequencies, the so called Mi(xed) Da(ta) S(ampling), or MIDAS, regression models. Regression models with mixed sampling frequencies and their advantages is still a relatively unexplored area. MIDAS was introduced in both filtering and regression context in a number of recent papers, including Ghysels et al. (2005, 2006) and Ghysels and Wright (2009), among others. These regressions deal with a situation often encountered in practice. For example, macroeconomic data is typically sampled at monthly or quarterly frequencies while financial time series are sampled at almost arbitrarily higher frequencies. Despite the fact that most economic time series are not sampled at the same frequency the typical practice of estimating econometric models involves aggregating all variables to the same (low) frequency using an equal weighting scheme. However, there is no a priori reason why one should (i) ignore the fact that the variables involved in empirical models are in fact generated from processes of different/mixed frequencies and (ii) estimate econometric models based on an aggregation scheme of equal weights. In fact

∗ Corresponding address: Department of Economics, University of North Carolina, Gardner Hall CB 3305, Chapel Hill, NC 27599-3305, USA. Tel.: +1 919 966 5325. E-mail addresses: [email protected] (E. Andreou), [email protected] (E. Ghysels), [email protected] (A. Kourtellos). 1 P.O. Box 537, CY 1678 Nicosia, Cyprus. 2 Gardner Hall CB 3305, Chapel Hill, NC 27599-3305, USA. 0304-4076/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2010.01.004

one would expect that for most time series declining weights would be a more appropriate aggregation scheme and that an equal weighting scheme may lead to information loss and thereby to inefficient and/or biased estimates. The contributions of the paper are as follows. We introduce a new decomposition for the MIDAS regression. Namely, we decompose the conditional mean of the MIDAS regression model into two terms: an aggregated term based on equal or flat weights and a nonlinear term which, involves weighted, higher order differences of the high frequency process. This allows us to link the MIDAS regression model with the traditional temporal aggregation literature, which omits the MIDAS term. We derive the asymptotic properties of the MIDAS Nonlinear Least Squares (NLS) estimator, denoted as MIDAS-NLS and compare them with the traditional Least Squares (LS) estimator that involves regression models with a flat aggregation scheme that we call FLAT-LS. Moreover, we use our decomposition to study MIDAS models with various high frequency processes. In addition, we propose new tests for examining the hypothesis of equal weights in aggregating the time series data involved in regression models. We assess the finite sample properties of our estimators using an extensive Monte Carlo simulation and provide an empirical application of the relevance of MIDAS regression in the context of the empirical Solow growth model. The paper shows how the MIDAS regression model is related to the traditional temporal aggregation literature — see for instance Sims (1971), Engle and Liu (1972), Phillips (1972, 1973, 1974), Tiao and Wei (1976), Geweke (1978), Hsiao (1979), Granger (1987) among many others. While both the temporal aggregation literature and the MIDAS regression model study the consequences

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

247

of temporal aggregation, the MIDAS regression models do not assume that all processes are sampled at the same (high) frequency and do not impose an equal weighting scheme. Furthermore, our approach does not infer the temporally aggregated (low frequency) model from the disaggregated (high frequency) process and does not assume that the high frequency process is closed under temporal aggregation. From the empirical perspective our approach does not have to specify the functional form of the high frequency process and is not confined to a window of lags defined over a specific temporal aggregation horizon. Instead, we consider regression models where the variables have different sampling frequencies such that the high frequency process is projected into the low frequency process with a parsimonious weighting scheme. MIDAS regressions share some features with distributed lag models but also have unique novel features. A stylized distributed lag model is a regression model of the following type: yt = β0 + W (L)xt + εt , where W (L) is some finite or infinite lag polynomial operator, usually parameterized by a small set of hyperparameters (see e.g. Dhrymes (1971) for a survey on distributed lag models). In a standard MIDAS regression model the regressor is sampled m times more frequently than the regressand where the latter has a sample of T observations.4 Our asymptotics are based on T for a given m. Therefore we study the asymptotic properties of the estimators assuming that the span of the data set T grows and the high frequency sample size of the regressors would be mT such that when T → ∞ then both the low and high frequency samples tend to become large. We start from a process sampled at some discrete (high) frequency and study the consequences of temporally aggregating the process. We derive the asymptotic distribution of estimators in the context of MIDAS regressions with alternative high frequency processes such as an i.i.d., Autoregressive (AR) and Autoregressive Conditionally Heteroskedastic (ARCH). We show that under certain conditions a model that ignores the mixed sampling frequencies and simply aggregates the data using equal weights suffers from inefficient and in some cases biased and inconsistent estimators. We find that even when the high frequency process is a martingale difference the FLAT-LS estimator is asymptotically inefficient relative to the MIDAS-NLS estimator. But when the high frequency process exhibits a linear temporal dependence then the FLATLS is both inefficient and inconsistent. In addition, we propose new tests to examine the null hypothesis of the flat temporal aggregation scheme. We explore the above theoretical aspects and verify them via an extensive Monte Carlo simulation study. We also illustrate the relevance of the MIDAS regression model in the context of empirical economic growth. In particular, we find that the MIDAS-NLS estimates of the Solow growth model provide a better explanation of the cross-country growth differences than the standard LS estimates using an aggregation based on an equal weighting scheme. The paper is organized as follows. In Section 2 we introduce the MIDAS regression model and compare this with a linear regression model based on temporal aggregation. Section 3.1 derives the asymptotic properties of the estimators of the MIDAS models. Section 4 presents the detailed results of specific MIDAS regression models with the alternative high frequency regressors. In Section 5 we propose two new tests for the flat aggregation scheme. The asymptotic and finite sample properties of the estimators for alternative sample sizes T and m are addressed in the simulation Section 6. Finally Section 7 provides an empirical example and Section 8 concludes the paper.

2. The MIDAS regression model

4 We only study projections of the high frequency data onto the low frequency. The reverse, namely that of projecting the low frequency onto the high frequency series is called reverse MIDAS and is studied in a time series context by Ghysels and Valkanov (2006).

different high frequency variables xj,t /m for j = 1, . . . , p are observed at the same high frequency (mj = m). 6 Note that x(m) refers to flow variables since stock variables do not involve

(m)

Consider the mixed data sampling process {yt , xt /m }, where the real-valued yt is observed at t

) = 1, . . . , T and x(t m /m =

(1, x(2m,t)/m , . . . , x(pm,t)/m )0 is a p-dimensional vector of the higher frequency data observed at most m times between t and t − 1.5 The data are assumed to be weakly dependent. The conventional approach in empirical macroeconomics and finance employs the linear regression based on the aggregated data, which are typically obtained by a pre-estimation procedure that computes simple averages to bring the high frequency data to a common (low) frequency. To put it differently the conventional approach employs the linear regression subject to the temporal aggregation restriction of an equal weighting scheme. Define the variable xAt = (m)

(m)

Lj/m xt /m that temporally aggregates the covariates xt /m using equal weights (simple average), where q is the number of (m) high frequency lags used in the temporal aggregation of xt /m such 1 q

Pq

j =1

that q ≥ m and Lj/m is the high frequency lag operator. Then, assuming q = m, the conventional approach estimates by LS the following linear regression model yt = xAt 0 β∗ + u∗t ,

q = 2, 3, . . .

(2.1)

where u∗t is a martingale difference process with respect to the (m)

sigma fields Ft −1 generated by {xt −j/m , ut −1−j : j ≥ 0} and

E (u∗t 2 ) = σ ∗2 < ∞. In this paper we argue that the implicit assumption in the model (2.1), namely that temporal aggregation is based on equal weights (m) of xt /m , is restrictive.6 Instead we propose a flexible, data-driven aggregation scheme. Hence we contrast Eq. (2.1) with a linear projection of the high frequency data onto y, where the projection is characterized by a high frequency lag polynomial, W (L1/m ; θ). This lag polynomial is parameterized by a low dimensional r × 1 vector of parameters θ. This approach can generalize the linear regression to the MIDAS regression model yt = β0 xt (θ) + ut ,

t = 1, 2, . . . , T ,

(2.2)

where ut is a martingale difference process with respect to the (m) sigma fields Ft −1 and E (u2t ) = σ 2 < ∞ and xt (θ) = (1, x2,t

(θ2 ) , . . . , x(pm,t) (θp ))0 is a nonlinear function that maps the higher frequency data into a common low frequency such that (m)

(m)

xk,t (θk ) = W (L1/m ; θk )xk,t /m =

q X

wj,k (θk )Lj/m x(km,t)/m ,

j =1

k = 2 , . . . , p.

Pq

We assume that wj,k (θk ) ∈ (0, 1) and j=1 wj,k (θk ) = 1, which allows the identification of the slope coefficient vector β. Eq. (2.2) is a nonlinear regression that allows both the weighting scheme of the temporal aggregation as parameterized by the vector θ and the slope coefficient vector β to be estimated. Following Ghysels et al. (2006) the lag coefficients, wj,k (θk ), which determine the weighting scheme used in the temporal aggregation, are assumed to be given by the two parameter exponential Almon lag polynomial

wj,k (θk,1 , θk,2 ) =

exp{θk,1 j + θk,2 j2 } m

P

.

(2.3)

exp{θk,1 j + θk,2 j2 }

j =1

5 Without loss of generality and to keep the notation simple, we assume that the (mj )

aggregation.

t /m

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E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

and w∗j (θ) = diag (1, w2∗,j (θ2 ) , . . . , wp,j θ∗p ) and wk∗,j (θk ) =




(wk,j (θk ) − 1/qk ) such that

Pq

j =1

wk∗,j (θk ) = 0. Eq. (3.1) can be

viewed as a linear regression of yt on xAt augmented by the term xBt (θ) that takes the form of a weighted sum of (higher order) differences of the high frequency variable, (3.2). The nonlinearity of xBt (θ) is due to the nonlinear weighting schemes of MIDAS regression models such as the exponential Almon lag polynomial in (2.3). Eq. (3.1) implies that for a general, non-flat weighting scheme the traditional temporal aggregation approach may result in an omitted variable bias if the omitted term, xBt (θ), is correlated with xAt . Specifically, given that β∗ = (E (xAt xAt 0 ))−1 E (xAt yt ) we can easily show that β∗ = β + 0(θ)β, where 0(θ) = (E (xAt xAt 0 ))−1 E (xAt xBt (θ)0 ) is a matrix of coefficients from an infeasible regression of the vector of the omitted term xBt (θ), on the vector of aggregated covariates xAt based on simple averaging. This implies that the FLAT-LS estimator ∗ b β will be generally inconsistent with the following asymptotic bias ∗

ABias(βˆ ; β) = 0(θ)β. Fig. 1. Alternative weighting schemes of the exponential Almon polynomial.

Note that for θk,1 = θk,2 = 0 we obtain the flat aggregation scheme (m)

and hence, xt (0) = xAt .7 Eq. (2.3) provides a parsimonious way to model the q coefficients of the lag polynomial W (L1/m ; θ). To see this consider weekly data aggregated to a quarterly frequency. An unrestricted but overparameterized MIDAS regression would involve 12p + 1 unknown parameters while a MIDAS regression would involve 3p + 1. It is also possible to obtain a further reduction in the number of estimated parameters if one is willing to make further assumptions. One appealing assumption for many economic processes is that of a decaying memory pattern so that we can restrict the exponential Almon lag polynomial to only one parameter and estimate only 2p + 1 parameters. Another assumption is that different covariates exhibit similar temporal aggregation patterns so that we can use the same lag polynomial for different covariates. Fig. 1 illustrates the various shapes of a two-parameter and one-parameter exponential Almon lag polynomial, respectively. 3. Asymptotic properties of MIDAS-NLS and FLAT-LS estimators In this section we focus on the asymptotic properties of estimators. In particular, we establish that the FLAT-LS estimator is inconsistent whereas the MIDAS-NLS estimator is efficient. We devote a subsection to each topic.

To assess the consequences of misspecification of flat temporal aggregation we decompose the MIDAS regression model in Eq. (2.2) into two terms: the aggregated term based on flat weights, xAt , and the nonlinear term, xBt (θ), such that yt = β0 xAt + β0 xBt (θ) + ut

(3.1)

where q−1 X

While this omitted variable bias resembles the standard omitted variable bias of the linear regression there are two differences. First, the omitted variable xBt (θ) has the same regression coefficient as the equally weighted aggregated variable, xAt , and second, the omitted variable bias depends on the shape of the weighting scheme, W (L1/m ; θ). For instance, declining weights imply an omitted variable that exhibits memory decay or mean reversion, which will be associated with higher bias than an omitted variable with a near-flat weighting scheme. Moreover, the bias will be zero in two cases: (i) When the omitted term xBt (θ) is orthogonal to xAt , E (xAt xBt (θ)0 ) = 0, even when the true model is the MIDAS regression (2.2). (ii) When the true weighting scheme is flat, θ = 0, and the true model is the linear regression (2.1). Next we investigate the asymptotic efficiency of the MIDAS-NLS vis-a-vis the FLAT-LS estimator. 3.2. Asymptotic efficiency of the MIDAS-NLS estimator Define α = (β, θ) and g (xt ; α) = β0 xt (θ). Assuming that α is identified and the derivative gα (x, α) = ∂ g(x, α)/∂α exists, then under conditional homoskedasticity, the MIDAS-NLS estimator b α d

is asymptotically Normal T 1/2 (b α − α) → N (0, σ 2 (E (gα,t g0α,t ))−1 ), where gα,t = gα (xt , α). Define further Q(θ) = E (xt (θ)xt (θ)0 ), G12 (θ) = E (xt (θ)g0θ,t ), and G22 (θ) = E (gθ,t g0θ,t ), where gθ is the partial derivative of g (xt ; α) with respect to θ. Then, the asymptotic variance of b β, AVar (b β) = σ 2 (Q(θ) − G12 (θ)G22 (θ)−1 G12 (θ)0 )−1 .

3.1. Inconsistency of the LS estimator

xBt (θ) = xt (θ) − xAt =

(3.3)

(m)

w∗j (θ)(∆q−j xt −(j−1)/m )

(3.4)

Eq. (3.4) shows that the asymptotic variance not only depends (m) on the high frequency process being aggregated, xt /m , but also on the true temporal aggregation scheme, which is specified by a weighting polynomial function W (L1/m ; θ) or θ, and its derivative.8 ∗

On the other hand, the FLAT-LS, βˆ , estimates the restricted model (2.1), where u∗t = yt − β∗0 xAt and the asymptotic variance ∗

∗

of βˆ is given by AVar (b β ) = (QA )−1 E (xAt xAt 0 u∗t 2 ) (QA )−1 , where A A A0 ∗B Q =E (xt xt ). Let xt (θ) = xBt (θ) − 0(θ)0 xAt be the regression error of the infeasible regression of xBt (θ) on xAt . Define ∗ (θ) =




(3.2)

j =1

7 Other polynomials have been suggested — see Ghysels et al. (2006) for details.

8 These results do not readily extend to nonstationary high frequency regressors because the MIDAS-NLS estimator would involve a different asymptotic distribution theory; see Phillips (1991). Note that the omitted term xBt (θ) will be stationary even if the high frequency process is characterized by an I (1) or a linear trend.

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

E (xAt xAt 0 (β0 x∗t B (θ))2 ), then under conditional homoskedasticity the ∗

asymptotic variance of βˆ is given by ∗

AVar (b β ) = (QA )−1 Ω ∗ (θ)(QA )−1 + σ 2 (QA )−1

(3.5)

and its asymptotic MSE (AMSE) is given by ∗

(3.6)

Given that the MIDAS-NLS, b β, is an unbiased estimator of β and ˆ β) = AVar (β), ˆ let us compare the two AMSEs therefore, AMSE (β; ∂ xB (θ)

t given by Eqs. (3.6) and (3.4). Recall that G12 (θ) = E (xAt β0 ∂θ )+

∂ xB (θ)

t E (xBt (θ)β0 ∂θ ). Then, in the special case of a flat weighting scheme, θ = 0 and E (xAt xBt (θ)) = 0 we can easily show that

∗

ˆ β) = σ (Q ) since (0) = 0, 0(0) AMSE (βˆ ; β) = AMSE (β; = 0, Ω ∗ (0) = 0, and G12 (0) = 0. However, when θ = 0 A −1

2

xBt

∗

ˆ β) because but E (xAt xBt (θ)) 6= 0 then AMSE (βˆ ; β) < AMSE (β; ∂ xB (θ)

t G12 (0) = E (xAt β0 ∂θ )|θ=0 6= 0. In the interesting cases of non-flat weighting schemes, θ 6= 0, it is difficult to assess the

∗

relative efficiency of βˆ and βˆ at this level of generality since their AMSE would involve nonlinear functions of θ, their derivatives, gθ,t , and second-order moments, E (xBt (θ)g0θ,t ) and E (gθ,t g0θ,t ).9 Therefore, in order to study the asymptotic properties of the FLATLS and MIDAS-NLS estimators we derive analytical results for MIDAS regression models for specific high frequency processes. This approach provides additional information about the role of the parameters of the high frequency process, the aggregation horizon, m, and the weighting scheme, θ. We focus on three stationary processes that can be thought as the building blocks of most stochastic processes, namely i.i.d., Autoregressive (AR) and Autoregressive Conditionally Heteroskedastic (ARCH). The next section derives the specifications of the MIDAS regression models for alternative high frequency processes, as well as the analytical expressions of the asymptotic bias and variance of the FLAT-LS and MIDAS-NLS estimators. 4. Examples of MIDAS regression models In this section we examine the asymptotic properties of MIDASNLS and FLAT-LS in three examples of regression models with high frequency regressors being: (i) i.i.d, (ii) ARCH(1), and (iii) AR(1). More precisely, we provide the analytical and numerical results for the asymptotic bias and variance of the estimators for each example. 4.1. MIDAS regression model with i.i.d. regressors We start with the simple case where the univariate process

) (m) 2 2 {x(t m /m } is i.i.d., xt /m = et /m , et /m ∼ i.i.d.(0, σe ) with E (et /m ) < P A ∞. In this case the simple average term becomes xt = m1 m j =1 P m ∗ et −(j−1)/m and the nonlinear term is given by xBt (θ) = j=1 wj (θ)

et −(j−1)/m . Then using (3.1) the MIDAS regression model with an i.i.d. regressor can be written as yt = β0 +

m β1 X

m j =1

et −(j−1)/m + β1

m X

1 where wj∗ (θ) = (wj (θ) − m ) such that j=1 wj∗ (θ) = 0. It is easy to show that the FLAT-LS estimator of β1 in the linear regression model based on xAt (i.e. which omits the second term in (4.1)) would be unbiased. From Eq. (3.3), 0 = (γ0 , γ1 )0 and γ1 = cov(xAt , xBt (θ))/v ar (xAt ) and

ABias(βˆ 1∗ ; β1 ) = γ1 β1 = β1 ((σe2 /m)

AMSE (βˆ ; β) = T 0(θ)ββ0 0(θ)0 + (QA )−1 ∗ (θ)(QA )−1

+ σ 2 (QA )−1 .

249

Pm

m X

wj∗ (θ))/(σe2 /m) = 0 (4.2)

j =1

since j=1 wj∗ (θ) = 0.10 From Eq. (3.5) the asymptotic variance of the FLAT-LS estimator for this model is given by

Pm

AVar (βˆ 1∗ ) = mσ 2 /σe2 .

(4.3)

Interestingly, we see that the asymptotic efficiency of the FLATLS estimator deteriorates with the aggregation horizon, m. The MIDAS-NLS estimator of model (4.1) is given in the following proposition and derived in Appendix A. (m)

Proposition 4.1. Let the high frequency regressors xt /m be an i.i.d. process and consider the MIDAS regression model in (4.1). Then the asymptotic variance of βˆ 1 under homoskedasticity is:

 AVar (βˆ 1 ) =

σ2 σe2

X  2  m 1 ∗  wj (θ) +  m  j=1 m P

−

wj∗ (θ) +

j =1


 ∂wj∗ (θ)  1 m

m  ∂w ∗ (θ) 2 P j j =1

∂θ

∂θ

!2 −1      

.

(4.4)

Eq. (4.4) shows that when the weighting scheme is flat, θ = 0, Pm the AVar (βˆ 1∗ ) = AVar (βˆ 1 ) since wj∗ (0) = 0 and ( j=1 (wj∗ (θ) +

P Pm ∂wj∗ (θ) 2 ∂wj∗ (θ) 2 )( ∂θ )) |θ=0 = m12 ( m j=1 ( ∂θ )) |θ=0 = 0, because j =1 wj∗ (θ) = 0 for all values of θ. However, when the weighting scheme is non-flat, θ 6= 0, it is generally difficult to compare the two asymptotic variances because AVar (βˆ 1 ) depends on the aggregation horizon, m, the weights, wj∗ (θ), which are nonlinear functions of θ, (see for example the exponential Almon 1 m

polynomial (2.3)) and their derivatives. Therefore, to obtain a deeper understanding of the problem we evaluate the relative efficiency numerically for different values of the aggregation horizon m and weighting schemes, θ. We consider m = 3 to 100 and three values of θ = {(0, −0.05), (0, −0.005), (0, −0.0005)} that correspond to the fast decaying, slow decaying, and the nearflat weights, respectively.11 Fig. 2 shows the relative asymptotic MSE, RMSE = AVar (βˆ 1∗ )/ AVar (βˆ 1 ), as a function of m for given θ. The three lines correspond to the three weighting schemes. The solid, dotted, and dashed lines correspond to fast decaying, slow decaying, and near-flat weights,

wj∗ (θ)et −(j−1)/m + ut , (4.1)

j =1

10 We present the properties of the slope estimator of β since we found similar 1 results for constant, β0 . 11 Note that for numerical simplicity we present here the single parameter 2 exponential Almon lag polynomial given byP wj = exp(θ j2 )/ P i=1 exp(θ i ) and P m m 2 2 2 2 2 ∂wj (θ)/∂θ = (exp(θ j2 )(j2 m i=1 exp(θ i ) − i=1 i exp(θ i )))/( j=1 exp(θ i )) . In the simulations section below we consider the more general two-parameter polynomial given by (2.3).

Pm

9 Even in the special case of independence of xA and x∗B (θ) (which implies that t t 0(θ) = 0 and ∗ (θ) = 0), it is difficult to assess the relative efficiency because A A0 B B0 B Q(θ) = E (xt xt ) + E (xt (θ)xt (θ)) and G12 (θ) = E (xt (θ)g0θ,t ) 6= 0.

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E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

which increases with the aggregation horizon, m and is a function of the high frequency ARCH parameters. The proposition below presents the asymptotic variance of the MIDAS-NLS estimator in (4.6) and is proved in Appendix B. (m)

Proposition 4.2. Let the high frequency regressors xt /m be an ARCH (1) process and consider the MIDAS regression model in (4.6). Then the asymptotic variance of βˆ 1 under homoskedasticity is:

  2 m  σ 2 (1 − c1 )  1 X ∗ ˆ AVar (β1 ) = wj (θ) +   j =1 c0 m  !  −1  ∂w∗ (θ)  2
j  wj∗ (θ) + m1 ∂θ    .   2 m ∗ P ∂wj (θ)   ∂θ

m P j=1

−

(4.8)

j=1

Fig. 2. Relative efficiency — MIDAS with IID or ARCH regressor.

respectively. We show that for all three weighting schemes the MIDAS-NLS appears to be relatively more efficient than FLAT-LS since RMSE > 1. Moreover, for fast decaying weights, the MIDASNLS exhibits efficiency gains even for very small values of m and these gains increase at a faster rate relative to the other two weighting schemes.

Eq. (4.8) shows that the MIDAS-NLS will depend on m and θ, and the unconditional variance of the ARCH process (4.5), c0 /(1 − c1 ). It is also evident that the relative efficiency considerations are exactly the same as in the case of the MIDAS model with i.i.d. high frequency regressors. 4.3. MIDAS regression model with AR(1) regressors (m)

Let the univariate process {xt /m } follow a stationary AR(1) given (m)

∼ i.i.d.(0, σe2 ). Then the MIDAS by xt = regression model with an AR(1) high frequency regressor is

4.2. MIDAS regression model with ARCH(1) regressors (m)

Let the high frequency univariate process {xt /m } follow an ARCH(1) process given by (m) xt / m

=

q

) (σt(/mm) )2 e(t m /m

where

(σt(/mm) )2

= c0 +

(m) c1 (xt −1/m )2

(4.5)

(m)

where c0 > 0, 0 < c1 < 1, c12 < 1/3, et /m ∼ N .i.i.d.(0, 1), (m)

(m)

(m)

and P (et /m /et −1/m , et −2/m , . . .) = 0, where P (./.) denotes the best linear predictor. Note that the weak ARCH(1) model is closed under temporal aggregation but the results below also hold for strong and semi-strong ARCH models. In this case the simple average term xAt 1

becomes xAt =

Pm

Pmm

j =1

m) (m) σt(−( j−1)/m et −(j−1)/m and the nonlinear term

(m)

(m)

∗ is xBt (θ) = j=1 wj (θ) σt −(j−1)/m et −(j−1)/m . Then using (3.1) the MIDAS regression model with an ARCH(1) regressor can be written (m) in terms of the volatility σt /m

yt = β0 + β1

+ β1

m 1 X (m) (m) σ e m j=1 t −(j−1)/m t −(j−1)/m

m X

(4.6)

( ) j=1 wj∗ (θ) = 0, given that j=1 wj∗ (θ) = 0. m The asymptotic variance of the FLAT-LS estimator is

= γ1 β1 =

AVar (b β1∗ ) = m(1 − c1 )σ 2 /c0 ,

xAt

=



1

c0

m

1−φ

+

1

m−

1 − φm 1−φ



 +

1 − φ j =1

1 − φm 1−φ

can be

 xt −(m−1)

!

m−1

X

(m) j=1 xt −(j−1)/m

Pm−1

1 − φ et −(j−1) j




(4.10)

and the nonlinear term as

   X 1 1 − φ m−i (θ) = wi (θ) − c0 m 1−φ i=1 m−1

xBt

m−i

)x(t −(m−1))/m +

X

! φ et −j−(i−1) . (4.11) j

j =0

j=1

Pm

(4.9) 1 m

where the simple average term xAt = expressed as

− (1 − φ

m) (m) wj∗ (θ) σt(−( j−1)/m et −(j−1)/m + ut ,

c0 1−c1

yt = β0 + β1 xAt + β1 xBt (θ) + ut

m−i−1

or in terms of the observed squared high frequency process when (m) (m) we substitute σt −(j−1)/m = (c0 + c1 (xt −(j−1)/m )2 )1/2 in Eq. (4.6). The FLAT-LS estimator of β1 in the linear regression model based on xAt would be asymptotically unbiased since ABias(βˆ 1∗ ; β1 ) β1

(m) c0 +φ xt −1/m + et /m , et /m

Pm

(4.7)

In contrast to the MIDAS models with i.i.d. and ARCH regressors discussed above, in the MIDAS regression with an AR regressor, given in (4.9), xAt is correlated with xBt (θ). Consequently, the next proposition shows that the FLAT-LS estimator of β1 in the linear regression which omits xBt (θ) in (4.9), will be asymptotically biased. (m)

Proposition 4.3. Let the regressors xt /m be an AR(1) process and consider the MIDAS regression model in (4.9). The ABias(βˆ 1∗ ; β1 ) = γ1 β1 , where γ1 is given in Box I. Proof. See Appendix C.



E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

m −

1−φ m σe (1−φ) 1−φ 2 2

m−1

wi (θ) −

P i =1

γ1 =



1 m




(1 − φ 2

1−φ m 1−φ

m−i

σe2 1−φ 2

) +


+

m−1

1

(1−φ)

σe2 (1−φ)2



P

1−φ

j

 j
P

j =1

251

(wi (θ) − )φ 1 m

i=1

j −i

!

m−1

P

(1 − φ j )2

j =1

Box I.

We evaluate numerically the analytical expression in Box I as a function of the aggregation horizon, m, for different values of the parameters, θ, φ , and σe2 , in order to gain further insights about the behavior of the asymptotic bias. Fig. 3(a) shows the asymptotic bias for a persistent AR(1) process, φ = 0.9 and σe2 = 1, over aggregation horizons, m = 3 to 100, for the different weighting schemes: θ = (0, −0.05), θ = (0, −0.005), and θ = (0, −0.0005). In all cases we find that the bias becomes negative and increases in magnitude with m. As m becomes large the bias appears to stabilize at some negative value, which depends on the weighting scheme. This value is larger in absolute terms for faster decaying weights. Turning now to Fig. 3(b) we find that at least for m > 30, the bias is the largest for the model with the strongest degree of persistence. Finally, in Fig. 3(c) we examine the role of σe2 to the asymptotic bias. We find that when σe2 = 0.05 and φ = 0.9 the bias exhibits a hump shaped behavior, being positive for m ≤ 30 and then negative and increasing for m > 30. In particular, for m > 30, the asymptotic bias appears to be the strongest for σe2 = 0.05. We now turn to the analysis of the asymptotic efficiency of the FLAT-LS vis-a-vis the MIDAS-NLS estimators of β1 . Under a flat weighting scheme the asymptotic variance of the FLAT-LS estimator is: AVar (βˆ 1∗ ) =

σ2 σ2 = 2 A σe v ar (xt )

m2 (1 − φ)2 (1 − φ 2 )





m(1 − φ 2 ) − 2φ + 2φ m+1

. (4.12)

a

b

c

Eq. (4.12) is derived in Appendix D. Eq. (4.12) shows that for large m the term φ m+1 will dominate since |φ| < 1.12 The asymptotic variance of the MIDAS-NLS slope estimator in model (4.9) is given by the following proposition: (m)

Proposition 4.4. Let the regressors xt /m be an AR(1) process and consider the MIDAS regression model in (4.9). Then the asymptotic variance of βˆ 1 under conditional homoskedasticity is AVar (βˆ 1 ) =

σ 2 E (x2θ ) Var (x(θ))E (x2θ ) − (E (xθ x(θ)))2

σe2 where E (xθ ) = 1 − φ2

X ∂wi (θ)

X i=1

Var (x(θ)) =

σe2 1 − φ2

∂θ

i=1

m−1

+σ

+

(1 − φ

m−i

)

X ∂wj+1 (θ) φ m−i−(j+1) ∂θ j =0

m−i−1

!2 ,

1−2

X

wi (θ)(1 − φ m−i )

E (xθ x(θ)) =

i =1

wi (θ) 1 − φ

m−i

!2


i=1

+σ

2 e

Fig. 3. (a)–(c): BIAS of FLAT-LS in the MIDAS regression with AR(1) regressor.

m−1

m−1

X

!2

m−1

2

2 e

(4.13)

m−1

m−i−1

X

X

i =1

j =0

−1 X σe2 m ∂wi (θ) (1 − φ m−i ) 2 1 − φ i=1 ∂θ ! m −1 X m−i × wi (θ)(1 − φ ) − 1

!2 wj+1 (θ)φ

m−j−i−1

,

+σ

2 e

i=1 m−1

m−i−1

X

X

i=1

j =0

× was not tractable and therefore we focus on the AVar (βˆ 1∗ ) of the well specified LS regression in Eq. (4.12). This is in fact the worst case scenario for the MIDAS-NLS estimator.

wj+1 (θ)φ

!! X  ∂wj+1 (θ)  m−i−j−1 φ . ∂θ j =0

m−i−1

12 The analytical AVar (βˆ ∗ ) in the case of the misspecified LS regression model 1

! m−i−j−1

Proof. See Appendix D.



252

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

a

b

θ = 0, in model (2.2), the MIDAS term xt (θ) becomes the

simple aggregated term xAt . This is made obvious in Eq. (3.1), which shows that the term xBt (θ), vanishes under this hypothesis and therefore one can simply average the data to a common low frequency. We also point out that although one needs to worry about identification issues in regression models with additive nonlinearity as in model (3.1), here we do not. The reason is that while xBt (θ) vanishes under the null, the regression slope parameter β is still identified because the terms xAt and xBt (θ) are restricted by construction in model (3.1) to have the same slope coefficient β. When one views the problem of testing for flat aggregation as a problem of testing for omitted variables, one can use variations of the standard tests for a conditional mean zero or a conditional homoskedasticity by using auxiliary regressions based on the high (m) frequency information set Ft −1 . Here we focus on two conditional mean tests to examine the H0 : θ = 0. First, we propose to test this hypothesis directly by applying a simple LM test LM

c

S0LS − S1M S0LS

d

→ χ 2 (r ),

∗ where S0LS = u∗0 t ut is the sum of the squared errors from the linear regression based on temporal aggregation with flat weights, (2.1), and S1M = u0t ut is the sum of the squared errors based on a MIDAS regression, which allows for a non-flat weighting scheme, (3.1). Second, we can test the flat aggregation scheme indirectly by viewing it as a problem of omitted variable bias in Eq. (2.1) and (m) test H0 : E (u∗t /Ft −1 ) = 0. We propose to examine this hypothesis using a Wu–Hausman type test with high frequency variables as instrumental variables. Consider the l × 1 vector of instruments (m) (m) (m) (m) zt = (xt , xt −1 , xt −2 , . . . , xt −k )0 , where l ≥ p and l = pk. Let us define the n × 1 vector y, the n × p matrix X A , n × l matrix Z by stacking yt , xAt and, zt , respectively. Let PZ = Z(Z0 Z)−1 Z0 be the projection matrix spanned by the columns of zt . Then, we can construct the instrumental variable estimator

βˆ IV = (XA0 PZ XA )−1 PZ XA0 y. Fig. 4. (a)–(c): Relative Efficiency of FLAT-LS in the MIDAS regression with AR(1) regressors.

Fig. 4(a)–(c) show the relative efficiency, RMSE = MSE (βˆ 1∗ )/

MSE (βˆ 1 ), as a function of m, for given θ, φ and σe2 , respectively. In Fig. 4(a) we set β1 = 1.5, φ = 0.9 and σe2 = 1 and find that

the MIDAS-NLS, βˆ 1 , appears to be relatively more efficient than the FLAT-LS, βˆ 1∗ , since RMSE > 1, for all three weighting schemes. The largest relative efficiency gains are obtained for the case of fast decaying weights (as shown by the solid line), especially for m > 10. In Fig. 4(b) we set θ = (0, −0.05) and σe2 = 1 and show that not only is RMSE > 1 for all φ = 0.5, 0.7, 0.9, but RMSE is relatively higher the stronger the persistence. Finally, Fig. 4(c) shows that RMSE for σe2 = 0.05 is much larger than those of σe2 = 0.5 and 1 for almost all m. 5. Testing the flat aggregation scheme In estimating a MIDAS regression model, the applied researcher is interested in examining whether the aggregation scheme based on the traditional approach of flat weights is empirically supported, albeit of the high frequency process being aggregated. Hence, under the hypothesis of a flat weighting scheme, H0 :

(5.1)

(5.2)

One version of the Wu–Hausman type tests can be applied as follows: First, we estimate the regressions xAt = 00 zt + et , where E (et | xAt ) = 0 and yt = β∗0 xAt + u∗t to obtain the residuals b et and b u∗t , respectively. In the second stage we estimate the auxiliary regression

b u∗t = γ0 xAt + δ0b et + ξt where E (ξt | xAt ,b et ) = 0

(5.3)

to test H0 : δ = 0 using a standard LM test d

WH (k) = TR2 → χ 2 (p).

(5.4)

These tests enjoy good size and power properties even for small m and the LM test has better power than the Wu–Hausman test.13 One issue with the Wu–Hausman test is the selection of instrumental variables. Ideally, we would like to choose those high frequency variables that maximize the power of the test. The selection of instruments for the purposes of maximizing power of a test statistic, as opposed to estimation efficiency, is very much an unresolved issue (see e.g. Ghysels and Hall, 1990). A practical recommendation for time series data is to use a small number of the most recent high frequency variables.

13 These simulation results are available upon request from the authors.

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

253

Table 1 Relative MSE efficiency of the slope coefficient of the MIDAS regression model. Aggregation horizon (m)

T

β1

3

40

100

0.6

1.5

3

0.6

1.5

3

0.6

1.5

3

Fast decaying weights IID

100 200 500

0.960 1.095 0.996

1.224 1.353 1.151

2.159 2.373 1.989

8.062 8.707 9.482

23.18 31.35 30.67

99.93 117.0 103.0

15.41 22.35 22.40

31.57 45.56 59.72

226 279 304

ARCH(1)

100 200 500

1.307 1.501 1.637

2.230 3.057 4.051

5.864 8.005 12.04

10.82 16.93 35.77

83.64 125.0 230.0

402 502 913

11.06 18.74 38.75

71.90 120.0 282.0

302 605 1360

AR(1)

100 200 500

1.038 1.114 0.975

1.219 1.260 1.003

1.551 1.566 1.519

198.0 467.0 1110.

1201 2856 6945

4882 11 438 28 124

437 1029 2555

1829 4837 14 976

10 608 26 199 64 899

IID

100 200 500

0.994 1.007 0.994

0.991 1.028 0.990

0.982 1.013 0.986

0.997 0.981 1.003

0.940 1.014 1.042

0.992 1.223 1.211

1.091 1.221 1.231

1.293 1.479 1.562

2.221 2.477 2.472

ARCH(1)

100 200 500

0.923 0.987 0.973

0.936 0.995 0.949

0.937 0.994 0.973

1.100 1.035 1.178

1.582 1.543 1.942

2.589 2.881 3.982

1.355 1.832 2.049

2.993 3.543 5.341

7.802 8.822 18.30

AR(1)

100 200 500

0.991 1.040 0.995

0.999 1.040 0.979

0.987 1.035 0.969

1.233 1.175 1.498

2.451 2.938 3.518

7.390 9.336 10.820

5.170 5.941 5.953

26.91 28.53 36.17

97.74 110.0 143.0

Near flat weights

Note: MSE refers to Median Square Error. Relative efficiency is measured by MSELS /MSENLS . The parameters for the exponential Almon lag polynomial in DGP are θ1 = 7×10−4 and θ2 = −5 × 10−2 .

6. Monte Carlo results This section examines the properties of the MIDAS-NLS estimator in the regression model given in (2.2) vis-a-vis the FLATLS estimator based on an equal weighting scheme given by (3.1), for different low frequency sample sizes, T , aggregation horizons, m, and weighting schemes W (L1/m ; θ). More precisely, we examine the AMSE efficiency of the MIDAS-NLS and FLAT-LS estimators for (m) a number of alternative high-frequency covariates, {xt /m } analyzed in Section 4. The Monte Carlo design is based on the following DGP of the MIDAS regression model yt = β0 + β1 xt (θ) + ut , 1/m

t = 1, 2, . . . , T

(6.1)

) ; θ)x(t m /m

where xt (θ) = W (L and ut ∼ N .i.i.d.(0, 0.125). We sample ut at a low frequency with a sample size, T , whereas the (m) covariate xt /m is sampled m times between t and t − 1 such that the high sampling frequency sample size is mT . The high(m) frequency data {xt /m } are projected onto the low frequency data, xt (θ), using a two-parameter exponential Almon lag polynomial given in (2.3). We investigate three alternative weighting schemes that correspond to fast decaying weights, θ ≡ (7 × 10−4 , −5 × 10−2 ), slow decaying weights, θ = (7 × 10−4 , −6 × 10−3 ), and near-flat weights, θ = (0, −5 × 10−4 ). For the high frequency (m) process {xt /m } we consider three alternative specifications: (m)

(i) xt /m ∼ N .i.i.d.(0, 1) (m)

q

) (m) 2 ) 2 (σt(/mm) )2 e(t m = c0 + c1 (x(t m /m , (σt /m ) −1/m ) , c0 = 0.25, c1 = 0.85, et /m ∼ N .i.i.d.(0, 1) (m) (m) (iii) xt /m = c + φ xt −1/m + et /m , c = 0.5, φ = 0.9, et /m ∼ N .i.i.d.(0, 1).

(ii) xt /m =

In the model (6.1) we set the intercept β0 = 0.5 and the slope coefficient, β1 = 0.6, 1.5, 3, which correspond to models of small, medium, and large Signal to Noise Ratios (SNR), respectively.14

14 Typically, a model with poor fit or low SNR is one with ratio of 1 to 1 and a model with good fit or high SNR is one where the signal is at least 5 times than the noise.

We consider three sample sizes T = 100, 200, 500 and three aggregation horizons m = 3, 40, 100. The simulated model in (6.1) is estimated using the LS and NLS methods. The LS method follows the traditional temporal aggregation approach, which aggregates the high frequency covariate using equal weights, 1/m. This approach yields the FLATLS estimators of β0 and β1 . The alternative approach studied in this paper involves mixing the two frequencies and estimating both the weighting or aggregation parameters, θ, and the regression parameters β0 and β1 , referred to as MIDAS-NLS. The efficiency of the MIDAS-NLS vis-a-vis the FLAT-LS estimators for model (6.1) is examined for the above high-frequency covariates for different sample sizes, T and aggregation horizons, m. For each estimator of β1 we obtain the square error between the estimated and true regression coefficients. The discussion of the simulation results refers to the relative Median Square Error (MSE) of the estimated slope coefficient β1 of the FLAT-LS relative to the MIDAS-NLS estimator, denoted by MSEFLAT –LS /MSEMIDAS–NLS . In addition, we show the simulated distribution of the MIDAS-NLS and FLAT-LS estimators of β1 . We present the results for the slope estimator of the regression model (6.1) but similar results were obtained for the constant. The Monte Carlo experiment was performed in MATLAB with 1000 simulations. Table 3 reports the relative MSEs with exponentially declining weights. In almost all DGPs the MSEFLAT –LS /MSEMIDAS–NLS are greater than one which provides simulation evidence of the relative efficiency of the MIDAS-NLS vis-a-vis the FLAT-LS estimator even for small aggregation horizons (e.g. m = 3 and 40) and for relative small sample sizes of T = 100. The efficiency gains are less pronounced when m = 3 especially for models with a poor SNR (β1 = 0.6), but this result mainly reflects the fact that these are models with a poor fit. In fact, the MIDAS-NLS estimator enjoys very large efficiency gains in all regression models as T and m increases. We obtain the largest relative efficiency gains in the MIDAS model with an AR high frequency regressor which is due to the bias effect as found in our asymptotic results of Section 4. Overall the results in Table 3 not only provide support for the asymptotic results derived in the paper, but also show that such results hold for relatively small samples of T = 100, and all the alternative high frequency processes.

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E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

a

b

c

d

e

f

Fig. 5. (a)–(f): MIDAS — IID model

Table 1 reports the relative MSEs for near-flat weights in order to assess if in this setup the MIDAS-NLS estimator offers no efficiency gains compared to the FLAT-LS estimator. Yet, we find that even for near-flat weights the MIDAS-NLS estimator is relatively more efficient than the FLAT-LS estimator when m ≥ 40 for all SNRs, high frequency processes and T sample sizes. As expected when the FLAT-LS estimator is unbiased (e.g. in the MIDAS models with i.i.d. and ARCH regressors), the relative MSEs are smaller compared to the case of the FLAT-LS estimator being biased (i.e. in the MIDAS model with an AR regressor). In general the results in Table 1 provide additional evidence of the relative MSE efficiency of the MIDAS-NLS estimator even when the weighting scheme is very close to being flat and when m ≥ 40. An alternative, informative comparison of the distribution of the MIDAS-NLS and FLAT-LS estimators, which complements the above MSE results can be found in Figs. 5(a)–(f) to 7(a)–(f). We present the distributions of the FLAT-LS and MIDAS-NLS estimators (m) of β1 in the MIDAS model (6.1) for different high frequency {xt /m } processes for 1000 simulations using the Gaussian kernel density estimates and Silverman’s bandwidth parameter. The distributions of the two estimators are compared (for the case of β1 = 1.5 for conciseness) across sample sizes, T and aggregation horizons, m. Two broad complementary results can be concluded from these figures: (m)

(i) When {xt /m } follows an i.i.d. or an ARCH process in the MIDAS regression model (6.1), then both the FLAT-LS and MIDAS-NLS estimators are unbiased as shown in Figs. 5(a)–(f) and 6(a)–(f). In contrast, Fig. 7(a)-(f) show that the FLAT-LS estimator of the slope coefficient is biased (compared to the unbiased MIDAS(m) NLS estimator) when {xt /m } follows an AR(1) for m > 3.

(ii) Irrespective of the high frequency regressor in the MIDAS models considered in the simulations, the variance of the MIDAS-NLS estimator of the slope parameter is smaller than the variance of the FLAT-LS estimator. The relative efficiency of the MIDAS-NLS estimator improves as T and m increase. Although the efficiency gains for large T are expected from standard NLS theory, our analysis shows that such relative efficiency improvements can also arise when m increases. We robustify the above results for other parameters θ that imply slow declining weights, θ = (7 × 10−4 , −6 × 10−3 ), and other near flat weights, θ = (10−5 , 10−5 ), as well as other shapes of weights (such as an inverse U shaped polynomial given by θ = (8 × 10−4 , −2 × 10−3 )). In addition, we use an alternative weighting polynomial based on the Beta function proposed in Ghysels et al. (2006), with parameter values given by θ = (1.5, 20) and (1.1, 5), that represent fast and slow decaying weights, respectively. We find that similar qualitative results apply in terms of the relative efficiency. In comparing the relative efficiency of the weighting schemes given by the Exponential Almon and Beta polynomials we find that the estimated parameters of the former are relatively more efficient. Moreover in evaluating the mean as well as the lower and upper quantiles (5% and 95%, respectively) of the square error, we find that these are stable across the SNRs and decrease as T and m increase. The main result is that the square errors of the MIDAS-NLS estimators are smaller than those of the FLAT-LS and in general the MIDAS-NLS estimators of exponential weights yield smaller square errors than those of the Beta weights.15

15 These results are not reported here for conciseness purposes but are available upon request from the authors.

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

a

b

c

d

e

f

255

Fig. 6. (a)–(f): MIDAS — ARCH(1) model

7. Empirical illustration: Revisiting the Solow growth model using MIDAS regressions In this section, we revisit the Solow growth model by Mankiw et al. (1992), which is regarded as a seminal study in growth empirics, in order to examine whether a MIDAS regression performs better in explaining the cross-country growth differences. The Solow growth model can be viewed as a natural application of MIDAS regression in the following sense. The Solow growth regression takes the form of a linear regression of long run growth on a set of growth determinants, which includes the initial income, the population growth rate and the saving rates of physical and human capital accumulation. With the exception of the data for human capital accumulation, which are observed every five years, the data are annual and typically cover the period 1965–94 for most countries. However, when one studies the long run growth, it is desirable to consider averages over the longest time horizon possible in order to eliminate the business cycle effects that likely dominate per capita income fluctuations at higher frequencies. So in cross-section regression the conventional practice temporally aggregates each determinant using flat weights (i.e. computes averages) in order to eliminate the time dimension. However, as we argued above this practice may give rise to substantial bias and inefficiency. At this point one may be wondering how this cross-sectional application is related to the time series models studied in the previous sections. In the cross country Solow growth regressions we can think of the time-series dimension as being finite and equivalent to the various growth periods (or aggregation horizons of 5- and 10-year periods), denoted by m in the paper. The role of the cross-sectional dimension, i, can be thought as being equivalent to t, noting that

the asymptotic analysis in this kind of application is in terms of the cross-section. Following Mankiw et al. (1992) and assuming that the per capita output in each country is determined by a common Cobb–Douglas production function we can write the Solow growth MIDAS regression for the aggregation horizon m as follows for gi,t = κ + β logi,t +β

−β −β

αK

αK + αH 1 − αK − αH

1 − αK − αH

αH

1 − αK − αH

log((nm i,t wi,t (θn ) + γ + δ))

log(sm k,i,t wi,t (θk )) log(sm h,i,t wi,t (θh )) + ei,t

(7.1)

where gi,t is real per capita growth in economy i from t to t + m, log yi,t is the logarithm of the initial real per capita income of the country i at period t, sm k,i,t denotes the savings rate for the physical capital accumulation out of the real output proxied by the average share of the real investment (including government investment) in real GDP, sm h,i,t denotes the savings rate for human capital accumulation out of real output proxied by the percentage of a country’s working age population in a secondary school observed every five years, and nm i,t is the annual population growth rate, ni,t , the rate of technical change, γ , and the depreciation of human and physical capital stocks, δ . Following standard practice we assume that (γ + δ) equals 0.05. With the exception of log yi,t all covariates are measured in a higher frequency. The high frequency covariates of population growth rates, investments and human capital accumulation are aggregated using time decaying weights, wi,t (θn ), wi,t (θk ), wi,t (θh ), respectively. These are based on the one parameter exponential polynomials that also allow for the case of

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E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

a

b

d

e

c

f

Fig. 7. (a)–(f): MIDAS — AR(1) model

equal weights.16 Notice that when the weights are flat then (7.1) becomes the standard Solow growth model. The coefficient of log yi,t , β , defines the β -convergence in the per capita income and it is readily interpretable in the context of the Solow growth model, since the property is implied by the local dynamics of the model. More precisely, β is defined as β ≡ −m−1 (1 − exp(−λm)), where λ is the speed of convergence for the growth rate of per capita income over the periods t and t + m. A negative β -coefficient is typically taken as evidence in the literature that poor countries are catching up with richer countries after controlling for heterogeneity. αK and αH denote the shares of the physical and human capital in the output, respectively. With the exception of the human capital data that is obtained from Bernanke and Gurkaynak (2001) all data is obtained from Penn World Table 6.2 and cover the years between 1965 and 1994. So we consider four aggregation horizons m = 30, 15, 10, 5. When m = 30 we have a cross-sectional MIDAS regression. When m = 15, m = 10, and m = 5 we have the pooled panel MIDAS regressions that correspond to the following panel datasets (i) 1965–79 and 1980–94; (ii) 1965–74, 1975–84, and 1985–94; and (iii) 1965–69, 1970–74, 1975–79, 1980–84, 1985–89, 1990–94, respectively. Table 2 presents the standard FLAT-LS estimates (columns 1–4) and MIDAS-NLS estimates (columns 5–8) for both the unrestricted model as well as the structural parameters of (7.1) for four

16 We also tried two parameter unrestricted exponential polynomials but we obtained similar results. However, we report the one-parameter case results due to parsimony.

aggregation horizons m = 30, 15, 10, 5. First of all we should note that both LS as well as MIDAS-NLS results replicate to a great extent the results in Mankiw et al. (1992). More precisely, we find that the coefficient to initial income per worker to be significant at the 1% level and negative. A negative coefficient on the log initial income per capita is typically taken as evidence in the literature that poorer countries are catching up with richer countries after controlling for heterogeneity. We also find that the coefficients of investment and schooling are positively significant while the coefficient of population growth rates is negative. Overall, our findings are therefore consistent with the results of Mankiw et al. (1992) and others in the existing ‘‘conditional convergence’’ literature. Interestingly, the MIDAS-NLS results appear to produce larger shares of physical capital and smaller shares of human capital than the corresponding FLAT-LS estimates. This difference is the largest when the aggregation horizon is the longest in our sample, m = 30. This finding is consistent with the analytical results in Section 3, which shows that the bias becomes larger with a longer aggregation horizon. As a robustness check for regional heterogeneity we also present sub-samples estimates of OECD vs. non OECD and SubSaharan Africa vs. non Sub-Saharan Africa (see Table 3) when m = 30. In unreported exercises we obtained similar results but weaker results when m = 15, 10, 5. The OECD subsample singles out the most developed countries while Sub-Saharan Africa aims at examining the set of countries with the worst performance in the sample. The results show that the speed of convergence appears to be larger in the MIDAS-NLS for the full sample than the corresponding FLAT-LS results. In contrast, the speed of convergence appears to be smaller in the MIDAS-NLS for sub-samples than the corresponding LS results. For instance, the speed of convergence for the

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

257

Table 2 Empirical Solow growth model for different aggregation horizons. Model

LS/NLS

Aggregation horizon

30 (1)

15 (2)

10 (3)

5 (4)

30 (5)

log(˜n)

−0.053*

−0.048* (0.008) 0.019* (0.003) 0.008* (0.002) −0.013* (0.002) 0.489* (0.044) 0.227* (0.041) 0.012* (0.002) 0.472

−0.043* (0.007) 0.021* (0.002) 0.007* (0.002) −0.013* (0.002) 0.515* (0.040) 0.206* (0.035) 0.012* (0.002) 0.388

−0.033* (0.005) 0.023* (0.001) 0.007* (0.001) −0.012* (0.001) 0.528* (0.028) 0.197* (0.024) 0.011* (0.001) 0.275

(0.011) 0.015* (0.004) 0.014* (0.003) −0.016* (0.002) 0.372* (0.067) 0.310* (0.062) 0.018* (0.004) 0.602

log(sk ) log(sh ) log(y0 )

αk αh λ R2adj

MIDAS-NLS

LM WH(1) WH(3)

15 (6)

10 (7)

5 (8)

−0.039* −0.044* (0.010) (0.008) 0.023* 0.020* (0.004) (0.002) 0.011* 0.010* (0.002) (0.002) −0.014* −0.014* (0.002) (0.002) 0.443* 0.514* (0.049) (0.048) 0.266* 0.208* (0.046) (0.044) 0.019* 0.013* (0.003) (0.002) 0.723 0.513 Testing flat aggregation

−0.039* (0.007) 0.021* (0.002) 0.009* (0.002) −0.013* (0.002) 0.527* (0.040) 0.197* (0.035) 0.012* (0.002) 0.422

−0.027* (0.006) 0.022* (0.001) 0.007* (0.001) −0.011* (0.001) 0.552* (0.029) 0.175* (0.025) 0.011* (0.001) 0.285

0.000 0.004 0.000

0.001 0.000 0.007

0.002 0.001 0.000

0.004 0.000 0.048

Notes: This table provides the LS and MIDAS-NLS estimates for Solow growth model (7.1) of the text for various aggregation horizons for a sample of 83 countries. The first four rows refer to the coefficient estimates of the unrestricted/reduced form of model, log investments, log schooling, and log initial income. The rows that correspond to αk , αh , λ refer to the NLS and MIDAS-NLS estimates of the structural parameters of the Solow growth model (7.1). Robust standard errors are given in parentheses. Time dummies (unreported) are included in each regression. The last three rows refer to the Flat Aggregation test using the LM test and the Hausman test with 1 and 3 lags as instrumental variables (WH(1) and WH(3)). * Significance at 1%.

Table 3 Empirical Solow growth model for different subsamples. Model

LS/NLS

Sample Sample size

Full sample (1) 83

SS Africa (2) 27

Non SS Africa (3) 56

OECD (4) 20

Non OECD (5) 63

Full sample (6) 83

log(˜n)

−0.053**

−0.075 (0.050) 0.008* (0.004) 0.014** (0.003) −0.019** (0.005) 0.187 (0.104) 0.372** (0.076) 0.024* (0.011) 0.427

−0.040** (0.010) 0.027** (0.004) 0.006 (0.004) −0.017** (0.003) 0.536** (0.065) 0.134* (0.064) 0.024** (0.005) 0.582

−0.026*

−0.070**

−0.039**

(0.011) 0.015* (0.006) 0.012* (0.005) −0.019** (0.003) 0.321** (0.099) 0.259* (0.104) 0.029** (0.006) 0.696

(0.021) 0.016** (0.004) 0.013** (0.003) −0.015** (0.003) 0.367** (0.085) 0.310** (0.064) 0.018** (0.005) 0.578

log(sk ) log(sh ) log(y0 )

αk αh λ R2adj

(0.011) 0.015** (0.004) 0.014** (0.003) −0.016** (0.002) 0.372** (0.067) 0.310** (0.062) 0.018** (0.004) 0.602

MIDAS-NLS

LM WH(1) WH(3)

SS Africa (7) 27

Non SS Africa (8) 56

OECD (9) 20

Non OECD (10) 63

−0.069 (0.010) (0.039) 0.023** 0.018** (0.004) (0.006) 0.011** 0.017** (0.002) (0.002) −0.014** −0.017** (0.002) (0.004) 0.443** 0.366** (0.049) (0.094) 0.266** 0.326** (0.046) (0.057) 0.019** 0.021** (0.003) (0.008) 0.723 0.640 Testing flat aggregation

−0.036** (0.010) 0.029** (0.004) 0.004 (0.004) −0.015** (0.002) 0.557** (0.077) 0.102 (0.073) 0.021** (0.004) 0.642

−0.045**

−0.051**

(0.011) 0.020** (0.004) 0.011* (0.005) −0.015** (0.002) 0.392** (0.071) 0.293** (0.066) 0.023** (0.003) 0.796

(0.017) 0.024** (0.004) 0.011** (0.003) −0.014** (0.002) 0.465** (0.066) 0.257** (0.050) 0.017** (0.003) 0.698

0.000 0.004 0.000

0.008 0.059 0.124

0.299 0.112 0.839

0.000 0.044 0.006

0.000 0.022 0.279

Notes: This table provides the LS and MIDAS NLS estimates for the Solow growth model (7.1) of the text for various subsamples and aggregation horizons (growth periods) m = 30. The first four rows refer to the coefficient estimates of the unrestricted / reduced form of model, log investments, log schooling, and log initial income. The rows that correspond to αk , αh , λ refer to the NLS and MIDAS-NLS estimates of the structural parameters of the Solow growth model (7.1). Robust standard errors are given in parentheses. Time dummies (unreported) are included in each regression. The last three rows refer to the Flat Aggregation test using the LM test and the Hausman test with 1 and 3 lags as instrumental variables (WH(1) and WH(3)). * Significance at 5%. ** Significance at 1%.

Sub-Sahara Africa countries is 0.029 in the case of LS while it is only 0.020 in the case of the MIDAS-NLS. This finding is consistent with the literature in that finds evidence for growth divergence. Overall we are forced to conclude that the MIDAS-NLS regressions appear to explain the cross-country growth differences better than the LS linear regression using aggregation based on an equal weighting scheme to estimate the Solow model at the same sampling frequency. Note that although it is possible that economic

theory can be informative about the shape of the weights, in this paper we do not attempt to attach any structural interpretation to the weights. Instead, we allow the data to determine the shape of the Almon lag polynomial, which yields the empirical weighting scheme. For example the evidence from the crosssection suggests that the annual Solow variables of the population growth rate and the saving rates of physical and human capital accumulation generally exhibit decaying weighting patterns and

258

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

that the determinants of average growth rate over a 30 year period gain a higher weight in more recent than distant information.17 Taking this evidence together, it seems that the Solow variables are endogenous rather than exogenous (or predetermined). The problem of endogeneity bias in cross-country growth regressions is widely recognized in the growth literature and it is typically addressed by instrumenting these variables using lags or initial periods. Although in principle one could control for endogeneity using a GMM approach in the context of MIDAS, the nonlinear nature of the MIDAS model suggests that one would need panel datasets with a much longer time span than the typical panels of 5 or 10-year periods for reliable finite sample inference. Additionally, the properties of the GMM for MIDAS models are unexplored, especially with respect to the choice of instruments with mixed frequencies, and hence, we do not pursue this issue further in this paper.

where wj∗ (θ) = (wj (θ) −

P ∗ ) such that m j=1 wj (θ) = 0 and P m 1 A B A B x(θ) = xt + xt (θ) where xt = m j=1 et −(j−1)/m , xt (θ) = ∗ Pm P m ∂wj (θ) ∗ B et −(j−1)/m . j=1 wj (θ)et −(j−1)/m and xθ := ∂ xt /∂θ = j=1 ∂θ In general the asymptotic variance of the NLS estimator of β1 is 1 m

given by: AVar (βˆ 1 ) = σ 2 E (x2θ )Var (xθ )/D where D = Var (xθ ) E (x(θ))2 E (x2θ ) − (E (xθ x(θ)))2



E (xθ ) = E (∂ 2

xBt

/∂θ) = E 2

m X ∂wj∗ (θ)

 m  X ∂wj∗ (θ) 2

=σ

Acknowledgements We would like to thank the Associate Editor and the two anonymous referees for their constructive comments and suggestions. We also thank the participants of the ‘Econometrics: Theory and Practice’ conference in honor of Professor Phoebus Dhrymes (June 2007), for helpful comments. Constantinos Kourouyiannis provided excellent research assistance. The authors acknowledge support from the Leventis Foundation UCY grant 3411-32021. The first author also acknowledges support of the European Community FP7/2008-2012 ERC grant 209116.

(A.3)

Then

2 e

We study regression models that involve data sampled at different frequencies and compare them with the traditional approach that involves temporal aggregation using an equal weighting scheme in order to estimate a model at the same sampling frequency. We derive the asymptotic properties of the MIDAS-NLS estimators of regression models with mixed frequencies and general weighting schemes and show that the FLAT-LS estimator is always relatively less efficient than the MIDAS-NLS estimator. Our analysis reveals the effects of the aggregation horizon on the properties of the MIDAS-NLS and FLATLS estimators. In particular, we show that the FLAT-LS estimator is asymptotically biased and inefficient when the high frequency regressor exhibits a linear temporal dependence as in the case of AR(1), while it is only inefficient in the case of i.i.d. and ARCH(1) high frequency processes. Furthermore, we propose two tests for the flat aggregation scheme: a simple LM test and a Wu–Hausman type test that use the high frequency variables as instruments. By means of an extensive simulation we verify all our theoretical results. We also illustrate our above results with an application in the context of empirical economic growth.



 2 − E (x(θ))E (x2θ ) − E (xθ x(θ))E (xθ ) .

j =1

8. Conclusion

(A.2)

∂θ

j =1

∂θ

!2 et −(j−1)/m

.

(A.4)

Xm  ∂wj∗ (θ) 2

since E (xθ ) ∂θ P = 0. Moreover, given that E (x(θ)) = E ( m1 m j=1 et −(j−1)/m + Pm ∗ (A.3) becomes D = Var (xθ ) w (θ) e ) = 0 then Eq. t −( j − 1 )/ m  j =1 j 2 2  E (x(θ)) E (xθ ) − (E (xθ x(θ)))2 and thereby the asymptotic variAlso Var (xθ ) = E (∂ xBt /∂θ)2 = σe2

j =1

ance in (A.2) simplifies to AVar (βˆ 1 ) =

σ 2 E (x2θ ) . E (x(θ))2 E (x2θ ) − (E (xθ x(θ)))2

(A.5)

We can also reparameterize P the MIDAS model in Eq. (A.1) in m terms of yt = β0 + β1 j=1 wj (θ )et −(j−1)/m + ut , where Pm Pm 1 ∗ x(θ) = j=1 wj (θ)et −(j−1)/m = j=1 (wj (θ) + m )et −(j−1)/m . To further evaluate D note that E (x(θ))2 = σe2

Pm

j =1

wj∗ (θ) +

1 2 m




,

∂wj∗ (θ) 2 j=1 ( ∂θ ) ,

) and j=1 (wj (θ) + Pm j 1 ∗ 2 (E (xθ x(θ))) = (σ ) ( j=1 (wj (θ) + m )( ∂θ )) . Therefore   E (x(θ))2 E (x2θ ) − (E (xθ x(θ)))2 2 X  m  m  X ∂wj∗ (θ) 2 1 2 2 ∗ = (σe ) wj (θ) + m ∂θ j =1 j =1  ∗  !2 m X ∂wj (θ) 1 2 2 ∗ − (σe ) (wj (θ) + ) . (A.6) m ∂θ j =1 E (x(θ)) E (xθ ) = (σ ) 2

2

2 2 e

2

Pm

∗

2 2 e

Pm

1 2 m ∂w∗ (θ)

Thus using the asymptotic variance representation of β1 for the MIDAS model with an i.i.d. regressor in Eq. (A.5), and its elements derived in Eqs. (A.4) and (A.6), the analytical expression of AVar (βˆ 1 ) is given by

 Appendix A. Proof of Proposition 4.1 for the MIDAS model with i.i.d. regressor Let the MIDAS regression model with an i.i.d. regressor yt = β0 + β1

m 1 X

m j =1

et −(j−1)/m + β1

m X

σ2 AVar (βˆ 1 ) = 2 σe

wj∗ (θ)et −(j−1)/m + ut (A.1)

m P

j =1

− 17 We should note that these findings are robust to the alternative families of polynomials that include the Beta polynomial as well as the one- and twoparameter exponential Almon polynomial.

 2 m  X 1  ∗ wj (θ) +   j =1 m 

j=1

!  −1  ∂w∗ (θ)  2
j  wj∗ (θ) + m1 ∂θ    .    2 m ∗ P ∂wj (θ)   ∂θ j=1

(A.7)

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

Appendix B. Proof of Proposition 4.2 for the MIDAS model with an ARCH(1) regressor Let (m)

xt /m

) {x(t m /m }

follow an ARCH(1) process given by

where c0 > 0, 0 < c1 <

(B.1)

∼ N .i.i.d.(0, 1), c12 < 1/3 = 0 where P (./.) denotes the best

and linear predictor. Note that the weak ARCH(1) model is closed under temporal aggregation but the results below hold for strong and semi-strong ARCH models. Then using (3.1) the MIDAS regression model with an ARCH(1) regressor can be written in terms of the (m) volatility σt /m yt = β0 + β1

m 1 X (m) (m) σ e m j=1 t −(j−1)/m t −(j−1)/m

m X

(B.2)

j =1

Pm

j =1

wj∗ (θ)

In general the AVar (βˆ 1 ) is given by Eqs. (A.2) and (A.3). Let



c0 +

∂ xBt /∂θ




Pm

=

(m) c1 (xt −(j−1)/m )2 m X ∂wj∗

E ( xθ ) = E

∂θ

j =1

1/2

j =1

∂wj∗ ∂θ

m) (m) σt(−( j−1)/m et −(j−1)/m =

(m) et −(j−1)/m .

Pm

∂wj∗

j=1

∂θ

Then

m) (m) σt(−( j−1)/m et −(j−1)/m

= 0.

 m  X ∂wj∗ (θ) 2

∂θ j =1 X   m ∂wj∗ (θ) 2

c0 1 − c1

∂θ

m j =1 m X

+

wj (θ) ∗



1 − c1

wj (θ) ∗

j =1

j =1

∂θ

(m)

2

c0



=

wj (θ) + ∗

j =1

1 m


 ∂wj∗ (θ) 

m

j =1

(m) σ (m) e ∂θ t −(j−1)/m t −(j−1)/m X  m  ∂wj∗ (θ) 2 j =1

∂θ

∂θ

m  ∂w ∗ (θ) 2 P j

∂θ

!2 −1      

. 

(B.6)

!

!  X m 1 ∗ 2 (wj (θ)) + .

m X ∂wj∗

1 − c1

(B.5)

 −1  X 2  m 1 ∗  wj (θ) +  1 − c1 m  j =1

j =1

m) (m) wj (θ) σt(−( j−1)/m et −(j−1)/m

1 − c1

c0

.

c0

m P

∗

c0

j =1

∂θ



Appendix C. Proof of Proposition 4.3 for the asymptotic bias FLAT-LS estimator in the MIDAS model with an AR(1) regressor

=0

In the case of an AR(1) high frequency regressor the MIDAS regression model can be expressed as m−1

yt = β0 + β1 xAt + β1

X

wi (θ) −

i=1



!2

1

.

thereby AVar (βˆ 1 ) in the MIDAS model with an ARCH regressor simplifies to that derived for the MIDAS with an i.i.d. regressor given in (A.5). Moreover,

E (x2θ ) = E

∂wj∗ (θ)

!  m m  X X ∂wj∗ (θ) 2 ∗ 2 + (wj (θ)) = 1 − c1 m ∂θ j =1 j=1 !2  2 X m ∂wj∗ (θ) c0 ∗ − wj (θ) . 1 − c1 ∂θ j =1 

E (xt −(j−1)/m )2

due to the martingale difference property of the ARCH. Then Eq. (A.3) becomes D = Var (xθ )[E (x(θ))2 E (x2θ ) − (E (xθ x(θ)))2 ] and

E (x(θ))2 =

(m)

E (xt −(j−1)/m )2

Therefore,

AVar (βˆ 1 ) = σ 2

j=1



!

∂θ

2 X m

c0

m 1 X (m) (m) σ e m j=1 t −(j−1)/m t −(j−1)/m

+

∂wj∗ (θ)

which implies

−

m X

(m)

E (xt −(j−1)/m )2

j =1

In addition, E (x(θ)) = E

.



Var (xθ ) = E (∂ xBt /∂θ)2 =

=

j =1

Finally, the AVar (βˆ 1 ) using Eqs. (B.3)–(B.5) for the MIDAS model with an ARCH regressor is

!

Hence



m

m X (wj∗ (θ))

∂θ

E (x(θ))2 E (x2θ ) − (E (xθ x(θ)))2

m) (m) B σt(−( j−1)/m et −(j−1)/m and xt (θ) =

Pm

1 m

+

 m  X ∂wj∗ (θ) 2



m) (m) σt(−( j−1)/m et −(j−1)/m .

xθ =

1 − c1

∗ m 1 X ∂wj (θ)

E (xθ x(θ)) =

(E (xθ x(θ))) =

m) (m) wj∗ (θ) σt(−( j−1)/m et −(j−1)/m + ut

1

In addition,

2

j =1

where xAt =

2

c0

j=1

(m) 1, et /m

(m) (m) (m) P (et /m /et −1/m , et −2/m , . . .)



×

q ) (m) 2 (m) 2 = (σt(/mm) )2 e(t m /m where (σt /m ) = c0 + c1 (xt −1/m )

+ β1

(E (x(θ))2 )(E (x2θ )) =

259

!2

  (B.3)

×

c0

1−φ

m−i



1−φ

+

X



m

− (1 − φ m−i )x(t −(m−1))/m !

m−i−1

!2

1

φ e(t −j−(i−1))/m + ut j

(C.1)

j=0

(B.4)

where wj∗ (θ) = (wj (θ) − process is given by

(m) xt /m

). The stationary high frequency AR(1) ) = c0 + φ xt(m −1/m + et /m . We can show that 1 m

260

E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

m −

1−φ m σe (1−φ) 1−φ 2 2

m−1

wi (θ) −

P

1 m

i=1

γ1 =






(1 − φ

2

1−φ m 1−φ

σe2 1−φ 2

m−i

+

)+

1

m−1

P

(1−φ)

σe2 (1−φ)2

m−1

P

1−φ

j =1

j

 j
P

(wi − )φ 1 m

i=1

!

j −i

(1 − φ j )2

j =1

Box II.

In general the AVar (βˆ 1 ) is given by Eqs. (A.2) and (A.3). Recall that xBt is given in Eq. (C.3) and hence

xAt can be written as xAt

=

c0

m

1−φ

+

1 − φm



1

m−

+

1−φ

X

1 − φ j =1

X ∂wi (θ)   1 − φ m−i  c0 (∂ /∂θ) = ∂θ 1−φ i=1 m−1

xBt

!

m−1

1

1 − φ m (m) x 1 − φ t −(m−1)/m



(1 − φ )et −(j−1)/m . j

xBt (θ) =

− (1 − φ

m−i

)x(t −(m−1))/m +

Therefore, E (xθ ) = E (∂ xBt /∂θ) =

wi (θ) −

i=1

 

φ e(t −j−(i−1))/m . j

1

c0

m

1−φ

m−i

1−φ

)x(t −(m−1))/m +

X

! φ e(t −j−(i−1))/m j

m−i ∂wi (θ ) (c0 ( 1−φ ) ∂θ 1−φ

− (1 − φ m−i )E (x(t −(m−1))/m )) or      m −1 X ∂wi (θ) 1 − φ m−i 1 − φ m−i E (xθ ) = c0 − c0 = 0. ∂θ 1−φ 1−φ i=1

 m−i

m−i−1

− (1 − φ

X j =0

Then xBt (θ) is m −1  X

!

m−i−1

(C.2)

(C.3)

Pm−1 i=1

In addition,

j =0

σe2 Var (xθ ) = E (xθ ) = 1 − φ2 2

and Cov(xAt , xBt (θ))

=

1 m

+

σ 1−φ (1 − φ) 1 − φ 2 2 e

m

−

m−1

1

X

(1 − φ)

1−φ

m−1

m−1

X

wi (θ) −

i=1

j 
X j

j =1

1 m



1

wi −



m

i =1

+σ

2 e

(1 − φ m−i )

Var (xAt ) =

1

1−φ

 m 2

1−φ

m2

(C.4)

σe2 1 − φ2

X ∂wj+1 (θ) φ m−i−j−1 ∂θ j =0

(C.5)

Let the MIDAS regression model with an AR high frequency representation be given in Eq. (C.1). The asymptotic variance of the FLAT-LS estimator is given by

σ = 2 σe



(1−φ m )2 1−φ 2 2

(D.8)

j =0

m−1

(m)

xt (θ) = xt −(m−1)/m +

X

+

P

(1 − φ )

− (1 − φ

m(1 − φ 2 ) − 2φ + 2φ m+1



.

m−i

)x(t −(m−1))/m +

X

! φ e(t −j−(i−1))/m (D.10) j

j=0

which implies that E (xt (θ)) =

j 2

   1 − φ m−i wi (θ) c0 1−φ 1−φ i =1   1 − φ m−i c0 − c0 = 1−φ 1−φ c0

m −1

+E

X

(D.11)

and

j =1 2

   1 − φ m−i wi (θ) c0 1−φ m−i−1

!

m−1

X

This facilitates the expression of x(θ) given by

m2 σ 2

m (1 − φ) (1 − φ ) 2

.

   1 − φ m−i wi (θ) c0 1−φ i=1 ! mX −i −1 m−i j − (1 − φ )x(t −(m−1))/m + φ e(t −j−(i−1))/m + ut . (D.9)

i=1

2

!2

m−1

(m)

Appendix D. Proof of Proposition 4.4 for the asymptotic variance of FLAT-LS and MIDAS-NLS estimators in the MIDAS regression model with an AR(1) regressor

σe2 (1−φ)2

)

σ 2 E (x2θ ) . Var (x(θ))E (x2θ ) − (E (xθ x(θ)))2

AVar (βˆ 1 ) =

yt = β0 + β1 xt −(m−1)/m + β1

Therefore, using Eqs. (C.4) and (C.5) the asymptotic bias in the MIDAS model with an AR(1) is given by ABias(βˆ 1∗ ; β1 ) = γ1 β1 where γ1 is given in Box II.

σ2 = v ar (xAt )

(1 − φ

The MIDAS model (4.9) can also be expressed in terms of the (m) high frequency process {xt /m }:

! m −1 X σe2 j 2 + (1 − φ ) . (1 − φ)2 j=1

AVar (βˆ 1∗ ) =

∂θ

i=1

m−i

Hence AVar (βˆ 1 ) simplifies to

.

The variance of xAt is



!2

m−i−1

X i=1

!! φ j −i

X ∂wi (θ)

m−1

(D.7)

σe2 σe2 Var (x(θ)) = + 2 1−φ 1 − φ2

m−1

X i=1

wi (θ) 1 − φ

m−i

!2


E. Andreou et al. / Journal of Econometrics 158 (2010) 246–261

+ σe2

m−1

m−i−1

X

X

i=1

j =0

 −2

σe2 1 − φ2

!2 wj+1 (θ)φ m−j−i−1

m −1 X

wi (θ)(1 − φ m−i ).

i =1

Finally, E (xθ x(θ)) =

σe2 1 − φ2

X ∂wi (θ)

m−1

i=1

∂θ

(1 − φ m−i ) !

m−1

×

X

wi (θ)(1 − φ

m−i

)

i =1

−

−1 X σe2 m ∂wi (θ) (1 − φ m−i ) 2 1 − φ i=1 ∂θ

+σ

2 e

m−1

m−i−1

X

X

i=1

j =0

! wj+1 (θ)φ

m−i−j−1

!! X  ∂wj+1 (θ)  m−i−j−1 φ .  ∂θ j =0

m−i−1

×

(D.12)

References Bernanke, B., Gurkaynak, R., 2001. Is growth exogenous? Taking Mankiw, Romer, and Weil Seriously. NBER Macroeconomics Annual 16, 11–57.

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Dhrymes, Phoebus, 1971. Distributed Lags: Problems of Estimation and Formulation, Holden-Day, San Francisco. Engle, R.F., Liu, T.-C., 1972. Effects of aggregation over time on dynamic characteristics of an econometric model. In: Hickman, B., Hickman, Bert G. (Eds.), Econometric Models of Cyclical Behavior. NBER, pp. 673–738. Geweke, J., 1978. Temporal aggregation in the multiple regression model. Econometrica 46, 643–661. Granger, C., 1987. Implications of aggregation with common factors. Econometric Theory 3, 208–222. Ghysels, E., Hall, A., 1990. Are consumption-based intertemporal capital asset pricing models structural? Journal of Econometrics 45, 121–139. Ghysels, E., Santa-Clara, P., Valkanov, R., 2005. There is a risk-return tradeoff after all. Journal of Financial Economics 76, 509–548. Ghysels, E., Sinko, A., Valkanov, R., 2006. MIDAS regressions: Further results and new directions. Econometric Reviews 26, 53–90. Ghysels, E., Valkanov, R., Granger causality tests with mixed data frequencies, Working Paper, 2006. Ghysels, E., Wright, J., 2009. Forecasting professional forecasters. Journal of Business and Economic Statistics 27, 504–516. Hsiao, C., 1979. Linear regression using both temporally aggregated and temporally disaggregated data. Journal of Econometrics 10 (2), 243–252. Mankiw, N.G., Romer, D., Weil, D., 1992. A contribution to the empirics of economic growth. Quarterly Journal of Economics 107 (2), 407–437. Phillips, P., 1972. The structural estimation of a stochastic differential equation system. Econometrica 40, 1021–1041. Phillips, P., 1973. The problem of identification in finite parameter continuous time models. Journal of Econometrics 1, 351–362. Phillips, P., 1974. The estimation of some continuous time models. Econometrica 42, 803–824. Phillips, P., 1991. Optimal inference in cointegrated systems. Econometrica 59 (2), 283–306. Sims, C.A., 1971. Discrete approximations to continuous time distributed lags in econometrics. Econometrica 39, 545–563. Tiao, G.C., Wei, W.S., 1976. Effect of temporal aggregation on the dynamic relationship of two time series variables. Biometrika 63, 513–523.